# MixedH2/H∞ Control for Itô-type Stochastic Time-Delay Systems with Applications to Clothing Hanging Device

Authors: Yan Qi; Min Zhang; Zhiguo Yan
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper deals with the problem of mixedH2/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. Third, two convex optimization problems with linear matrix inequality constraints are formulated to design the optimal mixed H2/H∞ controller which minimizes the guaranteed cost of the closed-loop 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.


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## Body

## 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∞ 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 time-delay systems was considered. The 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 [21–26]. On the contrary, H∞ control is an effective way to attenuate the disturbance, while H2 control can guarantee quadratic performance cost. By combining H2 control and H∞ control theory, the mixed H2/H∞ control theory is obtained. Owing to the fact that the mixed H2/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 H2/H∞ control for nonlinear stochastic systems with time-delay and state-dependent noise. H2/H∞ control problem of stochastic systems with random jumps was solved in [27]. Sathananthan et al. [28] studied guaranteed cost H∞ control of linear stochastic Markovian switching systems. Although the problem of H2/H∞ control has been investigated, there are few literature studies on Itô-type stochastic time-delay systems.Motivated by the abovementioned discussions, in this work, we aim to investigate the mixedH2/H∞ control for Itô-type stochastic time-delay systems. It is difficult to design state feedback H2/H∞ controller because of the complicated structure of the system. The main contributions of this paper are as follows. (i) The definition of H2/H∞ control for Itô-type stochastic time-delay systems is presented, which considers stability, H2 control performance index, and H∞ control performance index. (ii) The new sufficient conditions for the existence of state feedback H2/H∞ controller are provided in the form of linear matrix inequalities. (iii) An algorithm is given to optimize H2/H∞ performance index.The organization of this paper is as follows. Section2 is devoted to the problem statement, preliminaries, and lemmas. Section 3 provides the sufficient conditions for the existence of state feedback H2/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; trA indicates the trace of matrix A; A>0A≥0 indicates that A is a positive definite (positive semidefinite) matrix; In×n represents a n-dimensional identity matrix; ℛn 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 Itô-type stochastic time-delay system described by(1)dxt=A11xt+A12xt−τ+B11ut+B12vtdt+A21xt+A22xt−τ+B21utdwt,zt=C1xt+D1ut,xt=ϕt,∀t∈−τ,0,where xt is the state of the system, ut is the control input, zt is the control output, ϕt is the initial state function, and wt is a one-dimensional standard Wiener process defined on probability space (Ω, ℱ, ℱt, P). ℱt stands for the smallest σ-algebra generated by ws, 0≤s≤t, i.e., ℱt=σws0≤s≤t. τ>0 is the time delay. A11, A12, A21, A22, B11, B12, C1, and D1 are constant matrices with appropriate dimensions.Next, a new definition of the mean square stability for system (1) is given.Definition 1.
System (1) (ut≡0 and vt≡0) is said to be mean square stable, if(2)limt⟶∞Ext2=0.
Then, some lemmas for obtaining the main results are introduced.Lemma 1 (see [29]).
LetVt,x∈C1,2R+,Rn be a scalar function, and Vt,x>0, for the following stochastic system:(3)dxt=axdt+bxdwt.
The Itô formula ofVt,x is given as follows:(4)dVt,x=LVt,xdt+∂V′t,x∂xbxdwt,where(5)LVt,x=∂Vt,x∂t+∂V′t,x∂xax+12b′x∂2Vt,x∂x2bx.Lemma 2 (see [30]).
For givenx∈Rn, y∈Rm, N∈Rn×m, and ρ>0, then we have(6)2x′Ny≤ρx′x+1ρy′N′Ny.Lemma 3 (see [6]).
For some real matricesN,M′=MandR=R′>0, the following three conditions are equivalent:(7)M+NR−1N′<0,MNN′−R<0,M−N−N′−R<0.


## 3. MixedH2/H∞ Control for Stochastic Time-Delay Systems

In this section, a state feedbackH2/H∞ controller will be designed.We consider a state feedback controller for system (1) is(8)ut=Kxt,where K is the state feedback gain to be determined.The closed-loop system can be obtained by substituting (2) into (1):(9)dxt=A11xt+A12xt−τ+B11Kxt+B12vtdt+A21xt+A22xt−τ+B21Kxtdwt,zt=C1xt+D1Kxt,xt=ϕt,∀t∈−τ,0.Associated with system (1), the cost function is provided as follows:(10)Jsxt,ut=E∫0∞x′tTxt+u′tRutdt,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(11)Jsxt=E∫0∞x′tTxt+x′tK′RKxtdt.Based on the above analysis, the problem ofH2/H∞ control for stochastic time-delay systems is provided as follows.Definition 2.
For a given scalarγ>0, if there exist a positive scalar Js∗ and a state feedback controller (2) such that(i)
The closed-loop system (3) is asymptotically stable in mean square sense.(ii)
H2 cost function (5) satisfies Jsxt≤Js∗ under the condition of vt=0.(iii)
For any nonzero disturbancevt, the control output zt satisfies the following inequality with zero initial condition:(12)E∫0∞zt2ds<γ2E∫0∞vt2ds,then (2) is said to be a state feedback H2/H∞ controller for system (1).
The sufficient conditions for the existence of the state feedbackH2/H∞ controller (2) are given below. For this reason, an important lemma is first given.Lemma 4.
For a given scalarγ>0 and two symmetric positive definite matrices T and R, if there are two symmetric positive definite matrices P and Q such that(13)Γ11PA12+A21+B21K′PA22PB12∗−Q+A22′PA220∗∗−γ2I<0,hold, where Γ11=Q+A21+B21K′PA21+B21K+2A11+B11K′P+C1+D1K′C1+D1K+T+K′RK, then (2) is a mixed H2/H∞ controller of system (3), and the corresponding guaranteed cost for system (3) is Js∗=Ex′0Px0+∫−τ0x′sQxsds.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 Lemma3, condition (7) implies(14)Γ11+1γ2PB12B12′PPA12+A21+B21K′PA22∗−Q+A22′PA22<0.
Due toT>0, R>0, and γ>0, we can obtain C1+D1K′C1+D1K>0, 1/γ2PB12B12′P>0, and K′RK>0; then, (8) implies(15)Σ11PA12+A21+B21K′PA22∗−Q+A22′PA22<0,where Σ11=Q+A21+B21K′PA21+B21K+2A11+B11K′P.
Let a quadratic functionVxt,t=x′tPxt+∫−τ0x′t+sQxt+sds and the differential generation operator of system (3) be L1Vxt with v=0; then,(16)L1Vxt,t=x′tQxt−x′t−τQxt−τ+2x′tA11+B11K′+x′t−τA12′Pxt+x′tA21+B21K′+x′t−τA22′PA21+B21Kxt+A22xt−τ=x′tQxt−x′t−τQxt−τ+2x′tA11+B11K′+x′t−τA12′Pxt+x′t−τA22′PA22xt−τ+x′tA21+B21K′PA21+B21Kxt+2x′t−τA22′PA21+B21Kxt,that is,(17)L1V=xtxt−τ′Ω11PA12+A21+B21K′PA22∗−Q+A22′PA22xtxt−τ,where Ω11=Q+A11+B11K′P+PA11+B11K+A21+B21K′PA21+B21K.
In the light of (9), we can derive that L1Vxt,t<0, that is, the closed-loop system (3) is asymptotically stable in mean square sense.
Secondly, we prove that the control outputzt satisfies H∞ index for any nonzero disturbance vt under zero initial condition.
According to (7), T>0, and K′RK>0, we can obtain(18)Ψ11PA12+A21+B21K′PA22PB12∗−Q+A22′PA220∗∗−γ2I<0,where Ψ11=Q+A21+B21K′PA21+B21K+2A11+B11K′P+C1+D1K′C1+D1K.
Notice that(19)E∫0∞z′z−γ2v′vdt=E∫0∞x′tC1+D1K′C1+D1Kxt−γ2v′vdt+L2V−L2V≤E∫0∞x′tC1+D1K′C1+D1Kxt−γ2v′vdt+L2V,where L2Vxt is the infinitesimal operator of system (3) for any nonzero disturbance vt, and(20)L2Vxt,t=x′tQxt−x′t−τQxt−τ+2A11+B11Kxt+A12xt−τ+B12vt′Pxt+A21+B21Kxt+A22xt−τ′PA21+B21Kxt+A22xt−τ=x′tQxt−x′t−τQxt−τ+2x′tA11+B11K′Pxt+x′tA21+B21K′PA21+B21xt+2x′t−τA12′Pxt+2v′tB12′Pxt+2x′tA21+B21K′PA22xt−τ+x′t−τA22′PA22xt−τ.
Then, we can see that(21)E∫0∞z′z−γ2v′vdt≤E∫0∞xtxt−τvt′Ψ11Ψ12PB12∗−Q+A22′PA220∗∗−γ2Ixtxt−τvtdt,where Ψ12=PA12+A21+B21K′PA22.
Based on (12), we can see E∫0∞z′z−γ2v′vdt≤0, that is, (12) implies that E∫0∞zt2dt<γ2E∫0∞vt2dt. Therefore, system (3) satisfies H∞ index.
Thirdly, we prove that system (3) satisfies H2 index under the condition of vt=0.
Based on (13) and (14), C1+D1K′C1+D1K>0, and 1/γ2PB12B12′P>0, we obtain that(22)Θ11PA12+A21+B21K′PA22∗−Q+A22′PA22<0,holds, where Θ11=Ω11+T+K′RK.
Due to(23)E∫0∞x′tTxt+u′tRutdt=E∫0∞x′tT+K′RKxt+L1V−L1Vdt=E∫0∞x′tT+K′RKxt+L1Vdt−E∫0∞dV=E∫0∞x′tT+K′RKxt+L1Vdt−EVx∞,∞+EVx0,0=E∫0∞xtxt−τ′Θ11Ψ12∗Θ14xtxt−τdt+EVx0,0,where Θ14=−Q+A22′PA22.
In view of (22), we obtain(24)Θ11Ψ12∗Θ14<0.
According to (23) and (24), we can see(25)Js=E∫0∞x′tTxt+u′tRutdt<EVx0,0=Ex′0Px0+∫−τ0x′sQxsds=Js∗.
The proof is completed here.In order to solve the complex problem to seek the solution caused by the nonlinear terms in Lemma4, we give the following Lemma 5.Lemma 5.
For a given scalarγ>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(26)Ξ11Ξ12A12Q˜0P˜2P˜A21′2M′B21′B12∗Ξ22000000∗∗−Q˜3Q˜A22′0000∗∗∗−P˜0000∗∗∗∗−Q˜000∗∗∗∗∗−P˜00∗∗∗∗∗∗−P˜0∗∗∗∗∗∗∗−γ2I<0hold, where Ξ11=P˜A11′+A11P˜+M′B11′+B11M, Ξ12=3P˜C1′,P˜,3M′D1′,M′, and Ξ22=diag−I,−T˜,−I,−R˜; then, (8) is a mixed H2/H∞ controller of system (9), and the corresponding guaranteed cost for system (9) is Js∗=Ex′0P˜−1x0+∫−τ0x′sQ˜−1xsds. In this case, K=MP˜−1.Proof.
According to Lemma2 and (13), if the following inequality(27)ϒ11PA12PB12∗−Q+3A22′PA220∗∗−γ2I<0hold, where ϒ11=Q+4A21′PA21+4K′B21′PB21K+2A11′P+2K′B11′P+3C1′C1+3K′D1′D1K+T+K′RK, then (13) holds.
Using diagP−1,Q−1,I to premultiply and postmultiply inequality (27), we have(28)ϒ˜11A12Q−1B12∗−Q−1+3Q−1A22′PA22Q−10∗∗−γ2I<0,hold, where ϒ˜11=P−1QP−1+4P−1A21′PA21P−1+4P−1K′B21′PB21KP−1+2P−1A11′+2P−1K′B11′+3P−1C1′C1P−1+3P−1K′D1′D1KP−1+P−1TP−1+P−1K′RKP−1. Let P˜=P−1, M=KP˜, Q˜=Q−1, R˜=R−1, and T˜=T−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(29)minγ>0,α>0,W>0,P˜>0,Q˜>0,Mα+trWsubject to (26) and(30)−αx′0x0−P˜<0,(31)−WN′N−Q˜<0,has a solution α, W, P˜, Q˜, and M, then controller ut=MP˜−1xt is an optimal state feedback H2/H∞ controller which ensures the minimization of guaranteed cost Js∗=Ex′0P˜−1x0+∫−τ0x′sQ˜−1xsds for system (9), where ∫−τ0xsx′sds=NN′.Proof.
From Lemma5, the controller ut=MP˜−1xt is a guaranteed cost control law of system (9). (30) is equivalent to x′0P˜−1x0<α; (31) is equivalent to N′Q˜−1N<W.
Therefore, we can obtain(32)∫−τ0x′sQ˜xsds=∫−τ0trx′sQ˜−1xsds=trNN′Q˜−1=trN′Q˜−1N<trW.
Thus, we can obtainJs∗<α+trW.
Therefore, the minimization ofα+trW 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(33)Js∗=Ex′0P˜−1x0+∫−τ0x′sQ˜−1xsds=Etrx0x′0P˜−1+∫−τ0Etrxsx′sQ˜−1ds=trEx0x′0P˜−1+∫−τ0trExsx′sQ˜−1ds=trP˜−1+τ×trQ˜−1.
Therefore, we have the following optimization problem:(34)minγ>0,W1>0,W2>0,P˜>0,Q˜>0,MtrW1+τ×trW2,which subjects to (26) and(35)W1IIP˜>0,(36)W2IIQ˜>0.Theorem 2.
If there exist solution to (26), (34)–(36) then controller ut=MP˜−1xt is an optimal state feedback H2/H∞ controller which ensures the minimization of guaranteed cost (18) for system (1).Proof.
From Lemma5, the controller ut=MP˜−1xt is a H2/H∞ controller of system (9). We can see (34) is equivalent to 0<P˜−1<W1 and (35) is equivalent to 0<Q˜−1<W2 from Lemma 3. Therefore, the minimization of trW1+τ×trW2 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 ofα+trW in Theorem 1. The similar algorithm can also be applied to Theorem 2.By analyzing (26), (30), (31) in Theorem 1, we find that if (26), (30), (31) have no feasible solutions when γ takes the initial value, then (26), (30), (31) will have no feasible solutions for all γ>0. Next, we search for γ from the initial value that makes (26), (30), (31) have feasible solutions to optimize α+trW by using linear search algorithm. The specific algorithm is as follows.


## 5. Numerical Examples

The coefficient matrices of system (1) are given as follows:(37)A11=−3010−10−20,A12=0.10.200.1,A21=−2.70.80.9−1.6,A22=−0.2−0.5−0.3−1.4,B11=−43,B12=0.1−0.10.50.7,B21=−12,C1=2123,D1=−13,N=21,T=4−3−34,R=1,τ=1.First case: when the initial function is known andx0=12′, t∈−1,0. In order to find the minimum value of α+trW, we obtain the relationship between α+trW and γ by Algorithm 1, which is shown in Figure 1.Algorithm 1: Linear search algorithm.
Step 1: Given the values of τ.Step 2: Using linear search algorithm, if a series of γii=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 γi.Step 4: Solve the following minimization problem:mins.t.20,24,25α+trW.Step 5: Let i=i+1, if i+1>n, then turn to Step 6; otherwise, let γi=γi+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.Figure 1
Whenγ∈0,2, the minimum upper bound of α+trW.As can be seen from Figure1, α+trW decreases with the increase of γ, and minα+trW=38.4173 when γ=0.4, and minα+trW=29.7250 when γ=1.98.Takeγ=0.8, according to Theorem 1, we obtain that(38)P˜=0.84840.02150.02150.5896,Q˜=18.8352−4.2480−4.24801.0503,M=0.0629−0.0472,W=23.0449,α=7.7981.Therefore, the optimal state feedbackH2/H∞ controller is ut=0.0762−0.0828xt, and the guaranteed cost of closed-loop system is Js∗=30.8430.Take external disturbancevt=sint, then we can obtain the curves of x1 and x2 and Ext2 in Figure 2. From Figure 2, we can see that Ex02=5 and limt⟶∞Ext2=0, that is, closed-loop system (9) is mean square stable.Figure 2
Whent∈0,1, the response for Ext2.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 oftrW1+τ×trW2, we obtain the relationship between trW1+τ×trW2 and γ by Algorithm 1, which is shown in Figure 3.Figure 3
Whenγ∈0,2, the minimum upper bound of trW1+τ×trW2.As can be seen from Figure3, trW1+τ×trW2 decreases with the increase of γ, and mintrW1+τ×trW2=19.5450 when γ=0.4, mintrW1+τ×trW2=13.6648 when γ=1.98.Takeγ=0.8, according to Theorem 2, we obtain that(39)P˜=0.85200.02170.02170.5889,Q˜=20.5511−4.6765−4.67651.1567,M=0.0629−0.0472,trW1=2.8746,trW2=11.4205.Therefore, the optimal state feedbackH2/H∞ controller is ut=0.0759−0.0829xt, and the guaranteed cost of closed-loop system is Js∗=14.2951.


## 6. Conclusion

In this paper, the mixedH2/H∞ control problem for Itô-type stochastic time-delay systems is presented, and the description of H2/H∞ control problem for stochastic time-delay systems is given. On the basis of matrix transformation and convex optimization method, state feedback H2/H∞ controller is obtained to make the system satisfy H∞ performance index and H2 performance index. Moreover, an algorithm is given to solve state feedback controller and optimize H2/H∞ performance index. Finally, a numerical example is used to show the feasibility of the results. In the future work, we will investigate mixed H2/H∞ control for the more complex systems, such as, stochastic Markov jump systems with time delay.



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