Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 163769, 8 pages

http://dx.doi.org/10.1155/2013/163769

Research Article

Robust Control for Discrete-Time Stochastic Interval System with Time Delay

Shengchun Yu,¹ Guici Chen,1,2 and Yi Shen³

¹Hubei Province Key Laboratory of Systems Science in Metallurgical Process, Wuhan University of Science & Technology, Wuhan 430081, China

²School of Hydropower & Information Engineering, Huazhong University of Science & Technology, Wuhan 430074, China

³School of Automation, Huazhong University of Science & Technology, Wuhan 430074, China

Received 23 April 2013; Accepted 26 August 2013

Academic Editor: Bo Shen

Copyright © 2013 Shengchun Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The robust control problem for discrete-time stochastic interval system (DTSIS) with time delay is investigated in this paper. The stochastic interval system is equivalently transformed into a kind of stochastic uncertain time-delay system firstly. By constructing the appropriate Lyapunov-Krasovskii functional, the sufficient conditions for the existence of the robust controller for DTSIS are obtained in terms of linear matrix inequality (LMI) form, and the robust controller is designed. Finally, a numerical example with simulation is given to show the effectiveness and correctness of the designed robust controller.

1. Introduction

In the past few years, much research effort has paid to the robust control problems for stochastic systems which have come to play an important role in many fields including communication network, image processes, and mobile robot localization. So far, plenty of significant results also have been published; see, for example, [1–3] and the references therein. In the meantime, we all know that time delay arises naturally in many mathematical (non) linear models of real phenomena, such as communication, circuits theory, biology, mechanics, electronics, hydraulic, rolling mill, chemical systems, and computer controlled systems, and which is frequently one of the main sources of instability, oscillation and poor performance of control systems. So the stability analysis and robust control for dynamic time-delay systems have attracted a number of researchers; see, for example, [4–8] and the references therein.

Meanwhile, from Hinrichsen who presented the control problems for stochastic systems in 1998 [9], more and more experts begin to study the stochastic control problems [10–17]. In the view of dissipation, Berman develops a -type theory for a large class of time-continuous stochastic nonlinear systems. In particular, it introduces the notion of stochastic dissipative systems by analogy with the familiar notion of dissipation associated with the deterministic systems. The problem of output feedback control for uncertain stochastic systems with time-varying delay is discussed in [12], and the parameter uncertainties are assumed to be time-varying norm-bounded. Xu et al. [13] investigate the problems of robust stochastic stabilization and robust control for uncertain neutral stochastic time-delay systems.

On the other hand, when modeling real-time plants, the parameter uncertainties are unavoidable, which are very often the cause of instability and poor performance, such as modeling error, external perturbation, and parameter fluctuation during the physical implementation. As a result, the parameters of a system matrix are estimated only within certain closed intervals. We call this kind of system interval system or stochastic interval system (SIS) with the following form:

or

where , denote the lower and upper bounds of the interval for the coefficients, respectively. Interval system and stochastic interval system have been well known for their importance in practice applications. And in recent years, the stability analysis and stabilization problems of various interval systems have received plenty of research attention; see, for example, [18–24] and the references therein. Of course, there are a lot of research works on stability analysis and stabilization problems for stochastic interval system that have been reported; see, for example, [25–29] and the references therein. However, to the best of authors’ knowledge, so far little results on robust control problem for stochastic interval system with time-delay are available in the existing literature.

Inspired by the above discussions, in this paper we study the robust control problem for discrete-time stochastic interval system (DTSIS) with time delay. The stochastic interval system will be equivalently transformed into a kind of stochastic uncertain time-delay system firstly. By constructing appropriate Lyapunov-Krasovskii functional, the sufficient conditions for the existence of the robust controller for DTSIS are obtained in terms of linear matrix inequality (LMI) form, and the robust controller can be designed by MATLAB LMI control toolbox.

2. Problems Formulation and Preliminaries

Consider the following discrete-time stochastic interval system (DTSIS) with time delay:

whereis the state vector; is the time delay; is the control input;is the control output; is the exogenous disturbance input, which satisfies, where is the space of nonanticipatory square-summable stochastic process with respect to with the following norm:

is a scalar Brownian motion defined on a complete probability space with

In the system (3), is an interval matrix with appropriate dimension, which means

where , are determinate matrices.

Remark 1. It is not difficult to see that the matrices, can also be time-dependent as long as the mappings, , , , , and are continuous.

Set

whereis theth column ofidentification matrix, setting

where . Obviously, we have ; hereis theidentification matrix. So the interval matrixcan equivalently be written as

Similar to the above derivation, we can have

where; let

Then (9)–(11) can be rewritten as

So the system (3) can be readily derived as

For system (14), we design the following state feedback controller:

where the matrix is the controller gain, which is to be designed. Then the closed-loop stochastic control system can be rewritten as

where, , , , .

Definition 2. Discrete-time stochastic interval system (DTSIS) with time delay (3) is said to be stochastic exponentially stable in mean square, if there exist scalars, such that

In this paper, we aim to design the controller gain matrixin (16) such that the requirements are simultaneously satisfied.

(a)The zero-solution of the closed-loop stochastic control system (16) withis stochastic exponentially stable in mean square. (b)Under the zero-initial condition, the control outputsatisfies

for all nonzero.

Which is also said the closed-loop stochastic control system (16) is stochastic exponentially stable in mean square with disturbance attenuation level.

Lemma 3 (see [30]). For given symmetric matrix, and matriceswith appropriate dimension,withsatisfying holds if and only if holds for any.

3. Main Results

Theorem 4. If there exist two positive definite matricesand, and for a given positive constant, such that the following matrix inequality holds:

then the closed-loop stochastic control system (16) is stochastic exponentially stable in mean square with disturbance attenuation level.

Proof. Let; we construct the Lyapunov-Krasovskii functional as

Calculating the difference of along with the system (16) and taking the mathematical expectation, we have

where

It is clear formthat there exists a sufficient scalar, such that

Therefore, we have

We now in a position to analyze of the exponential stable in mean square for the stochastic interval delay system (16). According to the definition of , we have

where and . For any scalar , the earlier inequality, together with (23), implies that

where .

Furthermore, summing up both sides of the earlier inequality fromtowith respect to, we get

For ,

Then, from (27) and (28), it follows that

Let and ; we have

It also follows that

We can choose appropriate scalarsuch that

which implies

where . Hence the closed-loop stochastic control system (16) withis stochastic exponentially stable in mean square according to Definition 2.

Under the zero-initial condition, let , setting

where

So (19) implies that ; hence,. Thus, , such that (18) holds. The proof is completed.

To design the controller (15), matrix inequality (15) must be solvable and matrix inequality (15) in Theorem 4 must be inverted into LMI form. In the following section, we will find a way for the solution of (15).

Remark 5. The results obtained in the paper are based on the Lyapunov-Krasovskii functional and the corresponding technique used in the proof of Theorem 4, and it is easy to extend the results to stochastic system with time-varying delay or multiple delays. Details are omitted here due to page length consideration.

Theorem 6. If there exist two positive definite matrices , a matrix with appropriate dimension, and a positive constant , for a given positive constant , such that the following linear matrix inequality (LMI) holds

where , then the closed-loop stochastic control system (16) is stochastic exponentially stable in mean square with disturbance attenuation level with the controller designed as .

Proof. By (13) and (19), we have

where

By Lemma 3, we know that

Using Schur complement lemma and contragradient transformation, we can get the LMI (36) holds. The proof is completed.

Without considering of interval matrices, the system (3) changes into the following discrete-time stochastic time-delay system:

We can design the controller (15) by the following the corollary without proof.

Corollary 7. If there exist two positive definite matrices and a matrix with appropriate dimension, for a given positive constant , such that the following linear matrix inequality (LMI) holds

where is defined in (36), then the closed-loop stochastic control system (40) is stochastic exponentially stable in mean square with disturbance attenuation levelwith the controller designed as.

If we do not consider the stochastic disturbance in system (3), the system (3) can be written as

Then the robust controller can be designed by the following corollary.

Corollary 8. If there exist two positive definite matrices, a matrixwith appropriate dimension, and a positive constant , for a given positive constant, such that the following linear matrix inequality (LMI) holds

where is defined in (36), then the closed-loop stochastic control system (42) is exponentially stable with disturbance attenuation level with the controller designed as .

4. Numerical Example with Simulation

In this section, an example is given to show the usefulness of the designed controller. Consider the following discrete-time stochastic interval system with time delay:

Without considering the state-feedback control, we see that the open-loop stochastic interval system is unstable from Figure 1.

Figure 1: Response of open-loop DTSIS (44) to initial state.

We aim at designing a state-feedback controller for stochastic interval system (44), such that the closed-loop stochastic interval system is stochastic exponentially stable in mean square with disturbance attenuation level.

By using the method discussed in the previous section, we can calculate the matrices , , setting , solving the LMI (36) in Theorem 6 and then obtain the matrices as follows:

Note that in expression of Theorem 6, we can design the controller as

The response of the closed-loop stochastic interval system (44) is shown in Figures 2 and 3, which demonstrate that the designed controller for the stochastic interval system (44) is effective.

Figure 2: Response of closed-loop DTSIS (44) to initial state.

Figure 3: Response of control outputof closed-loop DTSIS (44).

5. Conclusions

In this paper, we have studied the robustcontrol problem for discrete-time stochastic interval system (DTSIS) with time delay. The stochastic interval system was equivalently transformed into a kind of stochastic uncertain time-delay system firstly. By constructing appropriate Lyapunov-Krasovskii functional, the sufficient conditions for the existence of the robustcontroller for DTSIS have been obtained in terms of linear matrix inequality (LMI) form, and the robustcontroller has been designed easily by MATLAB LMI control toolbox. A numerical example with simulation has been given to demonstrate the feasibility and effectiveness of the proposed method.

Of course, this method proposed in this paper also can be used to the the state estimation problem and the sampled-data synchronization control problem for the complex networks, which has been studied in [31, 32] and is one of our future research topics.

Acknowledgments

The authors thank the referee and the associate editor for their very helpful comments and suggestions. This work is supported by National Science Foundation of China (NSFC) under Grant nos. 61104127, 51079057, and 61134012, China Postdoctoral Science Foundation with Grant no. 2012M521428, and Hubei Province Key Laboratory of Systems Science in Metallurgical Process (Wuhan University of Science and Technology) under Grant no. Y201101.

References

X. Mao, N. Koroleva, and A. Rodkina, “Robust stability of uncertain stochastic differential delay equations,” Systems and Control Letters, vol. 35, no. 5, pp. 325–336, 1998.

Y. Niu, D. W. C. Ho, and J. Lam, “Robust integral sliding mode control for uncertain stochastic systems with time-varying delay,” Automatica, vol. 41, no. 5, pp. 873–880, 2005.

V. B. Kolmanovskii and A. D. Myshkis, Introduction To the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.

Z. Zeng, J. Wang, and X. Liao, “Global exponential stability of a general class of recurrent neural networks with time-varying delays,” IEEE Transactions on Circuits and Systems, vol. 50, no. 10, pp. 1353–1358, 2003.

Z. Zeng, J. Wang, and X. Liao, “Stability analysis of delayed cellular neural networks described using cloning templates,” IEEE Transactions on Circuits and Systems, vol. 51, no. 11, pp. 2313–2324, 2004.

Z. Zeng and J. Wang, “Improved conditions for global exponential stability of recurrent neural networks with time-varying delays,” IEEE Transactions on Neural Networks, vol. 17, no. 3, pp. 623–635, 2006.

Z. Zeng, J. Wang, and X. Liao, “Global asymptotic stability and global exponential stability of neural networks with unbounded time-varying delays,” IEEE Transactions on Circuits and Systems, vol. 52, no. 3, pp. 168–173, 2005.

X. X. Liao, J. Wang, and Z. Zeng, “Global asymptotic stability and global exponential stability of delayed cellular neural networks,” IEEE Transactions on Circuits and Systems, vol. 52, no. 7, pp. 403–409, 2005.

D. Hinrichsen and A. J. Pritchard, “Stochastic ,” SIAM Journal on Control and Optimization, vol. 36, no. 5, pp. 1504–1538, 1998.

N. Berman and U. Shaked, “-like control for nonlinear stochastic systems,” Systems and Control Letters, vol. 55, no. 3, pp. 247–257, 2006.

G. Wei, Z. Wang, and H. Shu, “Nonlinear control of stochastic time-delay systems with Markovian switching,” Chaos, Solitons and Fractals, vol. 35, no. 3, pp. 442–451, 2008.

S. Xu and T. Chen, “ output feedback control for uncertain stochastic systems with time-varying delays,” Automatica, vol. 40, no. 12, pp. 2091–2098, 2004.

S. Xu, P. Shi, Y. Chu, and Y. Zou, “Robust stochastic stabilization and control of uncertain neutral stochastic time-delay systems,” Journal of Mathematical Analysis and Applications, vol. 314, no. 1, pp. 1–16, 2006.

Z. Wang, F. Yang, D. W. C. Ho, and X. Liu, “Robust variance-constrained control for stochastic systems with multiplicative noises,” Journal of Mathematical Analysis and Applications, vol. 328, no. 1, pp. 487–502, 2007.

N. Berman and U. Shaked, “ control for discrete-time nonlinear stochastic systems,” IEEE Transactions on Automatic Control, vol. 51, no. 6, pp. 1041–1046, 2006.

B.-S. Chen and W. Zhang, “Stochastic H2/ control with state-dependent noise,” IEEE Transactions on Automatic Control, vol. 49, no. 1, pp. 45–57, 2004.

H. Gao, C. Wang, and J. Wang, “On performance analysis for continuous-time stochastic systems with polytopic uncertainties,” Circuits, Systems, and Signal Processing, vol. 24, no. 4, pp. 415–429, 2005.

Y.-T. Juang and C.-S. Shao, “Stability analysis of dynamic interval systems,” International Journal of Control, vol. 49, no. 4, pp. 1401–1408, 1989.

H. Zhang, Z. Wang, and D. Liu, “Robust stability analysis for interval Cohen-Grossberg neural networks with unknown time-varying delays,” IEEE Transactions on Neural Networks, vol. 19, no. 11, pp. 1942–1955, 2008.

K. D. Kim and N. K. Bose, “Vertex implications of stability for a class of delay-differential interval system,” IEEE Transactions on Circuits and Systems, vol. 37, no. 7, pp. 969–972, 1990.

X. Liao and J. Yu, “Robust stability for interval Hopfield neural networks with time delay,” IEEE Transactions on Neural Networks, vol. 9, no. 5, pp. 1042–1045, 1998.

S. Hu and J. Wang, “On stabilization of a new class of linear time-invariant interval systems via constant state feedback control,” IEEE Transactions on Automatic Control, vol. 45, no. 11, pp. 2106–2111, 2000.

C. Li, X. Liao, and R. Zhang, “Global robust asymptotical stability of multi-delayed interval neural networks: an LMI approach,” Physics Letters A, vol. 328, no. 6, pp. 452–462, 2004.

A. Chen, J. Cao, and L. Huang, “Global robust stability of interval cellular neural networks with time-varying delays,” Chaos, Solitons and Fractals, vol. 23, no. 3, pp. 787–799, 2005.

X. X. Liao and X. Mao, “Exponential stability of stochastic delay interval systems,” Systems and Control Letters, vol. 40, no. 3, pp. 171–181, 2000.

X. Mao, “Exponential stability of stochastic delay interval systems with Markovian switching,” IEEE Transactions on Automatic Control, vol. 47, no. 10, pp. 1604–1612, 2002.

X. Mao and C. Selfridge, “Stability of stochastic interval systems with time delays,” Systems and Control Letters, vol. 42, no. 4, pp. 279–290, 2001.

X. Mao, J. Lam, S. Xu, and H. Gao, “Razumikhin method and exponential stability of hybrid stochastic delay interval systems,” Journal of Mathematical Analysis and Applications, vol. 314, no. 1, pp. 45–66, 2006.

G. Wei, Z. Wang, H. Shu, and J. Fang, “Delay-dependent stabilization of stochastic interval delay systems with nonlinear disturbances,” Systems and Control Letters, vol. 56, no. 9-10, pp. 623–633, 2007.

L. Xie, “Output feedback control of systems with parameter uncertainty,” International Journal of Control, vol. 63, no. 4, pp. 741–750, 1996.

B. Shen, Z. D. Wang, and X. H. Liu, “Sampled-data synchronization control of dynamical networks with stochastic sampling,” IEEE Transactions on Automatic Control, vol. 57, no. 10, pp. 2644–2650, 2012.

D. Ding, Z. D. Wang, B. Shen, and H. Shu, “ state estimation for discrete-time complex networks with randomly occurring sensor saturations and randomly varying sensor delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 23, no. 5, pp. 725–736, 2012.