Chinese Journal of Aeronautics, (2016), 29(5): 1302–1312

Chinese Society of Aeronautics and Astronautics& Beihang University

Chinese Journal of Aeronautics

cja@buaa.edu.cnwww.sciencedirect.com

Roll-pitch-yaw autopilot design for nonlinear

time-varying missile using partial state observer

based global fast terminal sliding mode control

* Corresponding author. Tel.: +86 25 84303971.

E-mail address: hp.wang@njust.edu.cn (H. Wang).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2016.04.0201000-9361 � 2016 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Awad Ahmed, Wang Haoping *

Sino-French International Joint Laboratory of Automatic Control and Signal Processing (LaFCAS), School of Automation,Nanjing University of Science & Technology, Nanjing 210094, China

Received 9 November 2015; revised 18 January 2016; accepted 22 February 2016Available online 26 August 2016

KEYWORDS

Flight control system;

Global fast terminal sliding

mode control;

Integrated autopilot;

Nonlinear state observer;

Skid-to-turn missile

Abstract The acceleration autopilot design for skid-to-turn (STT) missile faces a great challenge

owing to coupling effect among planes, variation of missile velocity and its parameters, inexistence

of a complete state vector, and nonlinear aerodynamics. Moreover, the autopilot should be

designed for the entire flight envelope where fast variations exist. In this paper, a design of inte-

grated roll-pitch-yaw autopilot based on global fast terminal sliding mode control (GFTSMC) with

a partial state nonlinear observer (PSNLO) for STT nonlinear time-varying missile model, is

employed to address these issues. GFTSMC with a novel sliding surface is proposed to nullify

the integral error and the singularity problem without application of the sign function. The pro-

posed autopilot consisting of two-loop structure, controls STT maneuver and stabilizes the rolling

with a PSNLO in order to estimate the immeasurable states as an output while its inputs are missile

measurable states and control signals. The missile model considers the velocity variation, gravity

effect and parameters’ variation. Furthermore, the environmental conditions’ dynamics are mod-

eled. PSNLO stability and the closed loop system stability are studied. Finally, numerical simula-

tion is established to evaluate the proposed autopilot performance and to compare it with

existing approaches in the literature.� 2016 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an

open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The acceleration autopilot design for skid-to-turn missile is

still considered as one of the most attractive topics for controlengineers due to its enormous nonlinear dynamics, thecoupling effect between channels, and its rapid parameters’

variation.1 The most significant variation of missile parametersis its velocity which changes rapidly as a result of subjectingthe missile to a sudden acceleration during boosting phase

Roll-pitch-yaw autopilot design for nonlinear time-varying missile 1303

and deceleration during gliding phase due to aerodynamicdrag.2

Researches in this field have been started since 1944. One of

the commonly used autopilots was the three-loop autopilottopology.3 The conventional and linear quadratic regulatorsbased on the linearization of model dynamics around fixed

operating points were used. This technique is so-called ‘‘gainscheduling”. In Ref.4, a classical gain-scheduling design wasintroduced. In the 1990s, extensions of these techniques had

brought many developments, like guaranteed stability marginsand performance levels.5,6 The authors in Ref.7 presented a lin-ear quadratic Gaussian with loop transfer recovery techniqueto design gain scheduling autopilot. A gain scheduling based

autopilot in the presence of hidden coupling terms is illustratedin Ref.8 The combined optimal/classical approach was appliedto design the optimal controller in Ref.9 as well. Also, robust-

ness issues were introduced with suitable extensions of H1techniques.10,11 Clearly, the gain scheduling approach showsa good performance during the entire envelope, but the global

stability is guaranteed only in the case of slow variation ofboth the states and the missile parameters.

The development of linear parameter-varying (LPV) and

quasi-LPV approaches in the last two decades had pushedthe researchers towards a new systematic and strict methodol-ogy. For example, an acceleration autopilot design using theLPV reference model was presented for portable missile.12

Unfortunately, the disadvantage of these approaches, espe-cially for quasi-LPV autopilot, is the increment of conservativelevel.13 Also, most of these approaches except quasi-LPV

approach, were still based on the linearization of dynamicsaround operating points. Other drawbacks of LPV were thedifficulty in parameter variation recognition and the demand

of an additional filter for parameter estimation as well.14

The requirements of high maneuverability and the develop-ment of nonlinear control methods pushed the research

towards new control design approaches that consider essentialnonlinear dynamics. This led to the first generation of nonlin-ear autopilots which were based on both the inversion ofdynamics15 and the feedback linearization techniques.16 New

approaches were introduced in the last decades based on recentcontrol techniques, such as Lyapunov stabilization tech-niques17, sliding mode control (SMC)18,19, adaptive SMC20,

adaptive block dynamic surface control21, l1 adaptive con-trol22, simple adaptive control algorithm23, adaptive fussy slid-ing mode control24, robust hybrid control25, immersion and

invariance control26, backstepping control27, state-dependentRiccati equation (SDRE) approach28, adaptive SDRE withneural networks29, and fuzzy control.30 The approaches intro-duced in the nonlinear and/or adaptive context failed when

massive unstructured dynamics existed. Moreover, the strictrequirements needed for the response speed cannot often beachieved due to adaptation laws. Therefore, robust nonlinear

approaches based on the geometric theory as in Ref.31 andthe extensive use of Lyapunov direct criterion as in Ref.32 werepresented, demonstrating good performance at a high angle of

attack. It should be mentioned that the approaches based onthese methods have been only applied to simple single-input/single-output cases, disregarding the coupling and nonlineari-

ties occurring between pitch and yaw planes.A few works paid attention to presenting the integrated

autopilot to overcome the coupling effect between channels.For example, a robust backstepping approach has been

applied to multi-input/multi-output (MIMO) model to achieveboth bank-to-turn and skid-to-turn (STT) maneuvers1, a three-axis autopilot design using the classical three-loop autopilot

approach33, and an acceleration autopilot based on a linearrobust control scheme was presented to control roll, pitch,and yaw channels in an integrated way.34 These works showed

a good performance while considering the missile velocity con-stant, and the controller design is still based on linearizationexcept in Ref.1. Ref.35 presented a sliding mode based inte-

grated attitude control scheme considering velocity change.It showed good results, but the acceleration control was notpresented. In Ref.2, a sliding mode based roll-pitch-yaw inte-grated attitude and acceleration autopilot for a time-varying

velocity STT missile was proposed. It showed a good perfor-mance, but a complete state vector is essential and the velocityvariation was considered as a function of time. Likewise, grav-

ity effect, missile parameters’ variation, chattering phe-nomenon, and environmental dynamics were neglected.

Thus, to achieve a good tracking performance of the inte-

grated acceleration autopilot in presence of the above referredneglected factors without seeking for chattering elimination orcomplete state vector feedback, an integrated roll-pitch-yaw

autopilot using a partial state nonlinear observer (PSNLO)based global fast terminal sliding mode control (GFTSMC)approach is proposed for a skid-to-turn nonlinear time-varying (STTNTV) missile model. The missile model has taken

into account the coupling effect, gravity effect, missile param-eters’ variation, environmental conditions’ dynamics, and non-linear aerodynamics. In a similar manner, the missile velocity

and its height have been considered as a function of its states.GFTSMC with a suggested sliding surface is provided to avoidthe chattering phenomena of SMC, the singularity problem of

normal GFTSMC, and the demand for a relation between thesecond derivative of system states and its inputs. These slidingmode surfaces are suggested and used in the integrated two-

loop autopilot structure to nullify the integral error. PSNLOis presented to estimate the immeasurable states (angle ofattack a and side slip angle b) used to feedback the autopilot.The stability of closed loop system including PSNLO stability

is discussed.The remaining paper is organized as follows: In Section 2,

system description and modeling are provided. In Section 3,

integrated roll-pitch-yaw autopilot design, PSNLO designand its stability analysis, and the closed loop stability analysisare presented. The integrated autopilot design includes outer-

loop controller design based on GFTSMC with accelerationdynamics modeling and inner-loop controller design basedon GFTSMC. In Section 4, numerical simulations are pre-sented, and Section 5 is devoted to summary and concluding

remarks.

2. System description and modeling

The STTNTV missile model is aerodynamically controlled viacanard fins, and it has an axis-symmetric and cruciform shape.Thus, the next general assumptions can be considered:

(1) The moments of inertia Iyy(t) and Izz(t) are identical andproducts of inertia moments can be discarded.

(2) For short-range missiles, the Earth has been consideredflat and non-rotating.

1304 A. Awad, H. Wang

2.1. Mathematical modeling

The missile states, and its dynamic parameters measured in dif-

ferent coordinates; the orientation of the commonly used coor-dinate systems, and the related angles between them aredepicted in Fig. 1.Missile motion in space is described bymeansof the following differential equations. Solution of these equa-

tions gives missile linear velocity components (u,v,w) in (Xb,Yb,Zb) axes of body coordinate system (BCS), respectively, mis-sile angular rates (p,q, r) around (Xb,Yb,Zb) axes, respectively,

the Euler angles (u,h), (a,b), and the missile height h in Zi axisof inertial coordinate system (ICS). As mentioned in Refs.36–38,the differential equations, themissile velocityVm, dynamic pres-

sureQ(h,Vm), acceleration components (axb,ayb,azb) applied onthe missile in (Xb,Yb,Zb), respectively, and force components(Fxb,Fyb,Fzb) applied on the missile in (Xb,Yb,Zb), respectively,are developed and expressed as follows:

axb ¼ Fxb

mðtÞFxb ¼ TxðtÞ � sQðh;VmÞCx �mðtÞg sin h

8<: ð1Þ

ayb ¼ Fyb

mðtÞFyb ¼ �sQðh;VmÞðCbbþ CdydyÞ þmðtÞg cos h sin u

8<: ð2Þ

azb ¼ Fzb

mðtÞFzb ¼ �sQðh;VmÞðCaaþ CdpdpÞ þmðtÞg cos h cos u

8<: ð3Þ

_p ¼ sQðh;VmÞlIxxðtÞ Claaþ Clbbþ Cldrdr þ Clppl

2Vm

� �ð4Þ

_q ¼ sQðh;VmÞlIyyðtÞ Cmaaþ Cmdpdp þ Cmqql

2Vm

� �ð5Þ

_r ¼ sQðh;VmÞlIzzðtÞ Cnbbþ Cndydy þ Cnrrl

2Vm

� �ð6Þ

_a ¼ qþ 1

Vm cos bðazb cos a� axb sin aÞ � r sin a tan b

� p cos a tan b ð7Þ

_b ¼ �r cos aþ 1

Vm

ðayb cos b� axb cos a sin b

� azb sin a sin bÞ þ p sin a ð8Þ

Fig. 1 Coordinate systems orientation and their angular relations

_h ¼ Vm sinðh� aÞ ð9Þ

_u ¼ pþ q sin u tan hþ r cos u tan h ð10Þ

_h ¼ q cos u� r sin u ð11Þ

with

Vm ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ v2 þ w2

p

Qðh;VmÞ ¼ 0:5qðhÞV2m

(ð12Þ

_u ¼ axb þ rv� qw

_v ¼ ayb þ pw� ru

_w ¼ azb þ pvþ qu

8>><>>: ð13Þ

where s is the aerodynamic reference area, q the function of airdensity, l the missile characteristic length, g the gravity acceler-

ation inZi axis of ICS,Tx(t) the function of thrust component inXb axis, m(t) the function of missile mass, Ixx(t) the moments ofinertia function inXb axis, andCx the drag force derivative;Cla,

Clb, Clp, Cldr, Cma, Cmq, Cmdp, Cnb, Cnr, and Cndy are stabilityderivatives of the rolling, pitching and yawing moments; Ca,Cdp,Cb, andCdy are force derivatives of the pitching and yawing

forces; dr, dp, and dy are the fin angular deflections in roll, pitchand yaw planes, respectively; t is the missile flight time.

2.2. Aerodynamic coefficients modeling

Obviously, accurate estimation of the aerodynamic coefficientsis the corner stone of guidance and control system design.Furthermore, the aerodynamic coefficients evaluation via wind

tunnel tests is essential. The coefficients of aerodynamic forcesand moments obtained from aerodynamic coefficients’database based on experimental data, are functions of Mach

number Ma, a, b, dr, dp, and dy.

2.3. Environmental conditions dynamics modeling

The Lapse rate mathematical model for the troposphere hasbeen used to represent the dynamics of air density and thespeed of sound as a function of h as follows:

, c.g. is the missile center of gravity; w is the missile yaw angle.

Roll-pitch-yaw autopilot design for nonlinear time-varying missile 1305

qðhÞ ¼ q0 1� LT0h

� � gLR�1

VsðhÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficRðT0 � LhÞp

Ma ¼ Vm

VsðhÞ

8>>>>>><>>>>>>:

ð14Þ

where q0 is the air density at mean sea level, L the Lapse

rate, T0 the absolute temperature at mean sea level, Vs(h) thespeed function of sound at altitude h, and R the specific heatratio.

3. Autopilot design

In this section, the design process of roll-pitch-yaw autopilot

based on GFTSMC, the design process of PSNLO and its sta-bility analysis, and the discussion of closed loop system stabil-ity are presented as follows.

3.1. Roll-pitch-yaw autopilot design

As a result of autopilot application to missile model with rapidparameters’ variation, a GFTSMC based autopilot is supposed

as a robust autopilot. The normal GFTSMC makes theconvergence velocity to be finite without chatteringphenomenon.39,40 But it possess a singularity problem, and it

needs a relation between the second order derivative of systemstates and its inputs. To meet the nature of STTNLTV missilemodel and the structure of the proposed autopilot, aGFTSMC with novel sliding surface is presented. It has the

following advantages related to the other sliding surfaces inreference:

(1) Avoiding the usage of sign function which generates thechattering phenomena.

(2) Avoiding the singularity problem of normal GFTSMC.

(3) Avoiding the system output differentiation which isundesirable in the real system due to the noise existencein the system output measurements.

(4) Applicable for systems’ model which has a relationbetween the first order derivative of system states andits input.

(5) In general, the convergent characteristics of GFTSMC

are superior to those of the normal SMC.39

Since the missile performs STT maneuver, its outputs to be

controlled are (u, azb, and ayb). A scheme of the proposedautopilot based on GFTSMC with a PSNLO is shown inFig. 2, IMU is the inertial measuring unit. The proposed

autopilot consisting of a two-loop structure, controls pitchand yaw accelerations, and stabilizes the roll angle simultane-ously. The proposed outer-loop controller generates missile

roll, pitch, and yaw angular rate commands (pc,qc, rc)corresponding to roll angle command (uc) and pitch andyaw acceleration commands (azc,ayc). The inner-loop con-troller generates (dr,dp,dy) corresponding to (pc,qc, rc).

Remark 1. Merit of the two loop autopilot structure is thatsum of the relative degree of the states to be controlled and the

relative degree of the inner-loop coincides, and therefore,

unstable zero-dynamics is not observed as far as the inner-loopis stabilized.2

3.1.1. Outer-loop controller design based on GFTSMC

The derivative of the pitch acceleration can be described as

_azb ¼_Fzb

mðtÞ �_mðtÞm2ðtÞFzb

¼ � sCa

mðtÞQ _a� sCa

mðtÞ_Qa� sCdp

mðtÞ_Qdp � sCdp

mðtÞQ_dp

� g _u sin u cos h� g _h sin h cos u

þ _mðtÞmðtÞ g cos h cos u� _mðtÞ

mðtÞ azb ð15Þ

Substituting Eq. (7) into Eq. (15), we get

_azb ¼ Naqþ Na

Vm cos bðazb cos a� axb sin aÞ

� ðNa sin a tan bÞr� ðNa cos a tan bÞpþDaa

þDdpdp þNdp_dp � g _u sin u cos h

� g _h sin h cos uþ _mðtÞmðtÞ g cos h cos u� _mðtÞ

mðtÞ azb ð16Þ

where

Na ¼� sCamðtÞQ;Ndp ¼� sCdp

mðtÞQ;Da ¼� sCamðtÞ _Q;Ddp ¼� sCdp

mðtÞ _Q.

Similarly, yaw acceleration derivative is obtained asfollows:

_ayb ¼_Fyb

mðtÞ �_mðtÞm2ðtÞFyb

¼ � sCb

mðtÞQ_b� sCb

mðtÞ_Qb� sCdy

mðtÞ_Qdy � sCdy

mðtÞQ_dy

þ g _u cos u cos h� g _h sin h sin u

þ _mðtÞmðtÞ g cos h sin u� _mðtÞ

mðtÞ ayb ð17Þ

Substituting Eq. (8) into Eq. (17), we get

_ayb ¼ ð�Nb cos aÞrþ Nb

Vm

ðayb cos b� axb cos a sin b

� azb sin a sin bÞ þ ðNb sin aÞpþDbbþDdydy

þNdy_dy þ g _u cos u cos h� g _h sin h sin u

þ _mðtÞmðtÞ g cos h sin u� _mðtÞ

mðtÞ ayb ð18Þ

where

Nb ¼� sCb

mðtÞQ;Ndy ¼� sCdy

mðtÞQ;Db ¼� sCb

mðtÞ _Q;Ddy ¼� sCdy

mðtÞ _Q.

Re-arranging the equations for the derivative of u, azb, andayb into the affine matrix form, we get

½ _u; _azb; _ayb�T ¼ faða; b; x; ur; tÞ þ gaða; x; tÞua ð19Þwhere fað�Þ 2 R3�1 is smooth function in terms of a, b, x, ur,and t; gað�Þ 2 R3�3 is smooth function in terms of a, x, and t;

x ¼ ½p; q; r; h;u;Vm; azb; ayb; axb�T is the measured state vector;

Fig. 2 Block diagram of integrated roll-pitch-yaw autopilot based on GFTSMC with PSNLO.

faða; b; x; ur; tÞ ¼q sin u tan hþ r cos u tan h

D1 þ D2

D3 þ D4

264

375

D1 ¼ Na

Vm cos bðazb cos a� axb sin aÞ � ðNa sin a tan bÞr� ðNa cos a tan bÞpþDaa

D2 ¼ Ddpdp þNdp_dp � g _u sin u cos h� g _h sin h cos uþ _mðtÞ

mðtÞ g cos h cos u� _mðtÞmðtÞ azb

D3 ¼ Nb

Vm

ðayb cos b� axb cos a sin b� azb sin a sin bÞ þ ðNb sin aÞpþDbbþDdydy

D4 ¼ Ndy_dy þ g _u cos u cos h� g _h sin h sin uþ _mðtÞ

mðtÞ g cos h sin u� _mðtÞmðtÞ ayb

8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:

ð20Þ

1306 A. Awad, H. Wang

ua 2 R3�1 is the angular rate commands vector; ur is the inputvector. Note that the inverse of gað�Þ always exists.

gaða; x; tÞ ¼1 0 0

0 Na 0

0 0�Nb cos a

264

375

ua ¼ pc; qc; rc½ �T

8>>><>>>:

ð21Þ

The proposed sliding surfaces si that consider theintegral tracking error as a state, and the reaching laws for

the outer-loop controller based on GFTSMC, are defined asfollows:

si ¼ ei þ kiR t

0ei dsþ rið

R t

0ei dsÞqi0=pi0 i ¼ u; azb; ayb

eacc ¼eu

eazb

eayb

264

375 ¼

u� uc

azb � azc

ayb � ayc

264

375

8>>>><>>>>:

ð22Þ

_si ¼ �Kpisi � KsiðsiÞqi=pi ð23Þwhere ki, ri, Kpi, and Ksi are positive real designed values; qi0,pi0, qi, and pi are positive odd real designed values (qi0 < pi0,and qi < pi). For more details, see Ref.39,40.

Differentiating Eq. (22) with respect to time and applyingthe reaching law to Eq. (23), the outer-loop control law isobtained as follows:

ua ¼ ½pc; qc; rc�T ¼ �gaða; x; tÞ�1ðfaða; b; x; ur; tÞ þ Aea þ AraÞð24Þ

where

Aea¼kueuþruðqu0=pu0Þð

R t

0eudsÞðqu0=pu0Þ�1

eu� _uc

kazbeazbþrazbðqazb0=pazb0ÞðR t

0eazbdsÞðqazb0=pazb0Þ�1

eazb� _azc

kaybeaybþraybðqayb0=payb0ÞðR t

0eaybdsÞðqayb0=payb0Þ�1

eayb� _ayc

266664

377775

Ara¼KpusuþKsuðsuÞqu=pu

KpazbsazbþKsazbðsazbÞqazb=pazb

KpaybsaybþKsaybðsaybÞqayb=payb

2664

3775

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ð25Þ

Remark 2. The control law derived from the normal

GFTSMC has the part xðq=pÞ�11 x2 where q and p are positive

odd integers, which satisfy q < p. This part may cause asingularity problem if x2 – 0 when x1 ¼ 0.41 From Eqs. (24)

and (25), the part ðR t

0 ei dsÞðqi0=pi0Þ�1

ei; i ¼ u; azb; ayb will avoid

singularity problem because if ei – 0, thenR t

0 ei ds –0; i ¼ u; azb; ayb.

3.1.2. Inner-loop controller design based on GFTSMC

Re-arranging Eqs. (4)–(6) for derivatives of p, q, and r, respec-tively into the affine matrix form, we get

½ _p; _q; _r�T ¼ frða; b; x; tÞ þ grðx; tÞur ð26Þ

where frð�Þ 2 R3�1; grð�Þ 2 R3�3 are smooth functions in terms

of a, b, x, and t; ur 2 R3�1 is the input vector. The inverse ofgrð�Þ always exists.

Roll-pitch-yaw autopilot design for nonlinear time-varying missile 1307

frða; b; x; tÞ ¼

sQl

IxxðtÞ Claaþ Clbbþ Clppl

2Vm

� �sQl

IyyðtÞ Cmaaþ Cmqql

2Vm

� �sQl

IzzðtÞ Cnbbþ Cnrrl

2Vm

� �

266666664

377777775

grðx; tÞ ¼

sQlCldr

IxxðtÞ 0 0

0sQlCmdp

IyyðtÞ 0

0 0sQlCndy

IzzðtÞ

266666664

377777775

ur ¼ dr; dp; dy� �T

8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:

ð27Þ

Similarly, the sliding surfaces sj and the reaching laws of theinner-loop controller are defined as

sj ¼ ej þ kjR t

0ej dsþ rjð

R t

0ej dsÞqj0=pj0

_sj ¼ �Kpjsj � KsjðsjÞqj=pj j ¼ p; q; r

erate ¼ ½ep; eq; er�T ¼ ½p� pc; q� qc; r� rc�T

8>><>>: ð28Þ

Differentiating Eq. (28) with respect to time and applyingthe reaching law, the inner-loop control law is obtained asfollows:

ur ¼ ½dr; dp; dy�T ¼ �grðx; tÞ�1ðfrða; b; x; tÞ þ Aer þ ArrÞ ð29Þwhere

Aer ¼kpep þ rpðqp0=pp0Þð

R t

0ep dsÞðqp0=pp0Þ�1

ep � _pc

kqeq þ rqðqq0=pq0ÞðR t

0eq dsÞðqq0=pq0Þ�1

eq � _qc

krer þ rrðqr0=pr0ÞðR t

0er dsÞðqr0=pr0Þ�1

er � _rc

26664

37775

Arr ¼Kppsp þ KspðspÞqp=ppKpqsq þ KsqðsqÞqq=pqKprsr þ KsrðsqÞqr=pr

2664

3775

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð30Þ

Remark 3. Similarly as in Remark 2, from Eqs. (29) and (30),

the part ðR t

0 ei dsÞðqi0=pi0Þ�1

ei; i ¼ p; q; r will avoid singularity

problem because if ei – 0, thenR t

0 ei ds– 0; i ¼ p; q; r.

3.2. PSNLO design and its stability analysis

Unfortunately, a and b are unavailable to be measured in the

real missile system. Consequently, the partial state observer isnecessary. Devaud et al.36 presented a design of simple quad-ratic observer for a and b under some simplifying assumptions.

In this paper, an efficient PSNLO for a and b is presented. Theproposed PSNLO model equation is expressed as follows:with

_a_b

" #¼ K

azbðaÞ � azb

aybðbÞ � ayb

" #þ _aða; bÞ

_bða; bÞ

" #¼ K

azbðaÞ � azb

aybðbÞ � ayb

" #

þqþ 1

Vm cos bðazbðaÞ cos a� axb sin aÞ � r sin a tan b�

�r cos aþ 1Vm

ðaybðbÞ cos b� axb cos a sin b� azbðaÞ sin a

24

K ¼ diagðka; kbÞ ð32Þwhere a and b are the estimated angles of attack and side slip;ka and kb are positive real designed numbers.

In order to discuss stability of the proposed PSNLO, thecandidate Lyapunov function is chosen as follows:

Voðea; ebÞ ¼ 1

2e2a þ

1

2e2b ð33Þ

_Voðea; ebÞ ¼ eað _a� _aÞ þ ebð _b� _bÞ ð34Þwhere ea ¼ a� a; eb ¼ b� b.

For simplicity in stability analysis, the next assumption canbe used because of the smallness of a and b, especially in thecase of canard missile control.38

sin x � x; cos x � 1 if jxj 6 p6

Based on the above, the Lyapunov function derivative canbe presented as follows:

_Voðea; ebÞ ¼ eað1þ kaVmÞVm

ðazbðaÞ � azbÞ � earðab� abÞ

� e2aaxbVm

þ ebð1þ kbVmÞVm

ðaybðbÞ � aybÞ

� e2baxb

Vm

� ebVm

ðazbðaÞab� azbabÞ ð35Þ

Substituting Eqs. (2) and (3) into Eq. (35), we get

_Voðea; ebÞ ¼ �e2aBað1þ kaVmÞ � earðab� abÞ

� e2aaxbVm

� e2bBbð1þ kbVmÞ �e2baxb

Vm

� ebVm

ðazbðaÞab� azbabÞ ð36Þ

where Ba ¼ sCaQmðtÞVm

;Bb ¼ sCbQmðtÞVm

.

The smallness of r, a, and b and Vm � ab lead to

earðab� abÞ � 0;ebVm

ðazbðaÞab� azbabÞ � 0.

Based on the above, we get

_Voðea; ebÞ ¼ �e2aBað1þ kaVmÞ � e2aaxbVm

� e2bBbð1þ kbVmÞ �e2baxb

Vm

ð37Þ

Since Vm, m(t), Q, Ca, and Cb are bounded positive

functions for canard control missile, Ba;Bb;Vm >

0 8 t > 0; axb > 0 8 0 < t > Ts where Ts is the end time of

the sustaining phase.

p cos a tan b

sin bÞ þ p sin a

35 ð31Þ

1308 A. Awad, H. Wang

Then Eq. (37) shows that the proper selection of ka and

kb > 0 guarantee _Voðea; ebÞ < 0, i.e., the presented PSNLO is

asymptotically stable during all the flight envelope.

3.3. Closed loop stability analysis

In order to discuss the closed loop stability of integrated roll-pitch-yaw autopilot based on GFTSMC with PSNLO,designed for STTNTV missile model, the candidate Lyapunov

function is defined as

V ¼ Vs þ Vo; _V ¼ _Vs þ _Vo ð38Þ

Vs ¼X 1

2s2i ;

_Vs ¼X

si _si; i ¼ p; q; r;u; azb; ayb ð39Þ

Because of the smallness of a, cos a � 1, i.e., gaða; x; tÞ �gaðx; tÞ.

Then, the outer and inner loops control law of integratedroll-pitch-yaw autopilot based on GFTSMC with PSNLOi.e., Eqs. (24) and (29) becomes the following:

ua ¼ ½pc; qc; rc�T

¼ �gaðx; tÞ�1ðfaða; b; x; ur; tÞ þ Aea þ AraÞ ð40Þ

ur ¼ ½dr; dp; dy�T ¼ �grðx; tÞ�1ðfrða; b; x; tÞ þ Aer þ ArrÞ ð41ÞDifferentiating Eq. (22) with respect to time and applying

the outer-loop control law in Eq. (40), we get

_su

_sazb

_sayb

264

375 ¼ faða; b; x; ur; tÞ þ gaðx; tÞua

þ� _uc þ kueu þ ruðqu0=pu0Þð

R t

0eu dsÞðqu0=pu0Þ�1

eu

� _azc þ kazbeazb þ razbðqazb0=pazb0ÞðR t

0eazb dsÞðqazb0=pazb0Þ�1

eazb

� _ayc þ kaybeayb þ raybðqayb0=payb0ÞðR t

0eayb dsÞðqayb0=payb0Þ�1

eayb

26664

37775

¼ faða;b; x; ur; tÞ � faða; b;x; ur; tÞ � Ara

ð42ÞSimilarly, differentiating Eq. (28) with respect to time and

applying the inner-loop control law in Eq. (41), we get

_sp

_sq

_sr

2664

3775 ¼ frða; b; x; tÞ þ grðx; tÞur

þ� _pc þ kpep þ rpðqp0=pp0Þð

R t

0ep dsÞðqp0=pp0Þ�1

ep

� _qc þ kqeq þ rqðqq0=pq0ÞðR t

0eq dsÞðqq0=pq0Þ�1

eq

� _rc þ krer þ rrðqr0=pr0ÞðR t

0er dsÞðqr0=pr0Þ�1

er

266664

377775

¼ frða; b; x; tÞ � frða; b; x; tÞ � Arr

ð43Þ

with

faða; b; x; ur; tÞ � faða; b; x; ur; tÞ ¼ ½du; dazb; dayb�Tfrða; b; x; tÞ � frða; b; x; tÞ ¼ ½dp; dq; dr�T

(ð44Þ

Substituting Eqs. (42)–(44) into Eq. (39), we get

_Vs ¼X

si½di � Kpisi � KsiðsiÞqi=pi �¼

Xsidi � Kpis

2i � KsiðsiÞðqiþpiÞ=pi �;

i ¼ p; q; r;u; azb; ayb ð45ÞSince ea and eb are asymptotically stable and the initial val-

ues of a and b are zero, ea and eb are very small and bounded.

And fað�Þ and frð�Þ are smooth functions in terms of a, b, x, urand t. Then di; i ¼ p; q; r;u; azb; ayb are small and boundedbecause they are based on ea and eb values where di = 0 if eaand eb are zero, i.e.,

jdij 6 Di i ¼ p; q; r;u; azb; ayb ð46ÞBased on the above, since (p + q) is even number,

sidi � KsiðsiÞðqiþpiÞ=pi 6 0 should be satisfied, i.e.,

Ksi Pjdij

jsqi=pii jor Ksi P

Di

jsqi=pii j; i ¼ p; q; r;u; azb; ayb ð47Þ

Remark 4. As mentioned above, ea � 0; and eb � 0, then

Di � 0; i ¼ p; q; r;u; azb; ayb, i.e., the sufficient large value of

Ksi can satisfy the above condition in Eq. (47).

Eqs. (45) and (47) show that the proper selection of

Kpi P 0;Ksi P Di

jsqi=pii

j; i ¼ p; q; r;u; azb; ayb, guarantees

_Vs < 0. For more details, see Ref.39.

Finally, since _Vs < 0; _Vo < 0, then _V < 0.From Laypunov stability theorem, the appropriate choice

of ka; kb;Kpi;Ksi; i ¼ p; q; r;u; azb; ayb can guarantee asymp-

totic convergence of PSNLO estimation error and the slidingsurfaces si in case of the closed loop system. Since the only

way to nullify si is to enforce ei = 0, tracking is carried out,and ei is asymptotically stable. Moreover, it is assumed thatthe missile internal dynamics are stable and bounded, i.e.,the closed loop system is asymptotically stable.

4. Numerical simulation

To verify the proposed autopilot performance and to compare itwith the usual sliding mode control (USMC) in Ref.2 and theautopilot design in Ref.33, numerical simulation is carried out ina MATLAB�-Simulink� environment with fixed step time

0.02 s (applicable in the real system) for the whole flight time(8 s) considering all the missile parameters’ variations in evaluat-ing the proposedGFTSMCbased autopilotwithPSNLO.USMC

based autopilot considered the sliding surface si ¼ ei; and _si ¼�Kpisi � KsisignðsiÞ; i ¼ p; q; r;u; azb; ayb. The presented

autopilot inRef.33 is basedon classical three loop (CTL) techniquewith a little modification. The real data of man portable missiletype is chosen as a severe case because the missile should be

launched in a low-speed, resulting in a dramatic parameters’variation. Nevertheless, the fin deflection angles are limited dueto hardware constraints.12

4.1. Missile dynamical parameters

The parameters value describing the airframe and the environ-

mental conditions are listed in Table 1. In particular, ourunderlying missile is aerodynamically controlled using canardfins, and contains a two stage rocket motor with boosting

Roll-pitch-yaw autopilot design for nonlinear time-varying missile 1309

and sustaining phases where end time of the boosting phaseTb = 1.77 s, and end time of the sustaining phaseTs = 7.77 s. During the boosting phase, missile acceleration

is extremely high and its velocity increases from 27.4 m/s upto 420 m/s during 1.77 s as shown in Fig. 3. Obviously, exces-sive dynamics in missile velocity produces a great challenge to

the autopilot. The aerodynamic coefficients extracted from adatabase at each time step have a nonlinear dependence onboth Ma and the incidence angles. The moments of inertia

are approximated as a time function as long as the missile massis always known at each instant during its flight. For moredetails, see Ref.42.

In order to reflect the mechanical system physical restric-

tions, upper and lower limits of the actuator deflection wereconstrained as

maxfjdrjg 6 15�

maxfjdpjg 6 15�

maxfjdyjg 6 15�

8><>: ð48Þ

Initial values of the differential equations are chosen as real

initial values of the underlying missile as follows:

½uð0Þ; vð0Þ;wð0Þ; pð0Þ; qð0Þ; rð0Þ; hð0Þ;uð0Þ; að0Þ; bð0Þ; hð0Þ�T

¼ ½27:4; 0; 0; 0; 0; 0; 0:5; 0; 0; 0; 2�Tð49Þ

Fig. 3 Missile velocity profile.

Table 1 Missile and environmental conditions parameters.

Symbol Definition Value

Vm0 Velocity at t0 27.4 m/s

Tx(t0) Thrust at t0 2001.6 N

Tx(tf) Thrust at tf 0 N

m(t0) Mass at t0 9.12 kg

m(tf) Mass at tf 5.25 kg

Ixx(t0) Inertia moment at t0 0.4123 kg�m2

Iyy(t0),

Izz(t0)

Inertia moments at t0 1.3857 kg�m2

Ixx(tf) Inertia moment at tf 0.2371 kg�m2

Iyy(tf),

Izz(tf)

Inertia moments at tf 0.7959 kg�m2

l Diameter 0.071 m

s Reference area 0.0039 m2

q0 Air density at mean sea level 1.2255 kg/m3

T0 Absolute temperature at mean

sea level

288.15 K

L Lapse rate 0.0065 K/m

R Specific heat ratio 1.4

c Characteristic gas constant 287:05 J � ðkg �KÞ�1

Note: t0 and tf are the initial and final missile flight times,

respectively.

Moreover, to check robustness of the proposed autopilot,the white noise values are added to all of the measured feed-back signals in the numerical simulation.

4.2. Autopilot design parameters

Designed parameters of the proposed roll-pitch-yaw autopilot

based on GFTSMC with PSNLO are chosen as follows:ku = 20, kazb = 2400, kayb = 2400, ru = 1, razb = 1,rayb = 1, qu0 = 5, qazb0 = 5, qayb0 = 5, pu0 = 9, pazb0 = 9,

payb0 = 9, Kpu = 0.01, Kpazb = 0.8, Kpayb = 0.7, Ksu = 0.1,Ksazb = 0.1, Ksayb = 0.1, qu = 1, qazb = 1, qayb = 1, pu = 3,pazb = 3, payb = 3, kp = 10, kq = 20, kr = 20, rp = 1,

rq = 1, rr = 1, qp0 = 5, qq0 = 5, qr0 = 5, pp0 = 9, pq0 = 9,pr0 = 9, Kpp = 0.01, Kpq = 0.01, Kpr = 0.01, Ksp = 0.1,Ksq = 0.1, Ksr = 0.1, qp = 1, qq = 1, qr = 1, pp = 3, pq = 3,pr = 3, ka = 0.1, and kb = 0.1.

4.3. Simulation results

To investigate the presented autopilot performance against the

dynamic acceleration commands with roll stabilization duringthe entire flight time, the reference roll angle and the referenceyaw and pitch acceleration commands are considered as

½uc; ayc; azc�T ¼ ½0; aycðtÞ; azcðtÞ�T t P 0 ð50Þ

Fig. 4 Profile of roll angle, yaw acceleration and pitch

acceleration.

Fig. 6 Estimation errors of angle of attack and side slip angle

profile.

Fig. 7 Profile of roll, yaw and pitch fin deflection.

Fig. 5 Profile of angle of attack and side slip angle.

1310 A. Awad, H. Wang

The comparison of the tracking performance among thepresented GFTSMC based autopilot with PSNLO, USMC

based autopilot, and CTL based autopilot in roll, pitch andyaw planes is depicted in Fig. 4. Fig. 4 shows a good trackingperformance of the suggested autopilot, and a robustness

against the noise in all channels during the entire flight time.Clearly, only a little accepted error in uc tracking performanceoccurs. On the contrary, it shows that USMC based autopilot

and CTL based autopilot fail. The angles a and b and theirestimations are shown in Fig. 5. Fig. 5 shows a good perfor-mance of PSNLO during the entire flight time, and the small-ness of a and b which minimizes the aerodynamic non-linearity

and relaxes the coupling between planes of symmetry. The esti-mation errors of a and b are shown in Fig. 6 that shows a verysmall estimation error during the whole flight time. The related

roll, yaw and pitch fin deflections are shown in Fig. 7. Fig. 7shows a small and smooth fin deflection which relaxes theactuators. It is evident that overall simulation results of the

proposed autopilot have shown a good tracking performancein presence of all of the coupling effect, missile parameters’variation, environmental conditions’ dynamics, dynamic

acceleration commands, rapid velocity variation, and noises.

On the other hand, no demand for the inapplicable measure-ments of states in the real missile system.

5. Conclusions

In this paper, a global fast terminal sliding mode control withpartial state nonlinear observer based integrated roll-pitch-yawautopilot design has been presented for skid-to-turn nonlinear

time-varying missile to achieve a good performance in presenceof the commonly disregarded factors such as the couplingeffect, unavailability of the full state vector, chattering prob-

lem, and more missile model dynamics during the entire flighttime. The partial state nonlinear observer is designed to esti-mate the immeasurable states. The stability analysis is per-

formed. The numerical simulation is conducted to verifyperformance of the proposed autopilot design, and to compareit with an existing approach in literature. The results show

robustness and good tracking performance with relativesmooth fin deflections in all missile channels in the presenceof the aforesaid considerations. Also, it achieves a small angleof attack and side slip angle which minimize the aerodynamics

nonlinearity and the coupling effect. Conversely, the compared

Roll-pitch-yaw autopilot design for nonlinear time-varying missile 1311

techniques fail. The future work is suggested to consider alumped disturbances in the missile model and to design a dis-turbance observer which can be added to the proposed autopi-

lot to compensate the lumped disturbance effects. Moreover,the proposed sliding surface can be employed in different con-trol applications.

Acknowledgements

This work is co-supported by the National Natural ScienceFoundation of China (No. 61304077), International Science& Technology Cooperation Program of China (No.

2015DFA01710), the Natural Science Foundation of JiangsuProvince of China (No. BK20130765), the Chinese Ministryof Education Project of Humanities and Social Sciences(No. 13YJCZH171), the 11th Jiangsu Province Six Talent

Peaks of High Level Talents Project of China (No.2014_ZBZZ_005), and the Jiangsu Province Project Blue:Young Academic Leaders Project.

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Awad Ahmed is a Ph.D. candidate at Automation School, Nanjing

University of Science and Technology, China. He received his B.S. in

Cairo, Egypt in 2002; he was a technical support engineer until 2004,

and senior researcher until 2006. In 2010, he received the M.S. degree

in electric engineering, missile infrared seeker and flight simulation,

from Military Technical College, Cairo, Egypt. His research interests

are embedded systems and interfaces, real time systems, hardware-

in-loop simulation, visual tracking, and he mostly concerns missile

guidance and control.

Wang Haoping is a professor and Ph.D. supervisor at Automation

School, Nanjing University of Science and Technology, China. He

received his Ph.D. degree in automatic control from Lille University of

Science and Technology (LUST), France in 2008. He was research

fellows at Modeling Information and System Laboratory of Picardie

University and at Automatics, Computer Engineering and Signal

Processing Laboratory of LUST, France. His research interests include

the theory and applications of hybrid systems, visual servo control,

friction modeling and compensation, modeling and control of diesel

engines, biotechnological processes and wind turbine systems.