Finite-Time H∞ Control for Itô-Type Nonlinear Time-Delay Stochastic Systems

The finite-time <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> control problem for an Itô-type stochastic system with nonlinear perturbation and time delay is investigated. First, the finite-time <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> control problem for a nonlinear time-delay stochastic system is presented taking into consideration both the transient performance and the capability to attenuate the disturbance of a closed-loop system in a given finite-time interval. Second, using the Lyapunov–Krasoviskii functional method and the matrix inequality technique, some sufficient conditions for the existence of finite-time <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> state feedback controller and dynamic-output feedback controller for nonlinear time-delay stochastic systems are obtained. These conditions guarantee the mean-square finite-time bounded-ness of the closed-loop systems and determine the <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> control performance index. Third, this problem is transformed into an optimization problem with matrix inequality constraints, and the corresponding algorithms are given to optimize the <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> performance index and obtain the maximum time-delay. Finally, a numerical example is used to illustrate the effectiveness of the proposed method.


I. INTRODUCTION
In recent years, control problems of stochastic nonlinear systems have been receiving increased attention because of their extensive applications in many practical systems, such as liquid-level systems [1], chemical reactor systems [2], [3], and industrial and economic systems [4]. In addition, many excellent results have been published. For example, [5] proposed three stochastic nonlinear control schemes to study the global stabilization of stochastic nonlinear systems. In [6], finite-time stability for stochastic nonlinear systems was considered and a general Lyapunov theorem of stochastic finitetime stability was proved. A finite-time tracking problem of switched stochastic nonlinear uncertain systems was studied in [7]. Some other nice results can be referred to [8]- [10]. In considering the influence of time-delays on the system, much of the focus has been on a general model of stochastic nonlinear time-delay systems. To date, numerous results on these systems have been obtained. For instance, [11] studied The associate editor coordinating the review of this manuscript and approving it for publication was Bing Li . the stability of a class of nonlinear uncertain stochastic timedelay systems, and a sufficient delay-dependent criterion was established by constructing a new Lyapunov-Krasovskii function. The output feedback adaptive tracking control problem for a class of stochastic nonlinear time-delay systems was studied in [12], and an observer-based adaptive neural quantization tracking control scheme was proposed. For other excellent results, the reader is referred to [13]- [15] and references therein.
At present, most of the results of stochastic systems are based on the asymptotic stability in the Lyapunov sense, which only concerns the asymptotic behavior of the system in the limit of infinite time. However, the transient behavior is also significant in many practical systems. For example, a large transient voltage can destroy the normal operation of a power system [16]. To deal with this problem, the concept of finite-time stability was proposed, and many interesting results have been published, such as the finite-time stability of stochastic discrete-time-varying systems in [17], stochastic Markov jump systems in [18]- [22], stochastic time-delay systems in [23], and T-S fuzzy systems in [24]. In contrast, H ∞ control is one of the most crucial robust control methods because an external interference can be suppressed; many results have been reported. For instance, [25] and [26] are devoted to the robust H ∞ control problem for uncertain stochastic nonlinear systems with time-varying delays and stochastic nonlinear uncertain T-S fuzzy systems with timedelay. Robust H ∞ filtering and control for a class of linear systems with fractional stochastic noise were studied in [27]. Other nice results are to be found in [28]- [31]. Taking advantage of finite-time stability and H ∞ control, a new control method called the finite-time H ∞ control was proposed, and various results have been reported. For example, the finitetime H ∞ control problem for a singular Markovian jump system with an actuator fault was investigated in [32] with a sliding mode control approach being applied. The reference [33] addressed the problem of robust finite-time H ∞ control of singular stochastic systems and [34] studied the robust and resilient finite-time H ∞ control problem for uncertain discrete-time nonlinear systems with Markovian jump parameters.
Although the problem of finite-time H ∞ control has been investigated, there is a lack of literature on Itô-type stochastic nonlinear time-delay systems. To redress this issue, the finitetime H ∞ control these systems was investigated in our study. Because of the complexity of the systems considered, designing a H ∞ controller with state feedback and dynamic-output feedback is difficult. By using the Itô formula, the Gronwall inequality, and the matrix inequality technique, the above difficulties are overcome. The main contributions from this study are: (i) a precise statement of the finite-time H ∞ control problem for Itô-type stochastic nonlinear systems with time-delay that considers both the transient performance and the capability of attenuating the disturbances in the closedloop systems in a given finite-time interval; (ii) two new conditions developed from matrix inequalities concerning the sufficiency for the existence of two H ∞ controllers one providing state feedback and the other dynamic-output feedback; and (iii) the establishment of two algorithms that solve the gain parameter settings of the two controllers and that optimize the H ∞ performance index and maximum timedelay, simultaneously.
The rest of this paper is organized as follows. Section II gives some preliminaries, definitions, and lemmas. In Section III, we provide the conditions of sufficiency for the existence of the finite time H ∞ controllers for Itô-type stochastic nonlinear time-delay systems. Section IV provides the two algorithms that solve the theorems. Section V presents a numerical example to demonstrate the effectiveness of the proposed method. In the last section, our conclusions are stated.
Notation: A denotes the transpose of matrix A; tr(A) denotes the trace of matrix A; A>0(A 0) signifies that A is a positive definite (positive semi-definite) matrix; λ max (A)(λ min (A)) denotes the maximum (minimum) eigenvalue of matrix A; I n×n denotes the n-dimensional identity matrix; R n denotes an n-dimensional Euclidean space; E represents the mathematical expectation of a random process; an asterisk " * " in a matrix marks elements to be obtained by the symmetry of the matrix.

II. PRELIMINARIES
Consider an Itô-type stochastic nonlinear system with time delay described by where x(t) denotes the state of the system, y(t) the measurement output, u(t) the control input, z(t) the control output, φ(t) the initial state function, and w(t) a onedimensional standard Wiener process defined on probability space ( , F , F t , P). F t stands for the smallest σ -algebra generated by w(s), with 0≤s≤t, i.e., The nonlinear terms H 0 (x(t)) and H 1 (x(t)) satisfy where ε > 0. Next, a new definition of the mean-square finite-time bounded-ness for the system (1) is given.
Definition 1: For given 0 < c 1 < c 2 , R > 0, T > 0, system (1) (u(t) ≡ 0) is said to be the mean-square finitetime bounded with respect to (c 1 , for all t ∈ [0, T ] and v(t) ∈ . Remark 1: The mean-square finite-time bounded-ness reflects the transient performance of the system in a fixed time interval. That is, the average energy of the system does not exceed a given upper bound in the prescribed time-interval. The transient performance is also important in many practical systems. For example, a large transient voltage can destroy the normal operation of the power system.
Next, some lemmas for obtaining the main results are introduced.
Lemma 1 (Gronwall Inequality): [35] Let f (t) be a nonnegative function; if it satisfies VOLUME 8, 2020 for some constants a ≥ 0 and b ≥ 0, then the following inequality is established: Lemma 2 [36]: For given x ∈ R n , y ∈ R m , N ∈ R n×m and ρ > 0, then we have Lemma 3 [37]: Let V (t, x) ∈ C 1,2 (R + , R n ) be a scalar function, and V (t, x)>0; for the following stochastic system the Itô formula of V (t, x) is given as follows: where

III. MAIN RESULTS
The design of the state feedback finite-time H ∞ controller and that of the dynamic-output feedback finite-time H ∞ controller are described next.

A. STATE FEEDBACK FINITE-TIME H ∞ CONTROL
Consider the following state feedback controller where K is the controller gain matrix to be solved. A closed-loop system is obtained by substituting (4) into system (1) giving Next, the problem concerning the state feedback finitetime H ∞ controller is described.
Definition 2: For given scalars 0 < c 1 < c 2 , T > 0, d > 0, γ > 0, and a matrix R > 0, if there exists a state feedback controller (4) such that: (i) the closed-loop system (5) is mean-square finite-time bounded with respect to (c 1 , c 2 , T , R, d 2 ); (ii) for any non-zero disturbance v(t), the control output z(t) satisfies the following inequality with zero initial condition, then (4) is said to be a state feedback finite-time H ∞ controller for system (1).

Remark 2:
The definition considers both the attenuation level of the disturbance and the mean-square finite-time bounded-ness, which is widely applied in practical systems. For example, in the solar power supply system, if the load power is too large or the external interference is strong, the normal operation of the system will deteriorate.
The sufficient conditions for the existence of the state feedback finite-time H ∞ controller (4) are given below. For this purpose, an important lemma is first stated and proved.
Proof: The proof is divided into two parts. First, the closed-loop system (5) is proved to be mean-square finitetime bounded.
Next, we prove that the control output z(t) satisfies (6) for any non-zero disturbance v(t) imposing to the zero initial condition. From (7) and (18), we have Multiplying both sides of (26) by e −αt , we have From lemma 3, we have . (28) Combining (27) and (28), we find Integrating from 0 to t, taking the mathematical expectation on both sides of (29), and imposing the zero initial condition, we have Because then (30) implies that and the proof is complete.

VOLUME 8, 2020
Based on lemma 4, we next derive the following Theorem 1.

B. DYNAMIC OUTPUT FEEDBACK FINITE-TIME H ∞ CONTROL
The previous subsection assumes that the state variables are available, which does not always hold in practice. In this case, one should estimate x(t) from the measurement output y(t). As usual, consider the following dynamic-output feedback controller Substituting (42) into (1) and setting η(t)=[x (t)x (t)] , the following augmented closed-loop system is obtained The problem concerning the dynamic-output feedback finite-time H ∞ controller is stated next.
Definition 3: For given scalars 0<c 1 <c 2 , T >0, d>0, γ >0, and a positive definite matrixR, if there exists a dynamicoutput feedback controller (42) such that (i) the closed-loop system (43) is mean-square finite-time bounded with respect to (c 1 , c 2 , T , R, d 2 ), that is, (ii) for any non-zero disturbance v(t), the control output z(t) satisfies (6) with zero initial condition, then (42) is said to be a dynamic-output feedback finite-time H ∞ controller for system (1).
Next, a sufficient condition for the existence of the dynamic-output feedback finite-time H ∞ controller are given below. First, an important lemma is proved.
Proof: The proof is divided into two parts. First, the closed-loop system (43) is proved to be mean-square finite-time bounded.
In the second part of the proof, the control output z(t) is proved to satisfy (6) for any non-zero disturbance v(t) under zero initial condition. From (44) and (55), we have Multiplying both sides of (63) by e −αt , we obtain Using lemma 3, we have From (64) and (65), one sees that Integrating from 0 to t and taking the mathematical expectation on both sides of (66) applying the zero initial condition, we have Because The proof is complete. On the basis of the above analysis, we prove the following Theorem 2.
This completes the proof.

IV. NUMERICAL ALGORITHMS
In this section, two algorithms are presented that produced the results obtained above. One finds the minimum value of γ ; the other finds the maximum value of τ . By analyzing (32)- (35) in Theorem 1, we find that if these equations have no feasible solutions when α = 0, then they will have no feasible solutions for all α > 0. The specifics of the algorithm are as follows.

Algorithm 1
Step 1: Set the values of c 1 , c 2 , T , R, d, and τ .
Step 2: Using the linear search algorithm, if a series of α i (i = 1, · · · , n) can be found that ensure inequalities (32)- (35) have feasible solutions, then move to Step 3; otherwise, move to Step 7.
Step 3: Let i = 1, then we take α i .
Step 6: There are solutions to this problem; print data and then stop.
Step 7: There is no solution to this problem; stop.
By analyzing (32)- (35) in Theorem 1, we find that if these equations have no feasible solutions when α = 0 and τ = 0, then they will have no feasible solutions for all α > 0 and τ > 0. The specific algorithm is as follows.

Algorithm 2
Step 1: Set the values of c 1 , c 2 , T , R, d and γ .

V. NUMERICAL EXAMPLES
The matrix of coefficients of the system (1) is given as follows.

A. STATE FEEDBACK FINITE-TIME H ∞ CONTROL
To find the minimum value of γ , the relationship between γ and α is obtained using Algorithm 1 (see Fig. 1). From Fig. 1, γ increases with increasing α, and the minimum value of γ = 0.0315 is obtained when α = 0.
Setting α = 0, according to Theorem 1, we obtain Therefore, the state feedback controller is as follows: Setting γ = 0.1, then by Algorithm 2, the relationship between τ and α so as to find the maximum value of τ was obtained (see Fig. 2).  From Fig. 2, τ decreases with increasing α, and τ has a maximum value in the range of α; that is, τ = 9.13 when α = 0.

B. DYNAMIC OUTPUT FEEDBACK FINITE-TIME H ∞ CONTROL
To find the minimum value of γ , the relationship between γ and α is determined using Algorithm 1 (see Fig. 4).   From Fig. 4, γ decreases first as α increases, and then increases as α increases. The optimal solution γ = 0.1908 is obtained when α = 1.3.
Setting α = 1.3, then, according to Theorem 2, we derive a set of solutions Note, for B f = 0, C f = 0, the above solutions do not satisfy the requirements for the coefficients of the dynamicoutput feedback controller. Therefore, setting X > 0, the following solutions are obtained,    Setting γ = 0.2, the relationship between τ and α (Fig. 5) shows that τ decreases with increasing α. Its maximum value is τ = 2.3 when α = 0.

VI. CONCLUSION
We investigated the finite-time H ∞ control problem for the Itô-type stochastic nonlinear time-delay systems. Two kinds of controllers, state feedback and dynamic-output feedback finite-time H ∞ controllers, were designed. Moreover, their corresponding algorithms used in solving the state feedback controller and dynamic-output feedback controller were presented. They also simultaneously optimize the H ∞ performance index and determine the maximum time-delay τ .