# Robust Adaptive Fuzzy Control for a Class of Uncertain MIMO Nonlinear Systems with Input Saturation

Authors: Shenglin Wen; Ye Yan
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper studies the robust adaptive fuzzy control design problem for a class of uncertain multiple-input and multiple-output (MIMO) nonlinear systems in the presence of actuator amplitude and rate saturation. In the control scheme, fuzzy logic systems are used to approximate unknown nonlinear systems. To compensate the effect of input saturations, an auxiliary system is constructed and the actuator saturations then can be augmented into the controller. The modified tracking error is introduced and used in fuzzy parameter update laws. Furthermore, in order to deal with fuzzy approximation errors for unknown nonlinear systems and external disturbances, a robust compensation control is designed. It is proved that the closed-loop system obtainsH∞ tracking performance through Lyapunov analysis. Steady and transient modified tracking errors are analyzed and the bound of modified tracking errors can be adjusted by tuning certain design parameters. The proposed control scheme is applicable to uncertain nonlinear systems not only with actuator amplitude saturation, but also with actuator amplitude and rate saturation. Detailed simulation results of a rigid body satellite attitude control system in the presence of parametric uncertainties, external disturbances, and control input constraints have been presented to illustrate the effectiveness of the proposed control scheme.


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## Body

## 1. Introduction

In most practical control applications, such as those in robot manipulation and aerospace industry, the performance of the controller is directly related to the accuracy of the mathematical model and external disturbances. However, it is difficult to establish an appropriate mathematical model for a large number of nonlinear systems when the systems are complex and highly coupled nonlinear with structured uncertainties and external disturbances [1]. To tackle with this problem, fuzzy logic systems and neural networks have been extensively used in complex and ill-defined nonlinear systems due to their approximation ability of dealing with the nonlinear smooth functions [2]. Many adaptive fuzzy control and adaptive neural network control schemes have been developed for single-input and single-output (SISO) nonlinear systems [3–7], MIMO nonlinear systems [8–17], and SISO/MIMO nonlinear systems with immeasurable states [18–20], respectively. Generally, these adaptive fuzzy and neural network control approaches can achieve nice control performance without control saturation. If physical actuators saturation such as magnitude and rate constraints is considered, the adaptive intelligent control approaches mentioned above can not be implemented [21].As we know, in many practical dynamic systems, physical actuators saturation on hardware indicates an inevitable constraint of the magnitude and rate limitations of the control signal. For example, due to physical limitation, momentum exchange devices or thrusters as actuator for the satellite attitude control system fail to render infinite control torque and thus the actuator can only provide limited control torques within a limited rate [22]. Control saturation is one of the most common nonsmooth nonlinearity that should be explicitly considered in the control design. The controllers that ignore actuator limitations may give rise to undesirable inaccuracy, severely degrade the performance of system, or even damage the stability of system [23]. Hence, the controller design subjected to the control saturation while simultaneously achieving higher performance is a very practical problem.The design of tracking controllers for uncertain MIMO nonlinear systems with actuator constraints is a challenging problem. During the past decades, there have been extensive researches on the control of nonlinear systems with various constraints. Analysis and design of control systems with control saturation have been widely studied in [24–42]. Farrell et al. [24–26] have presented an adaptive backstepping approach for unknown nonlinear systems with known magnitude, rate, and bandwidth constraints on intermediate states or actuators without disturbance. To tackle with the physical saturation, an auxiliary system with the same order as that of the plant was constructed to compensate the effect of saturation. The control input saturation is investigated through online approximation based control for uncertain nonlinear systems in [26]. An adaptive control and the constrained adaptive control in combination with the backstepping technique are proposed in [29]. A direct adaptive fuzzy control approach for uncertain nonlinear systems with input saturation is presented in [30], in which a Nussbaum function is used to compensate for the nonlinear term arising from the input saturation. In [31], an adaptive fuzzy output feedback control algorithm for a class of output constrained uncertain nonlinear systems with input saturation is developed by employing a barrier Lyapunov function and an auxiliary system. Adaptive backstepping tracking control based on fuzzy neural networks is investigated for unknown chaotic systems in [32] and with control input constraints in [33]. Neural network based adaptive control schemes with external disturbances and actuator saturations are presented in [35], in which auxiliary systems are added to attenuate the effects of input saturation. It is apparent that the presence of input saturation constraint substantially increases the complexity of control system design for uncertain MIMO nonlinear systems. In the constrained adaptive control, the key problem is how to handle the constraint effect of the actuator’s physical constraints. To this end, we introduce an auxiliary design system to handle the constraint effect in this paper. Based on the states of the auxiliary design system, constrained adaptive control is investigated for a class of uncertain MIMO nonlinear systems with input constraints using robust fuzzy control technique.In this paper, a robust adaptive fuzzy tracking control scheme is presented to handle the external disturbances and actuator physical constraints for uncertain MIMO nonlinear systems. In control design, fuzzy logic systems are used to approximate unknown nonlinear systems. Note that input saturations are nonsmooth functions but the adaptive fuzzy control technique requires all functions differentiable [6]. To compensate the effect of input saturations, an auxiliary system is constructed and the actuator saturations then can be augmented into the controller. The modified tracking error is introduced and used in fuzzy parameter update laws. Besides, in order to deal with fuzzy approximation errors for unknown nonlinear systems and external disturbances, a robust compensation control is designed. It is proved that the proposed control approach can guarantee that all the signals of the resulting closed-loop system are bounded, and the closed-loop system obtains H∞ tracking performance through Lyapunov analysis. The transient modified tracking errors performance is derived to be explicit functions of design parameters and thus bounds of modified tracking errors can be adjusted by tuning design parameters.The rest of this paper is organized as follows. The description of the uncertain MIMO nonlinear system under consideration and necessary preliminaries are given in Section2. In Section 3, the robust adaptive fuzzy control without control saturation is firstly designed. When the actuators have physical limitations, this approach may not be able to be successfully implemented. In order to solve this problem, a robust adaptive fuzzy control scheme with input saturation is investigated. The simulation results of satellite attitude control are presented to demonstrate the effectiveness of proposed controller in Section 4. Section 5 contains the conclusion.


## 2. Problem Formulation and Preliminary

### 2.1. Problem Formulation

Consider a class of uncertain MIMO affine nonlinear system(1)x˙=fx+∑i=1mgixui+d′y1=h1x⋮ym=hmx,
where x=x1,…,xn∈Rn is the state vector available for measurement; u=u1,…,um∈Rm is the control vector with ui≤uimax⁡, where uimax⁡ denote the actuator amplitude; y=y1,…,ym∈Rm is the output vector; h1x,…,hmx are smooth functions defined on the open set of Rn. fx, g1x,…,gmx are continuous unknown but smooth vector fields. d′=d1′,…,dn′T∈Rn is external disturbance vector, where d1′ is unknown but bounded.Define thatLfhi is the Lie derivative of hi along vector field fx and Lfkhi is recursively as Lfkhi=LfLfk-1hi. A multivariable uncertain nonlinear system of the form (1) has a vector relative degree [r1,r2,…,rm] at a point x0 if
(2)LgjLfkhix=0∀1≤j≤m,k≤ri-1
for all x in a neighborhood of x0.Denote(3)Gx=Lg1Lfr1-1h1x⋯LgmLfr1-1h1xLg1Lfr2-1h2x⋯LgmLfr2-1h2x⋯⋱⋯Lg1Lfrm-1hmx⋯LgmLfrm-1hmx=g11x⋯g1mx⋮⋱⋮gm1x⋯gmmx.Note thatGx is nonsingular at x=x0.Using feedback linearization, the nonlinear system (1) can be transformed into the following form [43]:
(4)yr=Fx+Gxu+d,
where
(5)yr=y1r1,y2r2,…,ymrmT,(6)Fx=Lfr1h1xLfr2h2x⋮Lfrmhmx=f1xf2x⋮fmx(7)d=∑i=1r1Lx˙i-1Ld′Lfr1-ih1x∑i=1r2Lx˙i-1Ld′Lfr2-ih2x⋮∑i=1rmLx˙i-1Ld′Lfrm-ihmx=d1d2⋮dm(8)x=y1,…,y1r1,y2,…,y2r2,…,ym,…,ymrm.In system (4), d still represents unknown bounded external disturbance vector. The relative degree of the system is assumed to be equal to the order of the system and is expressed as ∑i=1mri=n, which implies that the system does not have any zero dynamics.Let the desired output trajectory be given by(9)yd=yd1,yd2,…,ydmT.This paper aims to develop a robust adaptive fuzzy tracking control scheme such that all closed-loop system signals asymptotically converge to a compact set in the presence of input saturation, system uncertainties, and external disturbances and ensure the system outputsy track the desired trajectory yd. For system (4) to be controllable, the following assumptions are made.Assumption 1.
The desired trajectoryydi and its nth order derivatives are known and bounded.Assumption 2.
Forx in certain controllability region Uc∈Rn, σGx≠0, where σGx denotes the minimum singular value of the matrix Gx.


### 2.2. Fuzzy Logic Systems

In this paper, fuzzy logic systems are used to approximate unknown nonlinear functionsFx and Gx. Following, the approximation presentation of fuzzy logic systems is given. Without loss of generality, the unknown uncertainty is assumed as fx. A fuzzy logic system consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine working on fuzzy rules, and the defuzzifier. The fuzzy inference engine uses fuzzy IF-THEN rules to perform a mapping from an input linguistic vector x=x1,x2,…,xnT∈Rn to an output variable y∈R. Then, the ith fuzzy rule can be represented as [3]
(10)Rl:Ifx1isF1land⋯andxnisFnl,thenyisGll=1,2,…,r,
where Fil and Gl are fuzzy sets characterized with the fuzzy membership functions μFilxi and μGly, respectively, and r is the number of fuzzy rules.Through singleton function, center average defuzzifier, and product inference, the fuzzy logic system can be expressed as follows:(11)yx=∑l=1ry-l∏i=1nμFilxi∑l=1r∏i=1nμFilxi,
where y-l=max⁡y∈RμGly.By introducing the concept of fuzzy basis function vector, the final output of the fuzzy logic system can be expressed as(12)yx=θTξx,
where θ=y-1,…,y-rT is the adjustable parameter vector and ξx=ξ1x,…,ξrxT is the fuzzy basis function vector. The fuzzy basis function is defined as follows:
(13)ξlx=∑l=1r∏i=1nμFilxi∑l=1r∏i=1nμFilxi.Lemma 3.
Letfx be a continuous function defined on a compact set Uc. Then, for any constants ε>0, there exists a fuzzy logic system such as [2]
(14)sup⁡x∈Uc⁡fx-θTξx≤ε.Define the optimal parameter vectorθ* as
(15)θ*=argmin⁡θ∈Ωfsup⁡x∈Ucfx-θTξx.Under the optimization parameter vector, the unknown functionfx can be written as
(16)fx=θ*Tξx+ε,
where ε is the fuzzy minimum approximation error.In this work,Fx and Gx are unknown vector and matrix, respectively, and the fuzzy logic systems are used to approximate the unknown functions. Let fix  i=1,…,m and gijx  (i,j=1,…,m) be approximated by fuzzy logic systems as follows:
(17)f^ix∣θfi=θfiTξfixi=1,…,mg^ijx∣θgij=θgijTξgijxi,j=1,…,m,
where θfi∈RMfi and θgij∈RMgij are parameter vectors, ξfix∈RMfi and ξgijx∈RMgij are fuzzy basis function vectors, and Mfi and Mgij are the corresponding dimensions of the basis vectors.Denote(18)F^x∣θF=f^1x∣θf1⋮f^mx∣θfmG^x∣θG=g^11x∣θg11⋯g^1mx∣θg1m⋮⋱⋮g^m1x∣θgm1⋯g^mmx∣θgmm.ThenF^x∣θF and G^x∣θG can be used as the approximation of Fx and Gx, respectively.Define the optimal approximation weight vectors forfii=1,…,m and giji,j=1,…,m as follows:
(19)θfi*=argmin⁡θfi∈ΩFsup⁡x∈Uc⁡f^ix∣θfi-fixθgij*=argmin⁡θgij∈ΩGsup⁡x∈Ucg^ijx∣θgij-gijx,
where ΩF, ΩG, and Uc denote the sets of suitable bounds on θfi, θgij, and x respectively. θfi*  i=1,…,m and θgij*  i,j=1,…,m are constants vectors.The unknown functionfi and gij can be expressed as
(20)fix=θfi*Tξfix+εfii=1,…,mgijx=θgij*Tξgijx+εgiji,j=1,…,m,
where εfi and εgij are the smallest approximation errors of the fuzzy logic systems.


## 2.1. Problem Formulation

Consider a class of uncertain MIMO affine nonlinear system(1)x˙=fx+∑i=1mgixui+d′y1=h1x⋮ym=hmx,
where x=x1,…,xn∈Rn is the state vector available for measurement; u=u1,…,um∈Rm is the control vector with ui≤uimax⁡, where uimax⁡ denote the actuator amplitude; y=y1,…,ym∈Rm is the output vector; h1x,…,hmx are smooth functions defined on the open set of Rn. fx, g1x,…,gmx are continuous unknown but smooth vector fields. d′=d1′,…,dn′T∈Rn is external disturbance vector, where d1′ is unknown but bounded.Define thatLfhi is the Lie derivative of hi along vector field fx and Lfkhi is recursively as Lfkhi=LfLfk-1hi. A multivariable uncertain nonlinear system of the form (1) has a vector relative degree [r1,r2,…,rm] at a point x0 if
(2)LgjLfkhix=0∀1≤j≤m,k≤ri-1
for all x in a neighborhood of x0.Denote(3)Gx=Lg1Lfr1-1h1x⋯LgmLfr1-1h1xLg1Lfr2-1h2x⋯LgmLfr2-1h2x⋯⋱⋯Lg1Lfrm-1hmx⋯LgmLfrm-1hmx=g11x⋯g1mx⋮⋱⋮gm1x⋯gmmx.Note thatGx is nonsingular at x=x0.Using feedback linearization, the nonlinear system (1) can be transformed into the following form [43]:
(4)yr=Fx+Gxu+d,
where
(5)yr=y1r1,y2r2,…,ymrmT,(6)Fx=Lfr1h1xLfr2h2x⋮Lfrmhmx=f1xf2x⋮fmx(7)d=∑i=1r1Lx˙i-1Ld′Lfr1-ih1x∑i=1r2Lx˙i-1Ld′Lfr2-ih2x⋮∑i=1rmLx˙i-1Ld′Lfrm-ihmx=d1d2⋮dm(8)x=y1,…,y1r1,y2,…,y2r2,…,ym,…,ymrm.In system (4), d still represents unknown bounded external disturbance vector. The relative degree of the system is assumed to be equal to the order of the system and is expressed as ∑i=1mri=n, which implies that the system does not have any zero dynamics.Let the desired output trajectory be given by(9)yd=yd1,yd2,…,ydmT.This paper aims to develop a robust adaptive fuzzy tracking control scheme such that all closed-loop system signals asymptotically converge to a compact set in the presence of input saturation, system uncertainties, and external disturbances and ensure the system outputsy track the desired trajectory yd. For system (4) to be controllable, the following assumptions are made.Assumption 1.
The desired trajectoryydi and its nth order derivatives are known and bounded.Assumption 2.
Forx in certain controllability region Uc∈Rn, σGx≠0, where σGx denotes the minimum singular value of the matrix Gx.


## 2.2. Fuzzy Logic Systems

In this paper, fuzzy logic systems are used to approximate unknown nonlinear functionsFx and Gx. Following, the approximation presentation of fuzzy logic systems is given. Without loss of generality, the unknown uncertainty is assumed as fx. A fuzzy logic system consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine working on fuzzy rules, and the defuzzifier. The fuzzy inference engine uses fuzzy IF-THEN rules to perform a mapping from an input linguistic vector x=x1,x2,…,xnT∈Rn to an output variable y∈R. Then, the ith fuzzy rule can be represented as [3]
(10)Rl:Ifx1isF1land⋯andxnisFnl,thenyisGll=1,2,…,r,
where Fil and Gl are fuzzy sets characterized with the fuzzy membership functions μFilxi and μGly, respectively, and r is the number of fuzzy rules.Through singleton function, center average defuzzifier, and product inference, the fuzzy logic system can be expressed as follows:(11)yx=∑l=1ry-l∏i=1nμFilxi∑l=1r∏i=1nμFilxi,
where y-l=max⁡y∈RμGly.By introducing the concept of fuzzy basis function vector, the final output of the fuzzy logic system can be expressed as(12)yx=θTξx,
where θ=y-1,…,y-rT is the adjustable parameter vector and ξx=ξ1x,…,ξrxT is the fuzzy basis function vector. The fuzzy basis function is defined as follows:
(13)ξlx=∑l=1r∏i=1nμFilxi∑l=1r∏i=1nμFilxi.Lemma 3.
Letfx be a continuous function defined on a compact set Uc. Then, for any constants ε>0, there exists a fuzzy logic system such as [2]
(14)sup⁡x∈Uc⁡fx-θTξx≤ε.Define the optimal parameter vectorθ* as
(15)θ*=argmin⁡θ∈Ωfsup⁡x∈Ucfx-θTξx.Under the optimization parameter vector, the unknown functionfx can be written as
(16)fx=θ*Tξx+ε,
where ε is the fuzzy minimum approximation error.In this work,Fx and Gx are unknown vector and matrix, respectively, and the fuzzy logic systems are used to approximate the unknown functions. Let fix  i=1,…,m and gijx  (i,j=1,…,m) be approximated by fuzzy logic systems as follows:
(17)f^ix∣θfi=θfiTξfixi=1,…,mg^ijx∣θgij=θgijTξgijxi,j=1,…,m,
where θfi∈RMfi and θgij∈RMgij are parameter vectors, ξfix∈RMfi and ξgijx∈RMgij are fuzzy basis function vectors, and Mfi and Mgij are the corresponding dimensions of the basis vectors.Denote(18)F^x∣θF=f^1x∣θf1⋮f^mx∣θfmG^x∣θG=g^11x∣θg11⋯g^1mx∣θg1m⋮⋱⋮g^m1x∣θgm1⋯g^mmx∣θgmm.ThenF^x∣θF and G^x∣θG can be used as the approximation of Fx and Gx, respectively.Define the optimal approximation weight vectors forfii=1,…,m and giji,j=1,…,m as follows:
(19)θfi*=argmin⁡θfi∈ΩFsup⁡x∈Uc⁡f^ix∣θfi-fixθgij*=argmin⁡θgij∈ΩGsup⁡x∈Ucg^ijx∣θgij-gijx,
where ΩF, ΩG, and Uc denote the sets of suitable bounds on θfi, θgij, and x respectively. θfi*  i=1,…,m and θgij*  i,j=1,…,m are constants vectors.The unknown functionfi and gij can be expressed as
(20)fix=θfi*Tξfix+εfii=1,…,mgijx=θgij*Tξgijx+εgiji,j=1,…,m,
where εfi and εgij are the smallest approximation errors of the fuzzy logic systems.


## 3. Adaptive Fuzzy Robust Control Designs

In this section, we first design the adaptive fuzzy approximation based control problem without control saturation and then consider the case where actuators have physical limitations, such as magnitude and rate constraints.


### 3.1. Adaptive Fuzzy Robust Control

The tracking errors are defined as(21)ei=ydi-yii=1,2,…,m.Defineuli  i=1,…,m as follows:
(22)uli=ydiri+∑j=1rikijeij-1i=1,…,m,
where ki1,…,kiri are parameters to be chosen such that the roots of the equation sri+kirisri-1+⋯+ki1=0 in the open left-half complex plane.IfFx and Gx are known, external disturbances are ignored; then, according to nonlinear dynamic inversion control techniques, the control law is given by
(23)u=G-1x-Fx+ul,
where ul=ul1,…,ulmT.BecauseFx and Gx are unknown, di≠0, the control law (23) cannot be implemented in practice. Using the approximation of Fx and Gx, and considering external disturbances, the controller is modified as follows:
(24)u=G^-1x∣θG-F^x∣θF+ul+ud,
where ud is the robust compensation term, which is used to attenuate the effect of external disturbances and approximation errors.Substituting (24) into system (4) yields
(25)yr=Fx-F^x∣θF+Gx-G^x∣θGu+ul+ud+d.Considering (22), the ith subsystem of (25) can be rewritten as
(26)e˙i=Aiei+Bi∑j=1mg^ijx∣θgij-gijxujf^ix∣θfi-fix+∑j=1mg^ijx∣θgij-gijxuj-Biudi-Bidi,
where udi is the ith element of ud and
(27)Ai=010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1-ki1-ki2-ki3⋯-kiriBi=00⋮01ei=eie˙i⋮eiri-2eiri-1.Define the minimum approximation error as(28)ωi′=f^ix∣θfi*-fix+∑j=1mg^ijx∣θgij*-gijxuj.According to fuzzy system theory, the following assumption is reasonable.Assumption 4.
The minimum approximation error is square integrable, that is,∫0Tωi′Tωi′dt<∞.Using the definition (28) and the optimal approximation for fix and gijx, (26) can be rewritten in the following form:
(29)e˙i=Aiei+Biθ~fiTξfix+∑j=1mθ~gijTξgijxuj-Biudi+Biωi,
where θ~fi=θfi-θfi*,  θ~gij=θgij-θgij*, and ωi=ωi′-di.Theorem 5.
Consider the uncertain MIMO nonlinear system presented by (4) with external disturbance. If the controller is chosen as  (24), the parameters update laws and udi are adopted as follows:
(30)θ˙fi=-γfiξfixBiTPieii=1,…,m(31)θ˙gij=-γgijξgijxBiTPieiuji,j=1,…,m(32)udi=12ρi2BiTPieii=1,…,m.Then the followingH∞ tracking performance can be obtained
(33)∫0TeTQedt≤eT0Pe0+∑i=1m1γfiθ~fiT0θ~fi0+∑i,j=1m1γgijθ~gijT0θ~gij0+∑i=1mρi2∫0Tωi2dt,
where γfi  i=1,…,m and γgij  i,j=1,…,m are positive adaptive scalar, ρi  i=1,…,m are positive parameters representing for prescribed disturbance attenuation levels, e=e1T,…,emTT, Q=diagQ1,…,Qm, and Qi∈Rm×m  i=1,…,m are arbitrary symmetric positive definite matrices, and P=diagP1,…,Pm and Pi∈Rm×m  i=1,…,m are symmetric positive definite solution of the following equations:
(34)PiAi+AiTPi=-Qi,
where Qi∈Rm×mi=1,…,m are arbitrary symmetric positive definite matrices.Proof.
For theith subsystem of (4), consider the following Lyapunov function candidate
(35)Vi=12eiTPiei+12γfiθ~fiTθ~fi+12∑i=1m1γgijθ~gijTθ~gij.
Differentiating (35) and considering (29) yield
(36)V˙i=12e˙iTPiei+12eiTPie˙i+1γfiθ~˙fiTθ~fi+∑i=1m1γgijθ~˙gijTθ~gij=12eiTPiAi+AiTPiei+12-udi+ωiBiTPiei+eiTPiBi+1γfiγfieiTPiBiξfiTx+θ~˙fiTθ~fi+∑i=1m1γgijγgijeiTPiBiξgijTxuj+θ~˙gijTθ~gij.
Substituting (30)–(32) into (36), we obtain
(37)V˙i=-12eiTQiei+12ωiBiTPiei+eiTPiBi-12ρi2eiTPiBiBiTPiei=-12eiTQiei+12ρi2ωi2-121ρieiTPiBi-ρiωi2≤-12eiTQiei+12ρi2ωi2.
Integrating both sides of the above inequality from 0 toT yields
(38)12∫0TeiTQieidt≤Vi0-ViT+ρi22∫0Tωi2dt.
SinceViT is nonnegative, according to the definition of Vi, the following inequality is obtained:
(39)∫0TeiTQieidt≤eiT0Piei0+1γfiθ~fiT0θ~fi0+∑i=1m1γgijθ~gijT0θ~gij0+ρi2∫0Tωi2dt.
Considering the Lyapunov functionV=∑i=1mVi, it is easy to obtain the H∞ tracking performance index (55). This completes the proof.The controller proposed by (24) is able to guarantee the Lyapunov stability of the closed-loop system and attenuate the effect of system uncertainties and external disturbances. However, no saturation on actuators is taken into account, which is rather important for practical applications (including satellite systems). Assume that the control input u is constraint by saturation functions; it can be readily shown that the control law (24), the parameters update laws (30), (31), and the robust compensation term (32) with saturation limits cannot guarantee the stability of the closed-loop system.It is expected that during saturation the magnitude of the tracking error will increase, since the control signal is not being achieved. This tracking error is not the result of function approximation error; therefore, we need to be careful so that the approximator does not cause “unlearning” during the period when the actuators are saturated. Clearly, the parameters update laws (30) and (31) depend on the tracking error ei, thus if the tracking error increases due to saturation, the parameters update laws may cause a significant change in the weights in response to the increase in tracking error. Next, we develop a robust fuzzy adaptive control scheme to address the input saturation problem.


### 3.2. Adaptive Fuzzy Control with Input Saturation

When the actuators have physical constraints such as the magnitude and rate limitations, the above approach may not be able to be successfully implemented. Considering the magnitude and rate limitations on the actuator, controller (24) is modified as
(40)uc=G^-1x∣θG-F^x∣θF+ul+uaw+udu=SMRuc,
where uc is obtained by certainty equivalence principle; uaw is the auxiliary control term which is used to compensate the effect of input saturation; SMR· is a function including the magnitude and rate constraints which can produce a limited output.The state space representation of each component ofuc is
(41)r˙i1r˙i2=ri2SRωiSMuci-ri1uiu˙i=ri1ri2,
where uci is the ith element of uc, and ui is the ith element of u; ωi is the natural frequency; and SM· and SR· are the saturation functions corresponding to magnitude and rate, respectively. The function SM· is defined as
(42)SM·=Mifx≥Mxifx<M-Mifx≤-M
and SR· has the same definition.To compensate the effects of input limitations, an auxiliary system is introduced as follows [35]:
(43)ξ˙i1=ξi2-λi1ξi1⋯ξ˙i,ri-1=ξiri-λi,ri-1ξi,ri-1ξ˙iri=-λiriξiri+∑j=1mg^ijx∣θgijuj-ucji=1,2,…,m,
where λij  i=1,…,m;j=1,…,ri are positive design parameters.Define the modified tracking error as(44)e-i=ydi-yi-ξi1i=1,2,…,m.From (4), (40), and (43), we obtain
(45)e-iri+∑j=1rikije-iri-1=f^ix∣θfi-fix+∑j=1mg^ijx∣θgij-gijxuj-uawi-udi-di-ξi1ri-∑j=1rikijξi1j-1+∑j=1mg^ijx∣θgijuj-ucj,
where uawi is the ith element of uaw.From (43), the term ξi1ri can be expressed as
(46)ξi1ri=-∑j=1riλijξijri-j+∑j=1mg^ijx∣θgijuj-ucj.Let(47)uawi=∑j=1riλijξijri-j-kijξijj-1.With (46) and (47), (45) becomes
(48)e-iri+∑j=1rikije-iri-1=f^ix∣θfi-fix+∑j=1mg^ijx∣θgij-gijxuj-udi-di.Definee-i=e-i,e-˙i,…,e-iri-1T and using the optimal approximation for fix and gijx, (48) can be rewritten as
(49)e-˙i=Aie-i+Biθ~fiTξfix+∑j=1mθ~gijTξgijxuj-Biudi+Biωi.Theorem 6.
Consider the uncertain MIMO nonlinear system presented by (4) with external disturbance and input saturation, if the controller is chosen as (40), the auxiliary control term uaw, the parameters update laws, and udi are adopted as follows:
(50)ξ˙i1=ξi2-λi1ξi1⋯ξ˙i,ri-1=ξiri-λi,ri-1ξi,ri-1ξ˙iri=-λiriξiri+∑j=1mg^ijx∣θgijuj-ucji=1,2,…,m(51)uaw=∑j=1r1λ1jξ1jr1-j-k1jξ1jj-1⋯∑j=1rmλmjξmjrm-j-kmjξmjj-1T(52)θ˙fi=-γfiξfixBiTPie-ii=1,…,m(53)θ˙gij=-γgijξgijxBiTPie-iuji,j=1,…,m(54)udi=12ρi2BiTPie-ii=1,…,m.Then the followingH∞ tracking performance can be obtained:
(55)∫0Te-TQe-dt≤e-T0Pe-0+∑i=1m1γfiθ~fiT0θ~fi0+∑i,j=1m1γgijθ~gijT0θ~gij0+∑i=1mρi2∫0Tωi2dt,
where γfi  i=1,…,m and γgij  i,j=1,…,m are positive adaptive scalar, ρi  i=1,…,m are positive parameters representing for prescribed disturbance attenuation levels, e-=e-1T,…,e-mTT, Q=diagQ1,…,Qm, and Qi∈Rm×m  i=1,…,m are arbitrary symmetric positive definite matrices, and P=diagP1,…,Pm and Pi∈Rm×mi=1,…,m are symmetric positive definite solution of the following equations:
(56)PiAi+AiTPi=-Qi.Proof.
For theith subsystem of (4), consider the following Lyapunov function candidate:
(57)Vi=12e-iTPie-i+12γfiθ~fiTθ~fi+12∑i=1m1γgijθ~gijTθ~gij.
Differentiating (57) and considering (49) yield
(58)V˙i=12e-˙iTPie-i+12e-iTPie-˙i+1γfiθ~˙fiTθ~fi+∑i=1m1γgijθ~˙gijTθ~gij=12e-iTPiAi+AiTPie-i+12-udi+ωiBiTPie-i+e-iTPiBi+1γfiγfie-iTPiBiξfiTx+θ~˙fiTθ~fi+∑i=1m1γgijγgije-iTPiBiξgijTxuj+θ~˙gijTθ~gij.
Substituting (52)–(54) into (58), we obtain
(59)V˙i=-12e-iTQie-i+12ωiBiTPie-i+e-iTPiBi-12ρi2e-iTPiBiBiTPie-i=-12e-iTQie-i+12ρi2ωi2-121ρie-iTPiBi-ρiωi2≤-12e-iTQie-i+12ρi2ωi2.
Integrating both sides of the above inequality from 0 toT yields
(60)12∫0Te-iTQie-idt≤Vi0-ViT+ρi22∫0Tωi2dt.
SinceViT is nonnegative, according to the definition of Vi, the following inequality is obtained:
(61)∫0Te-iTQie-idt≤e-iT0Pie-i0+1γfiθ~fiT0θ~fi0+∑i=1m1γgijθ~gijT0θ~gij0+ρi2∫0Tωi2dt.
Considering the Lyapunov functionV=∑i=1mVi, it is easy to obtain the H∞ tracking performance index (55). This completes the proof.From the above analysis, it is concluded that, in the case of no control saturations, the signalsξij  i=1,…,m;j=1,…,ri remain zeros and the control law becomes the same as the standard robust adaptive control law described in the previous section. In the presence of control saturations, ξij is nonzero, thus giving rise to a modified tracking error e-i=ydi-yi-ξi1, which is used in fuzzy parameter update laws. The auxiliary control term uaw in (40) will be used to compensate the effects of control saturations. Note that the fuzzy parameter update laws (52) and (53) are similar to the corresponding update laws (30) and (31) derived in the standard fuzzy approximation based control problem with tracking error ei being replaced by the modified tracking error e-i. The use of the modified tracking error in the fuzzy update laws is crucial in preventing actuator constraints in online approximation schemes.Corollary 7.
For theith subsystem of (4), it is assumed that ∫0Tdi2dt<∞. If the control law (40), the auxiliary control term (51), and the parameter update laws (52)–(54) are adopted, then the following statements hold.(i)
The closed loop system is stable, and the signalse-i, θfi, θgij, and ui are bounded.(ii)
The steady modified tracking error satisfieslimt→∞e-i=0, and the bound of the transient modified tracking error will be given as follows:
(62)e-i2≤21/γfiθ~fiT0θ~fi0+∑i=1m1/γgijθ~gijT0θ~gij0λmin⁡Qi+2e-iT0Pie-i0+ρi2∫0Tωi2dtλmin⁡Qi.Proof.
From (59), it can be obtained that
(63)V˙i≤-ciVi+μi,
where ci=min⁡λv,1/γfi,1/γgij, λv=min⁡inf⁡λmin⁡Qi/sup⁡λmax⁡Qi, Mi=max⁡θfi, Mij=max⁡θgij, μi=Mi2/2γfi+∑j=1m1/2γgijMij2+(1/2)ρi2ω-i2, ω-i=sup⁡ωi, and λmin⁡Qi and λmax⁡Qi are the minimum and maximum eigenvalue of Qi, respectively.
From inequality (63), all the signals e-i, θfi, θgij, and ui are bounded by using Barbalat’s lemma [8].
On the other hand, from (59), it can be obtained that
(64)V˙i≤-12λmin⁡Qie-i2+12ρi2ωi2.
From the above inequality, one obtains(65)e-i2≤e-i2≤-2V˙i+ρi2ωi2λmin⁡Q.
Hence,(66)e-i2=∫0Te-i2dt≤2Vi0-ViT+ρi2∫0Tωi2dtλmin⁡Qi≤2Vi0+ρi2∫0Tωi2dtλmin⁡Qi.
According to the definition ofVi, (62) is obtained. Assumption 4 implies ∫0Tωi′2dt<∞; then ∫0Tωi2dt=∫0Tωi′-di2dt<∞. Therefore, ωi∈L2. From (66) we can conclude that e-i∈L2. Using Barbalat’s lemma, it follows that limt→∞e-it=0. This completes the proof.Remark 8.
According to Theorem6, the ith subsystem of (4) achieves a H∞ tracking performance with a prescribed disturbance attenuation level ρi; that is, the L2 gain from ωi to the modified tracking error e-i is equal or less than ρi.Remark 9.
As indicated from Corollary7, the steady modified tracking error e-i will converge to zero. The bound of the transient modified tracking error e-i is an explicit function of the design parameters and the external disturbance and fuzzy approximation error ωi. The bound can be decreased by choosing the initial estimates θfi0, θgij0 closing to the true values θfi*, θgij*. The effects of parameter initial estimate errors on the transient tracking performance can be reduced by increasing the adaptive gain values γfi, γgij, and λmin⁡Qi. Furthermore, the effect of external disturbance and fuzzy approximation error ωi on the transient tracking performance can be reduced by decreasing ρi and increasing λmin⁡Qi. Small ρi implies high disturbance attenuation level.


## 3.1. Adaptive Fuzzy Robust Control

The tracking errors are defined as(21)ei=ydi-yii=1,2,…,m.Defineuli  i=1,…,m as follows:
(22)uli=ydiri+∑j=1rikijeij-1i=1,…,m,
where ki1,…,kiri are parameters to be chosen such that the roots of the equation sri+kirisri-1+⋯+ki1=0 in the open left-half complex plane.IfFx and Gx are known, external disturbances are ignored; then, according to nonlinear dynamic inversion control techniques, the control law is given by
(23)u=G-1x-Fx+ul,
where ul=ul1,…,ulmT.BecauseFx and Gx are unknown, di≠0, the control law (23) cannot be implemented in practice. Using the approximation of Fx and Gx, and considering external disturbances, the controller is modified as follows:
(24)u=G^-1x∣θG-F^x∣θF+ul+ud,
where ud is the robust compensation term, which is used to attenuate the effect of external disturbances and approximation errors.Substituting (24) into system (4) yields
(25)yr=Fx-F^x∣θF+Gx-G^x∣θGu+ul+ud+d.Considering (22), the ith subsystem of (25) can be rewritten as
(26)e˙i=Aiei+Bi∑j=1mg^ijx∣θgij-gijxujf^ix∣θfi-fix+∑j=1mg^ijx∣θgij-gijxuj-Biudi-Bidi,
where udi is the ith element of ud and
(27)Ai=010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1-ki1-ki2-ki3⋯-kiriBi=00⋮01ei=eie˙i⋮eiri-2eiri-1.Define the minimum approximation error as(28)ωi′=f^ix∣θfi*-fix+∑j=1mg^ijx∣θgij*-gijxuj.According to fuzzy system theory, the following assumption is reasonable.Assumption 4.
The minimum approximation error is square integrable, that is,∫0Tωi′Tωi′dt<∞.Using the definition (28) and the optimal approximation for fix and gijx, (26) can be rewritten in the following form:
(29)e˙i=Aiei+Biθ~fiTξfix+∑j=1mθ~gijTξgijxuj-Biudi+Biωi,
where θ~fi=θfi-θfi*,  θ~gij=θgij-θgij*, and ωi=ωi′-di.Theorem 5.
Consider the uncertain MIMO nonlinear system presented by (4) with external disturbance. If the controller is chosen as  (24), the parameters update laws and udi are adopted as follows:
(30)θ˙fi=-γfiξfixBiTPieii=1,…,m(31)θ˙gij=-γgijξgijxBiTPieiuji,j=1,…,m(32)udi=12ρi2BiTPieii=1,…,m.Then the followingH∞ tracking performance can be obtained
(33)∫0TeTQedt≤eT0Pe0+∑i=1m1γfiθ~fiT0θ~fi0+∑i,j=1m1γgijθ~gijT0θ~gij0+∑i=1mρi2∫0Tωi2dt,
where γfi  i=1,…,m and γgij  i,j=1,…,m are positive adaptive scalar, ρi  i=1,…,m are positive parameters representing for prescribed disturbance attenuation levels, e=e1T,…,emTT, Q=diagQ1,…,Qm, and Qi∈Rm×m  i=1,…,m are arbitrary symmetric positive definite matrices, and P=diagP1,…,Pm and Pi∈Rm×m  i=1,…,m are symmetric positive definite solution of the following equations:
(34)PiAi+AiTPi=-Qi,
where Qi∈Rm×mi=1,…,m are arbitrary symmetric positive definite matrices.Proof.
For theith subsystem of (4), consider the following Lyapunov function candidate
(35)Vi=12eiTPiei+12γfiθ~fiTθ~fi+12∑i=1m1γgijθ~gijTθ~gij.
Differentiating (35) and considering (29) yield
(36)V˙i=12e˙iTPiei+12eiTPie˙i+1γfiθ~˙fiTθ~fi+∑i=1m1γgijθ~˙gijTθ~gij=12eiTPiAi+AiTPiei+12-udi+ωiBiTPiei+eiTPiBi+1γfiγfieiTPiBiξfiTx+θ~˙fiTθ~fi+∑i=1m1γgijγgijeiTPiBiξgijTxuj+θ~˙gijTθ~gij.
Substituting (30)–(32) into (36), we obtain
(37)V˙i=-12eiTQiei+12ωiBiTPiei+eiTPiBi-12ρi2eiTPiBiBiTPiei=-12eiTQiei+12ρi2ωi2-121ρieiTPiBi-ρiωi2≤-12eiTQiei+12ρi2ωi2.
Integrating both sides of the above inequality from 0 toT yields
(38)12∫0TeiTQieidt≤Vi0-ViT+ρi22∫0Tωi2dt.
SinceViT is nonnegative, according to the definition of Vi, the following inequality is obtained:
(39)∫0TeiTQieidt≤eiT0Piei0+1γfiθ~fiT0θ~fi0+∑i=1m1γgijθ~gijT0θ~gij0+ρi2∫0Tωi2dt.
Considering the Lyapunov functionV=∑i=1mVi, it is easy to obtain the H∞ tracking performance index (55). This completes the proof.The controller proposed by (24) is able to guarantee the Lyapunov stability of the closed-loop system and attenuate the effect of system uncertainties and external disturbances. However, no saturation on actuators is taken into account, which is rather important for practical applications (including satellite systems). Assume that the control input u is constraint by saturation functions; it can be readily shown that the control law (24), the parameters update laws (30), (31), and the robust compensation term (32) with saturation limits cannot guarantee the stability of the closed-loop system.It is expected that during saturation the magnitude of the tracking error will increase, since the control signal is not being achieved. This tracking error is not the result of function approximation error; therefore, we need to be careful so that the approximator does not cause “unlearning” during the period when the actuators are saturated. Clearly, the parameters update laws (30) and (31) depend on the tracking error ei, thus if the tracking error increases due to saturation, the parameters update laws may cause a significant change in the weights in response to the increase in tracking error. Next, we develop a robust fuzzy adaptive control scheme to address the input saturation problem.


## 3.2. Adaptive Fuzzy Control with Input Saturation

When the actuators have physical constraints such as the magnitude and rate limitations, the above approach may not be able to be successfully implemented. Considering the magnitude and rate limitations on the actuator, controller (24) is modified as
(40)uc=G^-1x∣θG-F^x∣θF+ul+uaw+udu=SMRuc,
where uc is obtained by certainty equivalence principle; uaw is the auxiliary control term which is used to compensate the effect of input saturation; SMR· is a function including the magnitude and rate constraints which can produce a limited output.The state space representation of each component ofuc is
(41)r˙i1r˙i2=ri2SRωiSMuci-ri1uiu˙i=ri1ri2,
where uci is the ith element of uc, and ui is the ith element of u; ωi is the natural frequency; and SM· and SR· are the saturation functions corresponding to magnitude and rate, respectively. The function SM· is defined as
(42)SM·=Mifx≥Mxifx<M-Mifx≤-M
and SR· has the same definition.To compensate the effects of input limitations, an auxiliary system is introduced as follows [35]:
(43)ξ˙i1=ξi2-λi1ξi1⋯ξ˙i,ri-1=ξiri-λi,ri-1ξi,ri-1ξ˙iri=-λiriξiri+∑j=1mg^ijx∣θgijuj-ucji=1,2,…,m,
where λij  i=1,…,m;j=1,…,ri are positive design parameters.Define the modified tracking error as(44)e-i=ydi-yi-ξi1i=1,2,…,m.From (4), (40), and (43), we obtain
(45)e-iri+∑j=1rikije-iri-1=f^ix∣θfi-fix+∑j=1mg^ijx∣θgij-gijxuj-uawi-udi-di-ξi1ri-∑j=1rikijξi1j-1+∑j=1mg^ijx∣θgijuj-ucj,
where uawi is the ith element of uaw.From (43), the term ξi1ri can be expressed as
(46)ξi1ri=-∑j=1riλijξijri-j+∑j=1mg^ijx∣θgijuj-ucj.Let(47)uawi=∑j=1riλijξijri-j-kijξijj-1.With (46) and (47), (45) becomes
(48)e-iri+∑j=1rikije-iri-1=f^ix∣θfi-fix+∑j=1mg^ijx∣θgij-gijxuj-udi-di.Definee-i=e-i,e-˙i,…,e-iri-1T and using the optimal approximation for fix and gijx, (48) can be rewritten as
(49)e-˙i=Aie-i+Biθ~fiTξfix+∑j=1mθ~gijTξgijxuj-Biudi+Biωi.Theorem 6.
Consider the uncertain MIMO nonlinear system presented by (4) with external disturbance and input saturation, if the controller is chosen as (40), the auxiliary control term uaw, the parameters update laws, and udi are adopted as follows:
(50)ξ˙i1=ξi2-λi1ξi1⋯ξ˙i,ri-1=ξiri-λi,ri-1ξi,ri-1ξ˙iri=-λiriξiri+∑j=1mg^ijx∣θgijuj-ucji=1,2,…,m(51)uaw=∑j=1r1λ1jξ1jr1-j-k1jξ1jj-1⋯∑j=1rmλmjξmjrm-j-kmjξmjj-1T(52)θ˙fi=-γfiξfixBiTPie-ii=1,…,m(53)θ˙gij=-γgijξgijxBiTPie-iuji,j=1,…,m(54)udi=12ρi2BiTPie-ii=1,…,m.Then the followingH∞ tracking performance can be obtained:
(55)∫0Te-TQe-dt≤e-T0Pe-0+∑i=1m1γfiθ~fiT0θ~fi0+∑i,j=1m1γgijθ~gijT0θ~gij0+∑i=1mρi2∫0Tωi2dt,
where γfi  i=1,…,m and γgij  i,j=1,…,m are positive adaptive scalar, ρi  i=1,…,m are positive parameters representing for prescribed disturbance attenuation levels, e-=e-1T,…,e-mTT, Q=diagQ1,…,Qm, and Qi∈Rm×m  i=1,…,m are arbitrary symmetric positive definite matrices, and P=diagP1,…,Pm and Pi∈Rm×mi=1,…,m are symmetric positive definite solution of the following equations:
(56)PiAi+AiTPi=-Qi.Proof.
For theith subsystem of (4), consider the following Lyapunov function candidate:
(57)Vi=12e-iTPie-i+12γfiθ~fiTθ~fi+12∑i=1m1γgijθ~gijTθ~gij.
Differentiating (57) and considering (49) yield
(58)V˙i=12e-˙iTPie-i+12e-iTPie-˙i+1γfiθ~˙fiTθ~fi+∑i=1m1γgijθ~˙gijTθ~gij=12e-iTPiAi+AiTPie-i+12-udi+ωiBiTPie-i+e-iTPiBi+1γfiγfie-iTPiBiξfiTx+θ~˙fiTθ~fi+∑i=1m1γgijγgije-iTPiBiξgijTxuj+θ~˙gijTθ~gij.
Substituting (52)–(54) into (58), we obtain
(59)V˙i=-12e-iTQie-i+12ωiBiTPie-i+e-iTPiBi-12ρi2e-iTPiBiBiTPie-i=-12e-iTQie-i+12ρi2ωi2-121ρie-iTPiBi-ρiωi2≤-12e-iTQie-i+12ρi2ωi2.
Integrating both sides of the above inequality from 0 toT yields
(60)12∫0Te-iTQie-idt≤Vi0-ViT+ρi22∫0Tωi2dt.
SinceViT is nonnegative, according to the definition of Vi, the following inequality is obtained:
(61)∫0Te-iTQie-idt≤e-iT0Pie-i0+1γfiθ~fiT0θ~fi0+∑i=1m1γgijθ~gijT0θ~gij0+ρi2∫0Tωi2dt.
Considering the Lyapunov functionV=∑i=1mVi, it is easy to obtain the H∞ tracking performance index (55). This completes the proof.From the above analysis, it is concluded that, in the case of no control saturations, the signalsξij  i=1,…,m;j=1,…,ri remain zeros and the control law becomes the same as the standard robust adaptive control law described in the previous section. In the presence of control saturations, ξij is nonzero, thus giving rise to a modified tracking error e-i=ydi-yi-ξi1, which is used in fuzzy parameter update laws. The auxiliary control term uaw in (40) will be used to compensate the effects of control saturations. Note that the fuzzy parameter update laws (52) and (53) are similar to the corresponding update laws (30) and (31) derived in the standard fuzzy approximation based control problem with tracking error ei being replaced by the modified tracking error e-i. The use of the modified tracking error in the fuzzy update laws is crucial in preventing actuator constraints in online approximation schemes.Corollary 7.
For theith subsystem of (4), it is assumed that ∫0Tdi2dt<∞. If the control law (40), the auxiliary control term (51), and the parameter update laws (52)–(54) are adopted, then the following statements hold.(i)
The closed loop system is stable, and the signalse-i, θfi, θgij, and ui are bounded.(ii)
The steady modified tracking error satisfieslimt→∞e-i=0, and the bound of the transient modified tracking error will be given as follows:
(62)e-i2≤21/γfiθ~fiT0θ~fi0+∑i=1m1/γgijθ~gijT0θ~gij0λmin⁡Qi+2e-iT0Pie-i0+ρi2∫0Tωi2dtλmin⁡Qi.Proof.
From (59), it can be obtained that
(63)V˙i≤-ciVi+μi,
where ci=min⁡λv,1/γfi,1/γgij, λv=min⁡inf⁡λmin⁡Qi/sup⁡λmax⁡Qi, Mi=max⁡θfi, Mij=max⁡θgij, μi=Mi2/2γfi+∑j=1m1/2γgijMij2+(1/2)ρi2ω-i2, ω-i=sup⁡ωi, and λmin⁡Qi and λmax⁡Qi are the minimum and maximum eigenvalue of Qi, respectively.
From inequality (63), all the signals e-i, θfi, θgij, and ui are bounded by using Barbalat’s lemma [8].
On the other hand, from (59), it can be obtained that
(64)V˙i≤-12λmin⁡Qie-i2+12ρi2ωi2.
From the above inequality, one obtains(65)e-i2≤e-i2≤-2V˙i+ρi2ωi2λmin⁡Q.
Hence,(66)e-i2=∫0Te-i2dt≤2Vi0-ViT+ρi2∫0Tωi2dtλmin⁡Qi≤2Vi0+ρi2∫0Tωi2dtλmin⁡Qi.
According to the definition ofVi, (62) is obtained. Assumption 4 implies ∫0Tωi′2dt<∞; then ∫0Tωi2dt=∫0Tωi′-di2dt<∞. Therefore, ωi∈L2. From (66) we can conclude that e-i∈L2. Using Barbalat’s lemma, it follows that limt→∞e-it=0. This completes the proof.Remark 8.
According to Theorem6, the ith subsystem of (4) achieves a H∞ tracking performance with a prescribed disturbance attenuation level ρi; that is, the L2 gain from ωi to the modified tracking error e-i is equal or less than ρi.Remark 9.
As indicated from Corollary7, the steady modified tracking error e-i will converge to zero. The bound of the transient modified tracking error e-i is an explicit function of the design parameters and the external disturbance and fuzzy approximation error ωi. The bound can be decreased by choosing the initial estimates θfi0, θgij0 closing to the true values θfi*, θgij*. The effects of parameter initial estimate errors on the transient tracking performance can be reduced by increasing the adaptive gain values γfi, γgij, and λmin⁡Qi. Furthermore, the effect of external disturbance and fuzzy approximation error ωi on the transient tracking performance can be reduced by decreasing ρi and increasing λmin⁡Qi. Small ρi implies high disturbance attenuation level.


## 4. Simulation Examples

The attitude tracking control problem of a rigid body satellite system is simulated in this section to illustrate the effectiveness of the robust adaptive fuzzy controllers proposed in this paper. The mathematical model of the satellite attitude system can be reformulated to the general form of uncertain nonlinear MIMO system as follows [42, 44]:
(67)y¨=Fx+Gxu+d,
where x=q,ωT is the state vector, where q=q0,q1,q2,q3T is the attitude quaternion in the body-fixed reference frame relative to the inertial frame satisfying q02+q12+q22+q32=1, here q0 is chosen as q0=1-q12-q22-q32; ω=ωx,ωy,ωzT is the angular velocity of the body-fixed reference relative to the inertial frame. y=q1,q2,q3T is the output vector, u=u1,u2,u3T is the control torque vector, d′∈R3 denotes the bounded external disturbance torques, and d≜Gxd′. Fx and Gx are expressed as follows:
(68)f1x=-14q1ωx2+ωy2+ωz2+Ix-Iy2Izq2ωxωy+Ix-Iz2Iyq3ωxωz+Iy-Iz2Ixq0ωyωzf2x=-14q2ωx2+ωy2+ωz2+Iy-Ix2Izq1ωxωy+Iz-Ix2Iyq0ωxωz+Iy-Iz2Ixq3ωyωzf3x=-14q3ωx2+ωy2+ωz2+Ix-Iy2Izq0ωxωy+Iz-Ix2Iyq1ωxωz+Iz-Iy2Ixq2ωyωzGx=q02Ix-q32Iyq22Izq32Ixq02Iy-q12Iz-q22Ixq12Iyq02Iz,
where Ix, Iy, and Iz are the principal central moments of inertia of the satellite.Within this simulation, the nonlinear functionsFx and Gx are assumed completely unknown, that is, the fuzzy adaptive controllers do not require the knowledge of the system’s model. In fact, the dynamic model of the satellite attitude system is only required for simulation purpose.Since the components ofFx and Gx are assumed unknown, three fuzzy logic systems in the form of (12) are used to approximate the elements of Fx, and nine are used to approximate the elements of Gx. The fuzzy logic systems used to describe Fx have q1, q2, q3, ω1, ω2, and ω3 as inputs, and the ones used to describe Gx have q1, q2, and q3 as inputs. For each state variable x=q1,q2,q3,ω1,ω2,ω3T, we define seven Gaussian membership functions as
(69)μFi1xi=11+exp⁡-5xi+0.6,μFi2xi=exp⁡-0.5xi+0.42μFi3xi=exp⁡-0.5xi+0.22,μFi4xi=exp⁡-0.5xi2μFi5xi=exp⁡-0.5xi-0.22,μFi6xi=exp⁡-0.5xi-0.42μFi7xi=11+exp⁡-5xi-0.6,i=1,2,3,4,5,6.The fuzzy basis function vectors are chosen as follows:(70)ξfi=∏i=16μFi1xi∑j=17∏i=16μFijxi,…,∏i=16μFi7xi∑j=17∏i=16μFijxiTi=1,2,3ξgij=∏i=13μFi1xi∑j=17∏i=13μFijxi,…,∏i=13μFi7xi∑j=17∏i=13μFijxiTi,j=1,2,3.Then we obtain fuzzy logic systems(71)f^ix∣θfi=θfiTξfixi=1,2,3g^ijx∣θgij=θgijTξgijxi,j=1,2,3.Using (71) approximate the unknown functions fi and gij, respectively.For the given coefficientski1=0.25  i=1,2,3 and ki2=0.5  i=1,2,3, we have
(72)A1=A2=A3=01-0.25-0.5B1=B2=B3=01.SelectingQi=diag[0.5,0.5] and solving (34), we obtain
(73)Pi=1.6250.50.53i=1,2,3.The parameters of the adaptive fuzzy controllers are chosen asγfi=1×10-3, γgij=1×10-3, and ρi=0.5, i,j=1,2,3.The initial attitude quaternion isq0=0.9014,0.25,0.25,0.25T, and the initial value of the angular velocity is ω0=0.0050.0050.005T rad/s. The parameters of system are given as Ix=187.5 kg·m2, Iy=468.5 kg·m2, and Iz=468.5 kg·m2. The external disturbance torque vector is given by d′=0.3sin0.05t,0.3cos⁡0.05t,-0.3sin0.05tT N·m. The initial values of the parameters θfi and θgij are set to random values uniformly distributed between 0,1. The desired output trajectory is chosen as yd=0.5sin0.005t,0.5sin0.005t,0.5sin0.005tT. The control objective is to force the system output y to track the desired output trajectory yd.


### 4.1. Without Input Saturation

In this section, the tracking control problem is simulated to demonstrate the effectiveness of robust adaptive fuzzy controller (24) proposed in Section 3.1.Using the control law (24) and (30)–(32), simulation results are presented in Figures 1 and 2. Figure 1 shows the curves of the system outputs and its reference trajectories, which indicates that the robust adaptive fuzzy controller achieves a good performance in tracking control problem and the effects of fuzzy approximation errors and external disturbances on tracking errors are effectively attenuated. Figure 2 shows that the control inputs can be carried out feasibly without any constraints. The control signals are obtained by certainty equivalence principle. Therefore, they do not satisfy the control input limitations naturally.Figure 1
Trajectories of outputs without saturation.Figure 2
Trajectories of control inputs without saturation.


### 4.2. Actuator Amplitude Saturation

In order to demonstrate that the proposed adaptive control scheme (40) can work effectively under actuator amplitude saturation, numerical simulations have been performed and presented in this section.Consider satellite attitude model (67) with the same disturbances and system initial conditions mentioned above. The control input vector u=u1,u2,u3T has amplitude limits ui≤1 N·m, i=1,2,3. The auxiliary system is constructed as follows:
(74)ξ˙i1=ξi2-λi1ξi1ξ˙i2=-λi2ξi2+∑j=13g^ijx∣θgijuj-ucji=1,2,3,
where λij=1  i=1,2,3;j=1,2 and ξij0=0  i=1,2,3;j=1,2.Using the control law (40), the auxiliary control term (51), the parameters update laws (52)-(53), and the robust compensation term (54), we can get the simulations in Figures 3 and 4.Figure 3
Trajectories of outputs with amplitude saturation.Figure 4
Trajectories of control inputs with amplitude saturation.The system outputs and its reference trajectories are shown in Figure3, which indicate that the outputs track their reference trajectories well in spite of the external disturbances, system uncertainties, and actuator amplitude saturation. Figure 4 shows the trajectories of the control inputs with actuator amplitude saturation. From these simulations, it is obvious that proposed control scheme (40) can achieve a good tracking performance when the actuator amplitude limits are considered.


### 4.3. Actuator Amplitude and Rate Saturation

For further analysis, actuator amplitude and rate saturations are considered. The control input vectoru=u1,u2,u3T has the amplitude and rate limitation ui≤1, u˙i≤2, i=1,2,3. The other initial parameters are the same as mentioned in Section 4.2.According to (41), the dynamics of control inputs are expressed as follows:
(75)u˙i=S1ωiS1uci-uii=1,2,3,
where ωi=20.5  i=1,2,3.Using control law (40), the actuator amplitude and rate constraints (41), the auxiliary control term (51), the parameters update laws (52)-(53), and the robust compensation term (54), simulation results are presented in Figures 5–8. Figure 5 shows the curves of outputs and their reference trajectories, which indicate that a good tracking control performance is still achieved under actuator amplitude and rate constraint conditions. The control signals and their derivatives are given in Figures 6 and 7. It is observed that the control signals satisfy the amplitude and rate limitations. That is, the proposed robust control scheme for the satellite attitude control system can prevent the control signals from reaching amplitude and rate saturation limits. Figure 8 shows the signals uci  i=1,2,3. Obviously, they do not satisfy the control input limitations.Figure 5
Trajectories of outputs with amplitude and rate saturation.Figure 6
Trajectories of control inputs with amplitude and rate saturation.Figure 7
Derivatives of control inputs.Figure 8
u
c
1, uc2, and uc3.From the aforementioned simulations, it is demonstrated that the robust adaptive fuzzy tracking controller proposed in this paper not only can generate control inputs that satisfy actuator amplitude and rate saturations but also can effectively attenuate the effects of approximation error and extern disturbance on tracking errors. Thus, the proposed robust adaptive control scheme is valid for satellite attitude control system with actuator amplitude and rate saturation.


## 4.1. Without Input Saturation

In this section, the tracking control problem is simulated to demonstrate the effectiveness of robust adaptive fuzzy controller (24) proposed in Section 3.1.Using the control law (24) and (30)–(32), simulation results are presented in Figures 1 and 2. Figure 1 shows the curves of the system outputs and its reference trajectories, which indicates that the robust adaptive fuzzy controller achieves a good performance in tracking control problem and the effects of fuzzy approximation errors and external disturbances on tracking errors are effectively attenuated. Figure 2 shows that the control inputs can be carried out feasibly without any constraints. The control signals are obtained by certainty equivalence principle. Therefore, they do not satisfy the control input limitations naturally.Figure 1
Trajectories of outputs without saturation.Figure 2
Trajectories of control inputs without saturation.


## 4.2. Actuator Amplitude Saturation

In order to demonstrate that the proposed adaptive control scheme (40) can work effectively under actuator amplitude saturation, numerical simulations have been performed and presented in this section.Consider satellite attitude model (67) with the same disturbances and system initial conditions mentioned above. The control input vector u=u1,u2,u3T has amplitude limits ui≤1 N·m, i=1,2,3. The auxiliary system is constructed as follows:
(74)ξ˙i1=ξi2-λi1ξi1ξ˙i2=-λi2ξi2+∑j=13g^ijx∣θgijuj-ucji=1,2,3,
where λij=1  i=1,2,3;j=1,2 and ξij0=0  i=1,2,3;j=1,2.Using the control law (40), the auxiliary control term (51), the parameters update laws (52)-(53), and the robust compensation term (54), we can get the simulations in Figures 3 and 4.Figure 3
Trajectories of outputs with amplitude saturation.Figure 4
Trajectories of control inputs with amplitude saturation.The system outputs and its reference trajectories are shown in Figure3, which indicate that the outputs track their reference trajectories well in spite of the external disturbances, system uncertainties, and actuator amplitude saturation. Figure 4 shows the trajectories of the control inputs with actuator amplitude saturation. From these simulations, it is obvious that proposed control scheme (40) can achieve a good tracking performance when the actuator amplitude limits are considered.


## 4.3. Actuator Amplitude and Rate Saturation

For further analysis, actuator amplitude and rate saturations are considered. The control input vectoru=u1,u2,u3T has the amplitude and rate limitation ui≤1, u˙i≤2, i=1,2,3. The other initial parameters are the same as mentioned in Section 4.2.According to (41), the dynamics of control inputs are expressed as follows:
(75)u˙i=S1ωiS1uci-uii=1,2,3,
where ωi=20.5  i=1,2,3.Using control law (40), the actuator amplitude and rate constraints (41), the auxiliary control term (51), the parameters update laws (52)-(53), and the robust compensation term (54), simulation results are presented in Figures 5–8. Figure 5 shows the curves of outputs and their reference trajectories, which indicate that a good tracking control performance is still achieved under actuator amplitude and rate constraint conditions. The control signals and their derivatives are given in Figures 6 and 7. It is observed that the control signals satisfy the amplitude and rate limitations. That is, the proposed robust control scheme for the satellite attitude control system can prevent the control signals from reaching amplitude and rate saturation limits. Figure 8 shows the signals uci  i=1,2,3. Obviously, they do not satisfy the control input limitations.Figure 5
Trajectories of outputs with amplitude and rate saturation.Figure 6
Trajectories of control inputs with amplitude and rate saturation.Figure 7
Derivatives of control inputs.Figure 8
u
c
1, uc2, and uc3.From the aforementioned simulations, it is demonstrated that the robust adaptive fuzzy tracking controller proposed in this paper not only can generate control inputs that satisfy actuator amplitude and rate saturations but also can effectively attenuate the effects of approximation error and extern disturbance on tracking errors. Thus, the proposed robust adaptive control scheme is valid for satellite attitude control system with actuator amplitude and rate saturation.


## 5. Conclusions

In this work, a robust fuzzy tracking control approach has been presented for a class of uncertain nonlinear MIMO systems in the presence of input saturation and external disturbances. Fuzzy logic systems are used to approximate the unknown system nonlinear terms. An auxiliary system is constructed to compensate the effects of actuator saturations, and then the actuator saturations can be augmented into the controller. The modified tracking error is introduced and used in fuzzy parameter update laws. A robust compensation control is designed to attenuate the effects of external disturbances and fuzzy approximation errors. The stability properties and tracking performance of the closed-loop system are obtained through Lyapunov analysis. Steady and transient modified tracking errors are analyzed and the bound of modified tracking errors can be adjusted by tuning certain design parameters. The proposed control scheme is applicable to uncertain nonlinear systems not only with actuator amplitude saturation, but also with actuator amplitude and rate saturation. The simulation results of satellite attitude control system are presented to demonstrate the effectiveness of proposed controller. In this paper, the control system relies on the output derivatives, up tori-1 order as in (22), which is also practically restrictive for high order systems. Future research will be concentrated on an observer-based robust adaptive fuzzy control of uncertain nonlinear systems with actuator saturations based on the results of [18–20] and this paper.



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*Source: 561397-2015-03-02.xml*