Guaranteed cost control for discrete-time networked control systems

Journal of Systems Engineering and Electronics Vol. 22, No. 4, August 2011, pp.661–671 Available online at www.jseepub.com

Guaranteed cost control for discrete-time networked control systems with random Markov delays Li Qiu1,2 , Bugong Xu1 , and Shanbin Li1,∗ 1. School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, P. R. China; 2. College of Mechatroincs and Control Engineering, Shenzhen University, Shenzhen 518060, P. R. China

Abstract: The guaranteed cost control for a class of uncertain discrete-time networked control systems with random delays is addressed. The sensor-to-controller (S-C) and contraller-to-actuator (C-A) random network-induced delays are modeled as two Markov chains. The focus is on the design of a two-mode-dependent guaranteed cost controller, which depends on both the current S-C delay and the most recently available C-A delay. The resulting closed-loop systems are special jump linear systems. Sufficient conditions for existence of guaranteed cost controller and an upper bound of cost function are established based on stochastic Lyapunov-Krasovakii functions and linear matrix inequality (LMI) approach. A simulation example illustrates the effectiveness of the proposed method.

Keywords: networked control systems (NCSs), guaranteed cost control, random Markov delays, linear matrix inequality (LMI). DOI: 10.3969/j.issn.1004-4132.2011.04.016

1. Introduction Networked control systems (NCSs) are feedback control loops closed via a real time network [1,2]. In NCSs, communication networks are employed to transmit information and control signals (reference input, plant output, control input, etc.) among control system components (sensors, controller, actuators, etc.) [2]. The use of the communication networks brings many advantages such as low cost, reduced weight, simple installation and maintenance, as well as high efficiency, flexibility and reliability [2−4]. However, the communication networks in control loops make the analysis and design of NCSs complicated. One main issue is the network-induced delays (sensor-to-controller (S-C) and contraller-to-actuator (C-A) delays), which Manuscript received January 8, 2010. *Corresponding author. This work was supported by the NSFC-Guangdong Joint Foundation Key Project (U0735003), the Overseas Cooperation Foundation (60828006), the Scientific Research Foundation for Returned Overseas Chinese Scholars, State Education Ministry, the Fundamental Research Funds for the Central Universities (2009ZM0076) and the Natural Science Foundation of Guangdong Province (06105413).

occur when sensor, controllers, and actuators exchange data across the network [5]. Great efforts have been made on stability analysis, controller design for stabilization and performance of NCSs with time delays, as presented in [5−13]. The objective of guaranteed cost control is to design a controller not only robustly stabilizing an uncertain system, but also guaranteeing an adequate level of performance for any admissible value of uncertainty. Significant progress has been made in the study of guaranteed cost control for uncertain systems in [14−17] and NCSs with time-delay in [18−25]. In [18,19], sufficient conditions of guaranteed cost control law for NCSs with random communication time-delay are obtained by Lyapunov stability theory and linear matrix inequality (LMI) method. In [20,21], the guaranteed cost state feedback controller is designed for NCSs with interval time-varying delay by Lyapunov functions method. In [22,23], NCSs with larger time delays are stabilized respectively by Lyapunov stability theory and slack matrix variables approach. In [24,25], the guaranteed cost control for a class of NCSs with uncertain time delays is investigated based on Lyapunov stability theory and LMI method. However, in the aforementioned works, the developed NCSs are with interval time-varying delay, uncertain time delay, large time delay, or random time delay [18−25]. In order to reduce the conservativeness of the stabilization conditions of NCSs, it is desirable to incorporate both the S-C delay and the C-A delay into the design [6]. In this paper, two independent Markov chains dk and τk−1 are introduced to describe the current S-C and previous C-A random time delays. However, practically the previous CA delay τk−1 is not always available because the information about previous C-A delay τk−1 needs to be transmitted through the S-C communication link before reaching the controller, as shown in Fig. 1. To the best of the authors’ knowledge, involving two network-induced delay modes to design the guaranteed cost controller that simultane-

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Journal of Systems Engineering and Electronics Vol. 22, No. 4, August 2011

ously depends on both dk and τk−dk −1 has not been fully investigated, which motivates the present study. When considering both dk and τk−dk −1 , the resulting closed-loop systems can be modeled as special jump linear systems. The two-mode-dependent state feedback problem for such systems is formulated as a convex optimization over a set of linear matrix inequalities. Compared with some existing controller design methods, the controller designed by our method can not only robustly stabilize the uncertain systems but also guarantee an adequate level of performance for any admissible value of the uncertainty. Notation In the sequel, if not explicit, matrices are assumed to have appropriate dimensions. The notation A > () 0 denotes a symmetric positive definite (positive semidefinite) matrix. diag{A1 , . . . , An } refers to an n × n diagonal matrix with Ai as its ith diagonal entry. I and 0 denote identity matrix and zero matrix. The symbol ⊗ denotes the Kronecker product. sym{A} denotes the expression A + AT . In a matrix, ∗ denotes blocks that are readily inferred by symmetry of a symmetric matrix.

One way to model the delays τk and dk is using the finite state Markov chain as in [5,6,12,13]. The main advantages of the Markov model are: the dependencies between two delays are taken into account since in real networks and the current time delays are usually related with the previous delays [12]; and the packet dropout could be included naturally [13]. In this paper τk and dk are modeled as two homogeneous Markov chains that take ˜ = {0, 1, . . . , N }, ˜ = {0, 1, . . . , M } and N values in M ˜ ˜ , dmin = min N ˜, and τmin = min M , τmax = max M ˜ dmax = max N . π = {πij } and λ = {λmn } denote transition probability matrices of τk and dk respectively, with probabilities πij and λmn , which are defined by

2. Problem description

˜ and m, n ∈ N ˜. ∀i, j ∈ M It is very important that the delays information for the controller design in NCSs. The previous C-A delay (τk−1 ) is not available because the information about C-A delay needs to be transmitted through the S-C communication link before reaching the controller, as shown in Fig. 1. Consequently, the controller gain is designed depending on both dk and τk−dk −1 .

Consider the NCSs shown in Fig. 1.



πij = Pr{τk+1 = j|τk = i} λmn = Pr{dk+1 = n|dk = m}

where πij  0, λmn  0 and

M  j=0

πij = 1,

N 

(5)

λmn = 1,

n=0

uk = u ¯k−τk = Kdk ,τk−dk −1 xk−dk −τk

(6)

xk+1 = (A + ΔA)xk + (B + ΔB)uk

(1)

[ΔA ΔB] = DΔk [E1 E2 ]

(2)

ΔT k Δk  I

(3)

where Kdk ,τk−dk −1 is the state feedback controller gain. Clearly, the controller (6) is two-mode-dependent. By combining (1) and (6), the closed-loop system can be written as ⎧ ˆ ˆ ⎪ ⎨ xk+1 = Axk + BKdk ,τk−dk −1 xk−dk −τk (7) xk = φk , k = −dmax − τmax , ⎪ ⎩ −dmax − τmax + 1, . . . , 0

where xk ∈ R is the state and uk ∈ R is the control input. A, B, D, E1 and E2 are known constant matrices with appropriate dimensions. Δk is an uncertain time-varying matrix representing norm bounded parameter uncertainty. Random delays exist in S-C and C-A, as shown in Fig. 1. Here τk  0 is the random time delay from C-A, dk  0 is the random time delay from S-C, and the controller is to be designed. Here, it is assumed that both τk and dk are bounded, which are as follows:  0  τmin  τk  τmax . (4) 0  dmin  dk  dmax

ˆ = B + ΔB. where Aˆ = A + ΔA, B Remark 1 By applying the proposed two-modedependent controller in (6), the resulting closed-loop system in (7) can not be transformed into a standard Markov jump linear system, because the closed-loop system depends on τk , dk and τk−dk −1 . In addition, τk−dk −1 is related with both dk and τk . The delay model in this paper is similar to [6], which models the system with random delays as a special discrete-time Markov jump linear system based on extended space method. However, the result proposed therein can not be directly applied to the networked guaranteed cost control for system (7).

Fig. 1

Diagram of a networked control system

The uncertain discrete-time plant model is

where

n

m

Li Qiu et al.: Guaranteed cost control for discrete-time networked control systems with random Markov delays

Remark 2 The closed-loop system (7) is a class of special jump linear system governed by two Markov chains representing the random networked delays and the inherent uncertainties of the open-loop plant. Given positive definite symmetric matrices R1 and R2 , we consider the cost function: J=

∞ 

T E[xT k R1 xk + uk R2 uk |φ(·)].

(8)

k=0

Associated with the system (7), the cost function is as follows: J=

∞ 

E[XkT RXk |φ(·), d0 , τ−d0 −1 , τ0 ]

(9)

k=0

663

where Υ (x0 , φi , −dmax − τmax  i  0, d0 , τ−d0 −1 , τ0 ) is a nonnegative function of the system initial values satisfying Υ (0, 0, . . . , 0) = 0. Special jump linear system (7) is robust stochastically stable, if system (7) is stochastically stable for all admissible uncertainties ΔT k Δk  I. Definition 2 [26] Considering the uncertain system (7) and cost function (9), if there exist a control law uk and a positive scalar J ∗ such that for all admissible uncertainties, the closed-loop system is stable and the value of cost function (9) satisfies J  J ∗ , then J ∗ is a guaranteed cost and uk is a guaranteed cost controller for the uncertain system (7).

3. Robust performance

where R = diag{R1 , KdTk ,τk−d XkT =



xT k

k −1

R2 Kdk ,τk−dk −1 },

xT k−dk −τk



.

The objective of this paper is to consider the robust stabilization controller as (6) which achieves small value J ∗ as possible for the special jump linear system with uncertain parameters and random-delays. Associated with the cost function (9), the guaranteed cost controller for closedloop uncertain system (7) is defined as follows: Definition 1 [26] The system in (7) without uncertainties is stochastically stable if there exists a constant C such that ∞  E[||xk ||2 |x0 , d0 , τ−d0 −1 , τ0 ]  k=0

CΥ (x0 , φi , −dmax −τmax  i  0, d0 , τ−d0 −1 , τ0 ) (10) ⎡

Π dk+1

π0j2 πj2 j3 · · · πjdk+1 0 ⎢π π M M M    ⎢ 1j2 j2 j3 · · · πjdk+1 0 ⎢ = ··· .. ⎢ . jdk+1 =0 j3 =0 j2 =0 ⎣ πMj2 πj2 j3 · · · πjdk+1 0

For the system described by (7), the analysis of robust performance is based on Lyapunov functions. For the ease ˜ and of presentation, when the system is in mode m ∈ N ˜ (i.e., dk = m, τk−d −1 = i), Kd ,τ i ∈ M is k k k−dk −1 denoted as Km,i . The controller gains are related to the Markov mode delay τk−dk −1 , for which the multi-step jump of Markov chain is involved in the system (7). It is the transition probability matrix for the multi-step delay mode jump that is important for controller design. We give the lemma about the transition probability matrix for the multi-step jump mode as follows: Lemma 1 [6] If the transition probability matrix from τk−1 to τk is Π , the transition probability matrix from τk−dk+1 to τk is Π dk+1 , which is still a transition probability matrix of the Markov chain. Specially, when dk+1 = 0, the transition probability matrix is Π dk+1 = Π 0 = I, where

π0j2 πj2 j3 · · · πjdk+1 1 π1j2 πj2 j3 · · · πjdk+1 1 .. . πMj2 πj2 j3 · · · πjdk+1 1

⎤ · · · π0j2 πj2 j3 · · · πjdk+1 M · · · π1j2 πj2 j3 · · · πjdk+1 M ⎥ ⎥ ⎥ . (11) .. .. ⎥ . ⎦ . · · · πMj2 πj2 j3 · · · πjdk+1 M

Lemma 2 [26] For any vectors a, b ∈ Rn , positive matrix Z ∈ Rnz,nz , the following inequality holds

holds, if and only if there exists a scalar ε > 0 such that

2aT b  aT Za + bT Z −1 b.

Y + εHH T + ε−1 E T E T < 0.

Lemma 3 [2,26] Given Y = Y T , H and E are real matrices with appropriate dimensions, all F satisfying F T F T  I, the matrix inequality

Theorem 1 For system (7), cost function (9) and random but bounded scalar

Y + HF E + E T F T H T < 0

dk ∈ [dmin , dmax ]

and τk ∈ [τmin , τmax ] ,

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˜, i ∈ M ˜ , there exist a scalar ε > 0 if for each mode m ∈ N and matrices Pm,i > 0, Q1 > 0, Q2 > 0, Z1 > 0, Z2 > 0, ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

˜ , ∀i ∈ M ˜ satisUs , Ms , Ws (s = 1, 2, 3), Km,i , ∀m ∈ N fying the following matrix inequality:

−1 −P¯m,i + εDDT

√ ε α1 DDT

√ ε α1 DDT

∗

−Z1−1 + εα1 DDT

εα1 DDT

∗

∗

−Z2−1 + εα1 DDT

∗

∗

∗

where

Ω1

⎤

⎥ Ω2 ⎥ ⎥ 0. From (26), it is seen that for any t > 0

R1

⎢ ⎣ 0 0 k−1 

+

Mα2 Z1−1 MT 0 T R2 Km,i Km,i

0

+

E[V (k + 1)] − E[V (0)]  −β

W α1 Z2−1 W T −

⎤⎫ 0 ⎪ ⎬ ⎥ 0 ⎦ ξk − ⎪ ⎭ 0

t 

∞ 

E[xT k xk ] 

k=0

(ξkT W + ylT Z2 )Z2−1 (ξkT W + ylT Z2 )T .

l=k−dmax −τmax

(24) Then, since both Z1 > 0 and Z2 > 0, the last three terms are nonpositive in (24), so we have E[ΔV ]  ⎧⎡ ⎤ AˆT ⎪ ⎨  ⎢ ˆ T⎥ ¯ ˆ ˆ ξ T ⎣ (BK m,i ) ⎦ Pm,i A BKm,i 0 + ⎪ ⎩ 0 ⎤ ⎡ Λ12 Λ13 R1 + Λ11 ⎥ ⎢ T Km,i R2 Km,i + Λ22 Λ23 ⎦ + ⎣ ΛT 12 ΛT 23

J=

W2T

W3T

(29)

∞ 

E[XkT RXk |φ(0), d0 , τ−d0 −1 , τ0 ]  V (0). (30)

Then, (13) is yielded from (30). According to Definition 2, the result of Theorem 1 is true.  Remark 3 The upper bound (13) depends on the initial condition of system (7). In order to remove the dependence on the initial condition, the initial state of system (7) is supposed to be arbitrary but belongs to the set Z = {xl ∈ Rn : xl = T zl , zlT zl  1,

where T is a given matrix. Then according to Lemma 2, the cost bound (13) leads to

T

Λ11 , Λ12 , Λ13 , Λ22 , Λ23 , Λ33 are defined in Theorem 1. By Schur complement and Lemma 3, (12) guarantees Γ < 0. Therefore, E[ΔV ]  −λmin (−Γ )ξkT ξk  −βxT k xk

E[XlT RXl ]

l = −dmax − τmax , −dmax − τmax + 1, . . . , 0}

T

T MT 3

k−1 

k=0

U α1 Z1−1 U T + Mα2 Z1−1 MT + W α1 Z2−1 W T − ⎡ ⎤⎫ R1 0 0 ⎪ ⎬ ⎢ ⎥ T T R K 0 0 K (25) ⎣ ⎦ ξk = ξK Γ ξk m,i 2 m,i ⎪ ⎭ 0 0 0

MT 2

(28)

when k → ∞, (29) yields

0

U3T

E[V (k + 1) − V (k)]  −J.

l=0

⎤ ⎤T ⎡ (Aˆ − I)T (Aˆ − I)T ⎥ ⎢ ˆ ⎢ ˆ T⎥ ⎣ (BKm,i )T ⎦ α1 (Z1 + Z2 ) ⎣ (BK m,i ) ⎦ +

U2T

According to Definition 1, the closed-loop system in (7) is robust stochastically stable for all admissible uncertainties. On the other hand, combining (25) and Γ < 0, it yields

E[V (k)|φ(0)]  V (0) −

⎡

 U = U1T  M = MT 1  T W = W1

1 E[V (0)] < ∞. β

From (28), it yields

Λ33

0

1 E[V (0)] β

when t → ∞, then

l=k−dmax −τmax

where

E[xT k xk ] 

k=0

(ξkT M + ylT Z1 )Z1−1 (ξkT M + ylT Z1 )T −

ΛT 13

(27)

Furthermore,

l=k−dk −τk

k−1 

E[xT k xk ].

k=0

(ξkT U + ylT Z1 )Z1−1 (ξkT U + ylT Z1 )T −

k−d k −τk −1

t 

(26)

J  λmax (T T Pd0 ,τd0 −1 T ) + α1 λmax (T T Q2 T ) +   α2 (2α1 − α2 − 1) λmax (T T Q1 T ) + d0 + τ0 + 2 2α1 λmax (T T (Z1 + Z2 )T ) where α1 , α2 are defined in Theorem 1.

(31)

Li Qiu et al.: Guaranteed cost control for discrete-time networked control systems with random Markov delays

4. Synthesis of controller design

where

In terms of the feasible solutions to a set of LMIs, a parameterized formulation of the guaranteed cost control law is presented in this section. Theorem 2 For system (7), cost function (9) and random but bounded scalar dk ∈ [dmin , dmax ] and τk ∈ ˜, i ∈ M ˜ , suppose [τmin , τmax ], if for each mode m ∈ N that for tuning parameters δ1 > 0, δ2 > 0 there exist a ˆ 2 > 0, Zˆ1 > 0, ˆ 1 > 0, Q scalar ε > 0 and matrices Q ˆ s, ˆ ˆ ˆs , M Z2 > 0, X > 0, Ym,i , Pm,i > 0, X(m, i) > 0, U ˆ s (s = 1, 2, 3), ∀m ∈ N ˜ , ∀i ∈ M ˜ satisfying the LMIs: W ⎡

Θ11

⎢ ∗ ⎢ ⎢ ⎣ ∗ ∗

Θ12

Θ13

Θ22

Θ23

∗

Θ33

∗

∗

⎡

Θ44

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

−R1−1

Θ14

667

Θ11 = −X(m, i) + L(m, i)L(m, i)T ⊗ εDDT √ Θ12 = Θ13 = L(m, i) ⊗ ε α1 DDT Θ14 =



0 L(m, i) ⊗ AX

0

L(m, i) ⊗ BYm,i

Θ22 =

−δ12 Zˆ1

+ εα1 DD

T

Θ23 = εα1 DDT Θ33 = −δ22 Zˆ2 + εα1 DDT

Θ24 ⎥ ⎥ ⎥ 0 are known tuning parameters. Pre- and post-multiplying S T and S to (34) respectively, and the notations are as follows: ˆ = X T U X, U

ˆ = X T MX M

ˆ = X T W X, W

Zˆ1 = X T Z1 X

Zˆ2 = X T Z2 X,

ˆ 1 = X T Q1 X Q

ˆ 2 = X T Q2 X, Q

Pˆm,i = X T Pm,i X

Ym,i = Km,i X.

4

√  α1 Z2 (BKm,i + DΔk E2 Km,i ) 0 . . . 0

R1 + Λ11

6

4

√ Ξ2 = α1 Z1 (A − I + DΔk E1 )

 Ξ1 = χ(m, i) ⊗ (A + DΔk E1 ) ⎡



⎤ √ α1 W1 √ ⎥ α1 W2 ⎥ ⎥ √ α1 W3 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ −Z2

We have Theorem 2 by Schur complement and Lemma 3.  Some transformations are needed in order to formulate the main results of Theorem 2 into convex optimization problem. From (33), we establish the following inequalities: ! TT −γ1 I