H∞ Control for Stochastic Singular Systems With Time-Varying Delays

In this article, we study the <inline-formula> <tex-math notation="LaTeX">$\mathrm {H}_{\infty }$ </tex-math></inline-formula> control problems for stochastic singular systems with time-varying delays. Firstly, a new Lyapunov-Krasovskii functional is constructed, employing the free weighting matrix technique and Jensen inequality, the stochastic admissibility criteria in the mean square for stochastic singular time-varying delay systems are proposed on the basis of the auxiliary vector function. Secondly, the state feedback controller is designed such that the resulting closed-loop system meets regular, impulse-free, stochastically stable in the mean square and has <inline-formula> <tex-math notation="LaTeX">$\mathrm {H}_{\infty }$ </tex-math></inline-formula> performance <inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula>. In the proof process, the dual equation is used to derive the conditions of stochastic admissibility in the mean square. Finally, a practical example of DC motor model is presented to show the validity of our proposed theoretical results.


I. INTRODUCTION
In the past few decades, singular systems play an important role in many scientific fields, such as biologic systems, circuit systems and power systems. Therefore, the singular systems have been investigated by many researchers and a lot of important results relating to such systems have been reported (see [1]- [6]). In recent years, a more general model than singular systems is the stochastic singular systems, which play a more extensive role in model analysis than singular systems. As is well known, stochastic singular systems, also known as generalized stochastic systems, refer to the generalized dynamic systems under stochastic interference. It is widely found in many fields such as industry, social economy, power systems, financial economy systems, aerospace systems, etc. Due to the complexity of the internal structure of the stochastic singular systems, especially the co-existence of the impulse problem and the stochastic disturbance factors in the systems, it is difficult to study the theory of the stochastic singular systems. Therefore, considering the stochastic characteristics of the systems, many scholars began to pay attention to the study of singular systems with stochastic characteristics, such as stability analysis The associate editor coordinating the review of this manuscript and approving it for publication was Guangdeng Zong .
On the other hand, time delays inevitably exist in a variety of practical systems, which can frequently lead to instability or significantly deteriorated performance, and greatly increase the difficulty of stability analysis and controller design (see [18]- [25]). For singular systems, compared with the previously studied time-invariant delay systems, in recent years, more and more attention has been paid to the study of singular time-varying delay systems. It should be noted that the study of singular systems with time-varying delays is much more complex than that of normal systems with time-varying delays, because it requires consideration not only of stability, but also of regularity, impulse-free or causality (for discrete time singular systems) under time-varying delay case. For example, Yue and Han [26] investigated the delay-dependent robust H ∞ controller design for uncertain singular systems with time-varying discrete and distributed delays. Xia et al. [27] studied about the problem of filtering for nonlinear singular Markovian jumping systems with interval time-varying delays. Especially, it increases the difficulty of stability and related control analysis of the systems when the singular systems have both time-varying delays and random disturbance, and gradually attracts more and more attention. Xing et al. [28] researched the stability criteria for stochastic singular systems with time-varying delays and uncertain parameters. Li et al. [29] studied stability and stabilisation problems for a series of continuous stochastic singular systems with multiple time-varying delays via a delay-distribution-dependent Lyapunov functional, and so on. However, according to the author's grasp of the situation, the problem of H ∞ control for stochastic singular systems with time-varying delays has not been fully investigated yet. Therefore, the comprehensive and thorough study of the control problem of stochastic singular systems with time-varying delays is of great significance and necessity both in theory and practice, which motivates us to do this study.
In this work, we focus on studying the H ∞ control problem for stochastic singular systems with time-varying delays. In the first part, by constructing a new Lyaponov-Krasovskii functional, based on the auxiliary vector function, using the free weighting matrix technique and the improved Jensen inequality, we propose the stochastic admissibility criteria in the mean square for the systems consideration. In the second part, by designing the state feedback controller based on the stochastic admissibility criteria in the mean square, the corresponding closed-loop systems are regular, impulse-free and stochastically stable in the mean square. In the third part, in the process of designing the state feedback controller, the dual equation is adopted to derive the conditions of stochastic admissibility in the mean square of the resulting closed-loop systems. An example of DC motor model is given to show the effectiveness of the controller design method.
The main contributions of this study include: 1) We study the stability and H ∞ control for singular systems with time-varying delays and stochastic disturbances at the same time. Compared with singular systems [21], stochastic systems [36], the models we study are more general and have a wider range of applications. 2) A novel Lyapunov-Krasovskii functional is built, the stochastic admissibility conditions in the mean square of stochastic singular systems with time-varying delays are proposed. 3) We employ an auxiliary vector function and the new free-weighting-matrix approach to reduce the conservatism of the solution for the systems. 4) The dual equation is used in the proof process of designing the state feedback controller, which easily derive the conditions of stochastic admissibility in the mean square of the systems.
Notations. The following symbols will be used throughout the work. n represents the n-dimensional Euclidean space, and n×n denotes the set of all n × n real matrices. The symbol * represents transpose terms in a symmetric matrix and diag{. . .} stands for a block-diagonal matrix. A T is the transpose of matrix A. λ max (A) and λ min (A) are used to denote the maximum and minimum eigenvalue of A, respectively. E{·} denotes the expectation operator. I is the identity matrix with the appropriate dimensions. If the dimensions of matrices are not explicitly specified, they are assumed to be algebraically compatible.

II. PROBLEM FORMULATION AND PRELIMINARIES
Consider the stochastic singular time-varying delay system defined in a completely probability space ( , F,P) as follows where Next, the state feedback controller is designed as where K is the state feedback gain matrix. Then, the closedloop system is as follows Below, introduced some preliminary works, which will be the basis of the main research results of this article.
(II) The system (1) with u(t) = 0 and v(t) = 0 is stochastically stable in the mean square, for ∀σ > 0, there Definition 3: If the system (1) is regular, impulse-free and stochastically stable in the mean square, then the system (1) is said to be stochastically admissible in the mean square.
Assumption 1: Under Assumption 1, if rank(E) = r, without loss of generality, we can decompose matrices in the form (1) as follows and Furthermore, by the expression (4), system (1) with v(t) = 0 is restricted equivalent to the following dynamics decomposition form and Lemma 2: Consider the stochastic singular system Define an infinitesimal operator L, then, the stochastic derivative of V (x(t)) is given by where and E + is the generalized inverse of the matrix E.
, by the Itô differential formula, combining with the Eq. (6), we have where operator LV (x(t)) is defined as by the calculation, one gets so, (8) holds, the proof is completed.

III. MAIN RESULTS
In this section, the stochastic admissibility condition in the mean square is firstly derived for the system (1). At first, introduce an auxiliary vector function η 0 (t).
Using the above formula and system (1), we can get Then

Jx(s)dω(s). (11)
Theorem 1: For given scalars h 0 > 0, γ > 0, the system (1) is stochastically admissible in the mean square with the H ∞ performance index γ , if there exist matrices P > 0, Q > 0, Z > 0, S, S 0 , S 1 , S 2 , S 3 , S d such that the following matrix inequality hold. where R ∈ n×(n−r) is an arbitrary column full rank matrix satisfying E T R = 0. Proof: Firstly, we prove the system (1) with u(t) = 0 and v(t) = 0 is regular and impulse-free. VOLUME 8, 2020 From rank(E) = r ≤ n, there exist two invertible matrices G, H ∈ n×n such that It is noted that From (12), we have Because Q > 0, we have Before and after multiplying (14) by H T and H, respectively, it has Since and˜ are independent of the results discussed below, the real expressions of these two irrelevant are omitted. According to (15), we get A T 22 κS T 21 + S 21 κ T A 22 < 0, it is easy to verify A 22 is a nonsingular matrix. Thus, According to formula (16), it has det(sE − A) = 0, deg(det(sE − A) = rank(E). Thus, the pair (E, A) is regular and impulse-free.
Additionally, from (12), we can also obtain the following matrix inequality Then, before and after multiplying (17) by [I , I ] and [I , I ] T , respectively, we can obtain Using the same approach as above, we have From (18), we can easily see that This implies that the pair (E, A + A d ) is regular and impulse-free. According to Definition 2, the system (1) with u(t) = 0 and v(t) = 0 is regular and impulse-free.
Next, we prove the system (1) with u(t) = 0 and v(t) = 0 is stochastically stable in the mean square.
Constructing a Lyaponov-Krasovskii functional as follows x T (s)Qx(s)ds, By Lemma 2, computing the stochastic derivative of V (t) along the trajectory of the system (1), one has where and When u(t) = 0, from (9), one has So, there exist free weighting matrices S, S d , S 0 , S 1 , it has Because E T R = 0, from (11), there exist free weighting matrices S 2 , S 3 , we have From (20), (22) and (23), we obtain where η 0 (s)ds). (25) When v(t) = 0, let Then, we have For the condition (20), by the Schur complement lemma, we have < 0. Thus, Therefore, according to Definition 2, system (1) with u(t) = 0 and v(t) = 0 is stochastically stable in the mean square. It follows from Definition 3, we have the system (1) is stochastically admissible in the mean square. Next, the system (1) with the performance H ∞ index γ is analyzed. Set Let We have From (12), by Schur complement lemma, one has J T < 0. Therefore, the system (1) is stochastically admissible in the mean square and has H ∞ performance γ . This completes the proof.
Remark 2: When J = 0, the system (1) is reduced to the singular systems discussed by Wu et al. [22], however, they investigated the time-invariant delay systems. Here, we study time-varying delay systems. When E = 0, system (1) is reduced to the stochastic systems with time-varying delays studied by Xia et al. [36], the model we studied is more complex and has extensive applications.
Next, we give the state feedback controller design method of the system (3), in order to analyse conveniently, the closedloop system (3) can be written the following equivalent form among themẼ Therefore, we only need to prove the system (29) is stochastically admissible in the mean square and has H ∞ performance index γ . Theorem 2: For given two scalars h 0 > 0, γ > 0, the closed-loop system (29) is stochastically admissible in VOLUME 8, 2020 the mean square with H ∞ performance index γ . If there exist positive matrices P > 0, Q > 0, Z > 0, matrices X , Y , S, S 0 , S 2 , S 3 , such that the following matrix inequality holds.
is an arbitrary column full rank matrix satisfying E T R = 0.
Then the state feedback control law is given by Proof: Denotẽ where X ∈ n×n is an arbitrary nonsingular matrix. Thẽ P,Q,R,S,S d ,S 1 ,S 2 ,S 3 ,S 0 in (33) is used to replace the P, Q, R, S, S d , S 1 , S 2 , S 3 , S 0 in formula (12) of Theorem 1. By Schur complement lemma, and let λ → 0, we get We know, in terms of the regularity, the absence of impulse and stochastic stability of the stochastic singular systems, the system (29) can be written in the following dual equation form In Eq. (35), we useẼ T , B v , J , C, respectively, and let Y = KX , thus we can prove that the closed-loop system (29) is stochastically admissible in the mean square and has H ∞ performance index γ . This completes the proof.

IV. ILLUSTRATIVE EXAMPLES
Let's give a practical example to demonstrate the effectiveness of the proposed method.
where the state vector x 1 (t) = v(t) stands for speed, and the state vector x 2 (t) = i(t) refers to current. K v , K t , andR refer to electromotive force, torque constant, and electric resistor, respectively. Here, J i and b i are defined as follows  Set J m = 0.5kgm, J c1 = 50kgm, J c2 = 150kgm, b c1 = 100, b c2 = 240,R = 1 , b m = 1, K t = 3N · m/A, K v = 1Vs/rad, and n = 2.
According to the modeling idea in [37], and the time of the trigger is out of control will cause delay phenomenon in a DC input and output, then the system parameters are given as In addition, DC motor often be affected by temperature, humidity, the complexity and uncertainty of the external environment and other factors, there exist stochastic disturbance ω(t), external input disturbance v(t) in the DC motor model, the parameters are given as follows Design the state feedback controller (2), set h 0 = 0.3, µ = 0.1. For any delay 0 ≤ h(t) ≤ 0.3, by solving (31) in Theorem 2, the corresponding state feedback gain is given by K = −1.6245 −2.8310 .
When ω(t) = t − 0.3 * sin(t) is considered, we can obtain the following plot.
Figs. 1 and 2 plot state response x(t) of the closed-loop system (3), respectively, from which, we can see that the state x(t) meets stochastic admissibility in the mean square of the studied system.

V. CONCLUSION
In this work, we discuss the stochastic admissibility in the mean square for singular systems with stochastic disturbance and time-varying delays. By constructing a new Lyaponov-Krasovskii functional, depending on the auxiliary vector function and using the free weighting matrix technique and the improved Jensen inequality, the stochastic admissibility criteria in the mean square for the systems consideration are proposed. In the process of designing the controller, the dual equation is adopted to derive the stochastic admissibility conditions in the mean square easily. Finally, an example of DC motor model is given to verify the validity of the theoretical results. Our conclusion can be further extended to stochastic singular systems with time-varying delays and parameter uncertainties.