# Robust Adaptive Output Feedback Control for a Guided Spinning Rocket

Authors: Zhongjiao Shi; Liangyu Zhao; Yeqing Zhu
Journal: International Journal of Aerospace Engineering
 (2018)

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## Abstract

An adaptive autopilot is presented for the pitch and yaw channels of a guided spinning rocket. Firstly, the uncertain dynamic model of a guided spinning rocket is established, which is used to evaluate the performance of the proposed adaptive autopilot. Secondly, a robust adaptive output feedback autopilot containing a baseline component and an adaptive component is designed. The main challenge 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 to evaluate the robustness and tracking performance of the proposed robust adaptive autopilot. The simulation results showed that the robust adaptive output autopilot can achieve asymptotic command tracking with significant uncertainty in control effectiveness, moment coefficient, and measurement noise.


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## Body

## 1. Introduction

Traditional artillery ballistic gun-launched munitions cannot satisfy more and more stringent performance requirements about precision-strike capability, dispersion error reduction, and range augmentation required on modern battlefields. Complex guided systems such as missiles can meet these requirements, but they remain expensive due to the integration of high-quality actuators and sensors. The idea is to develop a guided rocket, which permits to reach a compromise between the low cost of ballistic projectiles and the necessity to have high-performance systems with efficient control algorithms. The guided rocket chosen in this work is a canard-controlled spinning configuration, which makes use of an existing multiple launch rocket system (MLRS). This one has the advantage of simplifying the structure of the control system, avoiding asymmetric ablation, relaxing the manufacturing error tolerance, and improving the penetration ability. It is also easy to implement and does not require the design of a new launch system. However, this kind of configuration has the disadvantages of cross coupling due to the spinning airframe. Besides, uncertainties in control effectiveness and moment coefficients are among the practical challenges in the control system design. Finally, the sensors are of limited performance, due to the low cost and small size specification.These disadvantages can be handled by developing a flight autopilot using modern multivariable control methods, such as robust control [1, 2] and gainscheduling control [3, 4]. For the dual-channel-controlled spinning rocket, various autopilots were designed, such as rate loop autopilot [5], attitude autopilot [6], acceleration autopilot [7], and three-loop autopilot [8]. However, these related works were carried out under the nominal condition without considering uncertainties which may be experienced during the whole flight trajectory. Additionally, it is difficult to design an autopilot for a guided spinning rocket with excellent performance using the traditional separate channel design method.Adaptive control is known as a proper method to deal with uncertainties and has been used in numerous applications. Therefore, this is aimed at investigating the potential of robust adaptive control for improving stability and performance of the spinning rocket autopilot. A generic transport aircraft autopilot was designed by employing a modified MRAC scheme to guarantee the transient performance [9]. A Lyapunov-based model reference adaptive PD/PID controller for Satellite Launch Vehicle (SLV) systems was proposed to improve the tracking performance and robustness under wind disturbances [10]. An adaptive integral feedback controller for pitch and yaw channels of an autonomous underwater vehicle (AUV) was designed to handle the actuator saturations [11]. These controllers require that the system state must be measurable [12], which may not always be possible. For this reason, there has been an increasing motivation to develop an adaptive output feedback controller. Existing classical methods of multi-input and multi-output (MIMO) output feedback adaptive control are applicable for square systems; that is, the plant has the same number of inputs and outputs [13, 14]. Recently, an output feedback adaptive spinning rocket autopilot was designed, but only the acceleration information was used to construct the autopilot [15]. In order to make full use of all the measurable information, the output feedback adaptive autopilot for a spinning rocket with four measurable outputs and two control inputs, which is a nonsquare system, needs to 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 the closed-loop system. These two properties were proved to be equivalent for a linear time-invariant system [16]. For the square system, necessary and sufficient conditions for passifiability of the linear system by output feedback were presented in [17, 18]. An observer-based method was also included in the design of controllers to guarantee the that closed-loop system is SPR [19]. For the nonsquare system, the squaring-down procedure was employed for passification design [20]. And necessary and sufficient conditions for passifiability of nonsquare systems by output feedback were given in [21].The main contribution of this paper is to combine the observer-based method and squaring-down method to design an adaptive output feedback autopilot for guided spinning rockets. First, a mixing matrixM is designed to make the modified error dynamic model a square system. Then, a Luenberger observer which also serves as the reference model is employed to make the error dynamic model satisfy the SPR property.The remainder of this paper is structured in the following manner. Section2 develops the uncertain dynamic model of a dual-channel-controlled spinning rocket. Section 3 presents the robust adaptive output-feedback autopilot design using robust SPR lemma. Section 4 demonstrates the performance of the robust adaptive output-feedback autopilot via numerical simulations. Finally, Section 5 concludes this paper.


## 2. Mathematical Model

The guided spinning rocket considered in this paper is an axially symmetric rolling airframe, as is shown in Figure1. Two pairs of canard rotating with the airframe are employed as control surfaces twisting and steering the rocket. The maneuver of the airframe requires an autopilot that is able to track different kinds of signals under a different dynamic environment, with slight or even no change of the autopilot architecture.Figure 1
Sketch of guided spinning projectile.


### 2.1. Dynamical Equation

According to Newton’s second law, the translational motion of center of gravity (cg) can be described in the nonrolling body coordinate system as follows [22]:
(1)mdVAdt=m∂VN∂t+ωr×VN=F+mg,where m is the mass of the spinning rocket, VA=x,y,zT is the velocity of cg represented in inertial coordinate, VN=u,v,wT is the velocity of cg in the nonrolling body coordinate, and ωr is the angular rate of the nonrolling body coordinate system with respect to the inertial coordinate system. F is the aerodynamic force, and g is the acceleration of gravity, which are all represented in the nonrolling body coordinate system. All the variables in (1) can be modeled as
(2)ωr=−rtanθ,q,rT,(3)F=FxFyFz=QS−Cx−CNββ+CNδδz−CNαα−CNδδy,(4)g=gxgygz=−gsinθ−0gcosθ,where q and r are angular rates along OYN and OZN defined in Figure 1; θ is the pitch angle of the spinning rocket; Q is the dynamic pressure; S is the reference area; Cx is the drag coefficient slope; CNα and CNβ are the normal force coefficient slopes, due to the character of symmetric CNα=CNβ, so as other coefficients related with α and β; CNδ is the control force coefficient slope; δy and δz represent the actuator deflection angles in the nonrolling coordinate system; and α and β are the angle of attack and the sideslip angle, respectively. The expressions of α and β are presented as
(5)α=arctanwu≈wV,β=arcsinvu≈vV.By substituting (2), (3), and (4) into (1), the force equations are obtained:
(6)u̇+wq−vr=Fxm+gx,v̇+ur−wrtanθ=Fym+gy,ẇ−uq−vrtanθ=Fzm+gz.According to the theorem of angular momentum, the rotational motion of the airframe can be described in the nonrolling body coordinate system as(7)dHdt=∂H∂t+ωr×H=M,where H is the angular momentum and M is the moment acting on the rocket. All the variables in (7) are described in the nonrolling body coordinate system as
(8)H=Ixp,Iyq,IzrT,(9)∂H∂t=Ixṗ,Iyq̇,IzṙT,where Ix, Iy, and Iz are moments of inertia, due to the character of symmetric Iy=Iz; p is the spinning rate in the nonrolling coordinate.For a canard-controlled spinning rocket, positiveδz and δy create positive moments in Mz and My, respectively. Therefore, the aerodynamic moment can be expressed as
(10)M=MxMyMz=QSlClδxδx−ClpplVCmαα−CmqqlV−CmpαβplV+Cmδδy−Cmαβ−CmqrlV−CmpααplV+Cmδδz,where l is the reference length; Clδx is the rolling moment coefficient due to the canted angle of tails, which is denoted as δx; Clp is the rolling damping moment coefficient; Cmα is the static moment coefficient; Cmq is the damping moment coefficient; Cmpα is the Magnus moment coefficient; and Cmδ is the control moment coefficient.Remark 1.
The guided spinning rocket considered in this paper is a fin-stabilized airframe. For a fin-stabilized rocket, the classic Magnus force is typically ignored in (3) since its effect is rather small for slowly rolling rockets. However, the Magnus moment with physical mechanisms specific to fin-stabilized rockets cannot be ignored in (10), which can be expressed as a dynamic side moment due to spin rate p and angle of attack α or sideslip angel β in the form of Cmpααpl/V [23].By substituting (8), (9), and (10) into (7), the moment equations are obtained:
(11)Ixṗ=Mx,Iyq̇+Ixpr+Iyr2tanθ=My,Iyṙ−Ixpq−Iyqrtanθ=Mz.Equations (6) and (11) are the nonlinear dynamical model of a spinning rocket, which represents the angular motion of the spinning rocket.


### 2.2. Linearized Lateral Equation

For flight control which corresponds to the tracking of the dual-canard-controlled spinning rocket acceleration command, only the lateral dynamic system is necessary. In order to simplify the design procedure of the autopilot, some common and reasonable assumptions are made to linearize the lateral dynamic model of the spinning projectile [7]:Assumption 1.
The velocity, roll rate, mass, and aerodynamic coefficients of the rocket remain constant over a short time period.Assumption 2.
For simplicity, the gravity and the small canard force are ignored. The lateral velocitiesv and w are small with respect to the axis velocity u, so that u≈V.Applying the above assumptions to (6) and (11), the linearized angular motion of the spinning rocket can be described as
(12)β̇=−r−cNαβ,α̇=q−cNαα,q̇=cmαα−cmpαPpβ−cmqq−Ppr+cmδδy,ṙ=−cmαβ−cmpαPpα−cmqr−Ppq+cmδδz,where the dimensionless coefficients are defined as cNα=QSCNα/mV, cmα=QSlCmα/Iy, cmq=QSl2Cmq/IyV, cmpα=QSl2Cmpα/IxV, cmδ=QSlCmδ/Iy, and Pp=pIx/Iy.Usually, not all the states of a spinning rocket in (12) can be measured by sensors, especially the angle of attack α and the sideslip angle β. Low-cost inertial measurement units (IMUs) are the general sensors installed in the spinning rocket, which can provide the accelerations ay, az, and angular rates q and r of the lateral motion. In this condition, the accelerations measured by IMU along the OYN and OZN axes can be expressed as
(13)ayaz=−cNαV00−cNαVβα.Hence, the overall dynamic equation of the spinning rocket can be rearranged to the following state-space form,(14)ẋpt=Apxpt+Bput,ypt=Cpxpt,zt=Czxpt,where xp=βαqrT∈ℝnp is the state vector, u=δyδzT∈ℝm is the control input vector, yp=ayazqrT∈ℝnp is the measurement output vector, and z=ayaz∈ℝr is the regulated output vector that is also measured. Ap∈ℝnp×np is the system matrix, Bp∈ℝnp×m is the input matrix, Cp∈ℝpp×np is the measurement output matrix, and Cz∈ℝr×np is the regulated output matrix, and they are all known matrices.


### 2.3. Uncertain Dynamic Equation

The dynamic model presented in (14) is the ideal case where all the matrices are known. In reality, these matrices are unknown and are obtained through various methods. The system matrix Ap and output matrices Cp and Cz can be determined through wind-tunnel tests fairly accurately. In contrast, the input matrix Bp may not be accurate, as control inputs are subjected to perturbations in the flight period. First, the orientation of the canards can lead to the dynamical parameters different from the trim condition as represented by the nominal model. This effect can be modeled as an additive term ΨpTxp. Second, canard failure caused by electronic circuit or control surface damage is another effect which may cause the inaccuracy of input matrices. This effect can be modeled as constant matched uncertainty weights Λ. The modified uncertain dynamic model is given as
(15)ẋpt=Apxpt+BpΛut+ΨpTxp,ypt=Cpxpt,zt=Czxpt,where Λ∈ℝm×m and Ψp∈ℝnp×m are all nonsingular matrices.


## 2.1. Dynamical Equation

According to Newton’s second law, the translational motion of center of gravity (cg) can be described in the nonrolling body coordinate system as follows [22]:
(1)mdVAdt=m∂VN∂t+ωr×VN=F+mg,where m is the mass of the spinning rocket, VA=x,y,zT is the velocity of cg represented in inertial coordinate, VN=u,v,wT is the velocity of cg in the nonrolling body coordinate, and ωr is the angular rate of the nonrolling body coordinate system with respect to the inertial coordinate system. F is the aerodynamic force, and g is the acceleration of gravity, which are all represented in the nonrolling body coordinate system. All the variables in (1) can be modeled as
(2)ωr=−rtanθ,q,rT,(3)F=FxFyFz=QS−Cx−CNββ+CNδδz−CNαα−CNδδy,(4)g=gxgygz=−gsinθ−0gcosθ,where q and r are angular rates along OYN and OZN defined in Figure 1; θ is the pitch angle of the spinning rocket; Q is the dynamic pressure; S is the reference area; Cx is the drag coefficient slope; CNα and CNβ are the normal force coefficient slopes, due to the character of symmetric CNα=CNβ, so as other coefficients related with α and β; CNδ is the control force coefficient slope; δy and δz represent the actuator deflection angles in the nonrolling coordinate system; and α and β are the angle of attack and the sideslip angle, respectively. The expressions of α and β are presented as
(5)α=arctanwu≈wV,β=arcsinvu≈vV.By substituting (2), (3), and (4) into (1), the force equations are obtained:
(6)u̇+wq−vr=Fxm+gx,v̇+ur−wrtanθ=Fym+gy,ẇ−uq−vrtanθ=Fzm+gz.According to the theorem of angular momentum, the rotational motion of the airframe can be described in the nonrolling body coordinate system as(7)dHdt=∂H∂t+ωr×H=M,where H is the angular momentum and M is the moment acting on the rocket. All the variables in (7) are described in the nonrolling body coordinate system as
(8)H=Ixp,Iyq,IzrT,(9)∂H∂t=Ixṗ,Iyq̇,IzṙT,where Ix, Iy, and Iz are moments of inertia, due to the character of symmetric Iy=Iz; p is the spinning rate in the nonrolling coordinate.For a canard-controlled spinning rocket, positiveδz and δy create positive moments in Mz and My, respectively. Therefore, the aerodynamic moment can be expressed as
(10)M=MxMyMz=QSlClδxδx−ClpplVCmαα−CmqqlV−CmpαβplV+Cmδδy−Cmαβ−CmqrlV−CmpααplV+Cmδδz,where l is the reference length; Clδx is the rolling moment coefficient due to the canted angle of tails, which is denoted as δx; Clp is the rolling damping moment coefficient; Cmα is the static moment coefficient; Cmq is the damping moment coefficient; Cmpα is the Magnus moment coefficient; and Cmδ is the control moment coefficient.Remark 1.
The guided spinning rocket considered in this paper is a fin-stabilized airframe. For a fin-stabilized rocket, the classic Magnus force is typically ignored in (3) since its effect is rather small for slowly rolling rockets. However, the Magnus moment with physical mechanisms specific to fin-stabilized rockets cannot be ignored in (10), which can be expressed as a dynamic side moment due to spin rate p and angle of attack α or sideslip angel β in the form of Cmpααpl/V [23].By substituting (8), (9), and (10) into (7), the moment equations are obtained:
(11)Ixṗ=Mx,Iyq̇+Ixpr+Iyr2tanθ=My,Iyṙ−Ixpq−Iyqrtanθ=Mz.Equations (6) and (11) are the nonlinear dynamical model of a spinning rocket, which represents the angular motion of the spinning rocket.


## 2.2. Linearized Lateral Equation

For flight control which corresponds to the tracking of the dual-canard-controlled spinning rocket acceleration command, only the lateral dynamic system is necessary. In order to simplify the design procedure of the autopilot, some common and reasonable assumptions are made to linearize the lateral dynamic model of the spinning projectile [7]:Assumption 1.
The velocity, roll rate, mass, and aerodynamic coefficients of the rocket remain constant over a short time period.Assumption 2.
For simplicity, the gravity and the small canard force are ignored. The lateral velocitiesv and w are small with respect to the axis velocity u, so that u≈V.Applying the above assumptions to (6) and (11), the linearized angular motion of the spinning rocket can be described as
(12)β̇=−r−cNαβ,α̇=q−cNαα,q̇=cmαα−cmpαPpβ−cmqq−Ppr+cmδδy,ṙ=−cmαβ−cmpαPpα−cmqr−Ppq+cmδδz,where the dimensionless coefficients are defined as cNα=QSCNα/mV, cmα=QSlCmα/Iy, cmq=QSl2Cmq/IyV, cmpα=QSl2Cmpα/IxV, cmδ=QSlCmδ/Iy, and Pp=pIx/Iy.Usually, not all the states of a spinning rocket in (12) can be measured by sensors, especially the angle of attack α and the sideslip angle β. Low-cost inertial measurement units (IMUs) are the general sensors installed in the spinning rocket, which can provide the accelerations ay, az, and angular rates q and r of the lateral motion. In this condition, the accelerations measured by IMU along the OYN and OZN axes can be expressed as
(13)ayaz=−cNαV00−cNαVβα.Hence, the overall dynamic equation of the spinning rocket can be rearranged to the following state-space form,(14)ẋpt=Apxpt+Bput,ypt=Cpxpt,zt=Czxpt,where xp=βαqrT∈ℝnp is the state vector, u=δyδzT∈ℝm is the control input vector, yp=ayazqrT∈ℝnp is the measurement output vector, and z=ayaz∈ℝr is the regulated output vector that is also measured. Ap∈ℝnp×np is the system matrix, Bp∈ℝnp×m is the input matrix, Cp∈ℝpp×np is the measurement output matrix, and Cz∈ℝr×np is the regulated output matrix, and they are all known matrices.


## 2.3. Uncertain Dynamic Equation

The dynamic model presented in (14) is the ideal case where all the matrices are known. In reality, these matrices are unknown and are obtained through various methods. The system matrix Ap and output matrices Cp and Cz can be determined through wind-tunnel tests fairly accurately. In contrast, the input matrix Bp may not be accurate, as control inputs are subjected to perturbations in the flight period. First, the orientation of the canards can lead to the dynamical parameters different from the trim condition as represented by the nominal model. This effect can be modeled as an additive term ΨpTxp. Second, canard failure caused by electronic circuit or control surface damage is another effect which may cause the inaccuracy of input matrices. This effect can be modeled as constant matched uncertainty weights Λ. The modified uncertain dynamic model is given as
(15)ẋpt=Apxpt+BpΛut+ΨpTxp,ypt=Cpxpt,zt=Czxpt,where Λ∈ℝm×m and Ψp∈ℝnp×m are all nonsingular matrices.


## 3. Robust Adaptive Autopilot Design

The fight sequence of guided spinning rockets is decomposed into three flight phases: boost phase, free flight phase, and guided phase. In this section, we mainly consider the guidance phase, during which the autopilot begins to work. In the guided phase, guidance law gives the desired commands which are usually in the form of acceleration, to the autopilot based on the target and rocket’s relative motion information. Therefore, designing an acceleration autopilot is more reasonable and effective. The underlying problem is to design an acceleration autopilot that is able to track different kinds of signals under different dynamic environments, with slight or even no change of the autopilot architecture. So, the output feedback adaptive control method is employed to design the acceleration autopilot.It should be noted that the output feedback adaptive autopilot requires the dynamical system in (15) as the controllable, observable, and minimum phases, which can be satisfied by the canard-controlled spinning rocket. The uncertainty Ψp is bound with Ψp<Ω<∞ and rankCpBp=m.


### 3.1. Addition of Integral Error

Command tracking and disturbance rejection are the main issues to be handled through the integral action [24, 25]. In this light, consider the regulated output signal tracking error which is defined as follows:
(16)ẋe=zcmd−z,where zcmd is a piecewise continuous command signal. An integral error state is defined as
(17)xe=∫0tẋedt=∫0tzcmd−zdt.Appending (17) to the plant in (15), the augmented open-loop dynamics is given by
(18)ẋpẋe=Ap0np×r−Cz0r×r⏟Axpxe⏟xBp0r×m⏟BΛu+ΨpTxp+0np×rIr×r⏟Bcmdzcmd,ypxe⏟y=Cp0np×r0r×npIr×r⏟Cxpxe.Therefore, (18) can be written more compactly as follows:
(19)ẋ=Ax+BΛut+ΨTx+Bcmdzcmd,y=Cx,where Ψ=ΨpT0m×rT is unknown. Define n=np+r, then x∈ℝn.


### 3.2. Control Architecture

Following the design procedure in [25], the controller is divided into two parts: a baseline controller and an adaptive controller. The control input u in (15) is described as
(20)u=ubl+uad,where uad is the adaptive component and ubl is the baseline component, defined as
(21)ubl=KxTxm,where Kx is designed by applying the linear quadratic regulator (LQR) technique on the nominal plant model, that is, when Λ=I and Ψ=0. xm is the state of the observer,
(22)ẋm=Amxm+Bcmdzcmd+Lym−y,ym=Cxm,where Am=A+BKxT, and KxT is selected such that Am is Hurwitz.Remark 2.
The observer (22) serves as three purposes:
(1)
Luenberger observer. MatrixL is equivalent to an observer gain, for which the purpose is to estimate the state vector x in plant (19). The estimated state vector helps to construct the baseline controller ubl .
(2)
Reference model. As our ultimate goal is to establish a model reference adaptive controller, (22) serves as a closed-loop reference model (CRM), which has been proved to result in improving transient properties [26–28].
(3)
Robust compensator. The output error feedback termLym−y can be treated as a robust compensator, making the error dynamics satisfy the SPR property.The adaptive controlleruad is defined as
(23)uad=ΘTtxm,where Θt is the estimated uncertain parameter, to be updated by a well-designed update law. Substituting the controller (20) into the augmented plant (19),
(24)ẋ=Ax+BΛKx+ΘtT+ΨTx+Bcmdzcmd,y=Cx.If the proposed control architecture can realize accurate command tracking, an ideal uncertain parameterΘ∗ must exists and satisfies the following matching condition,
(25)Am=A+BΨT+BΛΘ∗T+KxTwith Θ∗T=Λ−1−IKxT−Λ−1ΨT.The error dynamic model between the reference model (22) and the closed-loop plant (24) is resorted to accomplish the whole design procedure. The state error ex=x−xm and the estimate parameter error Θ~t=Θt−Θ∗ satisfy the dynamics
(26)ėx=A+LC+BΨTex+BΛΘ~Ttxm,ey=Cex,where ey is the measured output error. The problem of finding a stabilizing adaptive controller is equivalent to finding an observer gain L and an adaptive law for Θt in (26), so that the underlying transfer function matrix is SPR.The following adaptive law is employed to update the estimated uncertain parameter,(27)Θ̇t=−ΓxmeyTMTsgnΛ,where Γ, the adaptive gain, is a positive diagonal free design matrix, M∈ℝm×n is a mixing matrix to “square up” the transfer function, and sgnΛ represents the sign matrix of the input uncertainty Λ.


### 3.3. Robust SPR Design

The SPR property is usually employed to design a stable adaptive law for the error dynamic model (26) of the uncertain plant. However, the definition of SPR is restricted to square transfer functions. As the transfer function of (26) is nonsquare, a suitable mixing matrix M has to be chosen to make the SPR properties applicable to the error dynamic model, yielding
(28)Gs=MCsI−A−LC−BΨT−1B,where Gs is a square transfer function of the modified error dynamic model,
(29)ėx=A+LC+BΨTex+BΛΘ~Ttxm,em=MCex.Thus, the design of an output feedback adaptive controller is converted to selecting mixing matrixM∈ℝm×n and observer matrix L∈ℝn×n, such that the overall closed-loop system is stable.The choice of mixing matrixM is not unique. M can be computed as
(30)M=BTPC−1,which satisfies PB=CTMT.Lemma 1.
Given the strictly proper transfer matrixGswith stabilizable and detectable realization (A,B,C,D), whereA∈ℝn×nis asymptotically stable,B∈ℝn×m,C∈ℝm×n, andD∈ℝm×m; then,Gsis SPR if, and only if, there exists matricesP=P⊤>0,P∈ℝn×n,Q∈ℝn×m, andW∈ℝm×msuch that
(31)A⊤P+PA=−QQT,PB−CT=−QW,D+DT=WTW.Proof 1.
The complete proof on the Lemma1 can be found in [29] Lemma 3.1.Lemma 2.
LetHandEbe given matrices of appropriate dimensions, andFsatisfiesFFT<I; for anyε>0, there is
(32)HFE+HFET≤εHHT+ε−1ETE.Proof 2.
The complete proof on the Lemma2 can be found in [30].The output error feedback termLym−y can be treated as 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 form of robust control,
(33)ėx=A+ΔAex+B1w+B2um,em=Cmex,where ∆A=BΨT is defined as the uncertain state matrix; B1=B, B2=In, Cm=MC, w=ΛΘTtxm, and um=Lym−y=LCex are defined as the modified virtual control inputs for modified error dynamics.The modified error dynamics is SPR and stable if and only if Lemma1 is satisfied. Equation (29) is strictly proper; that is, D=0, so Lemma 1 cannot be used directly. For the case D=0, the above set of equations in 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 should be satisfied:
(34)A+ΔA+LCTP+PA+ΔA+LC<0,(35)PB1=CTMT.(36)XAT+AX+B2W+WTB2EXTB1∗−γ0∗∗−1γ<0,where E=ΩIn. The matrix in (36) is symmetric, and ∗ represents the transposition of the corresponding item in the matrix.Lemma 3.
Given an uncertain plant (33), the closed-loop system is stable and SPR, if and only if there exist symmetric positive matricesXandWand a constantγ>0satisfiesProof 3.
The state matrix uncertainty∆A can be described as follows:
(37)ΔA=B1ΨT=B1FE,where F=ΨT/Ω, and the inequality FFT<I holds.DefineP=X−1 and K=LC=WX−1, and multiply inequality (36) on both sides by the matrix diagP,I,I,
(38)A+B2KTP+A+B2KETPB1∗−γ0∗∗−1γ<0.By the Schur complement, the LMI defined in (36) is equivalent to the following inequality:
(39)A+B2KTP+PA+B2K+γPB1B1TP+1γETE<0.With Lemma2, we have
(40)PB1FE+PB1FET<γPB1B1TP+1γETE.Thus, inequality (39) is equivalent to the following inequality,
(41)A+B2K+B1FETP+PA+B2K+B1FE<0,which means A+∆A+LCTP+PA+∆A+LC<0. Combining PB=CTMT (34) holds. So, the modified error dynamics is stable and SPR.The observer gainL can be obtained by solving the feasible LMI with any widely available numerical LMI solver. It can be expressed as
(42)L=W∗X∗−1C−1,where W∗ and X∗ are the feasible solution of (36).Figure2 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),Figure 2
Control architecture.


### 3.4. Stability Analysis

Given the uncertain linear system in (19), the reference model in (22) with L as in (42), the control architecture in (20), and the update law in (27) result in global stability, with limt→∞ex=0.Consider the following Lyapunov function candidate,(43)Vex,Θ~=exTPex+trΛΘ~Γ−1Θ~,where Λ represents the absolute value of each entry of the matrix element, satisfying Λ=sgnΛΛ.The time derivative of (43) along the system trajectories is given by
(44)V̇=−exTQex+2exTPBΛΘ~Txm+2trΛΘ~TΓ−1Θ~̇,where A+BΨT+LCTP+PA+BΨT+LC=−Q<0. Substituting the update law given in (27) yields
(45)V̇=−exTQex+2exTPBΛΘ~Txm−2trΛΘ~TxmeyTMTsgnΛ=−exTQex+2exTPBΛΘ~Txm−2eyTMTΛΘ~Txm=−exTQex+2exTPBΛΘ~Txm−2exTCTMTΛΘ~Txm=−exTQex≤0,which implies that V is a Lyapunov function. Since V>0 and V̇≤0, then Vt≤V0<∞. Thus, Vt∈L∞, which means ex, Θ~∈L∞. Since zcmd, ex∈L∞, and the reference model are stable, xm∈L∞, which implies that xp∈L∞.Furthermore, asymptotic stability of the tracking errors is demonstrated by invoking LaSalle’s invariance principle, which states that, for a negative semidefinite Lyapunov system in the form of (45), all system trajectories are contained within the domain Ω0=ex,Θt∣Vex,Θt,t≤Vex,Θt,0, where the subscript 0 denotes the initial conditions, and the entire state space ex,Θt ultimately reaches the domain Ωf=Ω0∩Ωz, where Ωz denotes the domain defined by the Lyapunov derivative identical to zero. In other words, the state space ultimately reaches the domain defined by V̇ex,Θt,t≡0 [31, 32]. Because V̇ex,Θt,t is negative-definite in ex, the system ends with ex≡0. Thus, xt→xmt and the bound reference tracking of zcmd by z follow from the stability of the closed-loop system.Remark 3.
In this paper, the adaptive parameterΘt is not guaranteed to converge to its true unknown value Θ∗ nor is it assured to converge to constant value in any way. All that is known is that the unknown parameter remains uniformly bound in time. Sufficient conditions for parameter convergence are known as persistency of excitation.


## 3.1. Addition of Integral Error

Command tracking and disturbance rejection are the main issues to be handled through the integral action [24, 25]. In this light, consider the regulated output signal tracking error which is defined as follows:
(16)ẋe=zcmd−z,where zcmd is a piecewise continuous command signal. An integral error state is defined as
(17)xe=∫0tẋedt=∫0tzcmd−zdt.Appending (17) to the plant in (15), the augmented open-loop dynamics is given by
(18)ẋpẋe=Ap0np×r−Cz0r×r⏟Axpxe⏟xBp0r×m⏟BΛu+ΨpTxp+0np×rIr×r⏟Bcmdzcmd,ypxe⏟y=Cp0np×r0r×npIr×r⏟Cxpxe.Therefore, (18) can be written more compactly as follows:
(19)ẋ=Ax+BΛut+ΨTx+Bcmdzcmd,y=Cx,where Ψ=ΨpT0m×rT is unknown. Define n=np+r, then x∈ℝn.


## 3.2. Control Architecture

Following the design procedure in [25], the controller is divided into two parts: a baseline controller and an adaptive controller. The control input u in (15) is described as
(20)u=ubl+uad,where uad is the adaptive component and ubl is the baseline component, defined as
(21)ubl=KxTxm,where Kx is designed by applying the linear quadratic regulator (LQR) technique on the nominal plant model, that is, when Λ=I and Ψ=0. xm is the state of the observer,
(22)ẋm=Amxm+Bcmdzcmd+Lym−y,ym=Cxm,where Am=A+BKxT, and KxT is selected such that Am is Hurwitz.Remark 2.
The observer (22) serves as three purposes:
(1)
Luenberger observer. MatrixL is equivalent to an observer gain, for which the purpose is to estimate the state vector x in plant (19). The estimated state vector helps to construct the baseline controller ubl .
(2)
Reference model. As our ultimate goal is to establish a model reference adaptive controller, (22) serves as a closed-loop reference model (CRM), which has been proved to result in improving transient properties [26–28].
(3)
Robust compensator. The output error feedback termLym−y can be treated as a robust compensator, making the error dynamics satisfy the SPR property.The adaptive controlleruad is defined as
(23)uad=ΘTtxm,where Θt is the estimated uncertain parameter, to be updated by a well-designed update law. Substituting the controller (20) into the augmented plant (19),
(24)ẋ=Ax+BΛKx+ΘtT+ΨTx+Bcmdzcmd,y=Cx.If the proposed control architecture can realize accurate command tracking, an ideal uncertain parameterΘ∗ must exists and satisfies the following matching condition,
(25)Am=A+BΨT+BΛΘ∗T+KxTwith Θ∗T=Λ−1−IKxT−Λ−1ΨT.The error dynamic model between the reference model (22) and the closed-loop plant (24) is resorted to accomplish the whole design procedure. The state error ex=x−xm and the estimate parameter error Θ~t=Θt−Θ∗ satisfy the dynamics
(26)ėx=A+LC+BΨTex+BΛΘ~Ttxm,ey=Cex,where ey is the measured output error. The problem of finding a stabilizing adaptive controller is equivalent to finding an observer gain L and an adaptive law for Θt in (26), so that the underlying transfer function matrix is SPR.The following adaptive law is employed to update the estimated uncertain parameter,(27)Θ̇t=−ΓxmeyTMTsgnΛ,where Γ, the adaptive gain, is a positive diagonal free design matrix, M∈ℝm×n is a mixing matrix to “square up” the transfer function, and sgnΛ represents the sign matrix of the input uncertainty Λ.


## 3.3. Robust SPR Design

The SPR property is usually employed to design a stable adaptive law for the error dynamic model (26) of the uncertain plant. However, the definition of SPR is restricted to square transfer functions. As the transfer function of (26) is nonsquare, a suitable mixing matrix M has to be chosen to make the SPR properties applicable to the error dynamic model, yielding
(28)Gs=MCsI−A−LC−BΨT−1B,where Gs is a square transfer function of the modified error dynamic model,
(29)ėx=A+LC+BΨTex+BΛΘ~Ttxm,em=MCex.Thus, the design of an output feedback adaptive controller is converted to selecting mixing matrixM∈ℝm×n and observer matrix L∈ℝn×n, such that the overall closed-loop system is stable.The choice of mixing matrixM is not unique. M can be computed as
(30)M=BTPC−1,which satisfies PB=CTMT.Lemma 1.
Given the strictly proper transfer matrixGswith stabilizable and detectable realization (A,B,C,D), whereA∈ℝn×nis asymptotically stable,B∈ℝn×m,C∈ℝm×n, andD∈ℝm×m; then,Gsis SPR if, and only if, there exists matricesP=P⊤>0,P∈ℝn×n,Q∈ℝn×m, andW∈ℝm×msuch that
(31)A⊤P+PA=−QQT,PB−CT=−QW,D+DT=WTW.Proof 1.
The complete proof on the Lemma1 can be found in [29] Lemma 3.1.Lemma 2.
LetHandEbe given matrices of appropriate dimensions, andFsatisfiesFFT<I; for anyε>0, there is
(32)HFE+HFET≤εHHT+ε−1ETE.Proof 2.
The complete proof on the Lemma2 can be found in [30].The output error feedback termLym−y can be treated as 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 form of robust control,
(33)ėx=A+ΔAex+B1w+B2um,em=Cmex,where ∆A=BΨT is defined as the uncertain state matrix; B1=B, B2=In, Cm=MC, w=ΛΘTtxm, and um=Lym−y=LCex are defined as the modified virtual control inputs for modified error dynamics.The modified error dynamics is SPR and stable if and only if Lemma1 is satisfied. Equation (29) is strictly proper; that is, D=0, so Lemma 1 cannot be used directly. For the case D=0, the above set of equations in 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 should be satisfied:
(34)A+ΔA+LCTP+PA+ΔA+LC<0,(35)PB1=CTMT.(36)XAT+AX+B2W+WTB2EXTB1∗−γ0∗∗−1γ<0,where E=ΩIn. The matrix in (36) is symmetric, and ∗ represents the transposition of the corresponding item in the matrix.Lemma 3.
Given an uncertain plant (33), the closed-loop system is stable and SPR, if and only if there exist symmetric positive matricesXandWand a constantγ>0satisfiesProof 3.
The state matrix uncertainty∆A can be described as follows:
(37)ΔA=B1ΨT=B1FE,where F=ΨT/Ω, and the inequality FFT<I holds.DefineP=X−1 and K=LC=WX−1, and multiply inequality (36) on both sides by the matrix diagP,I,I,
(38)A+B2KTP+A+B2KETPB1∗−γ0∗∗−1γ<0.By the Schur complement, the LMI defined in (36) is equivalent to the following inequality:
(39)A+B2KTP+PA+B2K+γPB1B1TP+1γETE<0.With Lemma2, we have
(40)PB1FE+PB1FET<γPB1B1TP+1γETE.Thus, inequality (39) is equivalent to the following inequality,
(41)A+B2K+B1FETP+PA+B2K+B1FE<0,which means A+∆A+LCTP+PA+∆A+LC<0. Combining PB=CTMT (34) holds. So, the modified error dynamics is stable and SPR.The observer gainL can be obtained by solving the feasible LMI with any widely available numerical LMI solver. It can be expressed as
(42)L=W∗X∗−1C−1,where W∗ and X∗ are the feasible solution of (36).Figure2 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),Figure 2
Control architecture.


## 3.4. Stability Analysis

Given the uncertain linear system in (19), the reference model in (22) with L as in (42), the control architecture in (20), and the update law in (27) result in global stability, with limt→∞ex=0.Consider the following Lyapunov function candidate,(43)Vex,Θ~=exTPex+trΛΘ~Γ−1Θ~,where Λ represents the absolute value of each entry of the matrix element, satisfying Λ=sgnΛΛ.The time derivative of (43) along the system trajectories is given by
(44)V̇=−exTQex+2exTPBΛΘ~Txm+2trΛΘ~TΓ−1Θ~̇,where A+BΨT+LCTP+PA+BΨT+LC=−Q<0. Substituting the update law given in (27) yields
(45)V̇=−exTQex+2exTPBΛΘ~Txm−2trΛΘ~TxmeyTMTsgnΛ=−exTQex+2exTPBΛΘ~Txm−2eyTMTΛΘ~Txm=−exTQex+2exTPBΛΘ~Txm−2exTCTMTΛΘ~Txm=−exTQex≤0,which implies that V is a Lyapunov function. Since V>0 and V̇≤0, then Vt≤V0<∞. Thus, Vt∈L∞, which means ex, Θ~∈L∞. Since zcmd, ex∈L∞, and the reference model are stable, xm∈L∞, which implies that xp∈L∞.Furthermore, asymptotic stability of the tracking errors is demonstrated by invoking LaSalle’s invariance principle, which states that, for a negative semidefinite Lyapunov system in the form of (45), all system trajectories are contained within the domain Ω0=ex,Θt∣Vex,Θt,t≤Vex,Θt,0, where the subscript 0 denotes the initial conditions, and the entire state space ex,Θt ultimately reaches the domain Ωf=Ω0∩Ωz, where Ωz denotes the domain defined by the Lyapunov derivative identical to zero. In other words, the state space ultimately reaches the domain defined by V̇ex,Θt,t≡0 [31, 32]. Because V̇ex,Θt,t is negative-definite in ex, the system ends with ex≡0. Thus, xt→xmt and the bound reference tracking of zcmd by z follow from the stability of the closed-loop system.Remark 3.
In this paper, the adaptive parameterΘt is not guaranteed to converge to its true unknown value Θ∗ nor is it assured to converge to constant value in any way. All that is known is that the unknown parameter remains uniformly bound in time. Sufficient conditions for parameter convergence are known as persistency of excitation.


## 4. Case Study

In this section, numerical examples are performed to evaluate the performance of the output feedback adaptive control scheme applied to the autopilot design for the dual-canard-controlled spinning rocket. All the simulations are performed in MATLAB R2017b with a 64-bit processor, 16 GB memory, and 0.001 s time-step. Robust Control Toolbox 6.4 in MATLAB is employed to obtain a feasible solution of observer (36). And all the simulations are performed with an adaptive controller acting on the nonlinear model.The nominal model for autopilot design is the linearized lateral dynamics of the spinning rocket at the speedV of 581 m/s and altitude H of 5000 m. The numerical values for the 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 nominal plant is stable.We show the response of the closed-loop system with the baseline controller and adaptive controller, respectively. Square-wave commands with amplitude of 1 m/s2 and frequency of 1 Hz are applied to the pitch and yaw channel; there is a 0.1 s time lag between the two channels. It is observed in Figure 3 that both the baseline controller and the adaptive controller are able to guide the spinning rocket following the acceleration command and eventually achieved a zero tracking error. Figure 4 presents the behavior of control signals in the pitch and yaw channels. Only a small amount of canard deflection angles is used to achieve the command tracking. The angular rates of pitch and yaw channels are presented in Figure 5, which shows a good performance in both channels. Considering simulation results for this case, both the proposed adaptive controller and the baseline controller present a proper performance in the presence of the disturbances enforced due to the channel couplings.Figure 3
The nominal case tracking performance.Figure 4
The nominal case control signal.Figure 5
The nominal case angular rates.The same controllers were used in the presence of the following uncertainties:(46)Λ=0.9I,Ψ=000.120000000.1200.Such uncertainties stem from the fact that the model parameters are expected to vary significantly (by about 30% from the nominal A matrix) compared to those determined from wind channel tests. At this time, the eigenvalues of the system matrix have a positive real part, which means the system is unstable.Outcomes of the pitch and yaw channel autopilots in the same set-point are presented in Figure6. In this case, the baseline controller is not able to suppress the unstable pitch and yaw modes, whereas the adaptive controller is able. The behavior of canard angles in the pitch and yaw channels is presented in Figure 7. The amplitude of canard angles for the adaptive controller is slightly bigger than that of the nominal model, due to the uncertainty. The angular rates of pitch and yaw channels are presented in Figure 8, which implies that the adaptive controller shows a better performance.Figure 6
The uncertain case tracking performance.Figure 7
The uncertain case control signal.Figure 8
The uncertain case angular rates.In order to visualize the destructive effect of the control effectiveness loss, simulations with different control effectiveness are performed. According to Figures9 and 10, the proposed adaptive controller can autoadjust the canard angles to handle the loss of control effectiveness. Considering simulation results for this scenario, the proposed adaptive controller presents a proper performance in the presence of the disturbances enforced due to the channel couplings and control effectiveness loss.Figure 9
Control performance withΛ=0.9I.Figure 10
Control performance withΛ=0.6I.Measurement noise is also taken into account in order to carry out more investigations in the perturbed conditions. To this end, a worst-case condition is enforced; a white noise with a standard deviation of 0.25(°/s) is injected into the rate gyroscopes of each channel, which is corresponding to the noise of low-cost MEMS gyroscopes. Figure 11 shows that the adaptive controller was robust in the presence of measurement noise in the angular rate feedback loop.Figure 11
Adaptive controller performance with measurement noise.We analyze the robustness of the overall closed-loop system with the steady-state gainKxT+ΨT in what follows. The gain and phase margin for the baseline controller and the closed-loop adaptive controller under the uncertain model are calculated as in Appendix B:
(47)GMb1=−818.1dBPMb1=±51.9°,GMad=−14.231.6dBPMad=±58.3°.It can be seen that the adaptive controller, in general, is more robust than the baseline controller is.


## 5. Conclusion

This paper presents a robust adaptive output feedback autopilot for a guided spinning rocket to resistant external disturbance and maintain precision command tracking. The autopilot is composed of a baseline controller augmented with an adaptive component to accommodate control effectiveness uncertainty and matched plant uncertainty, and it makes use of the closed-loop reference model to improve the transient properties of the overall adaptive system. Adaptive control is applicable to nonsquare systems through designing mixing matrixM and the observer gain matrix L. This procedure only needs to solve a set of linear matrix inequalities, which is given by the robust SPR lemma. The performance of the proposed adaptive output feedback controller is evaluated by numerical simulations when applied to the lateral dynamic model of a guided spinning rocket. The simulation results showed that the robust adaptive output controller can achieve asymptotic command tracking with significant uncertainty in control effectiveness, moment coefficient, and measurement noise.



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*Source: 1427487-2018-05-14.xml*