Robust Output Feedback Control of Networked ...

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stabilization for uncertain networked control systems (NCSs) with random time delay via the output feedback control. A mode dependent controller is proposed ...
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, [email protected] ** Tarbiat Modares University, [email protected] *** K. N. Toosi University of Technology,[email protected]

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 LMIs. 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 network­ induced 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

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 network­ induced 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 of NCS 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 Lyapunov­ Krasovskii 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.

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 ...

{

flijll + � (ll) ( II Pr{r(t + ll) = jlr(t) = i} = l + fliiLJ. + 0 LJ.)

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) n

where x(t) E R 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

(3)

where flij ;::: 0, i j , flii = - I.f=l,j*iflij' In each mode in the Markov process, the corresponding delay is assumed to be time-varying which satisfies 0:::; dO, t):::; hi' diet):::;

1=

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 [mi nk {r�}, (nS

+

p(t) E [mi nl {r n , (na

+

l)hS + maxk{ r� +ns+ l} )' Vk E N l)h a + maxl{ r f+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 +

2.

� =1=!}

t

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 event­ driven 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 Network delay I I r fa I '-__--' I

Network delay

r;

,---

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:

BF; CX; -A; H; ° ° °

:::i

X; CTFtBT 8·= EAX; 1 EBF; CX;

__ ..J

5;

*

*

*

*

*

*

-EBI °

*

-EAI ° °

*

*

0, Ai > 0, if there exist matrices Xi > 0, Zi > 0, Mi > ° and H i > ° with appropriate dimensions, for any i= 1, ... , N, and positive scalars EA, EB > ° such that the following LMTs hold:

590

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 o

20

40

60

time[sec]

80

[6]

i

100

Fridman, E., Seuret, A, Richard, J.-P., "Robust

sampled-data

stabilization of linear systems: An input delay approach. " Automatica

120

40,1441-1446,2004 [7]

Lian, F.-L., Moyne, J., Tilbury, D., "Modelling and optimal controller design

of

networked

control

systems

with

multiple

delays,"

International Journal of Control 76,591- 606,2003 [8]

W.A Zhang, and L. Yu, "New approach to stabilization of networked control systems with time-varying delays," lET Control TheolY Appl., Vol. 2,no. 12,pp. 1094-1104,2008.

Fig. 3: Output of the closed-loop system

[9]

Wen-An Zhang, Li Yu, "Modeling and control of networked control systems

with

both

network-induced

delay

and

packet-dropout,"

Automatica, Volume 44,Issue 12,Pages 3206-3210,December 2008 [10] Kim, Y.H., Park, H.S., Kwon, W.H., "Stability and a scheduling method for network based control systems," In: Proceedings of the 100

time[sec]

12

LB i i II i : 1I· �II·illll·IIII·II�H!IIIII�· �· �· mllllllmIH 20

40

60

time[sec]

80

100

12

Fig. 5: Control input

'

o

20

40

60

time[sec]

80

[ I I] Park, H.S., Kim, Y.H., Kim, D.-S., Kwon, W.H., "A scheduling method for network based control systems," IEEE Transactions on

Fig. 4: State responses of the closed-loop system

o

1996 IEEE IECON 22nd International Conference on Industrial Electronics, Control, and Instrumentation, pp. 934-939 ,August 1996

100

120

Control Systems Technology 10,318-330,2002 [12] Halevi, Y., Ray, A, "Integrated communication and control systems: Part I - Analysis. Journal of Dynamic Systems," Measurement and

ControlllO, 367-373,1988 [13] Ray, A, Halevi, Y., "Integrated communication and control systems: - Design considerations," Journal of Dynamic Systems, Measurement and Control 110,374-381,1988

Part IT

[14] Walsh, G.c., Ye, H., Bushnell, L., "Stability analysis of networked control systems," In: Proceedings of the America Control Conference, San Diego,CA,USA,pp. 2876-2880,June 1999 [15] Goodwin, G.c., Haimovich, H., Quevedo, D.E., Welsh, 1.S., "A moving horizon approach to networked control system design," IEEE

Transactions on Automatic Control 49,1427-1445,2004 [16] Bertsekas, D., Gallager, R. Data Networks, 2nd edn. Prentice Hall, Englewood Cliffs, 1992 [17] 1. Nilsson and B. Bernhardsson, "LQR control over a Markov network," In Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, California, USA,

communication

Fig. 6: Delay mode transition

Atlantis,pp. 4586-4591,December 1997

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.

[18] R. Krtolica, U. Ozguner, H. Chan, H. Goktas, 1. Winkelman, and M. Liubakka, "Stability

of

linear

feedback

systems

with

random

communication delays," International Journal of Control, vol. 59,no. 4,1994,pp. 925- 953. [19] 1. Xiong and 1. Lam, "Stabilization of linear systems over networks with bounded packet loss," Automatica, vol. 43, no. 1, pp. 80-87, Jan. 2007. [20] L.

Xiao, A

Hassibi, and

1.

P.

How, "Control

with

random

communication delays via a discrete-time jump system approach," in

Proc. Amer. Control Conj, Chicago, IL, Jun, vol. 3, pp.2199-2204. 2000 [21] 1. nilsson, "Real time control systems with delay. " PhD thesis, Lund Institute of Technology, 1998. [22] D. Huang; S. K. Nguang; , "State feedback control of uncertain networked control systems with random time delays," IEEE Transactions on Automatic Control, vo1.53, no.3, pp.829-834, April 2008

References

[23] D. Huang and S. K. Nguang, "Dynamic Output Feedback Control for Uncertain

[ I]

Baruch, 1.E. F., Cox, MJ., "Remote control and robots: an Internet solution," Computing

[2]

[3]

[4]

& Control Engineering Journal 7,39-45,1996

Control

Systems

with

Random

Network­

Systems, pp 841-847,2009

Branicky, M.S., Phillips, SM., Zhang, W., "Stability of networked

[24] Liqian Zhang, Yang Shi, Tongwen Chen, and Biao Huang,"A New

control systems: Explicit analysis of delay," In: Proceedings of the America Control Conference, Chicago, IL, USA, pp. 2352-2357

Method for Stabilization of Networked ControlSystems With Random

(June 2000)

2005

W. Zhang, M. S. Branicky, and S. M. Phillips, "Stability of networked

Delays ", IEEE Trans. On

Automatic Control, Vol. 50, No. 8, Aug.

[25] Zhang W., Yu L., "Output feedback stabilization of networked control

control systems," IEEE Control Syst. Magzine, vol. 21, no. I, pp. 84-

systems with packet dropout ", IEEE Trans. on

99,Feb. 200 I.

pp. 1705-1710,2007

Gao, H., Chen, T., Lam, 1., "A new delay system approach to Yong Zhang, Huajing

Autom. Control, 52,

[26] MU S., CHU T., HAO F., WANG L., "Output feedback control of networked control systems ". Proc. IEEE Int. Conj on Syst., Man

network-based control," Automatica 44,39-52 (2008) [5]

Networked

induced Delays," International Journal of Control, Automation, and

Fang, "Stabilization of nonlinear networked

systems with sensor random packet dropout and time-varying delay,"

Applied Mathematical Modelling, Volume 35, Issue 5, Pages 22532264,May 20 I I.

Cybern., vol. I,pp. 211-216,2003 [27] Y. Shi and B. Yu, "Output

Feedback Stabilization of Networked

Control Systems With Random Delays Modeled by Markov Chains,"

IEEE Trans. on Autom. Control, VOL. 54, NO.7, JULY 2009

[28] Y. Sun, and Y. Huo, "Robust Static Output

Feedback Control for

Networked Control System with Random Time Delay," The 2nd

591

International Conference on Advanced Computer Control (ICACC). Shenyang,27-29 March 2010 [29] D. Huang and S.K. Nguang,Robust Control. for Uncertain Networked

Control Systems, LNCIS 386,springer,pp. 17-21 [30] E1-Kebir Boukas, Stochastic switching systems, analaysis and design, Birkhauser,2005 [31] E.-K. Boukas and Z.-K. Liu, Deterministic and stochastic time delay

systems. Boston: Birkhauser,2002. [32] X. Zhao and Q. Zeng, "Delay-dependent Hoo performance analysis for Markovian jump systems with mode-dependent time varying delays and

partially

known

transition

rates,"

International Journal of

Control, Automation and Systems, 2010

592