# State Tracking of MRAC Systems in the Presence of Controller Temporary Failure Based on a Switching Method

Authors: Caiyun Wu; Jun Zhao
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The state tracking problem for a class of model reference adaptive control (MRAC) systems in the presence of controller temporary failures is studied. Due to the controller temporary failure, the considered system is viewed as an error switched system. The properties of Lyapunov function candidates without switching are described. Then the notion of global practical stability of switched systems is presented, and sufficient conditions for global practical stability of the error system under the restrictions of controller failure frequency and unavailability rate are provided. An example is presented to demonstrate the feasibility and effectiveness of the proposed method.


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## Body

## 1. Introduction

There are often parameter, structural, and environment uncertainties in practical systems [1–4]. Model reference adaptive control (MRAC) has been used as an important control approach for such uncertain systems [5, 6]. Closed-loop signal boundedness and asymptotic tracking can be ensured by a state feedback controller and adaptive laws in MRAC systems [7].On the other hand, controller temporary failures are often encountered in real control systems due to various environment factors during operation. Some motivations of studying controller failures are summarized in [8]. The reasons can be roughly classified into two categories: passive and positive ones. A typical passive reason is that the signals are not transmitted perfectly or the controller itself is not available for some reasons. For instance, the packet dropout phenomenon in networked control systems leads to controller failure, which is inevitable because of unreliable transmission paths. In contrast to passive reasons, a typical positive reason is that the controller is purposefully suspended from time to time for an economic or system life consideration [9]. Apparently, controller failures may lead to severe performance deterioration of systems. Especially, for adaptive control systems, the controller failure may cause the tracking error divergence due to the uncertainties of systems. Therefore, it is both theoretically and practically important to develop some new techniques to deal with the case of controller temporary failure of adaptive systems.Recently, there are rapidly growing interests in switched systems and switching control in the control community [10–13]. In the study of stability of switched systems, one of the effective tools is the average dwell time approach [14–16]. Based on this approach, exponential stability is guaranteed if the unavailability rate of the controller is smaller than a specified constant and the average time interval between controller failures is large enough [9, 17, 18]. In [8], this result was further extended to symmetric linear time-invariant system. The concept of controller failure frequency was first introduced in [9], and the cases of controller temporary failure for a class of time-varying delay systems were analyzed in [19]. Interestingly, a nonswitched MRAC system in the presence of controller temporary failure can be viewed as a switched system with a switching signal depending on the time interval between controller failures. Thus, theories and methods of switched systems may be applicable to the study of the state tracking problem for nonswitched MRAC systems with controller temporary failure. However, this issue has been rarely explored so far.In this paper, we study the state tracking problem for MRAC systems in the presence of controller temporary failure. As in [8], the controller temporary failure means that the controller itself is not available or the controller signals are not transmitted perfectly within a certain time interval. Furthermore, we assume that the parameter estimation is “frozen” in the instant of the controller temporary failure until the controller works normally. There are two main issues to be addressed in this paper. One is to describe a tradition MRAC system in the presence of controller temporary failure as an error switched system with two subsystems: the normal error subsystem which stands for the case without controller failure and the unstable error subsystem which describes the case of controller failure. The other issue is the stability analysis for the error switched system. To address the second issue, we analyze the properties of Lyapunov function candidates without switching and introduce the notions of global practical stability, failure frequency, and unavailability rate.The results in this paper have three features. First of all, MRAC systems in the presence of controller temporary failure are first considered. Secondly, the state tracking problem is studied from a switched system point of view. Finally, the global practical stability criterion is given for the considered system under the condition of controller failure frequency and unavailability rate.The organization of the paper is as follows. The state tracking problem in the presence of controller failure is formulated in Section2. In Section 3, we present an error switched system. Section 4 gives three lemmas and the main result. An example is given to illustrate the effectiveness of the proposed method in Section 5. Finally, the conclusions are presented in Section 6.The notation is standard. Consider the following:λmax(A)(λmin(A))
: the largest (smallest) eigenvalue of matrixA;∥A∥=λmax(AAT): the norm of matrix A;∥x∥=(∑i=1n|xi|2): the norm of a vector x=(x1,x2,…,xn)T∈Rn;tr[A]: the trace of a square matrix A.


## 2. Problem Statement

Consider a system(1)x˙(t)=Θf(x)+Bu(t),
where B∈Rn×n is input matrix, x(t)∈Rn is the state, u(t)∈Rn is the control input, Θ∈Rn×p is an uncertain constant parameter matrix with a bounded δΘ, that is, ∥Θ∥≤δΘ, and f(x)∈Rp is a vector which can be described as f(x)=Fx+g(x), where F∈Rp×n is a constant parameter matrix and ∥g(x)∥≤M for some M≥0.The classical state tracking problem is to design a controller such that the statex(t) of the system (1) tracks a given reference state xm(t) generated from the reference model system
(2)x˙m(t)=Amxm(t)+Bmr(t),
where Am∈Rn×n is a constant Hurwitz matrix, Bm∈Rn×n is a constant input matrix, and r(t)∈Rn is a bounded and piecewise continuous reference input.Suppose that there exist matricesR, T, and W* such that the following matching equations are satisfied:
(3)Am=BR,Bm=BT,BW*=Θ,
where W* is an unknown matrix due to uncertain constant parameter matrix Θ.Define the tracking errore(t)=xm(t)-x(t). To solve the state tracking problem, we use the controller structure [20]
(4)u(t)=ψω,
where ψ=[RW^(t)T] and ω=[xT-fT(x)rT], W^(t) is the estimate of unknown matrix W*, and ∥W^(t)∥≤δW^.Apply a parameter projection adaptive law(5)W^˙(t)=-ΓBTP1e(t)fT(x)+Fs,
where Γ=diag{Γ1,Γ2,…,Γi,…,ΓM} with positive constants Γi, P1 is a symmetric positive definite matrix satisfying AmTP1+P1Am<0, and Fs is a vector satisfying
(6)Fs={0,if∥W^(t)∥≤δW^,ΓBTP1e(t)fT(t),if∥W^(t)∥>δW^.
Then, the closed-loop system is
(7)x˙(t)=Amx(t)+Bmr(t)+BW~(t)f(x),
where W~(t)=W*-W^(t).From [5, 20, 21], x(t) converges asymptotically to xm(t) under the controller (4) and the adaptive law (5), that is, limt→∞e(t)=0.We now consider the case of controller temporary failure depicted in Figure1. Controller failures occur when the controller (a) itself is not available or when the signals are not transmitted perfectly on the route (b). Suppose that the time interval of the controller failures is not more than a specified constant Ts, which means the designed controller can be recovered within a finite time interval [19]. Also, the failed controller implies the complete breakdown of the controller (u(t)=0) in its failure time interval [8]. Hence, the system (1) with the controller temporary failure is dominated by the following piecewise differential equations:
(8)x˙(t)={Θf(t)+Bu(t),whenthecontrollerworks,Θf(t),whenthecontrollerfails.Figure 1
Controller failures occur in MRAC systems.We introduce the following definitions which will play key roles in deriving our main results.Definition 1 (see [8]).
For anyT2>T1, denote T+(T1,T2) as the total time interval of controller failure during (T1,T2), and call the ratio T+(T1,T2)/(T2-T1) the unavailability rate of the controller in the system.Definition 2 (see [9]).
For anyT2>T1≥0, let Nf(T1,T2) denote the number of control failure in the time interval (T1,T2).Ff(T1,T2)=Nf(T1,T2)/(T2-T1) is referred to as the controller failure frequency in the time interval (T1,T2).In this paper, our objective is to develop conditions under whichx(t) tracks xm(t) subject to controller temporary failure.


## 3. Error Switched System

From the closed-loop system (7), we have a normal error system
(9)e˙(t)=Ame(t)-BW~(t)f(x).When controller fails, we obtain an unstable error system(10)e˙(t)=Ame(t)+Amx(t)+Bmr-Θf(x).In this condition, we choose the adaptive law(11)W^˙(t)=0.Remark 3.
When controller fails, because the adaptive parameterW^(t) has no influence on the tracking error e(t), it is proper that the parameter estimation W^(t) is “frozen” in the instants of the controller temporary failure until the controller works normally.Based on (5), (9), (10), and (11), e(t) is governed by the following error switched system:
(12)e˙(t)=Ame(t)+Φσ(t),
where σ(t):[0,+∞)→M={1,2}, Φ1=-BW~(t)f(x), and Φ2=Amx(t)Bmr-Θf(x).Meanwhile, we have a switching adaptive law of the following form:(13)W^˙(t)=(-ΓBTP1e(t)fT(x)+Fs)Ψσ,
where Ψ1=Ip×p and Ψ2=0p×p.Whenσ=1, the normal error subsystem is active, which corresponds to the case of no controller failure; when σ=2, the unstable error subsystem is active, which denotes that the controller fails.Therefore, the problem of state tracking in the presence of controller temporary failure can be handled by means of analyzing the stability of the error switched system (12) with the switching adaptive law (13).To analyze the stability of the error switched system (12), we introduce the following definition.Definition 4 (see [22]).
Consider system (12). Given a constant r*>0, the system (12) is said to be globally practically stable with respect to r* if there exist a switching law σ(t) and a constant T=T(e(t0),r*)≥0 which depends on e(t0) and r* such that e(t;t0,e(t0))∈S(r*)≜{e∣∥e∥≤r*} for t≥t0+T.Remark 5.
Unlike theε-practical stability concept [23], the initial error in Definition 4 is not required to be bounded. If the initial error is constrained by e(t0)∈B(r*), then the global practical stability, given by Definition 4, degenerates into ε-practical stability [22]. Global practical stability stated here expresses a global version of the existing practical stability concept. Obviously, Definition 4 covers the ε-practical stability as a special case.


## 4. Main Result

In this section, firstly, we give three lemmas to analyze the properties of Lyapunov function candidates without switching. Secondly, we present a theorem to give some conditions under which the error switched system (12) is globally practically stable. Let
(14)z(t)=[eT(t),w~1T(t),…w~nT(t),…w~pT(t)]T∈Rn+n×p,
where w~i(t)∈Rn is the ith column of W~(t),i=1,2,…,p; that is,
(15)W~(t)=[w~1(t),…,w~p(t)]∈Rn×p.According to [21], we have
(16)tr[W~TΓ-1Fs]≤0.Note that the parameter estimatesW^ are bounded; thus there exists a constant l>0 defined as
(17)l=max(tr[W~TΓ-1W~]).Consider the situation of the system (1) without controller failure. A Lyapunov functional candidate of the normal error subsystem of (12) is chosen as
(18)V1(z(t))=eT(t)P1e(t)+tr[W~T(t)Γ-1W~(t)].DifferentiatingV1 along the trajectory of the normal error subsystem of (12) and the adaptive law (13) gives
(19)V˙1(z(t))=eT(t)(AmTP1+P1Am)e(t)+2tr[W~TΓ-1Fs].The following lemma gives the estimate of the convergence rate ofV1 along the trajectory of the normal error subsystem of (12).Lemma 6.
Consider the normal error subsystem of (12). For any given r1>l/λmin(P1), denote S(r1)={e∣∥e∥<r1}. If e(t)∈Rn/S(r1), then the inequality
(20)V(z(t))≤exp(-2λ1(t-t0))V1(z(t0))
holds for any λ1 satisfying 0<λ1≤(λ1¯/2)(1-(l/r12λmin(P1))).Proof.
From (17) and (18), it is easy to get
(21)eT(t)P1e(t)≤V1(z(t))≤eT(t)P1e(t)+l.
It is obvious that(22)λmin(P1)∥e(t)∥2≤V1(z(t))≤λmax(P1)∥e(t)∥2+l.
From (16) and (19), it holds that
(23)V˙1(z(t))≤eT(AmTP1+P1Am)e<0.
SinceAm is Hurwitz matrix, there exists a scalar λ1¯>0 such that AmTP1+P1Am+λ1¯P1<0.
With the help of (21)– (23), we have
(24)V˙1(z(t))≤-λ1¯V1(z(t))+λ1¯l=-2λ1V1(z(t))-(λ1¯-2λ1)V1(z(t))+λ1¯l≤-2λ1V1(z(t))-λmin(P1)(λ1¯-2λ1)∥e(t)∥2+λ1¯l.
Given that r1>l/λmin(P1), when e(t)∈Rn/S(r1), applying (24) leads to V˙1(z(t))≤-2λ1V1(z(t)) that is, V(z(t))≤exp(-2λ1(t-t0))V1(z(t0)); for any λ1 satisfying 0<λ1≤(λ1¯/2)(1-(l/r12λmin(P1))).
This completes the proof.When the controller fails, for the unstable error subsystem of (12), we choose another Lyapunov functional candidate of the following form:
(25)V2(z(t))=eT(t)P2e(t)+tr[W~T(t)Γ-1W~(t)],
where P2 is a positive definite matrix.DifferentiatingV2 along the trajectory of the normal error subsystem of (12) and the adaptive law (13) gives
(26)V˙2(z(t))=e˙(t)P2e(t)+e(t)P2e˙(t).Then, in the following lemma, we estimate the divergence rate ofV2 along the trajectory of the unstable error subsystem of (12).Lemma 7.
Consider the unstable error subsystem of (12). For any given r2>0, denote S(r2)={e∣∥e∥<r2}. If e(t)∈Rn/S(r2), then the inequality
(27)V(z(t))≤exp(2λ2(t-t0))V2(z(t0))
holds for any λ2 satisfying 0<λ2≤(1/2λmin(P2))((β2/r2)+β1).Proof.
From (17), we have
(28)eT(t)P2e(t)≤V2(z(t))≤eT(t)P2e(t)+l.
It is obvious that(29)λmin(P2)∥e(t)∥2≤V2(z(t))≤λmax(P2)∥e(t)∥2+l.
From (10) and (26), we have
(30)V˙2(z(t))=eT(t)(AmTP2+P2Am)e(t)+2eT(t)P2(Amxm+Bmr(t)-Θf(x))=eT(t)(AmTP2-P2Am+2ΘF)e(t)+2eT(t)P2((Am-ΘF)xm+Bmr(t)-Θg(x))=eT(t)P3e(t)+eT(t)ξ1-eT(t)P2Θg(x),
where P3≜AmTP2-P2Am+2ΘF and ξ1≜2P2(Am-ΘF)xm+Bmr(t).
SinceΘ and r(t) are bounded, we have
(31)V˙2(z(t))≤∥e(t)∥2∥P3∥+∥e(t)∥∥ξ1∥+∥e(t)∥∥P2∥∥Θ∥∥g(x)∥≤∥e(t)∥2∥P3∥+∥e(t)∥∥ξ1∥+M∥e(t)∥∥P2∥∥Θ∥≤∥e(t)∥2∥P3∥+∥e(t)∥(∥ξ1∥+M∥P2∥∥Θ∥).
Then, it holds that(32)V˙2(z(t))≤β1∥e(t)∥2+β2∥e(t)∥,
where β1=∥P3∥ andβ2=∥ξ1∥+M∥P2∥∥Θ∥.
Therefore, for anyr2>0, when e(t)∈Rn/S(r2), then the inequality V˙2(z(t))≤2λ2V2(z(t)), that is, V(z(t))≤exp(2λ2(t-t0))V2(z(t0)), holds for any λ2 satisfying 0<λ2≤(1/2λmin(P2))((β2/r2)+β1).
This completes the proof.Based on Lemmas6 and 7, we have the following lemma.Lemma 8.
Consider the subsystems of(12). For anyr3>0, ife(t)∈Rn/S(r3), then the inequality(33)η1∥e(t)∥2≤Vi(z(t))≤η3∥e(t)∥2,
holds for η1=min{λmin(P1),λmin(P2)}, η2=max{λmax(P1),λmax(P2)}, and η3=η2+(l/r32).Proof.
Form (22) and (29), we have
(34)eT(t)Pie(t)≤Vi(z(t))≤eT(t)Pie(t)+l.
Denote thatη1=min{λmin(P1),λmin(P2)} and η2=max{λmax(P1),λmax(P2)}. Then, we have
(35)η1∥e(t)∥2≤Vi(z(t))≤η2∥e(t)∥2+l.
For anyr3>0, when e(t)∈Rn/S(r3), we can find a constant η3=η2+(l/r32) such that η1∥e(t)∥2≤Vi(z(t))≤η3∥e(t)∥2.
This completes the proof.Furthermore, according to Lemmas6–8, for any given r0′=max{r1,r2,r3} and t>t0, if e(t)∈Rn/S(r0′), it is true that
(36)V(z(t))≤{exp(-2λ1(t-t0))V1(z(t0))whenthecontrollerworks,exp(2λ2(t-t0))V2(z(t0))whenthecontrollerfails.For (18) and (25), it is obvious that there existsμ=η3/η1 such that
(37)Vi(z(t))≤μVj(z(t))(i,j=1,2).Without loss of generality, fork=0,1,…, we assume that the controller works during [t2k,t2k+1), which means that the first subsystem is active on [t2k,t2k+1), while the controller fails during [t2k+1,t2k+2), which denotes that the second subsystem is active on [t2k+1,t2k+2).Whene(t2k)∈S(r1), from (22), we have e(t2k+1)∈S(r1′) for any given r1′≥(r12λmax(P1)+l)/λmin(P1). It is obvious that (36) and (37) still hold for r0≥max{r1′,r0′}.Remark 9.
Because of the uncertainties of the systems, the properties of the Lyapunov function candidates are restricted outside the ball with the radiusr0 as described as in Lemmas 6–8.Now, we are in the position to give the main result of this paper.Theorem 10.
Consider the error switched system (12) with the adaptive control law (13). For k=0,1,…, if the switching law σ(t) satisfies the following two conditions:
Condition 1
(38)
T
+
(
t
2
k
,
t
)
(
t
-
t
2
k
)
≤
(
λ
1
-
λ
*
)
(
λ
1
+
λ
2
)
holds for some scalar λ*∈(0,λ1),
Condition 2
(39)
F
f
(
t
2
k
,
t
)
≤
F
f
*
=
λ
ln
μ
holds for some scalar λ∈(0,λ*), then, the error switched system (12) is globally practically stable.Proof.
For any givenr0≥max{r1′,r0′}, denote
(40)r*=exp[∥Θ∥∥F∥(Ts)]·r0+(κ+M∥Θ∥)[exp(∥Θ∥∥F∥(Ts))-1]∥Θ∥∥F∥,
where κ≥∥(Am-ΘF)xm(t)+Bmr(t)∥ for some κ>0. When the initial error e(t0)∈Rn, we will show that there exists a constant T=T(e(t0),r*)≥0, concerned with e(t0) and r* such that e(t;t0,e(t0))∈S(r*) for t≥t0+T under the switching law σ(t) satisfying Conditions 1-2. To this end, we will prove the theorem in three cases.(a)
Fore(t2k)∈S(r0), we will show that e(t2k+2)∈S(r*).(b)
Fore(t2k)∈Rn/S(r0), we will prove that there exists T2k=T2k(e(t2k),r0)≥0 such that ∥e(t2k+T2k)∥=r0 for e(t)∈Rn/S(r0) and t∈[t2k,T2k).(c)
When the initial errore(t0)∈Rn, we will show that there exists a constant T=T(e(t0),r*)≥0 such that e(t;t0,e(t0))∈S(r*) for t≥t0+T under the switching law σ(t) satisfying Conditions 1 and 2.
We first prove (a). Considere(t2k)∈S(r0). Because of the asymptotical stability of the normal subsystem of (12), it is obvious that e(t2k+1)∈S(r0). When t∈[t2k+1,t2k+2), the second subsystem is active. From (10), we have
(41)e˙(t)=ΘFe(t)+(Am-ΘF)xm+Bmr-Θg(x);
then, the trajectory of the error switched system (12) satisfies
(42)e(t2k+2)=exp[ΘF(t2k+2-t2k+1)]e(t2k+1)+∫t2k+1t2k+2{[(Am-ΘF)xm(t)+Bmr(t)-Θg(x(t))]exp[ΘF(t2k+2-t)]×[(Am-ΘF)xm(t)+Bmr(t)-Θg(x(t))]}dt.
With the help of∥eA∥≤e∥A∥, it is obvious that
(43)∥e(t2k+2)∥≤exp[∥Θ∥∥F∥·(t2k+2-t2k+1)]∥e(t2k+1)∥+∫t2k+1t2k+2{∥Θ∥∥g(x(τ))∥exp[∥Θ∥∥F∥·(t2k+2-τ)]×[∥(Am-ΘF)xm(τ)+Bmr(τ)∥+∥Θ∥∥g(x(τ))∥]}dτ≤exp[∥Θ∥∥F∥(t2k+2-t2k+1)]∥e(t2k+1)∥+∫t2k+1t2k+2{(Λ+M∥Θ∥)exp[∥Θ∥∥F∥(t2k+2-τ)]}dτ=exp[∥Θ∥∥F∥(t2k+2-t2k+1)]∥e(t2k+1)∥+(Λ+M∥Θ∥)[exp(∥Θ∥∥F∥(t2k+2-t2k+1))-1]∥Θ∥∥F∥.
Because Ts=maxk{t2k+2-t2k+1}, we have
(44)∥e(t2k+2)∥≤∥e(t2k+1+Ts)∥=exp[∥Θ∥∥F∥(Ts)]∥e(t2k+1)∥+(κ+M∥Θ∥)[exp(∥Θ∥∥F∥(Ts))-1]∥Θ∥∥F∥.
Note thatr0 is the maximum value of ∥e(t2k+1)∥; thus we have
(45)∥e(t2k+2)∥≤r*=exp[∥Θ∥∥F∥(Ts)]r0+(κ+M∥Θ∥)[exp(∥Θ∥∥F∥(Ts))-1]∥Θ∥∥F∥.
Because ofr*>r0, we have e(t2k+2)∈S(r*) when e(t2k)∈S(r0).
Then, we prove (b). Fore(t2k)∈Rn/S(r0), obviously, we only need to consider the trajectory e(t) being totally outside S(r0). We discuss two cases for t≥t2k. One is t∈[t2j,t2j+1), and the other is t∈[t2j+1,t2j+2), where j=k,k+1,….
Thus, from Lemmas6 and 7, it is true that
(46)V(z(τ))≤{exp(-2λ1(τ-t2k))V1(z(t2k)),ift2k≤τ<t2k+1exp(2λ2(τ-t2k+1))V2(z(t2k+1)),ift2k+1≤τ<t2k+2.
Ift∈[t2j,t2j+1), according to (37) and (46), it holds that
(47)V(z(t))≤exp(-2λ1(t-t2j))V1(z(t2j))≤exp(-2λ1(t-t2j))μV2(z(t2j))≤μexp(-2λ1(t-t2j))exp(2λ2(t2j-t2j-1))×V2(z(t2j-1))≤μ2exp(-2λ1(t-t2j))exp(2λ2(t2j-t2j-1))×V1(z(t2j-1))≤μ2exp(-2λ1(t-t2j))exp(2λ2(t2j-t2j-1))×exp(-2λ1(t2j-1-t2j-2))V1(z(t2j-2))≤⋯≤μ2(j-k)exp(-2λ1(t-t2k-T+(t2k,t)))×exp(2λ2T+(t2k,t))V1(z(t2k)).
Ift∈[t2j+1,t2j+2), again from (37) and (46), we have
(48)V(z(t))≤exp(2λ2(t-t2j+1))V2(z(t2j+1))≤exp(2λ2(t-t2j+1))μV1(z(t2j+1))≤μexp(2λ2(t-t2j+1))exp(-2λ1(t2j+1-t2j))×V1(z(t2j))≤μ2exp(2λ2(t-t2j+1))exp(-2λ1(t2j+1-t2j))×V2(z(t2j))≤μ2exp(2λ2(t-t2j+1))exp(-2λ1(t2j+1-t2j))×exp(2λ2(t2j-t2j-1))V2(z(t2j-1))≤⋯≤μ2(j-k)+1exp(-2λ1(t-t2k-T+(t2k,t)))×exp(2λ2T+(t2k,t))V1(z(t2k)).
By Definition2, we know Nf(t2k,t)=2(j-k) for t∈[t2j,t2j+1) and Nf(t2k,t)=2(j-k)+1 for t∈[t2j+1,t2j+2). Thus, for any t∈[t2j+1,t2(j+1)), from (47) and (48), we can obtain
(49)V(z(t))≤μNf(t2k,t)exp(-2λ1(t-t2k-T+(t2k,t)))×exp(2λ2T+(t2k,t))V1(z(t2k)).
With the help of Lemma 8, it holds that
(50)∥e(z(t))∥≤η2η1exp(Nf(t2k,t)lnμ)×exp(-λ1(t-t2k-T+(t2k,t)))×exp(λ2T+(t2k,t))∥e(z(t2k))∥.
Applying Condition 1 gives
(51)exp(-λ1(t-t2k-T+(t2k,t))+λ2T+(t2k,t))≤exp(-λ*(t-t2k)).
From Condition 2 and Definition 1, we have
(52)exp(Nf(t2k,t)lnμ)≤exp(λ(t-t2k)).
Using (49), (51), and (52) results in
(53)∥e(z(t;t2k,e(t2k)))∥≤exp(-(λ*-λ)(t-t2k))∥e(z(t2k))∥.
Therefore, whene(t2k)∈Rn/S(r0), there exists T2k=T2k(e(t2k),r0)≥0 such that ∥e(t2k+T2k)∥=r0 under the switching law σ(t) satisfying Conditions 1-2. Obviously, {T2k} is a decreasing sequence, and thus T0=maxk{T2k}.
Finally, we prove (c). Ife(t0)∈Rn/S(r*), by applying (a), (b), and r*>r0, there exists a positive constant T=T(e(t0),r*)<T0 such that e(t;t0,e(t0))∈S(r*) for t≥t0+T. If e(t0)∈S(r*), the result remains true with T=0.
This completes the proof.Remark 11.
When the initial errore(t0)∈S(r*), the error switched system is ε-practical stability [23].Remark 12.
The error switched system (12) is globally practically stable if the controller fails only for a short time interval and with a low frequency of occurrence.


## 5. Example

In this section, we present an example to demonstrate the effectiveness of the proposed method in this paper.Consider the system (1) with
(54)Θ=[1312],F=[1001],g(x)=[sinx1cosx2],B=[1002].The reference statexm is generated by the reference model (2) with Am=[-200-3], Bm=[11], and the reference input is r=1.ChooseR=[-200-1.5], T=[10.5], P1=P2=[0.4449000.2862], and Γ=[200020]. We have r*=6.8147 when r0=5. Then, according to (38) and (39), we obtain T+(t2k,t)/(t-t2k)≤0.0302 and Ff(t2k,t)≤Ff*=0.548. The switching signal is chosen as
(55)σ(t)={1,t∈Π,2,t∈[0,+∞)Π,
Where Π=[15k,15k+1.95)∪[15k+2,15k+4)∪[15k+4.05,15k+6.35)∪[15k+6.4,15k+8.85)∪[15k+8.9,15k+11.55)∪[15k+11.6,15k+14.55)∪[15k+14.6,15k+15), k=0,1,2,…, which is described in Figure 2. It is easy to verify that σ(t) satisfies Conditions 1-2 of Theorem 10.Figure 2
Switching signal.Whenxm(t0)=[0,0]T and x(t0)=[5,6]T, the norm of the tracking error of (12) with and without controller failures is shown in Figures 3 and 4, respectively.Figure 3
The norm of the tracking error of (12) when controller fails.Figure 4
The norm of the tracking error of (12) without controller failure.Simulations are carried out for∥e(t0)∥=7.81≥r0 and ∥e(t0)∥=2.236<r*. The results are depicted in Figures 5 and 6.Figure 5
The norm of the tracking error of (12) under σ(t)(∥e(t0)∥=7.81).Figure 6
The norm of the tracking error of (12) under σ(t)(∥e(t0)∥=2.236).From Figures5 and 6, we can conclude that whether ∥e(t0)∥≥r0 or not, the states of the system (1) with controller temporary failure track the reference signal xm(t) well under the switching signal σ(t), and the tracking error ∥e(t)∥ is small in the sense of r*. Simulation illustrates the effectiveness of the proposed method.


## 6. Conclusion

This paper has considered the state tracking problem for a class of MRAC systems in the presence of controller temporary failure. A key point is to describe such a system as an error switched system. The properties of Lyapunov function candidates without switching have been given. Then, the global practical stability of the error switched system can be ensured by the proposed scheme, providing that the controller suffers from failures only for a relatively short time interval and with a low frequency of occurrence. It is an interesting topic to extend the results for the output tracking problem of adaptive systems.



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*Source: 741216-2013-10-27.xml*