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Mixed [H.sub.2]/[H.sub.[infinity]] Control for Ito-type Stochastic Time-Delay Systems with Applications to Clothing Hanging Device.

1. 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 discrete-time stochastic nonlinear systems with external disturbances was considered. The 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 finite-time dissipative control for stochastic interval systems with time delay and Markovian switching was investigated. Some other nice results can be referred to [9-17] and the references therein.

At present, [H.sub.[infinity]] control has been receiving increased attention because it can suppress external interference, and many efforts have been devoted to extending the results for [H.sub.[infinity] control over the last few decades. For instance, Ma and Liu [18] investigated the finite-time [H.sub.[infinity]] 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.sub.[infinity]] control for stochastic time-delay systems was considered. The problems of robust stabilization and robust [H.sub.[infinity]] 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 [21-26]. On the contrary, [H.sub.[infinity]] control is an effective way to attenuate the disturbance, while [H.sub.2] control can guarantee quadratic performance cost. By combining [H.sub.2] control and HTO control theory, the mixed [H.sub.2]/[H.sub.[infinity]] control theory is obtained. Owing to the fact that the mixed [H.sub.2]/[H.sub.[infinity]] control can minimize a desired control performance and eliminate the effect of disturbance, it is more attractive than the sole HTO control in engineering practice. For example, Gao et al. [12] investigated the problem of [H.sub.2]/[H.sub.[infinity]] control for nonlinear stochastic systems with time-delay and state-dependent noise. [H.sub.2]/[H.sub.[infinity]] control problem of stochastic systems with random jumps was solved in [27]. Sathananthan et al. [28] studied guaranteed cost [H.sub.[infinity]] control of linear stochastic Markovian switching systems. Although the problem of [H.sub.2]/[H.sub.[infinity]] control has been investigated, there are few literature studies on Ito-type stochastic time-delay systems.

Motivated by the abovementioned discussions, in this work, we aim to investigate the mixed [H.sub.2]/[H.sub.[infinity]] control for Ito-type stochastic time-delay systems. It is difficult to design state feedback [H.sub.2]/[H.sub.[infinity]] controller because of the complicated structure of the system. The main contributions of this paper are as follows. (i) The definition of [H.sub.2]/[H.sub.[infinity]] control for Ito-type stochastic time-delay systems is presented, which considers stability, [H.sub.2] control performance index, and [H.sub.[infinity] control performance index. (ii) The new sufficient conditions for the existence of state feedback [H.sub.2]/[H.sub.[infinity]] controller are provided in the form of linear matrix inequalities. (iii) An algorithm is given to optimize [H.sub.2]/[H.sub.[infinity]] performance index.

The 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.sub.2]/[H.sub.[infinity]] controller for Ito-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 [greater than or equal to] 0) indicates that A is a positive definite (positive semidefinite) matrix; [I.sub.nxn] represents a n-dimensional identity matrix; [R.sup.n] shows n-dimensional Euclidean space; E represents the mathematical expectation of random process; and the asterisk "*" in the matrix indicates symmetry term.

2. Preliminaries

Consider the following Ito-type stochastic time-delay system described by

[mathematical expression not reproducible], (1)

where x(t) is the state of the system, u(t) is the control input, z(t) is the control output, [phi](t) is the initial state function, and w(t) is a one-dimensional standard Wiener process defined on probability space ([OMEGA], F, Ft, P). [F.sub.t] stands for the smallest ff-algebra generated by w (s), 0 [less than or equal to] s [less than or equal to] t, i.e., [F.sub.t] = [sigma]{w (s) | 0 [less than or equal to] s [less than or equal to] t}. t > 0 is the time delay. [A.sub.11], [A.sub.12], [A.sub.21], [A.sub.22], [B.sub.11], [B.sub.12], [C.sub.1], and [D.sub.1] are constant matrices with appropriate dimensions.

Next, a new definition of the mean square stability for system (1) is given.

Definition 1. System (1) (u (t) [equivalent to] 0 and v(t) = 0) is said to be mean square stable, if

[mathematical expression not reproducible]. (2)

Then, some lemmas for obtaining the main results are introduced.

Lemma 1 (see [29]). Let V (t, x) [member of] [C.sup.1,2] ([R.sub.+], [R.sup.n]) be a scalar function, and V (t, x) > 0, for the following stochastic system:

dx(t)= a(x)dt + b(x)dw(t). (3)

The Ito formula of V(t, x) is given as follows:

[mathematical expression not reproducible], (4)

where

[mathematical expression not reproducible]. (5)

Lemma 2 (see [30]). For given x [member of] [R.sup.n], y [member of] [R.sup.m], N [member of] [R.sup.nxm], and [rho] > 0, then we have

[mathematical expression not reproducible]. (6)

Lemma 3 (see [6]). For some real matrices N, [M.sup.r] = M and R = R' > 0, the following three conditions are equivalent:

[mathematical expression not reproducible], (7)

3. Mixed [H.sub.2]/[H.sub.[infinity]] Control for Stochastic Time-Delay Systems

In this section, a state feedback [H.sub.2]/[H.sub.[infinity]] controller will be designed.

We consider a state feedback controller for system (1) is

u(t) = Kx(t), (8)

where K is the state feedback gain to be determined.

The closed-loop system can be obtained by substituting (2) into (1):

[mathematical expression not reproducible], (9)

Associated with system (1), the cost function is provided as follows:

[mathematical expression not reproducible], (10)

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

[mathematical expression not reproducible], (11)

Based on the above analysis, the problem of [H.sub.2]/[H.sub.[infinity]] control for stochastic time-delay systems is provided as follows.

Definition 2. For a given scalar [gamma] > 0, if there exist a positive scalar [J.sup.*.sub.s] and a state feedback controller (2) such that

(i) The closed-loop system (3) is asymptotically stable in mean square sense.

(ii) [H.sub.2] cost function (5) satisfies [J.sub.s] (x(t)) [less than or equal to] [J.sup.*.sub.s] under the condition of v(t) = 0.

(iii) For any nonzero disturbance v(t), the control output z(t) satisfies the following inequality with zero initial condition:

[mathematical expression not reproducible], (12)

then (2) is said to be a state feedback [H.sub.2]/[H.sub.[infinity]] controller for system (1).

The sufficient conditions for the existence of the state feedback [H.sub.2]/[H.sub.[infinity]] controller (2) are given below. For this reason, an important lemma is first given.

Lemma 4. For a given scalar [gamma] > 0 and two symmetric positive definite matrices T and R, if there are two symmetric positive definite matrices P and Q such that

[mathematical expression not reproducible], (13)

hold, where [[GAMMA].sup.11] = Q + ([A.sub.21] + [B.sub.21] K)'P ([A.sub.21] + [B.sub.21] K) + 2 ([A.sub.11] + [B.sub.11] K)' P + ([C.sub.1] + [D.sub.1] K)' ([C.sub.1] + [D.sub.1] K) + T + K' RK, then (2) is a mixed [H.sub.2]/[H.sub.[infinity]] controller of system (3), and the corresponding guaranteed cost for system (3) is [mathematical expression not reproducible].

Proof. The 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

[mathematical expression not reproducible]. (14)

Due to T > 0, R > 0, and [gamma] > 0, we can obtain ([C.sub.1] + [D.sub.1]K)' ([C.sub.1] + [D.sub.1] K) > 0, (1/[[gamma].sup.2]) [PB.sub.12] [B.sub.12] P > 0, and K'RK > 0; then, (8) implies

[mathematical expression not reproducible], (15)

where [[summation[].sub.11] = Q + ([A.sub.21] + [B.sub.21]K)' P([A.sub.21] + [B.sub.21]K) + 2([A.sub.11] + [B.sub.11]K)' P.

Let a quadratic function [mathematical expression not reproducible], differential generation operator of system (3) be [L.sub.1] V(x(t)) with v = 0; then,

[mathematical expression not reproducible], (16)

that is,

[mathematical expression not reproducible], (17)

where [[OMEGA].sub.11] = Q + ([A.sub.11] + [B.sub.11] K)'P + P([A.sub.11] + [B.sub.11] K) + ([A.sub.21] + [B.sub.21] K)' P([A.sub.21] + [B.sub.21] K).

In the light of (9), we can derive that [L.sub.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.sub.[infinity]] index for any nonzero disturbance v(t) under zero initial condition.

According to (7), T > 0, and K'RK > 0, we can obtain

[mathematical expression not reproducible], (18)

where = [[PSI].sub.11] + ([A.sub.21] + [B.sub.21] K)' P([A.sub.21] + [B.sub.21] K) +2 ([A.sub.11] + [B.sub.11]K)' P + ([C.sub.1] + [D.sub.1]K)' ([C.sub.1] + [D.sub.1] K).

Notice that

[mathematical expression not reproducible], (19)

where [L.sub.2] V(x(t)) is the infinitesimal operator of system (3) for any nonzero disturbance v(t), and

[mathematical expression not reproducible]. (20)

Then, we can see that

[mathematical expression not reproducible], (21)

where = [[PSI].sub.12] + [PA.sub.12] + ([A.sub.21] + [B.sub.21] K)' [PA.sub.22].

Based on (12), we can see [mathematical expression not reproducible], that is, (12) implies that [mathematical expression not reproducible]. Therefore, system (3) satisfies [H.sub.[infinity] index.

Thirdly, we prove that system (3) satisfies [H.sub.2] index under the condition of v(t) = 0.

Based on (13) and (14), ([C.sub.1] + [D.sub.1] K)' ([C.sub.1] + [D.sub.1] K) > 0, and (1/[[gamma].sup.2]) [PB.sub.12] [B.sub.12]' P > 0, we obtain that

[mathematical expression not reproducible], (22)

holds, where [[THETA].sub.11] = [[OMEGA].sub.11] + T' + K' RK.

Due to

[mathematical expression not reproducible], (23)

where [[THETA]].sub.14] = -Q + [A'.sub.22] [PA.sub.22].

In view of (22), we obtain

[mathematical expression not reproducible]. (24)

According to (23) and (24), we can see

[mathematical expression not reproducible]. (25)

The proof is completed here.

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 [gamma] > 0 and two symmetric positive definite matrices [??] and [??], if there are two symmetric positive definite matrices [??] and [??] and a matrix M such that

[mathematical expression not reproducible]. (26)

hold, where [mathematical expression not reproducible], [mathematical expression not reproducible], and [[XI].sub.22] = diag {-I, -T, -I, -R}; then, (8) is a mixed [H.sub.2]/[H.sub.[infinity]] controller of system (9), and the corresponding guaranteed cost for system (9) is [mathematical expression not reproducible]. In this case, K = [MP.sup.-1].

Proof. According to Lemma 2 and (13), if the following inequality

[mathematical expression not reproducible] (27)

hold, where [Y.sub.11] = Q + 4[A'.sub.21] [PA.sub.21] + 4K, [B'.sub.21] [PB.sub.21] K + 2[A'.sub.11] P + 2K' [B'.sub.11] P + 3 [C.sub.1] [C.sub.1] + 3K' [D.sub.1] [D.sub.1] K + T + K'RK, then (13) holds.

Using diag {[P.sup.-1], [Q.sup.-1], I} to premultiply and postmultiply inequality (27), we have

[mathematical expression not reproducible], (28)

hold, where [mathematical expression not reproducible]. Let [mathematical expression not reproducible], and T = [T.sup.-1]; by Lemma 3, we obtain (26) from (28).

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

[mathematical expression not reproducible] (29)

subject to (26) and

[mathematical expression not reproducible], (30)

[mathematical expression not reproducible], (31)

has a solution a, W, [??], [??], and M, then controller u(t) = MP x(t) is an optimal state feedback [H.sub.2]/[H.sub.[infinity]] controller which ensures the minimization of guaranteed cost [mathematical expression not reproducible] for system (9), where[mathematical expression not reproducible].

Proof. From Lemma 5, the controller u(t) = [MP.sup.-1] x(t) is a guaranteed cost control law of system (9). (30) is equivalent to x' (0)[P.sup.-1] x(0) < [alpha]; (31) is equivalent to N' [Q.sup.-1] < W.

Therefore, we can obtain

[mathematical expression not reproducible]. (32)

Thus, we can obtain [J*.sub.s] < [alpha] + tr(W).

Therefore, the minimization of [alpha] + tr (W) implies the minimization of guaranteed cost for system (9).

The 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

[mathematical expression not reproducible]. (33)

Therefore, we have the following optimization problem:

[mathematical expression not reproducible], (34)

which subjects to (26) and

[mathematical expression not reproducible], (35)

[mathematical expression not reproducible]. (36)

Theorem 2. If there exist solution to (26), (34)-(36) then controller u(t) = [MP.sup.-1] x(t) is an optimal state feedback HH controller which ensures the minimization of guaranteed cost (18) for system (1).

Proof. From Lemma 5, the controller u(t) = MP 1x(t) is a [H.sub.2]/[H.sub.[infinity]] controller of system (9). We can see (34) is equivalent to 0 < [P.sup.-1] < [W.sub.1] and (35) is equivalent to 0 < [Q.sup.-1] < [W.sub.2] from Lemma 3. Therefore, the minimization of tr ([W.sub.1]) + t x tr([W.sub.2]) implies the minimization of the guaranteed cost for system (1).

The proof is complete.

4. Numerical Algorithms

In this section, an algorithm is presented in order to find the minimum value of a + tr(W) in Theorem '. The similar algorithm can also be applied to Theorem 2.

By analyzing (26), (30), (3') in Theorem ', we find that if (26), (30), (3') have no feasible solutions when [gamma] takes the initial value, then (26), (30), (3') will have no feasible

solutions for all [gamma] > 0. Next, we search for y from the initial value that makes (26), (30), (31) have feasible solutions to optimize [alpha] + tr (W) by using linear search algorithm. The specific algorithm is as follows.

5. Numerical Examples

The coefficient matrices of system (1) are given as follows:

[mathematical expression not reproducible]. (37)

First case: when the initial function is known and x(0) = [1 2]', t [member of] [-1,0]. In order to find the minimum value of [alpha] + tr(W), we obtain the relationship between a + tr(W) and [gamma] by Algorithm 1, which is shown in Figure 1.

As can be seen from Figure 1, [alpha] + tr(W) decreases with the increase of [gamma], and min[a + tr(W)] = 38.4173 when [gamma] = 0.4, and min [[alpha] + tr(W)] = 29.7250 when [gamma] = 1.98.

Take [gamma] = 0.8, according to Theorem 1, we obtain that

[mathematical expression not reproducible]. (38)

Therefore, the optimal state feedback [H.sub.2]/[H.sub.[infinity]] controller is u(t) = [0.0762 - 0.0828]*(t), and the guaranteed cost of closed-loop system is [J*.sub.s] = 30.8430.

Take external disturbance v(t) = sin(t), then we can obtain the curves of [x.sub.1] and [x.sub.2] and E [[parallel]x(t)[parallel].sup.2] in Figure 2. From Figure 2, we can see that E [[parallel]x(t)[parallel].sup.2] = 5 and lim [[parallel]x(t)[parallel].sup.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.sub.1]) + t x tr([W.sub.2]), we obtain the relationship between tr ([W.sub.1]) + t x tr ([W.sub.2]) and y by Algorithm 1, which is shown in Figure 3.

As can be seen from Figure 3, tr([W.sub.1]) + [tau] x tr([W.sub.2]) decreases with the increase of [gamma], and min[tr([W.sub.1]) + t x tr ([W.sub.2])] = 19.5450 when [gamma] = 0.4, min[tr ([W.sub.1] + t x tr([W.sub.2])] = 13.6648 when [gamma] = 1.98.

Take [gamma] = 0.8, according to Theorem 2, we obtain that

[mathematical expression not reproducible]. (39)
Algorithm 1: Linear search algorithm.

Step 1: Given the values of [tau].
Step 2: Using linear search algorithm, if a series of [[gamma].sub.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 [[gamma].sub.i].
Step 4: Solve the following minimization problem:
[mathematical expression not reproducible]. (W).
Step 5: Let i = i + 1, if i + 1 > n, then turn to Step 6; otherwise,
let [[gamma].sub.i] = [[gamma].sub.i] + 1, and turn to Step 4.
Step 6: There are solutions to this problem, printing data, and
then stop.
Step 7: There is no solution to this problem and stop.


Therefore, the optimal state feedback [H.sub.2]/[H.sub.[infinity]] controller is u(t) = [0.0759 - 0.0829]x(t), and the guaranteed cost of closed-loop system is [J*.sub.s] = 14.2951.

6. Conclusion

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

https://doi.org/10.1155/2020/4298230

Data Availability

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

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 61877062 and 61977043), China Postdoctoral Science Foundation (Grant no. 2017M610425), and Open Foundation of Key Laboratory of Pulp and Paper Science and Technology of Ministry of Education of China (Grant no. KF201419).

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Yan Qi, (1) Min Zhang, (2) and Zhiguo Yan [ID] (2)

(1) School of Fine Arts and Design, University of Jinan, Jinan 250022, China

(2) School of Electrical Engineering and Automation, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China

Correspondence should be addressed to Zhiguo Yan; yanzg500@sina.com

Received 16 April 2020; Accepted 20 May 2020; Published 20 June 2020

Guest Editor: Yi Qi

Caption: Figure 1: When [gamma] [member of] [0,2], the minimum upper bound of [alpha] + tr(W).

Caption: Figure 2: When t [member of] [0,1], the response for E [[parallel]x(t)[parallel].sup.2].

Caption: Figure 3: When y [member of] [0,2], the minimum upper bound of tr([W.sub.1]) + t x tr([W.sub.2]).
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Title Annotation:Research Article
Author:Qi, Yan; Zhang, Min; Yan, Zhiguo
Publication:Mathematical Problems in Engineering
Date:Jun 30, 2020
Words:4510
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