# Finite-TimeH2/H∞ Control for Linear Itô Stochastic Systems with x,u,v-Dependent Noise

Authors: Zhiguo Yan; Shiyu Zhong; Xingping Liu
Journal: Complexity
 (2018)

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## Abstract

This paper deals with the problem of theH2/H∞ control based on finite-time boundedness for linear stochastic systems. The motivation for investigating this problem comes from one observation that the H2/H∞ control does not involve systems’ transient performance. To express this problem clearly, a concept called finite-time H2/H∞ control is introduced. Moreover, state feedback and observer-based finite-time H2/H∞ controllers are designed, which guarantee finite-time boundedness, H2 performance index, and H∞ performance index of the closed-loop systems. Furthermore, an optimization algorithm on the finite-time H2/H∞ control is presented to obtain the minimum values of the H2 index and H∞ index. Finally, we use an example to show the validity of the obtained results.


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## Body

## 1. Introduction

Recently, stochastic systems have been receiving more and more attention, which is due to their applications in many practical fields, such as finance systems [1] and power systems [2]. A lot of results on stochastic systems have been obtained. For example, [3] gave some Razumikhin-type theorems on the pth moment input-to-state stability for impulsive stochastic delayed systems by utilizing the Razumikhin technique and average dwell-time approach. By using the interval matrix transformation method, the problems of finite-time dissipative control for stochastic interval systems are investigated in [4]. The literature [5] investigated the state prediction problem for nonlinear stochastic differential systems by utilizing the Carleman embedding technique. In addition, many other nice results on stochastic systems have also been obtained; see, e.g., Lyapunov stability conditions [6], reliable output feedback control [7] and distributed containment control [8], and the references therein.On the other hand, one of the important control methods in dealing with robust control problems is theH2/H∞ control method. Its concrete implication is to seek a controller which not only minimizes the H2 cost function but also restrains the effect of external disturbance. The H2/H∞ control method has been applied to many fields, such as communication systems [9] and synthetic gene network design [10]. Meanwhile, H2/H∞ control problems have been extended to many control system models, and many nice results have been obtained, such as stochastic systems [11], descriptor systems [12, 13], Markovian jump systems [14], and the recent monograph [15]. Nevertheless, most of the results on H2/H∞ control problems in existing literature are based on Lyapunov’s asymptotic stability, which do not reflect the transient performance of the systems. In some cases, large transient performance has a negative effect on the practical systems. For example, in power systems, a large transient current cannot be permitted, because it can damage the system [16]. In order to describe the phenomenon precisely, finite-time (FT) stability was introduced in the literature [17–19]. In the sequel, considering external disturbance, [20] extended FT stability to FT boundedness. Currently, the issues on FT stability and FT boundedness have been extensively studied for many kinds of system models; see, e.g. [21–30] and the references therein. Taking into account the advantages of FT stability and H2/H∞ control, the finite-time H2/H∞ control problem is proposed in this paper, which not only guarantees FTB but also satisfies the H2/H∞ performance indices. Up to date, there is almost no literature to consider the FT H2/H∞ control of stochastic systems.This paper will study the problems of FTH2/H∞ control for linear Itô stochastic systems. Because of the complexity of the problems, the appropriate controller design will be more difficult. By using the method of stochastic analysis, state feedback and observer-based FT H2/H∞ controllers are designed. The main contributions of this paper are as follows: (i) The definition of the FT H2/H∞ control for linear stochastic systems with x,u,v-dependent noise (which is called state-, control-, and disturbance-dependent noise) is first given, which simultaneously presents FT boundedness, H2 performance index, and H∞ performance index, respectively, of the closed-loop system. (ii) The two new sufficient conditions for the existence of the state feedback controller and observer-based controller are obtained in the form of linear matrix inequalities (LMIs). (iii) A parameter optimization algorithm is given to obtain the minimum values of the H2 performance index and H∞ performance index, simultaneously.The organization of this paper is summarized as follows: In Section2, we give some preliminaries. In Section 3, the state feedback FT H2/H∞ controller is designed. In Section 4, the observer-based FT H2/H∞ controller is designed. In Section 5, an optimization algorithm is provided for obtaining the minimum values of the H2 performance index and H∞ performance index, simultaneously. In Section 6, a numerical example is discussed, and some remarks are concluded in Section 7.Notation.
Ω,F,Ftt≥0,ℙ: a complete probability space with a filtration Ftt≥0. M′: transpose of a matrix M. M>0: M is a positive definite symmetric matrix. In×n: n×n identity matrix. LF2R+,Rl: the space of nonanticipative stochastic process ξt with respect to filtration Ft satisfying ξtL22≔E∫0∞ξt2dt<∞. λmaxM and λminM: the maximum and minimum eigenvalue of matrix M. E· is the mathematical expectation of the stochastic process. The asterisk “∗” in a matrix stands for the symmetry term. diag⋯ represents a diagonal matrix. The “wrt” denotes “with respect to.”


## 2. Preliminaries

We consider the linear stochastic system as follows:(1)dxt=A1xt+B1ut+G1vtdt+A2xt+B2ut+G2vtdwt,yt=C1xt,zt=C2xt+D1ut+D2vt,x0=x0∈Rn,where A1, A2, B1, B2, C1, C2, D1, D2, G1, and G2 are constant matrices with appropriate dimensions. xt∈Rn, ut∈Rm, vt∈Rp, yt∈Rq, and zt∈Rl are called state, control input, external disturbance, measurement output, and controlled output, respectively. x0 is the initial state. wt is a one-dimensional Wiener process, which is defined as Ω,F,Ftt≥0,ℙ. vt is the disturbance and satisfies the following set:
(2)Ω=vt:E∫0tv′svsds<h,h>0,t∈0,T.Next, the definition of FT stochastic boundedness for system (1) is introduced, which is the generalization of FT boundedness in [20, 31–33].Definition 1.
System (1) with ut=0 is FT stochastically bounded wrt d1,d2,T,R,h, if
(3)x′0Rx0≤d1⇒Ex′tRxt<d2,∀t∈0,T,∀vt∈Ω,where 0<d1<d2, T>0, and R>0.For the subsequent analysis, the following lemmas are useful for the FTH2/H∞ controller design.Lemma 1 [27].
Ifρt>0, f≥0, and g≥0 satisfy
(4)ρt≤f+g∫0tρsds,t∈0,T,then we obtain
(5)ρt≤fexpgt.Lemma 2 [34].
LetVt,x∈C1,2R+,R+×Rn be a scalar function and Vt,x>0. For the following stochastic system
(6)dxt=axdt+bxdwt,the Itô formula of Vt,x is given as follows:
(7)dVt,x=LVt,xdt+∂V′t,x∂xbxdwt,where
(8)LVt,x=∂Vt,x∂t+∂V′t,x∂xax+12b′x∂2Vt,x∂x2bx.


## 3. State Feedback Finite-TimeH2/H∞ Controller Design

This section gives a problem formulation of state-feedback finite-time (SFFT)H2/H∞ control for system (1). And also, the two sufficient conditions are given for the existence of a SF controller.Considering the following linear state-feedback (SF) controller(9)ut=Kxt,where K is the feedback gain matrix.Substituting the controller (9) into system (1) leads to the closed-loop system
(10)dxt=A~1xt+G1vtdt+A~2xt+G2vtdwt,yt=C1xt,zt=C~xt+D2vt,x0=x0∈Rn,where A~1=A1+B1K, A~2=A2+B2K, and C~=C2+D1K.Associated with system (1), the following cost function is provided:
(11)Jsxt,ut=E∫0Tx′tP1xt+u′tP2utdt,where P1>0 and P2>0 are the given positive scalars or given weighting matrices.Also, by substituting (9) into (11), the corresponding performance function (11) becomes
(12)Jsxt=E∫0Tx′tP1xt+x′tK′P2Kxtdt.For a givenγ>0, under the condition of zero initial value, the discretional nonzero disturbance vt and the control output zt satisfy the following form:
(13)E∫0Tz′tztdt<γ2E∫0Tv′tvtdt.Based on the above preparations, we are in a position to give the definition of the state-feedback finite-timeH2/H∞ control.Definition 2.
For some given positive scalarsγ, d1, d2, T, R, and h, considering system (1), cost function (11), and the inequality (13), if there exist a positive scalar Js∗ and a SF controller (9) such that
(i)
system (10) is FT stochastically bounded wrt d1,d2,T,R,h
(ii)
H2 cost function (12) satisfies Jsxt≤Js∗ under the condition of vt=0
(iii)
for any nonzero disturbancevt, under the zero initial condition, the inequality (13) is satisfiedthen (9) is a state-feedback finite-time H2/H∞ controller of the stochastic system (1).Remark 1.
Definition2 considers three aspects of actual systems: FT boundedness of systems’ states, minimum performance cost, and ability to suppress interference of the closed-loop systems, which is more complex than only considering the problems of the H2 performance index or H∞ performance index. In addition, these three aspects involved by the FT H2/H∞ control are often required by the actual systems. For example, in the power systems, a large transient current cannot be permitted, electric energy consumption is expected to be minimum, and the system is expected to have better ability to suppress interference [16].A sufficient condition for the existence of the SFFTH2/H∞ controller will be given by the following theorem.Theorem 1.
For the given positive scalarsγ, d1, d2, T, R, and h, if there exist a matrix M>0, a matrix K, and a scalar α≥0 such that
(14)T1G1+M~C~′D2M~A~2′∗−γ2I+D2′D2G2′∗∗−M~<0,(15)T2M~A~2′∗−M~<0,(16)d1λminM+hγ2<d2λmaxMe−αT,hold, where M~=R−1/2MR−1/2, T1=A~1M~+M~A~1′−αM~+M~C~′C~M~, and T2=A~1M~+M~A~1′−αM~+M~P1M~+M~K′P2KM~, then ut=Kxt is a SFFT H2/H∞ controller of system (1) and the upper bound of the H2 index can be given as Jstate∗=λmaxM−1d1eαT.Proof.
The proof will be divided into three steps.Step 1.
Prove that the condition (i) is satisfied.
It is noticed that(17)M~C~′C~M~M~C′~D20D2′C~M~D2′D20000=M~C~′D2′0M~C~′D2′0′≥0.
Therefore, the condition (14) implies that
(18)A~1M~+M~A~1′−αM~G1M~A~2′∗−γ2IG2′∗∗−M~<0.
LetVxt=x′tM~−1xt, the infinitesimal operator L1Vxt of system (10) is defined as follows:
(19)ℒ1Vxt=A1+B1Kxt+G1vt′M~−1xt+x′tM~−1A1+B1Kxt+G1vt+A2+B2Kxt+G2vt′M~−1A2+B2Kxt+G2vt=xtvt′Z¯1A~1′M~−1G2+M~−1G1∗G2′M~−1G2xtvt,where Z¯1=A~1′M~−1+M~−1A~1+A~2′M~−1A~2.
Pre- and postmultiplying the inequality (18) by diagM~−1,I,M~−1, it leads to the following inequality:
(20)A~1′M~−1+M~−1A~1−αM~−1M~−1G1A~2′M~−1∗−γ2IG2′M~−1∗∗−M~−1<0.
According to the Schur complement, (20) is equivalent to
(21)A~1′M~−1+M~−1A~1+A~2′M~−1A~2−αM~−1M~−1G1+A~2′M~−1G2∗G2′M~−1G2−γ2I<0.
Considering the conditions (19) and (21), it follows that
(22)ℒ1Vxt<αVxt+γ2v′tvt.
Integrating (22) from 0 to t and then taking the expectation, it follows that
(23)EVxt<Vx0+α∫0tEVxsds+γ2∫0tEv′svsds.
By Lemma1, we get
(24)EVxt<Vx0eαt+γ2eαt∫0tEv′svsds.
According to known conditions, it yields(25)EVxt=Ex′tR1/2M−1R1/2xt≥λminM−1Ex′tRxt,
(26)Vx0eαt=x′0R1/2M−1/R12x0eαt≤λmaxM−1x′0Rx0eαt≤λmaxM−1d1eαT,
(27)γ2eαt∫0tEv′svsds<eαThγ2.
From (24), (25), (26), and (27), it is easy to obtain
(28)Ex′tRxt<λmaxMeαTd1λminM+hγ2.
According to condition (16), it follows that (28) leads to Ex′tRxt<d2 for all t∈0,T. So, the closed-loop system (10) is FT stochastically bounded wrt d1,d2,T,R,h.
Step 2.
Prove that the condition (ii) is satisfied.
Under the condition ofvt=0, the infinitesimal operator L2Vxt of system (10) is as follows:
(29)ℒ2Vxt=A~1xt′M~−1xt+x′tM~−1A~1xt+A~2xt′M~−1A~2xt=x′tA~1′M~−1+M~−1A~1+A~2′M~−1A~2xt.
By Schur’s complement, it can be seen that (15) is equivalent to
(30)M~A~1′+A~1M~+M~A~2′M~−1A~2M~+M~P1M~+M~K′P2KM~−αM~<0.
Premultiplying and postmultiplying (30) by M~−1, we obtain
(31)A~1′M~−1+M~−1A~1+A~2′M~−1A~2+P1+K′P2K−αM~−1<0.
According to (29) and (31), we get
(32)ℒ2Vxt−αVxt+x′tP1+K′P2Kxt<0.
Integrating (32) from 0 to t, t∈0,T, and taking the expectation, it is obtained that
(33)E∫0tx′tP1+K′P2Kxtdt+EVxt<Vx0+αE∫0tVxtdt.
From (33), we get
(34)EVxt<Vx0+αE∫0tVxtdt,
(35)Jsxt<αE∫0tVxtdt+Vx0.
From (34), by Lemma 1, we obtain
(36)EVxt<Vx0eαt.
According to (35) and (36), it is obtained that
(37)Jsxt<αE∫0tx′0R1/2M−1R1/2x0eαtdt+x′0M~−1x0=x′0R1/2M−1R1/2x0eαt<λmaxM−1d1eαT=Js∗.
Step 3.
Prove that the condition (iii) is satisfied.
Premultiplying and postmultiplying (14) by diagM~−1,I,M~−1, and according to the Schur complement, we have
(38)T3A′~1M~−1G2+M~−1G1+C~′D2∗−γ2I+G2′M~−1G2+D2′D2<0,where T3=A~′M~−1+M~−1A~+A~1′M~−1A~1−αM~−1+P1+K′P2K+C~′C~.
According to (19) and (38), we have
(39)ℒ1Vxt<αVxt+γ2v′tvt−z′tzt.
Premultiplying and postmultiplying (39) by e−αt, we have
(40)e−αtℒ1Vxt<αe−αtVxt+e−αtγ2v′tvt−z′tzt.
By applying Lemma2, we can obtain that
(41)ℒ1e−αtVxt=−αe−αtVxt+e−αtℒ1Vxt.
According to (40) and (41), we get
(42)ℒ1e−αtVxt<e−αtγ2v′tvt−z′tzt.
Under the condition of zero initial value, integrating both sides of (42) from 0 to t, t∈0,T, taking the expectation, and after some calculations, we have
(43)e−αTEVxt<e−αTE∫0Tγ2v′svsds−z′szsds<γ2E∫0Tv′svsds−E∫0tz′szsds.
Becausee−αTEVxt>0, we can get
(44)E∫0Tz′szsds<γ2E∫0Tv′svsds.
This completes the proof.It should be noted that (14), (15), and (16) are not linear matrix inequalities and difficult to solve. To overcome the above difficulties, the following theorem is obtained.Theorem 2.
For the given positive scalarsγ, d1, d2, T, R, and h, if there exist a matrix M>0, a matrix Y, a scalar λ>0, and a scalar α≥0 such that
(45)T4G1M~A2′+Y′B′2M~C2′+Y′D1∗−γ2IG2′D2∗∗−M~0∗∗∗−I<0,(46)T4M~A2′+Y′B′2M~Y′∗−M~00∗∗−P1−10∗∗∗−P2−1<0,(47)hγ2−d2e−αTd1∗−λ<0,(48)λI<M<I,hold, where T4=A1M~+M~A1′−αM~+B1Y+Y′B1′, then ut=Kxt=YM~−1xt is a SFFT H2/H∞ controller and the upper bound of the H2 index can be given as Jstate∗=λ−1d1eαT.Proof.
LettingY=KM~, (45) and (46) can lead to inequalities (14) and (15), respectively, and it is easy to check that (16) in Theorem 1 can be guaranteed by inequalities (47) and (48). The proof is completed.Remark 2.
It is noted that the inequalities in Theorem2 are not LMIs, but if α is fixed, (45), (46), (47), and (48) can be regarded as LMIs.


## 4. Observer-Based Finite-TimeH2/H∞ Controller Design

In the above Section3, the SFFT H2/H∞ control problem of system (1) has been discussed. However, in many practical cases, it is difficult to measure the total states. Therefore, considering the problem of the observer-based finite-time (OBFT) H2/H∞ controller design is necessary. Typically, an observer-based (OB) controller is provided as follows:
(49)dx^t=A1x^t+B1ut+Lyt−C1x^tdt,ut=Kx^t,x^0=0,where x^t∈Rn is the estimation of xt and L is the estimator gain.Substituting (49) into system (1), we will obtain the closed-loop system
(50)dx~t=A¯1x~t+W1vtdt+A¯2x~t+W2vtdwt,z~t=T1x~t+D2vt,and the corresponding closed-loop cost function
(51)Jox~t=E∫0Tx~′tTx~tdt,where
(52)x~t=xtx^t,A¯1=A1B1KLC1A1+B1K−LC1,W1=G10,(53)A¯2=A2B2K00,W2=G20,T1=C2′K′D1′,T=P100K′P2K.For a givenγ>0, under the condition of zero initial value, the discretional nonzero disturbance vt and the control output zt satisfy the following form:
(54)E∫0Tz~′tz~tdt<γ2E∫0Tv′tvtdt.Then, the definition of the observer-based finite-timeH2/H∞ control is given as follows.Definition 3.
For the given positive scalarsγ, d1, d2, T, R, and h, considering system (1), cost function (11), and the inequality (13), if there exist an OB controller (49) and a positive scalar Jo∗ such that
(i)
system (50) is FT stochastically bounded wrt d1,d2,T,R,h
(ii)
H2 cost function (51) satisfies Jx~t≤Jo∗ under the condition of vt=0
(iii)
for any nonzero disturbance, under the zero initial condition, the inequality (54) is satisfiedthen (49) is said to be an observer-based finite-time H2/H∞ controller for the stochastic system (1).Remark 3.
From Definition3, it can be seen that the existence of the observer-based finite-time H2/H∞ controller (49) implies the existence of the SFFT H2/H∞ controller.By the results in Theorem2, we have designed a state-feedback finite-time H2/H∞ controller ut=Kxt. Next, a design condition of the observer-based finite-time H2/H∞ controller for the stochastic system (1) will be given.Theorem 3.
For the given positive scalarsγ, d1, d2, T, R, and h, if there exist a positive matrix Q and a scalar β≥0 such that
(55)T5Q~W1+T1′D2A¯2′Q~∗−γ2I+D2′D2W2′Q~∗∗−Q~<0,(56)A¯1′Q~+Q~A¯1+A¯2′Q~A¯2−βQ~+T<0,(57)λmaxQd1+hγ2<λminQd2e−βT,hold, where Q~=R1/2QR1/2 and T5=A¯1′Q~+Q~A¯1−βQ~+T1′T1, then (49) is an OBFT H2/H∞ controller and the upper bound of the H2 index can be given as Jo∗=λmaxQd1eβT.Proof.
The proof will be divided into three steps.Step 1.
Prove that the condition (i) is satisfied.
LetVx~t=x~′tQ~x~t with Q~>0 be solutions to (55), (56), and (57). The infinitesimal operator L3Vx~t of system (50) is defined as follows:
(58)ℒ3Vx~t=A¯1x~t+W1vt′Q~x~t+x~′tQ~A¯1x~t+W1vt+A¯2x~t+W2vt′Q~A¯2x~t+W2vt=x~′tA¯1′Q~+Q~A¯1+A¯2′Q~A¯2x~t+v′tW1′Q~+W2′Q~A¯2x~t+x~′tQ~W1+A¯2′Q~W2vt+v′tW2′Q~W2vt=x~tvtA¯1′Q~+Q~A¯1+A¯2′Q~A¯2Q~W1+A¯2′Q~W2∗W2′Q~W2x~tvt.
Note that(59)T1′T1T1′D20D2′T1D2′D20000=T1′D2′0T1′D2′0′≥0.
Therefore, the inequality (55) implies that
(60)A¯1′Q~+Q~A¯1−βQ~Q~W1A¯2′Q~∗−γ2IW2′Q~∗∗−Q~<0.
According to the Schur complement, (60) can be transformed into
(61)A¯1′Q~+Q~A¯1+A¯2′Q~A¯2−βQ~Q~W1+A¯2′Q~W2∗−γ2I+W2′Q~W2<0.
According to (58) and (61), we have
(62)ℒ3Vx~t<βVx~t+γ2v′tvt.
Integrating (62) from 0 to t and then taking the expectation, it follows that
(63)EVx~t<Vx~0+β∫0tEVx~sds+γ2∫0tEv′svsds.
According to Lemma1, it is obtained that
(64)EVx~t<Vx~0eβt+γ2eβt∫0tEv′svsds.
According to known conditions, it yields that(65)EVx~t=Ex~′tR1/2QR1/2x~t≥λminQEx~′tRx~t,(66)Vx~0eβt=x~′0R1/2QR1/2x~0eβt≤λmaxQx~′0Rx~0eβt≤λmaxQd1eβT,(67)γ2eβt∫0tEv′svsds<eβThγ2.
From (64), (65), (66), and (67), it is easy to obtain that
(68)Ex~′tRx~t<λmaxQd1eβT+hγ2eβTλminQ.
According to condition (57), it follows that (68) leads to Ex~′tRx~t<d2 for all t∈0,T. So, the closed-loop system (50) is FT stochastically bounded wrt d1,d2,T,R,h.
Step 2.
Prove that the condition (ii) is satisfied.
Under the condition ofvt=0, the infinitesimal operator L4Vx~t of system (50) is defined as follows:
(69)ℒ4Vx~t=A¯1x~t′Q~x~t+x~′tQ~A¯1x~t+A¯2x~t′Q~A¯2x~t=x~′tA¯1′Q~+Q~A¯1+A¯2′Q~A¯2x~t.
According to (56), we get
(70)ℒ4Vx~t−βVx~t+x~′tTx~t<0.
Integrating (70) from 0 to t and taking the expectation, the following results are obtained:
(71)EVx~t+E∫0tx~′tTx~tdt<Vx~0+βE∫0tVx~tdt.
From (71), we have
(72)EVx~t<Vx~0+βE∫0tVx~tdt,
(73)E∫0tx~′tTx~tdt<Vx~0+βE∫0tVx~tdt.
From (72), by Lemma 1, we obtain
(74)EVx~t<Vx~0eβt.
From (73) and (74), we have
(75)Jox~t<βE∫0tVx~tdt+Vx~0<βE∫0tVx~0eβtdt+Vx~0=Vx~0eβt<λmaxQd1eβT=Jo∗.
Step 3.
Prove that the condition (iii) is satisfied.
According to the Schur complement, it can be seen that (55) is equivalent to
(76)A¯1′Q~+Q~A¯1−βQ~+T1′T1+A¯2′Q~A¯2Q~W1+T1′D2+A¯2′Q~W2∗W2′Q~W2+D2′D2−γ2I<0.
According to (58) and (76), we get
(77)ℒ3Vx~t<βVx~t+γ2v′tvt−z~′tz~t.
Using the similar procedure to prove Step 3 in Theorem3, we can obtain
(78)E∫0Tz~′sz~sds<γ2E∫0Tv′svsds.
This completes the proof.We find that inequalities (55), (56), and (57) in Theorem 3 are not easy to solve, so the following theorem will transform (55), (56), and (57) into LMIs.Theorem 4.
For the given positive scalarsγ, d1, d2, T, R, and h, if there exist a scalar β>0, two matrices Q~11>0 and Q~22>0, a matrix N, and a scalar η>0 such that
(79)Σ11Σ12Σ13∗Σ22Σ23∗∗Σ33<0,(80)E11−βQ~11E12∗E22−βQ~22<0,(81)eβTηd1+hγ2−d2<0,(82)I<diagQ~11,Q~22<ηI,hold, where Σ11=A1′Q~11+Q~11A1+A2′Q~11A2+C2′C2−βQ~11, Σ12=C1′N′+Q~11B1K+A2′Q~11B2K+C2′D1K, Σ22=A1+B1K′Q~22+Q~22A1+B1K+K′B2′Q~11B2K−NC1−C1′N′+K′D1′D1K−βQ~22, Σ13=Q~11G1+C2′D2+A2′Q~11G2, Σ23=K′D1′D2+K′B2′Q11G2, Σ33=G2′Q~11G2+D2′D2−γ2I, E11=A1′Q~11+Q~11A1+A2′Q~11A2+P1, E12=C1′N′+Q~11B1K+A2′Q~11B2K, and E22=A1+B1K′Q~22+Q~22A1+B1K+K′B2′Q~11B2K−NC1−C1′N′+K′P2K, then (49) is an OBFT H2/H∞ controller and the upper bound of the H2 index can be given as Jo∗=ηd1eβT. In this case, an appropriate estimator gain matrix is given by L=Q~22−1N.Proof.
SettingQ~=diagQ~11,Q~22, substituting (52) and (53) into (55), and letting N=Q~22L, (79) can be obtained from (55). It is obvious that (80) can be obtained from (56). And then, (81) and (82) imply (57). The proof is completed.


## 5. Algorithm

In this section, an optimization algorithm on the FTH2/H∞ control is presented. The following algorithm is given for Theorem 2. Similarly, the algorithm can be utilized in Theorem 4.Analysis.
By Theorem2, let Js∗<ξ, and then minimize ξ, subject to (45), (46), (47), and (48), M>0 and
(83)Me−αTd1−ξ−1I>0,where 0<d1<d2, T>0, α≥0, and λ>0.By analyzing (45), (46), (47), and (48) in Theorem 2, we find the following: if (45), (46), (47), and (48) have no feasible solutions at the case of α=0, then (45), (46), (47), and (48) will have no feasible solutions for all α>0. Therefore, we seek α that makes (45), (46), (47), and (48) have feasible solutions from α=0 by linear searching. The detailed algorithm will be given as follows. One can use this algorithm to obtain the minimum value of ξ and γ2.Algorithm 1
Step 1: Given d1, d2, T, R, h, and a step size dα for α.
Step 2: Set i=1 and take α1=0.
Step 3: If αi makes the following optimization problem
mins.t.45,46,47,48,83,M>0ξ,
mins.t.45,46,47,48,83,M>0γ2,
solvable, then storeαi into Ui, ξmin into Fi, and γmin into Hi and let αi+1=αi+dα, loop.
Otherwise, go toStep 4.
Step 4: Exit.By running Algorithm1, we get the following results:
(1)
IfUi=empty, Fi=empty, and Hi=empty, then it shows that the above optimization problem (OP) is not solvable.
(2)
IfUi≠empty, Fi≠empty, and Hi≠empty, then it shows that the above optimization problem is solvable. Moreover, we can respectively plot the curves of Ui,Fi, Ui,Hi, and Fi,Hi. The three curves can help us choose a suitable finite-time H2/H∞ controller.
(3)
This algorithm can find the most suitableα to decrease the conservativeness of the conditions of Theorem 2.


## 6. Examples

This section provides an example to show the effectiveness of the obtained results. System (10) with parameters is as follows:
(84)A1=−10.1−10.810.7−10.2,A2=−0.810.4−1,B1=−2.15−4.13,B2=−0.81.2−11.5,G1=−0.50.4−0.1−0.5,G2=0.2−0.10.40.5,P1=4−3−34,P2=3223,C1=−0.8−1.51.1−3,C2=−1.9−0.11.5−2.5,x0=−0.70.7′,and d1=1, d2=5, D1=2, D2=0.2, T=1, R=I, and h=0.5.


### 6.1. SFFTH2/H∞ Controller Design

By running Algorithm1, the relations of α and ξ (Figure 1), α and γ (Figure 2), and γ and ξ (Figure 3) are obtained. From Figure 1, it can be seen that the bigger the number of α, the bigger ξ is. Also, the minimum value of the H2 performance index is ξ=1 when α=0, and the maximum value of the H2 performance index is ξ=4.4817 when α=1.5. The range of α is 0,1.5, and the range of ξ is 1,4.4817.Figure 1
ξ versus α.Figure 2
γ versus α.Figure 3
γ versus ξ.From Figure2, it can be seen that the bigger the number of α, the smaller γ is. Also, the maximum value of γ is 0.6757 when α=0, and the minimum value of γ is 0.6734 when α=1.5. The range of α is 0,1.5 and the range of γ is 1,4.4817.In fact, Figures1 and 2 imply Figure 3. To show the relations between γ and ξ clearly, Figure 3 is also drawn. From Figure 3, it can be seen that the bigger the number of ξ, the smaller γ is. Figures 1, 2, and 3 can provide us a method for how to select the suitable SFFT H2/H∞ controller. If we are mainly concerned with the H2 performance, a smaller α can be selected. If we are mainly concerned with the H∞ performance, a bigger α can be selected.Next, substitutingα=0 into Theorem 2 and then solving (45), (46), (47), and (48), we have
(85)M=0.65280.08100.08100.6935,Y=0.0746−0.06350.52360.3750,λ=0.3980.Therefore, we get the following controller gain matrix:(86)K=YM−1=0.1275−0.10640.74600.4536.Supposevt=sint∫01sin2tdt=0.275<h=0.5 uses the Euler-Maruyama method to simulate the standard Brownian motion, one can obtain the simulated curves of x′tRxt (30 curves) and Ex′tRxt of system (10) in Figure 4. From Figure 4, we see that system (10) is FT stochastically bounded wrt 1,5,1,I,0.5.Figure 4
The response forEx′tRxt.


### 6.2. OBFTH2/H∞ Controller Design

On the basis of the SF case, Algorithm1 can also be used to solve the similar OP for the OBFT H2/H∞ control. The relations of β and ξ (Figure 5), β and γ (Figure 6), and γ and ξ (Figure 7) are obtained. From Figure 5, it can be seen that the bigger the number of β, the bigger ξ is. Also, the minimum value of the H2 performance index is ξ=1 when β=0, and the maximum value of the H2 performance index is ξ=4.0552 when β=1.4. The range of α is 0,1.4, and the range of ξ is 1,4.0552.Figure 5
ξ versus β.Figure 6
γ versus β.Figure 7
γ versus ξ.From Figure6, it can be seen that the bigger the number of β, the smaller γ is. Also, the maximum value of γ is 0.8656 when β=0, and the minimum value of γ is 0.8413 when β=1.4. The range of β is 0,1.4, and the range of γ is 0.8413,0.8656.In fact, Figures5 and 6 also imply Figure 7. To show the relations between γ and ξ clearly, Figure 7 is also drawn. From Figure 7, it can be seen that the bigger the number of ξ, the smaller γ is. Figures 5–7 can provide us a method for how to select the suitable OBFT H2/H∞ controller. If we are mainly concerned with the H2 performance, a smaller β can be selected. If we are mainly concerned with the H∞ performance, a bigger β can be selected.Substitutingβ=0 into Theorem 4 and solving (79), (80), (81), and (82) yields
(87)Q11=1.9421−0.0442−0.04422.2252,Q22=3.19130.09000.09002.2214,N=4.2749−2.92181.65871.1962,η=3.6737Then, the corresponding observer gain matrix is obtained:(88)L=Q22−1N=1.3200−0.93180.69320.5762.Supposevt=sint∫01sin2tdt=0.275<h and uses the Euler-Maruyama method to simulate the standard Brownian motion, one can obtain the simulated curves of x~′tRx~t (30 curves) and Ex~′tRx~t of system (50) in Figure 8. From Figure 8, we see that system (10) is FT stochastically bounded wrt 1,5,1,I,0.5.Figure 8
The response forEx~′tRx~t.


## 6.1. SFFTH2/H∞ Controller Design

By running Algorithm1, the relations of α and ξ (Figure 1), α and γ (Figure 2), and γ and ξ (Figure 3) are obtained. From Figure 1, it can be seen that the bigger the number of α, the bigger ξ is. Also, the minimum value of the H2 performance index is ξ=1 when α=0, and the maximum value of the H2 performance index is ξ=4.4817 when α=1.5. The range of α is 0,1.5, and the range of ξ is 1,4.4817.Figure 1
ξ versus α.Figure 2
γ versus α.Figure 3
γ versus ξ.From Figure2, it can be seen that the bigger the number of α, the smaller γ is. Also, the maximum value of γ is 0.6757 when α=0, and the minimum value of γ is 0.6734 when α=1.5. The range of α is 0,1.5 and the range of γ is 1,4.4817.In fact, Figures1 and 2 imply Figure 3. To show the relations between γ and ξ clearly, Figure 3 is also drawn. From Figure 3, it can be seen that the bigger the number of ξ, the smaller γ is. Figures 1, 2, and 3 can provide us a method for how to select the suitable SFFT H2/H∞ controller. If we are mainly concerned with the H2 performance, a smaller α can be selected. If we are mainly concerned with the H∞ performance, a bigger α can be selected.Next, substitutingα=0 into Theorem 2 and then solving (45), (46), (47), and (48), we have
(85)M=0.65280.08100.08100.6935,Y=0.0746−0.06350.52360.3750,λ=0.3980.Therefore, we get the following controller gain matrix:(86)K=YM−1=0.1275−0.10640.74600.4536.Supposevt=sint∫01sin2tdt=0.275<h=0.5 uses the Euler-Maruyama method to simulate the standard Brownian motion, one can obtain the simulated curves of x′tRxt (30 curves) and Ex′tRxt of system (10) in Figure 4. From Figure 4, we see that system (10) is FT stochastically bounded wrt 1,5,1,I,0.5.Figure 4
The response forEx′tRxt.


## 6.2. OBFTH2/H∞ Controller Design

On the basis of the SF case, Algorithm1 can also be used to solve the similar OP for the OBFT H2/H∞ control. The relations of β and ξ (Figure 5), β and γ (Figure 6), and γ and ξ (Figure 7) are obtained. From Figure 5, it can be seen that the bigger the number of β, the bigger ξ is. Also, the minimum value of the H2 performance index is ξ=1 when β=0, and the maximum value of the H2 performance index is ξ=4.0552 when β=1.4. The range of α is 0,1.4, and the range of ξ is 1,4.0552.Figure 5
ξ versus β.Figure 6
γ versus β.Figure 7
γ versus ξ.From Figure6, it can be seen that the bigger the number of β, the smaller γ is. Also, the maximum value of γ is 0.8656 when β=0, and the minimum value of γ is 0.8413 when β=1.4. The range of β is 0,1.4, and the range of γ is 0.8413,0.8656.In fact, Figures5 and 6 also imply Figure 7. To show the relations between γ and ξ clearly, Figure 7 is also drawn. From Figure 7, it can be seen that the bigger the number of ξ, the smaller γ is. Figures 5–7 can provide us a method for how to select the suitable OBFT H2/H∞ controller. If we are mainly concerned with the H2 performance, a smaller β can be selected. If we are mainly concerned with the H∞ performance, a bigger β can be selected.Substitutingβ=0 into Theorem 4 and solving (79), (80), (81), and (82) yields
(87)Q11=1.9421−0.0442−0.04422.2252,Q22=3.19130.09000.09002.2214,N=4.2749−2.92181.65871.1962,η=3.6737Then, the corresponding observer gain matrix is obtained:(88)L=Q22−1N=1.3200−0.93180.69320.5762.Supposevt=sint∫01sin2tdt=0.275<h and uses the Euler-Maruyama method to simulate the standard Brownian motion, one can obtain the simulated curves of x~′tRx~t (30 curves) and Ex~′tRx~t of system (50) in Figure 8. From Figure 8, we see that system (10) is FT stochastically bounded wrt 1,5,1,I,0.5.Figure 8
The response forEx~′tRx~t.


## 7. Conclusions

In this study, state-feedback and observer-based finite-timeH2/H∞ control problems have been investigated. The problems have been transformed into optimization problems with the constraint of matrix inequalities, and a detailed optimization algorithm has been given to obtain the minimum values of the H2 index and H∞ index. An example has been given to demonstrate the results obtained.



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*Source: 1936021-2018-10-15.xml*