Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 896515, 13 pages
http://dx.doi.org/10.1155/2015/896515

Research Article

Fuzzy Control for Nonlinear Singular Markovian Jump Systems with Time Delay

Li Li,1,2 Qingling Zhang,² Yi Zhang,³ and Baoyan Zhu⁴

¹College of Mathematics and Physics, Bohai University, Jinzhou, Liaoning 121013, China
²Institute of Systems Science, Northeastern University, Shenyang, Liaoning 110819, China
³School of Science, Shenyang University of Technology, Shenyang, Liaoning 110870, China
⁴College of Science, Shenyang Jianzhu University, Shenyang, Liaoning 110168, China

Received 7 January 2015; Accepted 20 April 2015

Academic Editor: P. Balasubramaniam

Copyright © 2015 Li Li 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.

This paper investigates the problem of fuzzy control for a class of nonlinear singular Markovian jump systems with time delay. This class of systems under consideration is described by Takagi-Sugeno (T-S) fuzzy models. Firstly, sufficient condition of the stochastic stabilization by the method of the augmented matrix is obtained by the state feedback. And a designed algorithm for the state feedback controller is provided to guarantee that the closed-loop system not only is regular, impulse-free, and stochastically stable but also satisfies a prescribed performance for all delays not larger than a given upper bound in terms of linear matrix inequalities. Then fuzzy control for this kind of systems is also discussed by the static output feedback. Finally, numerical examples are given to illustrate the validity of the developed methodology.

1. Introduction

Singular systems, also known as descriptor systems, have been widely studied in the past several decades. They have broad applications and can be found in many practical systems, such as electrical circuits, power systems, network, economics, and other systems [1, 2]. Due to their extensive applications, many research topics on singular systems have been extensively investigated such as the stability and stabilization [3, 4] and control problem [5, 6]. A lot of attention has been paid to the investigation of Markovian jump systems (MJSs) over the past decades. Applications of such class of systems can be found representing many physical systems with random changes in their structures and parameters. Many important issues have been studied for this kind of physical systems, such as the stability analysis, stabilization, and control [7–10]. When singular systems experience abrupt changes in their structures, it is natural to model them as singular Markovian jump systems (SMJSs) [11–13]. Time delay is one of the instability sources for dynamical systems and is a common phenomenon in many industrial and engineering systems such as those in communication networks, manufacturing, and biology [14]. So the study of SMJSs with time delay is of theoretical and practical importance [15, 16].

The fuzzy control has been proved to be a powerful method for the control problem of complex nonlinear systems. Specially, the Takagi-Sugeno (T-S) fuzzy model has attracted much attention due to the fact that it provides an efficient approach to take full advantage of the linear control theory to the nonlinear control. In recent years, this fuzzy-model-based technique has been used to deal with nonlinear time delay systems [17, 18] and nonlinear MJSs [19, 20]. But singular Markovian jump fuzzy systems (SMJFSs) are not fully studied [21, 22], which motivates the main purpose of our study. In this paper, a new method using the augmented matrix will be given to the control of SMJFSs. By this method the number of LMIs will be decreased, so the complexity of the calculation will be greatly reduced when the number of fuzzy rulers is relatively large. And, at the same time, some new relaxation matrices added will reduce the conservation of control conditions compared with previous literatures. And when using the augmented matrix to design the static output feedback control, there are not any crossing terms between system matrices and controller gains, so assumptions for the output matrix [23], the equality constraint for the output matrix [24], and the bounding technique for crossing terms are not necessary; therefore, the conservatism brought by them will not exist.

In this paper, the fuzzy control problem for a class of nonlinear SMJSs with time delay which can be represented by T-S fuzzy models is considered. Our aim is to design fuzzy state feedback controllers and static output feedback controllers for SMJFSs with time delay, such that closed-loop systems are stochastically admissible (regular, impulse-free, and stochastically stable) with a prescribed performance . Sufficient criterions are presented in forms of LMIs which are simple and easy to implement compared with previous literatures. Finally, numerical examples are given to illustrate the merit and usability of the approach proposed in this paper.

Notations. Throughout this paper, notations used are fairly standard; for real symmetric matrices and , the notation () means that the matrix is positive semidefinite (positive definite). represents the transpose of the matrix , and represents the inverse of the matrix . is the maximal (minimal) eigenvalue of the matrix . stands for a block-diagonal matrix. is the unit matrix with appropriate dimensions, and, in a matrix, the term of symmetry is stated by the asterisk “.” Let stand for the -dimensional Euclidean space, is the set of all real matrices, and denotes the Euclidean norm of vectors. denotes the mathematics expectation of the stochastic process or vector. stands for the space of -dimensional square integrable functions on . denotes Banach space of continuous vector functions mapping the interval into with the norm .

2. Basic Definitions and Lemmas

Consider a SMJFS; its th fuzzy rule is given by

 : if is , is , and is , then

where is the state vector, is the control input, is the exogenous disturbance which belongs to , and is the controlled output. is a compatible vector-valued initial function, and is an unknown but constant delay satisfying . The scalar is the number of If-Then rules. are fuzzy sets. are premise variables. may be a singular matrix with . , , , , , , , and are known constant matrices with appropriate dimensions. is a continuous-time Markovian process with right continuous trajectories taking values in a finite set given by with the transition rate matrix satisfying

where , , and , for , is the transition rate from mode at time to at time and .

By fuzzy blending, the overall fuzzy model is inferred as follows:

where , . Letting , it follows that , .

For the notational simplicity, in the sequel, for each possible , , , , , and so on.

Definition 1 (see [15, 25]). (i) For a given scalar , the SMJS with time delay

is said to be regular and impulse-free for any constant time delay satisfying , if pairs and are regular and impulse-free.
(ii) System (4) is said to be stochastically stable if there exists a finite number such that the following inequality holds:

(iii) System (4) is said to be stochastically admissible if it is regular, impulse-free, and stochastically stable.

Lemma 2 (see [26]). Given matrices , if is nonsingular, there exist matrices , such that , where , such that , , , , and .

Lemma 3 (see [27]). For matrices , , and with appropriate dimensions, the following inequality holds:

Lemma 4 (see [28]). For any constant matrix , , scalar , and vector function such that the following integration is well defined; then

Lemma 5 (see [29]). Suppose there are piecewise continuous real square matrices , , and satisfying for all . Then the following conditions hold:

(i) and are nonsingular.(ii) for some .

Lemma 6 (see [30]). If the following conditions hold:

then the following parameterized matrix inequality holds:

where and .

Based on the parallel distributed compensation, the following state feedback controller will be considered here:

where () are local controller gains, such that the closed-loop system

is stochastically admissible.

3. The Design of the State Feedback Controller

Firstly, the sufficient condition will be given such that system (11) is stochastically admissible. Combining (4) and (10), fuzzy closed-loop system (11) can be rewritten in the following form:

where

Remark 7. For systems (11) and (12), it can be seen that

By and Definition 1, it can be obtained that the regularity and nonimpulse of system (11) are equal to the regularity and nonimpulse of system (12). So the stochastic admissibility of system (11) can be studied by system (12).

Theorem 8. For a prescribed scalar , there exists a state feedback controller (10) with such that system (11) when is stochastically admissible for any constant time delay satisfying , if there exist matrices , , , , , , , and , , such that

where , , , , , , , , is any matrix with full column rank and satisfies , and are nonsingular matrices that make .

Proof. From (15), it can be concluded that and are nonsingular matrices. Because ,

Denote ; from (17), it is easy to obtain that and is symmetric; then . So it can be concluded that and are nonsingular; furthermore, . Let . So is nonsingular. By Lemma 2, , where , , and is a matrix with full column rank and satisfies . Denote , , and . So

Denote , , and . By Lemma 3, it can be obtained that

Now pre- and postmultiplying (15) by and its transpose, by Schur complement lemma, and (18)-(19), it is easy to see that

where . Pre- and postmultiplying (16) by and its transposition by Schur complement lemma, it can be seen that

From (20), it can be concluded that

On the other hand, . Then

Denote ; from (22), it can be obtained that

for every , which implies that is nonsingular. Thus, the pair is regular and impulse-free for every . By (20), it is easy to see that

Pre- and postmultiplying (25) by and its transpose, it can be obtained that

Hence,

Equation (27) implies that the pair is regular and impulse-free for every . Thus, by Definition 1, system (12) is regular and impulse-free. By Remark 7, this implies that system (11) is regular and impulse-free.
Now, it will be shown that system (11) is stochastically stable. Define a new process by ; then is a Markovian process with the initial state . Now, for , choose the following stochastic Lyapunov-Krasovskii candidate for this system:

where

Let be the weak infinitesimal generator of the random process . Then, for each ,

From (21), it is clear that

From Lemma 4, it follows that

So it can be concluded that

where

Using (20), it is easy to see that there exists a scalar such that, for every , , where .
So, for , by Dynkin’s formula, it can be obtained that

which yields

Because , denote

By the regularity and nonimpulse of system (11), is nonsingular; for each , set . It is easy to obtain

where

Then, for each , system (11) is equal to

where .
For any , using Lemma 5, there exists a scalar such that , and , and ; it follows from (40) that

where

Then, for any ,

Applying the Gronwall-Bellman lemma, it can be obtained, for any , that

Thus,

It can be seen from (40) that

where , , . Hence,

where . Therefore,

Note that

Then, from (48) and (28), it can be obtained that there exists a scalar such that

This together with (36) and (48) implies that there exists a scalar such that

Considering this and Definition 1, system (11) is stochastically stable for any constant delay satisfying . Therefore, system (11) is stochastically admissible. This completes the proof.

In the following, a set of sufficient conditions will be developed under which the considered system is guaranteed to be stochastically admissible with an performance.

Definition 9. System (11) is said to be stochastically admissible with an performance , if it is stochastically admissible when , and under zero initial condition, for nonzero ,

The following result can be presented.

Theorem 10. For a prescribed scalar , there exists a state feedback controller (10) with such that system (11) is stochastically admissible with an performance for any constant time delay satisfying , if there exist matrices , , , , , , , and , , , such that (16) and

where

and the other notations are the same as in Theorem 8.

Proof. From Theorem 8 when system (11) is stochastically admissible. Let

Under zero initial condition, it is easy to see that

where

and notations of , , and are the same as in Theorem 8. Hence, by Schur complement lemma and using the similar method in the proof of Theorem 8, from (16) and (53), it can be obtained that for all . Therefore, for any nonzero , (52) holds. Hence, according to Definition 9, the system is stochastically admissible with an performance . This completes the proof.

Remark 11. Compared with methods in [21, 22], because of the method of the augmented matrix adopted in Theorems 8 and 10, the number of LMIs needed to solve is relatively small in this paper. When the value of is relatively large, the quality of the computation is greatly reduced. some new relaxation matrices added will reduce the conservatism of control conditions compared with previous literatures, which can be seen from Example 2.

4. The Design of the Static Output Feedback Controller

When , consider the overall SMJFS as follows:

where is the system output, () are known constant matrices with appropriate dimensions, and the other notations are the same as in (3).

The following static output feedback controller will be considered here:

where () are local controller gains, such that the closed-loop system is

It is difficult to drive LMI-based conditions of the stochastic stabilization by employing the static output feedback control approach due to the appearance of crossing terms between system matrices and control gains. And system (60) can be rewritten in the following form:

where

Remark 12. For systems (60) and (61), it can be seen that

As the discussion in Remark 7, the stochastic admissibility of system (60) can be studied by means of system (61).

Theorem 13. There exists an output feedback controller (59) with controller gains () such that system (60) with is stochastically admissible, if there exist matrices , , , , , , and , , , such that (16) and

where

, , , , is any matrix with full column rank and satisfies , are nonsingular matrices that make , and the other notations are the same as in Theorem 8.

Proof. Let . Using Lemma 6, the proof process is similar to Theorem 8.

Theorem 14. For a prescribed scalar , there exists an output feedback controller (59) with controller gains () such that system (60) is stochastically admissible with an performance for any constant time delay satisfying , if there exist matrices , , , , , , and , , , such that (16) and

where

and the other notations are the same as in Theorem 13.

Remark 15. Compared with the method in [31, 32], because of the augmented matrix adopted in Theorems 13 and 14, the number of LMIs needed to solve is greatly decreased. When the value of is relatively large, the computational complexity will be reduced. On the other hand, by the augmented matrix, there are not any crossing terms between system matrices and controller gains, so assumptions for the output matrix [23], the equality constraint for the output matrix [24], and the bounding technique for crossing terms are not necessary here; therefore, the conservatism brought by them will not happen.

5. Numerical Examples

Two examples will be given to illustrate the validity of developed methods.

Example 1. To illustrate the controller synthesis, the following nonlinear time delay system is considered:

The range of is assumed to satisfy , , , , , , , and is the control input. is the disturbance input. is a Markovian process taking values in a finite set , , , , and the output vector .

Choose the vector with , , and . Then, the system is described by

It can be expressed exactly by the following fuzzy singular Markovian jump form:

where

It is seen that , . Let , ; by solving (16) and (53) in Theorem 10, controller gains are given by

To demonstrate the effectiveness, assuming the initial condition , Figures 1 and 2 show state responses of the open-loop system and the closed-loop system controlled by (10), respectively. From Figure 1, it can be seen that the open-loop system is not stochastically admissible, and Figure 2 shows that when the controller obtained by Theorem 10 is exerted to this system it is stochastically admissible.

Figure 1: State responses of the open-loop system.

Figure 2: State responses of the closed-loop system.

Example 2. Consider the example without uncertainties in [6].
Mode 1: , , , , , , , and .
Mode 2: , , , , , , , and .
, , , , and in [6] , , but in this paper , are taken.

In Figure 3, “o” represents the range of the feasible solutions using Theorem 10 in this paper, and “” represents the range of the feasible solutions using Theorem   in [6]. This illustrates that the method obtained in this paper has less conservatism.

Figure 3: Comparison of the feasible regions.

6. Conclusions

In this paper, the problem of mode-dependent control for singular Markovian jump fuzzy systems with time delay is considered. This class of systems under consideration is described by T-S fuzzy models. The main contribution of this paper is to design state feedback controllers and static output feedback controllers which can guarantee that resulting closed-loop systems are stochastically admissible with an performance by the method of the augmented matrix. Finally, two examples are given to demonstrate the effectiveness of main results obtained here.

Conflict of Interests

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

Acknowledgments

This work was supported by The National Natural Science Foundation of China under Grants nos. 61304054, 61273003, and 61273008, respectively, and Science and Technology Research Fund of Liaoning Education Department under Grant no. L2013051.

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