Research ArticleAutopilot Design Method for the Blended MissileBased on Model Predictive Control

Baoqing Yang and Yuyu Zhao

Control and Simulation Center Harbin Institute of Technology Harbin 150001 China

Correspondence should be addressed to Baoqing Yang ybqhiteducn

Received 19 September 2014 Revised 24 January 2015 Accepted 26 January 2015

Academic Editor Mahmut Reyhanoglu

Copyright copy 2015 B Yang and Y ZhaoThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper develops a novel autopilot designmethod for blendedmissiles with aerodynamic control surfaces and lateral jets Firstlythe nonlinear model of blended missiles is reduced into a piecewise affine (PWA)model according to the aerodynamics propertiesSecondly based on the equivalence between the PWA model and mixed logical dynamical (MLD) model the MLD model ofblended missiles is proposed taking into account the on-off constraints of lateral pulse jets Thirdly a hybrid model predictivecontrol (MPC)method is employed to design autopilot Finally simulation results under different conditions are presented to showthe effectiveness of the proposed method which demonstrate that control allocation between aerodynamic control surfaces andlateral jets is realized by adjusting the weighting matrix in an index function

1 Introduction

The emergence of highly maneuverable targets has broughtnew challenges to guidance technology The improvementof homing guidance performance against highly maneu-verable targets in the future guided missiles requires thecontrol system to have faster response and wider operationrange [1] In the meantime as an important property ofadvanced missiles multiple actuators are often employed toenhance maneuverability as well as interception probabilityFor example both aerodynamic surfaces and reaction jets areemployed in the control system of PAC-3

With a higher angle of attack the missile dynamic modelis highly nonlinear and the coupling effects [2] as well as theuncertainties in both aerodynamic parameters and reactionjet thrust are oblivious Munson and Garbrick introducedamplification factors to describe the lateral jet interferenceeffect [3] Graham and Weinacht studied the interactionbetween the side jet and the external flow by a numericalmethod [4] It is obvious that the autopilot designed using thelinearized model around an operation point is usually unableto achieve satisfactory performance over a full flight envelopeRecently many nonlinear control methods are proposed forthe design of conventional aerofin autopilots For instance

the gain-scheduled approach was proposed in [5] wherelinear parameter varying transformations were adopted

On the other hand the autopilot design for blendedmissile has not been much reported which is more com-plicated than that of conventional aerofin autopilot due tothe heterogeneous actuation [1] Hirokawa et al [1] designedan autopilot for the case with aerodynamic surfaces andreaction jet using the coefficient diagram method (CDM)The feasibility of the autopilot based on variable structurewas discussed in [6] where a simple blending strategy isinvestigated aerodynamic control is used at low angles ofattack while reaction jet control is used when the missilemoves beyond the stall region Optimal control and staticcontrol allocation were combined in [7] to address the issueof dual control missile while a dynamic control allocationmethod was presented in [8]

In addition for the autopilot design of blended missiles(such as PAC-3) two aspects of control input constraintsshould be taken into account the saturation constraint onaerodynamic surfaces and the finite set constraint on reactionjets However they are not simultaneously considered inthe aforementioned work Besides the hybrid properties ofcontrol inputs (continuous aerodynamic surfaces and on-offreaction jets) are often neglected as in [6ndash8] Motivated by

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2015 Article ID 718036 13 pageshttpdxdoiorg1011552015718036

2 International Journal of Aerospace Engineering

these facts this paper attempts to design an autopilot usingexplicit hybrid MPC for blended missiles by noting thatMPC is a promising methodology for the control problem ofconstrained uncertain systems [9] and that the computationalburden of on-line optimization is effectively reduced by usingexplicit MPC instead of traditional MPC [10]

The remainder of this paper is as follows Section 2gives a mathematical model of blended missile including theconfiguration of reaction jets In Section 3 the piecewiseaffine model of blended missile is established followed by anMLD model which is obtained based on the equivalence ofpiecewise affine model and mixed logical dynamical modelIn Section 4 a hybrid MPC based method for autopilotdesign is proposed and an explicit control law is constructedIn Section 5 the effectiveness of the proposed methodis verified by simulation cases under different conditionsFinally several concluding remarks are given in Section 6

2 Mathematical Model of the Missile

The plane reference coordinate system 119874119880119881119882 the bodycoordinate system 119874119909119887119910119887119911119887 the trajectory coordinate sys-tem 119874119909119908119910119908119911119908 and velocity coordinate system 119874119909V119910V119911V areinvolved in this paper Figure 1 shows a missile with some keyvariables and identified axes The axes 119911119908 119911V are not givenwhose directions can be determined by the right hand rule

21 Missile Dynamic Model Thenonlinear motion equationsare given by

119898 = 119875 cos120572 cos120573 minus 119883119886 minus 119898119892 sin 120579 + 119865119886

119909119908

119898119881 120579 = 119875 (sin120572 cos 120574V + cos120572 sin120573 sin 120574V) + 119884119886 cos 120574V

minus 119898119892 cos 120579 + 119865119886

119910119908

119898119881 cos 120579V = minus 119875 (sin120572 sin 120574V minus cos120572 sin120573 cos 120574V)

minus 119884119886 sin 120574V minus 119885119886 cos 120574V minus 119865119886

119911119908

(1)

119869119909 = (119869119910 minus 119869119911) 119903119902 + 119872119909

119869119910 119902 = (119869119911 minus 119869119909) 119901119903 + 119872119910

119869119911 119903 = (119869119909 minus 119869119910) 119901119902 + 119872119911

(2)

where 119865119886

119909119908

119865119886

119910119908

119865119886

119911119908

are lateral forces in the 119909 119910 and 119911

directions of trajectory coordinate system respectively Forsimplicity we suppose that the body is symmetric about the119909-axis that is 119869119909119910 = 119869119910119911 = 119869119911119909 = 0 In (2) each of themoments119872119909119872119910 and119872119911 contains two components that are generatedby aerodynamic surface and the lateral pulse jets respectively

119872119909 = 119872119890119909 + 119872119886

119909119887

119872119910 = 119872119890119910 + 119872119886

119910119887

119872119911 = 119872119890119911 + 119872119886

119911119887

(3)

yw

y

V(yb)

q

120574 xw(x)

120573120572

120579O

r

W(zb)

Jets

U(xb)

120595 p

Figure 1 Key coordinate systems

where 119872119890119909 119872119890119910 and 119872119890119911 denote aerodynamic momentcomponents and 119872

119886

119909119887

119872119886119910119887

and 119872119886

119911119887

are pulse jet momentcomponents

22 Lateral Jet Forces and Moments Model As shown inFigure 1 the lateral jet force is generated by 180 pulse jetslocated in front of the center of mass of PAC-3 These jets aredivided into 10 rings and arranged in staggered positions (18pulse jets are included in each ring) In each ring these jetsare uniformly distributed and the central angle between twoneighboring jets is 20 degree Use 119894 (119894 = 1 2 10) and 119895

(119895 = 1 2 18) to denote the ringrsquos label and the jetrsquos labelin each ring respectivelyThe distance between ring 119894 and thecenter ofmass is denoted by 119897119894 while spacing of adjacent ringsis Δ119897 The layout scheme of pulse jets is shown in Figures 2and 3 Assume the force generated by each individual jet isa constant 119865119898 In body coordinate system the lateral forcegenerated by the (119894 119895) pulse jet is given by

[[

[

119865119894119895

119909119887

119865119894119895

119910119887

119865119894119895

119911119887

]]

]

=

[[[[[

[

0

119865119898 cos(2119895 minus 119894lowast

18120587)

minus119865119898 sin(2119895 minus 119894lowast

18120587)

]]]]]

]

(4)

The corresponding moment is given by

[[

[

119872119894119895

119909119887

119872119894119895

119910119887

119872119894119895

119911119887

]]

]

=

[[[[[

[

0

119865119898119897119894 sin(2119895 minus 119894lowast

18120587)

119865119898119897119894 cos(2119895 minus 119894lowast

18120587)

]]]]]

]

(5)

International Journal of Aerospace Engineering 3

12

3

4

5

6

7

8

910

11

12

13

14

15

16

17

18

yb

zb

(a) Odd-numbered rings

1

2

3

4

5

6

7

8

91011

12

13

14

15

16

17

18

yb

zb

(b) Even-numbered rings

Figure 2 The layout scheme of lateral pulse jets

where

119894lowast= 2 119894 is odd

119894lowast= 1 119894 is even

(6)

For the situation where all pulse jets are fired at the sametime the total force and moment are given by

[

[

119865119909119887

119865119910119887

119865119911119887

]

]

=

[[[[[[[[[[

[

0

119894=10

sum

119894=1

119895=18

sum

119895=1

119865119894119895

119910119887

119894=10

sum

119894=1

119895=18

sum

119895=1

119865119894119895

119911119887

]]]]]]]]]]

]

[

[

119872119909119887

119872119910119887

119872119911119887

]

]

=

[[[[[[[[[[

[

0

minus

119894=10

sum

119894=1

119895=18

sum

119895=1

119865119894119895

119911119887

119897119894

119894=10

sum

119894=1

119895=18

sum

119895=1

119865119894119895

119910119887

119897119894

]]]]]]]]]]

]

(7)

In order to avoid the coupling between the pitch momentand yaw moment each ring is divided into four controlregions positive pitch negative pitch positive yaw andnegative yaw control region as shown in Figure 4

The autopilot design of PAC-3 is more complicated thanthat of other conventional missiles which are controlled

12

88

99

1010

1111

1212

xb

zb

i

lil1

Δl

middot middot middotmiddot middot middot

O

Figure 3 The ring frames expansion of lateral pulse jets

only by aerodynamic surfaces due to the hybrid propertyof control inputs and the on-off property of pulse jet (thepulse jet can be fired only one time) To deal with thisproblem the work [7] proposed two-step design procedures(1) in the first step neglect the hybrid property (or on-offproperty) and design the expected force andmoment signals(2) in the other step the fire logic is derived by solving theretaliation problem of these signals with taking into accountthe hybrid property (or on-off property) Different fromthe above traditional procedures a novel procedure will bepresented in what follows where only one step is included Inpractical applications only a small number of jets (in a certainof rings) are activated over a finite time interval To make theidea of the following development clear we here consider asimple but representative situation where no more than two

4 International Journal of Aerospace Engineering

yb

zb

Negative pitchcontrol region

Negative yawcontrol region

Positive pitchcontrol region

Positive yawcontrol region

17

8

910

11

12

12

3

4

5

6

713

14

15

16

18

(a) Odd-numbered rings

1

2

3

4

5

6

7

8

910

11

12

13

14

15

16

17

18

yb

zb

Negative pitchcontrol region

Negative yawcontrol region

Positive pitchcontrol region

Positive yawcontrol region

(b) Even-numbered rings

Figure 4 Schematic of control regions

rings are allowed to be fired simultaneously and no morethan two jets are activated in each fired ring Meanwhile itshould be ensured that only odd rings or even rings are firedand the jets are fired symmetrically about the correspondingsymmetry axis of each control region

Take the positive pitch control region as an example Theforces provided by the jets (119894 1) (119894 2) (119894 3) (119894 17) and (119894 18)

in an odd ring are given by

119865119900 = [119865119898 119865119898 cos120587

9119865119898 cos

2120587

9119865119898 cos

2120587

9119865119898 cos

120587

9]

119879

(8)

Similarly the forces provided by the jets (119894 1) (119894 2) (119894 17)and (119894 18) in an even ring are given by

119865119890 = [119865119898 cos120587

18119865119898 cos

120587

6119865119898 cos

120587

6119865119898 cos

120587

18]

119879

(9)

where119865119900 and119865119890 denote the forces associated with an odd ringand an even ring respectively

Use 119865119894 to denote the lateral force generated by ring 119894

Clearly 119865119894 should satisfy the condition

1198651= 1198659 119865

3= 1198657 119865

2= 11986510 119865

4= 1198658 (10)

When odd ring is fired the lateral force and moment aregiven by

119865119910119887

isin 119865119898 2119865119898 cos120587

9 2119865119898 cos

2120587

9 2119865119898 4119865119898 cos

120587

9

4119865119898 cos2120587

9

119872119911119887

isin 1198651198981198975 21198651198981198975 cos120587

9 21198651198981198975 cos

2120587

9 21198651198981198975

41198651198981198975 cos120587

9 41198651198981198975 cos

2120587

9

(11)

While even ring is fired the lateral force and moment aregiven by

119865119910119887

isin 2119865119898 cos120587

18 2119865119898 cos

120587

6 4119865119898 cos

120587

18 4119865119898 cos

120587

6

119872119911119887

isin 21198651198981198976 cos120587

18 21198651198981198976 cos

120587

6 41198651198981198976 cos

120587

18

41198651198981198976 cos120587

6

(12)

Noting Δ119897 is quite small we consider 1198975 ≃ 1198976 = 119897 Inorder to ensure that the jets fire efficiency the jets resultingin small moment components along 119911119887 axis are not activatedAs a result the sets of forces and moments are respectivelygiven by

119880119910+

119865= 119865119898 2119865119898 cos

120587

9 2119865119898 cos

120587

18 2119865119898 4119865119898 cos

120587

9

4119865119898 cos120587

18

(13)

119880119910+

119872= 119865119898119897 2119865119898119897 cos

120587

9 2119865119898119897 cos

120587

18 2119865119898119897 4119865119898119897 cos

120587

9

4119865119898119897 cos120587

18

(14)

where 119865119910119887

isin 119880119910+

119865and119872119911

119887

isin 119880119910+

119872 During each control period

a control moment belonging to119880119910+

119872will be used as the input

The mutual interference between high-speed jet streamand air leads to lateral jet interference effect In orderto take into account this interference force and moment

International Journal of Aerospace Engineering 5

amplification factors 119870119865119910

119870119865119911

119870119872119910

and 119870119872119911

are introducedas in [3 4]Then the resulting lateral forces andmoments are

[[[

[

119865119886

119909119887

119865119886

119910119887

119865119886

119911119887

]]]

]

= [

[

0

119865119910119887

+ 119870119865119910

119865119910119887

119865119911119887

+ 119870119865119911

119865119911119887

]

]

[[[

[

119872119886

119909119887

119872119886

119910119887

119872119886

119911119887

]]]

]

= [

[

0

119872119910119887

+ 119870119872119910

119872119910119887

119872119911119887

+ 119870119872119911

119872119911119887

]

]

(15)

Remark 1 In fact each pulse jet can be fired only once so thelocation of the fired jet cannot provide force anymore Basedon this precondition elements of sets 119880

119910+

119865and 119880

119910+

119872will be

less and less over time In this paper quantity change of setsrsquoelements is not considered to simplify the problem

23 Attitude Control Model Some transformation and sim-plification are applied to themissile model for control designIt is assumed that themissilersquosmass is of a constant valueNotethat the goal is to establish the angle of attack and sideslipangle The attitude control model is given by

= 119903 + 119902 sin120572 tan120573 minus

119876119878 (119862120572

119910120572 + 119862

120575119911

119910120575119911) cos120572

119898119881 cos120573

minus

(119865119910119887

+ 119870119865119910

119865119910119887

) cos120572

119898119881 cos120573minus

119866119910 cos120572119898119881 cos120573

120573 = 119902 cos120572 +

(119876119878 (119862120573

119911120573 + 119862

120575119910

119911 120575119910) + 119865119911119887

+ 119870119865119911

119865119911119887

) cos120573

119898119881

+

(119876119878 (119862120572

119910120572 + 119862

120575119911

119910120575119911) + 119865119910

119887

+ 119870119865119910

119865119910119887

) sin120572 sin120573

119898119881

+119866119911 cos120573

119898119881+

119866119910 sin120572 sin120573

119898119881

119902 =119872119910119887

119869119910

+

119870119872119910

119872119910119887

119869119910

+

119876119878119871119898120573

119910120573

119869119910

+119876119878119871119898

120575119910

119910 120575119910

119869119910

+

119876119878119871119898119902

119910119902

119869119910

119903 =119872119911119887

119869119911

+119870119872119911

119872119911119887

119869119911

+119876119878119871119898

120572

119911120572

119869119911

+119876119878119871119898

120575119911

119911120575119911

119869119911

+119876119878119871119898

119903

119911119903

119869119911

(16)

where 119898120573

119910 119898120575119911

119910 119898119902

119910 119898120572

119911 119898120575119911

119911 and 119898

119903

119911are aerodynamic

parameters

3 Mixed Logical Dynamical Model ofBlended Missile

31 Piecewise Affine Model of Blended Missile To simplifyanalysis the gravity term and the channel coupling term are

Table 1 Missilersquos overall parameters

Missilersquos takeoff mass [kg] 255Missilersquos full-length [m] 486Missilersquos diameter [m] 0317Distance between missile head and center ofmass [m] 2569

Moment of inertia 119869119909 119869119910 119869119911 [kgsdotm2] 303 3063 3063

Lateral force provided by individual jets [N] 2200Distance between jet ring center and centerof mass [m] 126

Range of angle of attack [rad] minus120587

6le 120572 le

120587

6

Range of pitch angular velocity [rads] minus5120587

3le 119903 le

5120587

3

Range of elevator deflection [rad] minus120587

6le 120575119911 le

120587

6

ignoredWith (16) themissile attitude control model of pitchchannel is

= 119903 minus

119876119878 (119862120572

119910120572 + 119862

120575119911

119910120575119911) cos120572

119898119881minus

(1 + 119870119865119910

) 119865119910119887

cos120572

119898119881

119903 =

(1 + 119870119872119911

) 119865119910119887

119897

119869119911

+119876119878119871119898

120572

119911120572

119869119911

+119876119878119871119898

120575119911

119911120575119911

119869119911

+119876119878119871119898

119903

119911119903

119869119911

(17)

Choose 119909 = [120572 119903]119879 as system state and 119906 = [120575119911 119865119910

119887

]119879 as

control input The considered output is 119910 = 120572 Then (17) canbe rewritten into the following state space form

= 119891 (119909) + 119892 (119909) 119906

119910 = [1 0] 119909

(18)

where

119891 (119909) =

[[[[

[

119903 minus

119876119878119862120572

119910120572 cos120572

119898119881

119876119878119871 (119898120572

119911120572 + 119898

119903

119911119903)

119869119911

]]]]

]

119892 (119909) =

[[[[[[

[

minus

119876119878119862120575119911

119910cos120572

119898119881minus

(1 + 119870119865119910

) cos120572

119898119881

119876119878119871119898120575119911

119911

119869119911

(1 + 119870119872119911

) 119897

119869119911

]]]]]]

]

(19)

The missile parameters are presented in Table 1Generally the aerodynamic coefficients 119862

120572

119910 119862120575119911

119910 119898120572

119911

and 119898120575119911

119911and the amplification factors 119870119865

119910

119870119872119911

are mainlyaffected by the flight velocity 119881 and the angle of attack 120572

[11 12] Since the terminal guidance phase is consideredin this paper the flight time is quite short and the flightvelocity of the missile can be treated as a constant Thusthe aerodynamic coefficients and the amplification factors are

6 International Journal of Aerospace Engineering

mainly affected by the angle of attack 120572 The relationshipsbetween them are shown in Figure 5

In practical application since 120572 is the main factor thatleads to system nonlinearities the system model is usuallylinearized if 120572 varies in small range As seen from Figure 5curves of the relation between aerodynamic parametersand angle of attack can be expressed by six line segmentsapproximately Here we choose 120572 = minus037 rad minus0153 rad0 0153 rad and 037 rad as the operation points and dividethe whole operation region into six subregions As a resultthe original model (17) can be converted to the followingpiecewise affine models

=

1198861119909 + 1198871119906 + 1198901 [1 0] 119909 le minus037

1198862119909 + 1198872119906 + 1198902 minus037 lt [1 0] 119909 le minus0153

1198863119909 + 1198873119906 + 1198903 minus0153 lt [1 0] 119909 le 0

1198864119909 + 1198874119906 + 1198904 0 lt [1 0] 119909 le 0153

1198865119909 + 1198875119906 + 1198905 0153 lt [1 0] 119909 le 037

1198866119909 + 1198876119906 + 1198906 [1 0] 119909 gt 037

(20)

119910 = [1 0] 119909 (21)

where

119886119894 =120597119891(119909)

120597119909

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= [11988611

11989411988612

119894

11988621

11989411988622

119894

]

119887119894 = 119892 (1199091198940) = [11988711

11989411988712

119894

11988721

11989411988722

119894

] 119890119894 = [1198901

119894

1198902

119894

]

11988611

119894=

1205971198911 (119909)

120597120572

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= minus119876119878

119898119881(

120597119862120572

119910

120597120572

1003816100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1205721198940 cos1205721198940 + 119862120572

119910(1205721198940) cos1205721198940

minus 119862120572

119910(1205721198940) 1205721198940 sin1205721198940)

11988612

119894=

1205971198911(119909)

120597119903

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= 1

11988621

119894=

1205971198912 (119909)

120597120572

10038161003816100381610038161003816100381610038161003816119909=1199091198940

=119876119878119871

119869119911

(120597119898120572

119911

120597120572

100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1205721198940 + 119898120572

119911(1205721198940) +

120597119898119903

119911

120597120572

100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1199031198940)

11988622

119894=

1205971198912(119909)

120597119903

10038161003816100381610038161003816100381610038161003816119909=1199091198940

=119876119878119871

119869119911

119898120572

119911(1205721198940)

11988711

119894= minus

119876119878119862120575119911

119910(1205721198940) cos1205721198940119898119881

11988712

119894= minus

(1 + 119870119865119910

(1205721198940)) cos1205721198940119898119881

11988721

119894=

119876119878119871119898120575119911

119911(1205721198940)

119869119911

11988722

119894=

(1 + 119870119872119911

(1205721198940)) 119897

119869119911

1198901

119894= 11988611

1198941205721198940 + 119886

12

1198941199031198940

1198902

119894= 11988621

1198941205721198940 + 119886

22

1198941199031198940

(22)

where 119894 (119894 = 1 2 6) is the label corresponding to the 119894thregion

From Table 1 and Figure 5 we get the set of aerodynamicparameters at the point (119867119881) = (20 km 1000ms) as shownin Table 2

Choose the sampling period 119879119904 = 0025 s The discretestate-space expression is then given by

119909 (119896 + 1)

=

1198861119909 (119896) + 1119906 (119896) + 1198901 [1 0] 119909 (119896) le minus037

1198862119909 (119896) + 2119906 (119896) + 1198902 minus037 lt [1 0] 119909 (119896) le minus0153

1198863119909 (119896) + 3119906 (119896) + 1198903 minus0153 lt [1 0] 119909 (119896) le 0

1198864119909 (119896) + 4119906 (119896) + 1198904 0 lt [1 0] 119909 (119896) le 0153

1198865119909 (119896) + 5119906 (119896) + 1198905 0153 lt [1 0] 119909 (119896) le 037

1198866119909 (119896) + 6119906 (119896) + 1198906 [1 0] 119909 (119896) gt 037

(23)

119910 (119896) = [1 0] 119909 (119896) (24)

where

1198861 = [104 0025

022 0995] 1198862 = [

1051 0025

0218 0995]

1198863 = [1023 00252

0248 09951] 1198864 = [

09954 00248

02732 09954]

1198865 = [09697 00245

02913 09956] 1198866 = [

09594 00244

03137 09959]

1198901 = [minus00183

minus01123] 1198902 = [

minus00175

minus00792]

1198903 = [minus00131

minus00376] 1198904 = [

0

0]

1198905 = [minus00053

00454] 1198906 = [

minus00169

01193]

1 = [minus0028 159 times 10

minus6

minus2166 142 times 10minus4]

2 = [minus003 218 times 10

minus6

minus2224 187 times 10minus4]

International Journal of Aerospace Engineering 7

2

15

1

05

0

minus05

minus1

minus15minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SmV

)C120572 y

(a) Relation between 119862120572119910and 120572

0092

009

0088

0086

0084

0082

008

0078

0076

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SmV

)C120575119911y

(b) Relation between 119862120575119911119910 and 120572

125

12

115

11

105

10

95

9minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SLJz)m

120572 z

(c) Relation between119898120572119911and 120572

96

94

92

90

88

86

84minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (minusQSL

Jz)m

120575119911z

(d) Relation between119898120575119911119911 and 120572

1

09

08

07

06

05

04

03

02

01

Am

plifi

catio

n fa

ctorKF

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

(e) Relation between119870119865119910

and 120572

12

11

1

09

08

07

06

05

04

03

Am

plifi

catio

n fa

ctorKM119911

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

(f) Relation between119870119872119911

and 120572

Figure 5 Aerodynamic parameters as functions of angle of attack

8 International Journal of Aerospace Engineering

Table 2 Aerodynamic parameters of pitch channel

Aerodynamic parameter 119876119878119862120572

119910119898119881 119876119878119862

120575119911

119910119898119881 119876119878119871119898

120572

119911119869119911

119876119878119871119898119903

119911119869119911

119876119878119871119898120575119911

119911119869119911

120572 = minus053 minus1175 0076 947 minus032 minus8489

120572 = minus037 minus0944 0082 973 minus032 minus8924

120572 = minus0153 minus023 0082 1043 minus032 minus8938

120572 = 0 036 009 1101 minus032 minus9510

120572 = 0153 094 0082 1158 minus032 minus8934

120572 = 037 163 0082 1224 minus032 minus8918

3 = [minus003 268 times 10

minus6

minus2228 223 times 10minus4]

4 = [minus0032 277 times 10

minus6

minus2371 232 times 10minus4]

5 = [minus003 227 times 10

minus6

minus2228 198 times 10minus4]

6 = [minus0029 191 times 10

minus6

minus2224 173 times 10minus4]

(25)

32 Constraints Analysis Due to the symmetry of jet config-uration the set of possible negative pitch control force is givenby

119880119910minus

119865= minus119865119898 minus2119865119898 cos

120587

9 minus2119865119898 cos

120587

18 minus2119865119898 minus4119865119898 cos

120587

9

minus4119865119898 cos120587

18

(26)

By combining (13) and (26) we obtain the set of allpossible pitch control force

119880119910

119865= 119865119898 2119865119898 cos

120587

9 2119865119898 cos

120587

18 2119865119898 4119865119898 cos

120587

9

4119865119898 cos120587

18 minus119865119898 minus2119865119898 cos

120587

9 minus2119865119898 cos

120587

18 minus2119865119898

minus4119865119898 cos120587

9 minus4119865119898 cos

120587

18

(27)

Substituting the jet parameters shown in Table 1 into theforegoing set yields

119880119910

119865= 2200 4135 4333 4400 8269 8666 minus2200 minus4135

minus4333 minus4400 minus8269 minus8666

(28)

Noting that the lateral forces are discrete variable we willutilize the linear combination of logical variables to describethe lateral force In terms of piecewise affine model (20) we

introduce logical variables 120575119865119894

isin 0 1 119894 = 1 2 12 toexpress the lateral thrust 119865119910

119887

119865119910119887

= 22001205751198651

+ 41351205751198652

+ 43331205751198653

+ 44001205751198654

+ 82691205751198655

+ 86661205751198656

minus 22001205751198657

minus 41351205751198658

minus 43331205751198659

minus 440012057511986510

minus 826912057511986511

minus 866612057511986512

(29)

In (29) the logical variables should satisfy the constraints

12

sum

119894=1

120575119865119894

= 0 or 1 (30)

where 0 means that no lateral force is generated while 1

means that the applied lateral force equals to one element ofset 119880119910119865

Denote 1199061 = 120575119911 the control input 119906 in model (23) can berewritten as

119906 = [1199061 119865119910119887

]119879 (31)

As shown in Table 1 the constraints on system states andcontrol input are

119909min le 119909 (119896) le 119909max

1199061min le 1199061 (119896) le 1199061max(32)

where 119909min = [minus053 minus522]119879 119909max = [053 522]

119879 1199061min =

minus053 1199061max = 053According to (30) we have

12

sum

119894=1

120575119865119894

le 1 (33)

which shows the constraint on the logical control inputUse logical variables 120575119894(119896) isin 0 1 119894 = 1 2 5 to

describe the operation points which satisfy the constraints

[1 0] 119909 (119896) + 037 le 0 lArrrArr 1205751 (119896) = 1

[1 0] 119909 (119896) + 0153 le 0 lArrrArr 1205752 (119896) = 1

[1 0] 119909 (119896) le 0 lArrrArr 1205753 (119896) = 1

[1 0] 119909 (119896) minus 0153 le 0 lArrrArr 1205754 (119896) = 1

[1 0] 119909 (119896) minus 037 le 0 lArrrArr 1205755 (119896) = 1

(34)

International Journal of Aerospace Engineering 9

Equation (34) can be transformed into the followingequivalent mixed logical inequalities [13]

[1 0] 119909 (119896) + 037 ge 120576 + (1198981 minus 120576) 1205751 (119896)

[1 0] 119909 (119896) + 037 le 1198721 (1 minus 1205751 (119896))

[1 0] 119909 (119896) + 0153 ge 120576 + (1198982 minus 120576) 1205752 (119896)

[1 0] 119909 (119896) + 0153 le 1198722 (1 minus 1205752 (119896))

[1 0] 119909 (119896) ge 120576 + (1198983 minus 120576) 1205753 (119896)

[1 0] 119909 (119896) le 1198723 (1 minus 1205753 (119896))

[1 0] 119909 (119896) minus 0153 ge 120576 + (1198984 minus 120576) 1205754 (119896)

[1 0] 119909 (119896) minus 0153 le 1198724 (1 minus 1205754 (119896))

[1 0] 119909 (119896) minus 037 ge 120576 + (1198985 minus 120576) 1205755 (119896)

[1 0] 119909 (119896) minus 037 le 1198725 (1 minus 1205755 (119896))

(35)

where 1198981 = minus016 1198721 = 090 1198982 = minus0377 1198722 = 06831198983 = minus053 1198723 = 053 1198984 = minus0683 1198724 = 0377 1198985 =

minus0901198725 = 016 and 120576 = 10minus6

In addition we introduce the auxiliary logical variables120575119894(119896) isin 0 1 119894 = 6 9 as follows

1205756 (119896) = (1 minus 1205751 (119896)) 1205752 (119896)

1205757 (119896) = (1 minus 1205752 (119896)) 1205753 (119896)

1205758 (119896) = (1 minus 1205753 (119896)) 1205754 (119896)

1205759 (119896) = (1 minus 1205754 (119896)) 1205755 (119896)

(36)

With 1205751 1205756 1205757 1205758 1205759 and 1 minus 1205755 the six regions canbe presented Similarly (36) can be transformed into theequivalent mixed logical inequalities

minus1205751 (119896) + 1205752 (119896) minus 1205756 (119896) le 0

1205751 (119896) + 1205756 (119896) le 1

minus1205752 (119896) + 1205756 (119896) le 0

minus1205752 (119896) + 1205753 (119896) minus 1205757 (119896) le 0

1205752 (119896) + 1205757 (119896) le 1

minus1205753 (119896) + 1205757 (119896) le 0

minus1205753 (119896) + 1205754 (119896) minus 1205758 (119896) le 0

1205753 (119896) + 1205758 (119896) le 1

minus1205754 (119896) + 1205758 (119896) le 0

minus1205754 (119896) + 1205755 (119896) minus 1205759 (119896) le 0

1205754 (119896) + 1205759 (119896) le 1

minus1205755 (119896) + 1205759 (119896) le 0

(37)

To describe the state space model of each region weintroduce the following auxiliary continuous variables

1199111 (119896) = (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) 1205751 (119896)

1199112 (119896) = (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) 1205756 (119896)

1199113 (119896) = (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) 1205757 (119896)

1199114 (119896) = (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) 1205758 (119896)

1199115 (119896) = (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) 1205759 (119896)

1199116 (119896) = (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) (1 minus 1205755 (119896))

(38)

Equation (38) can be converted to the equivalent mixedlogical inequalities

1199111 (119896) ge (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198721198911 (1 minus 1205751 (119896))

1199111 (119896) le (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198981198911 (1 minus 1205751 (119896))

1199111 (119896) ge 11989811989111205751 (119896)

1199111 (119896) le 11987211989111205751 (119896)

1199112 (119896) ge (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198721198912 (1 minus 1205756 (119896))

1199112 (119896) le (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198981198912 (1 minus 1205756 (119896))

1199112 (119896) ge 11989811989121205756 (119896)

1199112 (119896) le 11987211989121205756 (119896)

1199113 (119896) ge (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198721198913 (1 minus 1205757 (119896))

1199113 (119896) le (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198981198913 (1 minus 1205757 (119896))

1199113 (119896) ge 11989811989131205757 (119896)

1199113 (119896) le 11987211989131205757 (119896)

1199114 (119896) ge (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198721198914 (1 minus 1205758 (119896))

1199114 (119896) le (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198981198914 (1 minus 1205758 (119896))

1199114 (119896) ge 11989811989141205758 (119896)

1199114 (119896) le 11987211989141205758 (119896)

1199115 (119896) ge (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198721198915 (1 minus 1205759 (119896))

1199115 (119896) le (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198981198915 (1 minus 1205759 (119896))

10 International Journal of Aerospace Engineering

1199115 (119896) ge 11989811989151205759 (119896)

1199115 (119896) le 11987211989151205759 (119896)

1199116 (119896) ge (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11987211989151205755 (119896)

1199116 (119896) le (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11989811989151205755 (119896)

1199116 (119896) ge 1198981198916 (1 minus 1205755 (119896))

1199116 (119896) le 1198721198916 (1 minus 1205755 (119896))

(39)

where1198721198911 = [073 1084]1198791198981198911 = [minus077 minus1106]

1198791198721198912 =[076 1166]

119879 1198981198912 = [minus079 minus 1256]119879 1198721198913 =

[077 1273]119879 1198981198913 = [minus078 minus 1280]

119879 1198721198914 =

[076 1403]119879 1198981198914 = [minus076 minus 1403]

119879 1198721198915 =

[0725 1291]119879 1198981198915 = [minus0736 minus 1205]

119879 1198721198916 =

[0696 1154]119879 and1198981198916 = [minus073 minus 1130]

119879Then the whole MLD model of the missile is given by

119909 (119896 + 1) =

6

sum

119894=1

119911119894 (119896)

119910 (119896) = [1 0] 119909 (119896)

st (28) (30) (32) (34) (36) (38)

(40)

where (29) (31) (33) (35) (37) and (39) represent all theconstraint inequalities

4 Autopilot Design UsingHybrid MPC Method

In general we expect that the output 119910 tracks its command 119910119888

as fast as possible with a small amount of fuel consumptionMotivated by this observation we consider the followingoptimization problem

119869lowast

= min119906(119896)119906(119896+1)120575(119896)120575(119896+1|119896)119911(119896)119911(119896+1|119896)

119873

sum

119894=1

(1003817100381710038171003817119910 (119896 + 119894 | 119896)

minus119910119888 (119896 + 119894)1003817100381710038171003817

2

119876

+ 119906(119896 + 119894)2

119877)

st MLD model (39)

1199061min le 1199061 (119896) 1199061 (119896 + 1) le 1199061max

119909min le 119909 (119896) 119909 (119896 + 1) le 119909max

(41)

where 119910119888 is the command for angle of attack 119910(119896 + 119894 | 119896) ispredictive value of angle of attack 119873 denotes the predictivehorizon 119876 and 119877 are the weighting matrices and 119877 =

diag(119877120575119911

119877119865119910

) 119877120575119911

119877119865119910

are weighted coefficients of aerody-namic control surfaces and lateral pulse jets respectivelyThe

ratio of 119877120575119911

and 119877119865119910

represents the control allocation betweenaerodynamic control surfaces and lateral pulse jets When119877119865119910

119877120575119911

is increased the requirement for lateral force will bedeclined that is the fuel consumption can be lessened Inaddition the deduction can be verified from the simulationresults of Case 1 and Case 2

Different control allocation between the dual actuators(aerodynamic control surfaces and lateral pulse jets) can beobtained by setting different 119876 and 119877 Since both logicaland continuous variables are involved in (41) the aboveon-line optimization problem is a mixed integer quadraticprogramming (MIQP) problemwhich can be solved by usingthe hybrid MPC toolbox of MATLAB [14 15]

Remark 2 In an explicit MPC controller the main factorsaffecting the number of subregions include the systemdimen-sion the predictive horizon and the number of constrains Inthis paper the blendedmissile with aerodynamic control sur-faces and lateral jets is investigated To deal with the discreteproperty of the lateral jet forces some logical variables areintroduced for MPC controller design which increases thenumber of subregions inevitably

5 Numerical Simulations

51 Case 1 Numerical simulations are performed to ver-ify the feasibility of the proposed method in this sectionThe involved parameters are given in Table 1 The weightmatrices are set as 119876 = 10 119877120575

119911

= 0025 and 119877119865119910

=

diag(0005 0005 0005) Suppose the initial state is 1205720 =0 1199030 = 0 In order to avoid excessive computational load wechoose predictive horizon119873 = 2 and control horizon119872 = 2The command for angle of attack is 120572119888 = 02 rad

Explicit form of the optimal controller is provided byhybrid toolbox which is not given here for simplicity Insteadwe present the partition results associated with the controllerin Figure 6 where 4190 subregions are obtained and for eachsubregion a control law of the form 119906 = 119870119909 + 119887 is givenAs a benefit of using explicit MPC the controller parameters119870 and 119887 for all the subregions are obtained simultaneouslyonce the optimization problem is solved This property isobviously different from that of gain-scheduled controller(where the controller parameters for different regions areseparately determined)

However by a simple calculation we conclude that thesesubregions donot require toomuch storage space (the neededstorage space is approximately 14 times 8 times 4190 = 450KB)Moreover it should be noted that when the algorithm isimplemented in practice much more time will be needed toidentify which subregion the current states enter into This isanother important factor that affects the computational costof the proposed algorithm

The simulation results are shown in Figures 7ndash9 It isseen from Figure 7 that the actual angle of attack tracksthe command in less than 01 seconds As shown in Figures8 and 9 the aerodynamic force converges to a constant astime increases while the jets are only activated during thebeginning period (when the tracking error is obvious)

International Journal of Aerospace Engineering 11

58

68

73

108

174

180

245

252

254

270

271

272

273

274

276

277

299

300

301

302

06

04

02

0

minus02

minus04

Elev

ator

defl

ectio

n (r

ad)

Angle of attack (rad)minus04 minus02 0 02 04 06

1634

1635

1638

1648

1649

1650

1664

1673

1677

1678

1679

1712

1726

1743

1744

1746

1748

1751

1752

1983

1985

1993

2011

Late

ral f

orce

(N)

Angle of attack (rad)minus04 minus02 0 02 04 06

1000

800

600

400

200

0

minus200

minus400

minus600

minus800

minus1000

Polyhedral partitionmdash4190 regions Polyhedral partitionmdash4190 regions

Figure 6 The partition associated with the explicit controller

025

02

015

01

005

0

ActualCommand

0 02 04 06 08 1

Time (s)

Ang

le o

f atta

ck (r

ad)

Figure 7 The response of angle of attack

52 Case 2 The results for the case with a different weight119877120575119911

= 0005 are shown in Figures 10ndash12 It is seen that adifferent control allocation result is obtained

04

02

0

minus02

minus04

minus060 02 04 06 08 1

Time (s)

Elev

ator

defl

ectio

n (r

ad)

Figure 8 The response of elevator deflection

53 Case 3 In this case a different command 120572119888 =

02 cos(05120587119905) rad is considered and the weight matrices arethe same as the ones for Case 1 The simulation results areshown in Figures 13ndash15

It is seen from Figure 13 that asymptotic tracking is alsoachievedThemethod proposed in this paper can realize both

12 International Journal of Aerospace Engineering

1

05

0

minus05

minus1

times104

0 02 04 06 08 1

Time (s)

Late

ral f

orce

(N)

Figure 9 The response of lateral force

025

02

015

01

005

0

Ang

le o

f atta

ck (r

ad)

0 02 04 06 08 1

Time (s)

ActualCommand

Figure 10 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 02 04 06 08 1

Time (s)

Figure 11 The response of elevator deflection

1

05

0

minus05

minus1

Late

ral f

orce

(N)

times104

0 02 04 06 08 1

Time (s)

Figure 12 The response of lateral force

03

025

02

015

01

005

0

minus005

minus01

minus015

minus02

Ang

le o

f atta

ck (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

ActualCommand

Figure 13 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

Figure 14 The response of elevator deflection

International Journal of Aerospace Engineering 13

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1

Late

ral f

orce

(N)

0 05 1 15 2 25 3 35 4

Time (s)

times104

Figure 15 The response of lateral force

fast tracking command and control allocation It is seen fromFigure 15 that the jets are activated while the tracking erroris obvious or command is varying rapidly

6 Conclusion

An autopilot design method for a missile with aerodynamiccontrol surfaces and lateral jets is presented in this paperThe nonlinear attitude control model is reduced to an MLDmodel Meanwhile the lateral force is described as linearcombination of logical variables due to the discrete values oflateral force Then the whole MLD model of attitude controlsystem is derived Autopilot design is accomplished usinghybrid MPC method By setting the related weighted coeffi-cients in index function the control allocation is obtainedMoreover numerical simulations are performed under thedifferent conditions the performance of tracking the attitudecommand and control allocation is verified and the explicitform of the control law can be obtained as well

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research presented in this document is supportedby the National Natural Science Foundation of Chinaunder Grant nos 61104193 61203191 and 61333001 theFundamental Research Funds for the Central Universities(HITNSRIF2012032) the Program for IBRSEM in HarbinInstitute of Technology under Grant HITIBRSEMA201415and the Foundation of Supporting Technology for Aerospaceunder Grant 2014-HT-HGD7

References

[1] R Hirokawa K Sato and S Manabe ldquoAutopilot design fora missile with reaction-jet using coefficient diagram methodrdquoin Proceedings of the AIAA Guidance Navigation and ControlConference and Exhibit 2001 August 2001

[2] P K Menon and V R Iragavarapu Adaptive Techniques forMultiple Actuator Blending Defense Technical InformationCenter 1998

[3] A G Munson and M W Garbrick ldquoJet interaction investiga-tionrdquo Report OR-9933 Martin Marietta Orlando Fla USA1969

[4] M Graham and P Weinacht ldquoNumerical simulation of lateralcontrol jetsrdquo in Proceedings of the 37th Aerospace SciencesMeeting and Exhibit 1999 AIAA Paper

[5] J S Shamma and J R Cloutier ldquoGain-scheduledmissile autopi-lot design using linear parameter varying transformationsrdquoJournal of Guidance Control and Dynamics vol 16 no 2 pp256ndash263 1993

[6] M Innocenti and A Thukral ldquoSimultaneous reaction jet andaerodynamic control of missile systemsrdquo AlAA 93-3739-CP1993

[7] D B Ridgely Y Lee and T Fanciullo ldquoDual aeropropulsivemissile control-optimal control and control allocationrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit Keystone Colo USA 2006

[8] D B Ridgely D Drake L Triplett and C Geise ldquoDynamiccontrol allocation of a missile with tails and reaction jetsrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 3158ndash3189 August 2007

[9] D Q Mayne J B Rawlings C V Rao and P O ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[10] A Bemporad F Borrelli and M Morari ldquoModel predictivecontrol based on linear programmingmdashthe explicit solutionrdquoIEEE Transactions on Automatic Control vol 47 no 12 pp1974ndash1985 2002

[11] C Jouannet and P Krus ldquoModelling of high angle of attackaerodynamicrdquo in Proceedings of the 25th AIAA Applied Aerody-namics Conference pp 1481ndash1516 June 2007

[12] S Venugopal and M Krishnamurthy ldquoMissile aerodynamics athigh angles of attack a prediction coderdquo Journal of Spacecraftand Rockets vol 32 no 2 pp 263ndash269 1995

[13] A Bemporad ldquoEfficient conversion of mixed logical dynamicalsystems into an equivalent piecewise affine formrdquo IEEE Trans-actions on Automatic Control vol 49 no 5 pp 832ndash838 2004

[14] F D Torrisi andA Bemporad ldquoHYSDELmdasha tool for generatingcomputational hybrid models for analysis and synthesis prob-lemsrdquo IEEE Transactions on Control Systems Technology vol 12no 2 pp 235ndash249 2004

[15] A Bemporad W P M H Heemels and B De SchutterldquoOn hybrid systems and closed-loop MPC systemsrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 863ndash8692002

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2 International Journal of Aerospace Engineering

these facts this paper attempts to design an autopilot usingexplicit hybrid MPC for blended missiles by noting thatMPC is a promising methodology for the control problem ofconstrained uncertain systems [9] and that the computationalburden of on-line optimization is effectively reduced by usingexplicit MPC instead of traditional MPC [10]

The remainder of this paper is as follows Section 2gives a mathematical model of blended missile including theconfiguration of reaction jets In Section 3 the piecewiseaffine model of blended missile is established followed by anMLD model which is obtained based on the equivalence ofpiecewise affine model and mixed logical dynamical modelIn Section 4 a hybrid MPC based method for autopilotdesign is proposed and an explicit control law is constructedIn Section 5 the effectiveness of the proposed methodis verified by simulation cases under different conditionsFinally several concluding remarks are given in Section 6

2 Mathematical Model of the Missile

The plane reference coordinate system 119874119880119881119882 the bodycoordinate system 119874119909119887119910119887119911119887 the trajectory coordinate sys-tem 119874119909119908119910119908119911119908 and velocity coordinate system 119874119909V119910V119911V areinvolved in this paper Figure 1 shows a missile with some keyvariables and identified axes The axes 119911119908 119911V are not givenwhose directions can be determined by the right hand rule

21 Missile Dynamic Model Thenonlinear motion equationsare given by

119898 = 119875 cos120572 cos120573 minus 119883119886 minus 119898119892 sin 120579 + 119865119886

119909119908

119898119881 120579 = 119875 (sin120572 cos 120574V + cos120572 sin120573 sin 120574V) + 119884119886 cos 120574V

minus 119898119892 cos 120579 + 119865119886

119910119908

119898119881 cos 120579V = minus 119875 (sin120572 sin 120574V minus cos120572 sin120573 cos 120574V)

minus 119884119886 sin 120574V minus 119885119886 cos 120574V minus 119865119886

119911119908

(1)

119869119909 = (119869119910 minus 119869119911) 119903119902 + 119872119909

119869119910 119902 = (119869119911 minus 119869119909) 119901119903 + 119872119910

119869119911 119903 = (119869119909 minus 119869119910) 119901119902 + 119872119911

(2)

where 119865119886

119909119908

119865119886

119910119908

119865119886

119911119908

are lateral forces in the 119909 119910 and 119911

directions of trajectory coordinate system respectively Forsimplicity we suppose that the body is symmetric about the119909-axis that is 119869119909119910 = 119869119910119911 = 119869119911119909 = 0 In (2) each of themoments119872119909119872119910 and119872119911 contains two components that are generatedby aerodynamic surface and the lateral pulse jets respectively

119872119909 = 119872119890119909 + 119872119886

119909119887

119872119910 = 119872119890119910 + 119872119886

119910119887

119872119911 = 119872119890119911 + 119872119886

119911119887

(3)

yw

y

V(yb)

q

120574 xw(x)

120573120572

120579O

r

W(zb)

Jets

U(xb)

120595 p

Figure 1 Key coordinate systems

where 119872119890119909 119872119890119910 and 119872119890119911 denote aerodynamic momentcomponents and 119872

119886

119909119887

119872119886119910119887

and 119872119886

119911119887

are pulse jet momentcomponents

22 Lateral Jet Forces and Moments Model As shown inFigure 1 the lateral jet force is generated by 180 pulse jetslocated in front of the center of mass of PAC-3 These jets aredivided into 10 rings and arranged in staggered positions (18pulse jets are included in each ring) In each ring these jetsare uniformly distributed and the central angle between twoneighboring jets is 20 degree Use 119894 (119894 = 1 2 10) and 119895

(119895 = 1 2 18) to denote the ringrsquos label and the jetrsquos labelin each ring respectivelyThe distance between ring 119894 and thecenter ofmass is denoted by 119897119894 while spacing of adjacent ringsis Δ119897 The layout scheme of pulse jets is shown in Figures 2and 3 Assume the force generated by each individual jet isa constant 119865119898 In body coordinate system the lateral forcegenerated by the (119894 119895) pulse jet is given by

[[

[

119865119894119895

119909119887

119865119894119895

119910119887

119865119894119895

119911119887

]]

]

=

[[[[[

[

0

119865119898 cos(2119895 minus 119894lowast

18120587)

minus119865119898 sin(2119895 minus 119894lowast

18120587)

]]]]]

]

(4)

The corresponding moment is given by

[[

[

119872119894119895

119909119887

119872119894119895

119910119887

119872119894119895

119911119887

]]

]

=

[[[[[

[

0

119865119898119897119894 sin(2119895 minus 119894lowast

18120587)

119865119898119897119894 cos(2119895 minus 119894lowast

18120587)

]]]]]

]

(5)

International Journal of Aerospace Engineering 3

12

3

4

5

6

7

8

910

11

12

13

14

15

16

17

18

yb

zb

(a) Odd-numbered rings

1

2

3

4

5

6

7

8

91011

12

13

14

15

16

17

18

yb

zb

(b) Even-numbered rings

Figure 2 The layout scheme of lateral pulse jets

where

119894lowast= 2 119894 is odd

119894lowast= 1 119894 is even

(6)

For the situation where all pulse jets are fired at the sametime the total force and moment are given by

[

[

119865119909119887

119865119910119887

119865119911119887

]

]

=

[[[[[[[[[[

[

0

119894=10

sum

119894=1

119895=18

sum

119895=1

119865119894119895

119910119887

119894=10

sum

119894=1

119895=18

sum

119895=1

119865119894119895

119911119887

]]]]]]]]]]

]

[

[

119872119909119887

119872119910119887

119872119911119887

]

]

=

[[[[[[[[[[

[

0

minus

119894=10

sum

119894=1

119895=18

sum

119895=1

119865119894119895

119911119887

119897119894

119894=10

sum

119894=1

119895=18

sum

119895=1

119865119894119895

119910119887

119897119894

]]]]]]]]]]

]

(7)

In order to avoid the coupling between the pitch momentand yaw moment each ring is divided into four controlregions positive pitch negative pitch positive yaw andnegative yaw control region as shown in Figure 4

The autopilot design of PAC-3 is more complicated thanthat of other conventional missiles which are controlled

12

88

99

1010

1111

1212

xb

zb

i

lil1

Δl

middot middot middotmiddot middot middot

O

Figure 3 The ring frames expansion of lateral pulse jets

only by aerodynamic surfaces due to the hybrid propertyof control inputs and the on-off property of pulse jet (thepulse jet can be fired only one time) To deal with thisproblem the work [7] proposed two-step design procedures(1) in the first step neglect the hybrid property (or on-offproperty) and design the expected force andmoment signals(2) in the other step the fire logic is derived by solving theretaliation problem of these signals with taking into accountthe hybrid property (or on-off property) Different fromthe above traditional procedures a novel procedure will bepresented in what follows where only one step is included Inpractical applications only a small number of jets (in a certainof rings) are activated over a finite time interval To make theidea of the following development clear we here consider asimple but representative situation where no more than two

4 International Journal of Aerospace Engineering

yb

zb

Negative pitchcontrol region

Negative yawcontrol region

Positive pitchcontrol region

Positive yawcontrol region

17

8

910

11

12

12

3

4

5

6

713

14

15

16

18

(a) Odd-numbered rings

1

2

3

4

5

6

7

8

910

11

12

13

14

15

16

17

18

yb

zb

Negative pitchcontrol region

Negative yawcontrol region

Positive pitchcontrol region

Positive yawcontrol region

(b) Even-numbered rings

Figure 4 Schematic of control regions

rings are allowed to be fired simultaneously and no morethan two jets are activated in each fired ring Meanwhile itshould be ensured that only odd rings or even rings are firedand the jets are fired symmetrically about the correspondingsymmetry axis of each control region

Take the positive pitch control region as an example Theforces provided by the jets (119894 1) (119894 2) (119894 3) (119894 17) and (119894 18)

in an odd ring are given by

119865119900 = [119865119898 119865119898 cos120587

9119865119898 cos

2120587

9119865119898 cos

2120587

9119865119898 cos

120587

9]

119879

(8)

Similarly the forces provided by the jets (119894 1) (119894 2) (119894 17)and (119894 18) in an even ring are given by

119865119890 = [119865119898 cos120587

18119865119898 cos

120587

6119865119898 cos

120587

6119865119898 cos

120587

18]

119879

(9)

where119865119900 and119865119890 denote the forces associated with an odd ringand an even ring respectively

Use 119865119894 to denote the lateral force generated by ring 119894

Clearly 119865119894 should satisfy the condition

1198651= 1198659 119865

3= 1198657 119865

2= 11986510 119865

4= 1198658 (10)

When odd ring is fired the lateral force and moment aregiven by

119865119910119887

isin 119865119898 2119865119898 cos120587

9 2119865119898 cos

2120587

9 2119865119898 4119865119898 cos

120587

9

4119865119898 cos2120587

9

119872119911119887

isin 1198651198981198975 21198651198981198975 cos120587

9 21198651198981198975 cos

2120587

9 21198651198981198975

41198651198981198975 cos120587

9 41198651198981198975 cos

2120587

9

(11)

While even ring is fired the lateral force and moment aregiven by

119865119910119887

isin 2119865119898 cos120587

18 2119865119898 cos

120587

6 4119865119898 cos

120587

18 4119865119898 cos

120587

6

119872119911119887

isin 21198651198981198976 cos120587

18 21198651198981198976 cos

120587

6 41198651198981198976 cos

120587

18

41198651198981198976 cos120587

6

(12)

Noting Δ119897 is quite small we consider 1198975 ≃ 1198976 = 119897 Inorder to ensure that the jets fire efficiency the jets resultingin small moment components along 119911119887 axis are not activatedAs a result the sets of forces and moments are respectivelygiven by

119880119910+

119865= 119865119898 2119865119898 cos

120587

9 2119865119898 cos

120587

18 2119865119898 4119865119898 cos

120587

9

4119865119898 cos120587

18

(13)

119880119910+

119872= 119865119898119897 2119865119898119897 cos

120587

9 2119865119898119897 cos

120587

18 2119865119898119897 4119865119898119897 cos

120587

9

4119865119898119897 cos120587

18

(14)

where 119865119910119887

isin 119880119910+

119865and119872119911

119887

isin 119880119910+

119872 During each control period

a control moment belonging to119880119910+

119872will be used as the input

The mutual interference between high-speed jet streamand air leads to lateral jet interference effect In orderto take into account this interference force and moment

International Journal of Aerospace Engineering 5

amplification factors 119870119865119910

119870119865119911

119870119872119910

and 119870119872119911

are introducedas in [3 4]Then the resulting lateral forces andmoments are

[[[

[

119865119886

119909119887

119865119886

119910119887

119865119886

119911119887

]]]

]

= [

[

0

119865119910119887

+ 119870119865119910

119865119910119887

119865119911119887

+ 119870119865119911

119865119911119887

]

]

[[[

[

119872119886

119909119887

119872119886

119910119887

119872119886

119911119887

]]]

]

= [

[

0

119872119910119887

+ 119870119872119910

119872119910119887

119872119911119887

+ 119870119872119911

119872119911119887

]

]

(15)

Remark 1 In fact each pulse jet can be fired only once so thelocation of the fired jet cannot provide force anymore Basedon this precondition elements of sets 119880

119910+

119865and 119880

119910+

119872will be

less and less over time In this paper quantity change of setsrsquoelements is not considered to simplify the problem

23 Attitude Control Model Some transformation and sim-plification are applied to themissile model for control designIt is assumed that themissilersquosmass is of a constant valueNotethat the goal is to establish the angle of attack and sideslipangle The attitude control model is given by

= 119903 + 119902 sin120572 tan120573 minus

119876119878 (119862120572

119910120572 + 119862

120575119911

119910120575119911) cos120572

119898119881 cos120573

minus

(119865119910119887

+ 119870119865119910

119865119910119887

) cos120572

119898119881 cos120573minus

119866119910 cos120572119898119881 cos120573

120573 = 119902 cos120572 +

(119876119878 (119862120573

119911120573 + 119862

120575119910

119911 120575119910) + 119865119911119887

+ 119870119865119911

119865119911119887

) cos120573

119898119881

+

(119876119878 (119862120572

119910120572 + 119862

120575119911

119910120575119911) + 119865119910

119887

+ 119870119865119910

119865119910119887

) sin120572 sin120573

119898119881

+119866119911 cos120573

119898119881+

119866119910 sin120572 sin120573

119898119881

119902 =119872119910119887

119869119910

+

119870119872119910

119872119910119887

119869119910

+

119876119878119871119898120573

119910120573

119869119910

+119876119878119871119898

120575119910

119910 120575119910

119869119910

+

119876119878119871119898119902

119910119902

119869119910

119903 =119872119911119887

119869119911

+119870119872119911

119872119911119887

119869119911

+119876119878119871119898

120572

119911120572

119869119911

+119876119878119871119898

120575119911

119911120575119911

119869119911

+119876119878119871119898

119903

119911119903

119869119911

(16)

where 119898120573

119910 119898120575119911

119910 119898119902

119910 119898120572

119911 119898120575119911

119911 and 119898

119903

119911are aerodynamic

parameters

3 Mixed Logical Dynamical Model ofBlended Missile

31 Piecewise Affine Model of Blended Missile To simplifyanalysis the gravity term and the channel coupling term are

Table 1 Missilersquos overall parameters

Missilersquos takeoff mass [kg] 255Missilersquos full-length [m] 486Missilersquos diameter [m] 0317Distance between missile head and center ofmass [m] 2569

Moment of inertia 119869119909 119869119910 119869119911 [kgsdotm2] 303 3063 3063

Lateral force provided by individual jets [N] 2200Distance between jet ring center and centerof mass [m] 126

Range of angle of attack [rad] minus120587

6le 120572 le

120587

6

Range of pitch angular velocity [rads] minus5120587

3le 119903 le

5120587

3

Range of elevator deflection [rad] minus120587

6le 120575119911 le

120587

6

ignoredWith (16) themissile attitude control model of pitchchannel is

= 119903 minus

119876119878 (119862120572

119910120572 + 119862

120575119911

119910120575119911) cos120572

119898119881minus

(1 + 119870119865119910

) 119865119910119887

cos120572

119898119881

119903 =

(1 + 119870119872119911

) 119865119910119887

119897

119869119911

+119876119878119871119898

120572

119911120572

119869119911

+119876119878119871119898

120575119911

119911120575119911

119869119911

+119876119878119871119898

119903

119911119903

119869119911

(17)

Choose 119909 = [120572 119903]119879 as system state and 119906 = [120575119911 119865119910

119887

]119879 as

control input The considered output is 119910 = 120572 Then (17) canbe rewritten into the following state space form

= 119891 (119909) + 119892 (119909) 119906

119910 = [1 0] 119909

(18)

where

119891 (119909) =

[[[[

[

119903 minus

119876119878119862120572

119910120572 cos120572

119898119881

119876119878119871 (119898120572

119911120572 + 119898

119903

119911119903)

119869119911

]]]]

]

119892 (119909) =

[[[[[[

[

minus

119876119878119862120575119911

119910cos120572

119898119881minus

(1 + 119870119865119910

) cos120572

119898119881

119876119878119871119898120575119911

119911

119869119911

(1 + 119870119872119911

) 119897

119869119911

]]]]]]

]

(19)

The missile parameters are presented in Table 1Generally the aerodynamic coefficients 119862

120572

119910 119862120575119911

119910 119898120572

119911

and 119898120575119911

119911and the amplification factors 119870119865

119910

119870119872119911

are mainlyaffected by the flight velocity 119881 and the angle of attack 120572

[11 12] Since the terminal guidance phase is consideredin this paper the flight time is quite short and the flightvelocity of the missile can be treated as a constant Thusthe aerodynamic coefficients and the amplification factors are

6 International Journal of Aerospace Engineering

mainly affected by the angle of attack 120572 The relationshipsbetween them are shown in Figure 5

In practical application since 120572 is the main factor thatleads to system nonlinearities the system model is usuallylinearized if 120572 varies in small range As seen from Figure 5curves of the relation between aerodynamic parametersand angle of attack can be expressed by six line segmentsapproximately Here we choose 120572 = minus037 rad minus0153 rad0 0153 rad and 037 rad as the operation points and dividethe whole operation region into six subregions As a resultthe original model (17) can be converted to the followingpiecewise affine models

=

1198861119909 + 1198871119906 + 1198901 [1 0] 119909 le minus037

1198862119909 + 1198872119906 + 1198902 minus037 lt [1 0] 119909 le minus0153

1198863119909 + 1198873119906 + 1198903 minus0153 lt [1 0] 119909 le 0

1198864119909 + 1198874119906 + 1198904 0 lt [1 0] 119909 le 0153

1198865119909 + 1198875119906 + 1198905 0153 lt [1 0] 119909 le 037

1198866119909 + 1198876119906 + 1198906 [1 0] 119909 gt 037

(20)

119910 = [1 0] 119909 (21)

where

119886119894 =120597119891(119909)

120597119909

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= [11988611

11989411988612

119894

11988621

11989411988622

119894

]

119887119894 = 119892 (1199091198940) = [11988711

11989411988712

119894

11988721

11989411988722

119894

] 119890119894 = [1198901

119894

1198902

119894

]

11988611

119894=

1205971198911 (119909)

120597120572

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= minus119876119878

119898119881(

120597119862120572

119910

120597120572

1003816100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1205721198940 cos1205721198940 + 119862120572

119910(1205721198940) cos1205721198940

minus 119862120572

119910(1205721198940) 1205721198940 sin1205721198940)

11988612

119894=

1205971198911(119909)

120597119903

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= 1

11988621

119894=

1205971198912 (119909)

120597120572

10038161003816100381610038161003816100381610038161003816119909=1199091198940

=119876119878119871

119869119911

(120597119898120572

119911

120597120572

100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1205721198940 + 119898120572

119911(1205721198940) +

120597119898119903

119911

120597120572

100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1199031198940)

11988622

119894=

1205971198912(119909)

120597119903

10038161003816100381610038161003816100381610038161003816119909=1199091198940

=119876119878119871

119869119911

119898120572

119911(1205721198940)

11988711

119894= minus

119876119878119862120575119911

119910(1205721198940) cos1205721198940119898119881

11988712

119894= minus

(1 + 119870119865119910

(1205721198940)) cos1205721198940119898119881

11988721

119894=

119876119878119871119898120575119911

119911(1205721198940)

119869119911

11988722

119894=

(1 + 119870119872119911

(1205721198940)) 119897

119869119911

1198901

119894= 11988611

1198941205721198940 + 119886

12

1198941199031198940

1198902

119894= 11988621

1198941205721198940 + 119886

22

1198941199031198940

(22)

where 119894 (119894 = 1 2 6) is the label corresponding to the 119894thregion

From Table 1 and Figure 5 we get the set of aerodynamicparameters at the point (119867119881) = (20 km 1000ms) as shownin Table 2

Choose the sampling period 119879119904 = 0025 s The discretestate-space expression is then given by

119909 (119896 + 1)

=

1198861119909 (119896) + 1119906 (119896) + 1198901 [1 0] 119909 (119896) le minus037

1198862119909 (119896) + 2119906 (119896) + 1198902 minus037 lt [1 0] 119909 (119896) le minus0153

1198863119909 (119896) + 3119906 (119896) + 1198903 minus0153 lt [1 0] 119909 (119896) le 0

1198864119909 (119896) + 4119906 (119896) + 1198904 0 lt [1 0] 119909 (119896) le 0153

1198865119909 (119896) + 5119906 (119896) + 1198905 0153 lt [1 0] 119909 (119896) le 037

1198866119909 (119896) + 6119906 (119896) + 1198906 [1 0] 119909 (119896) gt 037

(23)

119910 (119896) = [1 0] 119909 (119896) (24)

where

1198861 = [104 0025

022 0995] 1198862 = [

1051 0025

0218 0995]

1198863 = [1023 00252

0248 09951] 1198864 = [

09954 00248

02732 09954]

1198865 = [09697 00245

02913 09956] 1198866 = [

09594 00244

03137 09959]

1198901 = [minus00183

minus01123] 1198902 = [

minus00175

minus00792]

1198903 = [minus00131

minus00376] 1198904 = [

0

0]

1198905 = [minus00053

00454] 1198906 = [

minus00169

01193]

1 = [minus0028 159 times 10

minus6

minus2166 142 times 10minus4]

2 = [minus003 218 times 10

minus6

minus2224 187 times 10minus4]

International Journal of Aerospace Engineering 7

2

15

1

05

0

minus05

minus1

minus15minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SmV

)C120572 y

(a) Relation between 119862120572119910and 120572

0092

009

0088

0086

0084

0082

008

0078

0076

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SmV

)C120575119911y

(b) Relation between 119862120575119911119910 and 120572

125

12

115

11

105

10

95

9minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SLJz)m

120572 z

(c) Relation between119898120572119911and 120572

96

94

92

90

88

86

84minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (minusQSL

Jz)m

120575119911z

(d) Relation between119898120575119911119911 and 120572

1

09

08

07

06

05

04

03

02

01

Am

plifi

catio

n fa

ctorKF

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

(e) Relation between119870119865119910

and 120572

12

11

1

09

08

07

06

05

04

03

Am

plifi

catio

n fa

ctorKM119911

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

(f) Relation between119870119872119911

and 120572

Figure 5 Aerodynamic parameters as functions of angle of attack

8 International Journal of Aerospace Engineering

Table 2 Aerodynamic parameters of pitch channel

Aerodynamic parameter 119876119878119862120572

119910119898119881 119876119878119862

120575119911

119910119898119881 119876119878119871119898

120572

119911119869119911

119876119878119871119898119903

119911119869119911

119876119878119871119898120575119911

119911119869119911

120572 = minus053 minus1175 0076 947 minus032 minus8489

120572 = minus037 minus0944 0082 973 minus032 minus8924

120572 = minus0153 minus023 0082 1043 minus032 minus8938

120572 = 0 036 009 1101 minus032 minus9510

120572 = 0153 094 0082 1158 minus032 minus8934

120572 = 037 163 0082 1224 minus032 minus8918

3 = [minus003 268 times 10

minus6

minus2228 223 times 10minus4]

4 = [minus0032 277 times 10

minus6

minus2371 232 times 10minus4]

5 = [minus003 227 times 10

minus6

minus2228 198 times 10minus4]

6 = [minus0029 191 times 10

minus6

minus2224 173 times 10minus4]

(25)

32 Constraints Analysis Due to the symmetry of jet config-uration the set of possible negative pitch control force is givenby

119880119910minus

119865= minus119865119898 minus2119865119898 cos

120587

9 minus2119865119898 cos

120587

18 minus2119865119898 minus4119865119898 cos

120587

9

minus4119865119898 cos120587

18

(26)

By combining (13) and (26) we obtain the set of allpossible pitch control force

119880119910

119865= 119865119898 2119865119898 cos

120587

9 2119865119898 cos

120587

18 2119865119898 4119865119898 cos

120587

9

4119865119898 cos120587

18 minus119865119898 minus2119865119898 cos

120587

9 minus2119865119898 cos

120587

18 minus2119865119898

minus4119865119898 cos120587

9 minus4119865119898 cos

120587

18

(27)

Substituting the jet parameters shown in Table 1 into theforegoing set yields

119880119910

119865= 2200 4135 4333 4400 8269 8666 minus2200 minus4135

minus4333 minus4400 minus8269 minus8666

(28)

Noting that the lateral forces are discrete variable we willutilize the linear combination of logical variables to describethe lateral force In terms of piecewise affine model (20) we

introduce logical variables 120575119865119894

isin 0 1 119894 = 1 2 12 toexpress the lateral thrust 119865119910

119887

119865119910119887

= 22001205751198651

+ 41351205751198652

+ 43331205751198653

+ 44001205751198654

+ 82691205751198655

+ 86661205751198656

minus 22001205751198657

minus 41351205751198658

minus 43331205751198659

minus 440012057511986510

minus 826912057511986511

minus 866612057511986512

(29)

In (29) the logical variables should satisfy the constraints

12

sum

119894=1

120575119865119894

= 0 or 1 (30)

where 0 means that no lateral force is generated while 1

means that the applied lateral force equals to one element ofset 119880119910119865

Denote 1199061 = 120575119911 the control input 119906 in model (23) can berewritten as

119906 = [1199061 119865119910119887

]119879 (31)

As shown in Table 1 the constraints on system states andcontrol input are

119909min le 119909 (119896) le 119909max

1199061min le 1199061 (119896) le 1199061max(32)

where 119909min = [minus053 minus522]119879 119909max = [053 522]

119879 1199061min =

minus053 1199061max = 053According to (30) we have

12

sum

119894=1

120575119865119894

le 1 (33)

which shows the constraint on the logical control inputUse logical variables 120575119894(119896) isin 0 1 119894 = 1 2 5 to

describe the operation points which satisfy the constraints

[1 0] 119909 (119896) + 037 le 0 lArrrArr 1205751 (119896) = 1

[1 0] 119909 (119896) + 0153 le 0 lArrrArr 1205752 (119896) = 1

[1 0] 119909 (119896) le 0 lArrrArr 1205753 (119896) = 1

[1 0] 119909 (119896) minus 0153 le 0 lArrrArr 1205754 (119896) = 1

[1 0] 119909 (119896) minus 037 le 0 lArrrArr 1205755 (119896) = 1

(34)

International Journal of Aerospace Engineering 9

Equation (34) can be transformed into the followingequivalent mixed logical inequalities [13]

[1 0] 119909 (119896) + 037 ge 120576 + (1198981 minus 120576) 1205751 (119896)

[1 0] 119909 (119896) + 037 le 1198721 (1 minus 1205751 (119896))

[1 0] 119909 (119896) + 0153 ge 120576 + (1198982 minus 120576) 1205752 (119896)

[1 0] 119909 (119896) + 0153 le 1198722 (1 minus 1205752 (119896))

[1 0] 119909 (119896) ge 120576 + (1198983 minus 120576) 1205753 (119896)

[1 0] 119909 (119896) le 1198723 (1 minus 1205753 (119896))

[1 0] 119909 (119896) minus 0153 ge 120576 + (1198984 minus 120576) 1205754 (119896)

[1 0] 119909 (119896) minus 0153 le 1198724 (1 minus 1205754 (119896))

[1 0] 119909 (119896) minus 037 ge 120576 + (1198985 minus 120576) 1205755 (119896)

[1 0] 119909 (119896) minus 037 le 1198725 (1 minus 1205755 (119896))

(35)

where 1198981 = minus016 1198721 = 090 1198982 = minus0377 1198722 = 06831198983 = minus053 1198723 = 053 1198984 = minus0683 1198724 = 0377 1198985 =

minus0901198725 = 016 and 120576 = 10minus6

In addition we introduce the auxiliary logical variables120575119894(119896) isin 0 1 119894 = 6 9 as follows

1205756 (119896) = (1 minus 1205751 (119896)) 1205752 (119896)

1205757 (119896) = (1 minus 1205752 (119896)) 1205753 (119896)

1205758 (119896) = (1 minus 1205753 (119896)) 1205754 (119896)

1205759 (119896) = (1 minus 1205754 (119896)) 1205755 (119896)

(36)

With 1205751 1205756 1205757 1205758 1205759 and 1 minus 1205755 the six regions canbe presented Similarly (36) can be transformed into theequivalent mixed logical inequalities

minus1205751 (119896) + 1205752 (119896) minus 1205756 (119896) le 0

1205751 (119896) + 1205756 (119896) le 1

minus1205752 (119896) + 1205756 (119896) le 0

minus1205752 (119896) + 1205753 (119896) minus 1205757 (119896) le 0

1205752 (119896) + 1205757 (119896) le 1

minus1205753 (119896) + 1205757 (119896) le 0

minus1205753 (119896) + 1205754 (119896) minus 1205758 (119896) le 0

1205753 (119896) + 1205758 (119896) le 1

minus1205754 (119896) + 1205758 (119896) le 0

minus1205754 (119896) + 1205755 (119896) minus 1205759 (119896) le 0

1205754 (119896) + 1205759 (119896) le 1

minus1205755 (119896) + 1205759 (119896) le 0

(37)

To describe the state space model of each region weintroduce the following auxiliary continuous variables

1199111 (119896) = (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) 1205751 (119896)

1199112 (119896) = (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) 1205756 (119896)

1199113 (119896) = (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) 1205757 (119896)

1199114 (119896) = (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) 1205758 (119896)

1199115 (119896) = (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) 1205759 (119896)

1199116 (119896) = (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) (1 minus 1205755 (119896))

(38)

Equation (38) can be converted to the equivalent mixedlogical inequalities

1199111 (119896) ge (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198721198911 (1 minus 1205751 (119896))

1199111 (119896) le (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198981198911 (1 minus 1205751 (119896))

1199111 (119896) ge 11989811989111205751 (119896)

1199111 (119896) le 11987211989111205751 (119896)

1199112 (119896) ge (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198721198912 (1 minus 1205756 (119896))

1199112 (119896) le (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198981198912 (1 minus 1205756 (119896))

1199112 (119896) ge 11989811989121205756 (119896)

1199112 (119896) le 11987211989121205756 (119896)

1199113 (119896) ge (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198721198913 (1 minus 1205757 (119896))

1199113 (119896) le (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198981198913 (1 minus 1205757 (119896))

1199113 (119896) ge 11989811989131205757 (119896)

1199113 (119896) le 11987211989131205757 (119896)

1199114 (119896) ge (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198721198914 (1 minus 1205758 (119896))

1199114 (119896) le (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198981198914 (1 minus 1205758 (119896))

1199114 (119896) ge 11989811989141205758 (119896)

1199114 (119896) le 11987211989141205758 (119896)

1199115 (119896) ge (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198721198915 (1 minus 1205759 (119896))

1199115 (119896) le (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198981198915 (1 minus 1205759 (119896))

10 International Journal of Aerospace Engineering

1199115 (119896) ge 11989811989151205759 (119896)

1199115 (119896) le 11987211989151205759 (119896)

1199116 (119896) ge (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11987211989151205755 (119896)

1199116 (119896) le (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11989811989151205755 (119896)

1199116 (119896) ge 1198981198916 (1 minus 1205755 (119896))

1199116 (119896) le 1198721198916 (1 minus 1205755 (119896))

(39)

where1198721198911 = [073 1084]1198791198981198911 = [minus077 minus1106]

1198791198721198912 =[076 1166]

119879 1198981198912 = [minus079 minus 1256]119879 1198721198913 =

[077 1273]119879 1198981198913 = [minus078 minus 1280]

119879 1198721198914 =

[076 1403]119879 1198981198914 = [minus076 minus 1403]

119879 1198721198915 =

[0725 1291]119879 1198981198915 = [minus0736 minus 1205]

119879 1198721198916 =

[0696 1154]119879 and1198981198916 = [minus073 minus 1130]

119879Then the whole MLD model of the missile is given by

119909 (119896 + 1) =

6

sum

119894=1

119911119894 (119896)

119910 (119896) = [1 0] 119909 (119896)

st (28) (30) (32) (34) (36) (38)

(40)

where (29) (31) (33) (35) (37) and (39) represent all theconstraint inequalities

4 Autopilot Design UsingHybrid MPC Method

In general we expect that the output 119910 tracks its command 119910119888

as fast as possible with a small amount of fuel consumptionMotivated by this observation we consider the followingoptimization problem

119869lowast

= min119906(119896)119906(119896+1)120575(119896)120575(119896+1|119896)119911(119896)119911(119896+1|119896)

119873

sum

119894=1

(1003817100381710038171003817119910 (119896 + 119894 | 119896)

minus119910119888 (119896 + 119894)1003817100381710038171003817

2

119876

+ 119906(119896 + 119894)2

119877)

st MLD model (39)

1199061min le 1199061 (119896) 1199061 (119896 + 1) le 1199061max

119909min le 119909 (119896) 119909 (119896 + 1) le 119909max

(41)

where 119910119888 is the command for angle of attack 119910(119896 + 119894 | 119896) ispredictive value of angle of attack 119873 denotes the predictivehorizon 119876 and 119877 are the weighting matrices and 119877 =

diag(119877120575119911

119877119865119910

) 119877120575119911

119877119865119910

are weighted coefficients of aerody-namic control surfaces and lateral pulse jets respectivelyThe

ratio of 119877120575119911

and 119877119865119910

represents the control allocation betweenaerodynamic control surfaces and lateral pulse jets When119877119865119910

119877120575119911

is increased the requirement for lateral force will bedeclined that is the fuel consumption can be lessened Inaddition the deduction can be verified from the simulationresults of Case 1 and Case 2

Different control allocation between the dual actuators(aerodynamic control surfaces and lateral pulse jets) can beobtained by setting different 119876 and 119877 Since both logicaland continuous variables are involved in (41) the aboveon-line optimization problem is a mixed integer quadraticprogramming (MIQP) problemwhich can be solved by usingthe hybrid MPC toolbox of MATLAB [14 15]

Remark 2 In an explicit MPC controller the main factorsaffecting the number of subregions include the systemdimen-sion the predictive horizon and the number of constrains Inthis paper the blendedmissile with aerodynamic control sur-faces and lateral jets is investigated To deal with the discreteproperty of the lateral jet forces some logical variables areintroduced for MPC controller design which increases thenumber of subregions inevitably

5 Numerical Simulations

51 Case 1 Numerical simulations are performed to ver-ify the feasibility of the proposed method in this sectionThe involved parameters are given in Table 1 The weightmatrices are set as 119876 = 10 119877120575

119911

= 0025 and 119877119865119910

=

diag(0005 0005 0005) Suppose the initial state is 1205720 =0 1199030 = 0 In order to avoid excessive computational load wechoose predictive horizon119873 = 2 and control horizon119872 = 2The command for angle of attack is 120572119888 = 02 rad

Explicit form of the optimal controller is provided byhybrid toolbox which is not given here for simplicity Insteadwe present the partition results associated with the controllerin Figure 6 where 4190 subregions are obtained and for eachsubregion a control law of the form 119906 = 119870119909 + 119887 is givenAs a benefit of using explicit MPC the controller parameters119870 and 119887 for all the subregions are obtained simultaneouslyonce the optimization problem is solved This property isobviously different from that of gain-scheduled controller(where the controller parameters for different regions areseparately determined)

However by a simple calculation we conclude that thesesubregions donot require toomuch storage space (the neededstorage space is approximately 14 times 8 times 4190 = 450KB)Moreover it should be noted that when the algorithm isimplemented in practice much more time will be needed toidentify which subregion the current states enter into This isanother important factor that affects the computational costof the proposed algorithm

The simulation results are shown in Figures 7ndash9 It isseen from Figure 7 that the actual angle of attack tracksthe command in less than 01 seconds As shown in Figures8 and 9 the aerodynamic force converges to a constant astime increases while the jets are only activated during thebeginning period (when the tracking error is obvious)

International Journal of Aerospace Engineering 11

58

68

73

108

174

180

245

252

254

270

271

272

273

274

276

277

299

300

301

302

06

04

02

0

minus02

minus04

Elev

ator

defl

ectio

n (r

ad)

Angle of attack (rad)minus04 minus02 0 02 04 06

1634

1635

1638

1648

1649

1650

1664

1673

1677

1678

1679

1712

1726

1743

1744

1746

1748

1751

1752

1983

1985

1993

2011

Late

ral f

orce

(N)

Angle of attack (rad)minus04 minus02 0 02 04 06

1000

800

600

400

200

0

minus200

minus400

minus600

minus800

minus1000

Polyhedral partitionmdash4190 regions Polyhedral partitionmdash4190 regions

Figure 6 The partition associated with the explicit controller

025

02

015

01

005

0

ActualCommand

0 02 04 06 08 1

Time (s)

Ang

le o

f atta

ck (r

ad)

Figure 7 The response of angle of attack

52 Case 2 The results for the case with a different weight119877120575119911

= 0005 are shown in Figures 10ndash12 It is seen that adifferent control allocation result is obtained

04

02

0

minus02

minus04

minus060 02 04 06 08 1

Time (s)

Elev

ator

defl

ectio

n (r

ad)

Figure 8 The response of elevator deflection

53 Case 3 In this case a different command 120572119888 =

02 cos(05120587119905) rad is considered and the weight matrices arethe same as the ones for Case 1 The simulation results areshown in Figures 13ndash15

It is seen from Figure 13 that asymptotic tracking is alsoachievedThemethod proposed in this paper can realize both

12 International Journal of Aerospace Engineering

1

05

0

minus05

minus1

times104

0 02 04 06 08 1

Time (s)

Late

ral f

orce

(N)

Figure 9 The response of lateral force

025

02

015

01

005

0

Ang

le o

f atta

ck (r

ad)

0 02 04 06 08 1

Time (s)

ActualCommand

Figure 10 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 02 04 06 08 1

Time (s)

Figure 11 The response of elevator deflection

1

05

0

minus05

minus1

Late

ral f

orce

(N)

times104

0 02 04 06 08 1

Time (s)

Figure 12 The response of lateral force

03

025

02

015

01

005

0

minus005

minus01

minus015

minus02

Ang

le o

f atta

ck (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

ActualCommand

Figure 13 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

Figure 14 The response of elevator deflection

International Journal of Aerospace Engineering 13

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1

Late

ral f

orce

(N)

0 05 1 15 2 25 3 35 4

Time (s)

times104

Figure 15 The response of lateral force

fast tracking command and control allocation It is seen fromFigure 15 that the jets are activated while the tracking erroris obvious or command is varying rapidly

6 Conclusion

An autopilot design method for a missile with aerodynamiccontrol surfaces and lateral jets is presented in this paperThe nonlinear attitude control model is reduced to an MLDmodel Meanwhile the lateral force is described as linearcombination of logical variables due to the discrete values oflateral force Then the whole MLD model of attitude controlsystem is derived Autopilot design is accomplished usinghybrid MPC method By setting the related weighted coeffi-cients in index function the control allocation is obtainedMoreover numerical simulations are performed under thedifferent conditions the performance of tracking the attitudecommand and control allocation is verified and the explicitform of the control law can be obtained as well

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research presented in this document is supportedby the National Natural Science Foundation of Chinaunder Grant nos 61104193 61203191 and 61333001 theFundamental Research Funds for the Central Universities(HITNSRIF2012032) the Program for IBRSEM in HarbinInstitute of Technology under Grant HITIBRSEMA201415and the Foundation of Supporting Technology for Aerospaceunder Grant 2014-HT-HGD7

References

[1] R Hirokawa K Sato and S Manabe ldquoAutopilot design fora missile with reaction-jet using coefficient diagram methodrdquoin Proceedings of the AIAA Guidance Navigation and ControlConference and Exhibit 2001 August 2001

[2] P K Menon and V R Iragavarapu Adaptive Techniques forMultiple Actuator Blending Defense Technical InformationCenter 1998

[3] A G Munson and M W Garbrick ldquoJet interaction investiga-tionrdquo Report OR-9933 Martin Marietta Orlando Fla USA1969

[4] M Graham and P Weinacht ldquoNumerical simulation of lateralcontrol jetsrdquo in Proceedings of the 37th Aerospace SciencesMeeting and Exhibit 1999 AIAA Paper

[5] J S Shamma and J R Cloutier ldquoGain-scheduledmissile autopi-lot design using linear parameter varying transformationsrdquoJournal of Guidance Control and Dynamics vol 16 no 2 pp256ndash263 1993

[6] M Innocenti and A Thukral ldquoSimultaneous reaction jet andaerodynamic control of missile systemsrdquo AlAA 93-3739-CP1993

[7] D B Ridgely Y Lee and T Fanciullo ldquoDual aeropropulsivemissile control-optimal control and control allocationrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit Keystone Colo USA 2006

[8] D B Ridgely D Drake L Triplett and C Geise ldquoDynamiccontrol allocation of a missile with tails and reaction jetsrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 3158ndash3189 August 2007

[9] D Q Mayne J B Rawlings C V Rao and P O ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[10] A Bemporad F Borrelli and M Morari ldquoModel predictivecontrol based on linear programmingmdashthe explicit solutionrdquoIEEE Transactions on Automatic Control vol 47 no 12 pp1974ndash1985 2002

[11] C Jouannet and P Krus ldquoModelling of high angle of attackaerodynamicrdquo in Proceedings of the 25th AIAA Applied Aerody-namics Conference pp 1481ndash1516 June 2007

[12] S Venugopal and M Krishnamurthy ldquoMissile aerodynamics athigh angles of attack a prediction coderdquo Journal of Spacecraftand Rockets vol 32 no 2 pp 263ndash269 1995

[13] A Bemporad ldquoEfficient conversion of mixed logical dynamicalsystems into an equivalent piecewise affine formrdquo IEEE Trans-actions on Automatic Control vol 49 no 5 pp 832ndash838 2004

[14] F D Torrisi andA Bemporad ldquoHYSDELmdasha tool for generatingcomputational hybrid models for analysis and synthesis prob-lemsrdquo IEEE Transactions on Control Systems Technology vol 12no 2 pp 235ndash249 2004

[15] A Bemporad W P M H Heemels and B De SchutterldquoOn hybrid systems and closed-loop MPC systemsrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 863ndash8692002

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DistributedSensor Networks

International Journal of

International Journal of Aerospace Engineering 3

12

3

4

5

6

7

8

910

11

12

13

14

15

16

17

18

yb

zb

(a) Odd-numbered rings

1

2

3

4

5

6

7

8

91011

12

13

14

15

16

17

18

yb

zb

(b) Even-numbered rings

Figure 2 The layout scheme of lateral pulse jets

where

119894lowast= 2 119894 is odd

119894lowast= 1 119894 is even

(6)

For the situation where all pulse jets are fired at the sametime the total force and moment are given by

[

[

119865119909119887

119865119910119887

119865119911119887

]

]

=

[[[[[[[[[[

[

0

119894=10

sum

119894=1

119895=18

sum

119895=1

119865119894119895

119910119887

119894=10

sum

119894=1

119895=18

sum

119895=1

119865119894119895

119911119887

]]]]]]]]]]

]

[

[

119872119909119887

119872119910119887

119872119911119887

]

]

=

[[[[[[[[[[

[

0

minus

119894=10

sum

119894=1

119895=18

sum

119895=1

119865119894119895

119911119887

119897119894

119894=10

sum

119894=1

119895=18

sum

119895=1

119865119894119895

119910119887

119897119894

]]]]]]]]]]

]

(7)

In order to avoid the coupling between the pitch momentand yaw moment each ring is divided into four controlregions positive pitch negative pitch positive yaw andnegative yaw control region as shown in Figure 4

The autopilot design of PAC-3 is more complicated thanthat of other conventional missiles which are controlled

12

88

99

1010

1111

1212

xb

zb

i

lil1

Δl

middot middot middotmiddot middot middot

O

Figure 3 The ring frames expansion of lateral pulse jets

only by aerodynamic surfaces due to the hybrid propertyof control inputs and the on-off property of pulse jet (thepulse jet can be fired only one time) To deal with thisproblem the work [7] proposed two-step design procedures(1) in the first step neglect the hybrid property (or on-offproperty) and design the expected force andmoment signals(2) in the other step the fire logic is derived by solving theretaliation problem of these signals with taking into accountthe hybrid property (or on-off property) Different fromthe above traditional procedures a novel procedure will bepresented in what follows where only one step is included Inpractical applications only a small number of jets (in a certainof rings) are activated over a finite time interval To make theidea of the following development clear we here consider asimple but representative situation where no more than two

4 International Journal of Aerospace Engineering

yb

zb

Negative pitchcontrol region

Negative yawcontrol region

Positive pitchcontrol region

Positive yawcontrol region

17

8

910

11

12

12

3

4

5

6

713

14

15

16

18

(a) Odd-numbered rings

1

2

3

4

5

6

7

8

910

11

12

13

14

15

16

17

18

yb

zb

Negative pitchcontrol region

Negative yawcontrol region

Positive pitchcontrol region

Positive yawcontrol region

(b) Even-numbered rings

Figure 4 Schematic of control regions

rings are allowed to be fired simultaneously and no morethan two jets are activated in each fired ring Meanwhile itshould be ensured that only odd rings or even rings are firedand the jets are fired symmetrically about the correspondingsymmetry axis of each control region

Take the positive pitch control region as an example Theforces provided by the jets (119894 1) (119894 2) (119894 3) (119894 17) and (119894 18)

in an odd ring are given by

119865119900 = [119865119898 119865119898 cos120587

9119865119898 cos

2120587

9119865119898 cos

2120587

9119865119898 cos

120587

9]

119879

(8)

Similarly the forces provided by the jets (119894 1) (119894 2) (119894 17)and (119894 18) in an even ring are given by

119865119890 = [119865119898 cos120587

18119865119898 cos

120587

6119865119898 cos

120587

6119865119898 cos

120587

18]

119879

(9)

where119865119900 and119865119890 denote the forces associated with an odd ringand an even ring respectively

Use 119865119894 to denote the lateral force generated by ring 119894

Clearly 119865119894 should satisfy the condition

1198651= 1198659 119865

3= 1198657 119865

2= 11986510 119865

4= 1198658 (10)

When odd ring is fired the lateral force and moment aregiven by

119865119910119887

isin 119865119898 2119865119898 cos120587

9 2119865119898 cos

2120587

9 2119865119898 4119865119898 cos

120587

9

4119865119898 cos2120587

9

119872119911119887

isin 1198651198981198975 21198651198981198975 cos120587

9 21198651198981198975 cos

2120587

9 21198651198981198975

41198651198981198975 cos120587

9 41198651198981198975 cos

2120587

9

(11)

While even ring is fired the lateral force and moment aregiven by

119865119910119887

isin 2119865119898 cos120587

18 2119865119898 cos

120587

6 4119865119898 cos

120587

18 4119865119898 cos

120587

6

119872119911119887

isin 21198651198981198976 cos120587

18 21198651198981198976 cos

120587

6 41198651198981198976 cos

120587

18

41198651198981198976 cos120587

6

(12)

Noting Δ119897 is quite small we consider 1198975 ≃ 1198976 = 119897 Inorder to ensure that the jets fire efficiency the jets resultingin small moment components along 119911119887 axis are not activatedAs a result the sets of forces and moments are respectivelygiven by

119880119910+

119865= 119865119898 2119865119898 cos

120587

9 2119865119898 cos

120587

18 2119865119898 4119865119898 cos

120587

9

4119865119898 cos120587

18

(13)

119880119910+

119872= 119865119898119897 2119865119898119897 cos

120587

9 2119865119898119897 cos

120587

18 2119865119898119897 4119865119898119897 cos

120587

9

4119865119898119897 cos120587

18

(14)

where 119865119910119887

isin 119880119910+

119865and119872119911

119887

isin 119880119910+

119872 During each control period

a control moment belonging to119880119910+

119872will be used as the input

The mutual interference between high-speed jet streamand air leads to lateral jet interference effect In orderto take into account this interference force and moment

International Journal of Aerospace Engineering 5

amplification factors 119870119865119910

119870119865119911

119870119872119910

and 119870119872119911

are introducedas in [3 4]Then the resulting lateral forces andmoments are

[[[

[

119865119886

119909119887

119865119886

119910119887

119865119886

119911119887

]]]

]

= [

[

0

119865119910119887

+ 119870119865119910

119865119910119887

119865119911119887

+ 119870119865119911

119865119911119887

]

]

[[[

[

119872119886

119909119887

119872119886

119910119887

119872119886

119911119887

]]]

]

= [

[

0

119872119910119887

+ 119870119872119910

119872119910119887

119872119911119887

+ 119870119872119911

119872119911119887

]

]

(15)

Remark 1 In fact each pulse jet can be fired only once so thelocation of the fired jet cannot provide force anymore Basedon this precondition elements of sets 119880

119910+

119865and 119880

119910+

119872will be

less and less over time In this paper quantity change of setsrsquoelements is not considered to simplify the problem

23 Attitude Control Model Some transformation and sim-plification are applied to themissile model for control designIt is assumed that themissilersquosmass is of a constant valueNotethat the goal is to establish the angle of attack and sideslipangle The attitude control model is given by

= 119903 + 119902 sin120572 tan120573 minus

119876119878 (119862120572

119910120572 + 119862

120575119911

119910120575119911) cos120572

119898119881 cos120573

minus

(119865119910119887

+ 119870119865119910

119865119910119887

) cos120572

119898119881 cos120573minus

119866119910 cos120572119898119881 cos120573

120573 = 119902 cos120572 +

(119876119878 (119862120573

119911120573 + 119862

120575119910

119911 120575119910) + 119865119911119887

+ 119870119865119911

119865119911119887

) cos120573

119898119881

+

(119876119878 (119862120572

119910120572 + 119862

120575119911

119910120575119911) + 119865119910

119887

+ 119870119865119910

119865119910119887

) sin120572 sin120573

119898119881

+119866119911 cos120573

119898119881+

119866119910 sin120572 sin120573

119898119881

119902 =119872119910119887

119869119910

+

119870119872119910

119872119910119887

119869119910

+

119876119878119871119898120573

119910120573

119869119910

+119876119878119871119898

120575119910

119910 120575119910

119869119910

+

119876119878119871119898119902

119910119902

119869119910

119903 =119872119911119887

119869119911

+119870119872119911

119872119911119887

119869119911

+119876119878119871119898

120572

119911120572

119869119911

+119876119878119871119898

120575119911

119911120575119911

119869119911

+119876119878119871119898

119903

119911119903

119869119911

(16)

where 119898120573

119910 119898120575119911

119910 119898119902

119910 119898120572

119911 119898120575119911

119911 and 119898

119903

119911are aerodynamic

parameters

3 Mixed Logical Dynamical Model ofBlended Missile

31 Piecewise Affine Model of Blended Missile To simplifyanalysis the gravity term and the channel coupling term are

Table 1 Missilersquos overall parameters

Missilersquos takeoff mass [kg] 255Missilersquos full-length [m] 486Missilersquos diameter [m] 0317Distance between missile head and center ofmass [m] 2569

Moment of inertia 119869119909 119869119910 119869119911 [kgsdotm2] 303 3063 3063

Lateral force provided by individual jets [N] 2200Distance between jet ring center and centerof mass [m] 126

Range of angle of attack [rad] minus120587

6le 120572 le

120587

6

Range of pitch angular velocity [rads] minus5120587

3le 119903 le

5120587

3

Range of elevator deflection [rad] minus120587

6le 120575119911 le

120587

6

ignoredWith (16) themissile attitude control model of pitchchannel is

= 119903 minus

119876119878 (119862120572

119910120572 + 119862

120575119911

119910120575119911) cos120572

119898119881minus

(1 + 119870119865119910

) 119865119910119887

cos120572

119898119881

119903 =

(1 + 119870119872119911

) 119865119910119887

119897

119869119911

+119876119878119871119898

120572

119911120572

119869119911

+119876119878119871119898

120575119911

119911120575119911

119869119911

+119876119878119871119898

119903

119911119903

119869119911

(17)

Choose 119909 = [120572 119903]119879 as system state and 119906 = [120575119911 119865119910

119887

]119879 as

control input The considered output is 119910 = 120572 Then (17) canbe rewritten into the following state space form

= 119891 (119909) + 119892 (119909) 119906

119910 = [1 0] 119909

(18)

where

119891 (119909) =

[[[[

[

119903 minus

119876119878119862120572

119910120572 cos120572

119898119881

119876119878119871 (119898120572

119911120572 + 119898

119903

119911119903)

119869119911

]]]]

]

119892 (119909) =

[[[[[[

[

minus

119876119878119862120575119911

119910cos120572

119898119881minus

(1 + 119870119865119910

) cos120572

119898119881

119876119878119871119898120575119911

119911

119869119911

(1 + 119870119872119911

) 119897

119869119911

]]]]]]

]

(19)

The missile parameters are presented in Table 1Generally the aerodynamic coefficients 119862

120572

119910 119862120575119911

119910 119898120572

119911

and 119898120575119911

119911and the amplification factors 119870119865

119910

119870119872119911

are mainlyaffected by the flight velocity 119881 and the angle of attack 120572

[11 12] Since the terminal guidance phase is consideredin this paper the flight time is quite short and the flightvelocity of the missile can be treated as a constant Thusthe aerodynamic coefficients and the amplification factors are

6 International Journal of Aerospace Engineering

mainly affected by the angle of attack 120572 The relationshipsbetween them are shown in Figure 5

In practical application since 120572 is the main factor thatleads to system nonlinearities the system model is usuallylinearized if 120572 varies in small range As seen from Figure 5curves of the relation between aerodynamic parametersand angle of attack can be expressed by six line segmentsapproximately Here we choose 120572 = minus037 rad minus0153 rad0 0153 rad and 037 rad as the operation points and dividethe whole operation region into six subregions As a resultthe original model (17) can be converted to the followingpiecewise affine models

=

1198861119909 + 1198871119906 + 1198901 [1 0] 119909 le minus037

1198862119909 + 1198872119906 + 1198902 minus037 lt [1 0] 119909 le minus0153

1198863119909 + 1198873119906 + 1198903 minus0153 lt [1 0] 119909 le 0

1198864119909 + 1198874119906 + 1198904 0 lt [1 0] 119909 le 0153

1198865119909 + 1198875119906 + 1198905 0153 lt [1 0] 119909 le 037

1198866119909 + 1198876119906 + 1198906 [1 0] 119909 gt 037

(20)

119910 = [1 0] 119909 (21)

where

119886119894 =120597119891(119909)

120597119909

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= [11988611

11989411988612

119894

11988621

11989411988622

119894

]

119887119894 = 119892 (1199091198940) = [11988711

11989411988712

119894

11988721

11989411988722

119894

] 119890119894 = [1198901

119894

1198902

119894

]

11988611

119894=

1205971198911 (119909)

120597120572

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= minus119876119878

119898119881(

120597119862120572

119910

120597120572

1003816100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1205721198940 cos1205721198940 + 119862120572

119910(1205721198940) cos1205721198940

minus 119862120572

119910(1205721198940) 1205721198940 sin1205721198940)

11988612

119894=

1205971198911(119909)

120597119903

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= 1

11988621

119894=

1205971198912 (119909)

120597120572

10038161003816100381610038161003816100381610038161003816119909=1199091198940

=119876119878119871

119869119911

(120597119898120572

119911

120597120572

100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1205721198940 + 119898120572

119911(1205721198940) +

120597119898119903

119911

120597120572

100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1199031198940)

11988622

119894=

1205971198912(119909)

120597119903

10038161003816100381610038161003816100381610038161003816119909=1199091198940

=119876119878119871

119869119911

119898120572

119911(1205721198940)

11988711

119894= minus

119876119878119862120575119911

119910(1205721198940) cos1205721198940119898119881

11988712

119894= minus

(1 + 119870119865119910

(1205721198940)) cos1205721198940119898119881

11988721

119894=

119876119878119871119898120575119911

119911(1205721198940)

119869119911

11988722

119894=

(1 + 119870119872119911

(1205721198940)) 119897

119869119911

1198901

119894= 11988611

1198941205721198940 + 119886

12

1198941199031198940

1198902

119894= 11988621

1198941205721198940 + 119886

22

1198941199031198940

(22)

where 119894 (119894 = 1 2 6) is the label corresponding to the 119894thregion

From Table 1 and Figure 5 we get the set of aerodynamicparameters at the point (119867119881) = (20 km 1000ms) as shownin Table 2

Choose the sampling period 119879119904 = 0025 s The discretestate-space expression is then given by

119909 (119896 + 1)

=

1198861119909 (119896) + 1119906 (119896) + 1198901 [1 0] 119909 (119896) le minus037

1198862119909 (119896) + 2119906 (119896) + 1198902 minus037 lt [1 0] 119909 (119896) le minus0153

1198863119909 (119896) + 3119906 (119896) + 1198903 minus0153 lt [1 0] 119909 (119896) le 0

1198864119909 (119896) + 4119906 (119896) + 1198904 0 lt [1 0] 119909 (119896) le 0153

1198865119909 (119896) + 5119906 (119896) + 1198905 0153 lt [1 0] 119909 (119896) le 037

1198866119909 (119896) + 6119906 (119896) + 1198906 [1 0] 119909 (119896) gt 037

(23)

119910 (119896) = [1 0] 119909 (119896) (24)

where

1198861 = [104 0025

022 0995] 1198862 = [

1051 0025

0218 0995]

1198863 = [1023 00252

0248 09951] 1198864 = [

09954 00248

02732 09954]

1198865 = [09697 00245

02913 09956] 1198866 = [

09594 00244

03137 09959]

1198901 = [minus00183

minus01123] 1198902 = [

minus00175

minus00792]

1198903 = [minus00131

minus00376] 1198904 = [

0

0]

1198905 = [minus00053

00454] 1198906 = [

minus00169

01193]

1 = [minus0028 159 times 10

minus6

minus2166 142 times 10minus4]

2 = [minus003 218 times 10

minus6

minus2224 187 times 10minus4]

International Journal of Aerospace Engineering 7

2

15

1

05

0

minus05

minus1

minus15minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SmV

)C120572 y

(a) Relation between 119862120572119910and 120572

0092

009

0088

0086

0084

0082

008

0078

0076

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SmV

)C120575119911y

(b) Relation between 119862120575119911119910 and 120572

125

12

115

11

105

10

95

9minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SLJz)m

120572 z

(c) Relation between119898120572119911and 120572

96

94

92

90

88

86

84minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (minusQSL

Jz)m

120575119911z

(d) Relation between119898120575119911119911 and 120572

1

09

08

07

06

05

04

03

02

01

Am

plifi

catio

n fa

ctorKF

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

(e) Relation between119870119865119910

and 120572

12

11

1

09

08

07

06

05

04

03

Am

plifi

catio

n fa

ctorKM119911

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

(f) Relation between119870119872119911

and 120572

Figure 5 Aerodynamic parameters as functions of angle of attack

8 International Journal of Aerospace Engineering

Table 2 Aerodynamic parameters of pitch channel

Aerodynamic parameter 119876119878119862120572

119910119898119881 119876119878119862

120575119911

119910119898119881 119876119878119871119898

120572

119911119869119911

119876119878119871119898119903

119911119869119911

119876119878119871119898120575119911

119911119869119911

120572 = minus053 minus1175 0076 947 minus032 minus8489

120572 = minus037 minus0944 0082 973 minus032 minus8924

120572 = minus0153 minus023 0082 1043 minus032 minus8938

120572 = 0 036 009 1101 minus032 minus9510

120572 = 0153 094 0082 1158 minus032 minus8934

120572 = 037 163 0082 1224 minus032 minus8918

3 = [minus003 268 times 10

minus6

minus2228 223 times 10minus4]

4 = [minus0032 277 times 10

minus6

minus2371 232 times 10minus4]

5 = [minus003 227 times 10

minus6

minus2228 198 times 10minus4]

6 = [minus0029 191 times 10

minus6

minus2224 173 times 10minus4]

(25)

32 Constraints Analysis Due to the symmetry of jet config-uration the set of possible negative pitch control force is givenby

119880119910minus

119865= minus119865119898 minus2119865119898 cos

120587

9 minus2119865119898 cos

120587

18 minus2119865119898 minus4119865119898 cos

120587

9

minus4119865119898 cos120587

18

(26)

By combining (13) and (26) we obtain the set of allpossible pitch control force

119880119910

119865= 119865119898 2119865119898 cos

120587

9 2119865119898 cos

120587

18 2119865119898 4119865119898 cos

120587

9

4119865119898 cos120587

18 minus119865119898 minus2119865119898 cos

120587

9 minus2119865119898 cos

120587

18 minus2119865119898

minus4119865119898 cos120587

9 minus4119865119898 cos

120587

18

(27)

Substituting the jet parameters shown in Table 1 into theforegoing set yields

119880119910

119865= 2200 4135 4333 4400 8269 8666 minus2200 minus4135

minus4333 minus4400 minus8269 minus8666

(28)

Noting that the lateral forces are discrete variable we willutilize the linear combination of logical variables to describethe lateral force In terms of piecewise affine model (20) we

introduce logical variables 120575119865119894

isin 0 1 119894 = 1 2 12 toexpress the lateral thrust 119865119910

119887

119865119910119887

= 22001205751198651

+ 41351205751198652

+ 43331205751198653

+ 44001205751198654

+ 82691205751198655

+ 86661205751198656

minus 22001205751198657

minus 41351205751198658

minus 43331205751198659

minus 440012057511986510

minus 826912057511986511

minus 866612057511986512

(29)

In (29) the logical variables should satisfy the constraints

12

sum

119894=1

120575119865119894

= 0 or 1 (30)

where 0 means that no lateral force is generated while 1

means that the applied lateral force equals to one element ofset 119880119910119865

Denote 1199061 = 120575119911 the control input 119906 in model (23) can berewritten as

119906 = [1199061 119865119910119887

]119879 (31)

As shown in Table 1 the constraints on system states andcontrol input are

119909min le 119909 (119896) le 119909max

1199061min le 1199061 (119896) le 1199061max(32)

where 119909min = [minus053 minus522]119879 119909max = [053 522]

119879 1199061min =

minus053 1199061max = 053According to (30) we have

12

sum

119894=1

120575119865119894

le 1 (33)

which shows the constraint on the logical control inputUse logical variables 120575119894(119896) isin 0 1 119894 = 1 2 5 to

describe the operation points which satisfy the constraints

[1 0] 119909 (119896) + 037 le 0 lArrrArr 1205751 (119896) = 1

[1 0] 119909 (119896) + 0153 le 0 lArrrArr 1205752 (119896) = 1

[1 0] 119909 (119896) le 0 lArrrArr 1205753 (119896) = 1

[1 0] 119909 (119896) minus 0153 le 0 lArrrArr 1205754 (119896) = 1

[1 0] 119909 (119896) minus 037 le 0 lArrrArr 1205755 (119896) = 1

(34)

International Journal of Aerospace Engineering 9

Equation (34) can be transformed into the followingequivalent mixed logical inequalities [13]

[1 0] 119909 (119896) + 037 ge 120576 + (1198981 minus 120576) 1205751 (119896)

[1 0] 119909 (119896) + 037 le 1198721 (1 minus 1205751 (119896))

[1 0] 119909 (119896) + 0153 ge 120576 + (1198982 minus 120576) 1205752 (119896)

[1 0] 119909 (119896) + 0153 le 1198722 (1 minus 1205752 (119896))

[1 0] 119909 (119896) ge 120576 + (1198983 minus 120576) 1205753 (119896)

[1 0] 119909 (119896) le 1198723 (1 minus 1205753 (119896))

[1 0] 119909 (119896) minus 0153 ge 120576 + (1198984 minus 120576) 1205754 (119896)

[1 0] 119909 (119896) minus 0153 le 1198724 (1 minus 1205754 (119896))

[1 0] 119909 (119896) minus 037 ge 120576 + (1198985 minus 120576) 1205755 (119896)

[1 0] 119909 (119896) minus 037 le 1198725 (1 minus 1205755 (119896))

(35)

where 1198981 = minus016 1198721 = 090 1198982 = minus0377 1198722 = 06831198983 = minus053 1198723 = 053 1198984 = minus0683 1198724 = 0377 1198985 =

minus0901198725 = 016 and 120576 = 10minus6

In addition we introduce the auxiliary logical variables120575119894(119896) isin 0 1 119894 = 6 9 as follows

1205756 (119896) = (1 minus 1205751 (119896)) 1205752 (119896)

1205757 (119896) = (1 minus 1205752 (119896)) 1205753 (119896)

1205758 (119896) = (1 minus 1205753 (119896)) 1205754 (119896)

1205759 (119896) = (1 minus 1205754 (119896)) 1205755 (119896)

(36)

With 1205751 1205756 1205757 1205758 1205759 and 1 minus 1205755 the six regions canbe presented Similarly (36) can be transformed into theequivalent mixed logical inequalities

minus1205751 (119896) + 1205752 (119896) minus 1205756 (119896) le 0

1205751 (119896) + 1205756 (119896) le 1

minus1205752 (119896) + 1205756 (119896) le 0

minus1205752 (119896) + 1205753 (119896) minus 1205757 (119896) le 0

1205752 (119896) + 1205757 (119896) le 1

minus1205753 (119896) + 1205757 (119896) le 0

minus1205753 (119896) + 1205754 (119896) minus 1205758 (119896) le 0

1205753 (119896) + 1205758 (119896) le 1

minus1205754 (119896) + 1205758 (119896) le 0

minus1205754 (119896) + 1205755 (119896) minus 1205759 (119896) le 0

1205754 (119896) + 1205759 (119896) le 1

minus1205755 (119896) + 1205759 (119896) le 0

(37)

To describe the state space model of each region weintroduce the following auxiliary continuous variables

1199111 (119896) = (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) 1205751 (119896)

1199112 (119896) = (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) 1205756 (119896)

1199113 (119896) = (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) 1205757 (119896)

1199114 (119896) = (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) 1205758 (119896)

1199115 (119896) = (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) 1205759 (119896)

1199116 (119896) = (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) (1 minus 1205755 (119896))

(38)

Equation (38) can be converted to the equivalent mixedlogical inequalities

1199111 (119896) ge (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198721198911 (1 minus 1205751 (119896))

1199111 (119896) le (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198981198911 (1 minus 1205751 (119896))

1199111 (119896) ge 11989811989111205751 (119896)

1199111 (119896) le 11987211989111205751 (119896)

1199112 (119896) ge (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198721198912 (1 minus 1205756 (119896))

1199112 (119896) le (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198981198912 (1 minus 1205756 (119896))

1199112 (119896) ge 11989811989121205756 (119896)

1199112 (119896) le 11987211989121205756 (119896)

1199113 (119896) ge (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198721198913 (1 minus 1205757 (119896))

1199113 (119896) le (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198981198913 (1 minus 1205757 (119896))

1199113 (119896) ge 11989811989131205757 (119896)

1199113 (119896) le 11987211989131205757 (119896)

1199114 (119896) ge (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198721198914 (1 minus 1205758 (119896))

1199114 (119896) le (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198981198914 (1 minus 1205758 (119896))

1199114 (119896) ge 11989811989141205758 (119896)

1199114 (119896) le 11987211989141205758 (119896)

1199115 (119896) ge (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198721198915 (1 minus 1205759 (119896))

1199115 (119896) le (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198981198915 (1 minus 1205759 (119896))

10 International Journal of Aerospace Engineering

1199115 (119896) ge 11989811989151205759 (119896)

1199115 (119896) le 11987211989151205759 (119896)

1199116 (119896) ge (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11987211989151205755 (119896)

1199116 (119896) le (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11989811989151205755 (119896)

1199116 (119896) ge 1198981198916 (1 minus 1205755 (119896))

1199116 (119896) le 1198721198916 (1 minus 1205755 (119896))

(39)

where1198721198911 = [073 1084]1198791198981198911 = [minus077 minus1106]

1198791198721198912 =[076 1166]

119879 1198981198912 = [minus079 minus 1256]119879 1198721198913 =

[077 1273]119879 1198981198913 = [minus078 minus 1280]

119879 1198721198914 =

[076 1403]119879 1198981198914 = [minus076 minus 1403]

119879 1198721198915 =

[0725 1291]119879 1198981198915 = [minus0736 minus 1205]

119879 1198721198916 =

[0696 1154]119879 and1198981198916 = [minus073 minus 1130]

119879Then the whole MLD model of the missile is given by

119909 (119896 + 1) =

6

sum

119894=1

119911119894 (119896)

119910 (119896) = [1 0] 119909 (119896)

st (28) (30) (32) (34) (36) (38)

(40)

where (29) (31) (33) (35) (37) and (39) represent all theconstraint inequalities

4 Autopilot Design UsingHybrid MPC Method

In general we expect that the output 119910 tracks its command 119910119888

as fast as possible with a small amount of fuel consumptionMotivated by this observation we consider the followingoptimization problem

119869lowast

= min119906(119896)119906(119896+1)120575(119896)120575(119896+1|119896)119911(119896)119911(119896+1|119896)

119873

sum

119894=1

(1003817100381710038171003817119910 (119896 + 119894 | 119896)

minus119910119888 (119896 + 119894)1003817100381710038171003817

2

119876

+ 119906(119896 + 119894)2

119877)

st MLD model (39)

1199061min le 1199061 (119896) 1199061 (119896 + 1) le 1199061max

119909min le 119909 (119896) 119909 (119896 + 1) le 119909max

(41)

where 119910119888 is the command for angle of attack 119910(119896 + 119894 | 119896) ispredictive value of angle of attack 119873 denotes the predictivehorizon 119876 and 119877 are the weighting matrices and 119877 =

diag(119877120575119911

119877119865119910

) 119877120575119911

119877119865119910

are weighted coefficients of aerody-namic control surfaces and lateral pulse jets respectivelyThe

ratio of 119877120575119911

and 119877119865119910

represents the control allocation betweenaerodynamic control surfaces and lateral pulse jets When119877119865119910

119877120575119911

is increased the requirement for lateral force will bedeclined that is the fuel consumption can be lessened Inaddition the deduction can be verified from the simulationresults of Case 1 and Case 2

Different control allocation between the dual actuators(aerodynamic control surfaces and lateral pulse jets) can beobtained by setting different 119876 and 119877 Since both logicaland continuous variables are involved in (41) the aboveon-line optimization problem is a mixed integer quadraticprogramming (MIQP) problemwhich can be solved by usingthe hybrid MPC toolbox of MATLAB [14 15]

Remark 2 In an explicit MPC controller the main factorsaffecting the number of subregions include the systemdimen-sion the predictive horizon and the number of constrains Inthis paper the blendedmissile with aerodynamic control sur-faces and lateral jets is investigated To deal with the discreteproperty of the lateral jet forces some logical variables areintroduced for MPC controller design which increases thenumber of subregions inevitably

5 Numerical Simulations

51 Case 1 Numerical simulations are performed to ver-ify the feasibility of the proposed method in this sectionThe involved parameters are given in Table 1 The weightmatrices are set as 119876 = 10 119877120575

119911

= 0025 and 119877119865119910

=

diag(0005 0005 0005) Suppose the initial state is 1205720 =0 1199030 = 0 In order to avoid excessive computational load wechoose predictive horizon119873 = 2 and control horizon119872 = 2The command for angle of attack is 120572119888 = 02 rad

Explicit form of the optimal controller is provided byhybrid toolbox which is not given here for simplicity Insteadwe present the partition results associated with the controllerin Figure 6 where 4190 subregions are obtained and for eachsubregion a control law of the form 119906 = 119870119909 + 119887 is givenAs a benefit of using explicit MPC the controller parameters119870 and 119887 for all the subregions are obtained simultaneouslyonce the optimization problem is solved This property isobviously different from that of gain-scheduled controller(where the controller parameters for different regions areseparately determined)

However by a simple calculation we conclude that thesesubregions donot require toomuch storage space (the neededstorage space is approximately 14 times 8 times 4190 = 450KB)Moreover it should be noted that when the algorithm isimplemented in practice much more time will be needed toidentify which subregion the current states enter into This isanother important factor that affects the computational costof the proposed algorithm

The simulation results are shown in Figures 7ndash9 It isseen from Figure 7 that the actual angle of attack tracksthe command in less than 01 seconds As shown in Figures8 and 9 the aerodynamic force converges to a constant astime increases while the jets are only activated during thebeginning period (when the tracking error is obvious)

International Journal of Aerospace Engineering 11

58

68

73

108

174

180

245

252

254

270

271

272

273

274

276

277

299

300

301

302

06

04

02

0

minus02

minus04

Elev

ator

defl

ectio

n (r

ad)

Angle of attack (rad)minus04 minus02 0 02 04 06

1634

1635

1638

1648

1649

1650

1664

1673

1677

1678

1679

1712

1726

1743

1744

1746

1748

1751

1752

1983

1985

1993

2011

Late

ral f

orce

(N)

Angle of attack (rad)minus04 minus02 0 02 04 06

1000

800

600

400

200

0

minus200

minus400

minus600

minus800

minus1000

Polyhedral partitionmdash4190 regions Polyhedral partitionmdash4190 regions

Figure 6 The partition associated with the explicit controller

025

02

015

01

005

0

ActualCommand

0 02 04 06 08 1

Time (s)

Ang

le o

f atta

ck (r

ad)

Figure 7 The response of angle of attack

52 Case 2 The results for the case with a different weight119877120575119911

= 0005 are shown in Figures 10ndash12 It is seen that adifferent control allocation result is obtained

04

02

0

minus02

minus04

minus060 02 04 06 08 1

Time (s)

Elev

ator

defl

ectio

n (r

ad)

Figure 8 The response of elevator deflection

53 Case 3 In this case a different command 120572119888 =

02 cos(05120587119905) rad is considered and the weight matrices arethe same as the ones for Case 1 The simulation results areshown in Figures 13ndash15

It is seen from Figure 13 that asymptotic tracking is alsoachievedThemethod proposed in this paper can realize both

12 International Journal of Aerospace Engineering

1

05

0

minus05

minus1

times104

0 02 04 06 08 1

Time (s)

Late

ral f

orce

(N)

Figure 9 The response of lateral force

025

02

015

01

005

0

Ang

le o

f atta

ck (r

ad)

0 02 04 06 08 1

Time (s)

ActualCommand

Figure 10 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 02 04 06 08 1

Time (s)

Figure 11 The response of elevator deflection

1

05

0

minus05

minus1

Late

ral f

orce

(N)

times104

0 02 04 06 08 1

Time (s)

Figure 12 The response of lateral force

03

025

02

015

01

005

0

minus005

minus01

minus015

minus02

Ang

le o

f atta

ck (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

ActualCommand

Figure 13 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

Figure 14 The response of elevator deflection

International Journal of Aerospace Engineering 13

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1

Late

ral f

orce

(N)

0 05 1 15 2 25 3 35 4

Time (s)

times104

Figure 15 The response of lateral force

fast tracking command and control allocation It is seen fromFigure 15 that the jets are activated while the tracking erroris obvious or command is varying rapidly

6 Conclusion

An autopilot design method for a missile with aerodynamiccontrol surfaces and lateral jets is presented in this paperThe nonlinear attitude control model is reduced to an MLDmodel Meanwhile the lateral force is described as linearcombination of logical variables due to the discrete values oflateral force Then the whole MLD model of attitude controlsystem is derived Autopilot design is accomplished usinghybrid MPC method By setting the related weighted coeffi-cients in index function the control allocation is obtainedMoreover numerical simulations are performed under thedifferent conditions the performance of tracking the attitudecommand and control allocation is verified and the explicitform of the control law can be obtained as well

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research presented in this document is supportedby the National Natural Science Foundation of Chinaunder Grant nos 61104193 61203191 and 61333001 theFundamental Research Funds for the Central Universities(HITNSRIF2012032) the Program for IBRSEM in HarbinInstitute of Technology under Grant HITIBRSEMA201415and the Foundation of Supporting Technology for Aerospaceunder Grant 2014-HT-HGD7

References

[1] R Hirokawa K Sato and S Manabe ldquoAutopilot design fora missile with reaction-jet using coefficient diagram methodrdquoin Proceedings of the AIAA Guidance Navigation and ControlConference and Exhibit 2001 August 2001

[2] P K Menon and V R Iragavarapu Adaptive Techniques forMultiple Actuator Blending Defense Technical InformationCenter 1998

[3] A G Munson and M W Garbrick ldquoJet interaction investiga-tionrdquo Report OR-9933 Martin Marietta Orlando Fla USA1969

[4] M Graham and P Weinacht ldquoNumerical simulation of lateralcontrol jetsrdquo in Proceedings of the 37th Aerospace SciencesMeeting and Exhibit 1999 AIAA Paper

[5] J S Shamma and J R Cloutier ldquoGain-scheduledmissile autopi-lot design using linear parameter varying transformationsrdquoJournal of Guidance Control and Dynamics vol 16 no 2 pp256ndash263 1993

[6] M Innocenti and A Thukral ldquoSimultaneous reaction jet andaerodynamic control of missile systemsrdquo AlAA 93-3739-CP1993

[7] D B Ridgely Y Lee and T Fanciullo ldquoDual aeropropulsivemissile control-optimal control and control allocationrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit Keystone Colo USA 2006

[8] D B Ridgely D Drake L Triplett and C Geise ldquoDynamiccontrol allocation of a missile with tails and reaction jetsrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 3158ndash3189 August 2007

[9] D Q Mayne J B Rawlings C V Rao and P O ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[10] A Bemporad F Borrelli and M Morari ldquoModel predictivecontrol based on linear programmingmdashthe explicit solutionrdquoIEEE Transactions on Automatic Control vol 47 no 12 pp1974ndash1985 2002

[11] C Jouannet and P Krus ldquoModelling of high angle of attackaerodynamicrdquo in Proceedings of the 25th AIAA Applied Aerody-namics Conference pp 1481ndash1516 June 2007

[12] S Venugopal and M Krishnamurthy ldquoMissile aerodynamics athigh angles of attack a prediction coderdquo Journal of Spacecraftand Rockets vol 32 no 2 pp 263ndash269 1995

[13] A Bemporad ldquoEfficient conversion of mixed logical dynamicalsystems into an equivalent piecewise affine formrdquo IEEE Trans-actions on Automatic Control vol 49 no 5 pp 832ndash838 2004

[14] F D Torrisi andA Bemporad ldquoHYSDELmdasha tool for generatingcomputational hybrid models for analysis and synthesis prob-lemsrdquo IEEE Transactions on Control Systems Technology vol 12no 2 pp 235ndash249 2004

[15] A Bemporad W P M H Heemels and B De SchutterldquoOn hybrid systems and closed-loop MPC systemsrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 863ndash8692002

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4 International Journal of Aerospace Engineering

yb

zb

Negative pitchcontrol region

Negative yawcontrol region

Positive pitchcontrol region

Positive yawcontrol region

17

8

910

11

12

12

3

4

5

6

713

14

15

16

18

(a) Odd-numbered rings

1

2

3

4

5

6

7

8

910

11

12

13

14

15

16

17

18

yb

zb

Negative pitchcontrol region

Negative yawcontrol region

Positive pitchcontrol region

Positive yawcontrol region

(b) Even-numbered rings

Figure 4 Schematic of control regions

rings are allowed to be fired simultaneously and no morethan two jets are activated in each fired ring Meanwhile itshould be ensured that only odd rings or even rings are firedand the jets are fired symmetrically about the correspondingsymmetry axis of each control region

Take the positive pitch control region as an example Theforces provided by the jets (119894 1) (119894 2) (119894 3) (119894 17) and (119894 18)

in an odd ring are given by

119865119900 = [119865119898 119865119898 cos120587

9119865119898 cos

2120587

9119865119898 cos

2120587

9119865119898 cos

120587

9]

119879

(8)

Similarly the forces provided by the jets (119894 1) (119894 2) (119894 17)and (119894 18) in an even ring are given by

119865119890 = [119865119898 cos120587

18119865119898 cos

120587

6119865119898 cos

120587

6119865119898 cos

120587

18]

119879

(9)

where119865119900 and119865119890 denote the forces associated with an odd ringand an even ring respectively

Use 119865119894 to denote the lateral force generated by ring 119894

Clearly 119865119894 should satisfy the condition

1198651= 1198659 119865

3= 1198657 119865

2= 11986510 119865

4= 1198658 (10)

When odd ring is fired the lateral force and moment aregiven by

119865119910119887

isin 119865119898 2119865119898 cos120587

9 2119865119898 cos

2120587

9 2119865119898 4119865119898 cos

120587

9

4119865119898 cos2120587

9

119872119911119887

isin 1198651198981198975 21198651198981198975 cos120587

9 21198651198981198975 cos

2120587

9 21198651198981198975

41198651198981198975 cos120587

9 41198651198981198975 cos

2120587

9

(11)

While even ring is fired the lateral force and moment aregiven by

119865119910119887

isin 2119865119898 cos120587

18 2119865119898 cos

120587

6 4119865119898 cos

120587

18 4119865119898 cos

120587

6

119872119911119887

isin 21198651198981198976 cos120587

18 21198651198981198976 cos

120587

6 41198651198981198976 cos

120587

18

41198651198981198976 cos120587

6

(12)

Noting Δ119897 is quite small we consider 1198975 ≃ 1198976 = 119897 Inorder to ensure that the jets fire efficiency the jets resultingin small moment components along 119911119887 axis are not activatedAs a result the sets of forces and moments are respectivelygiven by

119880119910+

119865= 119865119898 2119865119898 cos

120587

9 2119865119898 cos

120587

18 2119865119898 4119865119898 cos

120587

9

4119865119898 cos120587

18

(13)

119880119910+

119872= 119865119898119897 2119865119898119897 cos

120587

9 2119865119898119897 cos

120587

18 2119865119898119897 4119865119898119897 cos

120587

9

4119865119898119897 cos120587

18

(14)

where 119865119910119887

isin 119880119910+

119865and119872119911

119887

isin 119880119910+

119872 During each control period

a control moment belonging to119880119910+

119872will be used as the input

The mutual interference between high-speed jet streamand air leads to lateral jet interference effect In orderto take into account this interference force and moment

International Journal of Aerospace Engineering 5

amplification factors 119870119865119910

119870119865119911

119870119872119910

and 119870119872119911

are introducedas in [3 4]Then the resulting lateral forces andmoments are

[[[

[

119865119886

119909119887

119865119886

119910119887

119865119886

119911119887

]]]

]

= [

[

0

119865119910119887

+ 119870119865119910

119865119910119887

119865119911119887

+ 119870119865119911

119865119911119887

]

]

[[[

[

119872119886

119909119887

119872119886

119910119887

119872119886

119911119887

]]]

]

= [

[

0

119872119910119887

+ 119870119872119910

119872119910119887

119872119911119887

+ 119870119872119911

119872119911119887

]

]

(15)

Remark 1 In fact each pulse jet can be fired only once so thelocation of the fired jet cannot provide force anymore Basedon this precondition elements of sets 119880

119910+

119865and 119880

119910+

119872will be

less and less over time In this paper quantity change of setsrsquoelements is not considered to simplify the problem

23 Attitude Control Model Some transformation and sim-plification are applied to themissile model for control designIt is assumed that themissilersquosmass is of a constant valueNotethat the goal is to establish the angle of attack and sideslipangle The attitude control model is given by

= 119903 + 119902 sin120572 tan120573 minus

119876119878 (119862120572

119910120572 + 119862

120575119911

119910120575119911) cos120572

119898119881 cos120573

minus

(119865119910119887

+ 119870119865119910

119865119910119887

) cos120572

119898119881 cos120573minus

119866119910 cos120572119898119881 cos120573

120573 = 119902 cos120572 +

(119876119878 (119862120573

119911120573 + 119862

120575119910

119911 120575119910) + 119865119911119887

+ 119870119865119911

119865119911119887

) cos120573

119898119881

+

(119876119878 (119862120572

119910120572 + 119862

120575119911

119910120575119911) + 119865119910

119887

+ 119870119865119910

119865119910119887

) sin120572 sin120573

119898119881

+119866119911 cos120573

119898119881+

119866119910 sin120572 sin120573

119898119881

119902 =119872119910119887

119869119910

+

119870119872119910

119872119910119887

119869119910

+

119876119878119871119898120573

119910120573

119869119910

+119876119878119871119898

120575119910

119910 120575119910

119869119910

+

119876119878119871119898119902

119910119902

119869119910

119903 =119872119911119887

119869119911

+119870119872119911

119872119911119887

119869119911

+119876119878119871119898

120572

119911120572

119869119911

+119876119878119871119898

120575119911

119911120575119911

119869119911

+119876119878119871119898

119903

119911119903

119869119911

(16)

where 119898120573

119910 119898120575119911

119910 119898119902

119910 119898120572

119911 119898120575119911

119911 and 119898

119903

119911are aerodynamic

parameters

3 Mixed Logical Dynamical Model ofBlended Missile

31 Piecewise Affine Model of Blended Missile To simplifyanalysis the gravity term and the channel coupling term are

Table 1 Missilersquos overall parameters

Missilersquos takeoff mass [kg] 255Missilersquos full-length [m] 486Missilersquos diameter [m] 0317Distance between missile head and center ofmass [m] 2569

Moment of inertia 119869119909 119869119910 119869119911 [kgsdotm2] 303 3063 3063

Lateral force provided by individual jets [N] 2200Distance between jet ring center and centerof mass [m] 126

Range of angle of attack [rad] minus120587

6le 120572 le

120587

6

Range of pitch angular velocity [rads] minus5120587

3le 119903 le

5120587

3

Range of elevator deflection [rad] minus120587

6le 120575119911 le

120587

6

ignoredWith (16) themissile attitude control model of pitchchannel is

= 119903 minus

119876119878 (119862120572

119910120572 + 119862

120575119911

119910120575119911) cos120572

119898119881minus

(1 + 119870119865119910

) 119865119910119887

cos120572

119898119881

119903 =

(1 + 119870119872119911

) 119865119910119887

119897

119869119911

+119876119878119871119898

120572

119911120572

119869119911

+119876119878119871119898

120575119911

119911120575119911

119869119911

+119876119878119871119898

119903

119911119903

119869119911

(17)

Choose 119909 = [120572 119903]119879 as system state and 119906 = [120575119911 119865119910

119887

]119879 as

control input The considered output is 119910 = 120572 Then (17) canbe rewritten into the following state space form

= 119891 (119909) + 119892 (119909) 119906

119910 = [1 0] 119909

(18)

where

119891 (119909) =

[[[[

[

119903 minus

119876119878119862120572

119910120572 cos120572

119898119881

119876119878119871 (119898120572

119911120572 + 119898

119903

119911119903)

119869119911

]]]]

]

119892 (119909) =

[[[[[[

[

minus

119876119878119862120575119911

119910cos120572

119898119881minus

(1 + 119870119865119910

) cos120572

119898119881

119876119878119871119898120575119911

119911

119869119911

(1 + 119870119872119911

) 119897

119869119911

]]]]]]

]

(19)

The missile parameters are presented in Table 1Generally the aerodynamic coefficients 119862

120572

119910 119862120575119911

119910 119898120572

119911

and 119898120575119911

119911and the amplification factors 119870119865

119910

119870119872119911

are mainlyaffected by the flight velocity 119881 and the angle of attack 120572

[11 12] Since the terminal guidance phase is consideredin this paper the flight time is quite short and the flightvelocity of the missile can be treated as a constant Thusthe aerodynamic coefficients and the amplification factors are

6 International Journal of Aerospace Engineering

mainly affected by the angle of attack 120572 The relationshipsbetween them are shown in Figure 5

In practical application since 120572 is the main factor thatleads to system nonlinearities the system model is usuallylinearized if 120572 varies in small range As seen from Figure 5curves of the relation between aerodynamic parametersand angle of attack can be expressed by six line segmentsapproximately Here we choose 120572 = minus037 rad minus0153 rad0 0153 rad and 037 rad as the operation points and dividethe whole operation region into six subregions As a resultthe original model (17) can be converted to the followingpiecewise affine models

=

1198861119909 + 1198871119906 + 1198901 [1 0] 119909 le minus037

1198862119909 + 1198872119906 + 1198902 minus037 lt [1 0] 119909 le minus0153

1198863119909 + 1198873119906 + 1198903 minus0153 lt [1 0] 119909 le 0

1198864119909 + 1198874119906 + 1198904 0 lt [1 0] 119909 le 0153

1198865119909 + 1198875119906 + 1198905 0153 lt [1 0] 119909 le 037

1198866119909 + 1198876119906 + 1198906 [1 0] 119909 gt 037

(20)

119910 = [1 0] 119909 (21)

where

119886119894 =120597119891(119909)

120597119909

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= [11988611

11989411988612

119894

11988621

11989411988622

119894

]

119887119894 = 119892 (1199091198940) = [11988711

11989411988712

119894

11988721

11989411988722

119894

] 119890119894 = [1198901

119894

1198902

119894

]

11988611

119894=

1205971198911 (119909)

120597120572

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= minus119876119878

119898119881(

120597119862120572

119910

120597120572

1003816100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1205721198940 cos1205721198940 + 119862120572

119910(1205721198940) cos1205721198940

minus 119862120572

119910(1205721198940) 1205721198940 sin1205721198940)

11988612

119894=

1205971198911(119909)

120597119903

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= 1

11988621

119894=

1205971198912 (119909)

120597120572

10038161003816100381610038161003816100381610038161003816119909=1199091198940

=119876119878119871

119869119911

(120597119898120572

119911

120597120572

100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1205721198940 + 119898120572

119911(1205721198940) +

120597119898119903

119911

120597120572

100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1199031198940)

11988622

119894=

1205971198912(119909)

120597119903

10038161003816100381610038161003816100381610038161003816119909=1199091198940

=119876119878119871

119869119911

119898120572

119911(1205721198940)

11988711

119894= minus

119876119878119862120575119911

119910(1205721198940) cos1205721198940119898119881

11988712

119894= minus

(1 + 119870119865119910

(1205721198940)) cos1205721198940119898119881

11988721

119894=

119876119878119871119898120575119911

119911(1205721198940)

119869119911

11988722

119894=

(1 + 119870119872119911

(1205721198940)) 119897

119869119911

1198901

119894= 11988611

1198941205721198940 + 119886

12

1198941199031198940

1198902

119894= 11988621

1198941205721198940 + 119886

22

1198941199031198940

(22)

where 119894 (119894 = 1 2 6) is the label corresponding to the 119894thregion

From Table 1 and Figure 5 we get the set of aerodynamicparameters at the point (119867119881) = (20 km 1000ms) as shownin Table 2

Choose the sampling period 119879119904 = 0025 s The discretestate-space expression is then given by

119909 (119896 + 1)

=

1198861119909 (119896) + 1119906 (119896) + 1198901 [1 0] 119909 (119896) le minus037

1198862119909 (119896) + 2119906 (119896) + 1198902 minus037 lt [1 0] 119909 (119896) le minus0153

1198863119909 (119896) + 3119906 (119896) + 1198903 minus0153 lt [1 0] 119909 (119896) le 0

1198864119909 (119896) + 4119906 (119896) + 1198904 0 lt [1 0] 119909 (119896) le 0153

1198865119909 (119896) + 5119906 (119896) + 1198905 0153 lt [1 0] 119909 (119896) le 037

1198866119909 (119896) + 6119906 (119896) + 1198906 [1 0] 119909 (119896) gt 037

(23)

119910 (119896) = [1 0] 119909 (119896) (24)

where

1198861 = [104 0025

022 0995] 1198862 = [

1051 0025

0218 0995]

1198863 = [1023 00252

0248 09951] 1198864 = [

09954 00248

02732 09954]

1198865 = [09697 00245

02913 09956] 1198866 = [

09594 00244

03137 09959]

1198901 = [minus00183

minus01123] 1198902 = [

minus00175

minus00792]

1198903 = [minus00131

minus00376] 1198904 = [

0

0]

1198905 = [minus00053

00454] 1198906 = [

minus00169

01193]

1 = [minus0028 159 times 10

minus6

minus2166 142 times 10minus4]

2 = [minus003 218 times 10

minus6

minus2224 187 times 10minus4]

International Journal of Aerospace Engineering 7

2

15

1

05

0

minus05

minus1

minus15minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SmV

)C120572 y

(a) Relation between 119862120572119910and 120572

0092

009

0088

0086

0084

0082

008

0078

0076

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SmV

)C120575119911y

(b) Relation between 119862120575119911119910 and 120572

125

12

115

11

105

10

95

9minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SLJz)m

120572 z

(c) Relation between119898120572119911and 120572

96

94

92

90

88

86

84minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (minusQSL

Jz)m

120575119911z

(d) Relation between119898120575119911119911 and 120572

1

09

08

07

06

05

04

03

02

01

Am

plifi

catio

n fa

ctorKF

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

(e) Relation between119870119865119910

and 120572

12

11

1

09

08

07

06

05

04

03

Am

plifi

catio

n fa

ctorKM119911

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

(f) Relation between119870119872119911

and 120572

Figure 5 Aerodynamic parameters as functions of angle of attack

8 International Journal of Aerospace Engineering

Table 2 Aerodynamic parameters of pitch channel

Aerodynamic parameter 119876119878119862120572

119910119898119881 119876119878119862

120575119911

119910119898119881 119876119878119871119898

120572

119911119869119911

119876119878119871119898119903

119911119869119911

119876119878119871119898120575119911

119911119869119911

120572 = minus053 minus1175 0076 947 minus032 minus8489

120572 = minus037 minus0944 0082 973 minus032 minus8924

120572 = minus0153 minus023 0082 1043 minus032 minus8938

120572 = 0 036 009 1101 minus032 minus9510

120572 = 0153 094 0082 1158 minus032 minus8934

120572 = 037 163 0082 1224 minus032 minus8918

3 = [minus003 268 times 10

minus6

minus2228 223 times 10minus4]

4 = [minus0032 277 times 10

minus6

minus2371 232 times 10minus4]

5 = [minus003 227 times 10

minus6

minus2228 198 times 10minus4]

6 = [minus0029 191 times 10

minus6

minus2224 173 times 10minus4]

(25)

32 Constraints Analysis Due to the symmetry of jet config-uration the set of possible negative pitch control force is givenby

119880119910minus

119865= minus119865119898 minus2119865119898 cos

120587

9 minus2119865119898 cos

120587

18 minus2119865119898 minus4119865119898 cos

120587

9

minus4119865119898 cos120587

18

(26)

By combining (13) and (26) we obtain the set of allpossible pitch control force

119880119910

119865= 119865119898 2119865119898 cos

120587

9 2119865119898 cos

120587

18 2119865119898 4119865119898 cos

120587

9

4119865119898 cos120587

18 minus119865119898 minus2119865119898 cos

120587

9 minus2119865119898 cos

120587

18 minus2119865119898

minus4119865119898 cos120587

9 minus4119865119898 cos

120587

18

(27)

Substituting the jet parameters shown in Table 1 into theforegoing set yields

119880119910

119865= 2200 4135 4333 4400 8269 8666 minus2200 minus4135

minus4333 minus4400 minus8269 minus8666

(28)

Noting that the lateral forces are discrete variable we willutilize the linear combination of logical variables to describethe lateral force In terms of piecewise affine model (20) we

introduce logical variables 120575119865119894

isin 0 1 119894 = 1 2 12 toexpress the lateral thrust 119865119910

119887

119865119910119887

= 22001205751198651

+ 41351205751198652

+ 43331205751198653

+ 44001205751198654

+ 82691205751198655

+ 86661205751198656

minus 22001205751198657

minus 41351205751198658

minus 43331205751198659

minus 440012057511986510

minus 826912057511986511

minus 866612057511986512

(29)

In (29) the logical variables should satisfy the constraints

12

sum

119894=1

120575119865119894

= 0 or 1 (30)

where 0 means that no lateral force is generated while 1

means that the applied lateral force equals to one element ofset 119880119910119865

Denote 1199061 = 120575119911 the control input 119906 in model (23) can berewritten as

119906 = [1199061 119865119910119887

]119879 (31)

As shown in Table 1 the constraints on system states andcontrol input are

119909min le 119909 (119896) le 119909max

1199061min le 1199061 (119896) le 1199061max(32)

where 119909min = [minus053 minus522]119879 119909max = [053 522]

119879 1199061min =

minus053 1199061max = 053According to (30) we have

12

sum

119894=1

120575119865119894

le 1 (33)

which shows the constraint on the logical control inputUse logical variables 120575119894(119896) isin 0 1 119894 = 1 2 5 to

describe the operation points which satisfy the constraints

[1 0] 119909 (119896) + 037 le 0 lArrrArr 1205751 (119896) = 1

[1 0] 119909 (119896) + 0153 le 0 lArrrArr 1205752 (119896) = 1

[1 0] 119909 (119896) le 0 lArrrArr 1205753 (119896) = 1

[1 0] 119909 (119896) minus 0153 le 0 lArrrArr 1205754 (119896) = 1

[1 0] 119909 (119896) minus 037 le 0 lArrrArr 1205755 (119896) = 1

(34)

International Journal of Aerospace Engineering 9

Equation (34) can be transformed into the followingequivalent mixed logical inequalities [13]

[1 0] 119909 (119896) + 037 ge 120576 + (1198981 minus 120576) 1205751 (119896)

[1 0] 119909 (119896) + 037 le 1198721 (1 minus 1205751 (119896))

[1 0] 119909 (119896) + 0153 ge 120576 + (1198982 minus 120576) 1205752 (119896)

[1 0] 119909 (119896) + 0153 le 1198722 (1 minus 1205752 (119896))

[1 0] 119909 (119896) ge 120576 + (1198983 minus 120576) 1205753 (119896)

[1 0] 119909 (119896) le 1198723 (1 minus 1205753 (119896))

[1 0] 119909 (119896) minus 0153 ge 120576 + (1198984 minus 120576) 1205754 (119896)

[1 0] 119909 (119896) minus 0153 le 1198724 (1 minus 1205754 (119896))

[1 0] 119909 (119896) minus 037 ge 120576 + (1198985 minus 120576) 1205755 (119896)

[1 0] 119909 (119896) minus 037 le 1198725 (1 minus 1205755 (119896))

(35)

where 1198981 = minus016 1198721 = 090 1198982 = minus0377 1198722 = 06831198983 = minus053 1198723 = 053 1198984 = minus0683 1198724 = 0377 1198985 =

minus0901198725 = 016 and 120576 = 10minus6

In addition we introduce the auxiliary logical variables120575119894(119896) isin 0 1 119894 = 6 9 as follows

1205756 (119896) = (1 minus 1205751 (119896)) 1205752 (119896)

1205757 (119896) = (1 minus 1205752 (119896)) 1205753 (119896)

1205758 (119896) = (1 minus 1205753 (119896)) 1205754 (119896)

1205759 (119896) = (1 minus 1205754 (119896)) 1205755 (119896)

(36)

With 1205751 1205756 1205757 1205758 1205759 and 1 minus 1205755 the six regions canbe presented Similarly (36) can be transformed into theequivalent mixed logical inequalities

minus1205751 (119896) + 1205752 (119896) minus 1205756 (119896) le 0

1205751 (119896) + 1205756 (119896) le 1

minus1205752 (119896) + 1205756 (119896) le 0

minus1205752 (119896) + 1205753 (119896) minus 1205757 (119896) le 0

1205752 (119896) + 1205757 (119896) le 1

minus1205753 (119896) + 1205757 (119896) le 0

minus1205753 (119896) + 1205754 (119896) minus 1205758 (119896) le 0

1205753 (119896) + 1205758 (119896) le 1

minus1205754 (119896) + 1205758 (119896) le 0

minus1205754 (119896) + 1205755 (119896) minus 1205759 (119896) le 0

1205754 (119896) + 1205759 (119896) le 1

minus1205755 (119896) + 1205759 (119896) le 0

(37)

To describe the state space model of each region weintroduce the following auxiliary continuous variables

1199111 (119896) = (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) 1205751 (119896)

1199112 (119896) = (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) 1205756 (119896)

1199113 (119896) = (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) 1205757 (119896)

1199114 (119896) = (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) 1205758 (119896)

1199115 (119896) = (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) 1205759 (119896)

1199116 (119896) = (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) (1 minus 1205755 (119896))

(38)

Equation (38) can be converted to the equivalent mixedlogical inequalities

1199111 (119896) ge (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198721198911 (1 minus 1205751 (119896))

1199111 (119896) le (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198981198911 (1 minus 1205751 (119896))

1199111 (119896) ge 11989811989111205751 (119896)

1199111 (119896) le 11987211989111205751 (119896)

1199112 (119896) ge (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198721198912 (1 minus 1205756 (119896))

1199112 (119896) le (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198981198912 (1 minus 1205756 (119896))

1199112 (119896) ge 11989811989121205756 (119896)

1199112 (119896) le 11987211989121205756 (119896)

1199113 (119896) ge (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198721198913 (1 minus 1205757 (119896))

1199113 (119896) le (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198981198913 (1 minus 1205757 (119896))

1199113 (119896) ge 11989811989131205757 (119896)

1199113 (119896) le 11987211989131205757 (119896)

1199114 (119896) ge (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198721198914 (1 minus 1205758 (119896))

1199114 (119896) le (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198981198914 (1 minus 1205758 (119896))

1199114 (119896) ge 11989811989141205758 (119896)

1199114 (119896) le 11987211989141205758 (119896)

1199115 (119896) ge (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198721198915 (1 minus 1205759 (119896))

1199115 (119896) le (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198981198915 (1 minus 1205759 (119896))

10 International Journal of Aerospace Engineering

1199115 (119896) ge 11989811989151205759 (119896)

1199115 (119896) le 11987211989151205759 (119896)

1199116 (119896) ge (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11987211989151205755 (119896)

1199116 (119896) le (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11989811989151205755 (119896)

1199116 (119896) ge 1198981198916 (1 minus 1205755 (119896))

1199116 (119896) le 1198721198916 (1 minus 1205755 (119896))

(39)

where1198721198911 = [073 1084]1198791198981198911 = [minus077 minus1106]

1198791198721198912 =[076 1166]

119879 1198981198912 = [minus079 minus 1256]119879 1198721198913 =

[077 1273]119879 1198981198913 = [minus078 minus 1280]

119879 1198721198914 =

[076 1403]119879 1198981198914 = [minus076 minus 1403]

119879 1198721198915 =

[0725 1291]119879 1198981198915 = [minus0736 minus 1205]

119879 1198721198916 =

[0696 1154]119879 and1198981198916 = [minus073 minus 1130]

119879Then the whole MLD model of the missile is given by

119909 (119896 + 1) =

6

sum

119894=1

119911119894 (119896)

119910 (119896) = [1 0] 119909 (119896)

st (28) (30) (32) (34) (36) (38)

(40)

where (29) (31) (33) (35) (37) and (39) represent all theconstraint inequalities

4 Autopilot Design UsingHybrid MPC Method

In general we expect that the output 119910 tracks its command 119910119888

as fast as possible with a small amount of fuel consumptionMotivated by this observation we consider the followingoptimization problem

119869lowast

= min119906(119896)119906(119896+1)120575(119896)120575(119896+1|119896)119911(119896)119911(119896+1|119896)

119873

sum

119894=1

(1003817100381710038171003817119910 (119896 + 119894 | 119896)

minus119910119888 (119896 + 119894)1003817100381710038171003817

2

119876

+ 119906(119896 + 119894)2

119877)

st MLD model (39)

1199061min le 1199061 (119896) 1199061 (119896 + 1) le 1199061max

119909min le 119909 (119896) 119909 (119896 + 1) le 119909max

(41)

where 119910119888 is the command for angle of attack 119910(119896 + 119894 | 119896) ispredictive value of angle of attack 119873 denotes the predictivehorizon 119876 and 119877 are the weighting matrices and 119877 =

diag(119877120575119911

119877119865119910

) 119877120575119911

119877119865119910

are weighted coefficients of aerody-namic control surfaces and lateral pulse jets respectivelyThe

ratio of 119877120575119911

and 119877119865119910

represents the control allocation betweenaerodynamic control surfaces and lateral pulse jets When119877119865119910

119877120575119911

is increased the requirement for lateral force will bedeclined that is the fuel consumption can be lessened Inaddition the deduction can be verified from the simulationresults of Case 1 and Case 2

Different control allocation between the dual actuators(aerodynamic control surfaces and lateral pulse jets) can beobtained by setting different 119876 and 119877 Since both logicaland continuous variables are involved in (41) the aboveon-line optimization problem is a mixed integer quadraticprogramming (MIQP) problemwhich can be solved by usingthe hybrid MPC toolbox of MATLAB [14 15]

Remark 2 In an explicit MPC controller the main factorsaffecting the number of subregions include the systemdimen-sion the predictive horizon and the number of constrains Inthis paper the blendedmissile with aerodynamic control sur-faces and lateral jets is investigated To deal with the discreteproperty of the lateral jet forces some logical variables areintroduced for MPC controller design which increases thenumber of subregions inevitably

5 Numerical Simulations

51 Case 1 Numerical simulations are performed to ver-ify the feasibility of the proposed method in this sectionThe involved parameters are given in Table 1 The weightmatrices are set as 119876 = 10 119877120575

119911

= 0025 and 119877119865119910

=

diag(0005 0005 0005) Suppose the initial state is 1205720 =0 1199030 = 0 In order to avoid excessive computational load wechoose predictive horizon119873 = 2 and control horizon119872 = 2The command for angle of attack is 120572119888 = 02 rad

Explicit form of the optimal controller is provided byhybrid toolbox which is not given here for simplicity Insteadwe present the partition results associated with the controllerin Figure 6 where 4190 subregions are obtained and for eachsubregion a control law of the form 119906 = 119870119909 + 119887 is givenAs a benefit of using explicit MPC the controller parameters119870 and 119887 for all the subregions are obtained simultaneouslyonce the optimization problem is solved This property isobviously different from that of gain-scheduled controller(where the controller parameters for different regions areseparately determined)

However by a simple calculation we conclude that thesesubregions donot require toomuch storage space (the neededstorage space is approximately 14 times 8 times 4190 = 450KB)Moreover it should be noted that when the algorithm isimplemented in practice much more time will be needed toidentify which subregion the current states enter into This isanother important factor that affects the computational costof the proposed algorithm

The simulation results are shown in Figures 7ndash9 It isseen from Figure 7 that the actual angle of attack tracksthe command in less than 01 seconds As shown in Figures8 and 9 the aerodynamic force converges to a constant astime increases while the jets are only activated during thebeginning period (when the tracking error is obvious)

International Journal of Aerospace Engineering 11

58

68

73

108

174

180

245

252

254

270

271

272

273

274

276

277

299

300

301

302

06

04

02

0

minus02

minus04

Elev

ator

defl

ectio

n (r

ad)

Angle of attack (rad)minus04 minus02 0 02 04 06

1634

1635

1638

1648

1649

1650

1664

1673

1677

1678

1679

1712

1726

1743

1744

1746

1748

1751

1752

1983

1985

1993

2011

Late

ral f

orce

(N)

Angle of attack (rad)minus04 minus02 0 02 04 06

1000

800

600

400

200

0

minus200

minus400

minus600

minus800

minus1000

Polyhedral partitionmdash4190 regions Polyhedral partitionmdash4190 regions

Figure 6 The partition associated with the explicit controller

025

02

015

01

005

0

ActualCommand

0 02 04 06 08 1

Time (s)

Ang

le o

f atta

ck (r

ad)

Figure 7 The response of angle of attack

52 Case 2 The results for the case with a different weight119877120575119911

= 0005 are shown in Figures 10ndash12 It is seen that adifferent control allocation result is obtained

04

02

0

minus02

minus04

minus060 02 04 06 08 1

Time (s)

Elev

ator

defl

ectio

n (r

ad)

Figure 8 The response of elevator deflection

53 Case 3 In this case a different command 120572119888 =

02 cos(05120587119905) rad is considered and the weight matrices arethe same as the ones for Case 1 The simulation results areshown in Figures 13ndash15

It is seen from Figure 13 that asymptotic tracking is alsoachievedThemethod proposed in this paper can realize both

12 International Journal of Aerospace Engineering

1

05

0

minus05

minus1

times104

0 02 04 06 08 1

Time (s)

Late

ral f

orce

(N)

Figure 9 The response of lateral force

025

02

015

01

005

0

Ang

le o

f atta

ck (r

ad)

0 02 04 06 08 1

Time (s)

ActualCommand

Figure 10 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 02 04 06 08 1

Time (s)

Figure 11 The response of elevator deflection

1

05

0

minus05

minus1

Late

ral f

orce

(N)

times104

0 02 04 06 08 1

Time (s)

Figure 12 The response of lateral force

03

025

02

015

01

005

0

minus005

minus01

minus015

minus02

Ang

le o

f atta

ck (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

ActualCommand

Figure 13 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

Figure 14 The response of elevator deflection

International Journal of Aerospace Engineering 13

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1

Late

ral f

orce

(N)

0 05 1 15 2 25 3 35 4

Time (s)

times104

Figure 15 The response of lateral force

fast tracking command and control allocation It is seen fromFigure 15 that the jets are activated while the tracking erroris obvious or command is varying rapidly

6 Conclusion

An autopilot design method for a missile with aerodynamiccontrol surfaces and lateral jets is presented in this paperThe nonlinear attitude control model is reduced to an MLDmodel Meanwhile the lateral force is described as linearcombination of logical variables due to the discrete values oflateral force Then the whole MLD model of attitude controlsystem is derived Autopilot design is accomplished usinghybrid MPC method By setting the related weighted coeffi-cients in index function the control allocation is obtainedMoreover numerical simulations are performed under thedifferent conditions the performance of tracking the attitudecommand and control allocation is verified and the explicitform of the control law can be obtained as well

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research presented in this document is supportedby the National Natural Science Foundation of Chinaunder Grant nos 61104193 61203191 and 61333001 theFundamental Research Funds for the Central Universities(HITNSRIF2012032) the Program for IBRSEM in HarbinInstitute of Technology under Grant HITIBRSEMA201415and the Foundation of Supporting Technology for Aerospaceunder Grant 2014-HT-HGD7

References

[1] R Hirokawa K Sato and S Manabe ldquoAutopilot design fora missile with reaction-jet using coefficient diagram methodrdquoin Proceedings of the AIAA Guidance Navigation and ControlConference and Exhibit 2001 August 2001

[2] P K Menon and V R Iragavarapu Adaptive Techniques forMultiple Actuator Blending Defense Technical InformationCenter 1998

[3] A G Munson and M W Garbrick ldquoJet interaction investiga-tionrdquo Report OR-9933 Martin Marietta Orlando Fla USA1969

[4] M Graham and P Weinacht ldquoNumerical simulation of lateralcontrol jetsrdquo in Proceedings of the 37th Aerospace SciencesMeeting and Exhibit 1999 AIAA Paper

[5] J S Shamma and J R Cloutier ldquoGain-scheduledmissile autopi-lot design using linear parameter varying transformationsrdquoJournal of Guidance Control and Dynamics vol 16 no 2 pp256ndash263 1993

[6] M Innocenti and A Thukral ldquoSimultaneous reaction jet andaerodynamic control of missile systemsrdquo AlAA 93-3739-CP1993

[7] D B Ridgely Y Lee and T Fanciullo ldquoDual aeropropulsivemissile control-optimal control and control allocationrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit Keystone Colo USA 2006

[8] D B Ridgely D Drake L Triplett and C Geise ldquoDynamiccontrol allocation of a missile with tails and reaction jetsrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 3158ndash3189 August 2007

[9] D Q Mayne J B Rawlings C V Rao and P O ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[10] A Bemporad F Borrelli and M Morari ldquoModel predictivecontrol based on linear programmingmdashthe explicit solutionrdquoIEEE Transactions on Automatic Control vol 47 no 12 pp1974ndash1985 2002

[11] C Jouannet and P Krus ldquoModelling of high angle of attackaerodynamicrdquo in Proceedings of the 25th AIAA Applied Aerody-namics Conference pp 1481ndash1516 June 2007

[12] S Venugopal and M Krishnamurthy ldquoMissile aerodynamics athigh angles of attack a prediction coderdquo Journal of Spacecraftand Rockets vol 32 no 2 pp 263ndash269 1995

[13] A Bemporad ldquoEfficient conversion of mixed logical dynamicalsystems into an equivalent piecewise affine formrdquo IEEE Trans-actions on Automatic Control vol 49 no 5 pp 832ndash838 2004

[14] F D Torrisi andA Bemporad ldquoHYSDELmdasha tool for generatingcomputational hybrid models for analysis and synthesis prob-lemsrdquo IEEE Transactions on Control Systems Technology vol 12no 2 pp 235ndash249 2004

[15] A Bemporad W P M H Heemels and B De SchutterldquoOn hybrid systems and closed-loop MPC systemsrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 863ndash8692002

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International Journal of

International Journal of Aerospace Engineering 5

amplification factors 119870119865119910

119870119865119911

119870119872119910

and 119870119872119911

are introducedas in [3 4]Then the resulting lateral forces andmoments are

[[[

[

119865119886

119909119887

119865119886

119910119887

119865119886

119911119887

]]]

]

= [

[

0

119865119910119887

+ 119870119865119910

119865119910119887

119865119911119887

+ 119870119865119911

119865119911119887

]

]

[[[

[

119872119886

119909119887

119872119886

119910119887

119872119886

119911119887

]]]

]

= [

[

0

119872119910119887

+ 119870119872119910

119872119910119887

119872119911119887

+ 119870119872119911

119872119911119887

]

]

(15)

Remark 1 In fact each pulse jet can be fired only once so thelocation of the fired jet cannot provide force anymore Basedon this precondition elements of sets 119880

119910+

119865and 119880

119910+

119872will be

less and less over time In this paper quantity change of setsrsquoelements is not considered to simplify the problem

23 Attitude Control Model Some transformation and sim-plification are applied to themissile model for control designIt is assumed that themissilersquosmass is of a constant valueNotethat the goal is to establish the angle of attack and sideslipangle The attitude control model is given by

= 119903 + 119902 sin120572 tan120573 minus

119876119878 (119862120572

119910120572 + 119862

120575119911

119910120575119911) cos120572

119898119881 cos120573

minus

(119865119910119887

+ 119870119865119910

119865119910119887

) cos120572

119898119881 cos120573minus

119866119910 cos120572119898119881 cos120573

120573 = 119902 cos120572 +

(119876119878 (119862120573

119911120573 + 119862

120575119910

119911 120575119910) + 119865119911119887

+ 119870119865119911

119865119911119887

) cos120573

119898119881

+

(119876119878 (119862120572

119910120572 + 119862

120575119911

119910120575119911) + 119865119910

119887

+ 119870119865119910

119865119910119887

) sin120572 sin120573

119898119881

+119866119911 cos120573

119898119881+

119866119910 sin120572 sin120573

119898119881

119902 =119872119910119887

119869119910

+

119870119872119910

119872119910119887

119869119910

+

119876119878119871119898120573

119910120573

119869119910

+119876119878119871119898

120575119910

119910 120575119910

119869119910

+

119876119878119871119898119902

119910119902

119869119910

119903 =119872119911119887

119869119911

+119870119872119911

119872119911119887

119869119911

+119876119878119871119898

120572

119911120572

119869119911

+119876119878119871119898

120575119911

119911120575119911

119869119911

+119876119878119871119898

119903

119911119903

119869119911

(16)

where 119898120573

119910 119898120575119911

119910 119898119902

119910 119898120572

119911 119898120575119911

119911 and 119898

119903

119911are aerodynamic

parameters

3 Mixed Logical Dynamical Model ofBlended Missile

31 Piecewise Affine Model of Blended Missile To simplifyanalysis the gravity term and the channel coupling term are

Table 1 Missilersquos overall parameters

Missilersquos takeoff mass [kg] 255Missilersquos full-length [m] 486Missilersquos diameter [m] 0317Distance between missile head and center ofmass [m] 2569

Moment of inertia 119869119909 119869119910 119869119911 [kgsdotm2] 303 3063 3063

Lateral force provided by individual jets [N] 2200Distance between jet ring center and centerof mass [m] 126

Range of angle of attack [rad] minus120587

6le 120572 le

120587

6

Range of pitch angular velocity [rads] minus5120587

3le 119903 le

5120587

3

Range of elevator deflection [rad] minus120587

6le 120575119911 le

120587

6

ignoredWith (16) themissile attitude control model of pitchchannel is

= 119903 minus

119876119878 (119862120572

119910120572 + 119862

120575119911

119910120575119911) cos120572

119898119881minus

(1 + 119870119865119910

) 119865119910119887

cos120572

119898119881

119903 =

(1 + 119870119872119911

) 119865119910119887

119897

119869119911

+119876119878119871119898

120572

119911120572

119869119911

+119876119878119871119898

120575119911

119911120575119911

119869119911

+119876119878119871119898

119903

119911119903

119869119911

(17)

Choose 119909 = [120572 119903]119879 as system state and 119906 = [120575119911 119865119910

119887

]119879 as

control input The considered output is 119910 = 120572 Then (17) canbe rewritten into the following state space form

= 119891 (119909) + 119892 (119909) 119906

119910 = [1 0] 119909

(18)

where

119891 (119909) =

[[[[

[

119903 minus

119876119878119862120572

119910120572 cos120572

119898119881

119876119878119871 (119898120572

119911120572 + 119898

119903

119911119903)

119869119911

]]]]

]

119892 (119909) =

[[[[[[

[

minus

119876119878119862120575119911

119910cos120572

119898119881minus

(1 + 119870119865119910

) cos120572

119898119881

119876119878119871119898120575119911

119911

119869119911

(1 + 119870119872119911

) 119897

119869119911

]]]]]]

]

(19)

The missile parameters are presented in Table 1Generally the aerodynamic coefficients 119862

120572

119910 119862120575119911

119910 119898120572

119911

and 119898120575119911

119911and the amplification factors 119870119865

119910

119870119872119911

are mainlyaffected by the flight velocity 119881 and the angle of attack 120572

[11 12] Since the terminal guidance phase is consideredin this paper the flight time is quite short and the flightvelocity of the missile can be treated as a constant Thusthe aerodynamic coefficients and the amplification factors are

6 International Journal of Aerospace Engineering

mainly affected by the angle of attack 120572 The relationshipsbetween them are shown in Figure 5

In practical application since 120572 is the main factor thatleads to system nonlinearities the system model is usuallylinearized if 120572 varies in small range As seen from Figure 5curves of the relation between aerodynamic parametersand angle of attack can be expressed by six line segmentsapproximately Here we choose 120572 = minus037 rad minus0153 rad0 0153 rad and 037 rad as the operation points and dividethe whole operation region into six subregions As a resultthe original model (17) can be converted to the followingpiecewise affine models

=

1198861119909 + 1198871119906 + 1198901 [1 0] 119909 le minus037

1198862119909 + 1198872119906 + 1198902 minus037 lt [1 0] 119909 le minus0153

1198863119909 + 1198873119906 + 1198903 minus0153 lt [1 0] 119909 le 0

1198864119909 + 1198874119906 + 1198904 0 lt [1 0] 119909 le 0153

1198865119909 + 1198875119906 + 1198905 0153 lt [1 0] 119909 le 037

1198866119909 + 1198876119906 + 1198906 [1 0] 119909 gt 037

(20)

119910 = [1 0] 119909 (21)

where

119886119894 =120597119891(119909)

120597119909

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= [11988611

11989411988612

119894

11988621

11989411988622

119894

]

119887119894 = 119892 (1199091198940) = [11988711

11989411988712

119894

11988721

11989411988722

119894

] 119890119894 = [1198901

119894

1198902

119894

]

11988611

119894=

1205971198911 (119909)

120597120572

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= minus119876119878

119898119881(

120597119862120572

119910

120597120572

1003816100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1205721198940 cos1205721198940 + 119862120572

119910(1205721198940) cos1205721198940

minus 119862120572

119910(1205721198940) 1205721198940 sin1205721198940)

11988612

119894=

1205971198911(119909)

120597119903

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= 1

11988621

119894=

1205971198912 (119909)

120597120572

10038161003816100381610038161003816100381610038161003816119909=1199091198940

=119876119878119871

119869119911

(120597119898120572

119911

120597120572

100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1205721198940 + 119898120572

119911(1205721198940) +

120597119898119903

119911

120597120572

100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1199031198940)

11988622

119894=

1205971198912(119909)

120597119903

10038161003816100381610038161003816100381610038161003816119909=1199091198940

=119876119878119871

119869119911

119898120572

119911(1205721198940)

11988711

119894= minus

119876119878119862120575119911

119910(1205721198940) cos1205721198940119898119881

11988712

119894= minus

(1 + 119870119865119910

(1205721198940)) cos1205721198940119898119881

11988721

119894=

119876119878119871119898120575119911

119911(1205721198940)

119869119911

11988722

119894=

(1 + 119870119872119911

(1205721198940)) 119897

119869119911

1198901

119894= 11988611

1198941205721198940 + 119886

12

1198941199031198940

1198902

119894= 11988621

1198941205721198940 + 119886

22

1198941199031198940

(22)

where 119894 (119894 = 1 2 6) is the label corresponding to the 119894thregion

From Table 1 and Figure 5 we get the set of aerodynamicparameters at the point (119867119881) = (20 km 1000ms) as shownin Table 2

Choose the sampling period 119879119904 = 0025 s The discretestate-space expression is then given by

119909 (119896 + 1)

=

1198861119909 (119896) + 1119906 (119896) + 1198901 [1 0] 119909 (119896) le minus037

1198862119909 (119896) + 2119906 (119896) + 1198902 minus037 lt [1 0] 119909 (119896) le minus0153

1198863119909 (119896) + 3119906 (119896) + 1198903 minus0153 lt [1 0] 119909 (119896) le 0

1198864119909 (119896) + 4119906 (119896) + 1198904 0 lt [1 0] 119909 (119896) le 0153

1198865119909 (119896) + 5119906 (119896) + 1198905 0153 lt [1 0] 119909 (119896) le 037

1198866119909 (119896) + 6119906 (119896) + 1198906 [1 0] 119909 (119896) gt 037

(23)

119910 (119896) = [1 0] 119909 (119896) (24)

where

1198861 = [104 0025

022 0995] 1198862 = [

1051 0025

0218 0995]

1198863 = [1023 00252

0248 09951] 1198864 = [

09954 00248

02732 09954]

1198865 = [09697 00245

02913 09956] 1198866 = [

09594 00244

03137 09959]

1198901 = [minus00183

minus01123] 1198902 = [

minus00175

minus00792]

1198903 = [minus00131

minus00376] 1198904 = [

0

0]

1198905 = [minus00053

00454] 1198906 = [

minus00169

01193]

1 = [minus0028 159 times 10

minus6

minus2166 142 times 10minus4]

2 = [minus003 218 times 10

minus6

minus2224 187 times 10minus4]

International Journal of Aerospace Engineering 7

2

15

1

05

0

minus05

minus1

minus15minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SmV

)C120572 y

(a) Relation between 119862120572119910and 120572

0092

009

0088

0086

0084

0082

008

0078

0076

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SmV

)C120575119911y

(b) Relation between 119862120575119911119910 and 120572

125

12

115

11

105

10

95

9minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SLJz)m

120572 z

(c) Relation between119898120572119911and 120572

96

94

92

90

88

86

84minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (minusQSL

Jz)m

120575119911z

(d) Relation between119898120575119911119911 and 120572

1

09

08

07

06

05

04

03

02

01

Am

plifi

catio

n fa

ctorKF

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

(e) Relation between119870119865119910

and 120572

12

11

1

09

08

07

06

05

04

03

Am

plifi

catio

n fa

ctorKM119911

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

(f) Relation between119870119872119911

and 120572

Figure 5 Aerodynamic parameters as functions of angle of attack

8 International Journal of Aerospace Engineering

Table 2 Aerodynamic parameters of pitch channel

Aerodynamic parameter 119876119878119862120572

119910119898119881 119876119878119862

120575119911

119910119898119881 119876119878119871119898

120572

119911119869119911

119876119878119871119898119903

119911119869119911

119876119878119871119898120575119911

119911119869119911

120572 = minus053 minus1175 0076 947 minus032 minus8489

120572 = minus037 minus0944 0082 973 minus032 minus8924

120572 = minus0153 minus023 0082 1043 minus032 minus8938

120572 = 0 036 009 1101 minus032 minus9510

120572 = 0153 094 0082 1158 minus032 minus8934

120572 = 037 163 0082 1224 minus032 minus8918

3 = [minus003 268 times 10

minus6

minus2228 223 times 10minus4]

4 = [minus0032 277 times 10

minus6

minus2371 232 times 10minus4]

5 = [minus003 227 times 10

minus6

minus2228 198 times 10minus4]

6 = [minus0029 191 times 10

minus6

minus2224 173 times 10minus4]

(25)

32 Constraints Analysis Due to the symmetry of jet config-uration the set of possible negative pitch control force is givenby

119880119910minus

119865= minus119865119898 minus2119865119898 cos

120587

9 minus2119865119898 cos

120587

18 minus2119865119898 minus4119865119898 cos

120587

9

minus4119865119898 cos120587

18

(26)

By combining (13) and (26) we obtain the set of allpossible pitch control force

119880119910

119865= 119865119898 2119865119898 cos

120587

9 2119865119898 cos

120587

18 2119865119898 4119865119898 cos

120587

9

4119865119898 cos120587

18 minus119865119898 minus2119865119898 cos

120587

9 minus2119865119898 cos

120587

18 minus2119865119898

minus4119865119898 cos120587

9 minus4119865119898 cos

120587

18

(27)

Substituting the jet parameters shown in Table 1 into theforegoing set yields

119880119910

119865= 2200 4135 4333 4400 8269 8666 minus2200 minus4135

minus4333 minus4400 minus8269 minus8666

(28)

Noting that the lateral forces are discrete variable we willutilize the linear combination of logical variables to describethe lateral force In terms of piecewise affine model (20) we

introduce logical variables 120575119865119894

isin 0 1 119894 = 1 2 12 toexpress the lateral thrust 119865119910

119887

119865119910119887

= 22001205751198651

+ 41351205751198652

+ 43331205751198653

+ 44001205751198654

+ 82691205751198655

+ 86661205751198656

minus 22001205751198657

minus 41351205751198658

minus 43331205751198659

minus 440012057511986510

minus 826912057511986511

minus 866612057511986512

(29)

In (29) the logical variables should satisfy the constraints

12

sum

119894=1

120575119865119894

= 0 or 1 (30)

where 0 means that no lateral force is generated while 1

means that the applied lateral force equals to one element ofset 119880119910119865

Denote 1199061 = 120575119911 the control input 119906 in model (23) can berewritten as

119906 = [1199061 119865119910119887

]119879 (31)

As shown in Table 1 the constraints on system states andcontrol input are

119909min le 119909 (119896) le 119909max

1199061min le 1199061 (119896) le 1199061max(32)

where 119909min = [minus053 minus522]119879 119909max = [053 522]

119879 1199061min =

minus053 1199061max = 053According to (30) we have

12

sum

119894=1

120575119865119894

le 1 (33)

which shows the constraint on the logical control inputUse logical variables 120575119894(119896) isin 0 1 119894 = 1 2 5 to

describe the operation points which satisfy the constraints

[1 0] 119909 (119896) + 037 le 0 lArrrArr 1205751 (119896) = 1

[1 0] 119909 (119896) + 0153 le 0 lArrrArr 1205752 (119896) = 1

[1 0] 119909 (119896) le 0 lArrrArr 1205753 (119896) = 1

[1 0] 119909 (119896) minus 0153 le 0 lArrrArr 1205754 (119896) = 1

[1 0] 119909 (119896) minus 037 le 0 lArrrArr 1205755 (119896) = 1

(34)

International Journal of Aerospace Engineering 9

Equation (34) can be transformed into the followingequivalent mixed logical inequalities [13]

[1 0] 119909 (119896) + 037 ge 120576 + (1198981 minus 120576) 1205751 (119896)

[1 0] 119909 (119896) + 037 le 1198721 (1 minus 1205751 (119896))

[1 0] 119909 (119896) + 0153 ge 120576 + (1198982 minus 120576) 1205752 (119896)

[1 0] 119909 (119896) + 0153 le 1198722 (1 minus 1205752 (119896))

[1 0] 119909 (119896) ge 120576 + (1198983 minus 120576) 1205753 (119896)

[1 0] 119909 (119896) le 1198723 (1 minus 1205753 (119896))

[1 0] 119909 (119896) minus 0153 ge 120576 + (1198984 minus 120576) 1205754 (119896)

[1 0] 119909 (119896) minus 0153 le 1198724 (1 minus 1205754 (119896))

[1 0] 119909 (119896) minus 037 ge 120576 + (1198985 minus 120576) 1205755 (119896)

[1 0] 119909 (119896) minus 037 le 1198725 (1 minus 1205755 (119896))

(35)

where 1198981 = minus016 1198721 = 090 1198982 = minus0377 1198722 = 06831198983 = minus053 1198723 = 053 1198984 = minus0683 1198724 = 0377 1198985 =

minus0901198725 = 016 and 120576 = 10minus6

In addition we introduce the auxiliary logical variables120575119894(119896) isin 0 1 119894 = 6 9 as follows

1205756 (119896) = (1 minus 1205751 (119896)) 1205752 (119896)

1205757 (119896) = (1 minus 1205752 (119896)) 1205753 (119896)

1205758 (119896) = (1 minus 1205753 (119896)) 1205754 (119896)

1205759 (119896) = (1 minus 1205754 (119896)) 1205755 (119896)

(36)

With 1205751 1205756 1205757 1205758 1205759 and 1 minus 1205755 the six regions canbe presented Similarly (36) can be transformed into theequivalent mixed logical inequalities

minus1205751 (119896) + 1205752 (119896) minus 1205756 (119896) le 0

1205751 (119896) + 1205756 (119896) le 1

minus1205752 (119896) + 1205756 (119896) le 0

minus1205752 (119896) + 1205753 (119896) minus 1205757 (119896) le 0

1205752 (119896) + 1205757 (119896) le 1

minus1205753 (119896) + 1205757 (119896) le 0

minus1205753 (119896) + 1205754 (119896) minus 1205758 (119896) le 0

1205753 (119896) + 1205758 (119896) le 1

minus1205754 (119896) + 1205758 (119896) le 0

minus1205754 (119896) + 1205755 (119896) minus 1205759 (119896) le 0

1205754 (119896) + 1205759 (119896) le 1

minus1205755 (119896) + 1205759 (119896) le 0

(37)

To describe the state space model of each region weintroduce the following auxiliary continuous variables

1199111 (119896) = (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) 1205751 (119896)

1199112 (119896) = (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) 1205756 (119896)

1199113 (119896) = (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) 1205757 (119896)

1199114 (119896) = (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) 1205758 (119896)

1199115 (119896) = (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) 1205759 (119896)

1199116 (119896) = (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) (1 minus 1205755 (119896))

(38)

Equation (38) can be converted to the equivalent mixedlogical inequalities

1199111 (119896) ge (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198721198911 (1 minus 1205751 (119896))

1199111 (119896) le (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198981198911 (1 minus 1205751 (119896))

1199111 (119896) ge 11989811989111205751 (119896)

1199111 (119896) le 11987211989111205751 (119896)

1199112 (119896) ge (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198721198912 (1 minus 1205756 (119896))

1199112 (119896) le (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198981198912 (1 minus 1205756 (119896))

1199112 (119896) ge 11989811989121205756 (119896)

1199112 (119896) le 11987211989121205756 (119896)

1199113 (119896) ge (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198721198913 (1 minus 1205757 (119896))

1199113 (119896) le (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198981198913 (1 minus 1205757 (119896))

1199113 (119896) ge 11989811989131205757 (119896)

1199113 (119896) le 11987211989131205757 (119896)

1199114 (119896) ge (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198721198914 (1 minus 1205758 (119896))

1199114 (119896) le (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198981198914 (1 minus 1205758 (119896))

1199114 (119896) ge 11989811989141205758 (119896)

1199114 (119896) le 11987211989141205758 (119896)

1199115 (119896) ge (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198721198915 (1 minus 1205759 (119896))

1199115 (119896) le (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198981198915 (1 minus 1205759 (119896))

10 International Journal of Aerospace Engineering

1199115 (119896) ge 11989811989151205759 (119896)

1199115 (119896) le 11987211989151205759 (119896)

1199116 (119896) ge (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11987211989151205755 (119896)

1199116 (119896) le (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11989811989151205755 (119896)

1199116 (119896) ge 1198981198916 (1 minus 1205755 (119896))

1199116 (119896) le 1198721198916 (1 minus 1205755 (119896))

(39)

where1198721198911 = [073 1084]1198791198981198911 = [minus077 minus1106]

1198791198721198912 =[076 1166]

119879 1198981198912 = [minus079 minus 1256]119879 1198721198913 =

[077 1273]119879 1198981198913 = [minus078 minus 1280]

119879 1198721198914 =

[076 1403]119879 1198981198914 = [minus076 minus 1403]

119879 1198721198915 =

[0725 1291]119879 1198981198915 = [minus0736 minus 1205]

119879 1198721198916 =

[0696 1154]119879 and1198981198916 = [minus073 minus 1130]

119879Then the whole MLD model of the missile is given by

119909 (119896 + 1) =

6

sum

119894=1

119911119894 (119896)

119910 (119896) = [1 0] 119909 (119896)

st (28) (30) (32) (34) (36) (38)

(40)

where (29) (31) (33) (35) (37) and (39) represent all theconstraint inequalities

4 Autopilot Design UsingHybrid MPC Method

In general we expect that the output 119910 tracks its command 119910119888

as fast as possible with a small amount of fuel consumptionMotivated by this observation we consider the followingoptimization problem

119869lowast

= min119906(119896)119906(119896+1)120575(119896)120575(119896+1|119896)119911(119896)119911(119896+1|119896)

119873

sum

119894=1

(1003817100381710038171003817119910 (119896 + 119894 | 119896)

minus119910119888 (119896 + 119894)1003817100381710038171003817

2

119876

+ 119906(119896 + 119894)2

119877)

st MLD model (39)

1199061min le 1199061 (119896) 1199061 (119896 + 1) le 1199061max

119909min le 119909 (119896) 119909 (119896 + 1) le 119909max

(41)

where 119910119888 is the command for angle of attack 119910(119896 + 119894 | 119896) ispredictive value of angle of attack 119873 denotes the predictivehorizon 119876 and 119877 are the weighting matrices and 119877 =

diag(119877120575119911

119877119865119910

) 119877120575119911

119877119865119910

are weighted coefficients of aerody-namic control surfaces and lateral pulse jets respectivelyThe

ratio of 119877120575119911

and 119877119865119910

represents the control allocation betweenaerodynamic control surfaces and lateral pulse jets When119877119865119910

119877120575119911

is increased the requirement for lateral force will bedeclined that is the fuel consumption can be lessened Inaddition the deduction can be verified from the simulationresults of Case 1 and Case 2

Different control allocation between the dual actuators(aerodynamic control surfaces and lateral pulse jets) can beobtained by setting different 119876 and 119877 Since both logicaland continuous variables are involved in (41) the aboveon-line optimization problem is a mixed integer quadraticprogramming (MIQP) problemwhich can be solved by usingthe hybrid MPC toolbox of MATLAB [14 15]

Remark 2 In an explicit MPC controller the main factorsaffecting the number of subregions include the systemdimen-sion the predictive horizon and the number of constrains Inthis paper the blendedmissile with aerodynamic control sur-faces and lateral jets is investigated To deal with the discreteproperty of the lateral jet forces some logical variables areintroduced for MPC controller design which increases thenumber of subregions inevitably

5 Numerical Simulations

51 Case 1 Numerical simulations are performed to ver-ify the feasibility of the proposed method in this sectionThe involved parameters are given in Table 1 The weightmatrices are set as 119876 = 10 119877120575

119911

= 0025 and 119877119865119910

=

diag(0005 0005 0005) Suppose the initial state is 1205720 =0 1199030 = 0 In order to avoid excessive computational load wechoose predictive horizon119873 = 2 and control horizon119872 = 2The command for angle of attack is 120572119888 = 02 rad

Explicit form of the optimal controller is provided byhybrid toolbox which is not given here for simplicity Insteadwe present the partition results associated with the controllerin Figure 6 where 4190 subregions are obtained and for eachsubregion a control law of the form 119906 = 119870119909 + 119887 is givenAs a benefit of using explicit MPC the controller parameters119870 and 119887 for all the subregions are obtained simultaneouslyonce the optimization problem is solved This property isobviously different from that of gain-scheduled controller(where the controller parameters for different regions areseparately determined)

However by a simple calculation we conclude that thesesubregions donot require toomuch storage space (the neededstorage space is approximately 14 times 8 times 4190 = 450KB)Moreover it should be noted that when the algorithm isimplemented in practice much more time will be needed toidentify which subregion the current states enter into This isanother important factor that affects the computational costof the proposed algorithm

The simulation results are shown in Figures 7ndash9 It isseen from Figure 7 that the actual angle of attack tracksthe command in less than 01 seconds As shown in Figures8 and 9 the aerodynamic force converges to a constant astime increases while the jets are only activated during thebeginning period (when the tracking error is obvious)

International Journal of Aerospace Engineering 11

58

68

73

108

174

180

245

252

254

270

271

272

273

274

276

277

299

300

301

302

06

04

02

0

minus02

minus04

Elev

ator

defl

ectio

n (r

ad)

Angle of attack (rad)minus04 minus02 0 02 04 06

1634

1635

1638

1648

1649

1650

1664

1673

1677

1678

1679

1712

1726

1743

1744

1746

1748

1751

1752

1983

1985

1993

2011

Late

ral f

orce

(N)

Angle of attack (rad)minus04 minus02 0 02 04 06

1000

800

600

400

200

0

minus200

minus400

minus600

minus800

minus1000

Polyhedral partitionmdash4190 regions Polyhedral partitionmdash4190 regions

Figure 6 The partition associated with the explicit controller

025

02

015

01

005

0

ActualCommand

0 02 04 06 08 1

Time (s)

Ang

le o

f atta

ck (r

ad)

Figure 7 The response of angle of attack

52 Case 2 The results for the case with a different weight119877120575119911

= 0005 are shown in Figures 10ndash12 It is seen that adifferent control allocation result is obtained

04

02

0

minus02

minus04

minus060 02 04 06 08 1

Time (s)

Elev

ator

defl

ectio

n (r

ad)

Figure 8 The response of elevator deflection

53 Case 3 In this case a different command 120572119888 =

02 cos(05120587119905) rad is considered and the weight matrices arethe same as the ones for Case 1 The simulation results areshown in Figures 13ndash15

It is seen from Figure 13 that asymptotic tracking is alsoachievedThemethod proposed in this paper can realize both

12 International Journal of Aerospace Engineering

1

05

0

minus05

minus1

times104

0 02 04 06 08 1

Time (s)

Late

ral f

orce

(N)

Figure 9 The response of lateral force

025

02

015

01

005

0

Ang

le o

f atta

ck (r

ad)

0 02 04 06 08 1

Time (s)

ActualCommand

Figure 10 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 02 04 06 08 1

Time (s)

Figure 11 The response of elevator deflection

1

05

0

minus05

minus1

Late

ral f

orce

(N)

times104

0 02 04 06 08 1

Time (s)

Figure 12 The response of lateral force

03

025

02

015

01

005

0

minus005

minus01

minus015

minus02

Ang

le o

f atta

ck (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

ActualCommand

Figure 13 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

Figure 14 The response of elevator deflection

International Journal of Aerospace Engineering 13

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1

Late

ral f

orce

(N)

0 05 1 15 2 25 3 35 4

Time (s)

times104

Figure 15 The response of lateral force

fast tracking command and control allocation It is seen fromFigure 15 that the jets are activated while the tracking erroris obvious or command is varying rapidly

6 Conclusion

An autopilot design method for a missile with aerodynamiccontrol surfaces and lateral jets is presented in this paperThe nonlinear attitude control model is reduced to an MLDmodel Meanwhile the lateral force is described as linearcombination of logical variables due to the discrete values oflateral force Then the whole MLD model of attitude controlsystem is derived Autopilot design is accomplished usinghybrid MPC method By setting the related weighted coeffi-cients in index function the control allocation is obtainedMoreover numerical simulations are performed under thedifferent conditions the performance of tracking the attitudecommand and control allocation is verified and the explicitform of the control law can be obtained as well

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research presented in this document is supportedby the National Natural Science Foundation of Chinaunder Grant nos 61104193 61203191 and 61333001 theFundamental Research Funds for the Central Universities(HITNSRIF2012032) the Program for IBRSEM in HarbinInstitute of Technology under Grant HITIBRSEMA201415and the Foundation of Supporting Technology for Aerospaceunder Grant 2014-HT-HGD7

References

[1] R Hirokawa K Sato and S Manabe ldquoAutopilot design fora missile with reaction-jet using coefficient diagram methodrdquoin Proceedings of the AIAA Guidance Navigation and ControlConference and Exhibit 2001 August 2001

[2] P K Menon and V R Iragavarapu Adaptive Techniques forMultiple Actuator Blending Defense Technical InformationCenter 1998

[3] A G Munson and M W Garbrick ldquoJet interaction investiga-tionrdquo Report OR-9933 Martin Marietta Orlando Fla USA1969

[4] M Graham and P Weinacht ldquoNumerical simulation of lateralcontrol jetsrdquo in Proceedings of the 37th Aerospace SciencesMeeting and Exhibit 1999 AIAA Paper

[5] J S Shamma and J R Cloutier ldquoGain-scheduledmissile autopi-lot design using linear parameter varying transformationsrdquoJournal of Guidance Control and Dynamics vol 16 no 2 pp256ndash263 1993

[6] M Innocenti and A Thukral ldquoSimultaneous reaction jet andaerodynamic control of missile systemsrdquo AlAA 93-3739-CP1993

[7] D B Ridgely Y Lee and T Fanciullo ldquoDual aeropropulsivemissile control-optimal control and control allocationrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit Keystone Colo USA 2006

[8] D B Ridgely D Drake L Triplett and C Geise ldquoDynamiccontrol allocation of a missile with tails and reaction jetsrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 3158ndash3189 August 2007

[9] D Q Mayne J B Rawlings C V Rao and P O ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[10] A Bemporad F Borrelli and M Morari ldquoModel predictivecontrol based on linear programmingmdashthe explicit solutionrdquoIEEE Transactions on Automatic Control vol 47 no 12 pp1974ndash1985 2002

[11] C Jouannet and P Krus ldquoModelling of high angle of attackaerodynamicrdquo in Proceedings of the 25th AIAA Applied Aerody-namics Conference pp 1481ndash1516 June 2007

[12] S Venugopal and M Krishnamurthy ldquoMissile aerodynamics athigh angles of attack a prediction coderdquo Journal of Spacecraftand Rockets vol 32 no 2 pp 263ndash269 1995

[13] A Bemporad ldquoEfficient conversion of mixed logical dynamicalsystems into an equivalent piecewise affine formrdquo IEEE Trans-actions on Automatic Control vol 49 no 5 pp 832ndash838 2004

[14] F D Torrisi andA Bemporad ldquoHYSDELmdasha tool for generatingcomputational hybrid models for analysis and synthesis prob-lemsrdquo IEEE Transactions on Control Systems Technology vol 12no 2 pp 235ndash249 2004

[15] A Bemporad W P M H Heemels and B De SchutterldquoOn hybrid systems and closed-loop MPC systemsrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 863ndash8692002

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6 International Journal of Aerospace Engineering

mainly affected by the angle of attack 120572 The relationshipsbetween them are shown in Figure 5

In practical application since 120572 is the main factor thatleads to system nonlinearities the system model is usuallylinearized if 120572 varies in small range As seen from Figure 5curves of the relation between aerodynamic parametersand angle of attack can be expressed by six line segmentsapproximately Here we choose 120572 = minus037 rad minus0153 rad0 0153 rad and 037 rad as the operation points and dividethe whole operation region into six subregions As a resultthe original model (17) can be converted to the followingpiecewise affine models

=

1198861119909 + 1198871119906 + 1198901 [1 0] 119909 le minus037

1198862119909 + 1198872119906 + 1198902 minus037 lt [1 0] 119909 le minus0153

1198863119909 + 1198873119906 + 1198903 minus0153 lt [1 0] 119909 le 0

1198864119909 + 1198874119906 + 1198904 0 lt [1 0] 119909 le 0153

1198865119909 + 1198875119906 + 1198905 0153 lt [1 0] 119909 le 037

1198866119909 + 1198876119906 + 1198906 [1 0] 119909 gt 037

(20)

119910 = [1 0] 119909 (21)

where

119886119894 =120597119891(119909)

120597119909

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= [11988611

11989411988612

119894

11988621

11989411988622

119894

]

119887119894 = 119892 (1199091198940) = [11988711

11989411988712

119894

11988721

11989411988722

119894

] 119890119894 = [1198901

119894

1198902

119894

]

11988611

119894=

1205971198911 (119909)

120597120572

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= minus119876119878

119898119881(

120597119862120572

119910

120597120572

1003816100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1205721198940 cos1205721198940 + 119862120572

119910(1205721198940) cos1205721198940

minus 119862120572

119910(1205721198940) 1205721198940 sin1205721198940)

11988612

119894=

1205971198911(119909)

120597119903

10038161003816100381610038161003816100381610038161003816119909=1199091198940

= 1

11988621

119894=

1205971198912 (119909)

120597120572

10038161003816100381610038161003816100381610038161003816119909=1199091198940

=119876119878119871

119869119911

(120597119898120572

119911

120597120572

100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1205721198940 + 119898120572

119911(1205721198940) +

120597119898119903

119911

120597120572

100381610038161003816100381610038161003816100381610038161003816120572=1205721198940

1199031198940)

11988622

119894=

1205971198912(119909)

120597119903

10038161003816100381610038161003816100381610038161003816119909=1199091198940

=119876119878119871

119869119911

119898120572

119911(1205721198940)

11988711

119894= minus

119876119878119862120575119911

119910(1205721198940) cos1205721198940119898119881

11988712

119894= minus

(1 + 119870119865119910

(1205721198940)) cos1205721198940119898119881

11988721

119894=

119876119878119871119898120575119911

119911(1205721198940)

119869119911

11988722

119894=

(1 + 119870119872119911

(1205721198940)) 119897

119869119911

1198901

119894= 11988611

1198941205721198940 + 119886

12

1198941199031198940

1198902

119894= 11988621

1198941205721198940 + 119886

22

1198941199031198940

(22)

where 119894 (119894 = 1 2 6) is the label corresponding to the 119894thregion

From Table 1 and Figure 5 we get the set of aerodynamicparameters at the point (119867119881) = (20 km 1000ms) as shownin Table 2

Choose the sampling period 119879119904 = 0025 s The discretestate-space expression is then given by

119909 (119896 + 1)

=

1198861119909 (119896) + 1119906 (119896) + 1198901 [1 0] 119909 (119896) le minus037

1198862119909 (119896) + 2119906 (119896) + 1198902 minus037 lt [1 0] 119909 (119896) le minus0153

1198863119909 (119896) + 3119906 (119896) + 1198903 minus0153 lt [1 0] 119909 (119896) le 0

1198864119909 (119896) + 4119906 (119896) + 1198904 0 lt [1 0] 119909 (119896) le 0153

1198865119909 (119896) + 5119906 (119896) + 1198905 0153 lt [1 0] 119909 (119896) le 037

1198866119909 (119896) + 6119906 (119896) + 1198906 [1 0] 119909 (119896) gt 037

(23)

119910 (119896) = [1 0] 119909 (119896) (24)

where

1198861 = [104 0025

022 0995] 1198862 = [

1051 0025

0218 0995]

1198863 = [1023 00252

0248 09951] 1198864 = [

09954 00248

02732 09954]

1198865 = [09697 00245

02913 09956] 1198866 = [

09594 00244

03137 09959]

1198901 = [minus00183

minus01123] 1198902 = [

minus00175

minus00792]

1198903 = [minus00131

minus00376] 1198904 = [

0

0]

1198905 = [minus00053

00454] 1198906 = [

minus00169

01193]

1 = [minus0028 159 times 10

minus6

minus2166 142 times 10minus4]

2 = [minus003 218 times 10

minus6

minus2224 187 times 10minus4]

International Journal of Aerospace Engineering 7

2

15

1

05

0

minus05

minus1

minus15minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SmV

)C120572 y

(a) Relation between 119862120572119910and 120572

0092

009

0088

0086

0084

0082

008

0078

0076

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SmV

)C120575119911y

(b) Relation between 119862120575119911119910 and 120572

125

12

115

11

105

10

95

9minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SLJz)m

120572 z

(c) Relation between119898120572119911and 120572

96

94

92

90

88

86

84minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (minusQSL

Jz)m

120575119911z

(d) Relation between119898120575119911119911 and 120572

1

09

08

07

06

05

04

03

02

01

Am

plifi

catio

n fa

ctorKF

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

(e) Relation between119870119865119910

and 120572

12

11

1

09

08

07

06

05

04

03

Am

plifi

catio

n fa

ctorKM119911

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

(f) Relation between119870119872119911

and 120572

Figure 5 Aerodynamic parameters as functions of angle of attack

8 International Journal of Aerospace Engineering

Table 2 Aerodynamic parameters of pitch channel

Aerodynamic parameter 119876119878119862120572

119910119898119881 119876119878119862

120575119911

119910119898119881 119876119878119871119898

120572

119911119869119911

119876119878119871119898119903

119911119869119911

119876119878119871119898120575119911

119911119869119911

120572 = minus053 minus1175 0076 947 minus032 minus8489

120572 = minus037 minus0944 0082 973 minus032 minus8924

120572 = minus0153 minus023 0082 1043 minus032 minus8938

120572 = 0 036 009 1101 minus032 minus9510

120572 = 0153 094 0082 1158 minus032 minus8934

120572 = 037 163 0082 1224 minus032 minus8918

3 = [minus003 268 times 10

minus6

minus2228 223 times 10minus4]

4 = [minus0032 277 times 10

minus6

minus2371 232 times 10minus4]

5 = [minus003 227 times 10

minus6

minus2228 198 times 10minus4]

6 = [minus0029 191 times 10

minus6

minus2224 173 times 10minus4]

(25)

32 Constraints Analysis Due to the symmetry of jet config-uration the set of possible negative pitch control force is givenby

119880119910minus

119865= minus119865119898 minus2119865119898 cos

120587

9 minus2119865119898 cos

120587

18 minus2119865119898 minus4119865119898 cos

120587

9

minus4119865119898 cos120587

18

(26)

By combining (13) and (26) we obtain the set of allpossible pitch control force

119880119910

119865= 119865119898 2119865119898 cos

120587

9 2119865119898 cos

120587

18 2119865119898 4119865119898 cos

120587

9

4119865119898 cos120587

18 minus119865119898 minus2119865119898 cos

120587

9 minus2119865119898 cos

120587

18 minus2119865119898

minus4119865119898 cos120587

9 minus4119865119898 cos

120587

18

(27)

Substituting the jet parameters shown in Table 1 into theforegoing set yields

119880119910

119865= 2200 4135 4333 4400 8269 8666 minus2200 minus4135

minus4333 minus4400 minus8269 minus8666

(28)

Noting that the lateral forces are discrete variable we willutilize the linear combination of logical variables to describethe lateral force In terms of piecewise affine model (20) we

introduce logical variables 120575119865119894

isin 0 1 119894 = 1 2 12 toexpress the lateral thrust 119865119910

119887

119865119910119887

= 22001205751198651

+ 41351205751198652

+ 43331205751198653

+ 44001205751198654

+ 82691205751198655

+ 86661205751198656

minus 22001205751198657

minus 41351205751198658

minus 43331205751198659

minus 440012057511986510

minus 826912057511986511

minus 866612057511986512

(29)

In (29) the logical variables should satisfy the constraints

12

sum

119894=1

120575119865119894

= 0 or 1 (30)

where 0 means that no lateral force is generated while 1

means that the applied lateral force equals to one element ofset 119880119910119865

Denote 1199061 = 120575119911 the control input 119906 in model (23) can berewritten as

119906 = [1199061 119865119910119887

]119879 (31)

As shown in Table 1 the constraints on system states andcontrol input are

119909min le 119909 (119896) le 119909max

1199061min le 1199061 (119896) le 1199061max(32)

where 119909min = [minus053 minus522]119879 119909max = [053 522]

119879 1199061min =

minus053 1199061max = 053According to (30) we have

12

sum

119894=1

120575119865119894

le 1 (33)

which shows the constraint on the logical control inputUse logical variables 120575119894(119896) isin 0 1 119894 = 1 2 5 to

describe the operation points which satisfy the constraints

[1 0] 119909 (119896) + 037 le 0 lArrrArr 1205751 (119896) = 1

[1 0] 119909 (119896) + 0153 le 0 lArrrArr 1205752 (119896) = 1

[1 0] 119909 (119896) le 0 lArrrArr 1205753 (119896) = 1

[1 0] 119909 (119896) minus 0153 le 0 lArrrArr 1205754 (119896) = 1

[1 0] 119909 (119896) minus 037 le 0 lArrrArr 1205755 (119896) = 1

(34)

International Journal of Aerospace Engineering 9

Equation (34) can be transformed into the followingequivalent mixed logical inequalities [13]

[1 0] 119909 (119896) + 037 ge 120576 + (1198981 minus 120576) 1205751 (119896)

[1 0] 119909 (119896) + 037 le 1198721 (1 minus 1205751 (119896))

[1 0] 119909 (119896) + 0153 ge 120576 + (1198982 minus 120576) 1205752 (119896)

[1 0] 119909 (119896) + 0153 le 1198722 (1 minus 1205752 (119896))

[1 0] 119909 (119896) ge 120576 + (1198983 minus 120576) 1205753 (119896)

[1 0] 119909 (119896) le 1198723 (1 minus 1205753 (119896))

[1 0] 119909 (119896) minus 0153 ge 120576 + (1198984 minus 120576) 1205754 (119896)

[1 0] 119909 (119896) minus 0153 le 1198724 (1 minus 1205754 (119896))

[1 0] 119909 (119896) minus 037 ge 120576 + (1198985 minus 120576) 1205755 (119896)

[1 0] 119909 (119896) minus 037 le 1198725 (1 minus 1205755 (119896))

(35)

where 1198981 = minus016 1198721 = 090 1198982 = minus0377 1198722 = 06831198983 = minus053 1198723 = 053 1198984 = minus0683 1198724 = 0377 1198985 =

minus0901198725 = 016 and 120576 = 10minus6

In addition we introduce the auxiliary logical variables120575119894(119896) isin 0 1 119894 = 6 9 as follows

1205756 (119896) = (1 minus 1205751 (119896)) 1205752 (119896)

1205757 (119896) = (1 minus 1205752 (119896)) 1205753 (119896)

1205758 (119896) = (1 minus 1205753 (119896)) 1205754 (119896)

1205759 (119896) = (1 minus 1205754 (119896)) 1205755 (119896)

(36)

With 1205751 1205756 1205757 1205758 1205759 and 1 minus 1205755 the six regions canbe presented Similarly (36) can be transformed into theequivalent mixed logical inequalities

minus1205751 (119896) + 1205752 (119896) minus 1205756 (119896) le 0

1205751 (119896) + 1205756 (119896) le 1

minus1205752 (119896) + 1205756 (119896) le 0

minus1205752 (119896) + 1205753 (119896) minus 1205757 (119896) le 0

1205752 (119896) + 1205757 (119896) le 1

minus1205753 (119896) + 1205757 (119896) le 0

minus1205753 (119896) + 1205754 (119896) minus 1205758 (119896) le 0

1205753 (119896) + 1205758 (119896) le 1

minus1205754 (119896) + 1205758 (119896) le 0

minus1205754 (119896) + 1205755 (119896) minus 1205759 (119896) le 0

1205754 (119896) + 1205759 (119896) le 1

minus1205755 (119896) + 1205759 (119896) le 0

(37)

To describe the state space model of each region weintroduce the following auxiliary continuous variables

1199111 (119896) = (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) 1205751 (119896)

1199112 (119896) = (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) 1205756 (119896)

1199113 (119896) = (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) 1205757 (119896)

1199114 (119896) = (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) 1205758 (119896)

1199115 (119896) = (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) 1205759 (119896)

1199116 (119896) = (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) (1 minus 1205755 (119896))

(38)

Equation (38) can be converted to the equivalent mixedlogical inequalities

1199111 (119896) ge (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198721198911 (1 minus 1205751 (119896))

1199111 (119896) le (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198981198911 (1 minus 1205751 (119896))

1199111 (119896) ge 11989811989111205751 (119896)

1199111 (119896) le 11987211989111205751 (119896)

1199112 (119896) ge (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198721198912 (1 minus 1205756 (119896))

1199112 (119896) le (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198981198912 (1 minus 1205756 (119896))

1199112 (119896) ge 11989811989121205756 (119896)

1199112 (119896) le 11987211989121205756 (119896)

1199113 (119896) ge (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198721198913 (1 minus 1205757 (119896))

1199113 (119896) le (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198981198913 (1 minus 1205757 (119896))

1199113 (119896) ge 11989811989131205757 (119896)

1199113 (119896) le 11987211989131205757 (119896)

1199114 (119896) ge (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198721198914 (1 minus 1205758 (119896))

1199114 (119896) le (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198981198914 (1 minus 1205758 (119896))

1199114 (119896) ge 11989811989141205758 (119896)

1199114 (119896) le 11987211989141205758 (119896)

1199115 (119896) ge (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198721198915 (1 minus 1205759 (119896))

1199115 (119896) le (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198981198915 (1 minus 1205759 (119896))

10 International Journal of Aerospace Engineering

1199115 (119896) ge 11989811989151205759 (119896)

1199115 (119896) le 11987211989151205759 (119896)

1199116 (119896) ge (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11987211989151205755 (119896)

1199116 (119896) le (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11989811989151205755 (119896)

1199116 (119896) ge 1198981198916 (1 minus 1205755 (119896))

1199116 (119896) le 1198721198916 (1 minus 1205755 (119896))

(39)

where1198721198911 = [073 1084]1198791198981198911 = [minus077 minus1106]

1198791198721198912 =[076 1166]

119879 1198981198912 = [minus079 minus 1256]119879 1198721198913 =

[077 1273]119879 1198981198913 = [minus078 minus 1280]

119879 1198721198914 =

[076 1403]119879 1198981198914 = [minus076 minus 1403]

119879 1198721198915 =

[0725 1291]119879 1198981198915 = [minus0736 minus 1205]

119879 1198721198916 =

[0696 1154]119879 and1198981198916 = [minus073 minus 1130]

119879Then the whole MLD model of the missile is given by

119909 (119896 + 1) =

6

sum

119894=1

119911119894 (119896)

119910 (119896) = [1 0] 119909 (119896)

st (28) (30) (32) (34) (36) (38)

(40)

where (29) (31) (33) (35) (37) and (39) represent all theconstraint inequalities

4 Autopilot Design UsingHybrid MPC Method

In general we expect that the output 119910 tracks its command 119910119888

as fast as possible with a small amount of fuel consumptionMotivated by this observation we consider the followingoptimization problem

119869lowast

= min119906(119896)119906(119896+1)120575(119896)120575(119896+1|119896)119911(119896)119911(119896+1|119896)

119873

sum

119894=1

(1003817100381710038171003817119910 (119896 + 119894 | 119896)

minus119910119888 (119896 + 119894)1003817100381710038171003817

2

119876

+ 119906(119896 + 119894)2

119877)

st MLD model (39)

1199061min le 1199061 (119896) 1199061 (119896 + 1) le 1199061max

119909min le 119909 (119896) 119909 (119896 + 1) le 119909max

(41)

where 119910119888 is the command for angle of attack 119910(119896 + 119894 | 119896) ispredictive value of angle of attack 119873 denotes the predictivehorizon 119876 and 119877 are the weighting matrices and 119877 =

diag(119877120575119911

119877119865119910

) 119877120575119911

119877119865119910

are weighted coefficients of aerody-namic control surfaces and lateral pulse jets respectivelyThe

ratio of 119877120575119911

and 119877119865119910

represents the control allocation betweenaerodynamic control surfaces and lateral pulse jets When119877119865119910

119877120575119911

is increased the requirement for lateral force will bedeclined that is the fuel consumption can be lessened Inaddition the deduction can be verified from the simulationresults of Case 1 and Case 2

Different control allocation between the dual actuators(aerodynamic control surfaces and lateral pulse jets) can beobtained by setting different 119876 and 119877 Since both logicaland continuous variables are involved in (41) the aboveon-line optimization problem is a mixed integer quadraticprogramming (MIQP) problemwhich can be solved by usingthe hybrid MPC toolbox of MATLAB [14 15]

Remark 2 In an explicit MPC controller the main factorsaffecting the number of subregions include the systemdimen-sion the predictive horizon and the number of constrains Inthis paper the blendedmissile with aerodynamic control sur-faces and lateral jets is investigated To deal with the discreteproperty of the lateral jet forces some logical variables areintroduced for MPC controller design which increases thenumber of subregions inevitably

5 Numerical Simulations

51 Case 1 Numerical simulations are performed to ver-ify the feasibility of the proposed method in this sectionThe involved parameters are given in Table 1 The weightmatrices are set as 119876 = 10 119877120575

119911

= 0025 and 119877119865119910

=

diag(0005 0005 0005) Suppose the initial state is 1205720 =0 1199030 = 0 In order to avoid excessive computational load wechoose predictive horizon119873 = 2 and control horizon119872 = 2The command for angle of attack is 120572119888 = 02 rad

Explicit form of the optimal controller is provided byhybrid toolbox which is not given here for simplicity Insteadwe present the partition results associated with the controllerin Figure 6 where 4190 subregions are obtained and for eachsubregion a control law of the form 119906 = 119870119909 + 119887 is givenAs a benefit of using explicit MPC the controller parameters119870 and 119887 for all the subregions are obtained simultaneouslyonce the optimization problem is solved This property isobviously different from that of gain-scheduled controller(where the controller parameters for different regions areseparately determined)

However by a simple calculation we conclude that thesesubregions donot require toomuch storage space (the neededstorage space is approximately 14 times 8 times 4190 = 450KB)Moreover it should be noted that when the algorithm isimplemented in practice much more time will be needed toidentify which subregion the current states enter into This isanother important factor that affects the computational costof the proposed algorithm

The simulation results are shown in Figures 7ndash9 It isseen from Figure 7 that the actual angle of attack tracksthe command in less than 01 seconds As shown in Figures8 and 9 the aerodynamic force converges to a constant astime increases while the jets are only activated during thebeginning period (when the tracking error is obvious)

International Journal of Aerospace Engineering 11

58

68

73

108

174

180

245

252

254

270

271

272

273

274

276

277

299

300

301

302

06

04

02

0

minus02

minus04

Elev

ator

defl

ectio

n (r

ad)

Angle of attack (rad)minus04 minus02 0 02 04 06

1634

1635

1638

1648

1649

1650

1664

1673

1677

1678

1679

1712

1726

1743

1744

1746

1748

1751

1752

1983

1985

1993

2011

Late

ral f

orce

(N)

Angle of attack (rad)minus04 minus02 0 02 04 06

1000

800

600

400

200

0

minus200

minus400

minus600

minus800

minus1000

Polyhedral partitionmdash4190 regions Polyhedral partitionmdash4190 regions

Figure 6 The partition associated with the explicit controller

025

02

015

01

005

0

ActualCommand

0 02 04 06 08 1

Time (s)

Ang

le o

f atta

ck (r

ad)

Figure 7 The response of angle of attack

52 Case 2 The results for the case with a different weight119877120575119911

= 0005 are shown in Figures 10ndash12 It is seen that adifferent control allocation result is obtained

04

02

0

minus02

minus04

minus060 02 04 06 08 1

Time (s)

Elev

ator

defl

ectio

n (r

ad)

Figure 8 The response of elevator deflection

53 Case 3 In this case a different command 120572119888 =

02 cos(05120587119905) rad is considered and the weight matrices arethe same as the ones for Case 1 The simulation results areshown in Figures 13ndash15

It is seen from Figure 13 that asymptotic tracking is alsoachievedThemethod proposed in this paper can realize both

12 International Journal of Aerospace Engineering

1

05

0

minus05

minus1

times104

0 02 04 06 08 1

Time (s)

Late

ral f

orce

(N)

Figure 9 The response of lateral force

025

02

015

01

005

0

Ang

le o

f atta

ck (r

ad)

0 02 04 06 08 1

Time (s)

ActualCommand

Figure 10 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 02 04 06 08 1

Time (s)

Figure 11 The response of elevator deflection

1

05

0

minus05

minus1

Late

ral f

orce

(N)

times104

0 02 04 06 08 1

Time (s)

Figure 12 The response of lateral force

03

025

02

015

01

005

0

minus005

minus01

minus015

minus02

Ang

le o

f atta

ck (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

ActualCommand

Figure 13 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

Figure 14 The response of elevator deflection

International Journal of Aerospace Engineering 13

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1

Late

ral f

orce

(N)

0 05 1 15 2 25 3 35 4

Time (s)

times104

Figure 15 The response of lateral force

fast tracking command and control allocation It is seen fromFigure 15 that the jets are activated while the tracking erroris obvious or command is varying rapidly

6 Conclusion

An autopilot design method for a missile with aerodynamiccontrol surfaces and lateral jets is presented in this paperThe nonlinear attitude control model is reduced to an MLDmodel Meanwhile the lateral force is described as linearcombination of logical variables due to the discrete values oflateral force Then the whole MLD model of attitude controlsystem is derived Autopilot design is accomplished usinghybrid MPC method By setting the related weighted coeffi-cients in index function the control allocation is obtainedMoreover numerical simulations are performed under thedifferent conditions the performance of tracking the attitudecommand and control allocation is verified and the explicitform of the control law can be obtained as well

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research presented in this document is supportedby the National Natural Science Foundation of Chinaunder Grant nos 61104193 61203191 and 61333001 theFundamental Research Funds for the Central Universities(HITNSRIF2012032) the Program for IBRSEM in HarbinInstitute of Technology under Grant HITIBRSEMA201415and the Foundation of Supporting Technology for Aerospaceunder Grant 2014-HT-HGD7

References

[1] R Hirokawa K Sato and S Manabe ldquoAutopilot design fora missile with reaction-jet using coefficient diagram methodrdquoin Proceedings of the AIAA Guidance Navigation and ControlConference and Exhibit 2001 August 2001

[2] P K Menon and V R Iragavarapu Adaptive Techniques forMultiple Actuator Blending Defense Technical InformationCenter 1998

[3] A G Munson and M W Garbrick ldquoJet interaction investiga-tionrdquo Report OR-9933 Martin Marietta Orlando Fla USA1969

[4] M Graham and P Weinacht ldquoNumerical simulation of lateralcontrol jetsrdquo in Proceedings of the 37th Aerospace SciencesMeeting and Exhibit 1999 AIAA Paper

[5] J S Shamma and J R Cloutier ldquoGain-scheduledmissile autopi-lot design using linear parameter varying transformationsrdquoJournal of Guidance Control and Dynamics vol 16 no 2 pp256ndash263 1993

[6] M Innocenti and A Thukral ldquoSimultaneous reaction jet andaerodynamic control of missile systemsrdquo AlAA 93-3739-CP1993

[7] D B Ridgely Y Lee and T Fanciullo ldquoDual aeropropulsivemissile control-optimal control and control allocationrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit Keystone Colo USA 2006

[8] D B Ridgely D Drake L Triplett and C Geise ldquoDynamiccontrol allocation of a missile with tails and reaction jetsrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 3158ndash3189 August 2007

[9] D Q Mayne J B Rawlings C V Rao and P O ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[10] A Bemporad F Borrelli and M Morari ldquoModel predictivecontrol based on linear programmingmdashthe explicit solutionrdquoIEEE Transactions on Automatic Control vol 47 no 12 pp1974ndash1985 2002

[11] C Jouannet and P Krus ldquoModelling of high angle of attackaerodynamicrdquo in Proceedings of the 25th AIAA Applied Aerody-namics Conference pp 1481ndash1516 June 2007

[12] S Venugopal and M Krishnamurthy ldquoMissile aerodynamics athigh angles of attack a prediction coderdquo Journal of Spacecraftand Rockets vol 32 no 2 pp 263ndash269 1995

[13] A Bemporad ldquoEfficient conversion of mixed logical dynamicalsystems into an equivalent piecewise affine formrdquo IEEE Trans-actions on Automatic Control vol 49 no 5 pp 832ndash838 2004

[14] F D Torrisi andA Bemporad ldquoHYSDELmdasha tool for generatingcomputational hybrid models for analysis and synthesis prob-lemsrdquo IEEE Transactions on Control Systems Technology vol 12no 2 pp 235ndash249 2004

[15] A Bemporad W P M H Heemels and B De SchutterldquoOn hybrid systems and closed-loop MPC systemsrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 863ndash8692002

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International Journal of

International Journal of Aerospace Engineering 7

2

15

1

05

0

minus05

minus1

minus15minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SmV

)C120572 y

(a) Relation between 119862120572119910and 120572

0092

009

0088

0086

0084

0082

008

0078

0076

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SmV

)C120575119911y

(b) Relation between 119862120575119911119910 and 120572

125

12

115

11

105

10

95

9minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (Q

SLJz)m

120572 z

(c) Relation between119898120572119911and 120572

96

94

92

90

88

86

84minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

Aero

dyna

mic

par

amet

er (minusQSL

Jz)m

120575119911z

(d) Relation between119898120575119911119911 and 120572

1

09

08

07

06

05

04

03

02

01

Am

plifi

catio

n fa

ctorKF

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

(e) Relation between119870119865119910

and 120572

12

11

1

09

08

07

06

05

04

03

Am

plifi

catio

n fa

ctorKM119911

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

Angle of attack (rad)

(f) Relation between119870119872119911

and 120572

Figure 5 Aerodynamic parameters as functions of angle of attack

8 International Journal of Aerospace Engineering

Table 2 Aerodynamic parameters of pitch channel

Aerodynamic parameter 119876119878119862120572

119910119898119881 119876119878119862

120575119911

119910119898119881 119876119878119871119898

120572

119911119869119911

119876119878119871119898119903

119911119869119911

119876119878119871119898120575119911

119911119869119911

120572 = minus053 minus1175 0076 947 minus032 minus8489

120572 = minus037 minus0944 0082 973 minus032 minus8924

120572 = minus0153 minus023 0082 1043 minus032 minus8938

120572 = 0 036 009 1101 minus032 minus9510

120572 = 0153 094 0082 1158 minus032 minus8934

120572 = 037 163 0082 1224 minus032 minus8918

3 = [minus003 268 times 10

minus6

minus2228 223 times 10minus4]

4 = [minus0032 277 times 10

minus6

minus2371 232 times 10minus4]

5 = [minus003 227 times 10

minus6

minus2228 198 times 10minus4]

6 = [minus0029 191 times 10

minus6

minus2224 173 times 10minus4]

(25)

32 Constraints Analysis Due to the symmetry of jet config-uration the set of possible negative pitch control force is givenby

119880119910minus

119865= minus119865119898 minus2119865119898 cos

120587

9 minus2119865119898 cos

120587

18 minus2119865119898 minus4119865119898 cos

120587

9

minus4119865119898 cos120587

18

(26)

By combining (13) and (26) we obtain the set of allpossible pitch control force

119880119910

119865= 119865119898 2119865119898 cos

120587

9 2119865119898 cos

120587

18 2119865119898 4119865119898 cos

120587

9

4119865119898 cos120587

18 minus119865119898 minus2119865119898 cos

120587

9 minus2119865119898 cos

120587

18 minus2119865119898

minus4119865119898 cos120587

9 minus4119865119898 cos

120587

18

(27)

Substituting the jet parameters shown in Table 1 into theforegoing set yields

119880119910

119865= 2200 4135 4333 4400 8269 8666 minus2200 minus4135

minus4333 minus4400 minus8269 minus8666

(28)

Noting that the lateral forces are discrete variable we willutilize the linear combination of logical variables to describethe lateral force In terms of piecewise affine model (20) we

introduce logical variables 120575119865119894

isin 0 1 119894 = 1 2 12 toexpress the lateral thrust 119865119910

119887

119865119910119887

= 22001205751198651

+ 41351205751198652

+ 43331205751198653

+ 44001205751198654

+ 82691205751198655

+ 86661205751198656

minus 22001205751198657

minus 41351205751198658

minus 43331205751198659

minus 440012057511986510

minus 826912057511986511

minus 866612057511986512

(29)

In (29) the logical variables should satisfy the constraints

12

sum

119894=1

120575119865119894

= 0 or 1 (30)

where 0 means that no lateral force is generated while 1

means that the applied lateral force equals to one element ofset 119880119910119865

Denote 1199061 = 120575119911 the control input 119906 in model (23) can berewritten as

119906 = [1199061 119865119910119887

]119879 (31)

As shown in Table 1 the constraints on system states andcontrol input are

119909min le 119909 (119896) le 119909max

1199061min le 1199061 (119896) le 1199061max(32)

where 119909min = [minus053 minus522]119879 119909max = [053 522]

119879 1199061min =

minus053 1199061max = 053According to (30) we have

12

sum

119894=1

120575119865119894

le 1 (33)

which shows the constraint on the logical control inputUse logical variables 120575119894(119896) isin 0 1 119894 = 1 2 5 to

describe the operation points which satisfy the constraints

[1 0] 119909 (119896) + 037 le 0 lArrrArr 1205751 (119896) = 1

[1 0] 119909 (119896) + 0153 le 0 lArrrArr 1205752 (119896) = 1

[1 0] 119909 (119896) le 0 lArrrArr 1205753 (119896) = 1

[1 0] 119909 (119896) minus 0153 le 0 lArrrArr 1205754 (119896) = 1

[1 0] 119909 (119896) minus 037 le 0 lArrrArr 1205755 (119896) = 1

(34)

International Journal of Aerospace Engineering 9

Equation (34) can be transformed into the followingequivalent mixed logical inequalities [13]

[1 0] 119909 (119896) + 037 ge 120576 + (1198981 minus 120576) 1205751 (119896)

[1 0] 119909 (119896) + 037 le 1198721 (1 minus 1205751 (119896))

[1 0] 119909 (119896) + 0153 ge 120576 + (1198982 minus 120576) 1205752 (119896)

[1 0] 119909 (119896) + 0153 le 1198722 (1 minus 1205752 (119896))

[1 0] 119909 (119896) ge 120576 + (1198983 minus 120576) 1205753 (119896)

[1 0] 119909 (119896) le 1198723 (1 minus 1205753 (119896))

[1 0] 119909 (119896) minus 0153 ge 120576 + (1198984 minus 120576) 1205754 (119896)

[1 0] 119909 (119896) minus 0153 le 1198724 (1 minus 1205754 (119896))

[1 0] 119909 (119896) minus 037 ge 120576 + (1198985 minus 120576) 1205755 (119896)

[1 0] 119909 (119896) minus 037 le 1198725 (1 minus 1205755 (119896))

(35)

where 1198981 = minus016 1198721 = 090 1198982 = minus0377 1198722 = 06831198983 = minus053 1198723 = 053 1198984 = minus0683 1198724 = 0377 1198985 =

minus0901198725 = 016 and 120576 = 10minus6

In addition we introduce the auxiliary logical variables120575119894(119896) isin 0 1 119894 = 6 9 as follows

1205756 (119896) = (1 minus 1205751 (119896)) 1205752 (119896)

1205757 (119896) = (1 minus 1205752 (119896)) 1205753 (119896)

1205758 (119896) = (1 minus 1205753 (119896)) 1205754 (119896)

1205759 (119896) = (1 minus 1205754 (119896)) 1205755 (119896)

(36)

With 1205751 1205756 1205757 1205758 1205759 and 1 minus 1205755 the six regions canbe presented Similarly (36) can be transformed into theequivalent mixed logical inequalities

minus1205751 (119896) + 1205752 (119896) minus 1205756 (119896) le 0

1205751 (119896) + 1205756 (119896) le 1

minus1205752 (119896) + 1205756 (119896) le 0

minus1205752 (119896) + 1205753 (119896) minus 1205757 (119896) le 0

1205752 (119896) + 1205757 (119896) le 1

minus1205753 (119896) + 1205757 (119896) le 0

minus1205753 (119896) + 1205754 (119896) minus 1205758 (119896) le 0

1205753 (119896) + 1205758 (119896) le 1

minus1205754 (119896) + 1205758 (119896) le 0

minus1205754 (119896) + 1205755 (119896) minus 1205759 (119896) le 0

1205754 (119896) + 1205759 (119896) le 1

minus1205755 (119896) + 1205759 (119896) le 0

(37)

To describe the state space model of each region weintroduce the following auxiliary continuous variables

1199111 (119896) = (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) 1205751 (119896)

1199112 (119896) = (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) 1205756 (119896)

1199113 (119896) = (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) 1205757 (119896)

1199114 (119896) = (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) 1205758 (119896)

1199115 (119896) = (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) 1205759 (119896)

1199116 (119896) = (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) (1 minus 1205755 (119896))

(38)

Equation (38) can be converted to the equivalent mixedlogical inequalities

1199111 (119896) ge (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198721198911 (1 minus 1205751 (119896))

1199111 (119896) le (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198981198911 (1 minus 1205751 (119896))

1199111 (119896) ge 11989811989111205751 (119896)

1199111 (119896) le 11987211989111205751 (119896)

1199112 (119896) ge (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198721198912 (1 minus 1205756 (119896))

1199112 (119896) le (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198981198912 (1 minus 1205756 (119896))

1199112 (119896) ge 11989811989121205756 (119896)

1199112 (119896) le 11987211989121205756 (119896)

1199113 (119896) ge (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198721198913 (1 minus 1205757 (119896))

1199113 (119896) le (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198981198913 (1 minus 1205757 (119896))

1199113 (119896) ge 11989811989131205757 (119896)

1199113 (119896) le 11987211989131205757 (119896)

1199114 (119896) ge (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198721198914 (1 minus 1205758 (119896))

1199114 (119896) le (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198981198914 (1 minus 1205758 (119896))

1199114 (119896) ge 11989811989141205758 (119896)

1199114 (119896) le 11987211989141205758 (119896)

1199115 (119896) ge (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198721198915 (1 minus 1205759 (119896))

1199115 (119896) le (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198981198915 (1 minus 1205759 (119896))

10 International Journal of Aerospace Engineering

1199115 (119896) ge 11989811989151205759 (119896)

1199115 (119896) le 11987211989151205759 (119896)

1199116 (119896) ge (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11987211989151205755 (119896)

1199116 (119896) le (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11989811989151205755 (119896)

1199116 (119896) ge 1198981198916 (1 minus 1205755 (119896))

1199116 (119896) le 1198721198916 (1 minus 1205755 (119896))

(39)

where1198721198911 = [073 1084]1198791198981198911 = [minus077 minus1106]

1198791198721198912 =[076 1166]

119879 1198981198912 = [minus079 minus 1256]119879 1198721198913 =

[077 1273]119879 1198981198913 = [minus078 minus 1280]

119879 1198721198914 =

[076 1403]119879 1198981198914 = [minus076 minus 1403]

119879 1198721198915 =

[0725 1291]119879 1198981198915 = [minus0736 minus 1205]

119879 1198721198916 =

[0696 1154]119879 and1198981198916 = [minus073 minus 1130]

119879Then the whole MLD model of the missile is given by

119909 (119896 + 1) =

6

sum

119894=1

119911119894 (119896)

119910 (119896) = [1 0] 119909 (119896)

st (28) (30) (32) (34) (36) (38)

(40)

where (29) (31) (33) (35) (37) and (39) represent all theconstraint inequalities

4 Autopilot Design UsingHybrid MPC Method

In general we expect that the output 119910 tracks its command 119910119888

as fast as possible with a small amount of fuel consumptionMotivated by this observation we consider the followingoptimization problem

119869lowast

= min119906(119896)119906(119896+1)120575(119896)120575(119896+1|119896)119911(119896)119911(119896+1|119896)

119873

sum

119894=1

(1003817100381710038171003817119910 (119896 + 119894 | 119896)

minus119910119888 (119896 + 119894)1003817100381710038171003817

2

119876

+ 119906(119896 + 119894)2

119877)

st MLD model (39)

1199061min le 1199061 (119896) 1199061 (119896 + 1) le 1199061max

119909min le 119909 (119896) 119909 (119896 + 1) le 119909max

(41)

where 119910119888 is the command for angle of attack 119910(119896 + 119894 | 119896) ispredictive value of angle of attack 119873 denotes the predictivehorizon 119876 and 119877 are the weighting matrices and 119877 =

diag(119877120575119911

119877119865119910

) 119877120575119911

119877119865119910

are weighted coefficients of aerody-namic control surfaces and lateral pulse jets respectivelyThe

ratio of 119877120575119911

and 119877119865119910

represents the control allocation betweenaerodynamic control surfaces and lateral pulse jets When119877119865119910

119877120575119911

is increased the requirement for lateral force will bedeclined that is the fuel consumption can be lessened Inaddition the deduction can be verified from the simulationresults of Case 1 and Case 2

Different control allocation between the dual actuators(aerodynamic control surfaces and lateral pulse jets) can beobtained by setting different 119876 and 119877 Since both logicaland continuous variables are involved in (41) the aboveon-line optimization problem is a mixed integer quadraticprogramming (MIQP) problemwhich can be solved by usingthe hybrid MPC toolbox of MATLAB [14 15]

Remark 2 In an explicit MPC controller the main factorsaffecting the number of subregions include the systemdimen-sion the predictive horizon and the number of constrains Inthis paper the blendedmissile with aerodynamic control sur-faces and lateral jets is investigated To deal with the discreteproperty of the lateral jet forces some logical variables areintroduced for MPC controller design which increases thenumber of subregions inevitably

5 Numerical Simulations

51 Case 1 Numerical simulations are performed to ver-ify the feasibility of the proposed method in this sectionThe involved parameters are given in Table 1 The weightmatrices are set as 119876 = 10 119877120575

119911

= 0025 and 119877119865119910

=

diag(0005 0005 0005) Suppose the initial state is 1205720 =0 1199030 = 0 In order to avoid excessive computational load wechoose predictive horizon119873 = 2 and control horizon119872 = 2The command for angle of attack is 120572119888 = 02 rad

Explicit form of the optimal controller is provided byhybrid toolbox which is not given here for simplicity Insteadwe present the partition results associated with the controllerin Figure 6 where 4190 subregions are obtained and for eachsubregion a control law of the form 119906 = 119870119909 + 119887 is givenAs a benefit of using explicit MPC the controller parameters119870 and 119887 for all the subregions are obtained simultaneouslyonce the optimization problem is solved This property isobviously different from that of gain-scheduled controller(where the controller parameters for different regions areseparately determined)

However by a simple calculation we conclude that thesesubregions donot require toomuch storage space (the neededstorage space is approximately 14 times 8 times 4190 = 450KB)Moreover it should be noted that when the algorithm isimplemented in practice much more time will be needed toidentify which subregion the current states enter into This isanother important factor that affects the computational costof the proposed algorithm

The simulation results are shown in Figures 7ndash9 It isseen from Figure 7 that the actual angle of attack tracksthe command in less than 01 seconds As shown in Figures8 and 9 the aerodynamic force converges to a constant astime increases while the jets are only activated during thebeginning period (when the tracking error is obvious)

International Journal of Aerospace Engineering 11

58

68

73

108

174

180

245

252

254

270

271

272

273

274

276

277

299

300

301

302

06

04

02

0

minus02

minus04

Elev

ator

defl

ectio

n (r

ad)

Angle of attack (rad)minus04 minus02 0 02 04 06

1634

1635

1638

1648

1649

1650

1664

1673

1677

1678

1679

1712

1726

1743

1744

1746

1748

1751

1752

1983

1985

1993

2011

Late

ral f

orce

(N)

Angle of attack (rad)minus04 minus02 0 02 04 06

1000

800

600

400

200

0

minus200

minus400

minus600

minus800

minus1000

Polyhedral partitionmdash4190 regions Polyhedral partitionmdash4190 regions

Figure 6 The partition associated with the explicit controller

025

02

015

01

005

0

ActualCommand

0 02 04 06 08 1

Time (s)

Ang

le o

f atta

ck (r

ad)

Figure 7 The response of angle of attack

52 Case 2 The results for the case with a different weight119877120575119911

= 0005 are shown in Figures 10ndash12 It is seen that adifferent control allocation result is obtained

04

02

0

minus02

minus04

minus060 02 04 06 08 1

Time (s)

Elev

ator

defl

ectio

n (r

ad)

Figure 8 The response of elevator deflection

53 Case 3 In this case a different command 120572119888 =

02 cos(05120587119905) rad is considered and the weight matrices arethe same as the ones for Case 1 The simulation results areshown in Figures 13ndash15

It is seen from Figure 13 that asymptotic tracking is alsoachievedThemethod proposed in this paper can realize both

12 International Journal of Aerospace Engineering

1

05

0

minus05

minus1

times104

0 02 04 06 08 1

Time (s)

Late

ral f

orce

(N)

Figure 9 The response of lateral force

025

02

015

01

005

0

Ang

le o

f atta

ck (r

ad)

0 02 04 06 08 1

Time (s)

ActualCommand

Figure 10 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 02 04 06 08 1

Time (s)

Figure 11 The response of elevator deflection

1

05

0

minus05

minus1

Late

ral f

orce

(N)

times104

0 02 04 06 08 1

Time (s)

Figure 12 The response of lateral force

03

025

02

015

01

005

0

minus005

minus01

minus015

minus02

Ang

le o

f atta

ck (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

ActualCommand

Figure 13 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

Figure 14 The response of elevator deflection

International Journal of Aerospace Engineering 13

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1

Late

ral f

orce

(N)

0 05 1 15 2 25 3 35 4

Time (s)

times104

Figure 15 The response of lateral force

fast tracking command and control allocation It is seen fromFigure 15 that the jets are activated while the tracking erroris obvious or command is varying rapidly

6 Conclusion

An autopilot design method for a missile with aerodynamiccontrol surfaces and lateral jets is presented in this paperThe nonlinear attitude control model is reduced to an MLDmodel Meanwhile the lateral force is described as linearcombination of logical variables due to the discrete values oflateral force Then the whole MLD model of attitude controlsystem is derived Autopilot design is accomplished usinghybrid MPC method By setting the related weighted coeffi-cients in index function the control allocation is obtainedMoreover numerical simulations are performed under thedifferent conditions the performance of tracking the attitudecommand and control allocation is verified and the explicitform of the control law can be obtained as well

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research presented in this document is supportedby the National Natural Science Foundation of Chinaunder Grant nos 61104193 61203191 and 61333001 theFundamental Research Funds for the Central Universities(HITNSRIF2012032) the Program for IBRSEM in HarbinInstitute of Technology under Grant HITIBRSEMA201415and the Foundation of Supporting Technology for Aerospaceunder Grant 2014-HT-HGD7

References

[1] R Hirokawa K Sato and S Manabe ldquoAutopilot design fora missile with reaction-jet using coefficient diagram methodrdquoin Proceedings of the AIAA Guidance Navigation and ControlConference and Exhibit 2001 August 2001

[2] P K Menon and V R Iragavarapu Adaptive Techniques forMultiple Actuator Blending Defense Technical InformationCenter 1998

[3] A G Munson and M W Garbrick ldquoJet interaction investiga-tionrdquo Report OR-9933 Martin Marietta Orlando Fla USA1969

[4] M Graham and P Weinacht ldquoNumerical simulation of lateralcontrol jetsrdquo in Proceedings of the 37th Aerospace SciencesMeeting and Exhibit 1999 AIAA Paper

[5] J S Shamma and J R Cloutier ldquoGain-scheduledmissile autopi-lot design using linear parameter varying transformationsrdquoJournal of Guidance Control and Dynamics vol 16 no 2 pp256ndash263 1993

[6] M Innocenti and A Thukral ldquoSimultaneous reaction jet andaerodynamic control of missile systemsrdquo AlAA 93-3739-CP1993

[7] D B Ridgely Y Lee and T Fanciullo ldquoDual aeropropulsivemissile control-optimal control and control allocationrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit Keystone Colo USA 2006

[8] D B Ridgely D Drake L Triplett and C Geise ldquoDynamiccontrol allocation of a missile with tails and reaction jetsrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 3158ndash3189 August 2007

[9] D Q Mayne J B Rawlings C V Rao and P O ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[10] A Bemporad F Borrelli and M Morari ldquoModel predictivecontrol based on linear programmingmdashthe explicit solutionrdquoIEEE Transactions on Automatic Control vol 47 no 12 pp1974ndash1985 2002

[11] C Jouannet and P Krus ldquoModelling of high angle of attackaerodynamicrdquo in Proceedings of the 25th AIAA Applied Aerody-namics Conference pp 1481ndash1516 June 2007

[12] S Venugopal and M Krishnamurthy ldquoMissile aerodynamics athigh angles of attack a prediction coderdquo Journal of Spacecraftand Rockets vol 32 no 2 pp 263ndash269 1995

[13] A Bemporad ldquoEfficient conversion of mixed logical dynamicalsystems into an equivalent piecewise affine formrdquo IEEE Trans-actions on Automatic Control vol 49 no 5 pp 832ndash838 2004

[14] F D Torrisi andA Bemporad ldquoHYSDELmdasha tool for generatingcomputational hybrid models for analysis and synthesis prob-lemsrdquo IEEE Transactions on Control Systems Technology vol 12no 2 pp 235ndash249 2004

[15] A Bemporad W P M H Heemels and B De SchutterldquoOn hybrid systems and closed-loop MPC systemsrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 863ndash8692002

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8 International Journal of Aerospace Engineering

Table 2 Aerodynamic parameters of pitch channel

Aerodynamic parameter 119876119878119862120572

119910119898119881 119876119878119862

120575119911

119910119898119881 119876119878119871119898

120572

119911119869119911

119876119878119871119898119903

119911119869119911

119876119878119871119898120575119911

119911119869119911

120572 = minus053 minus1175 0076 947 minus032 minus8489

120572 = minus037 minus0944 0082 973 minus032 minus8924

120572 = minus0153 minus023 0082 1043 minus032 minus8938

120572 = 0 036 009 1101 minus032 minus9510

120572 = 0153 094 0082 1158 minus032 minus8934

120572 = 037 163 0082 1224 minus032 minus8918

3 = [minus003 268 times 10

minus6

minus2228 223 times 10minus4]

4 = [minus0032 277 times 10

minus6

minus2371 232 times 10minus4]

5 = [minus003 227 times 10

minus6

minus2228 198 times 10minus4]

6 = [minus0029 191 times 10

minus6

minus2224 173 times 10minus4]

(25)

32 Constraints Analysis Due to the symmetry of jet config-uration the set of possible negative pitch control force is givenby

119880119910minus

119865= minus119865119898 minus2119865119898 cos

120587

9 minus2119865119898 cos

120587

18 minus2119865119898 minus4119865119898 cos

120587

9

minus4119865119898 cos120587

18

(26)

By combining (13) and (26) we obtain the set of allpossible pitch control force

119880119910

119865= 119865119898 2119865119898 cos

120587

9 2119865119898 cos

120587

18 2119865119898 4119865119898 cos

120587

9

4119865119898 cos120587

18 minus119865119898 minus2119865119898 cos

120587

9 minus2119865119898 cos

120587

18 minus2119865119898

minus4119865119898 cos120587

9 minus4119865119898 cos

120587

18

(27)

Substituting the jet parameters shown in Table 1 into theforegoing set yields

119880119910

119865= 2200 4135 4333 4400 8269 8666 minus2200 minus4135

minus4333 minus4400 minus8269 minus8666

(28)

Noting that the lateral forces are discrete variable we willutilize the linear combination of logical variables to describethe lateral force In terms of piecewise affine model (20) we

introduce logical variables 120575119865119894

isin 0 1 119894 = 1 2 12 toexpress the lateral thrust 119865119910

119887

119865119910119887

= 22001205751198651

+ 41351205751198652

+ 43331205751198653

+ 44001205751198654

+ 82691205751198655

+ 86661205751198656

minus 22001205751198657

minus 41351205751198658

minus 43331205751198659

minus 440012057511986510

minus 826912057511986511

minus 866612057511986512

(29)

In (29) the logical variables should satisfy the constraints

12

sum

119894=1

120575119865119894

= 0 or 1 (30)

where 0 means that no lateral force is generated while 1

means that the applied lateral force equals to one element ofset 119880119910119865

Denote 1199061 = 120575119911 the control input 119906 in model (23) can berewritten as

119906 = [1199061 119865119910119887

]119879 (31)

As shown in Table 1 the constraints on system states andcontrol input are

119909min le 119909 (119896) le 119909max

1199061min le 1199061 (119896) le 1199061max(32)

where 119909min = [minus053 minus522]119879 119909max = [053 522]

119879 1199061min =

minus053 1199061max = 053According to (30) we have

12

sum

119894=1

120575119865119894

le 1 (33)

which shows the constraint on the logical control inputUse logical variables 120575119894(119896) isin 0 1 119894 = 1 2 5 to

describe the operation points which satisfy the constraints

[1 0] 119909 (119896) + 037 le 0 lArrrArr 1205751 (119896) = 1

[1 0] 119909 (119896) + 0153 le 0 lArrrArr 1205752 (119896) = 1

[1 0] 119909 (119896) le 0 lArrrArr 1205753 (119896) = 1

[1 0] 119909 (119896) minus 0153 le 0 lArrrArr 1205754 (119896) = 1

[1 0] 119909 (119896) minus 037 le 0 lArrrArr 1205755 (119896) = 1

(34)

International Journal of Aerospace Engineering 9

Equation (34) can be transformed into the followingequivalent mixed logical inequalities [13]

[1 0] 119909 (119896) + 037 ge 120576 + (1198981 minus 120576) 1205751 (119896)

[1 0] 119909 (119896) + 037 le 1198721 (1 minus 1205751 (119896))

[1 0] 119909 (119896) + 0153 ge 120576 + (1198982 minus 120576) 1205752 (119896)

[1 0] 119909 (119896) + 0153 le 1198722 (1 minus 1205752 (119896))

[1 0] 119909 (119896) ge 120576 + (1198983 minus 120576) 1205753 (119896)

[1 0] 119909 (119896) le 1198723 (1 minus 1205753 (119896))

[1 0] 119909 (119896) minus 0153 ge 120576 + (1198984 minus 120576) 1205754 (119896)

[1 0] 119909 (119896) minus 0153 le 1198724 (1 minus 1205754 (119896))

[1 0] 119909 (119896) minus 037 ge 120576 + (1198985 minus 120576) 1205755 (119896)

[1 0] 119909 (119896) minus 037 le 1198725 (1 minus 1205755 (119896))

(35)

where 1198981 = minus016 1198721 = 090 1198982 = minus0377 1198722 = 06831198983 = minus053 1198723 = 053 1198984 = minus0683 1198724 = 0377 1198985 =

minus0901198725 = 016 and 120576 = 10minus6

In addition we introduce the auxiliary logical variables120575119894(119896) isin 0 1 119894 = 6 9 as follows

1205756 (119896) = (1 minus 1205751 (119896)) 1205752 (119896)

1205757 (119896) = (1 minus 1205752 (119896)) 1205753 (119896)

1205758 (119896) = (1 minus 1205753 (119896)) 1205754 (119896)

1205759 (119896) = (1 minus 1205754 (119896)) 1205755 (119896)

(36)

With 1205751 1205756 1205757 1205758 1205759 and 1 minus 1205755 the six regions canbe presented Similarly (36) can be transformed into theequivalent mixed logical inequalities

minus1205751 (119896) + 1205752 (119896) minus 1205756 (119896) le 0

1205751 (119896) + 1205756 (119896) le 1

minus1205752 (119896) + 1205756 (119896) le 0

minus1205752 (119896) + 1205753 (119896) minus 1205757 (119896) le 0

1205752 (119896) + 1205757 (119896) le 1

minus1205753 (119896) + 1205757 (119896) le 0

minus1205753 (119896) + 1205754 (119896) minus 1205758 (119896) le 0

1205753 (119896) + 1205758 (119896) le 1

minus1205754 (119896) + 1205758 (119896) le 0

minus1205754 (119896) + 1205755 (119896) minus 1205759 (119896) le 0

1205754 (119896) + 1205759 (119896) le 1

minus1205755 (119896) + 1205759 (119896) le 0

(37)

To describe the state space model of each region weintroduce the following auxiliary continuous variables

1199111 (119896) = (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) 1205751 (119896)

1199112 (119896) = (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) 1205756 (119896)

1199113 (119896) = (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) 1205757 (119896)

1199114 (119896) = (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) 1205758 (119896)

1199115 (119896) = (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) 1205759 (119896)

1199116 (119896) = (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) (1 minus 1205755 (119896))

(38)

Equation (38) can be converted to the equivalent mixedlogical inequalities

1199111 (119896) ge (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198721198911 (1 minus 1205751 (119896))

1199111 (119896) le (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198981198911 (1 minus 1205751 (119896))

1199111 (119896) ge 11989811989111205751 (119896)

1199111 (119896) le 11987211989111205751 (119896)

1199112 (119896) ge (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198721198912 (1 minus 1205756 (119896))

1199112 (119896) le (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198981198912 (1 minus 1205756 (119896))

1199112 (119896) ge 11989811989121205756 (119896)

1199112 (119896) le 11987211989121205756 (119896)

1199113 (119896) ge (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198721198913 (1 minus 1205757 (119896))

1199113 (119896) le (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198981198913 (1 minus 1205757 (119896))

1199113 (119896) ge 11989811989131205757 (119896)

1199113 (119896) le 11987211989131205757 (119896)

1199114 (119896) ge (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198721198914 (1 minus 1205758 (119896))

1199114 (119896) le (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198981198914 (1 minus 1205758 (119896))

1199114 (119896) ge 11989811989141205758 (119896)

1199114 (119896) le 11987211989141205758 (119896)

1199115 (119896) ge (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198721198915 (1 minus 1205759 (119896))

1199115 (119896) le (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198981198915 (1 minus 1205759 (119896))

10 International Journal of Aerospace Engineering

1199115 (119896) ge 11989811989151205759 (119896)

1199115 (119896) le 11987211989151205759 (119896)

1199116 (119896) ge (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11987211989151205755 (119896)

1199116 (119896) le (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11989811989151205755 (119896)

1199116 (119896) ge 1198981198916 (1 minus 1205755 (119896))

1199116 (119896) le 1198721198916 (1 minus 1205755 (119896))

(39)

where1198721198911 = [073 1084]1198791198981198911 = [minus077 minus1106]

1198791198721198912 =[076 1166]

119879 1198981198912 = [minus079 minus 1256]119879 1198721198913 =

[077 1273]119879 1198981198913 = [minus078 minus 1280]

119879 1198721198914 =

[076 1403]119879 1198981198914 = [minus076 minus 1403]

119879 1198721198915 =

[0725 1291]119879 1198981198915 = [minus0736 minus 1205]

119879 1198721198916 =

[0696 1154]119879 and1198981198916 = [minus073 minus 1130]

119879Then the whole MLD model of the missile is given by

119909 (119896 + 1) =

6

sum

119894=1

119911119894 (119896)

119910 (119896) = [1 0] 119909 (119896)

st (28) (30) (32) (34) (36) (38)

(40)

where (29) (31) (33) (35) (37) and (39) represent all theconstraint inequalities

4 Autopilot Design UsingHybrid MPC Method

In general we expect that the output 119910 tracks its command 119910119888

as fast as possible with a small amount of fuel consumptionMotivated by this observation we consider the followingoptimization problem

119869lowast

= min119906(119896)119906(119896+1)120575(119896)120575(119896+1|119896)119911(119896)119911(119896+1|119896)

119873

sum

119894=1

(1003817100381710038171003817119910 (119896 + 119894 | 119896)

minus119910119888 (119896 + 119894)1003817100381710038171003817

2

119876

+ 119906(119896 + 119894)2

119877)

st MLD model (39)

1199061min le 1199061 (119896) 1199061 (119896 + 1) le 1199061max

119909min le 119909 (119896) 119909 (119896 + 1) le 119909max

(41)

where 119910119888 is the command for angle of attack 119910(119896 + 119894 | 119896) ispredictive value of angle of attack 119873 denotes the predictivehorizon 119876 and 119877 are the weighting matrices and 119877 =

diag(119877120575119911

119877119865119910

) 119877120575119911

119877119865119910

are weighted coefficients of aerody-namic control surfaces and lateral pulse jets respectivelyThe

ratio of 119877120575119911

and 119877119865119910

represents the control allocation betweenaerodynamic control surfaces and lateral pulse jets When119877119865119910

119877120575119911

is increased the requirement for lateral force will bedeclined that is the fuel consumption can be lessened Inaddition the deduction can be verified from the simulationresults of Case 1 and Case 2

Different control allocation between the dual actuators(aerodynamic control surfaces and lateral pulse jets) can beobtained by setting different 119876 and 119877 Since both logicaland continuous variables are involved in (41) the aboveon-line optimization problem is a mixed integer quadraticprogramming (MIQP) problemwhich can be solved by usingthe hybrid MPC toolbox of MATLAB [14 15]

Remark 2 In an explicit MPC controller the main factorsaffecting the number of subregions include the systemdimen-sion the predictive horizon and the number of constrains Inthis paper the blendedmissile with aerodynamic control sur-faces and lateral jets is investigated To deal with the discreteproperty of the lateral jet forces some logical variables areintroduced for MPC controller design which increases thenumber of subregions inevitably

5 Numerical Simulations

51 Case 1 Numerical simulations are performed to ver-ify the feasibility of the proposed method in this sectionThe involved parameters are given in Table 1 The weightmatrices are set as 119876 = 10 119877120575

119911

= 0025 and 119877119865119910

=

diag(0005 0005 0005) Suppose the initial state is 1205720 =0 1199030 = 0 In order to avoid excessive computational load wechoose predictive horizon119873 = 2 and control horizon119872 = 2The command for angle of attack is 120572119888 = 02 rad

Explicit form of the optimal controller is provided byhybrid toolbox which is not given here for simplicity Insteadwe present the partition results associated with the controllerin Figure 6 where 4190 subregions are obtained and for eachsubregion a control law of the form 119906 = 119870119909 + 119887 is givenAs a benefit of using explicit MPC the controller parameters119870 and 119887 for all the subregions are obtained simultaneouslyonce the optimization problem is solved This property isobviously different from that of gain-scheduled controller(where the controller parameters for different regions areseparately determined)

However by a simple calculation we conclude that thesesubregions donot require toomuch storage space (the neededstorage space is approximately 14 times 8 times 4190 = 450KB)Moreover it should be noted that when the algorithm isimplemented in practice much more time will be needed toidentify which subregion the current states enter into This isanother important factor that affects the computational costof the proposed algorithm

The simulation results are shown in Figures 7ndash9 It isseen from Figure 7 that the actual angle of attack tracksthe command in less than 01 seconds As shown in Figures8 and 9 the aerodynamic force converges to a constant astime increases while the jets are only activated during thebeginning period (when the tracking error is obvious)

International Journal of Aerospace Engineering 11

58

68

73

108

174

180

245

252

254

270

271

272

273

274

276

277

299

300

301

302

06

04

02

0

minus02

minus04

Elev

ator

defl

ectio

n (r

ad)

Angle of attack (rad)minus04 minus02 0 02 04 06

1634

1635

1638

1648

1649

1650

1664

1673

1677

1678

1679

1712

1726

1743

1744

1746

1748

1751

1752

1983

1985

1993

2011

Late

ral f

orce

(N)

Angle of attack (rad)minus04 minus02 0 02 04 06

1000

800

600

400

200

0

minus200

minus400

minus600

minus800

minus1000

Polyhedral partitionmdash4190 regions Polyhedral partitionmdash4190 regions

Figure 6 The partition associated with the explicit controller

025

02

015

01

005

0

ActualCommand

0 02 04 06 08 1

Time (s)

Ang

le o

f atta

ck (r

ad)

Figure 7 The response of angle of attack

52 Case 2 The results for the case with a different weight119877120575119911

= 0005 are shown in Figures 10ndash12 It is seen that adifferent control allocation result is obtained

04

02

0

minus02

minus04

minus060 02 04 06 08 1

Time (s)

Elev

ator

defl

ectio

n (r

ad)

Figure 8 The response of elevator deflection

53 Case 3 In this case a different command 120572119888 =

02 cos(05120587119905) rad is considered and the weight matrices arethe same as the ones for Case 1 The simulation results areshown in Figures 13ndash15

It is seen from Figure 13 that asymptotic tracking is alsoachievedThemethod proposed in this paper can realize both

12 International Journal of Aerospace Engineering

1

05

0

minus05

minus1

times104

0 02 04 06 08 1

Time (s)

Late

ral f

orce

(N)

Figure 9 The response of lateral force

025

02

015

01

005

0

Ang

le o

f atta

ck (r

ad)

0 02 04 06 08 1

Time (s)

ActualCommand

Figure 10 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 02 04 06 08 1

Time (s)

Figure 11 The response of elevator deflection

1

05

0

minus05

minus1

Late

ral f

orce

(N)

times104

0 02 04 06 08 1

Time (s)

Figure 12 The response of lateral force

03

025

02

015

01

005

0

minus005

minus01

minus015

minus02

Ang

le o

f atta

ck (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

ActualCommand

Figure 13 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

Figure 14 The response of elevator deflection

International Journal of Aerospace Engineering 13

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1

Late

ral f

orce

(N)

0 05 1 15 2 25 3 35 4

Time (s)

times104

Figure 15 The response of lateral force

fast tracking command and control allocation It is seen fromFigure 15 that the jets are activated while the tracking erroris obvious or command is varying rapidly

6 Conclusion

An autopilot design method for a missile with aerodynamiccontrol surfaces and lateral jets is presented in this paperThe nonlinear attitude control model is reduced to an MLDmodel Meanwhile the lateral force is described as linearcombination of logical variables due to the discrete values oflateral force Then the whole MLD model of attitude controlsystem is derived Autopilot design is accomplished usinghybrid MPC method By setting the related weighted coeffi-cients in index function the control allocation is obtainedMoreover numerical simulations are performed under thedifferent conditions the performance of tracking the attitudecommand and control allocation is verified and the explicitform of the control law can be obtained as well

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research presented in this document is supportedby the National Natural Science Foundation of Chinaunder Grant nos 61104193 61203191 and 61333001 theFundamental Research Funds for the Central Universities(HITNSRIF2012032) the Program for IBRSEM in HarbinInstitute of Technology under Grant HITIBRSEMA201415and the Foundation of Supporting Technology for Aerospaceunder Grant 2014-HT-HGD7

References

[1] R Hirokawa K Sato and S Manabe ldquoAutopilot design fora missile with reaction-jet using coefficient diagram methodrdquoin Proceedings of the AIAA Guidance Navigation and ControlConference and Exhibit 2001 August 2001

[2] P K Menon and V R Iragavarapu Adaptive Techniques forMultiple Actuator Blending Defense Technical InformationCenter 1998

[3] A G Munson and M W Garbrick ldquoJet interaction investiga-tionrdquo Report OR-9933 Martin Marietta Orlando Fla USA1969

[4] M Graham and P Weinacht ldquoNumerical simulation of lateralcontrol jetsrdquo in Proceedings of the 37th Aerospace SciencesMeeting and Exhibit 1999 AIAA Paper

[5] J S Shamma and J R Cloutier ldquoGain-scheduledmissile autopi-lot design using linear parameter varying transformationsrdquoJournal of Guidance Control and Dynamics vol 16 no 2 pp256ndash263 1993

[6] M Innocenti and A Thukral ldquoSimultaneous reaction jet andaerodynamic control of missile systemsrdquo AlAA 93-3739-CP1993

[7] D B Ridgely Y Lee and T Fanciullo ldquoDual aeropropulsivemissile control-optimal control and control allocationrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit Keystone Colo USA 2006

[8] D B Ridgely D Drake L Triplett and C Geise ldquoDynamiccontrol allocation of a missile with tails and reaction jetsrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 3158ndash3189 August 2007

[9] D Q Mayne J B Rawlings C V Rao and P O ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[10] A Bemporad F Borrelli and M Morari ldquoModel predictivecontrol based on linear programmingmdashthe explicit solutionrdquoIEEE Transactions on Automatic Control vol 47 no 12 pp1974ndash1985 2002

[11] C Jouannet and P Krus ldquoModelling of high angle of attackaerodynamicrdquo in Proceedings of the 25th AIAA Applied Aerody-namics Conference pp 1481ndash1516 June 2007

[12] S Venugopal and M Krishnamurthy ldquoMissile aerodynamics athigh angles of attack a prediction coderdquo Journal of Spacecraftand Rockets vol 32 no 2 pp 263ndash269 1995

[13] A Bemporad ldquoEfficient conversion of mixed logical dynamicalsystems into an equivalent piecewise affine formrdquo IEEE Trans-actions on Automatic Control vol 49 no 5 pp 832ndash838 2004

[14] F D Torrisi andA Bemporad ldquoHYSDELmdasha tool for generatingcomputational hybrid models for analysis and synthesis prob-lemsrdquo IEEE Transactions on Control Systems Technology vol 12no 2 pp 235ndash249 2004

[15] A Bemporad W P M H Heemels and B De SchutterldquoOn hybrid systems and closed-loop MPC systemsrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 863ndash8692002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Aerospace Engineering 9

Equation (34) can be transformed into the followingequivalent mixed logical inequalities [13]

[1 0] 119909 (119896) + 037 ge 120576 + (1198981 minus 120576) 1205751 (119896)

[1 0] 119909 (119896) + 037 le 1198721 (1 minus 1205751 (119896))

[1 0] 119909 (119896) + 0153 ge 120576 + (1198982 minus 120576) 1205752 (119896)

[1 0] 119909 (119896) + 0153 le 1198722 (1 minus 1205752 (119896))

[1 0] 119909 (119896) ge 120576 + (1198983 minus 120576) 1205753 (119896)

[1 0] 119909 (119896) le 1198723 (1 minus 1205753 (119896))

[1 0] 119909 (119896) minus 0153 ge 120576 + (1198984 minus 120576) 1205754 (119896)

[1 0] 119909 (119896) minus 0153 le 1198724 (1 minus 1205754 (119896))

[1 0] 119909 (119896) minus 037 ge 120576 + (1198985 minus 120576) 1205755 (119896)

[1 0] 119909 (119896) minus 037 le 1198725 (1 minus 1205755 (119896))

(35)

where 1198981 = minus016 1198721 = 090 1198982 = minus0377 1198722 = 06831198983 = minus053 1198723 = 053 1198984 = minus0683 1198724 = 0377 1198985 =

minus0901198725 = 016 and 120576 = 10minus6

In addition we introduce the auxiliary logical variables120575119894(119896) isin 0 1 119894 = 6 9 as follows

1205756 (119896) = (1 minus 1205751 (119896)) 1205752 (119896)

1205757 (119896) = (1 minus 1205752 (119896)) 1205753 (119896)

1205758 (119896) = (1 minus 1205753 (119896)) 1205754 (119896)

1205759 (119896) = (1 minus 1205754 (119896)) 1205755 (119896)

(36)

With 1205751 1205756 1205757 1205758 1205759 and 1 minus 1205755 the six regions canbe presented Similarly (36) can be transformed into theequivalent mixed logical inequalities

minus1205751 (119896) + 1205752 (119896) minus 1205756 (119896) le 0

1205751 (119896) + 1205756 (119896) le 1

minus1205752 (119896) + 1205756 (119896) le 0

minus1205752 (119896) + 1205753 (119896) minus 1205757 (119896) le 0

1205752 (119896) + 1205757 (119896) le 1

minus1205753 (119896) + 1205757 (119896) le 0

minus1205753 (119896) + 1205754 (119896) minus 1205758 (119896) le 0

1205753 (119896) + 1205758 (119896) le 1

minus1205754 (119896) + 1205758 (119896) le 0

minus1205754 (119896) + 1205755 (119896) minus 1205759 (119896) le 0

1205754 (119896) + 1205759 (119896) le 1

minus1205755 (119896) + 1205759 (119896) le 0

(37)

To describe the state space model of each region weintroduce the following auxiliary continuous variables

1199111 (119896) = (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) 1205751 (119896)

1199112 (119896) = (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) 1205756 (119896)

1199113 (119896) = (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) 1205757 (119896)

1199114 (119896) = (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) 1205758 (119896)

1199115 (119896) = (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) 1205759 (119896)

1199116 (119896) = (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) (1 minus 1205755 (119896))

(38)

Equation (38) can be converted to the equivalent mixedlogical inequalities

1199111 (119896) ge (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198721198911 (1 minus 1205751 (119896))

1199111 (119896) le (1198861 (119896) 119909 (119896) + 1 (119896) 119906 (119896) + 1198901) minus 1198981198911 (1 minus 1205751 (119896))

1199111 (119896) ge 11989811989111205751 (119896)

1199111 (119896) le 11987211989111205751 (119896)

1199112 (119896) ge (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198721198912 (1 minus 1205756 (119896))

1199112 (119896) le (1198862 (119896) 119909 (119896) + 2 (119896) 119906 (119896) + 1198902) minus 1198981198912 (1 minus 1205756 (119896))

1199112 (119896) ge 11989811989121205756 (119896)

1199112 (119896) le 11987211989121205756 (119896)

1199113 (119896) ge (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198721198913 (1 minus 1205757 (119896))

1199113 (119896) le (1198863 (119896) 119909 (119896) + 3 (119896) 119906 (119896) + 1198903) minus 1198981198913 (1 minus 1205757 (119896))

1199113 (119896) ge 11989811989131205757 (119896)

1199113 (119896) le 11987211989131205757 (119896)

1199114 (119896) ge (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198721198914 (1 minus 1205758 (119896))

1199114 (119896) le (1198864 (119896) 119909 (119896) + 4 (119896) 119906 (119896) + 1198904) minus 1198981198914 (1 minus 1205758 (119896))

1199114 (119896) ge 11989811989141205758 (119896)

1199114 (119896) le 11987211989141205758 (119896)

1199115 (119896) ge (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198721198915 (1 minus 1205759 (119896))

1199115 (119896) le (1198865 (119896) 119909 (119896) + 5 (119896) 119906 (119896) + 1198905) minus 1198981198915 (1 minus 1205759 (119896))

10 International Journal of Aerospace Engineering

1199115 (119896) ge 11989811989151205759 (119896)

1199115 (119896) le 11987211989151205759 (119896)

1199116 (119896) ge (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11987211989151205755 (119896)

1199116 (119896) le (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11989811989151205755 (119896)

1199116 (119896) ge 1198981198916 (1 minus 1205755 (119896))

1199116 (119896) le 1198721198916 (1 minus 1205755 (119896))

(39)

where1198721198911 = [073 1084]1198791198981198911 = [minus077 minus1106]

1198791198721198912 =[076 1166]

119879 1198981198912 = [minus079 minus 1256]119879 1198721198913 =

[077 1273]119879 1198981198913 = [minus078 minus 1280]

119879 1198721198914 =

[076 1403]119879 1198981198914 = [minus076 minus 1403]

119879 1198721198915 =

[0725 1291]119879 1198981198915 = [minus0736 minus 1205]

119879 1198721198916 =

[0696 1154]119879 and1198981198916 = [minus073 minus 1130]

119879Then the whole MLD model of the missile is given by

119909 (119896 + 1) =

6

sum

119894=1

119911119894 (119896)

119910 (119896) = [1 0] 119909 (119896)

st (28) (30) (32) (34) (36) (38)

(40)

where (29) (31) (33) (35) (37) and (39) represent all theconstraint inequalities

4 Autopilot Design UsingHybrid MPC Method

In general we expect that the output 119910 tracks its command 119910119888

as fast as possible with a small amount of fuel consumptionMotivated by this observation we consider the followingoptimization problem

119869lowast

= min119906(119896)119906(119896+1)120575(119896)120575(119896+1|119896)119911(119896)119911(119896+1|119896)

119873

sum

119894=1

(1003817100381710038171003817119910 (119896 + 119894 | 119896)

minus119910119888 (119896 + 119894)1003817100381710038171003817

2

119876

+ 119906(119896 + 119894)2

119877)

st MLD model (39)

1199061min le 1199061 (119896) 1199061 (119896 + 1) le 1199061max

119909min le 119909 (119896) 119909 (119896 + 1) le 119909max

(41)

where 119910119888 is the command for angle of attack 119910(119896 + 119894 | 119896) ispredictive value of angle of attack 119873 denotes the predictivehorizon 119876 and 119877 are the weighting matrices and 119877 =

diag(119877120575119911

119877119865119910

) 119877120575119911

119877119865119910

are weighted coefficients of aerody-namic control surfaces and lateral pulse jets respectivelyThe

ratio of 119877120575119911

and 119877119865119910

represents the control allocation betweenaerodynamic control surfaces and lateral pulse jets When119877119865119910

119877120575119911

is increased the requirement for lateral force will bedeclined that is the fuel consumption can be lessened Inaddition the deduction can be verified from the simulationresults of Case 1 and Case 2

Different control allocation between the dual actuators(aerodynamic control surfaces and lateral pulse jets) can beobtained by setting different 119876 and 119877 Since both logicaland continuous variables are involved in (41) the aboveon-line optimization problem is a mixed integer quadraticprogramming (MIQP) problemwhich can be solved by usingthe hybrid MPC toolbox of MATLAB [14 15]

Remark 2 In an explicit MPC controller the main factorsaffecting the number of subregions include the systemdimen-sion the predictive horizon and the number of constrains Inthis paper the blendedmissile with aerodynamic control sur-faces and lateral jets is investigated To deal with the discreteproperty of the lateral jet forces some logical variables areintroduced for MPC controller design which increases thenumber of subregions inevitably

5 Numerical Simulations

51 Case 1 Numerical simulations are performed to ver-ify the feasibility of the proposed method in this sectionThe involved parameters are given in Table 1 The weightmatrices are set as 119876 = 10 119877120575

119911

= 0025 and 119877119865119910

=

diag(0005 0005 0005) Suppose the initial state is 1205720 =0 1199030 = 0 In order to avoid excessive computational load wechoose predictive horizon119873 = 2 and control horizon119872 = 2The command for angle of attack is 120572119888 = 02 rad

Explicit form of the optimal controller is provided byhybrid toolbox which is not given here for simplicity Insteadwe present the partition results associated with the controllerin Figure 6 where 4190 subregions are obtained and for eachsubregion a control law of the form 119906 = 119870119909 + 119887 is givenAs a benefit of using explicit MPC the controller parameters119870 and 119887 for all the subregions are obtained simultaneouslyonce the optimization problem is solved This property isobviously different from that of gain-scheduled controller(where the controller parameters for different regions areseparately determined)

However by a simple calculation we conclude that thesesubregions donot require toomuch storage space (the neededstorage space is approximately 14 times 8 times 4190 = 450KB)Moreover it should be noted that when the algorithm isimplemented in practice much more time will be needed toidentify which subregion the current states enter into This isanother important factor that affects the computational costof the proposed algorithm

The simulation results are shown in Figures 7ndash9 It isseen from Figure 7 that the actual angle of attack tracksthe command in less than 01 seconds As shown in Figures8 and 9 the aerodynamic force converges to a constant astime increases while the jets are only activated during thebeginning period (when the tracking error is obvious)

International Journal of Aerospace Engineering 11

58

68

73

108

174

180

245

252

254

270

271

272

273

274

276

277

299

300

301

302

06

04

02

0

minus02

minus04

Elev

ator

defl

ectio

n (r

ad)

Angle of attack (rad)minus04 minus02 0 02 04 06

1634

1635

1638

1648

1649

1650

1664

1673

1677

1678

1679

1712

1726

1743

1744

1746

1748

1751

1752

1983

1985

1993

2011

Late

ral f

orce

(N)

Angle of attack (rad)minus04 minus02 0 02 04 06

1000

800

600

400

200

0

minus200

minus400

minus600

minus800

minus1000

Polyhedral partitionmdash4190 regions Polyhedral partitionmdash4190 regions

Figure 6 The partition associated with the explicit controller

025

02

015

01

005

0

ActualCommand

0 02 04 06 08 1

Time (s)

Ang

le o

f atta

ck (r

ad)

Figure 7 The response of angle of attack

52 Case 2 The results for the case with a different weight119877120575119911

= 0005 are shown in Figures 10ndash12 It is seen that adifferent control allocation result is obtained

04

02

0

minus02

minus04

minus060 02 04 06 08 1

Time (s)

Elev

ator

defl

ectio

n (r

ad)

Figure 8 The response of elevator deflection

53 Case 3 In this case a different command 120572119888 =

02 cos(05120587119905) rad is considered and the weight matrices arethe same as the ones for Case 1 The simulation results areshown in Figures 13ndash15

It is seen from Figure 13 that asymptotic tracking is alsoachievedThemethod proposed in this paper can realize both

12 International Journal of Aerospace Engineering

1

05

0

minus05

minus1

times104

0 02 04 06 08 1

Time (s)

Late

ral f

orce

(N)

Figure 9 The response of lateral force

025

02

015

01

005

0

Ang

le o

f atta

ck (r

ad)

0 02 04 06 08 1

Time (s)

ActualCommand

Figure 10 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 02 04 06 08 1

Time (s)

Figure 11 The response of elevator deflection

1

05

0

minus05

minus1

Late

ral f

orce

(N)

times104

0 02 04 06 08 1

Time (s)

Figure 12 The response of lateral force

03

025

02

015

01

005

0

minus005

minus01

minus015

minus02

Ang

le o

f atta

ck (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

ActualCommand

Figure 13 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

Figure 14 The response of elevator deflection

International Journal of Aerospace Engineering 13

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1

Late

ral f

orce

(N)

0 05 1 15 2 25 3 35 4

Time (s)

times104

Figure 15 The response of lateral force

fast tracking command and control allocation It is seen fromFigure 15 that the jets are activated while the tracking erroris obvious or command is varying rapidly

6 Conclusion

An autopilot design method for a missile with aerodynamiccontrol surfaces and lateral jets is presented in this paperThe nonlinear attitude control model is reduced to an MLDmodel Meanwhile the lateral force is described as linearcombination of logical variables due to the discrete values oflateral force Then the whole MLD model of attitude controlsystem is derived Autopilot design is accomplished usinghybrid MPC method By setting the related weighted coeffi-cients in index function the control allocation is obtainedMoreover numerical simulations are performed under thedifferent conditions the performance of tracking the attitudecommand and control allocation is verified and the explicitform of the control law can be obtained as well

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research presented in this document is supportedby the National Natural Science Foundation of Chinaunder Grant nos 61104193 61203191 and 61333001 theFundamental Research Funds for the Central Universities(HITNSRIF2012032) the Program for IBRSEM in HarbinInstitute of Technology under Grant HITIBRSEMA201415and the Foundation of Supporting Technology for Aerospaceunder Grant 2014-HT-HGD7

References

[1] R Hirokawa K Sato and S Manabe ldquoAutopilot design fora missile with reaction-jet using coefficient diagram methodrdquoin Proceedings of the AIAA Guidance Navigation and ControlConference and Exhibit 2001 August 2001

[2] P K Menon and V R Iragavarapu Adaptive Techniques forMultiple Actuator Blending Defense Technical InformationCenter 1998

[3] A G Munson and M W Garbrick ldquoJet interaction investiga-tionrdquo Report OR-9933 Martin Marietta Orlando Fla USA1969

[4] M Graham and P Weinacht ldquoNumerical simulation of lateralcontrol jetsrdquo in Proceedings of the 37th Aerospace SciencesMeeting and Exhibit 1999 AIAA Paper

[5] J S Shamma and J R Cloutier ldquoGain-scheduledmissile autopi-lot design using linear parameter varying transformationsrdquoJournal of Guidance Control and Dynamics vol 16 no 2 pp256ndash263 1993

[6] M Innocenti and A Thukral ldquoSimultaneous reaction jet andaerodynamic control of missile systemsrdquo AlAA 93-3739-CP1993

[7] D B Ridgely Y Lee and T Fanciullo ldquoDual aeropropulsivemissile control-optimal control and control allocationrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit Keystone Colo USA 2006

[8] D B Ridgely D Drake L Triplett and C Geise ldquoDynamiccontrol allocation of a missile with tails and reaction jetsrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 3158ndash3189 August 2007

[9] D Q Mayne J B Rawlings C V Rao and P O ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[10] A Bemporad F Borrelli and M Morari ldquoModel predictivecontrol based on linear programmingmdashthe explicit solutionrdquoIEEE Transactions on Automatic Control vol 47 no 12 pp1974ndash1985 2002

[11] C Jouannet and P Krus ldquoModelling of high angle of attackaerodynamicrdquo in Proceedings of the 25th AIAA Applied Aerody-namics Conference pp 1481ndash1516 June 2007

[12] S Venugopal and M Krishnamurthy ldquoMissile aerodynamics athigh angles of attack a prediction coderdquo Journal of Spacecraftand Rockets vol 32 no 2 pp 263ndash269 1995

[13] A Bemporad ldquoEfficient conversion of mixed logical dynamicalsystems into an equivalent piecewise affine formrdquo IEEE Trans-actions on Automatic Control vol 49 no 5 pp 832ndash838 2004

[14] F D Torrisi andA Bemporad ldquoHYSDELmdasha tool for generatingcomputational hybrid models for analysis and synthesis prob-lemsrdquo IEEE Transactions on Control Systems Technology vol 12no 2 pp 235ndash249 2004

[15] A Bemporad W P M H Heemels and B De SchutterldquoOn hybrid systems and closed-loop MPC systemsrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 863ndash8692002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 International Journal of Aerospace Engineering

1199115 (119896) ge 11989811989151205759 (119896)

1199115 (119896) le 11987211989151205759 (119896)

1199116 (119896) ge (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11987211989151205755 (119896)

1199116 (119896) le (1198866 (119896) 119909 (119896) + 6 (119896) 119906 (119896) + 1198906) minus 11989811989151205755 (119896)

1199116 (119896) ge 1198981198916 (1 minus 1205755 (119896))

1199116 (119896) le 1198721198916 (1 minus 1205755 (119896))

(39)

where1198721198911 = [073 1084]1198791198981198911 = [minus077 minus1106]

1198791198721198912 =[076 1166]

119879 1198981198912 = [minus079 minus 1256]119879 1198721198913 =

[077 1273]119879 1198981198913 = [minus078 minus 1280]

119879 1198721198914 =

[076 1403]119879 1198981198914 = [minus076 minus 1403]

119879 1198721198915 =

[0725 1291]119879 1198981198915 = [minus0736 minus 1205]

119879 1198721198916 =

[0696 1154]119879 and1198981198916 = [minus073 minus 1130]

119879Then the whole MLD model of the missile is given by

119909 (119896 + 1) =

6

sum

119894=1

119911119894 (119896)

119910 (119896) = [1 0] 119909 (119896)

st (28) (30) (32) (34) (36) (38)

(40)

where (29) (31) (33) (35) (37) and (39) represent all theconstraint inequalities

4 Autopilot Design UsingHybrid MPC Method

In general we expect that the output 119910 tracks its command 119910119888

as fast as possible with a small amount of fuel consumptionMotivated by this observation we consider the followingoptimization problem

119869lowast

= min119906(119896)119906(119896+1)120575(119896)120575(119896+1|119896)119911(119896)119911(119896+1|119896)

119873

sum

119894=1

(1003817100381710038171003817119910 (119896 + 119894 | 119896)

minus119910119888 (119896 + 119894)1003817100381710038171003817

2

119876

+ 119906(119896 + 119894)2

119877)

st MLD model (39)

1199061min le 1199061 (119896) 1199061 (119896 + 1) le 1199061max

119909min le 119909 (119896) 119909 (119896 + 1) le 119909max

(41)

where 119910119888 is the command for angle of attack 119910(119896 + 119894 | 119896) ispredictive value of angle of attack 119873 denotes the predictivehorizon 119876 and 119877 are the weighting matrices and 119877 =

diag(119877120575119911

119877119865119910

) 119877120575119911

119877119865119910

are weighted coefficients of aerody-namic control surfaces and lateral pulse jets respectivelyThe

ratio of 119877120575119911

and 119877119865119910

represents the control allocation betweenaerodynamic control surfaces and lateral pulse jets When119877119865119910

119877120575119911

is increased the requirement for lateral force will bedeclined that is the fuel consumption can be lessened Inaddition the deduction can be verified from the simulationresults of Case 1 and Case 2

Different control allocation between the dual actuators(aerodynamic control surfaces and lateral pulse jets) can beobtained by setting different 119876 and 119877 Since both logicaland continuous variables are involved in (41) the aboveon-line optimization problem is a mixed integer quadraticprogramming (MIQP) problemwhich can be solved by usingthe hybrid MPC toolbox of MATLAB [14 15]

Remark 2 In an explicit MPC controller the main factorsaffecting the number of subregions include the systemdimen-sion the predictive horizon and the number of constrains Inthis paper the blendedmissile with aerodynamic control sur-faces and lateral jets is investigated To deal with the discreteproperty of the lateral jet forces some logical variables areintroduced for MPC controller design which increases thenumber of subregions inevitably

5 Numerical Simulations

51 Case 1 Numerical simulations are performed to ver-ify the feasibility of the proposed method in this sectionThe involved parameters are given in Table 1 The weightmatrices are set as 119876 = 10 119877120575

119911

= 0025 and 119877119865119910

=

diag(0005 0005 0005) Suppose the initial state is 1205720 =0 1199030 = 0 In order to avoid excessive computational load wechoose predictive horizon119873 = 2 and control horizon119872 = 2The command for angle of attack is 120572119888 = 02 rad

Explicit form of the optimal controller is provided byhybrid toolbox which is not given here for simplicity Insteadwe present the partition results associated with the controllerin Figure 6 where 4190 subregions are obtained and for eachsubregion a control law of the form 119906 = 119870119909 + 119887 is givenAs a benefit of using explicit MPC the controller parameters119870 and 119887 for all the subregions are obtained simultaneouslyonce the optimization problem is solved This property isobviously different from that of gain-scheduled controller(where the controller parameters for different regions areseparately determined)

However by a simple calculation we conclude that thesesubregions donot require toomuch storage space (the neededstorage space is approximately 14 times 8 times 4190 = 450KB)Moreover it should be noted that when the algorithm isimplemented in practice much more time will be needed toidentify which subregion the current states enter into This isanother important factor that affects the computational costof the proposed algorithm

The simulation results are shown in Figures 7ndash9 It isseen from Figure 7 that the actual angle of attack tracksthe command in less than 01 seconds As shown in Figures8 and 9 the aerodynamic force converges to a constant astime increases while the jets are only activated during thebeginning period (when the tracking error is obvious)

International Journal of Aerospace Engineering 11

58

68

73

108

174

180

245

252

254

270

271

272

273

274

276

277

299

300

301

302

06

04

02

0

minus02

minus04

Elev

ator

defl

ectio

n (r

ad)

Angle of attack (rad)minus04 minus02 0 02 04 06

1634

1635

1638

1648

1649

1650

1664

1673

1677

1678

1679

1712

1726

1743

1744

1746

1748

1751

1752

1983

1985

1993

2011

Late

ral f

orce

(N)

Angle of attack (rad)minus04 minus02 0 02 04 06

1000

800

600

400

200

0

minus200

minus400

minus600

minus800

minus1000

Polyhedral partitionmdash4190 regions Polyhedral partitionmdash4190 regions

Figure 6 The partition associated with the explicit controller

025

02

015

01

005

0

ActualCommand

0 02 04 06 08 1

Time (s)

Ang

le o

f atta

ck (r

ad)

Figure 7 The response of angle of attack

52 Case 2 The results for the case with a different weight119877120575119911

= 0005 are shown in Figures 10ndash12 It is seen that adifferent control allocation result is obtained

04

02

0

minus02

minus04

minus060 02 04 06 08 1

Time (s)

Elev

ator

defl

ectio

n (r

ad)

Figure 8 The response of elevator deflection

53 Case 3 In this case a different command 120572119888 =

02 cos(05120587119905) rad is considered and the weight matrices arethe same as the ones for Case 1 The simulation results areshown in Figures 13ndash15

It is seen from Figure 13 that asymptotic tracking is alsoachievedThemethod proposed in this paper can realize both

12 International Journal of Aerospace Engineering

1

05

0

minus05

minus1

times104

0 02 04 06 08 1

Time (s)

Late

ral f

orce

(N)

Figure 9 The response of lateral force

025

02

015

01

005

0

Ang

le o

f atta

ck (r

ad)

0 02 04 06 08 1

Time (s)

ActualCommand

Figure 10 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 02 04 06 08 1

Time (s)

Figure 11 The response of elevator deflection

1

05

0

minus05

minus1

Late

ral f

orce

(N)

times104

0 02 04 06 08 1

Time (s)

Figure 12 The response of lateral force

03

025

02

015

01

005

0

minus005

minus01

minus015

minus02

Ang

le o

f atta

ck (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

ActualCommand

Figure 13 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

Figure 14 The response of elevator deflection

International Journal of Aerospace Engineering 13

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1

Late

ral f

orce

(N)

0 05 1 15 2 25 3 35 4

Time (s)

times104

Figure 15 The response of lateral force

fast tracking command and control allocation It is seen fromFigure 15 that the jets are activated while the tracking erroris obvious or command is varying rapidly

6 Conclusion

An autopilot design method for a missile with aerodynamiccontrol surfaces and lateral jets is presented in this paperThe nonlinear attitude control model is reduced to an MLDmodel Meanwhile the lateral force is described as linearcombination of logical variables due to the discrete values oflateral force Then the whole MLD model of attitude controlsystem is derived Autopilot design is accomplished usinghybrid MPC method By setting the related weighted coeffi-cients in index function the control allocation is obtainedMoreover numerical simulations are performed under thedifferent conditions the performance of tracking the attitudecommand and control allocation is verified and the explicitform of the control law can be obtained as well

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research presented in this document is supportedby the National Natural Science Foundation of Chinaunder Grant nos 61104193 61203191 and 61333001 theFundamental Research Funds for the Central Universities(HITNSRIF2012032) the Program for IBRSEM in HarbinInstitute of Technology under Grant HITIBRSEMA201415and the Foundation of Supporting Technology for Aerospaceunder Grant 2014-HT-HGD7

References

[1] R Hirokawa K Sato and S Manabe ldquoAutopilot design fora missile with reaction-jet using coefficient diagram methodrdquoin Proceedings of the AIAA Guidance Navigation and ControlConference and Exhibit 2001 August 2001

[2] P K Menon and V R Iragavarapu Adaptive Techniques forMultiple Actuator Blending Defense Technical InformationCenter 1998

[3] A G Munson and M W Garbrick ldquoJet interaction investiga-tionrdquo Report OR-9933 Martin Marietta Orlando Fla USA1969

[4] M Graham and P Weinacht ldquoNumerical simulation of lateralcontrol jetsrdquo in Proceedings of the 37th Aerospace SciencesMeeting and Exhibit 1999 AIAA Paper

[5] J S Shamma and J R Cloutier ldquoGain-scheduledmissile autopi-lot design using linear parameter varying transformationsrdquoJournal of Guidance Control and Dynamics vol 16 no 2 pp256ndash263 1993

[6] M Innocenti and A Thukral ldquoSimultaneous reaction jet andaerodynamic control of missile systemsrdquo AlAA 93-3739-CP1993

[7] D B Ridgely Y Lee and T Fanciullo ldquoDual aeropropulsivemissile control-optimal control and control allocationrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit Keystone Colo USA 2006

[8] D B Ridgely D Drake L Triplett and C Geise ldquoDynamiccontrol allocation of a missile with tails and reaction jetsrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 3158ndash3189 August 2007

[9] D Q Mayne J B Rawlings C V Rao and P O ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[10] A Bemporad F Borrelli and M Morari ldquoModel predictivecontrol based on linear programmingmdashthe explicit solutionrdquoIEEE Transactions on Automatic Control vol 47 no 12 pp1974ndash1985 2002

[11] C Jouannet and P Krus ldquoModelling of high angle of attackaerodynamicrdquo in Proceedings of the 25th AIAA Applied Aerody-namics Conference pp 1481ndash1516 June 2007

[12] S Venugopal and M Krishnamurthy ldquoMissile aerodynamics athigh angles of attack a prediction coderdquo Journal of Spacecraftand Rockets vol 32 no 2 pp 263ndash269 1995

[13] A Bemporad ldquoEfficient conversion of mixed logical dynamicalsystems into an equivalent piecewise affine formrdquo IEEE Trans-actions on Automatic Control vol 49 no 5 pp 832ndash838 2004

[14] F D Torrisi andA Bemporad ldquoHYSDELmdasha tool for generatingcomputational hybrid models for analysis and synthesis prob-lemsrdquo IEEE Transactions on Control Systems Technology vol 12no 2 pp 235ndash249 2004

[15] A Bemporad W P M H Heemels and B De SchutterldquoOn hybrid systems and closed-loop MPC systemsrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 863ndash8692002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Aerospace Engineering 11

58

68

73

108

174

180

245

252

254

270

271

272

273

274

276

277

299

300

301

302

06

04

02

0

minus02

minus04

Elev

ator

defl

ectio

n (r

ad)

Angle of attack (rad)minus04 minus02 0 02 04 06

1634

1635

1638

1648

1649

1650

1664

1673

1677

1678

1679

1712

1726

1743

1744

1746

1748

1751

1752

1983

1985

1993

2011

Late

ral f

orce

(N)

Angle of attack (rad)minus04 minus02 0 02 04 06

1000

800

600

400

200

0

minus200

minus400

minus600

minus800

minus1000

Polyhedral partitionmdash4190 regions Polyhedral partitionmdash4190 regions

Figure 6 The partition associated with the explicit controller

025

02

015

01

005

0

ActualCommand

0 02 04 06 08 1

Time (s)

Ang

le o

f atta

ck (r

ad)

Figure 7 The response of angle of attack

52 Case 2 The results for the case with a different weight119877120575119911

= 0005 are shown in Figures 10ndash12 It is seen that adifferent control allocation result is obtained

04

02

0

minus02

minus04

minus060 02 04 06 08 1

Time (s)

Elev

ator

defl

ectio

n (r

ad)

Figure 8 The response of elevator deflection

53 Case 3 In this case a different command 120572119888 =

02 cos(05120587119905) rad is considered and the weight matrices arethe same as the ones for Case 1 The simulation results areshown in Figures 13ndash15

It is seen from Figure 13 that asymptotic tracking is alsoachievedThemethod proposed in this paper can realize both

12 International Journal of Aerospace Engineering

1

05

0

minus05

minus1

times104

0 02 04 06 08 1

Time (s)

Late

ral f

orce

(N)

Figure 9 The response of lateral force

025

02

015

01

005

0

Ang

le o

f atta

ck (r

ad)

0 02 04 06 08 1

Time (s)

ActualCommand

Figure 10 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 02 04 06 08 1

Time (s)

Figure 11 The response of elevator deflection

1

05

0

minus05

minus1

Late

ral f

orce

(N)

times104

0 02 04 06 08 1

Time (s)

Figure 12 The response of lateral force

03

025

02

015

01

005

0

minus005

minus01

minus015

minus02

Ang

le o

f atta

ck (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

ActualCommand

Figure 13 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

Figure 14 The response of elevator deflection

International Journal of Aerospace Engineering 13

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1

Late

ral f

orce

(N)

0 05 1 15 2 25 3 35 4

Time (s)

times104

Figure 15 The response of lateral force

fast tracking command and control allocation It is seen fromFigure 15 that the jets are activated while the tracking erroris obvious or command is varying rapidly

6 Conclusion

An autopilot design method for a missile with aerodynamiccontrol surfaces and lateral jets is presented in this paperThe nonlinear attitude control model is reduced to an MLDmodel Meanwhile the lateral force is described as linearcombination of logical variables due to the discrete values oflateral force Then the whole MLD model of attitude controlsystem is derived Autopilot design is accomplished usinghybrid MPC method By setting the related weighted coeffi-cients in index function the control allocation is obtainedMoreover numerical simulations are performed under thedifferent conditions the performance of tracking the attitudecommand and control allocation is verified and the explicitform of the control law can be obtained as well

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research presented in this document is supportedby the National Natural Science Foundation of Chinaunder Grant nos 61104193 61203191 and 61333001 theFundamental Research Funds for the Central Universities(HITNSRIF2012032) the Program for IBRSEM in HarbinInstitute of Technology under Grant HITIBRSEMA201415and the Foundation of Supporting Technology for Aerospaceunder Grant 2014-HT-HGD7

References

[1] R Hirokawa K Sato and S Manabe ldquoAutopilot design fora missile with reaction-jet using coefficient diagram methodrdquoin Proceedings of the AIAA Guidance Navigation and ControlConference and Exhibit 2001 August 2001

[2] P K Menon and V R Iragavarapu Adaptive Techniques forMultiple Actuator Blending Defense Technical InformationCenter 1998

[3] A G Munson and M W Garbrick ldquoJet interaction investiga-tionrdquo Report OR-9933 Martin Marietta Orlando Fla USA1969

[4] M Graham and P Weinacht ldquoNumerical simulation of lateralcontrol jetsrdquo in Proceedings of the 37th Aerospace SciencesMeeting and Exhibit 1999 AIAA Paper

[5] J S Shamma and J R Cloutier ldquoGain-scheduledmissile autopi-lot design using linear parameter varying transformationsrdquoJournal of Guidance Control and Dynamics vol 16 no 2 pp256ndash263 1993

[6] M Innocenti and A Thukral ldquoSimultaneous reaction jet andaerodynamic control of missile systemsrdquo AlAA 93-3739-CP1993

[7] D B Ridgely Y Lee and T Fanciullo ldquoDual aeropropulsivemissile control-optimal control and control allocationrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit Keystone Colo USA 2006

[8] D B Ridgely D Drake L Triplett and C Geise ldquoDynamiccontrol allocation of a missile with tails and reaction jetsrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 3158ndash3189 August 2007

[9] D Q Mayne J B Rawlings C V Rao and P O ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[10] A Bemporad F Borrelli and M Morari ldquoModel predictivecontrol based on linear programmingmdashthe explicit solutionrdquoIEEE Transactions on Automatic Control vol 47 no 12 pp1974ndash1985 2002

[11] C Jouannet and P Krus ldquoModelling of high angle of attackaerodynamicrdquo in Proceedings of the 25th AIAA Applied Aerody-namics Conference pp 1481ndash1516 June 2007

[12] S Venugopal and M Krishnamurthy ldquoMissile aerodynamics athigh angles of attack a prediction coderdquo Journal of Spacecraftand Rockets vol 32 no 2 pp 263ndash269 1995

[13] A Bemporad ldquoEfficient conversion of mixed logical dynamicalsystems into an equivalent piecewise affine formrdquo IEEE Trans-actions on Automatic Control vol 49 no 5 pp 832ndash838 2004

[14] F D Torrisi andA Bemporad ldquoHYSDELmdasha tool for generatingcomputational hybrid models for analysis and synthesis prob-lemsrdquo IEEE Transactions on Control Systems Technology vol 12no 2 pp 235ndash249 2004

[15] A Bemporad W P M H Heemels and B De SchutterldquoOn hybrid systems and closed-loop MPC systemsrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 863ndash8692002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 International Journal of Aerospace Engineering

1

05

0

minus05

minus1

times104

0 02 04 06 08 1

Time (s)

Late

ral f

orce

(N)

Figure 9 The response of lateral force

025

02

015

01

005

0

Ang

le o

f atta

ck (r

ad)

0 02 04 06 08 1

Time (s)

ActualCommand

Figure 10 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 02 04 06 08 1

Time (s)

Figure 11 The response of elevator deflection

1

05

0

minus05

minus1

Late

ral f

orce

(N)

times104

0 02 04 06 08 1

Time (s)

Figure 12 The response of lateral force

03

025

02

015

01

005

0

minus005

minus01

minus015

minus02

Ang

le o

f atta

ck (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

ActualCommand

Figure 13 The response of angle of attack

06

04

02

0

minus02

minus04

minus06

minus08

Elev

ator

defl

ectio

n (r

ad)

0 05 1 15 2 25 3 35 4

Time (s)

Figure 14 The response of elevator deflection

International Journal of Aerospace Engineering 13

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1

Late

ral f

orce

(N)

0 05 1 15 2 25 3 35 4

Time (s)

times104

Figure 15 The response of lateral force

fast tracking command and control allocation It is seen fromFigure 15 that the jets are activated while the tracking erroris obvious or command is varying rapidly

6 Conclusion

An autopilot design method for a missile with aerodynamiccontrol surfaces and lateral jets is presented in this paperThe nonlinear attitude control model is reduced to an MLDmodel Meanwhile the lateral force is described as linearcombination of logical variables due to the discrete values oflateral force Then the whole MLD model of attitude controlsystem is derived Autopilot design is accomplished usinghybrid MPC method By setting the related weighted coeffi-cients in index function the control allocation is obtainedMoreover numerical simulations are performed under thedifferent conditions the performance of tracking the attitudecommand and control allocation is verified and the explicitform of the control law can be obtained as well

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research presented in this document is supportedby the National Natural Science Foundation of Chinaunder Grant nos 61104193 61203191 and 61333001 theFundamental Research Funds for the Central Universities(HITNSRIF2012032) the Program for IBRSEM in HarbinInstitute of Technology under Grant HITIBRSEMA201415and the Foundation of Supporting Technology for Aerospaceunder Grant 2014-HT-HGD7

References

[1] R Hirokawa K Sato and S Manabe ldquoAutopilot design fora missile with reaction-jet using coefficient diagram methodrdquoin Proceedings of the AIAA Guidance Navigation and ControlConference and Exhibit 2001 August 2001

[2] P K Menon and V R Iragavarapu Adaptive Techniques forMultiple Actuator Blending Defense Technical InformationCenter 1998

[3] A G Munson and M W Garbrick ldquoJet interaction investiga-tionrdquo Report OR-9933 Martin Marietta Orlando Fla USA1969

[4] M Graham and P Weinacht ldquoNumerical simulation of lateralcontrol jetsrdquo in Proceedings of the 37th Aerospace SciencesMeeting and Exhibit 1999 AIAA Paper

[5] J S Shamma and J R Cloutier ldquoGain-scheduledmissile autopi-lot design using linear parameter varying transformationsrdquoJournal of Guidance Control and Dynamics vol 16 no 2 pp256ndash263 1993

[6] M Innocenti and A Thukral ldquoSimultaneous reaction jet andaerodynamic control of missile systemsrdquo AlAA 93-3739-CP1993

[7] D B Ridgely Y Lee and T Fanciullo ldquoDual aeropropulsivemissile control-optimal control and control allocationrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit Keystone Colo USA 2006

[8] D B Ridgely D Drake L Triplett and C Geise ldquoDynamiccontrol allocation of a missile with tails and reaction jetsrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 3158ndash3189 August 2007

[9] D Q Mayne J B Rawlings C V Rao and P O ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[10] A Bemporad F Borrelli and M Morari ldquoModel predictivecontrol based on linear programmingmdashthe explicit solutionrdquoIEEE Transactions on Automatic Control vol 47 no 12 pp1974ndash1985 2002

[11] C Jouannet and P Krus ldquoModelling of high angle of attackaerodynamicrdquo in Proceedings of the 25th AIAA Applied Aerody-namics Conference pp 1481ndash1516 June 2007

[12] S Venugopal and M Krishnamurthy ldquoMissile aerodynamics athigh angles of attack a prediction coderdquo Journal of Spacecraftand Rockets vol 32 no 2 pp 263ndash269 1995

[13] A Bemporad ldquoEfficient conversion of mixed logical dynamicalsystems into an equivalent piecewise affine formrdquo IEEE Trans-actions on Automatic Control vol 49 no 5 pp 832ndash838 2004

[14] F D Torrisi andA Bemporad ldquoHYSDELmdasha tool for generatingcomputational hybrid models for analysis and synthesis prob-lemsrdquo IEEE Transactions on Control Systems Technology vol 12no 2 pp 235ndash249 2004

[15] A Bemporad W P M H Heemels and B De SchutterldquoOn hybrid systems and closed-loop MPC systemsrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 863ndash8692002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Aerospace Engineering 13

1

08

06

04

02

0

minus02

minus04

minus06

minus08

minus1

Late

ral f

orce

(N)

0 05 1 15 2 25 3 35 4

Time (s)

times104

Figure 15 The response of lateral force

fast tracking command and control allocation It is seen fromFigure 15 that the jets are activated while the tracking erroris obvious or command is varying rapidly

6 Conclusion

An autopilot design method for a missile with aerodynamiccontrol surfaces and lateral jets is presented in this paperThe nonlinear attitude control model is reduced to an MLDmodel Meanwhile the lateral force is described as linearcombination of logical variables due to the discrete values oflateral force Then the whole MLD model of attitude controlsystem is derived Autopilot design is accomplished usinghybrid MPC method By setting the related weighted coeffi-cients in index function the control allocation is obtainedMoreover numerical simulations are performed under thedifferent conditions the performance of tracking the attitudecommand and control allocation is verified and the explicitform of the control law can be obtained as well

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research presented in this document is supportedby the National Natural Science Foundation of Chinaunder Grant nos 61104193 61203191 and 61333001 theFundamental Research Funds for the Central Universities(HITNSRIF2012032) the Program for IBRSEM in HarbinInstitute of Technology under Grant HITIBRSEMA201415and the Foundation of Supporting Technology for Aerospaceunder Grant 2014-HT-HGD7

References

[1] R Hirokawa K Sato and S Manabe ldquoAutopilot design fora missile with reaction-jet using coefficient diagram methodrdquoin Proceedings of the AIAA Guidance Navigation and ControlConference and Exhibit 2001 August 2001

[2] P K Menon and V R Iragavarapu Adaptive Techniques forMultiple Actuator Blending Defense Technical InformationCenter 1998

[3] A G Munson and M W Garbrick ldquoJet interaction investiga-tionrdquo Report OR-9933 Martin Marietta Orlando Fla USA1969

[4] M Graham and P Weinacht ldquoNumerical simulation of lateralcontrol jetsrdquo in Proceedings of the 37th Aerospace SciencesMeeting and Exhibit 1999 AIAA Paper

[5] J S Shamma and J R Cloutier ldquoGain-scheduledmissile autopi-lot design using linear parameter varying transformationsrdquoJournal of Guidance Control and Dynamics vol 16 no 2 pp256ndash263 1993

[6] M Innocenti and A Thukral ldquoSimultaneous reaction jet andaerodynamic control of missile systemsrdquo AlAA 93-3739-CP1993

[7] D B Ridgely Y Lee and T Fanciullo ldquoDual aeropropulsivemissile control-optimal control and control allocationrdquo inProceedings of the AIAA Guidance Navigation and ControlConference and Exhibit Keystone Colo USA 2006

[8] D B Ridgely D Drake L Triplett and C Geise ldquoDynamiccontrol allocation of a missile with tails and reaction jetsrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 3158ndash3189 August 2007

[9] D Q Mayne J B Rawlings C V Rao and P O ScokaertldquoConstrained model predictive control stability and optimal-ityrdquo Automatica vol 36 no 6 pp 789ndash814 2000

[10] A Bemporad F Borrelli and M Morari ldquoModel predictivecontrol based on linear programmingmdashthe explicit solutionrdquoIEEE Transactions on Automatic Control vol 47 no 12 pp1974ndash1985 2002

[11] C Jouannet and P Krus ldquoModelling of high angle of attackaerodynamicrdquo in Proceedings of the 25th AIAA Applied Aerody-namics Conference pp 1481ndash1516 June 2007

[12] S Venugopal and M Krishnamurthy ldquoMissile aerodynamics athigh angles of attack a prediction coderdquo Journal of Spacecraftand Rockets vol 32 no 2 pp 263ndash269 1995

[13] A Bemporad ldquoEfficient conversion of mixed logical dynamicalsystems into an equivalent piecewise affine formrdquo IEEE Trans-actions on Automatic Control vol 49 no 5 pp 832ndash838 2004

[14] F D Torrisi andA Bemporad ldquoHYSDELmdasha tool for generatingcomputational hybrid models for analysis and synthesis prob-lemsrdquo IEEE Transactions on Control Systems Technology vol 12no 2 pp 235ndash249 2004

[15] A Bemporad W P M H Heemels and B De SchutterldquoOn hybrid systems and closed-loop MPC systemsrdquo IEEETransactions on Automatic Control vol 47 no 5 pp 863ndash8692002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of