Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 426271, 9 pages

http://dx.doi.org/10.1155/2013/426271

Research Article

Control for the Continuous-Time Markovian Jump Linear Uncertain Systems with Partly Known Transition Rates and Input Quantization

Xin-Gang Zhao,¹ Dan Ye,1,2 and Jian-Da Han¹

¹State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China

²College of Information Science and Engineering, Northeastern University, Shenyang 110189, China

Received 22 May 2013; Accepted 19 July 2013

Academic Editor: Bochao Zheng

Copyright © 2013 Xin-Gang 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.

Abstract.

For a class of continuous-time Markovian jump linear uncertain systems with partly known transition rates and input quantization, the state-feedback control design is considered. The elements in the transition rates matrix include completely known, boundary known, and completely unknown ones. First, an cost index for Markovian jump linear uncertain systems is introduced; then by introducing a new matrix inequality condition, sufficient conditions are formulated in terms of linear matrix inequalities (LMIs) for the control of the Markovian jump linear uncertain systems. Less conservativeness is achieved than the result obtained with the existing technique. Finally, a numerical example is given to verify the validity of the theoretical results.

1. Introduction

Recently, much attention has been devoted to the study of the stochastic stability for the Markovian jump systems, and many important results have been published [1–4]. This is because the Markovian jump systems have been widely employed to model many practical systems, such as manufacturing systems, the power systems, and the economic systems in which they may experience abrupt changes in their structures and parameters [5, 6]. It is worth noticing that these results require that the transition probabilities/rates must be known a priori. However, in many practical engineering applications, the likelihood for obtaining the perfect information on all transition probabilities/rates elements is questionable, and the cost might be expensive in some cases. Therefore, the study of the stabilization of the Markovian jump systems with partly known transition probabilities/rates becomes interesting, and some well-known results have been published. The idea for the stochastic stability of the Markovian jump linear uncertain systems with partly known transition probabilities/rates is developed in Zhang et al. [7]. It is then applied to the control design in Zhang and Boukas [8]. In those papers, the feature of the information about the transition probabilities/rates matrix considered includes two kinds of elements: completely known and completely unknown ones. As a matter of fact, the transition probabilities matrix might involve completely known, completely unknown, and boundary known elements. In Shen and Yang [9], an state-feedback controller design method is proposed for the continuous-time Markovian jump linear uncertain systems with the three kinds of transition rates matrix elements. In order to yield the design condition for analysis and synthesis, a matrix inequality is introduced to present LMIs conditions. In this paper, less conservative design conditions will be formulated by introducing a new relaxed matrix inequality condition.

On the other hand, in many modern engineering practices, all kinds of information processing devices, such as analog-to-digital and digital-to-analog converters, have been widely used. By the utilization of such information processing devices, some advantages have been brought, for example, lower cost, reduced weight and power, simple installation, and maintenance. However, some new phenomena have also been induced, which might cause server deterioration of system performance or even lead to system instability. Signal quantization is one of the important aspects that should be fully considered in such cases, which always exists in computer-based control systems. Nowadays, many well-known results have been published on quantized feedback control. For example, the feedback stabilization problem is considered by utilizing dynamic quantizers [10–13] and static quantizers [14–17]. In addition, the filter design [18] and the control design [19] are also investigated. Specially, quantization errors have adverse effects on the network control systems which can often be modeled as Markovian jump systems. In Xiao et al. [20], the stabilization problem for single-input discrete Markovian jump linear uncertain systems via mode-dependent quantized state-feedback is addressed, but the transition rates are assumed to be completely known.

To the best of our knowledge, no result has been presented for the control design of the continuous-time Markovian jump linear uncertain systems with partly known transition rates and input signal quantization. In this paper, the control for a class of continuous-time Markovian jump linear uncertain systems with respect to partly known transition rates and input signal quantization is addressed. The structure of the controller consists of two parts: the nonlinear part is provided to eliminate the effect of input quantization, and the linear part is obtained by solving LMIs for achieving the performance against unknown transition rates and model uncertainties. In comparison with the design utilizing the LMIs technique in Shen and Yang [9], the design method has less conservativeness by introducing a relaxed inequality condition.

The rest of this paper is organized as follows. The problem statement and preliminaries are presented in Section 2. The main results are given in Section 3. In Section 4, a numerical example is presented to illustrate the effectiveness of the results, and the conclusions are drawn in Section 5.

Notations. Throughout this paper, the following notations are used. denotes the -dimensional Euclidean space; denotes the transpose of matrix ; and represent the identity matrix and a zero matrix in appropriate dimensions, respectively; denotes the mathematical expectation operator; , where and are symmetric matrices, means that is positive definite (positive semi-definite); denotes the -norm of the vector ; that is, , where . When , . For matrix , is used to present the matrix -norm: . The notation , in particular denotes the absolute value of a scalar, the standard Euclidean norm of a vector, and the induced norm of a matrix, respectively. In symmetric block matrices, an is used to represent a term that is induced by symmetry. Finally, the symbol is used to represent .

2. Problem Statement and Preliminaries

Consider a class of the continuous-time Markovian jump linear uncertain systems in the following probability space :

where is the system state and is the control input. is a continuous-time Markovian process with right continuous trajectories taking values in the finite set . It governs the switching among the different system modes with the following mode transition probabilities:

where , , and denote the switching rate from mode at time to mode at time and that for each .

In general, the Markovian process transition rates matrix is defined by:

In this paper, the transition rates of the jumping process are assumed to be partly available; that is, some elements in matrix have been exactly known, some ones have been merely known with lower and upper bounds, and others may have no information to use. For instance, for the system (1) with four operation modes, the transition rates matrix might be described by

where represents the completely unknown element of the transition rates matrix and the parameters and represent the elements with known lower and upper bounds. That is, , and , where , , , and are known parameters; denotes the precisely known element.

For clarity, we denote that ,  , with

Furthermore, let ; then one can obtain . If , it can be described as

Similarly, if , let us denote that

where denotes the th element in with the index in the th row of the matrix . denotes the th completely unknown element with the index in the th row of the matrix . and represent the number of elements in and , respectively. For example, considering the transition rates matrix (4), one can easily check that with , , , with ,  , and .

Remark 1. When the lower and upper bounds of the elements in are equal, the transition rates matrix is reduced to the considered case in Zhang et al. [7]. It is obvious that the solving method there can only treat as the completely unknown case which can result in some conservativeness.

For convenience, the notations , , , and are used for each possible value , , where and are known constant matrices with appropriate dimensions. Then, the system (1) can be described by

The following assumptions are assumed to be valid.

Assumption 1. The pair is controllable.

Assumption 2. Consider the following: , , and , where , , , and are known constant matrices with appropriate dimensions, and are time-varying uncertain matrices satisfying and , and the parameter satisfies .

In addition, the quantizer is defined by an operator function that rounds towards the nearest integer; that is,

where 0 is called a quantizing level of the quantizer. In computer-based control systems, the value of depends on the sampling accuracy and is known a priori. is the uniform quantizer with the fixed level . Define , since each component of is bounded by the half of the quantizing level ; thus, we have .

The objective of this paper is to design the state-feedback control law

such that the resulting closed-loop system is stochastically stable and obtains as small value of the cost index for the Markovian jump linear uncertain systems given in the following as possible

where and are positive definite matrices. The nonlinear part of the controller is designed against the effect of signal quantization, and the linear part is proposed to deal with model uncertainties and to unknown transition rates and achieve optimal performance.

Remark 2. When , , and , the above cost index is reduced to the following, (12) which is given in Yun et al. [17]:

Some useful lemmas are firstly presented before formulating the main result.

Lemma 3 (see [21]). Given a symmetric matrix and matrices and with appropriate dimensions, then for all satisfying , if and only if there exists a scalar such that the following inequality holds:

Lemma 4. For any given and matrices   , if there exists , then the following inequality holds:

Proof. According to and , one can obtain that , which further imply that inequality (14) holds.

Remark 5. In Shen and Yang [9], the inequality is introduced to obtain design conditions. It is clear that the utilization of Lemma 4 will result in less conservativeness since for any .

Lemma 6. For , , , and , the following inequality holds:

Lemma 7 (see [22]). For the symmetric and positive definite matrices and , if , then .

3. Main Results

Theorem 8. For the system (1) subject to Assumptions 1 and 2, suppose that there exist the symmetric positive definite matrices , general matrices , and positive scalars , , and such that

where

Then, the controller designed as

can drive the state trajectory to the origin asymptotically and can obtain the cost by the minimum of , where , , and .

Proof. Take the Lyapunov function candidate ; then, along the system trajectory of plant (1), the weak infinitesimal operator of the process for plant (8) at the point is given by Kushner [23] as follows:

According to Assumption 2, in Theorem 8, and Lemma 7, one can obtain that

It follows from (21) and (22) that

Furthermore, consider the following inequality:

Since and , the above inequality can be rewritten as

Using Lemma 3, it is equivalent to

That is,

Two cases will be considered.

Case 1 (). In this case, using Lemma 4, one can see that

Substituting into the first inequality in (22) and using the boundary information of the elements of the transition rates matrix, one can achieve that

Case 2 (; namely, it is completely unknown). In such case, let us take into (21); then, one can see that

Then, the stochastic stability can be guaranteed when

For Case 1, pre- and postmultiplying in the first inequality in (27) and using Lemma 7 in the second inequality in (21), one can get that

Let , , and ; then, we have

Applying the Schur complement formula, one can get (16).

For Case 2, pre- and postmultiplying in the first inequality in (28) and applying Lemma 7 to the second inequality in (28), one can obtain that

Let and ; then, one can see that

Thus, the LMIs in (17) are derived by using the Schur complement formula.

From the above proof, one can see that

It follows from Kushner [23] that

Since , by some simple calculation, one can achieve that

Therefore, the minimum cost can be obtained by minimizing . Thus, the proof is achieved.

Remark 9. The merit of the proposed results lies in that the transition rates of the jumping process are assumed to be more general, which means that some elements in the transition rates matrix have been exactly known, some ones have been merely known with lower and upper bounds, and others may have no information to use. Dealing with the unknown transition rates, a less conservative method is used. At the same time, the impact of the input signal quantization on the system is also considered. Finally, the controller design conditions are presented in the framework of LMIs.

In order to make comparison with the design method using the LMIs technique in Shen and Yang [9], we present the conditions designed by the utilization of Lemma 2 there.

Proposition 10. For the system (1) subject to Assumptions 1 and 2, suppose that there exist symmetric positive definite matrices , general matrices , and positive scalars , , and such that

where

Then, the controller designed as

can drive the state trajectory to the origin asymptotically and can obtain the cost by minimizing , where , , and .

Proof. The proof process is similar to that in Theorem 8 but with replacing , , and with in (22). The details are omitted here for space limitation.

4. Numerical Example

An example is presented to illustrate the effectiveness of the proposed method.

Consider the MJLSs with four operation modes as follows:

The considered transition rates matrix is given as follows:

where , , and . Choosing and solving (16)–(18) in Theorem 8 and (34)–(38) in Proposition 10, one can obtain the optimal performance indices shown in Table 1. It can be seen that less conservativeness is obtained by the proposed method in Theorem 8.

Table 1: Comparison of optimal .

Theorem 8
Proposition 10

1.4518
2.4567

For simulation, one can obtain the controller gains by solving the LMIs in Theorem 8 as follows:

, , , and and by considering the positive definite matrices

The following parameters are used in the simulation: , , , , , , , , and . In order to reduce the chattering effects induced by , is substituted by , where , , , and .

The switching mode, the control input, and the response curves of the system states are presented in Figures 1, 2, and 3, respectively. Among them, Figure 1 shows a possible system modes evolution which meets the transition rates given in this example. As shown in Figure 1, the system has 4 modes and is in different modes at a different time. Figure 2 shows the curve of the control input . With this controller, Figure 3 depicts the state response curves of the closed-loop system. It can be seen that the considered continuous-time Markovian jump linear uncertain system is stochastically stable in spite of mismatched uncertainty, the input signal quantization, and the partly known transition rates covering the completely known, the boundary unknown, and the completely unknown elements in the transition rates matrix.

Figure 1: Evolution of the system mode.

Figure 2: The curve of the control input .

Figure 3: The response curve of the state .

5. Conclusions

control design for a class of the continuous-time Markovian jump linear uncertain systems with partly unknown transition rates and input quantization has been investigated. The cost index for the Markovian jump systems is first introduced, and then sufficient conditions of the control for the Markovian jump linear uncertain systems with unknown transition rates have been presented. The controller is constructed by two parts. The nonlinear part is proposed to eliminate the effect of input quantization. The linear part is designed by solving the LMIs conditions for achieving the performance against model uncertainties and unknown transition rates. In comparison with the existing result in the literature, less conservativeness has been obtained by introducing new relaxed inequality conditions. Finally, a numerical example is given to show the effectiveness of the proposed design method.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grants nos. 61273155, 61273355, 61322312, and 61273356), the Foundation of State Key Laboratory of Robotics (Grant no. Z2013-06), the New Century Excellent Talents in the University (Grant no. NCET-11-0083), a Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant no. 201157), the Fundamental Research Funds for the Central Universities (Grant no. N120504003).

References

E.-K. Boukas, Stochastic Switching Systems: Analysis and Design, Control Engineering, Birkhäuser, Boston, Mass, USA, 2006.

S. Ma and E.-K. Boukas, “A singular system approach to robust sliding mode control for uncertain Markov jump systems,” Automatica, vol. 45, no. 11, pp. 2707–2713, 2009.

P. Shi, Y. Xia, G. P. Liu, and D. Rees, “On designing of sliding-mode control for stochastic jump systems,” IEEE Transactions on Automatic Control, vol. 51, no. 1, pp. 97–103, 2006.

L. Wu, P. Shi, and H. Gao, “State estimation and sliding-mode control of Markovian jump singular systems,” IEEE Transactions on Automatic Control, vol. 55, no. 5, pp. 1213–1219, 2010.

W. Li, H. Su, D. Wei, and K. Wang, “Global stability of coupled nonlinear systems with Markovian switching,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2609–2616, 2012.

E. F. Costa and J. B. R. do Val, “On the observability and detectability of continuous-time Markov jump linear systems,” SIAM Journal on Control and Optimization, vol. 41, no. 4, pp. 1295–1314, 2002.

L. Zhang, E.-K. Boukas, and J. Lam, “Analysis and synthesis of Markov jump linear systems with time-varying delays and partially known transition probabilities,” IEEE Transactions on Automatic Control, vol. 53, no. 10, pp. 2458–2464, 2008.

L. Zhang and E.-K. Boukas, “ control of a class of extended Markov jump linear systems,” IET Control Theory & Applications, vol. 3, no. 7, pp. 834–842, 2009.

M. Shen and G.-H. Yang, “ state feedback controller design for continuous Markov jump linear systems with partly known information,” International Journal of Systems Science, vol. 43, no. 4, pp. 786–796, 2012.

R. W. Brockett and D. Liberzon, “Quantized feedback stabilization of linear systems,” IEEE Transactions on Automatic Control, vol. 45, no. 7, pp. 1279–1289, 2000.

D. Liberzon, “Hybrid feedback stabilization of systems with quantized signals,” Automatica, vol. 39, no. 9, pp. 1543–1554, 2003.

B.-C. Zheng and G.-H. Yang, “Quantised feedback stabilisation of planar systems via switching-based sliding-mode control,” IET Control Theory & Applications, vol. 6, no. 1, pp. 149–156, 2012.

B.-C. Zheng and G.-H. Yang, “Decentralized sliding mode quantized feedback control for a class of uncertain large-scale systems with dead-zone input,” Nonlinear Dynamics, vol. 71, no. 3, pp. 417–427, 2013.

M. Fu and L. Xie, “The sector bound approach to quantized feedback control,” IEEE Transactions on Automatic Control, vol. 50, no. 11, pp. 1698–1711, 2005.

M. Fu and L. Xie, “Quantized feedback control for linear uncertain systems,” International Journal of Robust and Nonlinear Control, vol. 20, no. 8, pp. 843–857, 2010.

E. Tian, D. Yue, and X. Zhao, “Quantised control design for networked control systems,” IET Control Theory and Applications, vol. 1, no. 6, pp. 1693–1699, 2007.

S. W. Yun, Y. J. Choi, and P. Park, “ control of continuous-time uncertain linear systems with input quantization and matched disturbances,” Automatica, vol. 45, no. 10, pp. 2435–2439, 2009.

W.-W. Che and G.-H. Yang, “Quantised filter design for discrete-time systems,” International Journal of Control, vol. 82, no. 2, pp. 195–206, 2009.

C. Peng and Y.-C. Tian, “Networked control of linear systems with state quantization,” Information Sciences, vol. 177, no. 24, pp. 5763–5774, 2007.

N. Xiao, L. Xie, and M. Fu, “Stabilization of Markov jump linear systems using quantized state feedback,” Automatica, vol. 46, no. 10, pp. 1696–1702, 2010.

I. R. Petersen, “A stabilization algorithm for a class of uncertain linear systems,” Systems & Control Letters, vol. 8, no. 4, pp. 351–357, 1987.

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.

H. J. Kushner, Stochastic Stability and Control, Mathematics in Science and Engineering, vol. 33, Academic Press, New York, NY, USA, 1967.