20th Iranian Conference on Electrical Engineering, (ICEE2012), May 15-17,2012, Tehran, Iran

Robust Output Feedback Control of Networked Control Systems with Random Delay Modeled by Markov Chain

Masoumeh Azadegan *, Mohammad T. H. Beheshti **, and Babak Tavassoli *** * Tarbiat Modares University, azadegan@modares.ac.ir

** Tarbiat Modares University, m.behesht@modares.ac.ir *** K. N. Toosi University of Technology, tavassoli@kntu.ac.ir

Ahstract- This paper investigates the problem of robust

stabilization for uncertain networked control systems (NCSs) with random time delay via the output feedback control. A mode dependent controller is proposed by modeling the random delay as a Markov Chain. The resulting closed-loop system is expressed as a Markovianjump linear system (MJLS) with mode-dependent delay. Based on Lyapunov-Krasovskii method, robust stability condition and controller design method for such networked control systems with structured uncertainties are presented. The result is

formulated in terms of LMls. Output feedback gain can then be derived by using the feasible solution of LM Is. Simulation example demonstrates the feasibility and effectiveness of the proposed

approach.

Keywords: Networked Control System, Markov Chain, Lyapunov-Krasovskii, Robust Output feedback.

1. Introduction

Networked control systems (NCSs) are a type of distributed control systems where sensors, actuators, and controllers are interconnected through a communication network. This system setup has the advantages of low cost, flexibility, and less wiring. Such requirements are demanding in remote control systems [I]. Despite the advantages, there are some challenging problems with NCSs that need to be properly addressed to ensure the stability and performance of the closed-loop systems. One main issue is the network-induced delay, including sensor-to-controller and controller-to-actuator delay, which will degrade the system performance as well as stability. This delay depending on the network characteristics such as network load, topologies, routing schemes, etc. can be constant, time varying, or even random.

Many researchers have studied stability and controller design of NCS in the presence of network-induced delays. Stability analysis is one of the most concerned areas, therefore much effort has been devoted to this problem; for example, see [2-5]. Emphasis is mainly on the modeling of

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network-induced delays. Different control methods have been presented for NCS including sampled-data system/hybrid approach [6, 7], switched system approach [8, 9], sampling time scheduling approach [10, II], augmented deterministic discrete-time model approach [12, 13], perturbation approach [14], moving horizon approach [IS]. Since in random access networks such as Ethernet and Internet, network-induced delays are random processes [16], therefore the stochastic approaches are needed to model the behavior of the delays and this approach is more realistic to the nature of network delays. Therefore, we focus on the researches where the network delays are considered as a stochastic process. In [17], a Markov process is utilized to model these network delays. This definition using Markov property is realistic in industry network applications as network traffic and network load conditions are rather of random nature, either in spatial or temporal sense. Reference [18] modeled the random delays as Markov chains such that the closed-loop system is a jump linear system with one mode. But the state-feedback gain is mode-independent. In [19], a mode-independent state feedback controller is designed for NCSs subject to Markovian packet loss. [20] proposed the state feedback controller that only depends on the delay from sensor-to-controller. In [21], the delay is modeled as a Markov process and the effect of random delay is treated as an LQG problem. However, the network­induced random delay has to be less than one sampling interval. In [22] two Markov processes are used to model network-induced delays and a state feedback controller is designed for uncertain linear NCS and [23] presented a robust dynamic output feedback controller with the same modeling as [22]. It is noticed that both mentioned papers are based on the Lyapunov-Razumikhin method and their results are given in terms of the solvability of BMIs and an iterative algorithm is proposed for solving the BMI difficulties. In [24] the stabilization problem for networked control systems in the discrete-time domain with random delays is studied.

Generally speaking, the full state of the plant cannot be

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directly measurable. Therefore it is more significant to design the output feedback controller for the NCSs (see [25] and [26]). It is noticed that [26] assumed that there exist network only between sensors and controllers, whereas [25] assumed that network-induced delays are less than one sampling period. [27] designed output feedback controller for NCSs with discrete-time plant with two random delays modeled as two different Markov chains. [28] designed robust static output feedback controller for NCSs with random time delay and continuous-time plant, but it assumed that the controller to actuator transmission delay is constant.

To the best of the authors' knowledge, there are few literatures to investigate the robust static output-feedback stabilization problem of NCSs with random network­induced delays. However, in practical applications, the network-induced delays are mostly random because of using random access networks. Moreover, some uncertainties in the model of NCSs are inevitable due to environmental noise. Therefore the robust output-feedback stabilization problem of this kind ofNCSs is necessary in practice.

In this paper, we consider the robust stabilization problem ofNCS in the continuous-time domain. The overall random delay (sensor-to-controller and controller-to-actuator) is modeled as a Markov process. The resulting closed-loop system is expressed as a Markovian jump linear system with one mode-dependent delay. Based on the Lyapunov­Krasovskii method, a new methodology for designing a mode-dependent robust output feedback controller that stabilizes this class of systems is proposed. The sufficient condition on the existence of stabilizing controller is given in terms of LMT.

The rest of the paper is organized as follows: In section 2, problem formulation and preliminaries are introduced. Section 3 establishes sufficient stability condition for uncertain NCS in terms of LMT. Based on the stability condition, a mode-dependent controller design algorithm is proposed. An illustrative example is given in section 4. Finally, the conclusions are provided in section 5.

Notation: If a matrix is invertible, the superscript '-1' represents the matrix inverse. X> 0 means that X is a real symmetric and positive-definite matrix.

2. Problem Formulation and Preliminaries

Consider an uncertain linear system as the plant:

x(t) = (A + llA)x(t) + (B + llB)ft(t)

yet) = Cx(t) (1)

where x(t) E Rn is the state, ft(t) E Rm is the control input and yet) E RP is the measured output; A, Band C are known real matrices with appropriate dimensions. Matrices llA and

llB characterize the uncertainties in the system and satisfY the following assumption:

(2)

where DA,DB,EA and EB are known real constant matrices

588

of appropriate dimensions, and FA (t) and FB(t) are unknown matrix functions with Lebesgue-measurable

elements and satisfy F(t)TF(t) :::; I, in which I is the identity matrix of appropriate dimension.

The plant is interconnected by a controller over communication network, see Fig. 1. We use a homogenous stationary Markov chain {ret)} to model the overall network-induced delays (d(t) = rse(t) + reaCt)). ret) is a continuous-time discrete-state Markov process taking values in a finite set S := {1, 2, . . . , N} with a transition probability matrix given by

Pr{r(t + ll) = jlr(t) = i} = {fllijll + � (ll) ( II

) � 1=! (3)

+ fliiLJ. + 0 LJ. t = }

where flij ;::: 0, i 1= j, flii = -I.f=l,j*i flij' In each mode in

the Markov process, the corresponding delay is assumed to

be time-varying which satisfies 0:::; dO, t) :::; hi' diet) :::; fli' Ii = maxiES{h;}.

The stucture of NCS is depicted in Fig. 1. By defining

rse(t) and reaCt) as (3), Fig 1. can be reduced to the block diagram shown in Fig. 2 [29]. This system configuration will be investigated in this paper.

r(t) E [mink{ r�}, (nS + l)hS + maxk{ r�+ns+ l})' Vk E N

p(t) E [minl{ rn , (na + l)ha + maxl{ rf+na+1 }) ' VI E N (4)

where nS and na are the number of consecutive dropouts in sending sampled data through networks.

According to (1) and Fig. 2, the measurment from sensor to controller is given by

yCt) = yet - rse(t)) = Cx(t - rse(t)) (5)

and the output feedback control law has the form:

ft(t) = u(t - reaCt)) = F(r(t))y(t - reaCt)) (6)

Substituiting (5) and (6) in (1) results in the closed loop networked control system:

x(t) = (A + llA)x(t)

+ (B + llB)F(r(t))Cx(t - d(r(t), t)) (7)

where F(r(t)) is the mode-dependent controller gain which should be determined. System (7) is a Markovian jump linear system with random delay d(r(t), t). This definition

using Markov property is realistic in industry network applications as network traffic and network load conditions are rather of random nature, either in spatial or temporal sense.

Assume that the mode of the Markov process or state of the network load condition is accessible by the controller and the sensor. This assumption is reasonable and it is employed in [17]. The controller and the actuator are event­driven means that the control signal is calculated as soon as a new sensor data arrives at the controller and the control

signal is applied to the plant as soon as a new controller data arrives at the actuator.

--, I I Network delay Network delay I r fa I '--,---__ --' I

Fig. I: Structure of NCS with random delays

u(t - p(t)) yet)

r;

-------------------------I ,..--'--__ --, Network

__ ..J

--, I I Network delay Network delay I pet) I '---r-----l I

u ( t yCt - rCt))

Fig. 2: Block diagram of NCS with random delays

ret)

The following definitions and Lemma would be useful in deriving our results.

Definition 1: [30] Consider the Markovian jump system

i(t) = Ax(t) + Bx(t - d(r(t)) x(t)= cp(t) VtE[-f, O]

(8)

where x(t) E Rn is the state, and <p(t) is the initial condition. The system (8) is said to be stochastically stable if there exists a finite positive constant T(ra, <p(.)) such that the following holds for any initial condition (ra, <p(. )):

E [f" IIx(t) 112 dtl ra, cp(. )] � T(ra, cp(. ))

Lemma I: [31] For constant matrices Hand E with

appropriate dimensions, and scalar E > 0, and F satistying FT F � I , the following inequality holds:

HFE + ETFTHT � EHHT + E -IETE

3. Main Results

3. 1 Stability Analysis

Tn this subsection, a new delay-dependent robust stability condition for an uncertain NCS with random time delay is presented. The approach is derived by using the Lyapunov­Krasovskii functional. The following theorem shows that stochastic stability of the closed-loop system (7) can be guaranteed if there exists some matrices satistying certain LMT.

Theorem 1: The closed-loop system (7) is stochastically stable for a given controller gain matrix F; and for a given

589

h> 0, A; > 0, if there exist symmetric matrices X; > 0,

M; > ° and H; > ° with appropriate dimensions, for any i = 1, ... , N, and positive scalars EA , EB > ° such that the following LMls hold:

:::i BF; CX; * * *

X; CTFtBT -A; H; * * *

8·= 1 EAX; ° -EAI * * <0 (9)

EBF; CX; ° ° -EBI *

5; ° ° ° -X;

where

:::i = X;AT + AX; + lluX; + hM; + H; + EADADI + EBDBD�

X; = diag [Xv ... , X; -vX; +v ... , XN]

A; = 1 - 11; -I,f=1 Ilij hj

Proof: select a stochastic Lyapunov-Krasovskii functional candidate as:

where

V(x(t), ret)) = VI (x(t), ret)) + V2(x(t), ret)) + V3(x(t), r(t))

VI(x(t), r(t)) = xT(t)P(r(t))x(t),

V2(x(t), ret)) = f t xT(s)Q(r(t))x(s) ds,

t-d(r(t),t)

V3(x(t), ret)) = J O f t XT(S)QX(s) dsd8.

-h t+B where Pi , Q; , Q, i = 1, ... , N, are positive definite matrices with appropriate dimensions and

(10)

Let L(. ) be the weak infinitesimal generator of the random process {x(t), t}, then, for each ret) = i E 5, it can be shown that [32]:

LVI (x(t), i) = 2xT(t)P; [Ax(t) + BF; Cx(t - diet)] N

+ L llij XT(t)ljX(t) j=1

LV2 (x(t), i) = xT (t )Q; x(t)

- (1- d; (t)) xT(t - d; (t))Q; x(t - diet)) t N

+ f xT(S)(L Il; jQj) xes) ds t-d;(t) j=l N

+ L Il; j dj(t)xT(t - d; (t))Q; x(t-d; (t)) j=1

Combining (10) and (II) and considering the assumptions on the delay, we have:

LV(x(t), i) � xT(t)[(AT + EIFA(tYDDp; + PiCA N

+ DAFA(t)EA) + L Il; j lj j=l

+hQ + Qi] X(t)

+XT(t - di(t))[-AiQi] X(t-di(t))

+XT(t)[Pi(B + DBFB(t)EB)FiC] x(t - diet))

+XT(t - di(t)) [CTFt(BT + EIFB(t)TDf)Pi] x(t) < ° (12)

Pre and post multiplying (12) by Pi-1 and using lemma 1 yields:

N

LV(x(t), i) ::; XiAT + AXi + Xi I Ilij Xj-1 Xi + hMi + Hi

j=l + xT(t - di(t)) [-AiHi] x(t-di (t))

+ 2xT(t)BFiCXiX(t - diet))

+EADADI + Ei1XiEIEAXi + EBDBDI + Ei/xiCTFtEIEBFiCXi < °

(13)

Tn which

Xi = Pi-l, Mi = XiQXi, Hi = XiQiXi

Notice that

(14)

with Si and Xi keeping the same definition as (9). Substituting (14) into (13) and using Schur complement, (13) is equivalent to (9). Therefore

LV(x(t), i) ::; [XT(t) xT(t - di(t))] 8i[x(t) x(t -

di(t))f ::; -Amin(-8i)llx(t)112 ::; -�llx(t)112 (IS)

Tn which 8i is the matrix introduced in (9) and f3 = miniEd Amin (-8i)] > 0. Applying Dynkin's formula into (15), we have

E[V(x(t), rt)] - E[V(xo, ro)] = E [fLV(X(S), rJ dS] ::; -�E [f"X(S)112Ixo, ro dS]

which proves the stochastic stability of system (7) and the theorem is proved. By Theorem I, it is easy to obtain the following results.

3. 1 Controller Design

Tn this subsection, we concentrate on robust controller design for uncertain NCS described by (7).

Theorem 2: The closed-loop system (7) is robustly

stochastically stable for a given h > 0, Ai > 0, if there exist matrices Xi > 0, Zi > 0, Mi > ° and Hi > ° with appropriate

dimensions, for any i = 1, ... , N, and positive scalars

EA , EB > ° such that the following LMTs hold:

590

:::i BZiC * * *

cTzTBT -AiHi * * *

8·= 1 EAXi ° -EAI * * < 0 (16)

EBZiC ° ° -EBI *

Si ° ° ° -Xi

where

:::i = XiAT + AXi + lliiXi + hMi + Hi + +EADADI + EBDBDI

Si = [�Xi' ... , .J Ilii-l Xi, .J llii+1 Xi' ... , .jil;Xi]

Xi = diag[X1, ... 'Xi-1'Xi+1' ... , XN]

Ai = 1 - Ili -I.f=l Ilij hj

Y;C = CXi

And the controller gain is given by

(17)

Proof: Tf we let Fi = Ziy;-l and Y;C = CXi hold for every i E S, for some appropriate matrices that we have to determine, we see that LMT (9) holds if and only if LMT (16) holds. By Theorem I, we complete the proof.

3. Numerical Example

To illustrate the validation of the proposed method, we consider the following example taken from [3] and [22], where the plant parameters are considered as follows:

C = [11]

DA = [°01] , DB = [OOl] , EA = [1 O] , EB = -1 (18)

And assume that networked control system (7) has two modes. Tn each mode, the upper bound of random time delay and its derivative is considered as

h1 = 0.6, h2 = 1, 11 1 = 0.3, 112 = 0.5

We assume F(t) = sint, and it can be seen that IIF(t)11 ::; 1. The random time delay exist in S = {1, 2}, and the transition rate matrix of the random time delay is given by

0= [-1 2

By applying the proposed method and usmg the LMT Toolbox of the MATLAB, the gains are calculated as follows:

F1 = -1.1578 F2 = -1.4965

The initial condition of the system is arbitrarily chosen as Xo = [1 IV. Applying the obtained controller, the state trajectories and the output of the closed-loop system are shown in Fig. 3 and 4. As shown, the networked control system is stochastically stable by the proposed method. The control law u(t) is shown in Fig. 5. Fig. 6 Shows the mode transition of the corresponding delay during the simulation.

As you see, the simulation results confIrm the validity of the proposed control approach in this paper.

I:E iii i o 20 40 60 80 100 120

time[sec] Fig. 3: Output of the closed-loop system

time[sec] Fig. 4: State responses of the closed-loop system

100 12

LB i i II o 20 40 60 80 100 12

time[sec] Fig. 5: Control input

i ' : 1I··�II·illll·IIII·II�H!IIIII�··�··�··mllllllmIH o 20 40 60 80 100 120

time[sec] Fig. 6: Delay mode transition

Conclusion

Tn this paper, a Markov Chain is used to model the overall network-induced delays and networked control system is described as a Markovian Jump linear system. using Lyapunov-Krasovskii functional, a sufficient condition that guarantees robust stochastic stability of the NCS has been presented in terms of LMT. Based on stability condition, a mode-dependent robust output feedback controller for NCS with communication random time delays and structured uncertainties has been proposed. Numerical example verified the validity of the proposed method.

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