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Control of Singular Markovian Jump Systems with Bounded Transition Probabilities
This paper discusses control problems of continuous-time and discrete-time singular Markovian jump systems (SMJSs) with bounded transition probabilities. Improved sufficient conditions for continuous-time SMJSs to be regular, impulse free, and stochastically stable with -disturbance attenuation are established via less conservative inequality to estimate the transition jump rates, so are the discrete-time SMJSs. With the obtained conditions, the design of a state feedback controller which ensures the resulting closed-loop system to be stochastically admissible and with performance is given in terms of linear matrix inequalities (LMIs). Finally, illustrative examples are presented to show the effectiveness and the benefits of the proposed approaches.
Singular systems, also referred to as generalized systems, descriptor systems, implicit systems, differential-algebraic systems, or semistate systems , have been extensively studied in the past years due to the fact that singular systems better describe physical systems than regular ones. Many practical problems, such as those in Leontief’s dynamic system  and electrical and mechanical systems [3, 4], are modeled as singular systems. Considerable attention has been paid to investigate such continuous-time singular systems [5–8] and discrete-time singular systems [9–11], respectively. On the other hand, Markovian jump systems (MJSs) can be regarded as a special class of hybrid systems with finite operation modes whose structures are subject to random abrupt changes. The studies of MJSs are important in practical applications such as manufacturing systems, aircraft control, target tracking, robotics, solar receiver control, and power systems. Therefore, a great deal of attention has been devoted to the study of this class of systems in recent years such as [12–19].
When singular systems experience abrupt changes in their structure and parameters, it is natural to model them as SMJSs. Reference  considered the stochastic stability and robust stochastic stability conditions for continuous-time SMJSs. The delay-dependent control problem of continuous-time SMJSs was studied in  via Jensen’s integral inequality approach after introducing free weighting matrices, while  gave sufficient conditions for uncertain discrete-time SMJSs with mode-dependent time delays through transforming them into a standard linear system. Particularly,  proposed more general condition of control for discrete-time SMJSs, which does not require any decomposition of the original system, and the construction of controller gain does not need system decomposition. However, all the above-mentioned criteria on MJSs require the values of transition probabilities to be exactly know. Recently, Boukas discussed control of discrete-time normal state-space MJSs with bounded transition probabilities in  and stabilization of continuous-time SMJSs with full or partial knowledge of transition rates in .
Following the work of [24, 25], in this paper, the control problem is considered for both continuous-time and discrete-time SMJSs with bounded transition probabilities. Sufficient criteria guaranteeing SMJSs being stochastically admissible with an performance are presented in terms of LMIs, which are obtained by using an improved inequality to obtain less conservative estimation on transition probabilities. Based on these, a state feedback controller such that the resulting closed-loop system is stochastically admissible and possesses a prescribed performance is given. Numerical examples are provided to demonstrate the validity of developed methods.
Notation. denotes the dimensional Euclidean space. is the set of all real matrices. denotes the Euclidean norm. is the expectation operator with respect to some probability measure. In symmetric block matrices, we use “” as an ellipsis for the terms induced by symmetry, for a block-diagonal matrix, and .
2.1. Continuous-Time Markovian Jump Linear Systems
Consider the following continuous-time SMJS described by where is the state vector, is the control input, is the disturbance input, and is the control output. Matrix may be singular assumed . Parameter is the continuous-time Markov processes with right continuous trajectories taking values in a finite set with transition probabilities: where , , and the transition probability rate satisfies , for , and
For notational simplicity, in the sequel, for each possible , , the matrix will be denoted by , and so on.
Assumption 1. The transition probabilities for system (1) are assumed to be unknown but vary between two known bounds satisfying the following: where .
Definition 2 (see ). The nominal system (1) is said to be regular if is not identically zero for every .
The nominal system (1) is said to be impulse free if for every .
The nominal system in (1) is said to be stochastically stable if, when and , there exists a constant such that
The nominal system in (1) is said to be stochastically admissible if it is regular, impulse free, and stochastically stable.
Definition 3. Given , the nominal system in (1) is said to be stochastically admissible with -disturbance attenuation if it is stochastically admissible and satisfying the following such that holds for zero-initial condition and any nonzero .
In this paper, the controller such that the resulting closed-loop system is stochastically admissible with (6) is where is to be determined.
Lemma 4. Given a scalar , the unforced system (1) is stochastically admissible with performance if there exist matrices such that
2.2. Discrete-Time Markovian Jump Linear Systems
Consider the following discrete-time SMJS described by where is the state vector, is the control input, is the disturbance input, and is the control output. Matrix may be singular assumed . Parameter is a discrete-time, discrete-state Markovian chain taking values in a finite set with transition probabilities: where , for , and
Assumption 5. The transition probabilities for (10) are assumed to be unknown but vary between two known bounds satisfying the following:
From , we know that and may not equal 1. It is not necessary to exactly know the transition probabilities but only the bounds of , which are and .
Definition 6 (see ). The nominal system in (9) is said to be regular if is not identically zero for every .
The nominal system in (9) is said to be causal if for every .
The nominal system in (9) is said to be stochastically stable if, when and , there exists a constant such that
The nominal system in (9) is said to be stochastically admissible if it is regular, causal, and stochastically stable.
Definition 7. Given , the nominal system in (9) is said to be stochastically admissible with -disturbance attenuation such that holds for zero-initial condition and any nonzero .
The corresponding controller for system (9) is where is to be determined.
Lemma 8. Given a scalar , the unforced system (9) is stochastically admissible with performance if there exist matrices such that where .
Lemma 9 (see). Given any real matrices , , and with appropriate dimensions and such that and is symmetric, then, one has
3. Main Results
3.1. Continuous-Time Markovian Jump Linear Systems
Proof. From (3) and (4), we have
Then, by (8) and (21), we conclude that
Since and , and , we have
Via (22) and (23), we have (25) implying (22); that is, where
Let ; then pre- and postmultiply (25) by and its transpose. Taking into account (24) and via Schur complement, we obtain that (19) and (21) imply (8). This completes the proof.
Theorem 12. Given a scalar , there exists a state feedback controller in the form of (7) such that the resulting closed-loop system is stochastically admissible with (6), if there exist matrices , , and such that wherehold for all with constraint (4). The controller gain can be constructed as
Remark 13. It is seen that such conditions have equation constraint (28), which are not strict LMIs. In addition, the desired controller (7) is mode dependent, which requires its system mode to be available online. This makes the scope of application limited. In order to deal with this practical condition, mode-independent control is usually used. Based on the methods in [28, 29], such problems can be solved easily.
3.2. Discrete-Time Markovian Jump Linear Systems
Proof. From (16) and (17), similar to , we have
Since with , and (12), we have
From (12), we obtain that
Taking into account (34), (36), and (37), we have where
Then, by the Schur complement, we see that (32) is equivalent to (38). This completes the proof.
Remark 15. It should be remarked that when in (9),  gave important criteria under the constraint (12) via and , where must be positive-definite matrix which is not applicable in SMJSs. In SMJS (9), is only required to be nonsingular matrix. In the above proof, we remove this constraint via (36). In particular, if , then . Let ; we have ; thus, we have .
Theorem 16. Given a scalar , there exists a state feedback controller in the form of (15) such that the resulting closed-loop system (9) is stochastically admissible with (14), if there exist matrices , , , and such thatwhere hold for all with constraint (12). The controller gain can be constructed as
4. Numerical Examples
In this section, numerical examples are given to demonstrate the effectiveness of proposed theory.
Example 1. Consider the following continuous-time SMJS as (1) described as
Assume the bound transition probability matrices are
By the method in , we find that it is infeasible for this system no matter takes any values. However, for , via Theorem 12, we obtain the controller gain
Example 3. In order to do some comparison, consider the following discrete-time normal state-space MJS which can be obtained by (9) with and is described as
Assume the bound transition probability matrices are
By the method in , we have the minimum , while via Theorem 16, we obtain . By this example, it is seen that our method is less conservative.
In this paper, the problems of control for both continuous-time and discrete-time SMJSs with bounded transition probabilities have been investigated. Less conservative sufficient criteria for SMJSs to be stochastically admissible with performance are given by the LMI approach. Based on the obtained conditions, a kind of state feedback controller, such that not only stochastic admissibility but also a prescribed performance level is guaranteed, can be computed. Finally, numerical examples are provided to illustrate the advantage and effectiveness of the presented results in this paper.
This work was supported by Scientific Research Fund of Liaoning Provincial Education Department under Grant no. L2012464, Young Scientific Research Fund of Shenyang Institute of Engineering under Grant no. LGQN-1308, the National Natural Science Foundation of China under Grants nos. 61104066 and 61374043, and the Program for Liaoning Excellent Talents in University under Grant no. LJQ2013040.
- L. Dai, Singular Control Systems, vol. 118, Springer, Berlin, Germany, 1989.
- M. S. Silva and T. P. de Lima, “Looking for nonnegative solutions of a Leontief dynamic model,” Linear Algebra and Its Applications, vol. 364, no. 1, pp. 281–316, 2003.
- S. L. Campbell, Singular Systems of Dirential Equations, Pitman, London, UK, 1980.
- P. C. Muller, “Linear mechanical descriptor systems: identification, analysis and design,” in Proceedings of the of IFAC Conference on Control of Independent Systems, pp. 501–506, Belfort, France, 1997.
- S. Xu, P. van Dooren, R. Ştefan, and J. Lam, “Robust stability and stabilization for singular systems with state delay and parameter uncertainty,” IEEE Transactions on Automatic Control, vol. 47, no. 7, pp. 1122–1128, 2002.
- G. Lu and D. W. C. Ho, “Generalized quadratic stability for continuous-time singular systems with nonlinear perturbation,” IEEE Transactions on Automatic Control, vol. 51, no. 5, pp. 818–823, 2006.
- S. Zhu, C. Zhang, Z. Cheng, and J. Feng, “Delay-dependent robust stability criteria for two classes of uncertain singular time-delay systems,” IEEE Transactions on Automatic Control, vol. 52, no. 5, pp. 880–885, 2007.
- S. Ma, C. Zhang, and Z. Cheng, “Delay-dependent robust control for uncertain discrete-time singular systems with time-delays,” Journal of Computational and Applied Mathematics, vol. 217, no. 1, pp. 194–211, 2008.
- G. Zhang, Y. Xia, and P. Shi, “New bounded real lemma for discrete-time singular systems,” Automatica, vol. 44, no. 3, pp. 886–890, 2008.
- Q. L. Zhang, W. Q. Liu, and D. Hill, “A Lyapunov approach to analysis of discrete singular systems,” Systems & Control Letters, vol. 45, no. 3, pp. 237–247, 2002.
- S. Xu, C. Yang, Y. Niu, and J. Lam, “Robust stabilization for uncertain discrete singular systems,” Automatica, vol. 37, no. 5, pp. 769–774, 2001.
- F. Sun, H. Liu, K. He, and Z. Sun, “Reduced-order filtering for linear systems with Markovian jump parameters,” Systems & Control Letters, vol. 54, no. 8, pp. 739–746, 2005.
- H. Shen, X. Huang, J. Zhou, and Z. Wang, “Global exponential estimates for uncertain Markovian jump neural networks with reaction-diffusion terms,” Nonlinear Dynamics, vol. 69, no. 1-2, pp. 473–486, 2012.
- 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.
- S. Xu, T. Chen, and J. Lam, “Robust filtering for uncertain Markovian jump systems with mode-dependent time delays,” IEEE Transactions on Automatic Control, vol. 48, no. 5, pp. 900–907, 2003.
- E. K. Boukas and Z. K. Liu, “Robust control of discrete-time Markovian jump linear systems with mode-dependent time-delays,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 46, no. 12, pp. 1918–1924, 2001.
- S. P. He and F. Liu, “Robust stabilization of stochastic Markovian jumping systems via proportional integral control,” Signal Processing, vol. 91, no. 11, pp. 2478–2486, 2011.
- G. L. Wang and Q. L. Zhang, “Adaptive control of stochastic nonlinear systems with Markovian switching,” International Journal of Adaptive Control and Signal Processing, vol. 26, no. 9, pp. 848–860, 2012.
- G. L. Wang, Q. L. Zhang, and V. Sreeram, “Design of reduced-order filtering for Markovian jump systems with mode-dependent time delays,” Signal Processing, vol. 89, pp. 187–196, 2009.
- E. K. Boukas, “On robust stability of singular systems with random abrupt changes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 3, pp. 301–310, 2005.
- Z. Wu, H. Su, and J. Chu, “Delay-dependent control for singular Markovian jump systems with time delay,” Optimal Control Applications & Methods, vol. 30, no. 5, pp. 443–461, 2009.
- S. Ma and C. Zhang, “Robust stability and control for uncertain discrete Markovian jump singular systems with mode-dependent time-delay,” International Journal of Robust and Nonlinear Control, vol. 19, no. 9, pp. 965–985, 2009.
- J. Lam, Z. Shu, S. Xu, and E.-K. Boukas, “Robust control of descriptor discrete-time Markovian jump systems,” International Journal of Control, vol. 80, no. 3, pp. 374–385, 2007.
- E. K. Boukas, “ control of discrete-time Markov jump systems with bounded transition probabilities,” Optimal Control Applications & Methods, vol. 30, no. 5, pp. 477–494, 2009.
- E. K. Boukas, “On stability and stabilisation of continuous-time singular Markovian switching systems,” IET Control Theory & Applications, vol. 2, no. 10, pp. 884–894, 2008.
- S. Xu and J. Lam, Robust Control and Filtering of Singular Systems, vol. 332, Springer, Berlin, Germany, 2006.
- L. Xie and C. E. de Souza, “Robust control for linear systems with norm-bounded time-varying uncertainty,” IEEE Transactions on Automatic Control, vol. 37, no. 8, pp. 1188–1191, 1992.
- G. Wang and Q. Zhang, “Robust control of uncertain singular stochastic systems with Markovian switching via proportional-derivative state feedback,” IET Control Theory & Applications, vol. 6, no. 8, pp. 1089–1096, 2012.
- G. L. ] Wang, “Robust stabilization of singular Markovian jump systems with uncertain switching,” International Journal of Control, Automation, and Systems, vol. 11, no. 1, pp. 188–193, 2013.
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