H2/H‘ Control for Itô-type Stochastic Time-Delay Systems with Applications to Clothing Hanging Device

(is paper deals with the problem of mixed H2/H∞ control for Itô-type stochastic time-delay systems. First, the H2/H∞ control problem for stochastic time-delay systems is presented, which considers the mean square stability, H2 control performance index, and the ability of disturbance attenuation of the closed-loop systems. Second, by choosing an appropriate Lyapunov–Krasoviskii functional and using matrix inequality technique, some sufficient conditions for the existence of state feedback H2/H∞ controller for stochastic time-delay systems are obtained in the form of linear matrix inequalities. (ird, two convex optimization problems with linear matrix inequality constraints are formulated to design the optimalmixedH2/H∞ controller whichminimizes the guaranteed cost of the closedloop systems with known and unknown initial functions, and the corresponding algorithm is given to optimize H2/H∞ performance index. Finally, a numerical example is employed to show the effectiveness and feasibility of the proposed method.


Introduction
Over the past decades, there has been a rapid increase of interest in the study of stochastic systems due to the importance of stochastic models in science and engineering, such as finance systems [1] and power systems [2]. And a lot of excellent results have been obtained. For example, Zhu et al. [3] investigated the tracking control issue of stochastic systems subject to time-varying full state constraints and input saturation. In [4], the stability of a class of discretetime stochastic nonlinear systems with external disturbances was considered. e finite-time tracking control of a class of stochastic quantized nonlinear systems was studied in [5]. Furthermore, since stochasticity and time delay are the main sources resulting in the complexities of systems in reality, considerable interests have been focused on a general model of stochastic time-delay systems. For example, the problem of guaranteed cost robust stable control was considered via state feedback for a class of uncertain stochastic systems with time-varying delay in [6]. In [7], the mean square exponential stability of neutral-type linear stochastic time-delay systems with three different delays by using the Lyapunov-Krasovskii functionals was studied. In [8], the finitetime dissipative control for stochastic interval systems with time delay and Markovian switching was investigated. Some other nice results can be referred to [9][10][11][12][13][14][15][16][17] and the references therein.
At present, H ∞ control has been receiving increased attention because it can suppress external interference, and many efforts have been devoted to extending the results for H ∞ control over the last few decades. For instance, Ma and Liu [18] investigated the finite-time H ∞ control problem for singular Markovian jump system with actuator fault through the sliding mode control approach. In [19], the problem of nonfragile observer-based H ∞ control for stochastic timedelay systems was considered. e problems of robust stabilization and robust H ∞ control with maximal decay rate were investigated for discrete-time stochastic systems with time-varying norm-bounded parameter uncertainties in [20]. Some other nice results can be referred to [21][22][23][24][25][26]. On the contrary, H ∞ control is an effective way to attenuate the disturbance, while H 2 control can guarantee quadratic performance cost. By combining H 2 control and H ∞ control theory, the mixed H 2 /H ∞ control theory is obtained. Owing to the fact that the mixed H 2 /H ∞ control can minimize a desired control performance and eliminate the effect of disturbance, it is more attractive than the sole H ∞ control in engineering practice. For example, Gao et al. [12] investigated the problem of H 2 /H ∞ control for nonlinear stochastic systems with time-delay and state-dependent noise. H 2 /H ∞ control problem of stochastic systems with random jumps was solved in [27]. Sathananthan et al. [28]  e organization of this paper is as follows. Section 2 is devoted to the problem statement, preliminaries, and lemmas. Section 3 provides the sufficient conditions for the existence of state feedback H 2 /H ∞ controller for Itô-type stochastic time-delay systems. Section 4 gives an algorithm to solve the theorems. Section 5 presents a numerical example to demonstrate the effectiveness of the proposed method. Section 6 is our conclusions.
Notations: A ′ denotes the transpose of matrix A; tr(A) indicates the trace of matrix A; A > 0(A ≥ 0) indicates that A is a positive definite (positive semidefinite) matrix; I n×n represents a n-dimensional identity matrix; R n shows n-dimensional Euclidean space; E represents the mathematical expectation of random process; and the asterisk " * " in the matrix indicates symmetry term.

Preliminaries
Consider the following Itô-type stochastic time-delay system described by where x(t) is the state of the system, u(t) is the control input, z(t) is the control output, ϕ(t) is the initial state function, and w(t) is a one-dimensional standard Wiener process defined on probability space (Ω, F, F t , P). F t stands for the smallest σ-algebra generated by w(s), 0 ≤ s ≤ t, i.e., F t � σ w(s) | 0 ≤ s ≤ t { }. τ > 0 is the time delay. A 11 , A 12 , A 21 , A 22 , B 11 , B 12 , C 1 , and D 1 are constant matrices with appropriate dimensions.
Next, a new definition of the mean square stability for system (1) is given.
en, some lemmas for obtaining the main results are introduced.
Lemma 1 (see [29]). Let V(t, x) ∈ C 1,2 (R + , R n ) be a scalar function, and V(t, x) > 0, for the following stochastic system: e Itô formula of V(t, x) is given as follows: where Lemma 2 (see [30]). For given x ∈ R n , y ∈ R m , N ∈ R n×m , and ρ > 0, then we have Lemma 3 (see [6]). For some real matrices N, M ′ � M and R � R ′ > 0, the following three conditions are equivalent:

Mixed H 2 /H ' Control for Stochastic Time-Delay Systems
In this section, a state feedback H 2 /H ∞ controller will be designed.
We consider a state feedback controller for system (1) is where K is the state feedback gain to be determined. e closed-loop system can be obtained by substituting (2) into (1): Associated with system (1), the cost function is provided as follows: where T � T ′ > 0 and R � R ′ > 0 are the given positive scalars or given weighting matrices.
By substituting (2) into (4), we can obtain Based on the above analysis, the problem of H 2 /H ∞ control for stochastic time-delay systems is provided as follows.
Definition 2. For a given scalar c > 0, if there exist a positive scalar J * s and a state feedback controller (2) such that (i) e closed-loop system (3) is asymptotically stable in mean square sense.
output z(t) satisfies the following inequality with zero initial condition: then (2) is said to be a state feedback H 2 /H ∞ controller for system (1). e sufficient conditions for the existence of the state feedback H 2 /H ∞ controller (2) are given below. For this reason, an important lemma is first given.  (3), and the corresponding guaranteed cost for system (3) is Proof. e following proof is divided into three parts. First, it is proved that the closed-loop system (3) is mean square stable.
According to Lemma 3, condition (7) implies Due to T > 0, R > 0, and c > 0, we can obtain where that is, where In the light of (9), we can derive that L 1 V(x(t), t) < 0, that is, the closed-loop system (3) is asymptotically stable in mean square sense.
Secondly, we prove that the control output z(t) satisfies H ∞ index for any nonzero disturbance v(t) under zero initial condition.
According to (7), T > 0, and K ′ RK > 0, we can obtain where Notice that where L 2 V(x(t)) is the infinitesimal operator of system (3) for any nonzero disturbance v(t), and en, we can see that where Ψ 12 � PA 12 + (A 21 + B 21 K) ′ PA 22 . Based on (12), we can see irdly, we prove that system (3) satisfies H 2 index under the condition of v(t) � 0.

□
In order to solve the complex problem to seek the solution caused by the nonlinear terms in Lemma 4, we give the following Lemma 5.

Lemma 5.
For a given scalar c > 0 and two symmetric positive definite matrices T and R, if there are two symmetric positive definite matrices P and Q and a matrix M such that (8) is a mixed H 2 /H ∞ controller of system (9), and the corresponding guaranteed cost for system (9) Proof. According to Lemma 2 and (13), if the following inequality Mathematical Problems in Engineering hold, where (13) holds.
Summarizing the process, the proof is completed. □ Next, in order to get the least upper bound for cost function among all the possible solutions to inequality (26), the convex optimization problem is provided as follows.
Theorem 1. For system (9), if the following optimization problem min subject to (26) and has a solution α, W, P, Q, and M, then controller u(t) � MP − 1 x(t) is an optimal state feedback H 2 /H ∞ controller which ensures the minimization of guaranteed cost (9), Proof. From Lemma 5, the controller u(t) � MP − 1 x(t) is a guaranteed cost control law of system (9).
us, we can obtain J * s < α + tr(W). erefore, the minimization of α + tr(W) implies the minimization of guaranteed cost for system (9). e proof is completed here. □ Remark 1. It is an ideal case that the initial function is known. However, in general, the initial function of system (1) is not known, but the guaranteed cost depends on it. In order to avoid the dependence, we assume that the initial function is a white noise process with zero expectation function and unit covariance function.
When the initial function is not known, we have erefore, we have the following optimization problem: which subjects to (26) and

Numerical Algorithms
In this section, an algorithm is presented in order to find the minimum value of α + tr(W) in eorem 1. e similar algorithm can also be applied to eorem 2.

Numerical Examples
e coefficient matrices of system (1) are given as follows: First case: when the initial function is known and x(0) � 1 2 ′ , t ∈ [−1, 0]. In order to find the minimum value of α + tr(W), we obtain the relationship between α + tr(W) and c by Algorithm 1, which is shown in Figure 1.
As can be seen from Figure 1 (38) erefore, the optimal state feedback H 2 /H ∞ controller is u(t) � [0.0762 − 0.0828]x(t), and the guaranteed cost of closed-loop system is J * s � 30.8430. Take external disturbance v(t) � sin(t), then we can obtain the curves of x 1 and x 2 and E‖x(t)‖ 2 in Figure 2. From Figure 2, we can see that E‖x(0)‖ 2 � 5 and lim t⟶∞ E‖x(t)‖ 2 � 0, that is, closed-loop system (9) is mean square stable.
Second case: when the initial function is a white noise process with zero expectation function and unit covariance function, in order to find the minimum value of tr(W 1 ) + τ × tr(W 2 ), we obtain the relationship between tr(W 1 ) + τ × tr(W 2 ) and c by Algorithm 1, which is shown in Figure 3.
As can be seen from Figure 3

Conclusion
In this paper, the mixed H 2 /H ∞ control problem for Itôtype stochastic time-delay systems is presented, and the description of H 2 /H ∞ control problem for stochastic timedelay systems is given. On the basis of matrix transformation and convex optimization method, state feedback H 2 /H ∞ controller is obtained to make the system satisfy H ∞ performance index and H 2 performance index. Moreover, an algorithm is given to solve state feedback controller and optimize H 2 /H ∞ performance index. Finally, a numerical example is used to show the feasibility of the results. In the future work, we will investigate mixed H 2 /H ∞ control for the more complex systems, such as, stochastic Markov jump systems with time delay.

Data Availability
e data used to support the findings of this study are available from the corresponding upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.   Step 1: Given the values of τ.

References
Step 2: Using linear search algorithm, if a series of c i (i � 1, · · · , n) can be found to make inequalities (26), (30), (31) have feasible solutions, then turn to Step 3; otherwise, turn to Step 7.
Step 3: Let i � 1, then we take c i .
Step 5: Let i � i + 1, if i + 1 > n, then turn to Step 6; otherwise, let c i � c i+1 , and turn to Step 4.
Step 6: ere are solutions to this problem, printing data, and then stop.
Step 7: ere is no solution to this problem and stop.
ALGORITHM 1: Linear search algorithm. 8 Mathematical Problems in Engineering