Dynamic Output Feedback Fault-tolerant Control for Networked Control System with Random Delays

Proceedings of the 7th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes Barcelona, Spain, June 30 - July 3, 2009

Dynamic Output Feedback Fault-tolerant Control for Networked Control System with Random Delays Dexiao Xie*, Dengfeng Zhang**, Xiaodong Han*, He Huang*, Zhiquan Wang* * School of Automation, Nanjing University of Science and Technology, Nanjing 210094, P. R. China (e-mail: [email protected]). ** School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, P. R. China Abstract: In this paper, the problem of dynamic output feedback fault-tolerant control for networked control system with random delays and actuator fault is studied. Considering the stochastic time-delay occurs in two channels which between sensor and controller, and between controller and actuator, networked control system with possible actuator fault is modeled as finite states discrete-time Markovian jump linear system with two modes. Based on this model, the necessary and sufficient condition on the stochastic stability of networked control system with random delays and actuator fault is given, and the corresponding dynamic output controller design procedure is proposed via the cone complementarity linearization method. Finally, the numerical example and simulations have demonstrated the validity of the proposed results. Keywords: Networked control systems, fault-tolerant control, stochastically stable, dynamic output feedback, actuator faults. 1. INTRODUCTION Networked control systems(NCSs) are control loops closed through a shared communication network (Hespanha, Payam, and Xu, 2007; Zhang et al., 2001). The main advantages of such NCSs are lost cost, simple installation and maintenance, reduced system weight and volume, increased system agility, etc. Consequently, NCSs are applied to wide range of areas such as manufacturing plants, vehicles, aircrafts, and remote surgery (Walsh and Ye, 2001). Fault-tolerant control is a technique to guarantee system stability and satisfy some performance when system components arises fault. Therefore, with the increasing requirements on the reliability of engineering systems, the problem of fault-tolerant control has been the focus of much research literatures (Patton, 1997; Zhou, Frank, 1998). However, these literatures have not considered the problem of fault-tolerant control for NCSs. Simultaneously, the insertion of communication networks introduces new problems such as quantization, time-delays, packet dropouts, and band-limited channels, etc (AzimiSadjadi, 2003; Brockett and Liberzion, 2003; Elia and Mitter, 2001; Nilsson, Bernhardsson, and Wittenmark, 1998). These problems make system with sensor or actuator fault more fragile, and the analysis and design of fault-tolerant control more complex, otherwise traditional fault-tolerant control theory with many ideal assumptions must be re-evaluated before applying to NCSs. Therefore, research on faulttolerant control of NCSs has importance theoretical and applied significance so that it has obtained more attention (Li, Sauter, and Aubrun, 2007; Kong, Fang, 2005). Network-induced delay is one of the most important issues of NCSs. The delay may be constant, time-varying, and in most cases, random. It is known that the occurrence of delay 978-3-902661-46-3/09/$20.00 © 2009 IFAC

degrades the stability and control performance of closed-loop control systems. In recent years, NCSs with time-delay have been a hot research topic. A lot of useful results have been appeared. Nilsson, Bernhardsson, and Wittenmark.(1998) proposed a stochastic optimal controller and an optimal state estimator of NCS when the network-induced delay is random and less than one sampling time. Hu and Zhu(2003) have extended it to the case with longer delays. A hybrid systems approach was adopted to analyze the stability of NCS when the network-induced delay is deterministic and the controller gain is constant in Zhang et al. (2001); and the stability analysis and controller design of NCS were proposed by using a switched system approach, and controller gain is constant in Lin, Zhai, and Antsaklis.(2003). In Xie et al. (2008), the controller depending on the switching signal was designed to guarantee NCS asymptotical stability for reducing the conservativeness of the results in Lin, Zhai, and Antsaklis. (2003). It is noticed that in all of the aforementioned papers, the plant is in the continuous-time domain. For the discrete-time case, the network-induced random delays were modeled as Markov chains such that the closed-loop system is a jump linear system with one mode, the state-feedback gain only depends on the delay from sensor to controller in Lin, Hassibi, and How. (2000). Further more, Zhang, Shi, Chen, and Huang (2005) considered network-induced delays both between sensor and controller, and between controller and actuator, and the NCS was modeled as jump linear system with two modes, a necessary and sufficient condition on the stochastic stability of NCS was given. It is important to stabilize the NCS in the presence of actuator faults, but so far, all the aforementioned literatures have not considered the case of actuator fault. Therefore, we

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7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009

u f (k )

extend the results in Lin, Hassibi, How.(2000), and Zhang, Shi, Chen, and Huang (2005) to the NCS with random delay and actuator faults. This paper is concerned with the problem of dynamic output feedback fault-tolerant control for NCS with random delay. The delay from senor to controller and the delay from controller to actuator are modeled as two Markov chains. We adopt the dynamic output controller and consider the case of actuator fault for the NCS. By augmenting state variables, the NCS is modeled as Markovian jump linear system. The necessary and sufficient condition on the stochastic stability of NCS with random delays and actuator fault is proposed, and the corresponding dynamic output controller is designed. An inverted pendulum example is given to demonstrate the effectiveness of the proposed method. This paper is organized as follows. In section 2, a NCS with random delays and actuator fault is modeled as Markovian jump linear system. In section 3, according to the resulting faulty NCS model, the stochastic stability analysis and controller design of NCS with random delays and actuator fault are proposed. In section 4, a numerical example is given to illustrate the usefulness of the method. Finally, conclusions are given in section 5. n n× m Notation: Throughout the paper, and denote, respectively, the n-dimensional Euclidean space and the set of n-by-m real matrices. The superscript ‘T’ denotes the transpose for vectors or matrices. ||i|| refers to the Euclidean norm for vectors and induced 2-norm for matrices. E{i} stands for the mathematical expectation operator. λmin ( A) denotes the minimal eigenvalue of matrix A . "∗ " in matrices ⎡ A BT ⎤ ⎡ A ∗ ⎤ ⎡ A BT ⎤ indicates symmetric terms: ⎢ ⎥=⎢ ⎥. ⎥=⎢ ⎣B C ⎦ ⎣B C⎦ ⎣ ∗ C ⎦

2. PROBLEM FORMULATION We consider a class of NCSs with random delays and packet dropout shown in Figure 1, and suppose the physical plant is modeled by following linear discrete-time system: x(k + 1) = Ax(k ) + Bu f (k ) (1) y (k ) = Cx(k )

where x(k ) ∈

p

is the system state, u f (k ) ∈

q

is the

control input from the actuator that may be faulty, and y (k ) ∈ r is the control output. A , B , C are real constant matrices with appropriate dimensions. Considering all possible actuator faults, we adopt the following actuator fault model: u f ( t ) = Mu ( t ) (2) where u ( t ) ∈

q

is feedback control signal, M is actuator

faults matrix defined as (3). M = diag{m1 , m2 ,

, mq } ∈ Ξ

(3)

where Ξ = {M : M ≠ 0, 0 ≤ mil ≤ mi ≤ miu , mil < 1, miu ≥ 1, i = 1,

is a set of actuator faults matrix.

, q}

actuator

plant

sensor

y(k )

u (k ) Shared Communication Network (random delays, packet dropout)

u (k )

y (k ) controller

Fig.1 Networked control systems with random delays and actuator faults Remark 1: In the actuator faults matrix M , when mi = 1 , the corresponding actuator is in the normal case; when mi = 0 , the corresponding actuator is complete failure; when 0 ≤ mil < mi < miu , mil < 1 , miu ≥ 1 and mi ≠ 1 , the corresponding actuator would be in partial failure case. From above analysis and reference Zhang, Wang, and Hu. (2007). we know that the actuator fault model applied here can describe more fault modes for practical system. For the sake of simplicity, we introduce the decomposition of actuator faults matrix M with a similar manner in Zhang, Wang, and Hu(2007), which will be used for our main results. Define M 0 = diag[m01 , m02 , , m0 q ] , J = diag[ j1 , j2 , , jq ] , | L |= diag[| l1 |,| l2 |,

,| lq |] , where m0i = (mil + miu ) / 2 ,

ji = (miu − mil ) / (miu + mil ) , li = (mi − m0i ) / m0i . Thus, we have M = M 0 ( I + L) L ≤J≤I (4)

Consider a full order dynamic output feedback controller with state space representation xc (k + 1) = Ac xc (k ) + Bc y (k ) (5) u (k ) = Cc xc (k ) + Dc y (k ) where xc (k ) ∈ p is controller state, Ac , Bc , Cc and Dc are real constant matrices with appropriate dimensions, which will be determined later. We consider there are random delays in two communication channel shown in the Figure 1,where τ sc (k ) is random delay from sensor to controller, τ ca (k ) is random delay from controller to actuator. We assume that τ sc (k ) and τ ca (k ) are bounded, and satisfy that 0 ≤ τ sc (k ) ≤ τ 1 , 0 ≤ τ ca (k ) ≤ τ 2 (6) As we known, the current time delay is correlated with previous time delay in the practical communication system. Thus, we model τ sc (k ) and τ ca (k ) as two finite state homogeneous independent Markov chains (Zhang, Shi, Chen, and Huang, 2005), which take values in S1 = {0,1, ,τ 1} and S 2 = {0,1, ,τ 2 } respectively, and the corresponding transition probability matrix are Π1 = (π ij ) ∈ Rτ1 ×τ1 and

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7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009

Π 2 = (λmn ) ∈ Rτ 2 ×τ 2 , where π ij = P{τ sc (k + 1) = j | τ sc (k ) = i} ≥ 0

E3 (τ sc (k ),τ ca (k )) = [0

(7)

λmn = P{τ ca (k + 1) = n | τ ca (k ) = m} ≥ 0 and

τ1

∑ π ij = 1 , j =1

τ2

∑ λmn = 1 for all i, j ∈ S1 and m, n ∈ S2 ． n =1

Remark 2: From above analysis, we know that τ sc (k ) and τ ca (k ) sequences are modeled as two Markovian chains, it is said that this model is quite general (Lin, Hassibi, and How, 2000), communication packet dropout in the network can also be included naturally. We assume that the controller will always use the most recent data. Thus, if we have y (k ) = y (k − τ sc (k )) at step k , and there is no new packet coming at the step k + 1 , because of packet dropout or delays that is longer than 1, then the controller maintain the previous values at the step k + 1 , It means y (k + 1) = y (k − τ sc (k )) . Thus, the delay τ sc (k ) can increase at most by one each step, and is constrained with P{τ sc (k + 1) > τ sc (k ) + 1} = 0 . Then from(7), it means that π ij = 0 for all j > i + 1 . Similarly, we

have λmn = 0 , for all n > m + 1 . Considering the time-delay in the NCS, we have y (k ) = y (k − τ sc (k ))

(8)

u (k ) = u (k − τ ca (k ))

By using(1),(2),(5)and(8), we can have the following closedloop faulty NCS: x(k + 1) = Ax(k ) + BMCc xc (k − τ ca (k ))

(9)

+ BMDc Cx(k − τ ca (k ) − τ sc (k )) xc (k + 1) = Ac xc (k ) + Bc Cx(k − τ sc (k )) T

T

We augment system state vector as x (k ) = [ x (k ) x (k − 1) xT (k − τ 1 − τ 2 )]T T c

and augment controller state vector

T c

xcT (k − τ 2 )]T . Then, we let z (k )

xc (k ) = [ x (k ) x (k − 1) T

⎡⎣ x T (k ) xcT (k ) ⎤⎦ .Thus, the closed-loop faulty NCS in (9) can be rewritten as

⎡A ⎢I ⎢ A = ⎢0 ⎢ ⎢ ⎢⎣ 0 ⎡0 ⎢I ⎢ Ac 0 = ⎢ 0 ⎢ ⎢ ⎢⎣ 0

z (0) = [φ T (0), where

, φ T (−τ 1 − τ 2 ), ϕ T (0),

, ϕ T (−τ 2 )]T

(10)

T

Γ1 (τ ca (k )) = ⎡⎣ N1 0 ⎤⎦ BMCc [ 0 E1 (τ ca (k )) ]

0 I 0

0 0 I 0

0 0⎤ 0 0 ⎥⎥ 0 0⎥ ∈ ⎥ ⎥ I 0 ⎥⎦ 0 0⎤ 0 0 ⎥⎥ 0 0⎥ ∈ ⎥ ⎥ I 0 ⎥⎦

0] ∈

p × p (τ1 +τ 2 +1)

p (τ1 +τ 2 +1)× p (τ1 +τ 2 +1)

p (τ 2 +1)× p (τ 2 +1)

and E1 (τ ca (k )) , E2 (τ sc (k )) , E3 (τ sc (k ),τ ca (k )) have all elements being zeros except for the (τ ca (k ) + 1)th , (τ sc (k ) + 1)th , and (τ sc (k ) + τ ca (k ) + 1)th block being identity, respectively. Otherwise, function φ and ϕ are initial state function. Remark 3: By augmenting the state variable, the networked control system is modeled as system in(10) that is a discretetime Markovian jump linear system with two Markov parameters τ sc (k ) and τ ca (k ) , then we can apply the theory of jump linear system to analysis and design such NCS. Before proceeding, we need the following definition: Definition 1: The system(10) is stochastically stable if for every finite initial state z (0) , and initial Markov parameter τ sc (0) ∈ S1 , τ ca (0) ∈ S2 , there exists bounded number Ψ ( z (0),τ sc (0),τ ca (0)) > 0 , such that ⎧T ⎫ lim E ⎨∑ || z (k ) ||2 | z (0),τ sc (0),τ ca (0) ⎬ ≤ Ψ ( z (0),τ sc (0),τ ca (0)) T →∞ ⎩ k =0 ⎭ (11) The objective of this paper is to design dynamic output controller (5) such that the closed-loop faulty NCS in (10) with random delays is stochastically stable in the presence of actuator fault with pattern(3).

z (k + 1) = (Φ 0 + Γ1 (τ ca (k )) + Γ 2 (τ sc (k )) + Γ3 (τ sc (k ),τ ca (k ))) z (k ) Φ (τ sc (k ),τ ca (k )) z (k )

0

0 I

3. MAIN RESULT In this section, a necessary and sufficient condition on stochastically stable for the closed-loop system in (10) with random delays and actuator fault is given, and the corresponding fault-tolerant controller design technique is proposed.

T

Γ 2 (τ sc (k )) = ⎡⎣ 0 N 2 ⎤⎦ Bc C [ E2 (τ sc (k )) 0]

To start with, we introduce the following Lemma:

T

Γ3 (τ sc (k ),τ ca (k )) = ⎡⎣ N1 0 ⎤⎦ BMDc C [ E3 (τ sc (k ),τ ca (k )) 0] T

Φ 0 = diag{ A, Ac 0 } + [ 0 N 2 ] Ac [ 0 N 2 ] N1 = [ I

0 0

0] ∈

p × p (τ 1 +τ 2 +1)

N2 = [ I

0 0

0] ∈

p × p (τ 2 +1)

Lemma 1: Given any symmetric Y , and constant matrix R1 , R2 with appropriate dimension, if S = diag{s1 , sq } is a diagonal matrix with | S |≤ U , where | S |= diag{ s1 ,

, sq } ,

E1 (τ ca (k )) = [0

0 I

0] ∈

p× p (τ 2 +1)

U is a given positive diagonal matrix, then Y + R1ΣR2 + R2T ΣT R1T < 0 , If and only if there exists positive

E2 (τ sc (k )) = [0

0 I

0] ∈

p× p (τ1 +1)

constant β > 0 such that Y + β R1UR1T + β −1 R2T UR2 < 0 hold.

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7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009

Theorem 1:For given initial state z (0),τ sc (0),τ ca (0) , transition probability matrix Π1 = (π ij ) ∈ Rτ1 ×τ1 , Π 2 = (λmn ) ∈ Rτ 2 ×τ 2 , and the closed-loop faulty NCS in(10) is stochastically stable if and only if there exist P (i, m) > 0 , such that satisfy τ2

∑λ

mn

ΦT (i, m) P (i, n)Φ (i, m) − P (i, m) < 0

(12)

n=0

⎧ T = z T (t ) ⎨ E ∑ ΓT (k , t )Q(τ sc (k ),τ ca (k ))Γ(k , t ) | z (t ),τ sc (t ) = i, ⎩ k =t τ ca (t ) = m} z (t ) z T (t )Φ(T , t , i, m) z (t )

(15)

k

where Γ(k , t ) = ∏ Φ(τ sc (l ),τ ca (l )) , Q(τ sc (k ),τ ca (k )) > 0 . l =t

τ1

for all i ∈ S1 , m ∈ S2 , where P (i, n) = ∑ π ij P( j , n) . j =0

Proof: (Sufficient) Consider the following time-delay dependent Lyapunov function: V (k ) = z T (k ) P(τ sc (k ),τ ca (k )) z (k ) where P (τ sc (k ),τ ca (k )) > 0 is to be determined.

For any z (k ) ≠ 0 , we can see that Φ (T , t , i, m) is monotonically increasing as T increases. From(14), we know that Φ (T , t , i, m) is bounded above, thus for any z (t ) , we have lim Φ(T , t , i, m) = lim z T (t )Φ (T , t , i, m) z (t ) .Then we let T →∞

(13)

P (i, m)

E{Φ (T , t , i, m) − Φ (T , t + 1,τ sc (t + 1),τ ca (t + 1)) | z (t ),

τ1 ⎡ τ2 ⎤ = z T (k ) ⎢ ∑ λmn ΦT (i, m)∑ π ij P( j, n)Φ (i, m) − P(i, m) ⎥ z (k ) j =0 ⎣ n =0 ⎦ T z ( k )W (i, m) z ( k )

τ sc (t ) = i,τ ca (t ) = m} = zT (t )Q(i, m) z (t )

E{ΔV (k )} ≤ −λmin (−W (i, m)) || z (k ) ||2 ≤ −α || z (k ) ||2

Otherwise, by using(10) and (15), we have E{Φ (T , t + 1,τ sc (t + 1),τ ca (t + 1)) | z (t ),τ sc (t ) = i,τ ca (t ) = m} τ2

τ1

n=0

j =0

(18)

Thus, from (17) and (18), the following equation is holds.

where α = min λmin (−W (i, m)) . i∈S1 , m∈S2

τ2

τ1

n

j =0

z T (t )Φ(T , t , i, m) z (t ) − ∑ λmn ∑ π ij z T (t )ΦT (i, m)

Therefore, for any T > 0 , satisfy ⎧T ⎫ E{∑ ΔV (k )} ≤ −α E ⎨∑ || z (k ) ||2 ⎬ . k =0 ⎩ k =0 ⎭ T

(19)

T

Φ (T , t + 1,, j , n)Φ (i, m) z (t ) = z (t )Q(i, m) z (t )

For any z (t ) , let T → ∞ on each side of equation(19), and notice that (16) and Q(i, m) > 0 , we have

So we have ⎧T ⎫ E {V (T + 1) − V (0)} ≤ −α E ⎨∑ || z (k ) ||2 ⎬ . ⎩ k =0 ⎭ Then, we can obtained

τ2

τ1

n=0

j =0

P(i, m) − ∑ λmn ∑ π ij ΦT (i, m) P( j , n)Φ(i, m) = Q(i, m) > 0 Hence, the proof is complete.

⎧T ⎫ E ⎨∑ || z (k ) ||2 ⎬ ≤ α −1 z T (0) P(τ sc (0),τ ca (0)) z (0) ⎩ k =0 ⎭ = Ψ ( z (0),τ sc (0),τ ca (0))

Theorem 1 has given the necessary and sufficient condition on stochastic stability of the closed-loop faulty system in(10), but has not proposed the method to deign controller, the following theorem is given to deign the dynamic output faulttolerant controller(5)

⎧T ⎫ ⇒ lim E ⎨∑ || z (k ) ||2 ⎬ ≤ Ψ ( z (0),τ sc (0),τ ca (0)) T →∞ ⎩ k =0 ⎭

According to definition 1, we know system in (10) is stochastically stable. (Necessity) We assume that NCS in (10) is stochastically stable, then due to definition 1, we have T

(17)

= ∑ λmn ∑ π ij z T (t )ΦT (i, m)Φ (T , t + 1, j , n)Φ (i, m) z (t )

Thus, if (12) is holds, we have

lim ∑ || z (k ) ||2 ≤ Ψ ( z (0),τ sc (0),τ ca (0))

(16)

lim Φ (T , t , i, m)

T →∞

Notice that (15), we can obtain

From(10) and (13), we can get E{ΔV (k )} = E{V (k + 1) − V (k ) | τ sc (k ) = i,τ ca (k ) = m}

T →∞

T →∞

(14)

Theorem 2:For given initial state z (0),τ sc (0),τ ca (0) , transition probability matrix Π1 = (π ij ) ∈ Rτ1 ×τ1 , Π 2 = (λmn ) ∈ Rτ 2 ×τ 2 , and actuator faults matrix M ∈ Ξ , the closed-loop faulty NCS in(10) is stochastically stable If and only if there exist dynamic output fault-tolerant controller(5) with Ac , Bc , Cc , Dc ,and matrices P (i, m) > 0 , such that

k =0

* ⎡ − P (i, m) ⎢ 1/ 2 ˆ ϒ 0 (i, m) ⎢ λm 0 Φ (i, m) ⎢ ⎢ ˆ (i, m) λ 1/ 2 λ 1/ 2 Θ(i, m) ⎢ λm1/τ22 Φ mτ 2 m 0 ⎢ Σ2 0 ⎣⎢

Assume Q(τ sc (k ),τ ca (k )) > 0 , and let Φ (T , t , i, m) ⎧T ⎫ E ⎨∑ z T (k )Q(τ sc (k ),τ ca (k )) z (k ) | z (t ),τ sc (t ) = i,τ ca (t ) = m ⎬ ⎩ k =t ⎭

holds for all i ∈ S1 , m ∈ S2 .where 1052

*⎤ ⎥ * *⎥ ⎥ < 0 (20) ⎥ ϒτ 2 (i, m) * ⎥ ⎥ Σ1 ⎦⎥ 0 *

7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009

nonlinear minimization problem with LMI constraints. First, by introducing new variables X (i, s ) ( i ∈ S1 , s ∈ S 2 ), and let

ˆ (i, m) = Φ + Γˆ (m) + Γˆ (i ) + Γˆ (i, m) Φ 0 1 2 3 Γˆ 1 (m) = N1 BM 0 Cc E1 (m) , Γˆ 2 (i ) = N 2 Bc C [ E2 (i ) 0]

X (i, s ) = P −1 (i, s ) .According to the cone complementarity linearization method, the following iterative algorithm can be derived:

Γˆ 3 (i, m) = N1 BM 0 Dc CE3 (i, m) 1/ 2 1/ 2 ϒ s (i, m) = − P −1 (i, s ) + λms λms Θ(i, m)( s = 0,1, τ 2 )

Θ(i, m) = β m N1 BM 0 J ( N1 BM 0 )T +χ i , m N 2 BM 0 J ( N 2 BM 0 )T

Step1: Find a feasible solution satisfying LMI (20)＊, which is obtained by substituting P −1 (i, s) in (20) with X (i, s ) , set

Σ1 = diag{− β m I , − χ i , m I } T Σ 2 = ⎡⎢( GCc E1 (m) ) ⎣

( GDc CE3 (i, m) ) ⎤⎦⎥ T

T

as

T

E3 (i, m) = [ E3 (i, m) 0] , G = diag{ j1 , j2 ,

k =0

,

j =0

Step 2: Solve the following LMI optimization problem for variables ( P (i, m), X (i, s ), Ac , Bc , Cc , Dc , β m , χ i , m ) :

jq }

Proof: By Schur complement, inequality(12) is equivalent to ⎡ − P (i, m) * ⎤ <0 (21) ⎢ Σ Σ3 ⎥⎦ 4 ⎣ where , − P −1 (i,τ 2 )}

λmτ Φ(i, m)T ⎤⎦

let

P 0 (i, m) = ∑ π ij P 0 ( j , m) ;

N1 = ⎡⎣ N1 0 ⎤⎦ , N 2 = ⎡⎣ 0 N 2 ⎤⎦ , E1 (m) = [ 0 E1 (m) ] ,

Σ 4 = ⎡⎣ λm 0 Φ(i, m)T

and

τ1

T

Σ3 = diag{− P −1 (i, 0),

( P 0 (i, m), X 0 (i, s), Ac0 , Bc0 , Cc0 , Dc0 )

T

⎛ τ 2 τ1 ⎞ Minimize Trace ⎜ ∑∑ ( P k (i, s ) X (i, s ) + X k (i, s ) P (i, s )) ⎟ , ⎝ s = 0 i =1 ⎠ ⎡ P (i, s ) subject to LMI (20)＊, and. ⎢ ⎣ I s ∈ S 2 ),

⎤ ⎥ > 0 ( i ∈ S1 , X (i, s ) ⎦ I

Set P k +1 (i, s) = P(i, s) , X k +1 (i, s ) = X (i, s ) , Ack +1 = Ac ,

2

τ1

Considering the term of actuator fault matrix M in Φ (i, m) ,

j =0

and fault model(4), we substitute(4) into (21), have ⎡ − P (i, m) * ⎤ T T T T T T ⎢ ˆ ⎥ + R1 LR2 + R2 L R1 + R3 LR4 + R4 L R3 < 0 (22) Σ Σ 4 3⎦ ⎣ where

ˆ (i, m)T Σˆ 4 = ⎡⎣ λm 0 Φ

ˆ (i, m)T ⎤ λmτ Φ ⎦

Bck +1 = Bc , Cck +1 = Cc , Dck +1 = Dc , P k (i, s) = ∑ π ij P k ( j , s) ;

Step 3: If LMI (20) is satisfied, then exit. If LMI is (20) not satisfied and k < nt (max number of iteration), let k = k + 1 , then return to step 2. 4. NUMERICAL EXAMPLE

T

By using lemma 1, inequality (22) is holds for every L ≤ J ,

In this section, we validate the effectiveness of the proposed method by using the numerical example. We consider the cart and invert pendulum problem (Lin, Hassibi, and How. (2000); Zhang, Shi, Chen, and Huang (2005)), and the pendulum is modeled as model in(1), where ⎡1.0000 0.1000 -0.0166 -0.0005 ⎤ ⎢ 0 1.0000 -0.3374 -0.0166 ⎥⎥ A=⎢ ⎢ 0 0 1.0996 0.1033 ⎥ ⎢ ⎥ 0 2.0247 1.0996 ⎦ ⎣ 0

if and only if there exist scalar β m > 0 , χ i , m > 0 , such that

B = [ 0.0045 0.0896 −0.0068 −0.1377 ]

R1 = ⎡⎣ 0

2

T

λm 0 ( N1 BM 0 )

R2 = ⎡⎣Cc E1 (m) 0

λmτ ( N1 BM 0 ) ⎤⎦ T

T

2

0 ⎤⎦

R3 = ⎡⎣ 0 λm 0 ( N 2 BM 0 )T R4 = ⎡⎣ Dc CE3 (i, m) 0 0 ⎤⎦

λmτ ( N 2 BM 0 )T ⎤⎦

T

2

satisfy ⎡ − P (i, m) * ⎤ T −1 T T ⎢ ˆ ⎥ + β m R1 JR1 + β m R2 J R2 Σ Σ 4 3⎦ ⎣ + χ i , m R3 JR3T + χ i−, m1 R4T J T R4 < 0

(23)

By using Schur complement again, we know that inequality(20) is equivalent to(23). Thus, from (20), we can know inequality(12) holds which means the closed-loop faulty NCS in(10) is stochastically stable. The proof is complete. In fact, constraint condition in(20) is non-convex, we can not solve the inequality by Matlab tool directly. However, by using the cone complementarity linearization method (Ghaoui, Oustry, Aitrami, 1997), we can change it to

T

⎡1 0 0 0 ⎤ C=⎢ ⎥ ⎣0 0 1 0⎦

The actuator fault matrix M L = diag{0.5} , M U = diag{1.2} , with the simple calculation, we have M 0 = diag{0.85} , J = diag{0.412} .Then the random time-delays τ sc (k ) and τ ca (k ) are bounded as τ sc (k ) ∈ {0,1, 2} and τ ca (k ) ∈ {0,1} respectively, and their transition probability matrices are ⎡ 0.7 0.3 0 ⎤ ⎡ 0.5 0.5⎤ . given by Π1 = ⎢⎢ 0.6 0.3 0.1⎥⎥ , Π 2 = ⎢ 0.7 0.3⎥⎦ ⎣ ⎣⎢ 0.4 0.5 0.1⎦⎥ Then, by theorem 2 and iterative algorithm, we can get the dynamic output feedback as following:

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7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009

⎡ 0.0172 ⎢ 0.0259 Ac = ⎢ ⎢ 0.1149 ⎢ ⎣ -0.0098

0.1429 -0.2680 1.3415⎤ -1.0098 0.0219 0.0828⎥⎥ 0.2120 2.3787 0.6101 ⎥ ⎥ 2.0857 0.1572 -0.1882 ⎦

⎡ -0.1144 ⎢ 0.4503 Bc = ⎢ ⎢ 0.1766 ⎢ ⎣ 0.0561

-0.2100 ⎤ 1.1261 ⎥⎥ 2.1075 ⎥ ⎥ 0.0960 ⎦ Cc = [ -2.8023 6.0562 -2.778

1.3462]

Dc = [ -2.4502 15.9993]

We take the initial state x(0) = x(−1) = x(−2) = x(−3) = [0.1 0 0 0]T , the state trajectories of the system(1) are shown in Fig.2. From the simulation results, we know that NCS with random delays and actuator fault is stochastically stable. 0.3

x1 x2

0.2

x3

state

0.1

x4

0 -0.1 -0.2

0

100

200

300

400

500

k

Fig 2 System state trajectories 5. CONCLUSION In this paper, stochastic stability analysis and controller design method for the NCS with random delays and actuator fault have been given. Based on a more practical and general model of actuator continuous gain faults, the NCS which is modeled as Markovian jump linear system is stabilized by a dynamic output feedback controller. The necessary and sufficient condition on the stochastic stability of NCS is proposed in the form of nonlinear matrix inequality which is solved by using the cone complementary algorithm. Simulation results illustrate the feasibility and effectiveness of our methods. ACKNOWLEDGEMENTS This work was supported by National Natural Science Foundation of China (Grant No. 60574082 60804027) REFERENCES Hespanha, J. P., Payam. N., Xu, Y.G.(2007). A survey of recent results in networked control systems. Proceedings of the IEEE, 95(1): 138-162. Walsh, G. C., Ye, H.(2001). Scheduling of networked control systems. IEEE Control Systems Magazine, 21(1), 57–65.

Zhang, W., Branicky, M. S., Phillips, S. M. (2001) Stability of networked control systems, IEEE Control System Magazine, 21(1): 84–99. Patton, R. J. (1997). Fault tolerant control: the 1997 situation (survey). In Proceedings of the IFAC Symposium: SAFEPROCESS’’97, Hull, UK, 2:1033–1055. Zhou, D.H,. Frank, P.M.(1998). Fault diagnostics and fault tolerant control, IEEE Aerospace and Electronic Systems, 34(2): 420–427. Azimi-Sadjadi, B.(2003). stability of networked control systems in the presence of packet losses. In Proceedings of the conference on decision and control, Hawaii, USA, pp. 676-681. Brockett, R.W., Liberzon, D.(2000). Quantized feedback stabilization of linear systems, IEEE Transactions on Automatic Control, 45(7): 1279-1289. Elia, N., Mitter, S. K.(2001). Stabilization of linear systems with limited information. IEEE Transactions on Automatic Control, 46(9): 1384–1400. Nilsson, J., Bernhardsson, B., Wittenmark, B.(1998). Stochastic analysis and control of real-time systems with random time delays, Automatica, 34(1): 57–64. Hu, S., Zhu, Q.(2003). Stochastic optimal control and analysis of stability of networked control systems with long delay, Automatica, 39(11):1877–1884. Lin, H., Zhai, G., and Antsaklis, P. J.(2003). Robust stability and disturbance attenuation analysis of a class of networked control systems. In Proc 42nd IEEE Conf. Decision and Control, Maui, HI, pp. 1182–1187. Xie. D., Chen. X., Lv, L., Xu. N. (2008). Asymptotical stabilizability of networked control systems: time-delay switched system approach. IET Control Theory and Application, 2(9): 743–751. Lin, X., Hassibi, A., How, J. P.(2000).Control with random communication delays via a discrete-time jump linear system approach. In Proc. America Control Conference, 3:2199–2204. Zhang, L., Shi, Y., Chen, T. and Huang, B.(2005) A New Method for Stabilization of Networked Control Systems with Random Delays, IEEE Transactions on Automatic Control, 50(8): 1177-1181. Ghaoui, L. E., Oustry, F., Aitrami, M.(1997). A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Transactions on Automatic control,42(8): 1171 –1176. Li, S. B., Sauter, D., Aubrun, C., et al.(2007) Stability Guaranteed Active Fault Tolerant Control of Networked Control Systems. European Control Conference, Greece :EUCA. Kong, D., Fang, H. (2005).Stable Fault-tolerance Control for a Class of Networked Control Systems. ACTA Automatica Sinica, 31(2):267-273. Zhang D, Wang Z, Hu S.(2007) Robust Satisfactory Faulttolerant Control of Uncertain Linear Discrete-time Systems: an LMI Approach . International Journal of Systems Science, 38(2): 151-165.

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