Research Article | Open Access

# Robust Control for a Class of Discrete Time-Delay Stochastic Systems with Randomly Occurring Nonlinearities

**Academic Editor:**Hongli Dong

#### Abstract

In this paper, we consider the robust control problem for a class of discrete time-delay stochastic systems with randomly occurring nonlinearities. The parameter uncertainties enter all the system matrices; the stochastic disturbances are both state and control dependent, and the randomly occurring nonlinearities obey the sector boundedness conditions. The purpose of the problem addressed is to design a state feedback controller such that, for all admissible uncertainties, nonlinearities, and time delays, the closed-loop system is robustly asymptotically stable in the mean square, and a prescribed disturbance rejection attenuation level is also guaranteed. By using the Lyapunov stability theory and stochastic analysis tools, a linear matrix inequality (LMI) approach is developed to derive sufficient conditions ensuring the existence of the desired controllers, where the conditions are dependent on the lower and upper bounds of the time-varying delays. The explicit parameterization of the desired controller gains is also given. Finally, a numerical example is exploited to show the usefulness of the results obtained.

#### 1. Introduction

Various engineering systems, such as electrical networks, turbojet engines, microwave oscillators, nuclear reactors, and hydraulic systems, have the characteristics of time delay in signal transmissions. The existence of time delay is often a source of instability and poor performance. So far, the stability analysis and robust control for dynamic time-delay systems have attracted a number of researchers over the past years; see [1–4] and the references therein.

Since stochastic modeling has had extensive applications in control and communication problems, the stability analysis problem for linear stochastic time-delay systems has been studied by many authors. For example, in [5], the stability analysis problem for uncertain stochastic fuzzy systems with time delays has been considered. In [6], an LMI approach has been developed to cope with the robust control problem for linear uncertain stochastic systems with state delay. In [7], the robust energy-to-peak filtering problem has been dealt with for uncertain stochastic time-delay systems. In [8], the robust integral sliding mode control problem has been studied for uncertain stochastic systems with time-varying delays, and the performance has been analyzed in [9] for continuous-time stochastic systems with polytopic uncertainties.

On the other hand, it is well known that nonlinearity is ubiquitous and all pervading in the physical world and therefore nonlinear control has been an ever hot topic in the past few decades. It is worth mentioning that, among different descriptions of the nonlinearities, the so-called* sector nonlinearity* [10] has gained much attention for* deterministic* systems, and both the control analysis and model reduction problems have been investigated; see [11–13]. Recently, the control problem of nonlinear stochastic systems has stirred renewed research interests [14], and a variety of nonlinear stochastic systems have been investigated by different approaches, such as the minimax dynamic game approach [15], the input-to-state stabilization method [16, 17], the infinite-horizon risk-sensitive scheme [18], and the Lyapunov-based recursive design method [19]. Most recently, in [20–27], an -type theory has been developed for a large class of continuous- and discrete-time nonlinear stochastic systems. Notice that it is also quite common to describe nonlinearities as additive nonlinear disturbances. Such nonlinear disturbances may occur in a probabilistic way. For example, in a particular moment for networked control systems, the transmission channel for a large amount of packets may encounter severe network-induced congestions due to the bandwidth limitations, and the resulting phenomenon could be reflected by certain randomly occurring nonlinearities where the occurrence probability can be estimated via statistical tests. Recently some initial work has been reported on the dynamics for the systems with randomly occurring nonlinearities; see [27, 28] and the references therein. As far as we know, to date, however, the robust control problem for stochastic systems with time delays and randomly occurring nonlinearities has not been fully investigated, especially for discrete-time cases, which motivates us to shorten such a gap in the present investigation.

The main contribution of this paper can be summarized as follows. (1) A general framework is established to cope with robust control problem for a class of uncertain discrete-time stochastic systems involving randomly occurring sector nonlinearities and time-varying delays. (2) An effective LMI approach is proposed to design the state feedback controllers such that, for all the randomly occurring nonlinearities and time-delays, the overall uncertain closed-loop system is robustly asymptotically stable in the mean square and a prescribed disturbance rejection attenuation level is guaranteed. We first establish the sufficient conditions for the uncertain nonlinear stochastic time-delay systems to be stable in the mean square and then derive the explicit expression of the desired controller gains. (3) Sector nonlinearity technique is utilized to reduce the conservativeness for the main results.

*Notation*. Throughout this paper, and stand for the natural numbers and the nonnegative integer set, respectively; , and denote, respectively, the set of real numbers, the dimensional Euclidean space, and the set of all real matrices. The superscript represents the transpose for a matrix; the notation (resp., ), where and are symmetric matrices, means that is positive semidefinite (resp., positive definite). denotes the standard Euclidean norm. If is a square matrix, denote by (resp., ) the largest (resp., the smallest) eigenvalue of . Matrices, if not explicitly specified, are assumed to have compatible dimensions. Sometimes, the arguments of a function will be omitted in the analysis when no confusion can arise. In an underlying probability space , denotes the mean for a random variable, and means the occurrence probability of the event.

#### 2. Problem Formulation

Consider, on a probability space , the following uncertain discrete nonlinear stochastic system with time delays of the form: where is the state vector; is the control input; is the controlled output; is a white process noise with and are mutually independent Bernoulli-distributed white sequences and also assumed to be independent of .

Let distribution laws of and be given by It is well known that , , and the variances of and are, respectively, and . Notice that and characterize the random occurrence of nonlinear functions.

For the exogenous disturbance signal , it is assumed that , where is the space of nonanticipatory square-summable stochastic process with respect to with the following norm:

In the system (), the positive integer denotes the time-varying delay satisfying where and are known positive integers. The are the initial conditions independent of the process .

In , and are known real constant matrices. The matrices , , , , , , , and are time-varying matrices of the following form: Here, , , , , , , , and are known real constant matrices; , , , , , , , and are unknown matrices representing time-varying parameter uncertainties, which are assumed to satisfy the following admissible condition: where and are known real constant matrices, and is the unknown time-varying matrix-valued function subject to the following condition: Furthermore, the vector-valued functions and satisfy the sector nonlinearity condition: where are known real constant matrices.

*Remark 1. *It is customary that the nonlinear functions are said to belong to sectors , , respectively [10]. The nonlinear descriptions in (9)-(10) are quite general descriptions that include the usual Lipschitz conditions as a special case. Note that both the control analysis and model reduction problems for systems with sector nonlinearities have been intensively studied; see for example [11–13].

Substituting the state feedback to system () gives the following closed-loop system: where

In this paper, we aim to solve the robust control problem for the uncertain discrete nonlinear stochastic systems with time-varying delays. More specifically, a state feedback controller of the form is to be designed such that the following two requirements are met simultaneously.(1)The closed-loop system () with is mean-square asymptotically stable for all admissible uncertainties.(2)Under zero initial condition, the closed-loop system satisfies for any nonzero .

#### 3. Main Results

In this section, we begin with the robust stabilization problem for the closed-loop system (). A sufficient condition is derived in the form of an LMI in order to guarantee the robustly mean-square asymptotic stability for the closed-loop system () with .

Theorem 2. *Let be a given constant feedback gain matrix. The closed-loop system with is robustly asymptotically stable in the mean square if there exist two positive definite matrices , and two positive constant scalars , such that the following LMI holds:
**
where *

*Proof. *For the stability analysis of the system (), we construct the following Lyapunov-Krasovskii functional:
where
with and .

By calculating the difference of along the system () with and taking the mathematical expectation, we have
where
with
Substituting (26)-(27) into (25) leads to
where

Notice that (9) is equivalent to
where are defined in (15) and, similarly, it follows from (10) that
where are defined in (16).

It follows from (28), (30), and (31) that
where

From Lyapunov stability theory, in order to ensure the stability of the closed-loop system () with , we just need to show

By* Schur complement*, (34) is equivalent to
where

Therefore, it remains to show . Let , and where is defined in (18).

Obviously, is equivalent to . Now, we can rewrite as follows:
where where and are defined in (19).

According to* Schur complement*, is equivalent to
where

Noticing (7), we can rearrange as follows:
where
with

From (44), it follows
and then we obtain from (42)–(46)
where and , , and are defined in (18)–(22).

Using* Schur complement* once again, it is not difficult to see that (i.e., the condition (13)) is equivalent to that the right-hand side of (47) is negative definite; that is, . Thus, we arrive at , which completes the proof of the theorem.

Next we will analyze the performance of the closed-loop system ().

Theorem 3. *Let be a given constant feedback gain matrix and a given positive constant. Then, the closed-loop system is robustly mean-square asymptotically stable for and satisfies under the zero initial condition for any nonzero if there exist two positive definite matrices , and three scalars , , such that the following LMI holds
**
where **
and , , , , , , , and are defined as in Theorem 2.*

* Proof. *It is not difficult to verify that implies (i.e., (13)). According to Theorem 2, the closed-loop system () is robustly asymptotically stable in the mean square.

Next, we shall deal with the performance of the closed-loop system. As in Theorem 2, define Lyapunov-Krasovskii functional candidate for the system () as follows:
where

And we also introduce
where is nonnegative integer. Under the zero initial condition, one has
Along the similar lines to the proof of Theorem 2, we can derive that . Letting , we obtain , which concludes the proof of the theorem.

Finally, let us consider the design of control for the system . Based on Theorem 3, we have the following result.

Theorem 4. *Let be a given positive constant. Then, for the nonlinear stochastic system , there exists a state feedback controller such that the closed-loop system is robustly mean-square asymptotically stable for and satisfies under the zero initial condition for any nonzero if there exist two positive definite matrices and , a matrix , and three scalars , , such that the following LMI holds:
**
where **where , , , , , , , , , , and are defined as in Theorems 2 and 3. In this case, the state feedback gain matrix can be designed as
*

*Remark 5. *In Theorem 4, the robust controller design problem is reduced to the solvability of an LMI, which could be easily checked by utilizing the LMI Matlab toolbox. It is also worth pointing out that, in LMI framework, the sector nonlinearity condition is superior to commonly used Lipschitz condition in reducing the possible conservativeness.

#### 4. Numerical Example

In this section, a numerical example is presented to demonstrate the usefulness of the developed method on the design of robust control for the discrete uncertain nonlinear stochastic systems with time-varying delays.

Consider the system () with the following parameters:

The performance level is taken as . With the above parameters and by using the Matlab LMI Toolbox, we solve the LMI (56) and obtain

Therefore, the state feedback gain matrix can be designed as

#### 5. Conclusions

In this paper, we have studied the robust control problem for a class of uncertain discrete-time stochastic systems involving randomly occurring nonlinearities and time-varying delays. An effective linear matrix inequality (LMI) approach has been proposed to design the state feedback controllers such that the overall uncertain closed-loop system is robustly asymptotically stable in the mean square and a prescribed disturbance rejection attenuation level is guaranteed. We have first investigated the sufficient conditions for the uncertain stochastic time-delay systems under consideration to be stable in the mean square and then derived the explicit expression of the desired controller gains. A numerical example has been provided to show the usefulness and effectiveness of the proposed design method. It should be pointed out that the main results of this paper can be extended to the case where the parameter uncertainties are of the polytopic type [9, 29] and the case where system measurements suffer from quantization effects.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants 61374010, 61074129, and 61175111, the Natural Science Foundation of Jiangsu Province of China under Grant BK2012682, the Qing Lan Project of Jiangsu Province (2010), the 333 Project of Jiangsu Province (2011), and the Six Talents Peak Project of Jiangsu Province (2012).

#### References

- E. K. Boukas and Z.-K. Liu,
*Deterministic and Stochastic Time-Delay Systems*, Birkhauser, Boston, Mass, USA, 2002. - S. Elmadssia, K. Saadaoui, and M. Benrejeb, “New delay-dependent stability conditions for linear systems with delay,”
*Systems Science and Control Engineering*, vol. 1, no. 1, pp. 2–11, 2013. View at: Publisher Site | Google Scholar - J. Shen and J. Lam, “Decay rate constrained stability analysis for positive systems with discrete and distributed delays,”
*Systems Science and Control Engineering*, vol. 2, no. 1, pp. 7–12, 2014. View at: Publisher Site | Google Scholar - M. Kermani and A. Sakly, “Stability analysis for a class of switched nonlinear time-delay systems,”
*Systems Science and Control Engineering: An Open Access Journal*, vol. 2, no. 1, pp. 80–89, 2014. View at: Google Scholar - Z. Wang, D. W. C. Ho, and X. Liu, “A note on the robust stability of uncertain stochastic fuzzy systems with time-delays,”
*IEEE Transactions on Systems, Man, and Cybernetics A*, vol. 34, no. 4, pp. 570–576, 2004. View at: Publisher Site | Google Scholar - S. Xu and T. Chen, “Robust ${H}_{\infty}$ control for uncertain stochastic systems with state delay,”
*IEEE Transactions on Automatic Control*, vol. 47, no. 12, pp. 2089–2094, 2002. View at: Publisher Site | Google Scholar | MathSciNet - H. Gao, J. Lam, and C. Wang, “Robust energy-to-peak filter design for stochastic time-delay systems,”
*Systems and Control Letters*, vol. 55, no. 2, pp. 101–111, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - 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. View at: Publisher Site | Google Scholar | MathSciNet - H. Gao, C. Wang, and J. Wang, “On
*H*_{α}performance analysis for continuous-time stochastic systems with polytopic uncertainties,”*Circuits, Systems and Signal Processing*, vol. 24, no. 4, pp. 415–429, 2005. View at: Google Scholar - H. K. Khalil,
*Nonlinear Systems*, Prentice-Hall, Upper Saddle River, NJ, USA, 1996. - Q.-L. Han, “Absolute stability of time-delay systems with sector-bounded nonlinearity,”
*Automatica*, vol. 41, no. 12, pp. 2171–2176, 2005. View at: Publisher Site | Google Scholar | MathSciNet - S. J. Kim and I. J. Ha, “A state-space approach to analysis of almost periodic nonlinear systems with sector nonlinearities,”
*IEEE Transaction on Automatic Control*, vol. 44, no. 1, pp. 66–70, 1999. View at: Publisher Site | Google Scholar | MathSciNet - J. Lam, H. Gao, S. Xu, and C. Wang, “${H}_{\infty}$ and ${L}_{2}={L}_{\mathrm{\infty}}$ model reduction for system input with sector nonlinearities,”
*Journal of Optimization Theory and Applications*, vol. 125, no. 1, pp. 137–155, 2005. View at: Publisher Site | Google Scholar | MathSciNet - W. M. McEneaney, G. Yin, and Q. Zhang, Eds.,
*Stochastic Analysis, Control, Optimization and Applications*, Systems and Control: Foundations and Applications series, Birkhäuser, Boston, Mass, USA, 1999. - C. D. Charalambous, “Stochastic nonlinear minimax dynamic games with noisy measurements,”
*IEEE Transactions on Automatic Control*, vol. 48, no. 2, pp. 261–266, 2003. View at: Publisher Site | Google Scholar | MathSciNet - H. Deng and M. Krstić, “Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance,”
*Systems & Control Letters*, vol. 39, no. 3, pp. 173–182, 2000. View at: Publisher Site | Google Scholar | MathSciNet - H. Deng, M. Krstic, and R. J. Williams, “Stabilization of stochastic nonlinear systems driven by noise of unknown covariance,”
*IEEE Transactions on Automatic Control*, vol. 46, no. 8, pp. 1237–1253, 2001. View at: Publisher Site | Google Scholar | MathSciNet - Y. Liu, Z. Pan, and S. Shi, “Output feedback control design for strict-feedback stochastic nonlinear systems under a risk-sensitive cost,”
*IEEE Transactions on Automatic Control*, vol. 48, no. 3, pp. 509–513, 2003. View at: Publisher Site | Google Scholar | MathSciNet - S. Xie and L. Xie, “Decentralized stabilization of a class of interconnected stochastic nonlinear systems,”
*IEEE Transactions on Automatic Control*, vol. 45, no. 1, pp. 132–137, 2000. View at: Publisher Site | Google Scholar | MathSciNet - N. Berman and U. Shaked, “${H}_{\infty}$ control for discrete-time nonlinear stochastic systems,”
*IEEE Transactions on Automatic Control*, vol. 51, no. 6, pp. 1041–1046, 2006. View at: Publisher Site | Google Scholar | MathSciNet - H. Shu and G. Wei, “${H}_{\infty}$ analysis of nonlinear stochastic time-delay systems,”
*Chaos, Solitons and Fractals*, vol. 26, no. 2, pp. 637–647, 2005. View at: Publisher Site | Google Scholar - W. Zhang, B. Chen, and C. Tseng, “Robust ${H}_{\infty}$ filtering for nonlinear stochastic systems,”
*IEEE Transactions on Signal Processing*, vol. 53, no. 2, part 1, pp. 589–598, 2005. View at: Publisher Site | Google Scholar | MathSciNet - A. J. Shaiju and I. R. Petersen, “Discrete-time robust ${H}_{\infty}$ control of a class of nonlinear uncertain systems,”
*International Journal of Robust and Nonlinear Control*, vol. 23, no. 14, pp. 1629–1641, 2013. View at: Publisher Site | Google Scholar | MathSciNet - S. Zhang, Z. Wang, and D. Ding, “${H}_{\infty}$ fuzzy control with randomly occurring infinite distributed delays and channel fadings,”
*IEEE Transactions on Fuzzy Systems*, vol. 22, no. 1, pp. 189–200, 2014. View at: Google Scholar - X. Chang and G. Yang, “New results on output feedback ${H}_{\infty}$ control for linear discrete-time systems,”
*IEEE Transactions on Automatic Control*, vol. 59, no. 5, pp. 1355–1359, 2014. View at: Publisher Site | Google Scholar | MathSciNet - Z. Wang, D. Ding, H. Dong, and H. Shu, “${H}_{\infty}$ consensus control for multi-agent systems with missing measurements: the finite-horizon case,”
*Systems and Control Letters*, vol. 62, no. 10, pp. 827–836, 2013. View at: Publisher Site | Google Scholar - L. Wang, G. Wei, and W. Li, “Probability-dependent ${H}_{\infty}$ synchronization control for dynamical networks with randomly varying nonlinearities,”
*Neurocomputing*, vol. 133, pp. 369–376, 2014. View at: Publisher Site | Google Scholar - H. Dong, Z. Wang, and H. Gao, “Distributed H∞ filtering for a class of markovian jump nonlinear time-delay systems over lossy sensor networks,”
*IEEE Transactions on Industrial Electronics*, vol. 60, no. 10, pp. 4665–4672, 2013. View at: Publisher Site | Google Scholar - H. Gao, J. Lam, L. Xie, and C. Wang, “New approach to mixed ${H}_{2}={H}_{\infty}$ filtering for polytopic discrete-time systems,”
*IEEE Transactions on Signal Processing*, vol. 53, no. 8, part 2, pp. 3183–3192, 2005. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Yamin Wang 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.