research article adaptive integrated guidance and...

Research ArticleAdaptive Integrated Guidance and Fault Tolerant ControlUsing Backstepping and Sliding Mode

Mohsen Fathi Jegarkandi Asghar Ashrafifar and Reza Mohsenipour

Department of Aerospace Engineering Sharif University of Technology PO Box 11365-11155 Tehran Iran

Correspondence should be addressed to Asghar Ashrafifar ashrafimutgmailcom

Received 2 June 2015 Revised 7 September 2015 Accepted 21 September 2015

Academic Editor Mahmut Reyhanoglu

Copyright copy 2015 Mohsen Fathi Jegarkandi et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A newmethod of integrated guidance and control for homingmissiles with actuator fault against manoeuvring targets is proposedModel of the integrated guidance and control system in the pitch plane with actuator fault and some uncertainty is developed Acontrol law using combination of adaptive backstepping and sliding mode approaches is designed to achieve interception in thepresence of bounded uncertainties and actuator fault Simulation results show that new approach has better performance thanadaptive backstepping and has good performance in the presence of actuator fault

1 Introduction

The traditional way to design a missile guidance and controlsystem is to form subsystems separately followed by inte-grating them This method presents successful results anddemonstrates some remarkable performance in designingseveral missile guidance and control systems However sincein this method interactive relationships between the cooper-ating subsystems cannot be completely exploited the overallsystem performance may be constrained Therefore a designmethod named as integrated guidance and control (IGC)system is provided by incorporating some control theories toenhance the performance of the whole system

There are a variety of methods to solve IGC problem thatsomeof themare presented in the following In [1] an adaptivenonlinear IGC approach is proposed by adopting a back-stepping scheme for missile-target engagement model withuncertainties without fault Reference [2] has addressed IGCproblem using nonlinear optimal control technique whichis called 120579-119863 method In work [3] an integrated guidanceand autopilot system is designed for homing missiles againstground fixed targets To this intent an integrated model inthe pitch plane is formulated and then the adaptive nonlinearcontrol law is designed by adopting the sliding mode control

approach Also other control theories have been employedin the design of guidance laws including 119867

infincontrol [4]

variable structure control [5] feedback linearization scheme[6] slidingmode control [7] and adaptive fuzzy slidingmodecontrol [8]

A fault tolerant control system is capable of controlling asystem with satisfactory performance even if one or severalfaults occur in the system Fault tolerant control systems canbe classified into two main families passive fault tolerantcontrollers and active fault tolerant controllers In a passivefault tolerant controller deviations of the plant parametersfrom their actual values or deviations of the actuators fromtheir expected position may be efficiently compensated bya fixed robust feedback controller [9] Many control designlaws described by fault tolerant control have been proposedin [10ndash12] for different practical systems In [13] some faulttolerant controllers designed guidance strategy in the pres-ence of actuator faults Furthermore authors of [14] haveproposed a fault tolerant control for induction motors basedon backstepping strategy

In this paper first IGC system in the pitch plane withan actuator fault and some uncertainties is modelled Sincethe actuator fault uncertainty is multiplied by input wemust use some scheme of such sliding mode Then a control

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2015 Article ID 253478 7 pageshttpdxdoiorg1011552015253478

2 International Journal of Aerospace Engineering

Missile

Target

LOS

R

aM120601M

120582

VT

aT120601T

VM

Figure 1 Two-dimensional missile-target engagement geometry

law using combination of adaptive backstepping and slidingmode is designed to achieve the interception in the presenceof bounded uncertainties and actuator fault The proposedcontrol law is compared with control law of [1] for a freefault system Then the performance of designed law controlis investigated for the faulty system

The paper is organized as follows In Section 2 missile-target engagement geometry will be derived in the pitchplane In Section 3 actuator fault is modelled In Section 4control law is derived using the proposed new approach InSection 5 control law is derived for the faulty system Sim-ulation results and their analysis are presented in Section 6Finally a conclusion is made in Section 7

2 Engagement of Model Derivation

In this section the integrated guidance and control modelfor two-dimensional point mass missile-target engagementgeometry is driven in the pitch plane [1] In order to establisha better understanding of the concept engagement strategy isillustrated in Figure 1 Consider an inertial coordinate systemfixed to the surface of a flat-earth model The correspondingkinematic equations of motion are given as follows

= 119881119879cos (120582 minus 120601

119879) minus 119881119872cos (120582 minus 120601

119872)

= minus119881119879

119877sin (120582 minus 120601

119879) +

119881119872

119877sin (120582 minus 120601

119872)

(1)

where 119877 is the range along the line of sight (LOS) 120582 is theLOS angle 119881

119872and 119881

119879are the speed of missile and target

respectively that both are assumed to be constant 120601119872

and120601119879are missile and targetrsquos flight path angles respectively

Differentiating yields

= minus2119881119877

119877 +

119886119879120582

119877minus cos (120582 minus 120601

119872)119886119872

119877 (2)

where 119881119877= 119886

119872and 119886119879are normal acceleration of missile

and target respectively and 119886119879120582

is the projection of boundedtarget acceleration perpendicular to the line of sight

For modelling of planar missile dynamics proposedmodel in [1 7] is used as follows

120579 = 120572 + 120601119872997904rArr

= 120579 minus119871 (120572 120575)

119898119881119872

120579=119902

997888997888997888rarr

= 119902 minus119871 (120572 120575

119911)

119898119881119872

119902 =119872 (120572 119902 120575

119911)

119869119911

(3)

Therefore

= 119902 minus119871 (120572 120575

119911)

119898119881119872

120579 = 119902

119902 =119872 (120572 119902 120575

119911)

119869119911

(4)

where 120579 is the pitch attitude angle and 119902 is the missile pitchrate 120575

119911is deflection angle for pitch control and 119898 and 119869

119911are

the mass and moment of inertia respectively Also119872 is thepitch moment and 119871 is the lift forces where

119871

119898= (

119871120572

119898)120572 + Δ

119871= 119897120572120572 + Δ

119871

119872

119869119911

=1

119869119911

(119872120572120572 +119872

119902119902 +119872

120575119911

120575119911) + Δ

119872

= 119898120572120572 + 119898

119902119902 + 119898

120575119911

120575119911+ Δ119872

(5)

Therefore

= 119902 minus119897120572

119881119872

120572 + Δ120572

120579 = 119902

119902 = 119898120572120572 + 119898

119902119902 + 119898

120575119911

120575119911+ Δ119872

(6)

where 119897120572= 120597119897120597120572119898

119909= 120597119898120597119909 and Δ

120572= minusΔ119871119881119872in which

Δ119871and Δ

119872are bounded uncertainties On the other hand

missile acceleration can be written as

119886119872= 119881119872

120601119898= 119897120572120572 minus 119881119872Δ120572 (7)

By substituting (7) into (2) and defining 1199091= 119909

2= 120572 and

1199093= 119902 one can obtain

1= 119886111199091+ cos (120582 + 119909

2minus 120579) 119886

121199092+ Δ119902

2= 119886221199092+ 1199093+ Δ120572

3= 119886321199092+ 119886331199093+ 119887119906 + Δ

119872

(8)

International Journal of Aerospace Engineering 3

where

11988611= minus2

119881119877

119877

11988612= minus

119897120572

119877

11988622= minus

119897120572

119881119872

11988632= 119898120572

11988633= 119898119902

119887 = 119898120575119911

Δ119902=

119886119879119902

119877+cos (120582 minus 120601

119872) 119881119872Δ120572

119877

119906 = 120575119911

(9)

The above closed-loop system dynamics are actuator fault-free In next section closed-loop system dynamics withactuator fault is driven

3 Modeling of Actuator Fault

A loss of effectiveness actuator fault is considered in thiswork For this the desired control input 120575

119911was disconnected

and replaced by a faulty control signal 120575119911that takes control

over the plant The true input of the plant can be written as[9]

119906 = 120575119911+ 120590 (120575

119911minus 120575119911) (10)

where

120590 =

1 if actuator fails

0 otherwise(11)

The faulty control signal 120575119911can be expressed as follows

120575119911= 119891120575119911 (12)

where 119891 is a random constant that 03 le 119891 lt 1 If 119891 becomesless than 03 the actuator is saturated Therefore a faultysystem can be written

1= 119886111199091+ cos (120582 + 119909

2minus 120579) 119886

121199092+ Δ119902

2= 119886221199092+ 1199093+ Δ120572

3= 119886321199092+ 119886331199093+ 119887119891119906 + Δ

119872

(13)

In the following sections first a control law using newapproach is derived for a free fault system so one can be ableto compare the results of the new approach with the results of[1] Next the given approach is applied to the faulty system

4 Integrated Guidance and ControlDesign Using Combination of SlidingControl and Backstepping

System (8) is not in the standard form of common backstep-ping procedure because 119909

2appeared in the nonlinear term

of cos(119902 + 1199092minus 120579)119886

12 In order to deal with this situation

[1] has introduced an adaptive backstepping control schemeto design an IGC law that can zero LOS rate and maintainstability of the overall system

In this paper sliding mode control is used to improvethe performance of proposed IGC in [1] and we can consideractuator fault To this purpose suppose that virtual controls1205721(1199091) and 120572

2(1199091 1199092) in two backsteps are obtained such that

they keep 1199091around zero and guarantee 119909

1and 119909

2= 1199092minus

1205721to be input-to-state stable with respect to uncertainties

respectively [1] consider

1199092= 1205721(1199091) = minus

1

11988612

1198961199091

1199093= 1205722(1199091 1199092) = minus(119888

2+

1

212057521

)1199092

minus cos (120582 + 1199092minus 120579) 119886

121199091minus 119886221199092+1205971205721

120597119905

+1205971205721

1205971199091

(119886111199091+ cos (120582 + 119909

2minus 120579) 119886

121199092

minus1

212057522

1199092

1205971205721

1205971199091

)

(14)

where 119896 1198882 12057512 and the expressions of partial derivatives

12059712057211205971199091and 120597120572

1120597119905 can be found in [1]

Now instead of using of backstep again for calculating 119906sliding mode control is used To this purpose sliding surface119878 is selected as

119878 = 1199093minus 1205722(1199091 1199092) (15)

It is obvious that if a dynamic state feedback control law isdesigned such that the trajectories of the closed-loop systemare driven on the sliding surface and kept along it then theguidance strategy can be achieved Therefore differentiationof 119878must be zero

119878 = 3minus 2(1199091 1199092)

= 119886321199092+ 119886331199093+ 119887119906 + Δ

119872minus 2(1199091 1199092)

(16)

Then

119906eq = minus119887minus1

[119886321199092+ 119886331199093minus 2(1199091 1199092)] (17)

With the given sliding surface the control law is obtained as

119906 = 119906eq +119872

119887sign (119878)

= minus119887minus1

[119886321199092+ 119886331199093minus 2(1199091 1199092)] +

119872

119887sign (119878)

(18)

where |Δ119872| le 119872


0 5 10 15 20 25 30minus100

minus50

0

50

100

150

Time (s)

Missile acceleration

BacksteppingSliding + backstepping

0 5 10 15 20 25 30minus08

minus06

minus04

minus02

0

02

04

06

Time (s)

Missile deflection command

Backstepping [1]Sliding mode + backstepping

120575(r

ad)

g(m

s2)

Figure 2 Comparing missile acceleration and deflection angle for free fault system

In order to prevent chattering one can write

119906 = minus119887minus1

[119886321199092+ 119886331199093minus 2(1199091 1199092)]

+119872

119887tanh (120576119878) 120576 ≫ 0

(19)

5 Fault Tolerant Control Design

For the faulty system sliding surface 119878 is selected as beforeDifferentiation of 119878must be zero then

119878 = 119886321199092+ 119886331199093+ 119865119906 minus

2(1199091 1199092) = 0

where 119865 = 119887119891

(20)

Equation (20) can be rewritten as

119878 = 119886321199092+ 119886331199093+ 119865119906 + (119865 minus 119865) 119906 + Δ

119872

minus 2(1199091 1199092) = 0 where 119865 = 119887119891

(21)

where 119891 is the nominal value of 119891 By considering the inputas

119906 = 119906eq + V (22)

(21) can be expressed as follows

119878 = V + Δ (119909 V) (23)

where

Δ (119909 V) = Δ119872+ (119865 minus 119865) (119906eq +

V119865) (24)

119906eq =1

119865[minus119886321199092minus 119886331199093+ 2(1199091 1199092)] (25)

By substituting (25) in (24)

Δ (119909 V) = Δ119872minus(119865 minus 119865)

119865119886321199092minus(119865 minus 119865)

119865119886331199093

+(119865 minus 119865)

119865

10038161003816100381610038162 (1199091 1199092)1003816100381610038161003816 +

(119865 minus 119865)

119865V

(26)

From (26) it follows that

Δ (119909 V) le 119872 + 1198961

100381610038161003816100381611990911003816100381610038161003816 + 1198962

100381610038161003816100381611990921003816100381610038161003816 + 1198963

100381610038161003816100381611990931003816100381610038161003816 + 119896 |V| (27)

where11989611205722

le 1198961

100381610038161003816100381610038161003816100381610038161003816

119865 minus 119865

119865(11988632+ 11989621205722

)

100381610038161003816100381610038161003816100381610038161003816

le 1198962

100381610038161003816100381610038161003816100381610038161003816

119865 minus 119865

11986511988632

100381610038161003816100381610038161003816100381610038161003816

le 1198963

100381610038161003816100381610038161003816100381610038161003816

119865 minus 119865

119865

100381610038161003816100381610038161003816100381610038161003816

le 119896

10038161003816100381610038162 (1199091 1199092)1003816100381610038161003816 le 11989611205722

100381610038161003816100381611990911003816100381610038161003816 + 11989621205722

100381610038161003816100381611990921003816100381610038161003816

(28)

Values 11989611205722

and 11989621205722

can be calculated from [1] easily Bydefining

120573 (119909) = 119872 + 1198961

100381610038161003816100381611990911003816100381610038161003816 + 1198962

100381610038161003816100381611990921003816100381610038161003816 + 1198963

100381610038161003816100381611990931003816100381610038161003816 + 1198870 119887

0gt 0 (29)

control law is obtained as

119906 = 119906eq +120573 (119909)

119865 (1 minus 119896)tanh (120576119878) (30)

6 Numerical Results and Analysis

To evaluate performance of the proposed approach in thissection first the proposed approach is compared with the


0 5 10 15 20 25 30minus100

minus50

0

50

100

150

200 Missile acceleration

Time (s)0 5 10 15 20 25 30

minus20

minus10

0

10

20

30

40

50

Time (s)


Missile deflection command (deg)

120575(d

eg)

g(m

s2)

Figure 3 Missile acceleration and deflection angle for faulty system with 119891 = 09

0 5 10 15 20 25 300

500

1000

1500

2000

2500

3000

3500

4000Distance (m)

Time (s)

R (m

)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500Kinematic engagement geometric

MissileTarget

y1

(m)

x1 (m)

Figure 4 Relative range and interception trajectory for faulty system with 119891 = 09

0 5 10 15 20 25 30 35 40minus80

minus60

minus40

minus20

0

20

40

60

80Missile acceleration

Time (s)0 5 10 15 20 25 30 35 40

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Time (s)



120575(d

eg)

g(m

s2)

Figure 5 Missile acceleration and deflection for faulty system with 119891 = 03


0 5 10 15 20 25 30 35 400

500

1000

1500

2000

2500

3000

3500

4000 Distance (m)

Time (s)

R (m

)

0 2000 4000 6000 8000 10000 12000 14000minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500


MissileTarget

y1

(m)

x1 (m)


results of [1] for free fault system This comparison is shownin Figure 2 As it can be seen the proposed control law withless acceleration applied to the missile causes smaller missdistance This shows efficiency of the proposed approach touse for faulty system

MDBackstepping = 7 times 10minus4m

MDSliding + Backstepping = 2 times 10minus4m

(31)

The initial missile and target flight path angles their initialvelocities and the relative motion parameters can be foundin [1]

Next performance of the proposed control law is inves-tigated for faulty system For this purpose actuator effec-tiveness is selected between 30 and 100 in other words03 le 119891 lt 1 It is assumed that the actuator commands aretaken as a first-order lag system with a time constant 0001and limited by |120575

119911| lt 45 deg

As can be seen in Figures 3ndash6 the controllers are robustto achieve the missile interception in the presence of thetarget acceleration and actuator fault It is clear that the lessefficiency of actuator increases the time to go interceptiondistance and deflection angle for pitch control Also itincreases the control effort (Figure 7)

7 Conclusion

In this paper we introduce a new approach to design ofguidance laws for two-dimensional point mass missile-targetengagement in the presence of bounded uncertainties andactuator fault In proposed approach sliding mode controlis combined with adaptive backstepping to improve theperformance Numerical results show that the new controllaw has better performance than adaptive backstepping andhas good performance in faulty system

0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

16

18

20

Time (s)

f = 03f = 04f = 05f = 06

f = 07f = 08f = 09f = 1

intu2

Figure 7 Control effort comparison

Conflict of Interests

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

References

[1] H Yan XWang B Yu andH Ji ldquoAdaptive integrated guidanceand control based on backstepping and input-to-state stabilityrdquoAsian Journal of Control vol 16 no 2 pp 602ndash608 2014

[2] M Xin S N Balakrishnan and E J Ohlmeyer ldquoIntegratedguidance and control of missiles with methodrdquo IEEE Transac-tions on Control Systems Technology vol 14 no 6 pp 981ndash9922006


[3] M Hou and G Duan ldquoIntegrated guidance and control ofhoming missiles against ground fixed targetsrdquo Chinese Journalof Aeronautics vol 21 no 2 pp 162ndash168 2008

[4] C D Yang and H Y Chen ldquoNonlinear H-infinity robustguidance law for homingmissilesrdquo Journal of Guidance Controland Dynamics vol 21 no 6 pp 882ndash890 1998

[5] J Moon K Kim and Y Kim ldquoDesign of missile guidance lawvia variable structure controlrdquo Journal of Guidance Control andDynamics vol 24 no 4 pp 659ndash664 2001

[6] P K Menon and E J Ohlmeyer ldquoIntegrated design of agilemissile guidance and autopilot systemsrdquo Control EngineeringPractice vol 9 no 10 pp 1095ndash1106 2001

[7] T Shima M Idan and O M Golan ldquoSliding-mode controlfor integrated missile autopilot guidancerdquo Journal of GuidanceControl and Dynamics vol 29 no 2 pp 250ndash260 2006

[8] C-M Lin and C-F Hsu ldquoGuidance law design by adaptivefuzzy sliding-mode controlrdquo Journal of Guidance Control andDynamics vol 25 no 2 pp 248ndash256 2002

[9] G J J Ducard Fault-Tolerant Flight Control and GuidanceSystems Practical Methods for Small Unmanned Aerial VehiclesSpringer 2009

[10] M Benosman and K-Y Lum ldquoPassive actuatorsrsquo fault-tolerantcontrol for affine nonlinear systemsrdquo IEEE Transactions onControl Systems Technology vol 18 no 1 pp 152ndash163 2010

[11] Y Zhang and J Jiang ldquoFault tolerant control system designwith explicit consideration of performance degradationrdquo IEEETransactions on Aerospace and Electronic Systems vol 39 no 3pp 838ndash848 2003

[12] J Cieslak D Henry A Zolghadri and P Goupil ldquoDevelopmentof an active fault-tolerant flight control strategyrdquo Journal ofGuidance Control and Dynamics vol 31 no 1 pp 135ndash1472008

[13] Z Zhu Y Xia M Fu and S Wang ldquoFault tolerant controlfor missile guidance laws with finite-time convergencerdquo inProceedings of the Chinese Control and Decision Conference(CCDC rsquo11) pp 1288ndash1293 IEEE Mianyang China May 2011

[14] N Djeghali M Ghanes S Djennoune and J-P BarbotldquoBackstepping fault tolerant control based on second ordersliding mode observer application to induction motorsrdquo inProceedings of the 50th IEEEConference onDecision and Controland European Control Conference (CDC-ECC rsquo11) pp 4598ndash4603 Orlando Fla USA December 2011

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Missile

Target

LOS

R

aM120601M

120582

VT

aT120601T

VM

Figure 1 Two-dimensional missile-target engagement geometry

law using combination of adaptive backstepping and slidingmode is designed to achieve the interception in the presenceof bounded uncertainties and actuator fault The proposedcontrol law is compared with control law of [1] for a freefault system Then the performance of designed law controlis investigated for the faulty system

The paper is organized as follows In Section 2 missile-target engagement geometry will be derived in the pitchplane In Section 3 actuator fault is modelled In Section 4control law is derived using the proposed new approach InSection 5 control law is derived for the faulty system Sim-ulation results and their analysis are presented in Section 6Finally a conclusion is made in Section 7

2 Engagement of Model Derivation

In this section the integrated guidance and control modelfor two-dimensional point mass missile-target engagementgeometry is driven in the pitch plane [1] In order to establisha better understanding of the concept engagement strategy isillustrated in Figure 1 Consider an inertial coordinate systemfixed to the surface of a flat-earth model The correspondingkinematic equations of motion are given as follows

= 119881119879cos (120582 minus 120601

119879) minus 119881119872cos (120582 minus 120601

119872)

= minus119881119879

119877sin (120582 minus 120601

119879) +

119881119872

119877sin (120582 minus 120601

119872)

(1)

where 119877 is the range along the line of sight (LOS) 120582 is theLOS angle 119881

119872and 119881

119879are the speed of missile and target

respectively that both are assumed to be constant 120601119872

and120601119879are missile and targetrsquos flight path angles respectively

Differentiating yields

= minus2119881119877

119877 +

119886119879120582

119877minus cos (120582 minus 120601

119872)119886119872

119877 (2)

where 119881119877= 119886

119872and 119886119879are normal acceleration of missile

and target respectively and 119886119879120582

is the projection of boundedtarget acceleration perpendicular to the line of sight

For modelling of planar missile dynamics proposedmodel in [1 7] is used as follows

120579 = 120572 + 120601119872997904rArr

= 120579 minus119871 (120572 120575)

119898119881119872

120579=119902

997888997888997888rarr

= 119902 minus119871 (120572 120575

119911)

119898119881119872

119902 =119872 (120572 119902 120575

119911)

119869119911

(3)

Therefore

= 119902 minus119871 (120572 120575

119911)

119898119881119872

120579 = 119902

119902 =119872 (120572 119902 120575

119911)

119869119911

(4)

where 120579 is the pitch attitude angle and 119902 is the missile pitchrate 120575

119911is deflection angle for pitch control and 119898 and 119869

119911are

the mass and moment of inertia respectively Also119872 is thepitch moment and 119871 is the lift forces where

119871

119898= (

119871120572

119898)120572 + Δ

119871= 119897120572120572 + Δ

119871

119872

119869119911

=1

119869119911

(119872120572120572 +119872

119902119902 +119872

120575119911

120575119911) + Δ

119872

= 119898120572120572 + 119898

119902119902 + 119898

120575119911

120575119911+ Δ119872

(5)

Therefore

= 119902 minus119897120572

119881119872

120572 + Δ120572

120579 = 119902

119902 = 119898120572120572 + 119898

119902119902 + 119898

120575119911

120575119911+ Δ119872

(6)

where 119897120572= 120597119897120597120572119898

119909= 120597119898120597119909 and Δ

120572= minusΔ119871119881119872in which

Δ119871and Δ

119872are bounded uncertainties On the other hand

missile acceleration can be written as

119886119872= 119881119872

120601119898= 119897120572120572 minus 119881119872Δ120572 (7)

By substituting (7) into (2) and defining 1199091= 119909

2= 120572 and

1199093= 119902 one can obtain

1= 119886111199091+ cos (120582 + 119909

2minus 120579) 119886

121199092+ Δ119902

2= 119886221199092+ 1199093+ Δ120572

3= 119886321199092+ 119886331199093+ 119887119906 + Δ

119872

(8)


where

11988611= minus2

119881119877

119877

11988612= minus

119897120572

119877

11988622= minus

119897120572

119881119872

11988632= 119898120572

11988633= 119898119902

119887 = 119898120575119911

Δ119902=

119886119879119902

119877+cos (120582 minus 120601

119872) 119881119872Δ120572

119877

119906 = 120575119911

(9)







119906 = 120575119911+ 120590 (120575

119911minus 120575119911) (10)

where

120590 =

1 if actuator fails

0 otherwise(11)


120575119911= 119891120575119911 (12)


1= 119886111199091+ cos (120582 + 119909

2minus 120579) 119886

121199092+ Δ119902

2= 119886221199092+ 1199093+ Δ120572

3= 119886321199092+ 119886331199093+ 119887119891119906 + Δ

119872

(13)





of cos(119902 + 1199092minus 120579)119886






1and 119909

2= 1199092minus



1199092= 1205721(1199091) = minus

1

11988612

1198961199091

1199093= 1205722(1199091 1199092) = minus(119888

2+

1

212057521

)1199092

minus cos (120582 + 1199092minus 120579) 119886

121199091minus 119886221199092+1205971205721

120597119905

+1205971205721

1205971199091

(119886111199091+ cos (120582 + 119909

2minus 120579) 119886

121199092

minus1

212057522

1199092

1205971205721

1205971199091

)

(14)


12059712057211205971199091and 120597120572

1120597119905 can be found in [1]


119878 = 1199093minus 1205722(1199091 1199092) (15)


119878 = 3minus 2(1199091 1199092)

= 119886321199092+ 119886331199093+ 119887119906 + Δ

119872minus 2(1199091 1199092)

(16)

Then


[119886321199092+ 119886331199093minus 2(1199091 1199092)] (17)


119906 = 119906eq +119872

119887sign (119878)

= minus119887minus1

[119886321199092+ 119886331199093minus 2(1199091 1199092)] +

119872

119887sign (119878)

(18)

where |Δ119872| le 119872


0 5 10 15 20 25 30minus100

minus50

0

50

100

150

Time (s)



0 5 10 15 20 25 30minus08

minus06

minus04

minus02

0

02

04

06

Time (s)



120575(r

ad)

g(m

s2)



119906 = minus119887minus1

[119886321199092+ 119886331199093minus 2(1199091 1199092)]

+119872

119887tanh (120576119878) 120576 ≫ 0

(19)



119878 = 119886321199092+ 119886331199093+ 119865119906 minus

2(1199091 1199092) = 0

where 119865 = 119887119891

(20)


119878 = 119886321199092+ 119886331199093+ 119865119906 + (119865 minus 119865) 119906 + Δ

119872

minus 2(1199091 1199092) = 0 where 119865 = 119887119891

(21)


119906 = 119906eq + V (22)


119878 = V + Δ (119909 V) (23)

where

Δ (119909 V) = Δ119872+ (119865 minus 119865) (119906eq +

V119865) (24)

119906eq =1

119865[minus119886321199092minus 119886331199093+ 2(1199091 1199092)] (25)


Δ (119909 V) = Δ119872minus(119865 minus 119865)

119865119886321199092minus(119865 minus 119865)

119865119886331199093

+(119865 minus 119865)

119865

10038161003816100381610038162 (1199091 1199092)1003816100381610038161003816 +

(119865 minus 119865)

119865V

(26)


Δ (119909 V) le 119872 + 1198961

100381610038161003816100381611990911003816100381610038161003816 + 1198962

100381610038161003816100381611990921003816100381610038161003816 + 1198963

100381610038161003816100381611990931003816100381610038161003816 + 119896 |V| (27)

where11989611205722

le 1198961

100381610038161003816100381610038161003816100381610038161003816

119865 minus 119865

119865(11988632+ 11989621205722

)

100381610038161003816100381610038161003816100381610038161003816

le 1198962

100381610038161003816100381610038161003816100381610038161003816

119865 minus 119865

11986511988632

100381610038161003816100381610038161003816100381610038161003816

le 1198963

100381610038161003816100381610038161003816100381610038161003816

119865 minus 119865

119865

100381610038161003816100381610038161003816100381610038161003816

le 119896

10038161003816100381610038162 (1199091 1199092)1003816100381610038161003816 le 11989611205722

100381610038161003816100381611990911003816100381610038161003816 + 11989621205722

100381610038161003816100381611990921003816100381610038161003816

(28)

Values 11989611205722

and 11989621205722


120573 (119909) = 119872 + 1198961

100381610038161003816100381611990911003816100381610038161003816 + 1198962

100381610038161003816100381611990921003816100381610038161003816 + 1198963

100381610038161003816100381611990931003816100381610038161003816 + 1198870 119887

0gt 0 (29)


119906 = 119906eq +120573 (119909)

119865 (1 minus 119896)tanh (120576119878) (30)




0 5 10 15 20 25 30minus100

minus50

0

50

100

150


Time (s)0 5 10 15 20 25 30

minus20

minus10

0

10

20

30

40

50

Time (s)



120575(d

eg)

g(m

s2)


0 5 10 15 20 25 300

500

1000

1500

2000

2500

3000

3500

4000Distance (m)

Time (s)

R (m

)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0


MissileTarget

y1

(m)

x1 (m)


0 5 10 15 20 25 30 35 40minus80

minus60

minus40

minus20

0

20

40

60


Time (s)0 5 10 15 20 25 30 35 40

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Time (s)



120575(d

eg)

g(m

s2)



0 5 10 15 20 25 30 35 400

500

1000

1500

2000

2500

3000

3500

4000 Distance (m)

Time (s)

R (m

)

0 2000 4000 6000 8000 10000 12000 14000minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500


MissileTarget

y1

(m)

x1 (m)





(31)



119911| lt 45 deg


7 Conclusion


0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

16

18

20

Time (s)

f = 03f = 04f = 05f = 06

f = 07f = 08f = 09f = 1

intu2




References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










where

11988611= minus2

119881119877

119877

11988612= minus

119897120572

119877

11988622= minus

119897120572

119881119872

11988632= 119898120572

11988633= 119898119902

119887 = 119898120575119911

Δ119902=

119886119879119902

119877+cos (120582 minus 120601

119872) 119881119872Δ120572

119877

119906 = 120575119911

(9)







119906 = 120575119911+ 120590 (120575

119911minus 120575119911) (10)

where

120590 =

1 if actuator fails

0 otherwise(11)


120575119911= 119891120575119911 (12)


1= 119886111199091+ cos (120582 + 119909

2minus 120579) 119886

121199092+ Δ119902

2= 119886221199092+ 1199093+ Δ120572

3= 119886321199092+ 119886331199093+ 119887119891119906 + Δ

119872

(13)





of cos(119902 + 1199092minus 120579)119886






1and 119909

2= 1199092minus



1199092= 1205721(1199091) = minus

1

11988612

1198961199091

1199093= 1205722(1199091 1199092) = minus(119888

2+

1

212057521

)1199092

minus cos (120582 + 1199092minus 120579) 119886

121199091minus 119886221199092+1205971205721

120597119905

+1205971205721

1205971199091

(119886111199091+ cos (120582 + 119909

2minus 120579) 119886

121199092

minus1

212057522

1199092

1205971205721

1205971199091

)

(14)


12059712057211205971199091and 120597120572

1120597119905 can be found in [1]


119878 = 1199093minus 1205722(1199091 1199092) (15)


119878 = 3minus 2(1199091 1199092)

= 119886321199092+ 119886331199093+ 119887119906 + Δ

119872minus 2(1199091 1199092)

(16)

Then


[119886321199092+ 119886331199093minus 2(1199091 1199092)] (17)


119906 = 119906eq +119872

119887sign (119878)

= minus119887minus1

[119886321199092+ 119886331199093minus 2(1199091 1199092)] +

119872

119887sign (119878)

(18)

where |Δ119872| le 119872


0 5 10 15 20 25 30minus100

minus50

0

50

100

150

Time (s)



0 5 10 15 20 25 30minus08

minus06

minus04

minus02

0

02

04

06

Time (s)



120575(r

ad)

g(m

s2)



119906 = minus119887minus1

[119886321199092+ 119886331199093minus 2(1199091 1199092)]

+119872

119887tanh (120576119878) 120576 ≫ 0

(19)



119878 = 119886321199092+ 119886331199093+ 119865119906 minus

2(1199091 1199092) = 0

where 119865 = 119887119891

(20)


119878 = 119886321199092+ 119886331199093+ 119865119906 + (119865 minus 119865) 119906 + Δ

119872

minus 2(1199091 1199092) = 0 where 119865 = 119887119891

(21)


119906 = 119906eq + V (22)


119878 = V + Δ (119909 V) (23)

where

Δ (119909 V) = Δ119872+ (119865 minus 119865) (119906eq +

V119865) (24)

119906eq =1

119865[minus119886321199092minus 119886331199093+ 2(1199091 1199092)] (25)


Δ (119909 V) = Δ119872minus(119865 minus 119865)

119865119886321199092minus(119865 minus 119865)

119865119886331199093

+(119865 minus 119865)

119865

10038161003816100381610038162 (1199091 1199092)1003816100381610038161003816 +

(119865 minus 119865)

119865V

(26)


Δ (119909 V) le 119872 + 1198961

100381610038161003816100381611990911003816100381610038161003816 + 1198962

100381610038161003816100381611990921003816100381610038161003816 + 1198963

100381610038161003816100381611990931003816100381610038161003816 + 119896 |V| (27)

where11989611205722

le 1198961

100381610038161003816100381610038161003816100381610038161003816

119865 minus 119865

119865(11988632+ 11989621205722

)

100381610038161003816100381610038161003816100381610038161003816

le 1198962

100381610038161003816100381610038161003816100381610038161003816

119865 minus 119865

11986511988632

100381610038161003816100381610038161003816100381610038161003816

le 1198963

100381610038161003816100381610038161003816100381610038161003816

119865 minus 119865

119865

100381610038161003816100381610038161003816100381610038161003816

le 119896

10038161003816100381610038162 (1199091 1199092)1003816100381610038161003816 le 11989611205722

100381610038161003816100381611990911003816100381610038161003816 + 11989621205722

100381610038161003816100381611990921003816100381610038161003816

(28)

Values 11989611205722

and 11989621205722


120573 (119909) = 119872 + 1198961

100381610038161003816100381611990911003816100381610038161003816 + 1198962

100381610038161003816100381611990921003816100381610038161003816 + 1198963

100381610038161003816100381611990931003816100381610038161003816 + 1198870 119887

0gt 0 (29)


119906 = 119906eq +120573 (119909)

119865 (1 minus 119896)tanh (120576119878) (30)




0 5 10 15 20 25 30minus100

minus50

0

50

100

150


Time (s)0 5 10 15 20 25 30

minus20

minus10

0

10

20

30

40

50

Time (s)



120575(d

eg)

g(m

s2)


0 5 10 15 20 25 300

500

1000

1500

2000

2500

3000

3500

4000Distance (m)

Time (s)

R (m

)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0


MissileTarget

y1

(m)

x1 (m)


0 5 10 15 20 25 30 35 40minus80

minus60

minus40

minus20

0

20

40

60


Time (s)0 5 10 15 20 25 30 35 40

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Time (s)



120575(d

eg)

g(m

s2)



0 5 10 15 20 25 30 35 400

500

1000

1500

2000

2500

3000

3500

4000 Distance (m)

Time (s)

R (m

)

0 2000 4000 6000 8000 10000 12000 14000minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500


MissileTarget

y1

(m)

x1 (m)





(31)



119911| lt 45 deg


7 Conclusion


0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

16

18

20

Time (s)

f = 03f = 04f = 05f = 06

f = 07f = 08f = 09f = 1

intu2




References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20 25 30minus100

minus50

0

50

100

150

Time (s)



0 5 10 15 20 25 30minus08

minus06

minus04

minus02

0

02

04

06

Time (s)



120575(r

ad)

g(m

s2)



119906 = minus119887minus1

[119886321199092+ 119886331199093minus 2(1199091 1199092)]

+119872

119887tanh (120576119878) 120576 ≫ 0

(19)



119878 = 119886321199092+ 119886331199093+ 119865119906 minus

2(1199091 1199092) = 0

where 119865 = 119887119891

(20)


119878 = 119886321199092+ 119886331199093+ 119865119906 + (119865 minus 119865) 119906 + Δ

119872

minus 2(1199091 1199092) = 0 where 119865 = 119887119891

(21)


119906 = 119906eq + V (22)


119878 = V + Δ (119909 V) (23)

where

Δ (119909 V) = Δ119872+ (119865 minus 119865) (119906eq +

V119865) (24)

119906eq =1

119865[minus119886321199092minus 119886331199093+ 2(1199091 1199092)] (25)


Δ (119909 V) = Δ119872minus(119865 minus 119865)

119865119886321199092minus(119865 minus 119865)

119865119886331199093

+(119865 minus 119865)

119865

10038161003816100381610038162 (1199091 1199092)1003816100381610038161003816 +

(119865 minus 119865)

119865V

(26)


Δ (119909 V) le 119872 + 1198961

100381610038161003816100381611990911003816100381610038161003816 + 1198962

100381610038161003816100381611990921003816100381610038161003816 + 1198963

100381610038161003816100381611990931003816100381610038161003816 + 119896 |V| (27)

where11989611205722

le 1198961

100381610038161003816100381610038161003816100381610038161003816

119865 minus 119865

119865(11988632+ 11989621205722

)

100381610038161003816100381610038161003816100381610038161003816

le 1198962

100381610038161003816100381610038161003816100381610038161003816

119865 minus 119865

11986511988632

100381610038161003816100381610038161003816100381610038161003816

le 1198963

100381610038161003816100381610038161003816100381610038161003816

119865 minus 119865

119865

100381610038161003816100381610038161003816100381610038161003816

le 119896

10038161003816100381610038162 (1199091 1199092)1003816100381610038161003816 le 11989611205722

100381610038161003816100381611990911003816100381610038161003816 + 11989621205722

100381610038161003816100381611990921003816100381610038161003816

(28)

Values 11989611205722

and 11989621205722


120573 (119909) = 119872 + 1198961

100381610038161003816100381611990911003816100381610038161003816 + 1198962

100381610038161003816100381611990921003816100381610038161003816 + 1198963

100381610038161003816100381611990931003816100381610038161003816 + 1198870 119887

0gt 0 (29)


119906 = 119906eq +120573 (119909)

119865 (1 minus 119896)tanh (120576119878) (30)




0 5 10 15 20 25 30minus100

minus50

0

50

100

150


Time (s)0 5 10 15 20 25 30

minus20

minus10

0

10

20

30

40

50

Time (s)



120575(d

eg)

g(m

s2)


0 5 10 15 20 25 300

500

1000

1500

2000

2500

3000

3500

4000Distance (m)

Time (s)

R (m

)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0


MissileTarget

y1

(m)

x1 (m)


0 5 10 15 20 25 30 35 40minus80

minus60

minus40

minus20

0

20

40

60


Time (s)0 5 10 15 20 25 30 35 40

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Time (s)



120575(d

eg)

g(m

s2)



0 5 10 15 20 25 30 35 400

500

1000

1500

2000

2500

3000

3500

4000 Distance (m)

Time (s)

R (m

)

0 2000 4000 6000 8000 10000 12000 14000minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500


MissileTarget

y1

(m)

x1 (m)





(31)



119911| lt 45 deg


7 Conclusion


0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

16

18

20

Time (s)

f = 03f = 04f = 05f = 06

f = 07f = 08f = 09f = 1

intu2




References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20 25 30minus100

minus50

0

50

100

150


Time (s)0 5 10 15 20 25 30

minus20

minus10

0

10

20

30

40

50

Time (s)



120575(d

eg)

g(m

s2)


0 5 10 15 20 25 300

500

1000

1500

2000

2500

3000

3500

4000Distance (m)

Time (s)

R (m

)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0


MissileTarget

y1

(m)

x1 (m)


0 5 10 15 20 25 30 35 40minus80

minus60

minus40

minus20

0

20

40

60


Time (s)0 5 10 15 20 25 30 35 40

minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Time (s)



120575(d

eg)

g(m

s2)



0 5 10 15 20 25 30 35 400

500

1000

1500

2000

2500

3000

3500

4000 Distance (m)

Time (s)

R (m

)

0 2000 4000 6000 8000 10000 12000 14000minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500


MissileTarget

y1

(m)

x1 (m)





(31)



119911| lt 45 deg


7 Conclusion


0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

16

18

20

Time (s)

f = 03f = 04f = 05f = 06

f = 07f = 08f = 09f = 1

intu2




References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20 25 30 35 400

500

1000

1500

2000

2500

3000

3500

4000 Distance (m)

Time (s)

R (m

)

0 2000 4000 6000 8000 10000 12000 14000minus3000

minus2500

minus2000

minus1500

minus1000

minus500

0

500


MissileTarget

y1

(m)

x1 (m)





(31)



119911| lt 45 deg


7 Conclusion


0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

16

18

20

Time (s)

f = 03f = 04f = 05f = 06

f = 07f = 08f = 09f = 1

intu2




References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article adaptive integrated guidance and...

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