# Finite-TimeH2/H∞ Control Design for Stochastic Poisson Systems with Applications to Clothing Hanging Device

Authors: Yan Qi; Shiyu Zhong; Zhiguo Yan
Journal: Complexity
 (2020)

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## Abstract

In this paper, the design of finite-timeH2/H∞ controller for linear Itô stochastic Poisson systems is considered. First, the definition of finite-time H2/H∞ control is proposed, which considers the transient performance, H2 index, and H∞ index simultaneously in a predetermined finite-time interval. Then, the state feedback and observer-based finite-time H2/H∞ controllers are presented and some new sufficient conditions are obtained. Moreover, an algorithm is given to optimize H2 and H∞ index, simultaneously. Finally, a simulation example indicates the effectiveness of the results.


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## Body

## 1. Introduction

It is known to all that stochastic systems have been studied extensively and applied to biological network [1], power systems [2], financial systems [3, 4], and other fields. There are also many other applications of stochastic systems (see, e.g., [5–7]). In the past few decades, stochastic systems driven by Wiener noise have been widely investigated. For example, Shaikin [8] solved the optimization problem for multiplicative stochastic systems with several external disturbances and vector Wiener processes. Xiang et al. [9] introduced the finite-time properties and state feedback H∞ control problem for switched stochastic systems with Wiener noise. Yan et al. [10] were concerned with finite-time H2 control of the Markovian stochastic systems with Wiener noise. However, in the real world, an actual physical system is inevitably affected by Wiener noise and Poisson jump noise. At present, some achievements have been made in the research of stochastic Poisson systems (see, e.g., [11–14]).On the other hand,H2/H∞ optimization control is one of the most important problems in the controlled system. The H2 optimal control system has good system performance, while the H∞ control theory can deal with the system robustness problem well. In view of this, Bernstein and Haddad [15] proposed the H2/H∞ mixed control problem, which can solve both the problems of system performance and robustness. Since then, the H2/H∞ control has been developed and used extensively (see, e.g., [16–19]). Besides, in some engineering research, such as communication system [20–23], robotic operating system [24], and industrial production system [25], more attention should be paid to the system transient performance. In order to describe system transient performance clearly, the concepts of finite-time stability (FTS) and finite-time boundedness (FTB) are proposed, which reflect the specific system behavior in a relatively short time interval. Nowadays, the problems of FTS and FTB have been deeply investigated (see, e.g., [26–36]). In consideration of the merits of FTB and H2/H∞ control, the finite-time H2/H∞ control for stochastic systems with Wiener noise is first presented in [37], which satisfies both FTB and H2/H∞ performance index. However, in many practical systems, it is not only disturbed by Wiener noise, but also by Poisson noise. So far, there are few literature studies to investigate this problem of stochastic Poisson systems affected by both Wiener and Poisson noises.Motivated by aforementioned discussions, the problems of finite-timeH2/H∞ control for stochastic Poisson systems with both Wiener noise and Poisson noise are considered in this paper. The main work of this paper consists of the following three aspects:(i)
Unlike the model considered in [37], this paper studies the model of stochastic Poisson systems with Wiener and Poisson noises. The former considers only Wiener noise, and the latter considers both Wiener and Poisson noises. Moreover, in the former model, the measurement output yt is composed of only the state, but the measurement output considered in the latter model is composed of both the state and external interference. The latter model is more general than the former model in [37], which is used to model many real systems.(ii)
The two theorems (Theorems2 and 4) are obtained to guarantee the existence of state feedback finite-time (SFFT) and observer-based finite-time (OBFT) H2/H∞ controllers, respectively. The two theorems (Theorems 2 and 4) contain the parameters both α and Poisson jump intensity λ, which are complex than the corresponding conditions in [37]. By adjusting the two parameters, the most satisfying finite-time H2/H∞ controllers will be designed.(iii)
A new optimization algorithm constrained by matrix inequality is proposed to demonstrate the relationships amongα, λ, and optimal H2/H∞ index, which is more complex than that in [37].Notations: the notations presented in this work are standard. For specific contents, one can refer to [37].


## 2. Preliminaries

Consider a continuous-time stochastic Poisson system(1)dxt=A11xt+B11vt+F1rtdt+A12xt+B12vt+F2rtdWt+A13xt+B13vt+F3rtdNt,yt=C11xt+D11rt,zt=C12xt+D12vt,x0=x0∈Rn,where A11, A12, A13, B11, B12, B13, C11, C12, D11, D12, F1, F2, and F3 are known constant matrices. xt∈ℛl, yt∈ℛq, zt∈ℛs, and vt∈ℛn are the state vector, measurement output, control output, and control input, respectively. x0 is the initial condition of the system. Wt presents one-dimensional standard Wiener process and Nt is the marked Poisson process with Poisson jump intensity λ. rt∈ℛp is the disturbance input which satisfies the following equation:(2)ℰ∫0tr′srsds<f,f>0.Next, the definition of mean-square FTB of system (1) is introduced.Definition 1.
Given some scalarsb2>b1>0 and T>0 and a matrix R>0, the above stochastic system (1) with vt≡0 is mean-square FTB w.r.t. b1,b2,T,R,f, if(3)ℰx′0Rx0≤b1⟹ℰx′tRxt<b2,∀t∈0,T.Remark 1.
From Definition1, we can know that the concept of FTB describes the specific behavior of the stochastic system (1) in a prescribed time interval.Lemma 1 (see [38]).
LetV¯t,x∈C2R1,Rn and V¯t,x>0. Consider the following system(4)dxt=A1xdt+A2xdWt+A3xdNt,its stochastic differential of V¯t,x is given by(5)dV¯t,x=ℒV¯t,xdt+∂V¯′t,x∂xA2xdWt+V¯t,x+A3x−V¯t,xdNt,where(6)ℒV¯t,x=∂V¯t,x∂t+∂V¯t,x∂xA1x+12A2′x∂2V¯t,x∂x2A2x+λV¯t,x+A3x−V¯t,x.


## 3. Design of SFFTH2/H∞ Controller

In this section, a SFFTH2/H∞ controller for system (1) is designed. Consider a linear SF controller(7)vt=Kxt,where K is the required SF gain matrix.Substituting (7) into (1), the following closed-loop system is obtained:(8)dxt=A˜11xt+F1rtdt+A˜12xt+F2rtdWt+A˜13xt+F3rtdNt,yt=C11xt+D11rt,zt=C˜xt,x0=x0∈Rn,where A˜11=A11+B11K, A˜12=A12+B12K, A˜13=A13+B13K, and C˜=C12+D12K.Next, we choose the followingH2 cost function:(9)J1xt,vt=ℰ∫0Tx′tG1xt+v′tG2vtdt,where G1>0 and G2>0 are known weighting scalars or positive matrices.Similarly, substituting the SF controller (7) into (9), the following formula is obtained:(10)J1xt=ℰ∫0Tx′tG1xt+x′tK′G2Kxtdt.Givenγ>0 and assuming zero initial condition, the control output zt and the disturbance input rt satisfy the following equation:(11)ℰ∫0Tz′tztdt<γ2ℰ∫0Tr′trtdt.Based on the above preparations, the definition of the SFFTH2/H∞ controller is introduced.Definition 2.
Given positive scalarsb1, b2, T, and f and a matrix R>0. If a positive scalar J1∗ exists, a SF controller (7) can be designed to make the following conditions hold:(i)
The closed-loop system (8) is mean-square FTB w.r.t. b1,b2,T,R,f(ii)
TheH2 cost function (10) meets J1xt≤J1∗ under rt=0 condition(iii)
Assuming that the initial state is zero and the nonzero disturbance input and the control output satisfy inequality (11); then (7) is the SFFT H2/H∞ controller for system (1)Remark 2.
Definition3 implies that a SFFT H2/H∞ controller not only makes the closed-loop system FTB, but also gets minimum performance cost and better interference suppression capability. In actual systems, these three aspects really need to be considered. For example, in industrial steel rolling heating furnace, excessive instantaneous furnace temperature cannot be permitted. Moreover, it is hoped that the fuel consumption is less and the anti-interference ability is stronger in the rolling furnace.
Next, the following theorem is given for obtaining the SFFTH2/H∞ controller.Theorem 1.
Given positive scalarsb1, b2, T, and f and a matrix R>0, if there exist a nonnegative scalar α and two matrices N>0 and K such that(12)T1F1N˜A˜12′λN˜A˜13+I′∗−γ2IF2′λF3′∗∗−N˜0∗∗∗−N˜<0,(13)T2N˜A˜12′λN˜A˜13+I′∗−N˜0∗∗−N˜<0,(14)b1λminN+fγ2<b2λmaxNe−αT,hold, where N˜=R−1/2NR−1/2,T1=A˜11N˜+N˜A˜11′−λN˜−αN˜+N˜′C˜′C˜N˜, and T2=A˜11N˜+N˜A˜11′−λN˜−αN˜+N˜G1N˜+N˜K′G2KN˜, then vt=Kxt is said to be a SFFT H2/H∞ controller and we can get the upper bound of H2 index, that is, Jstate∗=λmaxN−1b1eαT.Proof.
Here are three steps to prove Theorem1.Step 1: prove that system (3) is mean-square FTB.Obviously,(15)N˜C˜′C˜N˜000000000000000=N˜C˜′000N˜C˜′000′≥0.Therefore, condition (12) means(16)T1˜F1N˜A˜12′λN˜A˜13+I′∗−γ2IF2′λF3′∗∗−N˜0∗∗∗−N˜<0,whereT1˜=A˜11N˜+N˜A˜11′−λN˜−αN˜.LetV¯xt=x′tN˜−1xt, and applying Lemma 1 for V¯xt, the ℒ1V¯xt of system (8) is given by(17)ℒ1V¯xt=xtrt′Z¯1Z¯2∗Z¯3xtrt,whereZ¯1=A˜11′N˜−1+N˜−1A˜11+A˜12′N˜−1A˜12+λA˜13+I′N˜−1A˜13+I−λN˜−1, Z¯2=λA˜13+I′N˜−1F3+A˜12′N˜−1F2+N˜−1F1, and Z¯3=λF3′N˜−1F3+F2′N˜−1F2.Pre- and postmultiplying (16) by diagN˜−1,I,N˜−1,N˜−1, we can get the following inequality:(18)ℐ1N˜−1F1A˜12′N˜−1ℐ2∗−γ2IF2′N˜−1λF3′N˜−1∗∗−N˜−10∗∗∗−N˜−1<0,whereℐ1=A˜11′N˜−1+N˜−1A˜11−λN˜−1−αN˜−1 and ℐ2=λA˜13+I′N˜−1.By utilizing Schur complement, (18) is equivalent to(19)Z¯1−αN˜−1Z¯2∗Z¯3−γ2I<0.Taking conditions (17) and (19) into consideration, it follows(20)ℒ1V¯xt<αV¯xt+γ2r′trt.Integrating from 0 tot on both sides of (20), then taking mathematical expectation, one has(21)ℰV¯xt<ℰV¯x0+α∫0tℰV¯xsds+γ2∫0tℰr′srsds.Utilizing Gronwall inequality in [26], it follows(22)ℰV¯xt<ℰV¯x0eαt+γ2eαt∫0tℰr′srsds.On the basis of above conditions, we have(23)ℰV¯xt=ℰx′tR1/2N−1R1/2xt≥λminN−1Εx′tRxt,(24)ℰV¯x0eαt=ℰx′0R1/2N−1R1/2x0eαt≤λmaxN−1Εx′0Rx0eαt≤λmaxN−1b1eαT,(25)γ2eαt∫0tℰr′srsds<eαTfγ2.From (22) to (25), the following inequality is obtained:(26)ℰx′tRxt<λmaxNeαTb1λminN+fγ2.According to condition (14), we get that (26) leads to ℰx′tRxt<b2 for all t∈0,T. So, system (8) is mean-square FTB w.r.t. b1,b2,T,R,f.Step 2: prove that theH2 cost function (10) satisfies J1xt≤J1∗ under rt=0 condition.Whenrt=0, we get that the ℒ2V¯xt of system (8) is given by(27)ℒ2V¯xt=x′tA˜11′N˜−1+N˜−1A˜11+A˜12′N˜−1A˜12+λA˜13+I′N˜−1A˜13+I−λN˜−1xt.By Schur complement, the equivalent condition of (13) is given by(28)N˜A˜11′+A˜11N˜+N˜A˜12′N˜−1A˜12N˜+λN˜A˜13+I′N˜−1A˜13+IN˜+N˜G1N˜+N˜K′G2KN˜−λN˜−αN˜<0.Pre- and postmultiplying (28) by N˜−1, it yields(29)A˜11′N˜−1+N˜−1A˜11+A˜12′N˜−1A˜12+G1+K′G2K+λA˜13+I′N˜−1A˜13+I−λN˜−1−αN˜−1<0.According to (27) and (29), we get(30)ℒ2V¯xt+x′tG1+K′G2Kxt−αV¯xt<0.Integrating from 0 tot on both sides of (30), then taking mathematical expectation, the following inequality is obtained:(31)ℰ∫0tx′tG1+K′G2Kxtdt+ℰV¯xt<ℰV¯x0+αℰ∫0tV¯xtdt.From (31), we get(32)ℰV¯xt<ℰV¯x0+αℰ∫0tV¯xtdt,(33)J1xt<αℰ∫0tV¯xtdt+ℰV¯x0.From (32), by Gronwall inequality, one has(34)ℰV¯xt<ℰV¯x0eαt.Combining (33) and (34), it is obtained that(35)J1xt<αℰ∫0tx′0R1/2N−1R1/2x0eαtdt+ℰx′0N˜−1x0=ℰx′0R1/2N−1R1/2x0eαt<λmaxN−1b1eαT=J1∗.Step 3: prove that the nonzero disturbance and the control output satisfy inequality (11).Pre- and postmultiplying (12) respectively by diagN˜−1,I,N˜−1,N˜−1, and then using Schur complement, we have(36)T3Z¯2∗−γ2I+Z¯2<0,whereT3=A˜11′N˜−1+N˜−1A˜11+A˜12′N˜−1A˜12+λA˜13+I′N˜−1A˜13+I+C˜′C˜−λN˜−1−αN˜−1.Combining (17), (36), we get(37)ℒ1V¯xt<αV¯xt+γ2r′trt−z′tzt.Pre- and postmultiplying (37) by e−αt, one has(38)e−αtℒ1V¯xt<αe−αtV¯xt+e−αtγ2r′trt−z′tzt.By applying Lemma1, we obtain(39)ℒ1e−αtV¯xt=−αe−αtV¯xt+e−αtℒ1V¯xt.According to (38) and (39), it yields(40)ℒ1e−αtV¯xt<e−αtγ2r′trt−z′tzt.Becausee−αt is between 0 and 1, for (40), we have(41)ℒ1e−αtV¯xt<γ2r′trt−z′tzt.Integrating from 0 tot on both sides of (41), then taking mathematical expectation, the following inequality can be obtained under zero initial condition:(42)e−αtℰV¯xt<γ2ℰ∫0tr′srsds−ℰ∫0tz′szsds.We know thate−αtℰV¯xt>0, so it yields(43)ℰ∫0tz′szsds<γ2ℰ∫0tr′srsds.This completes the proof.It is obvious that conditions (12)–(14) are not linear matrix inequalities. In order to simplify the solving process, the following theorem is given.Theorem 2.
Given positive scalarsb1, b2, T, and f and a matrix R>0, if there exist two scalars m>0 and α≥0 and two matrices N>0 and Y such that(44)T4F1T5T6T7∗−γ2IF2′F3′0∗∗−N˜00∗∗∗−N˜0∗∗∗∗−I<0,(45)T4T5T6N˜Y′∗−N˜000∗∗−N˜00∗∗∗−G1−10∗∗∗∗−G2−1<0,(46)fγ2−b2e−αTb1∗−m<0,(47)mI<N<I,hold, where T4=A11N˜+N˜A11′−λN˜−αN˜+B11Y+Y′B11′, T5=N˜A12′+Y′B12′, T6=λN˜A13′+Y′B13′+N˜, and T7=N˜C12′+Y′D12′, then vt=Kxt=YN˜−1xt is said to be a SFFT H2/H∞ controller and we can get the upper bound of H2 index, that is, J1∗=m−1b1eαT.Proof.
LetY=KN˜, inequalities (12) and (13) can be obtained from (44) and (45), respectively, and (14) in Theorem 1 can be obtained from (46) and (47) easily. This ends the proof.Remark 3.
In Theorem2, when α is fixed, (44)–(47) can be treated as LMIs which are easy to solve.


## 4. Design of OBFTH2/H∞ Controller

In some practical cases, not all states can be measured directly. Therefore, the design of OBFTH2/H∞ controller is necessary. Typically, an OB dynamic controller is given by(48)dx^t=A11x^t+B11vt+Lyt−C11x^tdt,vt=Kx^t,x^0=0,where x^t∈Rn is the estimation of xt and L is the desired estimator gain.Substituting the OB controller (48) into system (1), we will obtain the following closed-loop system:(49)dx˜t=A¯11x˜t+W1rtdt+A¯12x˜t+W2rtdWt+A¯13x˜t+W3rtdNt,z˜t=Hx˜t,and then we get the closed-loop cost function(50)J2x˜t=ℰ∫0Tx˜′tΞx˜tdt,where(51)x˜t=xtx^t,A¯11=A11B11KLC11A11+B11K−LC11,W1=F1LD11,A¯2=A12B12K00,(52)W2=F20,A¯13=A13B13K00,W3=F30,H=C12′K′D12′′,Ξ=G100K′G2K.Assuming that the initial state is zero, the control outputz˜t and the arbitrary nonzero disturbance input rt satisfy the following equation:(53)ℰ∫0Tz˜′tz˜tdt<γ2ℰ∫0Tr′trtdt.Then, we give the definition of OBFTH2/H∞ control.Definition 3.
Given positive scalarsb1, b2, T, and f and a matrix R˜>0. If a positive scalar J2∗ exists, an OBFT controller (48) can be designed to make the following conditions hold:(i)
System (49) is mean-square FTB w.r.t. b1,b2,T,R˜,f, that is, ℰx′0R˜x0≤b1⟹ℰx′tR˜xt<b2, where 0<b1<b2, T>0 and R˜=R00R(ii)
TheH2 cost function (50) meets J2x˜t≤J2∗ under rt=0 condition(iii)
Assuming that the initial state is zero, the nonzero disturbance input and the control output satisfy inequality (53); then (48) is an OBFT H2/H∞ controller for system (1)
Next, the following theorem is given for obtaining the OBFTH2/H∞ controller for system (1).Theorem 3.
Given positive scalarsb1, b2, T, and f and a matrix R˜>0, if there exist a nonnegative scalar β and a positive matrix P such that(54)ℋ1P˜W1A¯12′P˜λA¯13+I′P˜∗−γ2IW2′P˜λW3′P˜∗∗−P˜0∗∗∗−P˜<0,(55)A¯11′P˜+P˜A¯11+A¯12′P˜A¯12+λA¯13+I′P˜A¯13+I−λP˜−βP˜+Ξ<0,(56)λmaxPb1+fγ2<λminPb2e−βT,hold, where P˜=R˜1/2PR˜1/2andℋ1=A¯11′P˜+P˜A¯11+H′H−βP˜−λP˜, then (48) is said to be an OBFT H2/H∞ controller and we can get the upper bound of H2 index, that is, J2∗=λmaxPb1eβT.Proof.
Here are three steps to prove the theorem.Step 1: prove that system (49) is mean-square FTB.LetV¯x˜t=x˜′tP˜x˜t where P˜>0. Applying generalized Itô formula for V¯x˜t, the ℒ3V¯x˜t of system (49) is given by(57)ℒ3V¯x˜t=x˜trt′Z˜1Z˜2∗Z˜3x˜trt,whereZ˜1=A¯11′P˜+P˜A¯11+A′¯12P˜A¯12+λA¯13+I′P˜A¯13+I−λP˜, Z˜2=P˜W1+A′¯12P˜W2+λA¯13+I′P˜W3, and Z˜3=W2′P˜W2+λW3′P˜W3.Note that(58)H′H000000000000000=H′000H′000′≥0.Therefore, inequality (54) means(59)ℋ2P˜W1A¯12′P˜λA¯13+I′P˜∗−γ2IW2′P˜λW3′P˜∗∗−P˜0∗∗∗−P˜<0,whereℋ2=A¯11′P˜+P˜A¯11−βP˜−λP˜.By utilizing Schur complement, (59) can be converted into(60)Z˜1−βP˜Z˜2∗−γ2I+Z˜3<0.Combining (57) and (60), we get(61)ℒ3V¯x˜t<βV¯x˜t+γ2r′trt.Integrating from 0 tot on both sides of (61), then taking mathematical expectation, the following inequality is obtained:(62)ℰV¯x˜t<ℰV¯x˜0+β∫0tℰV¯x˜sds+γ2∫0tℰr′srsds.According to Gronwall inequality, it yields(63)ℰV¯x˜t<ℰV¯x˜0eβt+γ2eβt∫0tℰr′srsds.According to known conditions, it yields(64)ℰV¯x˜t=ℰx˜′tR˜1/2PR˜1/2x˜t≥λminPℰx˜′tR˜x˜t,(65)ℰV¯x˜0eβt=ℰx˜′0R˜1/2PR˜1/2x˜0eβt≤λmaxPℰx˜′0R˜x˜0eβt≤λmaxPb1eβT,(66)γ2eβt∫0tℰr′srsds<eβTfγ2.From (63) to –(66), we obtain(67)ℰx˜′tR˜x˜t<λmaxPb1eβT+fγ2eβTλminP.According to (56) and (67), we get ℰx˜′tR˜x˜t<b2 for all t∈0,T. So, system (49) is FTB w.r.t. b1,b2,T,R˜,f.Step 2: prove that theH2 cost function (50) satisfies J2x˜t≤J2∗ under rt=0 condition.Whenrt=0, we get that the ℒ4V¯x˜t of system (49) is given by(68)ℒ4V¯x˜t=x˜′tA¯11′P˜+P˜A¯11+A¯12′P˜A¯12+λA¯13+I′P˜A¯13+I−λP˜x˜t.According to (55), we have(69)ℒ4V¯x˜t−βV¯x˜t+x˜′tΞx˜t<0.Integrating from 0 tot on both sides of (69), then taking mathematical expectation, it yields(70)ℰV¯x˜t+ℰ∫0tx˜′tΞx˜tdt<ℰV¯x˜0+βℰ∫0tV¯x˜tdt.From (70), we have(71)ℰV¯x˜t<ℰV¯x0+βℰ∫0tV¯x˜tdt,(72)ℰ∫0tx˜′tΞx˜tdt<ℰV¯x˜0+βℰ∫0tV¯x˜tdt.Using Gronwall inequality for (71), one has(73)ℰV¯x˜t<ℰV¯x˜0eβt.From (72) and (73), we have(74)J2x˜t<βℰ∫0tV¯x˜tdt+ℰV¯x˜0<βℰ∫0tV¯x˜0eβtdt+ℰV¯x˜0=ℰV¯x˜0eβt<λmaxPb1eβT=J2∗.Step 3: prove that the nonzero disturbance and the control output satisfy the inequality (53).
By using Schur complement, we can obtain the following equivalent conditions of (54):(75)Z˜1+H′H−βP˜Z˜2∗Z˜3−γ2I<0.
According to (57) and (75), we get(76)ℒ3V¯x˜t<βV¯x˜t+γ2r′trt−z′tzt.
Repeating the proof process of Step 3 in Theorem1, it yields(77)ℰ∫0tz′szsds<γ2ℰ∫0tr′srsds.
This completes the proof.
Because the nonlinear problem of inequalities (54)–(56) in Theorem 3 is difficult to solve, we transform the inequalities (54)–(56) into LMIs.Theorem 4.
Given positive scalarsb1, b2, T, and f, if there exist two positive scalars β and ζ and three matrices P˜11>0, P˜22>0, and M such that(78)Σ11Σ12Σ13∗Σ22Σ23∗∗Σ33<0,(79)ϒ11−β+λP˜11ϒ12∗ϒ22−β+λP˜22<0,(80)eβTζb1+fγ2−b2<0,(81)I<diagP˜11,P˜22<ζI,hold, where Σ11=A11′P˜11+P˜11A11+A12′P˜11A12+C12′C12+λA13′+IP˜11A13+I−β+λP˜11,Σ12=C11′M′+P˜11B11K+A12′P˜11B12K+C12′D12K+λA13′+IP˜11B13K,Σ22=A11+B11K′P˜22+P˜22A11+B11K+K′B12′P˜11B12K+λK′B13′P˜11B13K+λP˜22−MC11−C11′M′+K′D12′D12K−β+λP˜22,Σ13=P˜11F1+A12′P˜11F2+λA13′+IP˜11F3,Σ23=MD11+K′B12′P11F2+λK′B13′P11F3,Σ33=F2′P˜11F2+λF3′P˜11F3−γ2I,ϒ11=A11′P˜11+P˜11A11+A12′P˜11A12+λA13′+IP˜11A13+I+G1,ϒ12=C11′M′+P˜11B11K+A12′P˜11B12K+λA13′+IP˜11B13K,ϒ22=A11+B11K′P˜22+P˜22A11+B11K+K′B12′P˜11B12K−MC11−C11′M′+λK′B13′P˜11B13K+λP˜22+K′G2K, then (48) is said to be an OBFT H2/H∞ controller and we can get the upper bound of H2 index, that is, J2∗=ζb1eβT. Furthermore, the estimator gain matrix L=P˜22−1M is obtained.Proof.
LetP˜=diagP˜11,P˜22 and M=P˜22L, by substituting (51) and (52) into (54) and (55), (78) and (79) can be easily derived, respectively. From (80) and (81), we can deduce that (49) holds. This ends the proof.


## 5. Algorithm

In this section, we propose an algorithm to optimizeH2 index and H∞ index.Analysis: in Theorem2, let J1∗<ξ, the following inequality is derived:(82)Ne−αTb1−ξ−1I>0,where 0<b1<b2, T>0, α≥0, and m>0.The main purpose of the algorithm is to check whether inequalities (44)–(47) in Theorem 2 have feasible solutions by changing the value of α. If there exist feasible solutions, then ξ and γ2 are optimized to get the minimum values. The detailed algorithm will be given as follows (Algorithm 1).Algorithm 1: Optimization algorithm.
Step 1: givenb1, b2, R, T, f, and λ.Step 2: take an appropriate step sizedα for α, and then the values of α are expressed as αi.Step 3: leti=1.Step 4: ifαi makes the following problems mins.t.37−40,74,N>0ξandmins.t.37−40,74,N>0γ2 feasible, then store αi into Ui, ξmin into Vi, and γmin into Wi, and let αi+1=αi+dα, loop. Otherwise, go to Step 5.Step 5: exit.


## 6. Examples

In this section, system (8) can be used to simulate a clothing hanging device and the parameters are as follows:(83)A11=−15−98−12,A12=−0.611.3−1.2,A13=−1.811.4−1.5,B11=−95′,B12=8.72.3′,B13=−2.81′,F1=−0.60.5′,F2=0.3−0.2′,F3=0.4−0.2′,C11=−0.9−1.5,C12=−1.8−2.5,x0=−0.70.7′,and G1=5, G2=4, D11=8, D12=10, b1=1, b2=4, T=1, R=I, f=0.4, and λ=2.5.


### 6.1. Design of SFFTH2/H∞ Controller

By using the above algorithm in Section5, the relationships of α and ξ (Figure 1), α and γ (Figure 2), and ξ and γ (Figure 3) are derived, respectively. It can be seen from Figure 1 that the value of ξ increases with the increase of α. Besides, it is obvious that ξ=1 when α=0 and ξ=3.0344 when α=1.11, that is, the minimum and maximum values of H2 performance index are 1 and 3.0344, respectively. Also, the range of α is [0, 1.11].Figure 1
ξ versus α.Figure 2
γ versus α.Figure 3
γ versus ξ.As you can see from Figure2, the value of γ decreases first and then increases with the increase of α. When α=0.85, γ can take the minimum value of 0.8688; at this point, we can get the optimal value of H∞ performance index. When α=1.11, γ can take the minimum value of 0.8822. Also, α can be taken within [0, 1.11].In fact, Figure3 reflects the relation between ξ and γ. As shown in Figure 3, with the increase of ξ, the value of γ decreases first, and at the point of ξ=2.3397, the value of γ begins to increase. From Figures 1 to 3, we can see how to choose the right state feedback finite-time H2/H∞ controller. If the cost problem is mainly considered, a smaller α can be selected. If the ability to suppress interference is mainly considered, we need to refer to Figure 2 to select the appropriate α.Next, substitutingα=0 into Theorem 2, we get(84)N=0.8673−0.0135−0.01350.8946,Y=−0.03660.0770,m=0.5237.Then, we get the controller gain matrix as follows:(85)K=−0.04090.0854.Because the statext in this example is two-dimensional, we make xt=x1tx2t′. Figure 4 describes the trajectories of x1t, x2t, and Ex′tRxt with stochastic fluctuation driven by both Wiener and Poisson noises in Figures 5 and 6 versus the dimensionless time λt. From Figure 4, we can see that the trajectory of Ex′tRxt does not exceed b2=4 in the time interval λT=2.5. Obviously, when the time interval is T=1, the trajectory does not exceed the given range, so we conclude that system (8) is mean-square FTB w.r.t. (1, 4, 1, I, 0.4). Among them, we assume that rt=sint∫01sin2tdt<f=0.4.Figure 4
The trajectory forEx′tRxt.Figure 5
The time evolution of Wiener process.Figure 6
The time evolution of Poisson counting process.


### 6.2. Design of OBFTH2/H∞ Controller

As in the case of state feedback, similar results can be obtained in the case of observer-based finite-timeH2/H∞ control. The relationships of β and ξ (Figure 7), β and γ (Figure 8), and ξ and γ (Figure 9) are derived, respectively. It can be seen from Figure 7 that the value of ξ increases with the increase of β. Besides, it is obvious that ξ=1 when β=0 and ξ=2.5857 when β=0.95, that is, the minimum and maximum values of H2 performance index are 1 and 2.5857, respectively. Also, the range of β is [0, 0.95].Figure 7
ξ versus β.Figure 8
γ versus β.Figure 9
γ versus ξ.As you can see from Figure8, the value of γ decreases first and then increases with the increase of β. When β=0.72, γ can take the minimum value of 1.1456, and at this point, we can get the optimal value of H∞ performance index. Besides, the maximum value of H∞ performance index is 1.1650 when β=0.95. Also, the range of β is [0, 0.95].In fact, Figure9 reflects the relation between ξ and γ. As shown in Figure 9, with the increase of ξ, the value of γ decreases first, and at the point of ξ=2.0544, the value of γ begins to increase. From Figures 7 to 9, we can see how to choose the right OBFT H2/H∞ controller. If the cost problem is mainly considered, select the smaller β with reference to Figure 7. If the ability to suppress interference is mainly considered, we need to refer to Figure 8 to select the appropriate β.Substitutingβ=0 into Theorem 4, we have(86)P11=1.33260.04580.04581.5739,P22=1.4821−0.0298−0.02981.7362,M=−0.03260.1348,ζ=2.7271.Then, we obtain the following observer gain matrix:(87)L=−0.02050.0773.Because the statex˜ in this example is two-dimensional, we make x˜=x˜1x˜2′. Figure 10 describes the trajectories of x˜1, x˜2, and Ex′tRx˜t with stochastic fluctuation driven by both Wiener and Poisson noises in Figures 5 and 6 versus the dimensionless time λt. From Figure 10, it is obvious that the trajectory of Ex′tRx˜t does not exceed b2=4 in the time interval λT=2.5. Obviously, when the time interval is T=1, the trajectory does not exceed the given range, so we draw a conclusion that system (8) is mean-square FTB w.r.t. (1, 4, 1, I, 0.4). Among them, we assume that rt=sint∫01sin2tdt<f=0.4.Figure 10
The trajectory forEx′tRx˜t.


## 6.1. Design of SFFTH2/H∞ Controller

By using the above algorithm in Section5, the relationships of α and ξ (Figure 1), α and γ (Figure 2), and ξ and γ (Figure 3) are derived, respectively. It can be seen from Figure 1 that the value of ξ increases with the increase of α. Besides, it is obvious that ξ=1 when α=0 and ξ=3.0344 when α=1.11, that is, the minimum and maximum values of H2 performance index are 1 and 3.0344, respectively. Also, the range of α is [0, 1.11].Figure 1
ξ versus α.Figure 2
γ versus α.Figure 3
γ versus ξ.As you can see from Figure2, the value of γ decreases first and then increases with the increase of α. When α=0.85, γ can take the minimum value of 0.8688; at this point, we can get the optimal value of H∞ performance index. When α=1.11, γ can take the minimum value of 0.8822. Also, α can be taken within [0, 1.11].In fact, Figure3 reflects the relation between ξ and γ. As shown in Figure 3, with the increase of ξ, the value of γ decreases first, and at the point of ξ=2.3397, the value of γ begins to increase. From Figures 1 to 3, we can see how to choose the right state feedback finite-time H2/H∞ controller. If the cost problem is mainly considered, a smaller α can be selected. If the ability to suppress interference is mainly considered, we need to refer to Figure 2 to select the appropriate α.Next, substitutingα=0 into Theorem 2, we get(84)N=0.8673−0.0135−0.01350.8946,Y=−0.03660.0770,m=0.5237.Then, we get the controller gain matrix as follows:(85)K=−0.04090.0854.Because the statext in this example is two-dimensional, we make xt=x1tx2t′. Figure 4 describes the trajectories of x1t, x2t, and Ex′tRxt with stochastic fluctuation driven by both Wiener and Poisson noises in Figures 5 and 6 versus the dimensionless time λt. From Figure 4, we can see that the trajectory of Ex′tRxt does not exceed b2=4 in the time interval λT=2.5. Obviously, when the time interval is T=1, the trajectory does not exceed the given range, so we conclude that system (8) is mean-square FTB w.r.t. (1, 4, 1, I, 0.4). Among them, we assume that rt=sint∫01sin2tdt<f=0.4.Figure 4
The trajectory forEx′tRxt.Figure 5
The time evolution of Wiener process.Figure 6
The time evolution of Poisson counting process.


## 6.2. Design of OBFTH2/H∞ Controller

As in the case of state feedback, similar results can be obtained in the case of observer-based finite-timeH2/H∞ control. The relationships of β and ξ (Figure 7), β and γ (Figure 8), and ξ and γ (Figure 9) are derived, respectively. It can be seen from Figure 7 that the value of ξ increases with the increase of β. Besides, it is obvious that ξ=1 when β=0 and ξ=2.5857 when β=0.95, that is, the minimum and maximum values of H2 performance index are 1 and 2.5857, respectively. Also, the range of β is [0, 0.95].Figure 7
ξ versus β.Figure 8
γ versus β.Figure 9
γ versus ξ.As you can see from Figure8, the value of γ decreases first and then increases with the increase of β. When β=0.72, γ can take the minimum value of 1.1456, and at this point, we can get the optimal value of H∞ performance index. Besides, the maximum value of H∞ performance index is 1.1650 when β=0.95. Also, the range of β is [0, 0.95].In fact, Figure9 reflects the relation between ξ and γ. As shown in Figure 9, with the increase of ξ, the value of γ decreases first, and at the point of ξ=2.0544, the value of γ begins to increase. From Figures 7 to 9, we can see how to choose the right OBFT H2/H∞ controller. If the cost problem is mainly considered, select the smaller β with reference to Figure 7. If the ability to suppress interference is mainly considered, we need to refer to Figure 8 to select the appropriate β.Substitutingβ=0 into Theorem 4, we have(86)P11=1.33260.04580.04581.5739,P22=1.4821−0.0298−0.02981.7362,M=−0.03260.1348,ζ=2.7271.Then, we obtain the following observer gain matrix:(87)L=−0.02050.0773.Because the statex˜ in this example is two-dimensional, we make x˜=x˜1x˜2′. Figure 10 describes the trajectories of x˜1, x˜2, and Ex′tRx˜t with stochastic fluctuation driven by both Wiener and Poisson noises in Figures 5 and 6 versus the dimensionless time λt. From Figure 10, it is obvious that the trajectory of Ex′tRx˜t does not exceed b2=4 in the time interval λT=2.5. Obviously, when the time interval is T=1, the trajectory does not exceed the given range, so we draw a conclusion that system (8) is mean-square FTB w.r.t. (1, 4, 1, I, 0.4). Among them, we assume that rt=sint∫01sin2tdt<f=0.4.Figure 10
The trajectory forEx′tRx˜t.


## 7. Conclusions

In this paper, state feedback and observer-based finite-timeH2/H∞ controllers for stochastic Poisson systems have been designed, respectively. Two sufficient conditions for guaranteeing the existence of controllers have been proposed and converted to matrix inequality constrained optimization problems, and an algorithm for all Theorems has been provided to derive the optimal H2 index and H∞ index under the condition of the finite-time boundedness.



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