Fuzzy Control for Nonlinear Singular Markovian Jump Systems with Time Delay

This paper investigates the problem ofH ∞ 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 H ∞ performance for all delays not larger than a given upper bound in terms of linear matrix inequalities. Then H ∞ 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.


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][8][9][10].When singular systems experience abrupt changes in their structures, it is natural to model them as singular Markovian jump systems (SMJSs) [11][12][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  −1 represents the inverse of the matrix . max  ( min ) is the maximal (minimal) eigenvalue of the matrix .diag{⋅} 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 R  stand for the -dimensional Euclidean space, R × is the set of all  ×  real matrices, and ‖ ⋅ ‖ denotes the Euclidean norm of vectors.E{⋅} denotes the mathematics expectation of the stochastic process or vector.  2 [0, ∞) stands for the space of -dimensional square integrable functions on [0, ∞). , = ([−, 0], R  ) denotes Banach space of continuous vector functions mapping the interval [−, 0] into R  with the norm ‖‖  = sup −≤≤0 ‖()‖.
(ii) System ( 4) is said to be stochastically stable if there exists a finite number  ((),  0 ) such that the following inequality holds: (iii) System ( 4) is said to be stochastically admissible if it is regular, impulse-free, and stochastically stable.
Based on the parallel distributed compensation, the following state feedback controller will be considered here: where is stochastically admissible.

The Design of the State Feedback 𝐻 ∞ 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 Mathematical Problems in Engineering Remark 7.For systems (11) and ( 12), it can be seen that By rank  = rank Ẽ 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).
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 () = 0, and under zero initial condition, for nonzero The following result can be presented.
Proof.From Theorem 8 when () = 0 system (11) is stochastically admissible.Let Under zero initial condition, it is easy to see that where and notations of Υ 1 , Υ 2 , and Υ 3 are the same as in Theorem 8. Hence, by Schur complement lemma and using the similar method in the proof of Theorem 8, from ?? and (53), it can be obtained that   () < 0 for all  > 0.
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.

The Design of the Static Output Feedback Controller
When   =  ∈ S, consider the overall SMJFS as follows: where () ∈ R  1 is the system output,  , ( ∈ S) 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   ( ∈ S,  ∈ T) 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: As the discussion in Remark 7, the stochastic admissibility of system (60) can be studied by means of system (61).
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.

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: ( where To demonstrate the effectiveness, assuming the initial condition () = [−1.20.8 −0.5]  , 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.
Example 2. Consider the example without uncertainties in [6].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 3 in [6].This illustrates that the method obtained in this paper has less conservatism.

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.

Figure 2 :Figure 3 :
Figure 2: State responses of the closed-loop system.