# Observer-Based Robust Adaptive Fuzzy Control for MIMO Nonlinear Uncertain Systems with Delayed Output

Authors: Chiang Cheng Chiang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

An observer-based robust adaptive fuzzy control scheme is presented to tackle the problem of the robust stability and the tracking control for a class of multiinput multioutput (MIMO) nonlinear uncertain systems with delayed output. Because the nonlinear system functions and the uncertainties of the controlled system including structural uncertainties are supposed to be unknown, fuzzy logic systems are utilized to approximate these nonlinear system functions and the upper bounded functions of the uncertainties. Moreover, the upper bound of uncertainties caused by these fuzzy modeling errors is also estimated. In addition, the state observer based on state variable filters is designed to estimate all states which are not available for measurement in the controlled system. By constructing an appropriate Lyapunov function and using strictly positive-real (SPR) stability theorem, the proposed robust adaptive fuzzy controller not only guarantees the robust stability of a class of multivariable nonlinear uncertain systems with delayed output but also maintains a good tracking performance. Finally, some simulation results are illustrated to verify the effectiveness of the proposed control approach.


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## Body

## 1. Introduction

Controller design for nonlinear systems has been given a lot of attention in the control community during the last two decades. Recently, the methods of feedback linearization have been successfully applied in the development of controllers for a class of nonlinear systems [1, 2]. However, the above techniques can be only applied to nonlinear systems whose dynamics are exactly known. Actually, it is difficult to obtain the exact construction of the systems in most of industrial and engineering systems because of the complexity of systems. In addition, the existence of uncertainties such as parameter uncertainties, modeling errors, and external disturbances may lead to poor performance and instability [3–8]. Hence, the study of robust stability of nonlinear uncertain systems in the presence of uncertainties is an important topic for the control design engineer.Recently, fuzzy control technique has been considered extensively in the control problems of complex and ill-defined nonlinear systems in the presence of incomplete knowledge of the plant [3, 9–15]. However, the above adaptive fuzzy controllers are only limited to the systems under the conservative assumption that system states are available for measurement. In order to relax this restriction, an observer-based fuzzy adaptive output feedback control for the SISO nonlinear systems is presented to tackle the nonlinear systems whose states are not available [16–23]. So far, few research results have been extended to MIMO nonlinear systems [24, 25]. The common feature of most previous results [16, 19, 24, 25] is the assumption that the controlled systems are free of uncertainties. Although the above restrictive assumption can be relaxed in [17, 18, 20–23], the uncertainties are assumed to be a bounded external disturbance. Therefore, the motivation of this paper is to synthesize an observer-based robust adaptive fuzzy control scheme to deal with the tracking control problem for a class of MIMO nonlinear uncertain systems with delayed output in the presence of uncertainties including the structural uncertainty.On the other hand, the design problem of nonlinear time-delay systems has received considerable attention in [26–31] because time-delay characteristic often encountered in various engineering systems may not only cause instability but also lead to serious deterioration in the performance of the plants. By employing the input-output approach and the scaled small gain theorem, the filtering problem for discrete-time T-S fuzzy systems with time-varying delay has been studied [26]. The robust H∞ control problem for stochastic systems with a delay in the state is investigated in [28], and the results are further extended to the stochastic time-delay systems with parameter uncertainties. The problems of stability analysis and robust control for uncertain systems with input delay were examined in [29, 31]. In [27], the design scheme of output feedback controller for a class of SISO nonlinear systems with delayed output was proposed to construct a delay-dependent controller making the closed-loop system globally asymptotically stable. In [30], a new approach for the construction of a state observer for SISO nonlinear systems with delayed output was presented to ensure the global exponential convergence to zero of the observation error. It should be pointed out that the uncertainty was not taken into account in [27, 30]. Therefore, the robust control problem of MIMO nonlinear systems with delayed output in the presence of uncertainties will be considered in this paper.In this paper, the problem of controller design for a class of MIMO nonlinear uncertain output-delay systems whose states are not available is considered. The main features of the proposed observer-based robust adaptive fuzzy controller are summarized as follows. (i) Fuzzy logic systems with some appropriate learning laws are applied to approximate the nonlinear system functions and the upper bounded functions of the uncertainties including the structural uncertainty. (ii) The unknown upper bound of the uncertainties caused by approximation (or fuzzy modeling) error is estimated by a simple adaptive law. (iii) The state observer based on state variable filters is designed to estimate all states which are not available for measurement in the controlled system. (iv) By constructing an appropriate Lyapunov function and using strictly positive-real (SPR) stability theorem, the proposed robust adaptive fuzzy output feedback controller can not only guarantee the robust stability of the whole closed-loop system but also obtain the good tracking performance.This paper is organized as follows: the description of the system and the concept of fuzzy logic systems are given together in Section2. Section 3 proposes an observer-based robust adaptive fuzzy output feedback controller to achieve the purpose of asymptotic stabilization and output tracking performance of the whole closed-loop MIMO nonlinear uncertain systems with delayed output. In Section 4, a series of simulation results are illustrated to show the effectiveness of the proposed control scheme. Finally, a conclusion is given in Section 5.


## 2. Problem Formulation and Preliminaries

### 2.1. Problem Formulation

Consider a class of MIMO nonlinear uncertain systems with delayed output in the following form:(1)x˙11(t)=x12(t),x˙12(t)=x13(t),⋮x˙1(n1-1)(t)=x1n1(t),x˙1n1(t)=f1(x(t))+∑j=1pg1j(x(t))uj(t)+Δϕ1(x(t)),⋮x˙p1(t)=xp2(t),x˙p2(t)=xp3(t),⋮x˙p(np-1)(t)=xpnp(t),x˙pnp(t)=fp(x(t))+∑j=1Pgpj(x(t))uj(t)+Δϕp(x(t)),y1(t)=x11(t-τ),⋮yp(t)=xp1(t-τ),
or equivalently
(2)x˙(t)=Ax(t)+B[F(x(t))+G(x(t))u(t)+ΔΦ(x(t))],y(t)=Cx(t-τ),
where
(3)F(x)=[f1(x),…,fp(x)]T∈Rp,G(x)=[G1(x),…,Gp(x)]∈Rp×p,Gi(x)=[g1i(x),…,gpi(x)]T∈Rp,i=1,2,…,p,A=diag[A1,…,Ap]∈Rn×n,B=diag[B1,…,Bp]∈Rn×p,C=diag[C1,…,Cp]∈Rp×n,Ai=[010⋯0001⋯0⋮⋮⋮⋯1000⋯0]∈Rni×ni,Bi=[00⋮1]∈Rni,CiT=[10⋮0]∈Rni,
where x=[x11,x12,…,x1n1,…,xp1,xp2,…,xpnp]T∈Rn is the system state vector which is assumed to be unavailable for measurement, u=[u1,…,up]T∈Rp is the control input, y=[y1,…,yp]T∈Rp is the output vector, n1+n2+⋯+np=n, τ>0 is a known time delay of the system, fi(x(t)) and gij(x(t)) are the unknown nonlinear system functions, and ΔΦ(x(t))  =  [Δϕ1(x(t),Δϕ2(x(t),…,Δϕp(x(t)]T∈Rp is the vector of uncertainties.Now let the output of the system and its derivatives be expressed as follows:(4)yi(t)=xi1(t-τ),y˙i(t)=x˙i1(t-τ)=xi2(t-τ),⋮yi(ni-1)(t)=x˙i(ni-1)(t-τ)=xini(t-τ),yi(ni)(t)=x˙ini(t-τ)=fi(x(t-τ))+∑j=1pgij(x(t-τ))uj(t-τ)+Δϕi(x(t-τ)).
Then, let yd=[y1d,…,ypd]T∈Rp be the given bounded output desired signal and contain finite derivatives up to the nth order. The tracking errors of the system can be defined as eij(t)=yi(j-1)(t)-yid(j-1)(t), for i=1,2,…,p and j=1,2,…,ni, and its derivatives can be obtained as
(5)e˙i1(t)=y˙i(t)-y˙id(t)=ei2(t),e˙i2(t)=yi(2)(t)-yid(2)(t)=ei3(t),⋮e˙i(ni-1)(t)=yi(ni-1)(t)-yid(ni-1)(t)=e1ni(t),e˙ini(t)=yi(ni)(t)-yid(ni)(t)=fi(x(t-τ))+∑j=1pgij(x(t-τ))uj(t-τ)+Δϕi(x(t-τ))-yid(ni)(t)=fi(x(t-τ))+∑j=1pgij(x(t-τ))uj(t)-yid(ni)(t)+Δψi(x,u),
where the uncertainty Δψi(x,u) including the structural uncertainty is defined as
(6)Δψi(x,u)=∑j=1pgij(x(t-τ))[uj(t-τ)-uj(t)]+Δϕi(x(t-τ)).
Thus, (5) can be expressed as
(7)e˙i(t)=Aiei(t)+Bi[fi(x(t-τ))+∑j=1pgij(x(t-τ))uj(t)-yid(ni)(t)+Δψi(x,u)∑j=1pgij(x(t-τ))].
From (7), it is easy to verify that the tracking error dynamics of the system (2) can be represented in the following form:
(8)e˙(t)=Ae(t)+B[Yd_F(x(t-τ))+G(x(t-τ))u(t)-Yd_(t)+ΔΨ(x,u)].e1_(t)=Ce(t),
where
(9)e=[e1T,e2T,…,epT]T=[e11,…,e1n1,…,ep1,…,epnp]T∈Rn,e1_=[e11,e21,…,ep1]T∈Rp,ΔΨ=[Δψ1,Δψ2,…,Δψp]T∈Rp,ei=[ei1,ei2,…,eini]T∈Rni,Yd_=[y1d(n1),y2d(n2),…,ypd(np)]T∈Rp,Yd=[y1d,…,y1d(n1-1),…,ypd,…,ypd(np-1)]T∈Rn.Based on the above discussion, the following assumptions should be made for the controller design.Assumption 1.
The matrixG(x) is nonsingular (i.e., G(x)-1 exists) for all x∈Ux where Ux is a compact set in Rn.Assumption 2.
∥
Δ
Ψ
(
x
,
u
)
∥
≤
h
1
(
x
)
+
h
2
(
x
(
t
-
τ
)
), where h1(x) and h2(x(t-τ)) are the unknown positive smooth continuous functions.Remark 3.
Compared with the previous results [27, 30], the uncertainties including the structural uncertainty are taken into account in this paper. In addition, the uncertainties considered in this paper are not supposed to be a bounded external disturbance unlike the previous results [17, 18, 20–23].The control objective of this paper is to design a control lawu(t) such that y(t) can follow a given desired reference signal ym(t) and guarantee that all the signals involved in the whole closed-loop system are bounded.


### 2.2. Description of Fuzzy Logic Systems

The basic configuration of the fuzzy logic system consists of four main components: fuzzy rule base, fuzzy inference engine, fuzzifier and defuzzifier [12, 16]. The fuzzy logic system performs a mapping from U⊂Rn to V⊂R. Let U=U1×⋯×Un where Ui⊂R, i=1,2,…,n. The fuzzifier maps a crisp point in U into a fuzzy set in U. The fuzzy rule base consists of a collection of fuzzy IF-THEN rules:
(10)R(l):IFx1isF1l,andx2isF2l,…,xnisFnlTHENyjisGjl,j=1,…,p,l=1,…,N,
in which x=[x1,x2,…,xn]T∈U and y∈V⊂R are the input and output of the fuzzy logic system, Fil and Gjl are fuzzy sets in Ui and Vj, respectively. N is the number of rules. The fuzzy inference engine performs a mapping from fuzzy sets in U to fuzzy sets in V, based upon the fuzzy IF-THEN rules in the fuzzy rule base and the compositional rule of inference. The defuzzifier maps a fuzzy set in V to a crisp point in V. Through singleton fuzzification, center average defuzzification, and product inference, the output of the fuzzy logic system can be expressed as
(11)yj=θjTξ(x),j=1,…,p,
where θjT=[θj1,…,θjN] with each variable θjl as the point at which the fuzzy membership function of Gjl achieves the maximum value, and ξ(x)=[ξ1(x),…,ξN(x)]T with each variable ξl as the fuzzy basis function defined as
(12)ξl(x)=∏i=1nμFil(xi)∑l=1N(∏i=1nμFil(xi)),
where μFil(xi) is the membership function of the fuzzy set.Now let MIMO fuzzy logic systems be expressed as follows:(13)y=Φ(x)θ,
where
(14)θT=[θ1T,…,θpT],θiT=[θi1,…,θiN]fori=1,2,…,p,Φ(x)=diag[ξT(x),…,ξT(x)].


## 2.1. Problem Formulation

Consider a class of MIMO nonlinear uncertain systems with delayed output in the following form:(1)x˙11(t)=x12(t),x˙12(t)=x13(t),⋮x˙1(n1-1)(t)=x1n1(t),x˙1n1(t)=f1(x(t))+∑j=1pg1j(x(t))uj(t)+Δϕ1(x(t)),⋮x˙p1(t)=xp2(t),x˙p2(t)=xp3(t),⋮x˙p(np-1)(t)=xpnp(t),x˙pnp(t)=fp(x(t))+∑j=1Pgpj(x(t))uj(t)+Δϕp(x(t)),y1(t)=x11(t-τ),⋮yp(t)=xp1(t-τ),
or equivalently
(2)x˙(t)=Ax(t)+B[F(x(t))+G(x(t))u(t)+ΔΦ(x(t))],y(t)=Cx(t-τ),
where
(3)F(x)=[f1(x),…,fp(x)]T∈Rp,G(x)=[G1(x),…,Gp(x)]∈Rp×p,Gi(x)=[g1i(x),…,gpi(x)]T∈Rp,i=1,2,…,p,A=diag[A1,…,Ap]∈Rn×n,B=diag[B1,…,Bp]∈Rn×p,C=diag[C1,…,Cp]∈Rp×n,Ai=[010⋯0001⋯0⋮⋮⋮⋯1000⋯0]∈Rni×ni,Bi=[00⋮1]∈Rni,CiT=[10⋮0]∈Rni,
where x=[x11,x12,…,x1n1,…,xp1,xp2,…,xpnp]T∈Rn is the system state vector which is assumed to be unavailable for measurement, u=[u1,…,up]T∈Rp is the control input, y=[y1,…,yp]T∈Rp is the output vector, n1+n2+⋯+np=n, τ>0 is a known time delay of the system, fi(x(t)) and gij(x(t)) are the unknown nonlinear system functions, and ΔΦ(x(t))  =  [Δϕ1(x(t),Δϕ2(x(t),…,Δϕp(x(t)]T∈Rp is the vector of uncertainties.Now let the output of the system and its derivatives be expressed as follows:(4)yi(t)=xi1(t-τ),y˙i(t)=x˙i1(t-τ)=xi2(t-τ),⋮yi(ni-1)(t)=x˙i(ni-1)(t-τ)=xini(t-τ),yi(ni)(t)=x˙ini(t-τ)=fi(x(t-τ))+∑j=1pgij(x(t-τ))uj(t-τ)+Δϕi(x(t-τ)).
Then, let yd=[y1d,…,ypd]T∈Rp be the given bounded output desired signal and contain finite derivatives up to the nth order. The tracking errors of the system can be defined as eij(t)=yi(j-1)(t)-yid(j-1)(t), for i=1,2,…,p and j=1,2,…,ni, and its derivatives can be obtained as
(5)e˙i1(t)=y˙i(t)-y˙id(t)=ei2(t),e˙i2(t)=yi(2)(t)-yid(2)(t)=ei3(t),⋮e˙i(ni-1)(t)=yi(ni-1)(t)-yid(ni-1)(t)=e1ni(t),e˙ini(t)=yi(ni)(t)-yid(ni)(t)=fi(x(t-τ))+∑j=1pgij(x(t-τ))uj(t-τ)+Δϕi(x(t-τ))-yid(ni)(t)=fi(x(t-τ))+∑j=1pgij(x(t-τ))uj(t)-yid(ni)(t)+Δψi(x,u),
where the uncertainty Δψi(x,u) including the structural uncertainty is defined as
(6)Δψi(x,u)=∑j=1pgij(x(t-τ))[uj(t-τ)-uj(t)]+Δϕi(x(t-τ)).
Thus, (5) can be expressed as
(7)e˙i(t)=Aiei(t)+Bi[fi(x(t-τ))+∑j=1pgij(x(t-τ))uj(t)-yid(ni)(t)+Δψi(x,u)∑j=1pgij(x(t-τ))].
From (7), it is easy to verify that the tracking error dynamics of the system (2) can be represented in the following form:
(8)e˙(t)=Ae(t)+B[Yd_F(x(t-τ))+G(x(t-τ))u(t)-Yd_(t)+ΔΨ(x,u)].e1_(t)=Ce(t),
where
(9)e=[e1T,e2T,…,epT]T=[e11,…,e1n1,…,ep1,…,epnp]T∈Rn,e1_=[e11,e21,…,ep1]T∈Rp,ΔΨ=[Δψ1,Δψ2,…,Δψp]T∈Rp,ei=[ei1,ei2,…,eini]T∈Rni,Yd_=[y1d(n1),y2d(n2),…,ypd(np)]T∈Rp,Yd=[y1d,…,y1d(n1-1),…,ypd,…,ypd(np-1)]T∈Rn.Based on the above discussion, the following assumptions should be made for the controller design.Assumption 1.
The matrixG(x) is nonsingular (i.e., G(x)-1 exists) for all x∈Ux where Ux is a compact set in Rn.Assumption 2.
∥
Δ
Ψ
(
x
,
u
)
∥
≤
h
1
(
x
)
+
h
2
(
x
(
t
-
τ
)
), where h1(x) and h2(x(t-τ)) are the unknown positive smooth continuous functions.Remark 3.
Compared with the previous results [27, 30], the uncertainties including the structural uncertainty are taken into account in this paper. In addition, the uncertainties considered in this paper are not supposed to be a bounded external disturbance unlike the previous results [17, 18, 20–23].The control objective of this paper is to design a control lawu(t) such that y(t) can follow a given desired reference signal ym(t) and guarantee that all the signals involved in the whole closed-loop system are bounded.


## 2.2. Description of Fuzzy Logic Systems

The basic configuration of the fuzzy logic system consists of four main components: fuzzy rule base, fuzzy inference engine, fuzzifier and defuzzifier [12, 16]. The fuzzy logic system performs a mapping from U⊂Rn to V⊂R. Let U=U1×⋯×Un where Ui⊂R, i=1,2,…,n. The fuzzifier maps a crisp point in U into a fuzzy set in U. The fuzzy rule base consists of a collection of fuzzy IF-THEN rules:
(10)R(l):IFx1isF1l,andx2isF2l,…,xnisFnlTHENyjisGjl,j=1,…,p,l=1,…,N,
in which x=[x1,x2,…,xn]T∈U and y∈V⊂R are the input and output of the fuzzy logic system, Fil and Gjl are fuzzy sets in Ui and Vj, respectively. N is the number of rules. The fuzzy inference engine performs a mapping from fuzzy sets in U to fuzzy sets in V, based upon the fuzzy IF-THEN rules in the fuzzy rule base and the compositional rule of inference. The defuzzifier maps a fuzzy set in V to a crisp point in V. Through singleton fuzzification, center average defuzzification, and product inference, the output of the fuzzy logic system can be expressed as
(11)yj=θjTξ(x),j=1,…,p,
where θjT=[θj1,…,θjN] with each variable θjl as the point at which the fuzzy membership function of Gjl achieves the maximum value, and ξ(x)=[ξ1(x),…,ξN(x)]T with each variable ξl as the fuzzy basis function defined as
(12)ξl(x)=∏i=1nμFil(xi)∑l=1N(∏i=1nμFil(xi)),
where μFil(xi) is the membership function of the fuzzy set.Now let MIMO fuzzy logic systems be expressed as follows:(13)y=Φ(x)θ,
where
(14)θT=[θ1T,…,θpT],θiT=[θi1,…,θiN]fori=1,2,…,p,Φ(x)=diag[ξT(x),…,ξT(x)].


## 3. Observer-Based Robust Adaptive Fuzzy Controller Design and Stability Analysis

According to the description of the fuzzy logic systems presented in Section2.2, we can construct the following fuzzy logic systems, over a compact set Ωx, to approximate the unknown nonlinear functions fi(x(t)) and gij(x(t)) and the unknown upper bounded functions h1(x), h2(x(t-τ)) as follows:
(15)f^i(x∣θf1)=θfiTξ(x),i=1,…,p,g^ij(x∣θgij)=θgijTξ(x),i,j=1,…,p,h^1(x)=θh1Tξ(x),h^2(x(t-τ))=θh2Tξ(x(t-τ)).
Based on the above statements, it can be easily shown that
(16)F^(x∣θ1)=[f^1(x∣θf1),…,f^p(x∣θfp)]T=Φ(x)θ1,G^(x∣θ2)=[g^11(x∣θg11),…,g^1p(x∣θg1p)⋮⋱⋮g^p1(x∣θgp1),…,g^pp(x∣θgpp)]=Φ(x)θ2,
where
(17)θ1T=[θf1T,…,θfpT],θfiT=[θfi1,…,θfiN],θ2=[θg11,…,θg1p⋮⋱⋮θgp1,…,θgpp],θgijT=[θgij1,…,θgijN],θh1T=[θh11,…,θh1N],θh2T=[θh21,…,θh2N],Φ(x)=diag[ξT(x),…,ξT(x)].
Suppose the state variables of the controlled system (1) are available for measurement, then the following robust adaptive fuzzy controller u(t) can be adopted to let the system achieve the above control objective and can be defined as follows:
(18)u(t)=G^(x(t-τ))-1[-F^(x(t-τ))-Kce(t)+Yd_(t)+us(t)Yd_],
where Kc=diag[Kc1,…,Kcp]∈Rp×n, where Kci=[ki1c,…,kinic]∈R1×ni for i=1,2,…,p, is the control gain matrix such that the characteristic polynomial of A-BKc is Hurwitz. The robust compensator us(t) will be designed to compensate the fuzzy approximation errors and the uncertainties.However, the state variables of the system are unavailable for measurement in many engineering systems. In addition, although the tracking error vectore can be obtained ideally by successive differentiation of ei1(i=1,2,…,p), ideal differentiators are physically unrealizable. Obviously, we must employ a state observer to estimate x and e and let x^ and e^ be the estimates of x and e, respectively. Then replacing F^(x∣θ1), G^(x∣θ2), h^1(x∣θh), h^2(x(t-τ)), x, and e by F^(x^∣θ1), G^(x^∣θ2), h^1(x^∣θh), h^2(x^(t-τ)), x^, and e^, respectively, the robust adaptive fuzzy controller (18) cannot be used to control the nonlinear system (2) and will be modified as the following form:
(19)u(t)=G^(x^(t-τ))-1[-F^(x^(t-τ))-Kce^(t)+Yd_(t)+us(t)Yd_],
where
(20)F^(x^∣θ1)=[f^1(x^∣θf1),…,f^p(x^∣θfp)]T=Φ(x^)θ1,(21)G^(x^∣θ2)=[g^11(x^∣θg11),…,g^1p(x^∣θg1p)⋮⋱⋮g^p1(x^∣θgp1),…,g^pp(x^∣θgpp)]=Φ(x^)θ2,(22)h^1(x^)=θh1Tξ(x^);h^2(x^(t-τ))=θh2Tξ(x^(t-τ)).
Substituting (19) into (8) yields
(23)e˙(t)=Ae(t)-BKce^(t)+B{F(x(t-τ))-F^(x^(t-τ)∣θ1)+[G(x(t-τ))-G^(x^(t-τ)∣θ2)]u(t)+us(t)+ΔΨ(x,u)F^},e1_(t)=Ce(t).
In order to estimate the output tracking error vector, we design the observer as follows:
(24)e^˙(t)=(A-BKc)e^(t)+Ko(e1_(t)-e^1_(t)),e^1_(t)=Ce^(t),
where Ko=diag[Ko1,…,Kop]∈Rn×p, where Koi=[k1io,…,kniio]T for i=1,2,…,p, is the observer gain matrix such that the characteristic polynomial of A-KoC is Hurwitz. Let us define the observation error vector as
(25)e~(t)=e(t)-e^(t).
Then by (23) and (24), we obtain
(26)e~˙(t)=(A-KoC)e~(t)+B{F(x(t-τ))-F^(x^(t-τ)∣θ1)+[G(x(t-τ))-G^(x^(t-τ)∣θ2)]u(t)+us(t)+ΔΨ(x,u)F^},e~1_(t)=Ce~(t).
It is assumed that θ1,θ2,θh1, and θh2belong to compact sets Ωθ1,  Ωθ2,  Ωθh1, and Ωθh2, respectively, which are defined as
(27)Ωθ1={θ1∈RN×p:∥θ1∥≤Mθ1<∞},Ωθ2={θ2∈RpN×p:∥θ2∥≤Mθ2<∞},Ωθh1={θh1∈RN:∥θh1∥≤Mθh1<∞},Ωθh2={θh2∈RN:∥θh2∥≤Mθh2<∞},
where Mθ1, Mθ2, Mθh1, and Mθh2 are the designed parameters by the designer, and N is the number of fuzzy inference rules.Now let us define the optimal parameter vectorsθ1*, θ2*, θh1*, and θh2* as follows:
(28)θ1*=argminθ1∈Ωθ1{supx∈Ωx,x^∈Ωx^|F(x)-F^(x^∣θ1)|},θ2*=argminθ2∈Ωθ2{supx∈Ωx,x^∈Ωx^|G(x)-G^(x^∣θ2)|},θh1*=argminθh1∈Ωθh1{supx∈Ωx,x^∈Ωx^|h1(x)-h^1(x^∣θh1)|},θh2*=argminθh2∈Ωθh2{supx∈Ωx,x^∈Ωx^|h^2h2(x(t-τ))-h^2(x^(t-τ))∣θh2)|supx∈Ωx,x^∈Ωx^},
where θ1*, θ2*, θh1*, and θh2* are bounded in the suitable closed sets Ωθ1, Ωθ2, Ωθh1, and Ωθh2, respectively. Also the parameter estimation errors are defined as
(29)θ~1=θ1*-θ1,(30)θ~2=θ2*-θ2,(31)θ~h1=θh1*-θh1;θ~h2=θh2*-θh2.
Then, the minimum approximation errors which correspond to the optimal parameter vectors are defined as
(32)ω1=F(x(t-τ))-F^(x^(t-τ)∣θ1*)+(G(x(t-τ))-G^(x^(t-τ)∣θ2*))u,(33)ω2=h1(x)-h^1(x^∣θh1*)+h2(x(t-τ))-h^2(x^(t-τ)∣θh2*).
Applying (29), (30), and (32), (26) can be rewritten as
(34)e~˙(t)=(A-KoC)e~(t)+B{Φ(x^)θ~1+Φ(x^)θ~2u(t)+F(x(t-τ))-F^(x^(t-τ)∣θ1*)+[G(x(t-τ))-G^(x^(t-τ)∣θ2*)]u(t)+us(t)+ΔΨ(x,u)θ~2}=(A-KoC)e~(t)+B[Φ(x^)θ~1+Φ(x^)θ~2u(t)+ω1+us(t)+ΔΨ(x,u)θ~1],e~1_(t)=Ce~(t).
Then, the output error dynamics in (34) can be expressed as follows:
(35)e~1_=H(s)[Φ(x^)θ~1+Φ(x^)θ~2u(t)+ω1+us(t)+ΔΨ(x,u)θ~1],
where
(36)H(s)=C(sI-(A-KoC))-1B.
Obviously, the transfer function H(s) is a known stable transfer function matrix. In order to utilize the SPR-Lyapunov design approach, (35) can be represented as
(37)e~1_=H(s)L(s)[ΨL(s)-1Φ(x^)θ~1+L(s)-1Φ(x^)θ~2u(t)+L(s)-1(ω1+us(t)+ΔΨ(x,u))],
with
(38)L(s)=diag[Li(s),…,Lp(s)],Li(s)=smi+bi1smi-1+⋯+bimi(mi=ni-1),i=1,2,…,p.
It is worth noting that L(s) is chosen to be a proper stable transfer function matrix and to make H(s)L(s) to be a proper SPR transfer function matrix. Then the state-space realization of (37) can be written as
(39)e~˙c(t)=(A-KoC)e~c+Bc[Φf(x^)θ~1+Φf(x^)θ~2u(t)+ω1f+usf(t)+ΔΨf(x,u)Φf(x^)θ~1+Φf(x^)θ~2u(t)],e~1_(t)=Ce~c(t),
with
(40)Bc=diag[Bc1,…,Bcp]∈Rn×p,(41)Bci=[1,bi1,bi2,…,bimi]T∈Rni×1,i=1,2,…,p,(42a)Φf(x^)=L(s)-1Φ(x^),(42b)ω1f=L(s)-1ω1,(42c)usf=L(s)-1us,(42d)ΔΨf(x,u)=L(s)-1ΔΨ(x,u).We define
(43)∥ω1f∥+|ω2|≤ωf,ω~f=ωf-ω^f,
where ω^f is the estimate of an unknown positive constant ωf.Remark 4.
SinceL(s) is chosen as a proper stable transfer function matrix, it is obvious that the norm of ΔΨf(x,u) also satisfies the inequality in Assumption 2. Moreover, the upper bound functions will be replaced by h^1(x^∣θh) and h^2(x^(t-τ)) because the state variables are unavailable for measurement.
Based on the Lyapunov stability theorem, we can obtain the robust compensatorusf(t) as follows:
(44)usf(t)=u1(t)+u2(t),
where
(45)u1(t)=-BcTP1e~c∥e~cTP1Bc∥×[ω^f+h^1(x^∣θh1)+h^2(x^(t-τ)∣θh2)],u2(t)=-BcTP1e~c∥e~cTP1Bc∥2[e^TP2Koe~1_],
where Kc is the control gain matrix, Ko is the observer gain matrix, and P1 and P2 are the symmetric positive definite matrices and will be solved later.
Then, the parameter adaptive learning laws are chosen as(46)θ˙1=γ1ΦfT(x^)BcTP1e~c,(47)θ˙2=γ2ΦfT(x^)BcTP1e~cuT,
(48a)θ˙h1=γ3∥e~cTP1Bc∥ξ(x^),(48b)θ˙h2=γ4∥e~cTP1Bc∥ξ(x^(t-τ)),
(49)ω^˙f=γ5∥e~cTP1Bc∥,
where γ1>0,  γ2>0,  γ3>0,  γ4>0, and γ5>0 are the positive adaptive gain constants and can be chosen by the designer.Remark 5.
Without loss of generality, the adaptive laws used in this paper are assumed such that the parameter vectors are within the constraint sets or on the boundaries of the constraint sets but moving toward the inside of the constraint sets. If the parameter vectors are on the boundaries of the constraint sets but moving toward the outside of the constraint sets, we have to use the projection algorithm [12] to modify the adaptive laws such that the parameter vectors will remain inside of the constraint sets. The proposed adaptive laws (46)–(49) can be modified as the following form:
(50)θ˙1={γ1ΦfT(x^)BcTP1e~c,if(∥θ1∥<Mθ1)or(∥θ1∥=Mθ1,θ1TΦfT(x^)BcTP1e~c≤0),P{γ1ΦfT(x^)BcTP1e~c},if(∥θ1∥=Mθ1,θ1TΦfT(x^)BcTP1e~c>0),
where P{γ1ΦfT(x^)BcTP1e~c} is defined as(51)P{γ1ΦfT(x^)BcTP1e~c}=γ1ΦfT(x^)BcTP1e~c-γ1θ1θ1T∥θ1∥2ΦfT(x^)BcTP1e~c,(52)θ˙2={γ2ΦfT(x^)BcTP1e~cuT,if(∥θ2∥<Mθ2)or(∥θ2∥=Mθ2,θ2TΦfT(x^)BcTP1e~cuT≤0),P{γ2ΦfT(x^)BcTP1e~cuT},if(∥θ2∥=Mθ2,θ2TΦfT(x^)BcTP1e~cuT>0),where P{γ2ΦfT(x^)BcTP1e~cuT} is defined as
(53)P{γ2ΦfT(x^)BcTP1e~cuT}=γ2ΦfT(x^)BcTP1e~cuT-γ2θ2θ2T∥θ2∥2ΦfT(x^)BcTP1e~cuT,(54a)θ˙h1={γ3∥e~cTP1Bc∥ξ(x^),if(∥θh1∥<Mθh1)or(∥θh1∥=Mθh1,∥e~cTP1Bc∥·θh1Tξ(x^)≤0),P{γ3∥e~cTP1Bc∥ξ(x^)},if(∥θh1∥=Mθh1,∥e~cTP1Bc∥·θh1Tξ(x^)>0),where P{γ3∥e~cTP1Bc∥ξ(x^)} is defined as
(54b)P{γ3∥e~cTP1Bc∥ξ(x^)}=γ3∥e~cTP1Bc∥ξ(x^)-γ3θh1θh1T∥θh1∥2∥e~cTP1Bc∥ξ(x^),(55a)θ˙h2={γ4∥e~cTP1Bc∥ξ(x^(t-τ)),if(∥θh2∥<Mθh2)or(∥θh2∥=Mθh2,∥e~cTP1Bc∥·θh2Tξ(x^(t-τ))≤0Mθh2),P{γ4∥e~cTP1Bc∥ξ(x^)},if(∥θh2∥=Mθh2,∥e~cTP1Bc∥·θh2Tξ(x^(t-τ))>0Mθh2),where P{γ4∥e~cTP1Bc∥ξ(x^(t-τ))} is defined as
(55b)P{γ4∥e~cTP1Bc∥ξ(x^(t-τ))}=γ4∥e~cTP1Bc∥ξ(x^(t-τ))-γ4θh2θh2T∥θh2∥2∥e~cTP1Bc∥ξ(x^(t-τ)).
The main result of the proposed observer-based robust adaptive fuzzy control scheme is summarized in the following theorem.Theorem 6.
Consider the MIMO nonlinear system (2) in the presence of output delay and the uncertainties subject to Assumptions 1-2. The observer-based robust adaptive fuzzy controller is defined by (19) and (44)-(45) with adaptation laws given by (46)–(49). For the given positive definite matrices Q1 and Q2, if there exist symmetric positive definite matrices P1 and P2 such that the following Lyapunov equations
(56)(A-KoC)TP1+P1(A-KoC)=-Q1,(57)(A-BKc)TP2+P2(A-BKc)=-Q2,(58)P1Bc=CT,
are satisfied, then all the closed-loop signals are bounded, and the tracking errors converge to a neighborhood of zero.Proof.
Consider the Lyapunov function candidate(59)V=12e~cTP1e~c+12e^TP2e^+12γ1(θ~1Tθ~1)+12γ2tr(θ~2Tθ~2)+12γ3(θ~h1Tθ~h1)+12γ4(θ~h2Tθ~h2)+12γ5ω~f2.
By the time derivative of V and the facts that θ~˙1=-θ˙1, θ~˙2=-θ˙2, θ~˙h1=-θ˙h1, θ~˙h2=-θ˙h2  and ω~˙f=-ω^˙f, it can be easily shown from (24) and (39) that
(60)V˙=12e~˙cTP1e~c+12e~cTP1e~˙c+12e^˙TP2e^+12e^TP2e^˙-1γ1(θ~1Tθ˙1)-1γ2tr(θ~2Tθ˙2)-12γ3(θ~h1Tθ˙h1)-12γ4(θ~h2Tθ˙h2)-1γ5ω~fω^˙f=12e~cT[(A-KoC)TP1+P1(A-KoC)]e~c+12e^T[(A-BKc)TP2+P2(A-BKc)]e^+e~cTP1Bc[Φf(x^)θ~1+Φf(x^)θ~2u(t)+ω1f+usf(t)+ΔΨf(x,u)θ~2]+e^TP2Koe~1_-1γ1(θ~1Tθ˙1)-1γ2tr(θ~2Tθ˙2)-12γ3(θ~h1Tθ˙h1)-12γ4(θ~h2Tθ˙h2)-1γ5ω~fω^˙f≤12e~cT[(A-KoC)TP1+P1(A-KoC)]e~c+12e^T[(A-BKc)TP2+P2(A-BKc)]e^+e~cTP1BcΦf(x^)θ~1+e~cTP1BcΦf(x^)θ~2u+e~cTP1Bcusf+∥e~cTP1Bc∥∥ω1f∥+∥e~cTP1Bc∥·∥ΔΨf(x,u)∥+e^TP2Koe~1_-1γ1(θ~1Tθ˙1)-1γ2tr(θ~2Tθ˙2)-12γ3(θ~h1Tθ˙h1)-12γ4(θ~h2Tθ˙h2)-1γ5ω~fω^˙f.
Applying Assumption 2, (56), and (57), it yields
(61)V˙≤-12e~cTQ1e~c-12e^TQ2e^+e~cTP1BcΦf(x^)θ~1+e~cTP1BcΦf(x^)θ~2u+e~cTP1Bcusf+∥e~cTP1Bc∥∥ω1f∥+∥e~cTP1Bc∥·(h1(x)+h2(x(t-τ)))+e^TP2Koe~1_-1γ1(θ~1Tθ˙1)-1γ2tr(θ~2Tθ˙2)-12γ3(θ~h1Tθ˙h1)-12γ4(θ~h2Tθ˙h2)-1γ5ω~fω^˙f=-12e~cTQ1e~c-12e^TQ2e^+e~cTP1BcΦf(x^)θ~1+e~cTP1BcΦf(x^)θ~2u+e~cTP1Bcusf+∥e~cTP1Bc∥∥ω1f∥+∥e~cTP1Bc∥·[h1(x)-h^1(x^∣θh1*)+h^1(x^∣θh1*)-h^1(x^∣θh1)+h^1(x^∣θh1)]+∥e~cTP1Bc∥·[h2(x(t-τ))-h^2(x^(t-τ)∣θh2*)+h^2(x^(t-τ)∣θh2*)-h^2(x^(t-τ)∣θh2)+h^2((t-τ)x^∣θh2)]+e^TP2Koe~1_-1γ1(θ~1Tθ˙1)-1γ2tr(θ~2Tθ˙2)-12γ3(θ~h1Tθ˙h1)-12γ4(θ~h2Tθ˙h2)-1γ5ω~fω^˙f.
According to (33) and (43), we have
(62)V˙≤-12e~cTQ1e~c-12e^TQ2e^+e~cTP1BcΦf(x^)θ~1+e~cTP1BcΦf(x^)θ~2u+e~cTP1Bcusf+∥e~cTP1Bc∥∥ω1f∥+∥e~cTP1Bc∥·h^1(x^∣θh1)+∥e~cTP1Bc∥·θ~h1Tξ(x^)+∥e~cTP1Bc∥·h^2(x^(t-τ)∣θh2)+∥e~cTP1Bc∥·θ~h2Tξ(x^(t-τ))+∥e~cTP1Bc∥·|ω2|+e^TP2Koe~1_-1γ1(θ~1Tθ˙1)-1γ2tr(θ~2Tθ˙2)-12γ3(θ~h1Tθ˙h1)-12γ4(θ~h2Tθ˙h2)-1γ5ω~fω^˙f≤-12e~cTQ1e~c-12e^TQ2e^+e~cTP1BcΦf(x^)θ~1+e~cTP1BcΦf(x^)θ~2u+e~cTP1Bcusf+∥e~cTP1Bc∥ω^f+∥e~cTP1Bc∥·h^1(x^∣θh1)+∥e~cTP1Bc∥·θ~h1Tξ(x^)+∥e~cTP1Bc∥·h^2(x^(t-τ)∣θh2)+∥e~cTP1Bc∥·θ~h2Tξ(x^(t-τ))+∥e~cTP1Bc∥·ω~f+e^TP2Koe~1_-1γ1(θ~1Tθ˙1)-1γ2tr(θ~2Tθ˙2)-12γ3(θ~h1Tθ˙h1)-12γ4(θ~h2Tθ˙h2)-1γ5ω~fω^˙f=-12e~cTQ1e~c-12e^TQ2e^+e~cTP1Bcusf+∥e~cTP1Bc∥ω^f+∥e~cTP1Bc∥·h^1(x^∣θh1)+∥e~cTP1Bc∥·h^2(x^(t-τ)∣θh2)+e^TP2Koe~1_+[e~cTP1BcΦf(x^)θ~1-1γ1(θ~1Tθ˙1)]+[e~cTP1BcΦf(x^)θ~2u-1γ2tr(θ~2Tθ˙2)]+[∥e~cTP1Bc∥·θ~h1Tξ(x^)-1γ3(θ~h1Tθ˙h1)]+[∥e~cTP1Bc∥·θ~h2Tξ(x^(t-τ))-1γ4(θ~h2Tθ˙h2)]+[∥e~cTP1Bc∥ω~f-1γ5ω~fω^˙f].
By employing (46)–(49) and using the control laws (44)-(45), we can obtain
(63)V˙≤-12e~cTQ1e~c-12e^TQ2e^.
Therefore, it can be concluded that V˙≤0 from (63), and the output tracking error of the closed-loop system converges asymptotically to a neighborhood of zero based on the Lyapunov synthesis approach. This completes the proof.


## 4. An Example and Simulation Results

In this paper, a numerical example is illustrated to verify the performance of the proposed observer-based robust adaptive fuzzy controller. Consider the following MIMO nonlinear uncertain system with delayed output:(64)x˙11(t)=x12(t),x˙12(t)=f1(x)+g11(x)u1(t)+g12(x)u2(t)+Δϕ1(x),x˙21(t)=x22(t),x˙22(t)=f2(x)+g21(x)u1(t)+g22(x)u2(t)+Δϕ2(x),y1(t)=x11(t-τ),y2(t)=x21(t-τ).
According to (2), the above equation can be rewritten as the following compact form:
(65)[x˙11(t)x˙12(t)x˙21(t)x˙22(t)]=[0100000000010000][x11(t)x12(t)x21(t)x22(t)]+[00100001]×([f1(x)f2(x)]+[g11(x)g12(x)g21(x)g22(x)][u1(t)u2(t)]+[Δϕ1(x)Δϕ2(x)]),[y1(t)y2(t)]=[10000010][x11(t-τ)x12(t-τ)x21(t-τ)x22(t-τ)],
where A1=A2=[0100], B1=B2=[01], C1=C2=[10], the nonlinear system functions f1(x)=(1-e-x11)/(1+e-x11), f2(x)=(1-e-x21)/(1+e-x21), g12(x)=g21(x)=0, and g11(x)=g22(x)=1, and the uncertainties ΔΦ=[0.1x21sin(t)0.1x12cos(t)] and the delay time is τ=0.1. In the simulation, these nonlinear system functions and uncertainties are assumed to be unknown and are estimated by the fuzzy logic systems. The fuzzy membership functions are chosen as follows:
(66)μFi1(x^i)=11+exp(5(x^i+0.8)),μFi2(x^i)=exp(-(x^i+0.6)2),μFi3(x^i)=exp(-(x^i+0.4)2),μFi4(x^i)=exp(-x^i2),μFi5(x^i)=exp(-(x^i-0.4)2),μFi6(x^i)=exp(-(x^i-0.6)2),μFi7(x^i)=11+exp(-5(x^i-0.8)),i=1,2,3,4.
Let
(67)ξl(x^)=μF1l(x^1)μF2l(x^2)μF3l(x^3)μF4l(x^4)∑l=17μF1l(x^1)μF2l(x^2)μF3l(x^3)μF4l(x^4),ξ(x^)=(ξ1(x^),ξ2(x^),ξ3(x^),ξ4(x^),ξ5(x^),ξ6(x^),ξ7(x^))T.
In this section, we apply the observer-based robust adaptive fuzzy controller proposed in Section 3 to deal with the output tracking control problem of an MIMO nonlinear uncertain system with delayed output in (65).The control objective is to force the system output to track the given desired trajectoriesy1d=y2d=0.5sin(2π/10). First, we select the control and observer gain matrices as Kc=[17040000018030] and Ko=[50100000075100]T, using (20) and (21) to approximate the unknown F(x) and G(x), respectively. Given the positive definite matrices Q1=diag[1,1,1,1] and Q2=diag[1,1,1,1], and solving the Lyapunov equations (56), we can get
(68)P1=[0.01010.005000.0051.2600000.00670.005000.0051.0483],P2=[0.1302-0.500-0.52.137500000.1-0.500-0.53.0167].
The initial conditions are chosen as x(0)=[-0.5,-0.5,0.5,0.5]T, e^(0)=[0.3,0.3,0.3,0.3]T, θh(0)=0, θf1(0)=θf2(0)=0, γ1=γ2=0.13, γ3=0.1, γ4=0.01, θg11(0)=1, θg12(0)=0, θg21(0)=0, θg22(0)=1. The Simulation results are shown in Figures 1, 2, 3, 4, 5, 6, and 7. Figures 1 and 2 show the output tracking performance of the system output. Figures 3, 4, 5, and 6 show the trajectories of the system states and their estimation states. Figure 7 shows the control signal. Obviously, the proposed controller can not only ensure that all the signals of the resulting closed-loop system are bounded but also obtain that the good tracking performance of an MIMO nonlinear uncertain output-delay system is achieved.Figure 1
The trajectories ofy1 and y1d.Figure 2
The trajectories ofy2 and y2d.Figure 3
The trajectories ofx11 and x^11.Figure 4
The trajectories ofx12 and x^12.Figure 5
The trajectories ofx21 and x^21.Figure 6
The trajectories ofx22 and x^22.Figure 7
The control signalsu1 and u2.


## 5. Conclusion

For a class of multiinput multioutput (MIMO) nonlinear systems with output delay and uncertainties, a robust adaptive fuzzy output feedback controller is proposed to deal with the problem of robust stability and output tracking control. Since the system states are unavailable for measurement, the elements of the observation error vector cannot be obtained completely. Hence, by means of the strictly positive-real (SPR) theory, all elements of the observation error vector will be obtained completely. Moreover, the fuzzy logic systems and some parameter adaptive laws are used to approximate the unknown nonlinear functions and the unknown upper bounds of the uncertainties including the structural uncertainty. By constructing an appropriate Lyapunov function, the proposed observer-based robust adaptive fuzzy controller can not only guarantee that all the signals in the whole closed-loop system are bounded but also obtain that the output tracking performance of the controlled system with delayed output and uncertainties can be achieved. Finally, some simulation results are provided to illustrate the validity of the proposed approach.



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*Source: 215201-2013-03-25.xml*