Fault estimation and accommodation for switched systems with time-varying delay

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Fault estimation and accommodation for switched systems with time-varying delay

This is the Published version of the following publication

Du, Dongsheng, Jiang, Bin and Shi, Peng (2011) Fault estimation and accommodation for switched systems with time-varying delay. International Journal of Control, Automation and Systems, 9 (3). pp. 442-451. ISSN 1598- 6446

The publisher’s official version can be found at http://dx.doi.org/10.1007/s12555-011-0303-3

Note that access to this version may require subscription.

Downloaded from VU Research Repository https://vuir.vu.edu.au/7898/

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Fault Estimation and Accommodation for Switched Systems with Time-varying Delay

Dongsheng Du, Bin Jiang, and Peng Shi

Abstract:This paper considers the problem of fault estimation and accommodation for a class of switched systems with time-varying delay. An adaptive fault estimation algorithm is proposed to estimate the fault, moreover, constant or time-varying fault can be estimated.

Meanwhile, a delay-dependent criteria is obtained with the purpose of reducing the conservatism of the fault estimation algorithm design. On the basis of fault estimation, an observer- based fault tolerant controller is designed to guarantee the stability of the closed-loop system.

Additionally, simulation results are presented to illustrate the efficiency of the proposed results.

Keywords:Accommodation, fault estimation, switched systems, time-varying delay.

1. INTRODUCTION

In order to increase the reliability and safety of dynamics systems, fault diagnosis (FD) and fault tolerant control (FTC) have been intensively investigated in re- cent years. Generally speaking, FTC includes two main approaches: passive and active. In passive FTC systems, which use feedback control laws that are robust with respect to some fixed faults [1, 2]. However, it may be failed for other accidental faults. Under this case, we must resort to active FTC technique, which relies on a fault detection and isolation (FDI) process to identify the fault-induced changes [3]. The control law is reconfig- ured online in response to the FDI decisions and hence has better fault-tolerance capability. It can be seen that FDI process is the first step in active FTC to monitor the system and determine the location of the fault. Then, fault estimation is utilized to on-line generate magnitude of the fault. Therefore, FD is a very meaningful and challenging topic, which has attracted many researches to devote themselves into this study domain. During the past decades, various effective methods, such as sliding This work was supported in part by the National Natural Science Foundation of P.R. China (Nos. 60874051, 60804011, 60904001), Funding for Outstanding Doctoral Dissertation in Nanjing University of Aeronautics and Astronautics (BCXJ10- 03), and the Engineering and Physical Sciences Research Council, UK (EP/F029195).

Dongsheng Du and Bin Jiang are with the College of Automa- tion Engineering, Nanjing University of Aeronautics and Astro- nautics, Nanjing 210016, Jiangsu, and Aeronautical Science and Technology Key Lab. of Aeronautical Test and Evaluation, P. R.

China. (e-mail: [email protected]).

Peng Shi is with the Department of Computing and Math- ematical Sciences, University of Glamorgan, Pontypridd, CF37 1DL, UK, School of Engineering and Science, Victoria Univer- sity, Melbourne 8001 Vic, Australia, and College of Automation, Harbin University of Science and Technology, China. (e-mail:

[email protected]).

mode observer approach using equivalent output injec- tion signal [4], adaptive technique [5, 6], and learning method based on neural network [7, 8] and so on, have been developed for the fault estimation problem. Among them, adaptive fault diagnosis observer has been proved to be an effective approach. The advantage of the adaptive fault diagnosis observer is that the state vector estimation and actuator fault estimation can be obtained simultaneously.

On the other hand, relatively few FD work was done for switched system compared with the plentiful FD achievements for general dynamic systems. FD for switched systems was considered in [9, 10], passive and active FTC for nonlinear switched systems with actuator faults were separately investigated in [11] and [12].

Sensor fault case for switched systems was studied in [13]. Switched system belongs to hybrid system, which consists a finite number of subsystems and an associated switching signal governing the switching among them. Many physical or man-made systems can be modeled as switched systems, including chemical pro- cesses, computer controlled systems, switched circuits and so on. During the past three decades, lots of stability theoretic results have been reported for switched systems [14]- [16]. On the other hand, time delay is the inherent features of many physical process and the big sources of instability and poor performances.

Switched systems with time delay have strong engineering background, such as in network control systems [17] and power systems [18]. Existing stability criteria of delay systems can be classified into two types:

delay-dependent and delay-independent methods. In re- cent years, much attention has been drawn to the development of delay-dependent conditions aimed at reducing the conservatism, and many theoretical studies have been conducted for switched systems with time de-

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lay [19]- [20]. However, to the best of our knowledge, the problem of fault estimation and accommodation for switched systems with time-varying delay has not been considered yet, which motivates us to investigate this meaningful issue.

In this work, based on adaptive fault diagnosis observer method and average dwell-time technique, the problem of fault estimation and accommodation for switched systems with time-varying delay is solved. A novel adaptive estimate law is proposed by solving some constrained linear matrix inequalities. An efficient algorithm is developed to solve these constrained LMIs and simulation results prove the effectiveness of the proposed method. The contributions of this work can be summarized as the following two aspects:

1:) An adaptive estimate law is developed for switched system with time-varying delay. By utilizing piecewise Lyapunov function and average dwell-time technique, a novel adaptive fault estimation algorithm is designed. Moreover, the obtained result is delay-dependent, which makes further efforts to degrade the conservativeness. On the other hand, many fault estimate law is confined to estimate constant fault. The advantage of such proposed estimation algorithm can not only estimate the fault fast and exactly, but also is adaptive to estimate two type of fault: constant fault case and time-varying fault case.

2:) To review the development of FD for switched systems, the fault estimation and accommodation for switched system with time-varying delay is not considered yet. Additionally, the time-delay is time- varying and constrained in an interval set. This condition is more practical than constant time delay case. This paper investigates this issue and efficient results are given. Another point should be pointed out that average dwell-time technique is utilized to stabilize the switched system, which is more general and flexible than dwell time switching and arbitrary switching signal.

The paper is organized as follows. Section 2 gives the model description. Section 3 presents the fault estimation algorithm, and an observer-based fault tolerant controller is designed. An example is illustrated in Section 4 to show the usefulness and applicability of the proposed approaches, and the paper is concluded in Section 5.

2. MODEL DESCRIPTION The switched system is described as follows :

˙

x(t) = A_σ(t)x(t) +A_dσ(t)x(t−d(t))

+B_σ(t)(u(t) +f(t)) (1)

y(t) = C_σ(t)x(t) (2)

wherex(t)∈Rⁿis the state vector,y(t)∈R^mis the output vector, u(t)∈R^p is the control input, f(t)∈R^l is the actuator fault. σ(t):[0,+∞)→ψ ={1,· · ·,N}is the switching signal, N >1 is the number of subsystems. At an arbitrary continuous timet, σ(t), denoted byσ for simplicity, is dependent ont orx(t), or both, or other switching rules. A_σ(t), A_dσ(t),B_σ_(t), andC_σ(t) are constant matrices with appropriate dimensions for allσ(t)∈ψ. When the i-subsystem is activated, we denote the matrices associated withσ(t) =ibyA_σ_(t)=A_i, A_dσ_(t)=A_di,B_σ(t)=B_i, andC_σ(t)=C_i.d(t)is the time delay and assumed to satisfy the following condition:

C1: d(t)is differentiable and bounded with a constant delay-derivative bound:

h₁≤d(t)≤h₂, d(t)˙ ≤d<1 (3) f(t) can be thought of as an additional signal expressed as f(t) =ν(t−t₀)f₀(t), the function ν(t−t₀)is given by

ν(t−t₀) =

½ 0, t≤t₀

1, t>t₀ (4)

where t0 is the time of fault occurring. That is, f(t) is zero prior to the failure time t ≤t₀ and is f₀(t) af- ter the failure occurs t >t₀. It is assumed that the derivation f₀(t)with respect to time is norm bounded, i.e. kf₀(t)k ≤ f₁, kf˙₀(t)k ≤ f₂, where 0≤ f₁ <∞, 0≤ f₂<∞.

The following lemmas are added for the convenes of later proof.

Lemma 1: ( [21]) Given matricesW,X andY of appropriate dimensions and withW symmetrical, then

W +XF(t)Y +Y^TF^T(t)X^T <0

holds for allF(t)satisfyingF^T(t)F(t)≤Iif and only if for someε>0,

W +εX X^T+ε⁻¹Y^TY <0

Lemma 2: ( [22]) For any real vectors a, b and matrix G>0 of compatible dimensions, the following inequality holds:

a^Tb+b^Ta≤a^TGa+b^TG⁻¹b, a,b∈Rⁿ Lemma 3: ( [23]) Consider the switched system

˙

x(t) = fσ(x(t)), σ ∈ ψ and let α > 0, µ > 1 be given constants. Suppose that there exist functions V_σ_(t) : Rⁿ →R, and two class functions β1 and β2

such that β1(|x(t)|)≤V_i(x(t))≤β2(|x(t)|), ˙V_i(x(t))≤

−αVi(x(t)),∀i∈ψ, and ˙V_i(x(t))≤µVj(x(t)),∀(i,j)∈ ψ×ψ then the system is globally uniformly asymptotically stable for any switching signal with average dwell time

τa>τ_a^∗=lnµ/α

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3. MAIN RESULTS 3.1. Fault diagnosis observer design

To diagnose the actuator fault, the following fault diagnosis observer is designed:

˙ˆ

x(t) =A_σ(t)x(tˆ ) +A_dσ_(t)x(tˆ −d(t))

+B_σ_(t)(u(t) +fˆ(t))−L_σ_(t)(y(t)ˆ −y(t))(5) ˆ

y(t) =C_σ_(t)x(t)ˆ (6)

where ˆx(t) ∈ Rⁿ is the state vector of the observer, ˆ

y(t)∈R^m is the output vector of the observer, ˆf(t) is an estimate of f(t),L_σ_(t)is the observer gain.

Denote

e(t) =x(t)ˆ −x(t); r(t) =y(tˆ )−y(t);

f˜(t) =fˆ(t)−f(t); (7) then the error dynamics is described by

˙

e(t) = A_σ(t₎e(t) +A_dσ_(t)e(t−d(t)) +B_σ(t)f˜(t)(8)

r(t) = C_σ(t)e(t) (9)

whereA_σ(t)=A_σ(t)−L_σ_(t)C_σ_(t). As a result, the purpose of fault estimation is to find a diagnostic algorithm for

fˆ(t)such that

t→∞lime(t) =0; lim

t→∞fˆ(t) =f(t) (10)

3.2. Fault estimation algorithm design

In the sequel, a convergent adaptive fault diagnostic algorithm to estimate the fault f(t) is given, which is obtained from the residualr(t). Consequently, ˙f(t)6=0 implies that the residualr(t)with respect to time is

˙

r(t) = f˙ˆ(t)−f˙(t) (11) Theorem 1: For the time delay d(t) satisfying the condition C1 and a given scalarα>0, if there exist positive definite matricesP_i,X_i,R_i,Y_i,U_i,V_i,Q_i1,Q_i2,Q_i3, Z_i1,Z_i2,G, matricesWi,Fi, N_i1,N_i2,M_i1,M_i2,S_i1,S_i2, and a scalarτ>0, such that

· P_i I I X_i

¸

≥0, P_iX_i=I, i=1,2,· · ·,N. (12)

· R_i X_i X_i V_i

¸

≥0,Y_iR_i=I,U_iV_i=I, i=1,2,· · ·,N.(13)

Ψ_i=

· Πi Γi

Γ^T_i Ω_i

¸

<0 (14)

where

U_i = h₂Z_i1+hZ¯ _i2, h¯=h₂−h₁,

Ω_i = diag{−Y_i,−h₂e^−αh²Z_i1,−h₂e^−αh²Z_i2,

−he¯ ^−α^h²(Z_i1+Z_i2),−τI,−τI},

Π_i =







ϕ11 ϕ12 S_i1

∗ ϕ22 S_i2

∗ ∗ −e^−αh¹Q_i1

∗ ∗ ∗

→

−M_i1 ϕ15

−Mi2 A^T_diC_i^TF_i^T

0 0

−e^−αh²Q_i2 0

∗ −2F_iC_iB_i+G





,

Γ_i =







A^T_i P_i−hC˜ _i^TW_i^T h₂N_i1 hS¯ _i1 A^T_diP_i h₂N_i2 hS¯ _i2

0 0 0

→

hM¯ _i1 C_i^TW_i^T 0 hM¯ _i2 0 0

0 0 0

0 0 F_iC_i





,

ϕ11 = PiAi+A^T_iPi−WiCi−C_i^TW_i^T+αPi+Ni1

+N_i1^T+

∑

3 j=1

Q_i,

ϕ12 = P_iA_di−N_i1+M_i1−S_i1+N_i2^T, ϕ₁₅ = PiBi−C^T_i F_i^T−A^T_iC^T_i F_i^T,

ϕ22 = (d−1)e^−αh²Q_i3−N_i2−N_i2^T+M_i2+M_i2^T

−S_i2−S^T_i2.

then, for the switching signal σ(t) with average dwell time satisfying

τa≥τ_a^∗=lnµ

α (15)

where

µ≥1with P_i≤µP_j,Q_i1≤µQj1,Q_i2≤µQ_j2, Qi3≤µQj3,Zi1≤µZj1,Zi2≤µZj2. (16) the following diagnosis algorithm

f˙ˆ(t) =−ΓF_i(˙r(t) +r(t)) (17) can realize

t→∞lime(t) =0; lim

t→∞

fˆ(t) =f(t)

whereWi=PiLi,∗denotes the symmetric elements in a symmetric matrix, andΓ=Γ^T >0 is a given weighting matrix.

Proof. Construct the following piecewise Lyapunov function candidate as

V_σ_(t)(t) = e^T(t)P_σ(t)e(t) +

Z _t

t−d(t)e^T(s)Q_σ(t)3e^α^(s−t)e(s)ds

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+

∑

2 j=1

Z _t

t−h_je^T(s)Q_σ(t)je^α(s−t)e(s)ds +

Z ₀

−h₂

Z _t

t+θe˙^T(s)Z_σ(t)1e^α^(s−t)e(s)dsdθ˙ +

Z _−h

1

−h₂

Z _t

t+θe˙^T(s)Z_σ(t)2e^α(s−t)e(s)dsdθ˙ +f˜^T(t)Γ⁻¹f˜(t) (18) Setσ(t) =iand taking the derivative of (18), we obtain

V˙i(t) ≤ 2e^T(t)Pie(t˙ ) +e^T(t)Qi3e(t)−(1−d)× e^T(t−d(t))Q_i3e^−αh²e(t−d(t)) +

∑

2 j=1

[e^T(t)Q_{i j}e(t)−e^T(t−h_j)Q_{i j}× e^−α^h^je(t−hj)] +e˙^T(t)(h2Zi1+hZ¯ i2)e(t˙ )

− Z _t

t−h₂e˙^T(s)Z_i1e^−αh²e(s)ds˙

− Z _t−h

1

t−h₂ e˙^T(s)Z_i2e^−α^h²e(s)ds˙

+αe^T(t)P_ie(t)−αVi(t)−2 ˜f^T(t)F_iC_ie(t)˙

−2 ˜f^T(t)F_iC_ie(t)−2 ˜f^T(t)Γ⁻¹f˙(t) (19) From the Newton-Leibniz formula, the following equations are true for any matricesN_i1, N_i2, M_i1, M_i2, S_i1, andS_i2with appropriate dimensions.

2[e^T(t)N_i1+e^T(t−d(t))N_i2]×

·

e(t)−e(t−d(t))− Z _t

t−d(t)e(s)ds˙

¸

=0 (20) 2[e^T(t)M_i1+e^T(t−d(t))M_i2] [e(t−d(t))

−e(t−h2)− Z _t−d(t)

t−h₂ e(s)ds˙

¸

=0 (21)

2[e^T(t)S_i1+e^T(t−d(t))S_i2] [e(t−h₁)

−e(t−d(t))− Z _t−h

1 t−d(t)e(s)ds˙

¸

=0 (22)

Moreover, the following equalities hold:

Z _t

t−h₂e˙^T(s)Z_i1e^−αh²e(s)ds˙

=

Z _t−d(t)

t−h₂ e˙^T(s)Zi1e^−αh²e(s)ds˙ +

Z _t

t−d(t)e˙^T(s)Zi1e^−αh²e(s)ds˙ Z _t−h

1

t−h₂ e˙^T(s)Z_i1e^−αh²e(s)ds˙

=

Z _t−d(t)

t−h₂ e˙^T(s)Z_i2e^−αh²e(s)ds˙ +

Z _t−h

1

t−d(t)e˙^T(s)Zi2e^−αh²e(s)ds˙ (23)

Thus, from (20) and (23), we can obtain that

− Z _t

t−d(t)e˙^T(s)Z_i1e^−αh²e(s)ds˙ −2ξ^T(t)N_i× Z _t

t−d(t)e(s)ds˙ ≤ − Z _t

t−d(t)[ξ^T(t)N_i+e(s)Z˙ _i1e^−αh²]

×(Zi1e^−αh²)⁻¹[ξ^T(t)Ni+e(s)Z˙ i1e^−αh²]ds

+h₂ξ^T(t)N_i(Z_i1e^−αh²)⁻¹N_iξ(t) (24) where

ξ(t) =





 e(t) e(t−h) e(t−h₁) e(t−h₂)

f˜(t)





, N_i=





 Ni1

N_i2 0 0 0





. (25)

By using the same methods as (24), from (21)-(23), we obtain

− Z _t−d(t)

−2ξ^T(t)M_i Z_t−d(t)

t−h₂ e(s)ds˙

≤ − Z _t−d(t)

t−h₂ [ξ^T(t)M_i+e(s)(Z˙ _i1+Z_i2)e^−α^h²][(Z_i1 +Z_i2)e^−α^h²]⁻¹[ξ^T(t)M_i+e(s)(Z˙ _i1+Z_i2)e^−αh²]ds +hξ¯ ^T(t)M_i[(Z_i1+Z_i2)e^−αh²]⁻¹M_iξ(t) (26) and

− Z_t−h

1

t−d(t)e˙^T(s)Z_i2e^−αh²e(s)ds˙

−2ξ^T(t)S_i Z _t−h

1 t−d(t)e(s)ds˙

≤ − Z_t

t−d(t)[ξ^T(t)S_i+e(s)Z˙ _i2e^−αh²](Z_i2e^−α^h²)⁻¹

×[ξ^T(t)S_i+e(s)Z˙ _i2e^−αh²]ds

+hξ¯ ^T(t)S_i(Z_i2e^−αh²)⁻¹S_iξ(t) (27) where

Mi=£

M_i1^T M_i2^T 0 0 0 ¤_T , S_i=£

S^T_i1 S_i2^T 0 0 0 ¤_T

. (28)

By Lemma 2, one can get that

−2 ˜f^T(t)Γ⁻¹f˙(t) ≤ f˜^T(t)Gf˜(t)

+ f₂²λmax(Γ⁻¹G⁻¹Γ⁻¹) (29)

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From (19), (24), (26), (27), and (29), one can get that V˙i(t) +αVi(t)

≤ ξ^T(t){Ξi+Ae^T_i (h2Zi1+hZ¯ i2)Aei

+h₂N_i(Z_i1e^−αh²)⁻¹N_i^T +hM¯ _i[(Z_i1+Z_i2)e^−α^h²]⁻¹M_i^T

+hS¯ _i(Z_i2e^−αh²)⁻¹S^T_i }ξ(t) +δ (30) where

Ξ_i =







ϕ11 ϕ12 Si1

∗ ϕ22 S_i2

∗ ∗ −e^−αh¹Q_i1

∗ ∗ ∗

→

−Mi1 ϕ¯₁₅

−M_i2 A^T_diC^T_i F_i^T

0 0

−e^−αh²Q_i2 0

∗ −2F_iC_iB_i+G





,

Ae = £

A_i A_di B_i 0 0 ¤ , ϕ¯₁₅ = PiBi−C_i^TF_i^T−A^T_iC_i^TF_i^T,

δ = f₂²λmax(Γ⁻¹G⁻¹Γ⁻¹).

SetW_i=P_iL_iand use Schur complement lemma, one can get that matrixΞ_iis equivalent to the following formula

Πi+





 0 0 0 0 F_iC_i





P_i⁻¹£

W_iC_i 0 0 0 0 ¤

+





 C_i^TW_i^T

0 0 0 0





P_i⁻¹£

0 0 0 0 C_i^TF_i^T ¤ (31)

Setτ=min{ε,ε⁻¹}and use Lemma 1, if the matrixP_i satisfies (12), one can see that (14) implies(31)<0.

By using (13), (14), and (30), one obtains that V˙_i(t) +αVi(t)≤ −εkξ(t)k²+δ (32) whereε is the minimum eigenvalue of−Ψi. It follows that

V˙_i(t) +αVi(t)<0 f or εkξ(t)k²>δ (33) witch means thatξ(t)converges to a small set according to Lyapunov stability theory.

Integrating the inequalities (33) gives that V_i(t)≤ e^−α(t−t^k⁾V_i(t_k)for any givent∈[t_k,t_k+1). From (16) and (18), at the switching instantt_k, it follows that

V˙_σ(t_k₎(t_k)≤µV_σ(t−

k)(t_k⁻), k=0,1,2,· · · (34)

Thus, using (34) andN_σ(t₀,t)≤^t−t_τ ⁰

a , one can get V_i(t) ≤ e^−α(t−t^k⁾µV_σ(t−

k)(t_k⁻)≤ · · ·

≤ e^−α(t−t⁰⁾µ^N^σ^(t⁰^,t)V_σ(t₀₎(t₀)

= e^−α(t−t⁰⁾e^N^σ^(t⁰^,t)ln^µV_σ_(t₀₎(t₀)

≤ e^−α(t−t⁰⁾e^t−t^τ^a⁰^lnµV_σ(t₀₎(t₀)

= e^−(α−^ln^τa^µ^)(t−t⁰⁾V_σ_(t₀₎(t0)

= e^−λ(t−t⁰⁾V_σ_(t₀₎(t₀) (35) whereλ=¹₂(α−^ln_τ^µ

a ).

Set

a = minλmin(P_i),

b = maxλmax(P_i) +h₁maxλmax(Q_i1) +h₂maxλmax(Q_i2) +h₂maxλmax(Q_i3) +h²₂

2 maxλmax(Z_i1) +h²₂−h²₁

2 maxλmax(Z_i2(36)).

It follows (35) and (36) that

a²ke(t)k²≤b²e^−2λ^(t−t⁰⁾ket₀k²_c (37) That is

ke(t)k ≤b

ae^−λ^(t−t⁰⁾ket₀kc (38) witch means that the state error e(t)→0 and the fault error ˜f(t)→0. Then the proof is completed.

Remark 1: From the adaptive fault estimation algorithm (17), one can get that it contains the derivative of r(t)and ˙r(t). It is feasible when ˙r(t)can be obtained.

But if the signal ˙r(t)can not be easily obtained from cer- tain systems, we should resort to other alternative methods. In order to deal with this problem, ˙rf(t) is intro- duced to be a substitute for ˙r(t)[24]. The relationship is defined as follows:

˙

r_f(t) =−1

ε(r_f(t)−r(t)) (39) From (39), one can get that under zero initial condition, using Laplace transform yields

˙

r_f(t) = 1

εs+1r(t)˙ (40) Therefore, it it easy to show that the substitute ˙r_f(t)can approximate to ˙r(t)with any desired accuracy asε→0.

Meanwhile, whens→0, that ist→∞, ˙rf(t)asymptotically converges to ˙r(t).

Remark 2: In the proving process, we set ˙˜f(t) = f˙ˆ(t)−f˙(t). If the fault is a constant, which means that f˙(t) =0. Under this case, the fault estimate algorithm

(7)

will be changed into ˙ˆf(t) = −ΓF_ir(t), which is only˙ adaptive to estimate constant faults. From the above dis- cussion, one can get that the estimate algorithm in (17) is not only adaptive to estimate constant faults, but also adaptive to estimate time-varying faults.

Remark 3: It can be seen that the conditions (12) and (13) are not strict LMI formation due to the equation PiXi=I,RiYi=I, andUiVi=I, which can not be solved directly by Matlab linear matrix inequality Control Tool- box. However, we can solve this nonconvex feasibility problem by formulating it into a special sequential op- timization problem subject to LMI constraints. In the following, a specific algorithm is given by utilizing the result in [25].

Algorithm 1:

Step 1:Find a feasible set{P_i⁽⁰⁾,X_i⁽⁰⁾,R⁽⁰⁾_i ,Y_i⁽⁰⁾,U_i⁽⁰⁾, V_i⁽⁰⁾,Q⁽⁰⁾_i1 ,Q⁽⁰⁾_i2 ,Q⁽⁰⁾_i3 ,Z⁽⁰⁾_i1 ,Z_i2⁽⁰⁾,W_i⁽⁰⁾,F_i⁽⁰⁾,G⁽⁰⁾,N_i1⁽⁰⁾, N_i2⁽⁰⁾,M_i1⁽⁰⁾,M_i2⁽⁰⁾,S⁽⁰⁾_i1 ,S⁽⁰⁾_i2 ,τ⁽⁰⁾,} satisfying (12), (13) and (14). Setk=0.

Step 2:Solve the following LMI problem mintr

ÃN i=1

∑

³

P_iX_i^(k)+P_i^(k)X_i+R_iY_i^(k)

+R^(k)_i Y_i+U_iV_i^(k)+U_i^(k)V_i

´´

subject to (12), (13) and (14).

Step 3: Substitute the obtained matrix variables {P_i,X_i,R_i,Y_i,U_i,V_i,Q_i1,Q_i2,Q_i3,Z_i1,Z_i2,W_i,F_i,G,N_i1, N_i2,M_i1,M_i2,S_i1,S_i2,τ,}into (12), (13) and (14). If the condition (12) is satisfied with

|tr(

∑

N i=1

(PiXi+RiYi+UiVi))−3(N+1)n|<ρ for some sufficient small scalarρ >0, then output the feasible solution{Pi,Xi,Ri,Yi,Ui,Vi,Q_i1,Q_i2,Q_i3,Z_i1, Z_i2,W_i,F_i,G,N_i1,N_i2,M_i1,M_i2,S_i1,S_i2,τ,},EXIT.

Step 4: Ifk>NwhereN is the maximum number of iterations allowed,EXIT.

Step 5:Setk=k+1,{P_i^(k),X_i^(k),R^(k)_i ,Y_i^(k),U_i^(k),V_i^(k), Q^(k)_i1 ,Q^(k)_i2 ,Q^(k)_i3 ,Z_i1^(k),Z^(k)_i2 ,W_i^(k),F_i^(k),G^(k),N_i1^(k),N_i2^(k), M_i1^(k),M_i2^(k),S^(k)_i1 ,S_i2^(k),τ^(k),}={P_i,X_i,R_i,Y_i,U_i,V_i,Q_i1, Q_i2,Q_i3,Z_i1,Z_i2,W_i,F_i,G,N_i1,N_i21,M_i1,M_i21,S_i1,S_i2,τ,}, and go toStep 2.

3.3. Fault accommodation

Since the statex(t)is unavailable, the estimation value ˆ

x(t)is substituted forx(t). Therefore, the observer-based normal controller is given

u_r(t) =−K_σ_(t)x(t) +ˆ d(t) (41) whereK_σ(t) is the feedback gain matrix andd(t)is the reference input.

Once a fault occurs, based on the accurate and rapid estimation of the fault , the following observer-based fault-tolerant controller is activated to compensate for the fault.

u(t) =u_r(t)−fˆ(t) (42) Assuming d(t) =0 and substituting (42) into (1), one obtains





˙

x(t) = (A_σ(t)−B_σ(t)K_σ(t))x(t) +A_dσ(t)x(t−d(t)) +ρ(t) y(t) =C_σ_(t)x(t)

(43) whereρ(t) =−B_σ(t)K_σ(t)e(t)−B_σ(t)f˜(t).

From the result of Theorem 1, one can get thate(t)→ 0 and ˜f(t)→0 whent →∞. The signal ρ(t) can be treated as a disturbance of the system (43). So, if only the feedback gain K_i can ensure that the following system is asymptotically stable.





˙

x(t) = (A_σ(t)−B_σ(t)K_σ(t))x(t) +A_dσ_(t)x(t−d(t)) y(t) =C_σ_(t)x(t)

(44) Construct the corresponding piecewise Lyapunov function as

V_σ(t)(t) = x^T(t)P_σ_(t)x(t) +

Z _t

t−d(t)x^T(s)Q_σ(s)e^α(s−t)x(s)ds(45) where P_σ_(t) and Q_σ(t) are positive definite matrices.

Thus, we have the following theorem.

Theorem 2: For the time delay d(t) satisfying the condition C1 and a given scalarα>0, if there exist positive definite matricesX_i,T_i, matrixJ_i, such that



 (1,1) A_hiT_i X_i

∗ −(1−d)e^−αh²T_i 0

∗ ∗ −Ti



<0 (46) where (1,1) =A_iX_i+X_iA^T_i −B_iJ_i−J_i^TB^T_i +αXi, J_i= KiXi. Then, for the switching signalσ(t)with average dwell time satisfying

τa≥τ_a^∗=lnµ

α (47)

where

µ≥1with P_i≤µP_j, Q_i≤µQj. (48) such that the system (44) is asymptotically stable.

Proof.Taking the derivative ofV_σ(t)along the trajec- tories of the system in (44) is

V˙i(t) ≤ 2x^T(t)Pi(Ai−BiKi)x(t) +x^T(t)Qix(t)

+2x^T(t)P_iA_dix(t−d(t)) +αx^T(t)P_ix(t)−αVi

−(1−d)x^T(t−d(t))Qie^−αh²x(t−d(t))

= η^T(t)ϒ_iη(t)−αVi (49)