Research ArticleRobust Adaptive Output Feedback Control for a GuidedSpinning Rocket

Zhongjiao Shi ,1,2 Liangyu Zhao ,1,2 and Yeqing Zhu1,2

1Beijing Institute of Technology, Beijing 100081, China2Key Laboratory of Dynamics and Control of Flight Vehicle, Ministry of Education, Beijing 100081, China

Correspondence should be addressed to Liangyu Zhao; zhaoly@bit.edu.cn

Received 25 October 2017; Revised 5 April 2018; Accepted 18 April 2018; Published 14 May 2018

Academic Editor: Kenneth M. Sobel

Copyright © 2018 Zhongjiao Shi et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An adaptive autopilot is presented for the pitch and yaw channels of a guided spinning rocket. Firstly, the uncertain dynamic modelof a guided spinning rocket is established, which is used to evaluate the performance of the proposed adaptive autopilot. Secondly, arobust adaptive output feedback autopilot containing a baseline component and an adaptive component is designed. The mainchallenge that needs to be addressed is the determination of a corresponding square and strictly positive real transfer function.A simple design procedure based on linear matrix inequality is proposed that allows the realization of such a transfer function,thereby allowing a globally stable adaptive output feedback law to be generated. Finally, numerical simulations are performed toevaluate the robustness and tracking performance of the proposed robust adaptive autopilot. The simulation results showed thatthe robust adaptive output autopilot can achieve asymptotic command tracking with significant uncertainty in controleffectiveness, moment coefficient, and measurement noise.

1. Introduction

Traditional artillery ballistic gun-launched munitions cannotsatisfy more and more stringent performance requirementsabout precision-strike capability, dispersion error reduction,and range augmentation required on modern battlefields.Complex guided systems such as missiles can meet theserequirements, but they remain expensive due to the integra-tion of high-quality actuators and sensors. The idea is todevelop a guided rocket, which permits to reach a compro-mise between the low cost of ballistic projectiles and thenecessity to have high-performance systems with efficientcontrol algorithms. The guided rocket chosen in this workis a canard-controlled spinning configuration, which makesuse of an existing multiple launch rocket system (MLRS).This one has the advantage of simplifying the structure ofthe control system, avoiding asymmetric ablation, relaxingthe manufacturing error tolerance, and improving the pene-tration ability. It is also easy to implement and does notrequire the design of a new launch system. However, this

kind of configuration has the disadvantages of cross couplingdue to the spinning airframe. Besides, uncertainties in con-trol effectiveness and moment coefficients are among thepractical challenges in the control system design. Finally,the sensors are of limited performance, due to the low costand small size specification.

These disadvantages can be handled by developing aflight autopilot using modern multivariable control methods,such as robust control [1, 2] and gainscheduling control[3, 4]. For the dual-channel-controlled spinning rocket, var-ious autopilots were designed, such as rate loop autopilot[5], attitude autopilot [6], acceleration autopilot [7], andthree-loop autopilot [8]. However, these related works werecarried out under the nominal condition without consideringuncertainties which may be experienced during the wholeflight trajectory. Additionally, it is difficult to design an auto-pilot for a guided spinning rocket with excellent performanceusing the traditional separate channel design method.

Adaptive control is known as a proper method to dealwith uncertainties and has been used in numerous

HindawiInternational Journal of Aerospace EngineeringVolume 2018, Article ID 1427487, 12 pageshttps://doi.org/10.1155/2018/1427487

applications. Therefore, this is aimed at investigating thepotential of robust adaptive control for improving stabilityand performance of the spinning rocket autopilot. A generictransport aircraft autopilot was designed by employing amodified MRAC scheme to guarantee the transient perfor-mance [9]. A Lyapunov-based model reference adaptivePD/PID controller for Satellite Launch Vehicle (SLV) sys-tems was proposed to improve the tracking performanceand robustness under wind disturbances [10]. An adaptiveintegral feedback controller for pitch and yaw channels ofan autonomous underwater vehicle (AUV) was designed tohandle the actuator saturations [11]. These controllersrequire that the system state must be measurable [12], whichmay not always be possible. For this reason, there has been anincreasing motivation to develop an adaptive output feed-back controller. Existing classical methods of multi-inputand multi-output (MIMO) output feedback adaptive controlare applicable for square systems; that is, the plant has thesame number of inputs and outputs [13, 14]. Recently, anoutput feedback adaptive spinning rocket autopilot wasdesigned, but only the acceleration information was used toconstruct the autopilot [15]. In order to make full use of allthe measurable information, the output feedback adaptiveautopilot for a spinning rocket with four measurable outputsand two control inputs, which is a nonsquare system, needsto be designed. The main challenge is to make the closed-loop transfer functions satisfy the strictly positive real(SPR) property or guarantee the strict passivity of theclosed-loop system. These two properties were proved to beequivalent for a linear time-invariant system [16]. For thesquare system, necessary and sufficient conditions for pas-sifiability of the linear system by output feedback were pre-sented in [17, 18]. An observer-based method was alsoincluded in the design of controllers to guarantee the thatclosed-loop system is SPR [19]. For the nonsquare system,the squaring-down procedure was employed for passificationdesign [20]. And necessary and sufficient conditions for pas-sifiability of nonsquare systems by output feedback weregiven in [21].

The main contribution of this paper is to combine theobserver-based method and squaring-down method to

design an adaptive output feedback autopilot for guidedspinning rockets. First, a mixing matrix M is designed tomake the modified error dynamic model a square system.Then, a Luenberger observer which also serves as the ref-erence model is employed to make the error dynamicmodel satisfy the SPR property.

The remainder of this paper is structured in the follow-ing manner. Section 2 develops the uncertain dynamicmodel of a dual-channel-controlled spinning rocket. Sec-tion 3 presents the robust adaptive output-feedback autopi-lot design using robust SPR lemma. Section 4 demonstratesthe performance of the robust adaptive output-feedbackautopilot via numerical simulations. Finally, Section 5 con-cludes this paper.

2. Mathematical Model

The guided spinning rocket considered in this paper is anaxially symmetric rolling airframe, as is shown in Figure 1.Two pairs of canard rotating with the airframe are employedas control surfaces twisting and steering the rocket. Themaneuver of the airframe requires an autopilot that is ableto track different kinds of signals under a differentdynamic environment, with slight or even no change ofthe autopilot architecture.

2.1. Dynamical Equation. According to Newton’s second law,the translational motion of center of gravity (cg) can bedescribed in the nonrolling body coordinate system as fol-lows [22]:

mdVA

dt=m

∂VN

∂t+ ωr ×VN = F +mg, 1

where m is the mass of the spinning rocket, VA = x, y, z T

is the velocity of cg represented in inertial coordinate,VN = u, v,w T is the velocity of cg in the nonrolling bodycoordinate, and ωr is the angular rate of the nonrolling bodycoordinate system with respect to the inertial coordinate sys-tem. F is the aerodynamic force, and g is the acceleration of

Canards

u

V

O

q

v

r

wXN

YN

ZN

�훼�훽

Figure 1: Sketch of guided spinning projectile.

2 International Journal of Aerospace Engineering

gravity, which are all represented in the nonrolling bodycoordinate system. All the variables in (1) can be modeled as

ωr = −r tan θ, q, r T , 2

F =Fx

Fy

Fz

=QS

−Cx

−CNββ + CNδδz

−CNαα − CNδδy

, 3

g =gx

gy

gz

=−g sin θ

−0g cos θ

, 4

where q and r are angular rates along OYN and OZN definedin Figure 1; θ is the pitch angle of the spinning rocket;Q is thedynamic pressure; S is the reference area; Cx is the drag coef-ficient slope; CNα and CNβ are the normal force coefficientslopes, due to the character of symmetric CNα = CNβ, so asother coefficients related with α and β; CNδ is the controlforce coefficient slope; δy and δz represent the actuatordeflection angles in the nonrolling coordinate system; and αand β are the angle of attack and the sideslip angle, respec-tively. The expressions of α and β are presented as

α = arctan wu

≈wV,

β = arcsin vu

≈vV

5

By substituting (2), (3), and (4) into (1), the force equa-tions are obtained:

u +wq − vr = Fx

m+ gx,

v + ur −wr tan θ =Fy

m+ gy,

w − uq − vr tan θ = Fz

m+ gz

6

According to the theorem of angular momentum, therotational motion of the airframe can be described in thenonrolling body coordinate system as

dHdt

= ∂H∂t

+ ωr ×H =M, 7

whereH is the angular momentum andM is the moment act-ing on the rocket. All the variables in (7) are described in thenonrolling body coordinate system as

H = Ixp, Iyq, IzrT , 8

∂H∂t

= Ixp, Iyq, IzrT , 9

where Ix, Iy, and Iz are moments of inertia, due to the char-acter of symmetric Iy = Iz ; p is the spinning rate in the non-rolling coordinate.

For a canard-controlled spinning rocket, positive δz andδy create positive moments in Mz and My, respectively.Therefore, the aerodynamic moment can be expressed as

M =Mx

My

Mz

=QSl

Clδxδx − ClpplV

Cmαα − CmqqlV

− CmpαβplV

+ Cmδδy

−Cmαβ − CmqrlV

− CmpααplV

+ Cmδδz

,

10

where l is the reference length; Clδx is the rolling momentcoefficient due to the canted angle of tails, which is denotedas δx ; Clp is the rolling damping moment coefficient; Cmα isthe static moment coefficient; Cmq is the damping momentcoefficient; Cmpα is the Magnus moment coefficient; andCmδ is the control moment coefficient.

Remark 1. The guided spinning rocket considered in thispaper is a fin-stabilized airframe. For a fin-stabilized rocket,the classic Magnus force is typically ignored in (3) since itseffect is rather small for slowly rolling rockets. However,the Magnus moment with physical mechanisms specificto fin-stabilized rockets cannot be ignored in (10), whichcan be expressed as a dynamic side moment due to spinrate p and angle of attack α or sideslip angel β in the formof Cmpαα pl/V [23].

By substituting (8), (9), and (10) into (7), the momentequations are obtained:

Ixp =Mx,Iyq + Ixpr + Iyr

2 tan θ =My ,Iyr − Ixpq − Iyqr tan θ =Mz

11

Equations (6) and (11) are the nonlinear dynamicalmodel of a spinning rocket, which represents the angularmotion of the spinning rocket.

2.2. Linearized Lateral Equation. For flight control whichcorresponds to the tracking of the dual-canard-controlledspinning rocket acceleration command, only the lateraldynamic system is necessary. In order to simplify the designprocedure of the autopilot, some common and reasonableassumptions are made to linearize the lateral dynamic modelof the spinning projectile [7]:

Assumption 1. The velocity, roll rate, mass, and aerody-namic coefficients of the rocket remain constant over ashort time period.

Assumption 2. For simplicity, the gravity and the smallcanard force are ignored. The lateral velocities v and w aresmall with respect to the axis velocity u, so that u ≈V .

3International Journal of Aerospace Engineering

Applying the above assumptions to (6) and (11), thelinearized angular motion of the spinning rocket can bedescribed as

β = −r − cNαβ,α = q − cNαα,q = cmαα − cmpαPpβ − cmqq − Ppr + cmδδy ,r = −cmαβ − cmpαPpα − cmqr − Ppq + cmδδz ,

12

where the dimensionless coefficients are defined as cNα =QSCNα/mV , cmα =QSlCmα/Iy, cmq =QSl2Cmq/IyV , cmpα =QSl2Cmpα/IxV , cmδ =QSlCmδ/Iy, and Pp = p Ix/Iy .

Usually, not all the states of a spinning rocket in (12) canbe measured by sensors, especially the angle of attack α andthe sideslip angle β. Low-cost inertial measurement units(IMUs) are the general sensors installed in the spinningrocket, which can provide the accelerations ay, az , and angu-lar rates q and r of the lateral motion. In this condition, theaccelerations measured by IMU along the OYN and OZNaxes can be expressed as

ay

az=

−cNαV 00 −cNαV

β

α13

Hence, the overall dynamic equation of the spinningrocket can be rearranged to the following state-space form,

xp t = Apxp t + Bpu t ,yp t = Cpxp t ,z t = Czxp t ,

14

where xp = β α q r T ∈ℝnp is the state vector, u =δyδz

T ∈ℝm is the control input vector, yp =ay az q r T ∈ℝnp is the measurement output vector,

and z = ayaz ∈ℝr is the regulated output vector that is alsomeasured. Ap ∈ℝnp×np is the system matrix, Bp ∈ℝnp×m is theinput matrix, Cp ∈ℝpp×np is the measurement output matrix,and Cz ∈ℝr×np is the regulated output matrix, and they are allknown matrices.

2.3. Uncertain Dynamic Equation. The dynamic model pre-sented in (14) is the ideal case where all the matrices areknown. In reality, these matrices are unknown and areobtained through various methods. The system matrix Ap

and output matrices Cp and Cz can be determined throughwind-tunnel tests fairly accurately. In contrast, the inputmatrix Bp may not be accurate, as control inputs are sub-jected to perturbations in the flight period. First, the orienta-tion of the canards can lead to the dynamical parametersdifferent from the trim condition as represented by the nom-inal model. This effect can be modeled as an additive termΨT

p xp. Second, canard failure caused by electronic circuit or

control surface damage is another effect which may causethe inaccuracy of input matrices. This effect can be modeledas constant matched uncertainty weights Λ. The modifieduncertain dynamic model is given as

xp t = Apxp t + Bp Λu t +ΨTp xp ,

yp t = Cpxp t ,z t = Czxp t ,

15

where Λ ∈ℝm×m and Ψp ∈ℝnp×m are all nonsingularmatrices.

3. Robust Adaptive Autopilot Design

The fight sequence of guided spinning rockets is decom-posed into three flight phases: boost phase, free flightphase, and guided phase. In this section, we mainly con-sider the guidance phase, during which the autopilotbegins to work. In the guided phase, guidance law givesthe desired commands which are usually in the form ofacceleration, to the autopilot based on the target androcket’s relative motion information. Therefore, designingan acceleration autopilot is more reasonable and effective.The underlying problem is to design an acceleration auto-pilot that is able to track different kinds of signals underdifferent dynamic environments, with slight or even nochange of the autopilot architecture. So, the output feed-back adaptive control method is employed to design theacceleration autopilot.

It should be noted that the output feedback adaptiveautopilot requires the dynamical system in (15) as thecontrollable, observable, and minimum phases, whichcan be satisfied by the canard-controlled spinning rocket.The uncertainty Ψp is bound with Ψp <Ω <∞ andrank CpBp =m.

3.1. Addition of Integral Error. Command tracking anddisturbance rejection are the main issues to be handledthrough the integral action [24, 25]. In this light, considerthe regulated output signal tracking error which is definedas follows:

xe = zcmd − z, 16

where zcmd is a piecewise continuous command signal. Anintegral error state is defined as

xe =t

0xedt =

t

0zcmd − z dt 17

Appending (17) to the plant in (15), the augmentedopen-loop dynamics is given by

4 International Journal of Aerospace Engineering

xp

xe=

Ap 0np×r

−Cz 0r×rA

xp

xex

Bp

0r×mB

Λu +ΨTp xp

+0np×rIr×rBcmd

zcmd ,

yp

xey

=Cp 0np×r0r×np Ir×r

C

xp

xe

18

Therefore, (18) can be written more compactly as follows:

x = Ax + B Λu t +ΨTx + Bcmdzcmd ,y = Cx,

19

where Ψ = ΨTp 0m×r

Tis unknown. Define n = np + r, then

x ∈ℝn.

3.2. Control Architecture. Following the design procedurein [25], the controller is divided into two parts: a baselinecontroller and an adaptive controller. The control input uin (15) is described as

u = ubl + uad , 20

where uad is the adaptive component and ubl is the baselinecomponent, defined as

ubl = KTx xm, 21

where Kx is designed by applying the linear quadratic regula-tor (LQR) technique on the nominal plant model, that is,when Λ = I and Ψ = 0. xm is the state of the observer,

xm = Amxm + Bcmdzcmd + L ym − y ,ym = Cxm,

22

where Am = A + BKTx , and KT

x is selected such that Am isHurwitz.

Remark 2. The observer (22) serves as three purposes:

(1) Luenberger observer. Matrix L is equivalent to anobserver gain, for which the purpose is to estimatethe state vector x in plant (19). The estimated statevector helps to construct the baseline controller ubl .

(2) Reference model. As our ultimate goal is to establisha model reference adaptive controller, (22) serves asa closed-loop reference model (CRM), which hasbeen proved to result in improving transient prop-erties [26–28].

(3) Robust compensator. The output error feedback termL ym − y can be treated as a robust compensator,making the error dynamics satisfy the SPR property.

The adaptive controller uad is defined as

uad =ΘT t xm, 23

where Θ t is the estimated uncertain parameter, to beupdated by a well-designed update law. Substituting the con-troller (20) into the augmented plant (19),

x = Ax + B Λ Kx +Θ t T +ΨTx + Bcmdzcmd ,

y = Cx24

If the proposed control architecture can realize accuratecommand tracking, an ideal uncertain parameter Θ∗ mustexists and satisfies the following matching condition,

Am = A + BΨT + BΛ Θ∗T + KTx 25

with Θ∗T = Λ−1 − I KTx −Λ−1ΨT .

The error dynamic model between the reference model(22) and the closed-loop plant (24) is resorted to accomplishthe whole design procedure. The state error ex = x − xmand the estimate parameter error Θ~ t =Θ t −Θ∗ satisfythe dynamics

ex = A + LC + BΨT ex + BΛΘTt xm,

ey = Cex,26

where ey is the measured output error. The problem of find-ing a stabilizing adaptive controller is equivalent to findingan observer gain L and an adaptive law for Θ t in (26), sothat the underlying transfer function matrix is SPR.

The following adaptive law is employed to update theestimated uncertain parameter,

Θ t = −ΓxmeTy MT sgn Λ , 27

where Γ, the adaptive gain, is a positive diagonal freedesign matrix, M ∈ℝm×n is a mixing matrix to “square up”the transfer function, and sgn Λ represents the sign matrixof the input uncertainty Λ.

3.3. Robust SPR Design. The SPR property is usuallyemployed to design a stable adaptive law for the errordynamic model (26) of the uncertain plant. However, the def-inition of SPR is restricted to square transfer functions. Asthe transfer function of (26) is nonsquare, a suitable mixingmatrixM has to be chosen to make the SPR properties appli-cable to the error dynamic model, yielding

G s =MC sI − A − LC − BΨT −1B, 28

where G s is a square transfer function of the modified errordynamic model,

5International Journal of Aerospace Engineering

ex = A + LC + BΨT ex + BΛΘTt xm,

em =MCex29

Thus, the design of an output feedback adaptive control-ler is converted to selecting mixing matrix M ∈ℝm×n andobserver matrix L ∈ℝn×n, such that the overall closed-loopsystem is stable.

The choice of mixing matrix M is not unique. M can becomputed as

M = BTPC−1, 30

which satisfies PB = CTMT .

Lemma 1. Given the strictly proper transfer matrix G s withstabilizable and detectable realization (A, B, C,D), where A∈ℝn×n is asymptotically stable, B ∈ℝn×m, C ∈ℝm×n, and D∈ℝm×m; then, G s is SPR if, and only if, there exists matricesP = P⊤ > 0, P ∈ℝn×n, Q ∈ℝn×m

, and W ∈ℝm×m such that

A⊤P + PA = −QQT ,PB − CT = −QW,D +DT =WTW

31

Proof 1. The complete proof on the Lemma 1 can be foundin [29] Lemma 3.1.

Lemma 2. Let H and E be given matrices of appropriatedimensions, and F satisfies FFT < I; for any ε > 0, there is

HFE + HFE T ≤ εHHT + ε−1ETE 32

Proof 2. The complete proof on the Lemma 2 can be foundin [30].

The output error feedback term L ym − y can be treatedas a robust compensator um for the error dynamic model,making the error dynamics satisfy the SPR property. First,rewrite the error dynamic model (29) into the canonical formof robust control,

ex = A + ΔA ex + B1w + B2um,em = Cmex,

33

where ΔA = BΨT is defined as the uncertain statematrix; B1 = B, B2 = In, Cm =MC, w =ΛΘT t xm, andum = L ym − y = LCex are defined as the modified virtualcontrol inputs for modified error dynamics.

The modified error dynamics is SPR and stable if andonly if Lemma 1 is satisfied. Equation (29) is strictlyproper; that is, D = 0, so Lemma 1 cannot be useddirectly. For the case D = 0, the above set of equationsin Lemma 1 reduces to the first two equations with W = 0[29]. The observer gain L is designed by finding a state-feedback control of (33), and the following inequality shouldbe satisfied:

A + ΔA + LC TP + P A + ΔA + LC < 0, 34

PB1 = CTMT 35

XAT + AX + B2W +WTB2 EX T B1∗ −γ 0∗ ∗ −

1γ

< 0, 36

where E =ΩIn. The matrix in (36) is symmetric, and ∗ rep-resents the transposition of the corresponding item in thematrix.

Lemma 3. Given an uncertain plant (33), the closed-loopsystem is stable and SPR, if and only if there exist symmetricpositive matrices X and W and a constant γ > 0 satisfies

Proof 3. The state matrix uncertainty ΔA can be describedas follows:

ΔA = B1ΨT = B1FE, 37

where F =ΨT /Ω, and the inequality FFT < I holds.

Define P = X−1 and K = LC =WX−1, and multiplyinequality (36) on both sides by the matrix diag P, I, I ,

A + B2KTP + A + B2K ET PB1

∗ −γ 0∗ ∗ −

1γ

< 0 38

By the Schur complement, the LMI defined in (36) isequivalent to the following inequality:

A + B2KTP + P A + B2K + γPB1B

T1 P + 1

γETE < 0 39

With Lemma 2, we have

PB1FE + PB1FET < γPB1B

T1 P + 1

γETE 40

Thus, inequality (39) is equivalent to the followinginequality,

A + B2K + B1FETP + P A + B2K + B1FE < 0, 41

which means A + ΔA + LC TP + P A + ΔA + LC < 0. Com-bining PB = CTMT (34) holds. So, the modified error dynam-ics is stable and SPR.

The observer gain L can be obtained by solving the feasi-ble LMI with any widely available numerical LMI solver. Itcan be expressed as

L =W∗X∗−1C−1, 42

where W∗ and X∗ are the feasible solution of (36).Figure 2 shows the structure of the control loop, where

the plant is the dynamical model of the spinning rocket in(14), the reference model is (22), and the adaptive law is (23),

6 International Journal of Aerospace Engineering

3.4. Stability Analysis. Given the uncertain linear system in(19), the reference model in (22) with L as in (42), the controlarchitecture in (20), and the update law in (27) result inglobal stability, with limt→∞ex = 0.

Consider the following Lyapunov function candidate,

V ex,Θ = eTx Pex + tr Λ ΘΓ−1Θ , 43

where Λ represents the absolute value of each entry of thematrix element, satisfying Λ = sgn Λ Λ .

The time derivative of (43) along the system trajectories isgiven by

V = −eTx Qex + 2eTx PBΛΘTxm + 2tr Λ ΘTΓ−1Θ , 44

where A + BΨT + LCTP + P A + BΨT + LC = −Q < 0.

Substituting the update law given in (27) yields

V = −eTx Qex + 2eTx PBΛΘTxm − 2tr Λ ΘT

xmeTy M

T sgn Λ

= −eTx Qex + 2eTx PBΛΘTxm − 2eTy MTΛΘT

xm

= −eTx Qex + 2eTx PBΛΘTxm − 2eTx CTMTΛΘT

xm= −eTx Qex ≤ 0,

45

which implies that V is a Lyapunov function. Since V > 0 andV ≤ 0, then V t ≤V 0 <∞. Thus, V t ∈ L∞, which meansex, Θ ∈ L∞. Since zcmd , ex ∈ L∞, and the reference model arestable, xm ∈ L∞, which implies that xp ∈ L∞.

Furthermore, asymptotic stability of the tracking errors isdemonstrated by invoking LaSalle’s invariance principle,which states that, for a negative semidefinite Lyapunov sys-tem in the form of (45), all system trajectories are containedwithin the domain Ω0 = ex,Θ t ∣V ex,Θ t , t ≤V ex,Θ t , 0 , where the subscript 0 denotes the initial condi-tions, and the entire state space ex,Θ t ultimately reachesthe domain Ωf =Ω0 ∩Ωz , where Ωz denotes the domaindefined by the Lyapunov derivative identical to zero. In otherwords, the state space ultimately reaches the domain definedby V ex,Θ t , t ≡ 0 [31, 32]. Because V ex,Θ t , t isnegative-definite in ex, the system ends with ex ≡ 0. Thus,x t → xm t and the bound reference tracking of zcmd byz follow from the stability of the closed-loop system.

Remark 3. In this paper, the adaptive parameter Θ t is notguaranteed to converge to its true unknown value Θ∗ nor isit assured to converge to constant value in any way. All thatis known is that the unknown parameter remains uniformlybound in time. Sufficient conditions for parameter conver-gence are known as persistency of excitation.

4. Case Study

In this section, numerical examples are performed toevaluate the performance of the output feedback adaptivecontrol scheme applied to the autopilot design for thedual-canard-controlled spinning rocket. All the simulationsare performed in MATLAB R2017b with a 64-bit processor,16GB memory, and 0.001 s time-step. Robust Control Tool-box 6.4 in MATLAB is employed to obtain a feasible solutionof observer (36). And all the simulations are performed withan adaptive controller acting on the nonlinear model.

The nominal model for autopilot design is the linearizedlateral dynamics of the spinning rocket at the speed V of581m/s and altitude H of 5000m. The numerical values forthe linear system matrices can be found in Appendix A.

All the eigenvalues of the nominal plant system matrixAp have a negative real part, which means the nominalplant is stable.

We show the response of the closed-loop system with thebaseline controller and adaptive controller, respectively.Square-wave commands with amplitude of 1m/s2 and fre-quency of 1Hz are applied to the pitch and yaw channel;there is a 0.1 s time lag between the two channels. It isobserved in Figure 3 that both the baseline controller andthe adaptive controller are able to guide the spinning rocketfollowing the acceleration command and eventually achieved

Plant

1AUX

Referencemodel

Adaptive law

ey

ym

yp

xmTKx

xe

s

zcmd

z

y

uad

ubl u

−−

++

++

Figure 2: Control architecture.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

az (Ada)az (Base)az (Cmd)

ay (Ada)ay (Base)ay (Cmd)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10Time (s)

−0.5

0

0.5

1

1.5

az (m

/s2 )

−0.5

0

0.5

1

1.5

ay (m

/s2 )

Figure 3: The nominal case tracking performance.

7International Journal of Aerospace Engineering

a zero tracking error. Figure 4 presents the behavior of con-trol signals in the pitch and yaw channels. Only a smallamount of canard deflection angles is used to achieve thecommand tracking. The angular rates of pitch and yawchannels are presented in Figure 5, which shows a goodperformance in both channels. Considering simulationresults for this case, both the proposed adaptive controllerand the baseline controller present a proper performancein the presence of the disturbances enforced due to thechannel couplings.

The same controllers were used in the presence of thefollowing uncertainties:

Λ = 0 9I, Ψ =0 0 0 12 0 0 00 0 0 0 12 0 0

46

Such uncertainties stem from the fact that the modelparameters are expected to vary significantly (by about

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

�훿z (Ada)�훿z (Base)

�훿y (Ada)�훿y (Base)

−10

−5

0

5

10

�훿y (d

eg)

−10

−5

0

5

10

�훿z (d

eg)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10Time (s)

Figure 4: The nominal case control signal.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

r (Ada)r (Base)

q (Ada)q (Base)

−4

−2

0

2

4

q (d

eg/s

)

−4

−2

0

2

4

r (de

g/s)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10Time (s)

Figure 5: The nominal case angular rates.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

az (Base)az (Cmd)

az (Ada)

ay (Base)ay (Cmd)

ay (Ada)

−0.5

0

0.5

1

1.5

ay (m

/s2 )

−0.5

0

0.5

1

1.5

az (m

/s2 )

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10Time (s)

Figure 6: The uncertain case tracking performance.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

�훿y (Base)�훿y (Ada)

�훿z (Base)�훿z (Ada)

−10

−5

0

5

10

�훿y (d

eg)

−10

−5

0

5

10

�훿z (d

eg)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10Time (s)

Figure 7: The uncertain case control signal.

8 International Journal of Aerospace Engineering

30% from the nominal A matrix) compared to those deter-mined from wind channel tests. At this time, the eigenvaluesof the system matrix have a positive real part, which meansthe system is unstable.

Outcomes of the pitch and yaw channel autopilots inthe same set-point are presented in Figure 6. In this case,the baseline controller is not able to suppress the unstablepitch and yaw modes, whereas the adaptive controller isable. The behavior of canard angles in the pitch and yawchannels is presented in Figure 7. The amplitude of canardangles for the adaptive controller is slightly bigger thanthat of the nominal model, due to the uncertainty. Theangular rates of pitch and yaw channels are presented inFigure 8, which implies that the adaptive controller showsa better performance.

In order to visualize the destructive effect of the con-trol effectiveness loss, simulations with different controleffectiveness are performed. According to Figures 9 and 10,the proposed adaptive controller can autoadjust the canardangles to handle the loss of control effectiveness. Consideringsimulation results for this scenario, the proposed adaptivecontroller presents a proper performance in the presence ofthe disturbances enforced due to the channel couplings andcontrol effectiveness loss.

Measurement noise is also taken into account in order tocarry out more investigations in the perturbed conditions. Tothis end, a worst-case condition is enforced; a white noisewith a standard deviation of 0.25(°/s) is injected into the rate

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

r (Base)r (Ada)

q (Base)q (Ada)

−5

0

5

q (d

eg/s

)

−5

0

5

r (de

g/s)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10Time (s)

Figure 8: The uncertain case angular rates.

0 0.2 0.4 0.6 0.8 1−10

0

10

0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

�훿y, �훿

z (d

eg)

ay, a

z (m

/s2 )

Time (s)

−5

0

5

q, r

(deg

/s)

�훿z

�훿y

r

q

az

ayazcmd

aycmd

Figure 9: Control performance with Λ = 0 9I.

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

�훿y�훿z

qr

ayaz

aycmd

azcmd

−20

0

20

�훿y, �훿

z (d

eg)

−0.5

0

0.5

1

1.5

ay, a

z (m

/s2 )

−5

0

5

q, r

(deg

/s)

0.2 0.4 0.6 0.8 10Time (s)

Figure 10: Control performance with Λ = 0 6I.

9International Journal of Aerospace Engineering

gyroscopes of each channel, which is corresponding to thenoise of low-cost MEMS gyroscopes. Figure 11 shows thatthe adaptive controller was robust in the presence of mea-surement noise in the angular rate feedback loop.

We analyze the robustness of the overall closed-loop sys-tem with the steady-state gain KT

x +ΨT in what follows. Thegain and phase margin for the baseline controller and theclosed-loop adaptive controller under the uncertain modelare calculated as in Appendix B:

GMb1 = −818 1 dB PMb1 = ±51 9°,GMad = −14 231 6 dB PMad = ±58 3°

47

It can be seen that the adaptive controller, in general, ismore robust than the baseline controller is.

5. Conclusion

This paper presents a robust adaptive output feedback auto-pilot for a guided spinning rocket to resistant external distur-bance and maintain precision command tracking. Theautopilot is composed of a baseline controller augmentedwith an adaptive component to accommodate control effec-tiveness uncertainty and matched plant uncertainty, and itmakes use of the closed-loop reference model to improvethe transient properties of the overall adaptive system. Adap-tive control is applicable to nonsquare systems through

designing mixing matrix M and the observer gain matrix L.This procedure only needs to solve a set of linear matrixinequalities, which is given by the robust SPR lemma. Theperformance of the proposed adaptive output feedback con-troller is evaluated by numerical simulations when appliedto the lateral dynamic model of a guided spinning rocket.The simulation results showed that the robust adaptive out-put controller can achieve asymptotic command trackingwith significant uncertainty in control effectiveness, momentcoefficient, and measurement noise.

Appendix

A. System Matrices

The nominal dynamical plant matrices for a guided spinningrocket of velocity V = 581m/s, altitude H = 5000m, andspinning rate p = 25 4 rad/s are

Ap =

−0 35 0 0 −10 −0 35 1 0

1 28 −54 89 −0 63 −0 0754 89 1 28 0 07 −0 63

,

Bp =

0 00 0

12 15 00 12 15

,

Cp =

−206 59 0 0 00 −206 59 0 00 0 1 00 0 0 1

,

Cz =−206 59 0 0 0

0 −206 59 0 0

A 1

The dynamical system Ap, Bp, Cp, 0 is controllableand observable. There is no transmission zero in the plant.The linear control design parameters are given as Qlqr =diag 0 5, 0 5,0,0,0 5, 0 5 and Rlqr = diag 0 005,0 005 .Following the design procedure, the baseline controller gainmatrix can be computed using the MATLAB command lqr:

KTx =

0 12 −133 98 −4 64 0 0 02 −10−133 98 0 12 0 −4 64 10 0

A 2

Robust Control Toolbox 6.4 is used to solve the LMI in(36), resulting in mixing matrix M and observer L

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

q

r

ayaz

aycmd

azcmd

�훿y�훿z

−10

0

10

�훿y, �훿

z (d

eg)

−0.5

0

0.5

1

1.5

ay, a

z (m

/s2 )

−5

0

5

q, r

(deg

/s)

0.2 0.4 0.6 0.8 10Time (s)

Figure 11: Adaptive controller performance with measurementnoise.

10 International Journal of Aerospace Engineering

M =0 0 0 0467 0 0 00 0 0 0 0467 0 0

,

L =

0 0 −0 64 −26 9 −103 29 00 0 26 9 −0 64 0 −103 290 −0 13 0 0 0 0

0 13 0 0 0 0 00 5 0 0 0 −0 5 00 0 5 0 0 0 −0 5

A 3

B. Multivariable Gain and Phase Margins

The multivariable gain and phase margin was calculatedalong the lines with [25], where the gain and phase marginsare defined using the return difference matrix σ I + Lu sand the stability robustness matrix σ I + Lu

−1 s .The operator σ corresponds to the minimum singular

values; Lu s denotes the input loop transfer function.First, the minimum value of the return difference and the

stability robustness transfer function over all frequency ω aredefined as ασ and βσ, respectively:

ασ =minω

σ I + Lu s ,

βσ =minω

σ I + Lu−1 s

B 1

Then, the gain and phase margin of the return differencematrices GMασ and PMασ and stability robustness matricesGMβσ and PMβσ are calculated:

GMασ = 1 + ασ −1 1 − ασ −1 ,GMβσ = 1 − βσ + 1βσ ,

PMασ = ±2 sin−1 ασ

2 ,

PMβσ = ±2 sin−1 βσ

2

B 2

The union of these gain and phase margins in (B.2) yieldsthe multivariable gain margin GM and phase margin PM,which can be written compactly as

GM= GMLGMU ,PM = PMLPMU

B 3

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The grant support from the National Natural Science Foun-dation of China (nos. 11202023 and 11532002) is greatlyacknowledged.

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