## Robust Control, Optimization, and Applications to Markovian Jumping Systems

View this Special IssueResearch Article | Open Access

# Finite-Time Control for a Class of Discrete-Time Markov Jump Systems with Actuator Saturation via Dynamic Antiwindup Design

**Academic Editor:**Hao Shen

#### Abstract

We deal with the finite-time control problem for discrete-time Markov jump systems subject to saturating actuators. A finite-state Markovian process is given to govern the transition of the jumping parameters. A controller designed for unconstrained systems combined with a dynamic antiwindup compensator is given to guarantee that the resulting system is mean-square locally asymptotically finite-time stabilizable. The proposed conditions allow us to find dynamic anti-windup compensator which stabilize the closed-loop systems in the finite-time sense. All these conditions can be expressed in the form of linear matrix inequalities and therefore are numerically tractable, as shown in the example included in the paper.

#### 1. Introduction

It is well known that more and more attention has been paid to the study of actuator saturation due to its practical and theoretical importance. Therefore, various approaches were investigated to handle systems with actuator saturation and dynamic antiwindup approach which is one of the most effective ways to deal with it. To this end, a great number of results have been reported in the literature; see, for example, [1, 2]. Furthermore, the stabilization problem of singular Markovian jump systems with discontinuities and saturation inputs was presented in [3]. Via dynamic antiwindup fuzzy design, the robust stabilization problem of state delayed T-S fuzzy systems with input saturation was proposed in [4].

On the other hand, Markov jump is frequently encountered in many practical systems. Therefore, the study of Markov jump systems has been a hot research topic due to its importance, and many results have been proposed based on various control techniques, such as robust control [5–9], control [10, 11], Passivity-based control [12–14], fuzzy dissipative control [15], and neural networks control [14, 16]. Furthermore, observer based finite-time control problem of discrete-time Markov jump systems was studied [17].

As it is well known, when dealing with the stability of s system, a distinction should have been made between classical Lyapunov stability and finite-time stability (FTS). Conversely, a system is said to be finite-time stable if, once we fix a time-interval, its state does not exceed some bounds during this time-interval. Some results on FTS have been carried out; see, [18, 19]. Furthermore, finite-time filtering problem of time-delay stochastic jump systems with unbiased estimation was proposed in [20]. By applying dynamic observer-based state feedback and the Lyapunov-Krasovskii functional approach, the finite-time control problem for time-delay nonlinear jump systems was addressed in the work of He and Liu [21]. However, to the best of our knowledge, the problem of finite-time stabilization of discrete-time stochastic systems subject to actuator saturation has not been fully investigated and it is the main purpose of our study.

In this paper, the attention is focused on the finite-time control problem of discrete-time Markov jump systems with actuator saturation based on dynamic antiwindup approach. A controller designed for unconstrained systems combined with a dynamic antiwindup compensator is given to ensure the stochastic finite-time boundedness and stochastic finite-time stabilization of the resulting closed-loop system for all admissible disturbances. The desired compensator can be designed via solving a convex optimization problem. Finally, a numerical example is employed to show the effectiveness of the proposed method.

*Notation.* Throughout the paper, for symmetric matrices and , the notation (resp., ) means that the matrix is positive semidefinite (resp., positive definite). is the identity matrix with appropriate dimension. The notation represents the transpose of the matrix (resp., ) means the largest (resp., smallest) eigenvalue of the matrix ; is a probability space; is the sample space; is the -algebra of subsets of the sample space and is the probability measure on ; denotes the expectation operator with respect to some probability measure . Matrices, if not explicitly stated, are assumed to have compatible dimensions. The symbol is used to denote a matrix which can be inferred by symmetry. .

#### 2. Preliminaries and Problem Description

Consider the following discrete-time Markov jump system in the probability space : where is the state vector, is the control input, and is the saturated control input. is the external disturbances, is the measurement output, and is the performance output. is a discrete-time Markov process and takes values from a finite set with transition probabilities given by where , for , and . Moreover, the transition rates matrix of the system is defined by The plant inputs are supposed to be bounded as follows: For the system , to simplify the notation, we denote for each , and the other symbols are similarly denoted. Assume that a linear controller is designed for any ; then, where is the controller state and is the controller output; and will be used for antiwindup augmentation. In absence of actuator saturation, the unconstrained closed-loop is formed by setting the following:

*Assumption 1. *The unconstrained closed-loop system (1)–(5) is well posed and internally stable.

In the presence of actuator saturation, the relation between and is that . To minimize performance degradation caused by saturation, the following antiwindup compensator is designed for the closed-loop systems: where . The resulting nonlinear closed-loop system (1), (5), (7) is depicted in Figure 1 and can be represented in the following compact form: where For this system, we introduce the following definitions and assumption.

*Assumption 2 (see [17]). *The external disturbance is varying and satisfies the following constraint condition:

*Definition 3 (see [17]). *The resulting closed-loop system (8) is stochastic finite-time stable (SFTB) with respect to with , and , if

*Definition 4 (see [17]). *The resulting closed-loop system (8) is said to be stochastic finite-time stable with respect to with , , , and , if the system (8) is SFTB with respect to , and under the zero-initial condition, the output satisfies
for any nonzero which satisfies (10), where is a prescribed positive scalar.

#### 3. Main Results

In this section, we investigate the stabilization analysis of the unconstrained systems and the antiwindup controller design of the resulting closed-loop system. Some sufficient conditions in terms of LMI are given. Before presenting the main results, we give some lemmas as follows.

Lemma 5 (see [4]). *For the closed-loop systems (8) with the matrix , the appropriate matrix is given, if is in the set , where is defined as follows:
**
then for any diagonal positive matrix , one has the following:
*

Lemma 6 (see [12]). *For the given symmetric matrix ,
**
where , , and , the following conditions are equivalent: *(1)*;
*(2)*;
*(3)*. *

##### 3.1. Design of Controller

In this section, we design the controller for the unconstrained systems with and . Combining system (1) with controller (5), we have where

Theorem 7. *For each , the unconstrained system (16) is SFTB with respect to with , if there exist scalars , , , and the given , two sets of mode-dependent symmetric positive-defined matrices and , such that the following conditions hold:
**
where
*

*Proof. *Define the following Lyapunov function for each :
It is readily obtained that
where
By using of Schur complement lemma to (18), and note that and , we derive ; then, we have
It follows that
It is shown that
Then we have
Since , it is easily found that
Letting
and noting that
it can be verified that
Similarly, for all , we can obtain
Then, it is not difficult to find that
which implies that
Then, one can obtain that
Setting
it is easy to see that
It is obvious that (40) is equivalent to (19).

This completes the proof.

##### 3.2. Design of Dynamic Antiwindup Compensator

Theorem 8. *For each , with antiwindup compensator (7), such that the resulting closed-loop system (10) is with respect to with , if there exist scalars , , and , three sets of mode-dependent symmetric positive-defined matrices , and diag positive-defined matrices , and two sets of mode-dependent matrices and , such that the following conditions hold:
**
where
**
with
*

*Proof. *Define the following Lyapunov function for each :
It is readily obtained that
where
Then, by pre- and postmultiplying (41) by with , we have
By using of Schur complement lemma, we derive
It follows that
Since , we get
It is shown that
Then, we have
Since , it is easily found that
The following proof is similar to the process of Theorem 7.

Based on Lemma 5, it is easy to obtain that
then pre- and post-multiply (58) by which implies (43). This completes the proof.

Theorem 9. *For each , with antiwindup compensator (7), such that the resulting closed-loop system (10) is said to be Stochastic finite-time stable via state feedback with respect to , if there exist three scalars , , and , two sets of mode-dependent symmetric positive-defined matrices and diag matrices , and two sets of mode-dependent matrices and , such that the following conditions hold:
**
with
*

*Proof. *Choose the similar Lyapunov function as Theorem 7 and denote
Thus, in light of Lemma 5, we have
Then pre- and postmultiply (59) by , and considering Schur complement lemma and (65), we derive that
holds for all . According to (66), one can obtain that
Then, we have
Under the zero-value initial condition and noting that , for all , it is shown that
Since and from (69), we have
The following proof is similar to the process of Zhang and Liu [17].

Since , it follows that
and then pre- and post-multiply (71) by and its transpose, respectively; we derive condition (61). This completes the proof.

#### 4. Illustrative Examples

In this section, a numerical example is provided to demonstrate the effectiveness of the proposed method. Consider the following systems with four operation modes.

Mode 1 is Mode 2 is Mode 3 is Mode 4 is With the given designed controllers, The transition rate matrix is given by the following: In this case, we choose the initial values for , , and ; Theorem 7 yields to , and the bounds of the input saturation .

Based on Theorem 9, we derive

*Remark 10. *Figures 2, 3, and 4 are given on the last page. Figure 1 is of the jump rates, Figure 2 and Figure 3 are state response of open and closed-loop system. Based on the figures provided, the controller and the compensator we designed guarantee that the resulting closed-loop systems are mean-square locally asymptotically finite-time stabilizable.

#### 5. Conclusions and Future Work

In this paper, the finite-time stabilization problem for a class of discrete-time Markov jump systems with input saturation has been investigated. Based on stochastic finite-time stability analysis, a controller designed for the unconstrained system with a dynamic antiwindup compensator subject to actuator saturation is given to guarantee the stochastic finite-time boundedness and stochastic finite-time stabilization of the considered closed-loop system for all admissible disturbances. Finally, the effectiveness of the proposed approach has been illustrated by simulation results. The finite-time stabilization problem of Markov jump systems with constrained input and time-delay will be considered in the future work.

#### Conflict of Interests

The authors declare no conflict of interests.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 61203047, the Natural Science Foundation of Anhui Province under Grant 1308085QF119, the Key Foundation of Natural Science for Colleges and Universities in Anhui province under Grant KJ2012A049.

#### References

- B. M. Chen, T. H. Lee, K. Peng, and V. Venkataramanan, “Composite nonlinear feedback control for linear systems with input saturation: theory and an application,”
*IEEE Transactions on Automatic Control*, vol. 48, no. 3, pp. 427–439, 2003. View at: Publisher Site | Google Scholar | MathSciNet - T. Hu, A. R. Teel, and L. Zaccarian, “Anti-windup synthesis for linear control systems with input saturation: achieving regional, nonlinear performance,”
*Automatica*, vol. 44, no. 2, pp. 512–519, 2008. View at: Publisher Site | Google Scholar | MathSciNet - J. Raouf and E. K. Boukas, “Stabilization of singular Markovian jump systems with discontinuities and saturation inputs,”
*IET Control Theory and Applications*, vol. 42, no. 5, pp. 767–780, 2011. View at: Google Scholar - X. Song, J. Lu, S. Xu, H. Shen, and J. Lu, “Robust stabilization of state delayed T-S fuzzy systems with input saturation via dynamic anti-windup fuzzy design,”
*International Journal of Innovative Computing, Information and Control*, vol. 7, no. 12, pp. 6665–6676, 2011. View at: Google Scholar - H. Liu, F. Sun, and E.-K. Boukas, “Robust control of uncertain discrete-time Markovian jump systems with actuator saturation,”
*International Journal of Control*, vol. 79, no. 7, pp. 805–812, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Li, B. Chen, Q. Zhou, and W. Qian, “Robust stability for uncertain delayed fuzzy Hopfield neural networks with Markovian jumping parameters,”
*IEEE Transactions on Systems, Man, and Cybernetics B*, vol. 39, no. 1, pp. 94–102, 2009. View at: Publisher Site | Google Scholar - H. Shen, J. H. Park, L. Zhang, and Z. Wu, “Robust extended dissipative control for sampled-data Markov jump systems,”
*International Journal of Control*, 2013. View at: Publisher Site | Google Scholar - H. Shen, X. Song, and Z. Wang, “Robust fault-tolerant control of uncertain fractional-order systems against actuator faults,”
*IET Control Theory and Applications*, vol. 7, no. 9, pp. 1233–1241, 2013. View at: Publisher Site | Google Scholar | MathSciNet - Z. Wu, P. Shi, H. Su, and J. Chu, “Stochastic synchronization of Markovian jump neural networks with time-varying delay using sampled-data,”
*IEEE Transactions on Cybernetics*, vol. 43, no. 6, pp. 1796–1806, 2013. View at: Google Scholar - S. Ma and E. K. Boukas, “Robust ${H}_{\infty}$ filtering for uncertain discrete Markov jump singular systems with mode-dependent time delay,”
*IET Control Theory and Applications*, vol. 3, no. 3, pp. 351–361, 2009. View at: Publisher Site | Google Scholar | MathSciNet - H. Shen, S. Xu, J. Zhou, and J. Lu, “Fuzzy ${H}_{\infty}$ filtering for nonlinear Markovian jump neutral systems,”
*International Journal of Systems Science*, vol. 42, no. 5, pp. 767–780, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Shen, S. Xu, X. Song, and G. Shi, “Passivity-based control for Markovian jump systems via retarded output feedback,”
*Circuits, Systems, and Signal Processing*, vol. 31, no. 1, pp. 189–202, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet - H. Shen, S. Xu, J. Lu, and J. Zhou, “Passivity-based control for uncertain stochastic jumping systems with mode-dependent round-trip time delays,”
*Journal of the Franklin Institute*, vol. 349, no. 5, pp. 1665–1680, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Wu, P. Shi, H. Su, and J. Chu, “Asynchronous ${I}_{2}$-${I}_{\infty}$ filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities,”
*Automatica*, vol. 50, no. 1, pp. 180–186, 2013. View at: Google Scholar - H. Shen, Z. Wang, X. Huang, and J. Wang, “Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback,”
*Journal of the Franklin Institute*, 2013. View at: Publisher Site | Google Scholar - G. Wang, J. Cao, and J. Liang, “Exponential stability in the mean square for stochastic neural networks with mixed time-delays and Markovian jumping parameters,”
*Nonlinear Dynamics*, vol. 57, no. 1-2, pp. 209–218, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Zhang and C. Liu, “Observer-based finite-time ${H}_{\infty}$ control of discrete-time Markovian jump systems,”
*Applied Mathematical Modelling*, vol. 37, no. 6, pp. 3748–3760, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Hong, J. Wang, and D. Cheng, “Adaptive finite-time control of nonlinear systems with parametric uncertainty,”
*IEEE Transactions on Automatic Control*, vol. 51, no. 5, pp. 858–862, 2006. View at: Publisher Site | Google Scholar | MathSciNet - S. He and F. Liu, “Observer-based finite-time control of time-delayed jump systems,”
*Applied Mathematics and Computation*, vol. 217, no. 6, pp. 2327–2338, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. He and F. Liu, “Finite-time ${H}_{\infty}$ filtering of time-delay stochastic jump systems with unbiased estimation,”
*Proceedings of the Institution of Mechanical Engineers I*, vol. 224, no. 8, pp. 947–959, 2010. View at: Publisher Site | Google Scholar - S. He and F. Liu, “Finite-time ${H}_{\infty}$ fuzzy control of nonlinear jump system with time delays via dynamic observer-based state feedback,”
*IEEE Transactions on Fuzzy Systems*, vol. 20, no. 4, pp. 605–614, 2012. View at: Google Scholar

#### Copyright

Copyright © 2014 Junjie Zhao 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.