# Fuzzy Observer-BasedH2/H∞ Output-Feedback Control for Stochastic Nonlinear Systems with Multiplicative Noise

Authors: Wenxuan Yang; Ting Hou
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

A design method is established for the mixedH2/H∞ output-feedback control of stochastic nonlinear systems with multiplicative noise. Firstly, using T-S fuzzy rules, we obtain a fuzzy model to approximate the original nonlinear system. Then, by Schur’s complement, the suboptimal H2/H∞ output-feedback control design is transformed into a two-step convex optimization problem. A numerical example is given to show the effectiveness of the proposed method.


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## Body

## 1. Introduction

One of the objectives of system control is to design a controller for the object model so that the closed-loop system achieves good performance while ensuring internal stability [1–4]. H2 control and H∞ control have been attractive subjects since they are of great practical significance in the field of engineering [5–9]. H2 control has a high request of model accuracy and generally does not consider the influence of model error. However, in practical control systems, the system could not exclude the implication of uncertain factors. Being put forward by Zames in 1981, nowadays the H∞ design idea has grown into an important robust control theory to eliminate external interference [10]. Accordingly, the combination of H2 and H∞ control design methods will ensure the robustness and optimality of the controlled system at the same time (we refer the readers to [11–14]).It is noticeable that randomness is ubiquitous in the real world [15]. For example, there is a great deal of randomness in financial risk managements. Correspondingly, stochastic H2/H∞ control has been an attractive subject in recent decades. Chen et al. [16] implemented a detailed study on stochastic H2/H∞ control problems for linear systems with state-dependent noise. Subsequently, [17–20] reported research progress on H2/H∞ control of Markov jump systems.On the other hand, nonlinearity is a universal phenomenon existing in engineering systems [21, 22]. Giving an example, a buck-boost circuit is rich in nonlinear dynamics. Generally speaking, control problems of nonlinear systems are more complicated than those of linear systems [23–25]. Linearizing the nonlinear system has become mature technology to treat the nonlinear problems. [26] introduced a suitable linear model gained by the T-S fuzzy rule to approximate a nonlinear system. [27] designed a mixed H2/H∞ controller for nonlinear systems based on fuzzy observer.According to all above, robust control for stochastic nonlinear systems is definitely worthy both from the theoretical and practical application views. Compared with [27], in which the considered system model does not contain multiplicative noise, it is clear that H2/H∞ control for nonlinear systems with multiplicative noise has broader application prospects. The other contribution of this paper is that the suboptimal H2/H∞ output-feedback control design is transformed into a two-step convex optimization problem, which is convenient for solving by MATLAB efficiently.This article is organized as follows. Section2 builds up an approximate model of the original nonlinear system by the T-S fuzzy rule. Section 3 designs a fuzzy observer-based H2/H∞ output-feedback controller by solving a two-step convex optimization problem. A numerical example is given to illustrate the efficiency of the proposed design method in Section 4. Section 5 gives the summary of this paper.For convenience, we adopt the following notations:trA: the trace of matrix AAT: the transpose of matrix AA≥0A>0: a positive semidefinite (positive definite) matrix AI: the identity matrixx: the Euclidean 2-norm of the n-dimensional real vector x


## 2. Problem Description

Consider the following nonlinear random perturbation system:(1)dxt=fxt+gxtutdt+Ctxtdw1,yt=hxt+vt,where xt=x1t,x2t,…,xntT∈Rn×1 is the state vector, ut is the input of the system, and yt is the measured output. We assume that w1 is a one-dimensional standard Wiener process. fxt, gxt, and hxt are supposed to be smooth functions. System (1) is influenced by vt which is a bounded measurement noise, that is, E∫0Tvv′dt=R0≥0.Using T-S fuzzy rule, we establish a linear fuzzy model for the stochastic model (1). Specifically, by the fuzzy rule, Ri: if m1t is Fi1, …, mgt is Fig, i=1,2,…,L, then we have(2)dxt=Aitxt+Bitutdt+Ctxtdw1,yt=Ditxt+vt,where Ri represents the i th rule, L demotes the rule number, mt=m1t,…,mgtT are the measurable prerequisite variables, Fij is the fuzzy set, and Ai, Bi, Dii=1,2,…,L are matrices with right dimensions.By using the single point blur method, the product of reasoning, and the average weighted fuzzification, the following form of fuzzy model is obtained:(3)dxt=∑i=1LμimtAitxt+Bitutdt∑i=1Lμimt+Ctxtdw1=∑i=1LhimtAitxt+Bitutdt+Ctxtdw1,yt=∑i=1LμimtDitxt∑i=1Lμimt+vt=∑i=1LhimtDitxt+vt,where μimt=∏j=1gFijmjt, himt=μimt/∑i=1Lμimt, mt=m1t,…,mgtT, and Fijmjt is the grade of membership of mjt in Fij. We suppose that μimt≥0, i=1,2,…,L. It is easy to see(4)himt≥0,∑i=1Lhimt=1.Thus, system (1) is equivalent to the following system:(5)dxt=∑i=1LhimtAitxt+Bitutdt+Ctxtdw1+Δf+Δgdt,yt=∑i=1LhimtDitxt+vt+Δh,where(6)Δf=fxt−∑i=1LhimtAixt,Δg=gxt−∑i=1LhimtBiut,Δh=hxt−∑i=1LhimtDixtrepresent the approximate error between system (3) and nonlinear model (1).Select the finite-dimension compensator shown below:(7)dx^t=∑i=1LhimtAitx^t+Bitut+Liy−∑i=1LhimtDitx^tdt,ut=∑j=1Lhjmtkjx^t,x^0=x^0,where ut=∑j=1Lhjmtkjx^t is the fuzzy controller and kjj=1,2,…,L is the control parameter.Settingx˜=x−x^ and x¯=x^,x˜T, we get the following closed-loop system:(8)dx¯=∑i=1L∑j=1LhimthjmtA¯ijx¯+B¯ivdt+C¯x¯dw1+Δf¯+Δg¯+Δh¯dt,where(9)A¯ij=Ai+BikjLiDi0Ai−LiDi,B¯i=Li−Li,C¯=00CC,Δf¯=0Δf,Δg¯=0Δg,Δh¯=∑i=1LhimtLiΔh−∑i=1LhimtLiΔh.Next we considerH∞ control performance. Given γ>0 and weighting matrix Q1>0, if x¯0=0,(10)E∫0Tx¯TQ1x¯dt<γ2E∫0TvTvdt≜γ2trR0,then we call the H∞ performance is satisfied.H∞ control aims to eliminate the influence of external interference, but the performance of the closed-loop system may not be ideal. Therefore, a mixed H2/H∞ control design based on fuzzy observer will be implemented. H2 performance is defined as follows:(11)Jx¯,u=E∫0Tx¯TQ2x¯+uTR1udt,where Q2>0 and R1>0.


## 3. Output-Feedback Control Design Based on Fuzzy Observer

In the previous work, using the T-S fuzzy rule, we got a fuzzy model (3) and the approximate error between nonlinear system (1) and the fuzzy model. This section attempts to design an output-feedback control satisfying H∞ performance and H2 performance for fuzzy model (3).Let the following inequalities be true:(12)Δf≤ΔAxt,Δg≤∑i=1LhimtΔBkjx^t,∑i=1LhimtLiΔh≤∑i=1LhimtLiΔDxt.By computation and the above inequalities, we have(13)Δf¯TΔf¯=ΔfTΔf=fxt−∑i=1LhimtAixtT×fxt−∑i=1LhimtAixt≤ΔAxtTΔAxt=ΔAx^t+ΔAx˜tT×ΔAx^t+ΔAx˜t=ΔA,ΔAx¯tT×ΔA,ΔAx¯t=Φx¯tTΦx¯t,Δg¯TΔg¯=ΔgTΔg≤∑j=1LhjmtΔBkjx^tT×∑j=1LhjmtΔBkjx^t=∑j=1LhjmtΔBkj,0x¯tT×∑j=1LhjmtΔBkj,0x¯t=∑j=1LhjmtΩjx¯tT∑j=1LhjmtΩjx¯t≤∑j=1Lhjmtx¯TtΩjTΩjx¯t,Δh¯TΔh¯=2∑i=1LhimtLiΔhT×∑i=1LhimtLiΔh≤2∑i=1LhimtLiΔDxtT×∑i=1LhimtLiΔDxt=2∑i=1LhimtLiΔD,LiΔDx¯tT×∑i=1LhimtLiΔD,LiΔDx¯t=2∑i=1LhimtΞix¯tT∑i=1LhimtΞix¯t≤2∑i=1Lhimtx¯TtΞiTΞix¯t,where Φ=ΔA,ΔA, Ωj=ΔBkj,0, Ξi=LiΔD,LiΔD, j=1,2,…,L,i=1,2,…,L.For the smooth progress of subsequent work, let us choose a Lyapunov function for system (8):(14)Vx¯t=x¯TtPx¯t,where P is a weighted matrix with appropriate dimensions.By integrating (14), we have(15)E∫0TdVx¯t=E∫0Tdx¯TPx¯=∑i=1L∑j=1LhmthjmtEi∫0TA¯ijx¯+Δf¯+Δg¯+Δh¯TPx¯+x¯TPA¯ijx¯+Δf¯+Δg¯+Δh¯+x¯TC¯TPC¯x¯+vTB¯iTPx¯+x¯TPB¯ivdt≤∑i=1L∑j=1LhimthjmtE∫0Tx¯T3P2+PA¯ij+A¯ijTP+C¯TPC¯+ΦTΦ+ΩjTΩj+2ΞiTΞix¯dt+∑i=1LhimtE∫0TvTB¯iTPx¯+x¯TPB¯ivdt=∑i=1L∑j=1LhimthjmtE∫0Tx¯T3P2+PA¯ij+A¯ijTP+C¯TPC¯+ΦTΦ+ΩjTΩj+2ΞiTΞix¯dt−E∫0Tγ−1x¯TP∑i=1LhimtB¯i−γvT⋅γ−1x¯TP∑i=1LhimtB¯i−γvTTdt+E∫0Tγ−2x¯TP∑i=1LhimtB¯i⋅∑i=1LhimtB¯iTPx¯+γ2vTvdt≤∑i=1L∑j=1LhimthjmtE∫0Tx¯T3P2+PA¯ij+A¯ijTP+C¯TPC¯+ΦTΦ+ΩjTΩj+2ΞiTΞi+γ−2PB¯iB¯iTx¯dt+E∫0Tγ2vTvdt.Based on (15), we can derive the following theorem.Theorem 1.
If there exists aP>0 satisfying the following inequalities:(16)3P2+PA¯ij+A¯ijTP+C¯TPC¯+γ−2PB¯iB¯iTP+Q1+ΦTΦ+ΩjTΩj+2ΞiTΞi<0,(17)4P2+PA¯ij+A¯ijTP+C¯TPC¯+k˜jTR1k˜j+Q2+ΦTΦ+ΩjTΩj+2ΞiTΞi<0,where k˜j=kj,0, then(a)
H∞ control performance (10) is fulfilled.(b)
H2 performance (11) has a upper bound, that is,(18)Jx¯,u=kjx^≤x¯02trP+tr∑i=1LhimtB¯iR0∑i=1LhimtB¯iT.Proof.
For givenQ1>0, by Schur’s complement and (15), one can see that(19)E∫0Tx¯TQ1x¯dt−γ2E∫0TvTvdt≤E∫0Tx¯TQ1x¯−γ2vTvdt+∑i=1L∑j=1LhimthjmtE∫0Tx¯TPA¯ijx¯+Δf¯+Δg¯+Δh¯+A¯ijx¯+Δf¯+Δg¯+Δh¯TPx¯+x¯TPB¯iv+vTB¯iTPx¯+x¯TC¯TPC¯x¯dt≤∑i=1L∑j=1LhimthjmtE∫0Tx¯T3P2+A¯ijTP+PA¯ij+C¯TPC¯+Q1+ΦTΦ+ΩjTΩj+2ΞiTΞix¯+x¯TPB¯iv+vTB¯iTPx¯−γ2vTvdt=∑i=1L∑j=1LhimthjmtE∫0Tx¯vTXPB¯iB¯iTP−γ2Ix¯vdt<0,where X=3P2+PA¯ij+A¯ijTP+C¯TPC¯+Q1+ΦTΦ+ΩjTΩj+2ΞiTΞi. Therefore, E∫0Tx¯TQ1x¯dt<γ2E∫0TvTvdt is directly derived, i.e., conclusion (a) is valid.
Now let us prove (b). Under the constraint of (8), with the help of the method of completing square, we assert that(20)Jx¯,u=kjx^=E∫0Tx¯TQ2x¯+uTR1udt≤∑i=1L∑j=1LhimthjmtE∫0Tx¯TQ2x¯+kjx^TR˜2kjx^+x¯TPA¯ijx¯+Δf¯+Δg¯+Δh¯+A¯ijx¯+Δf¯+Δg¯+Δh¯TPx¯+x¯TPB¯iv+vTB¯iTPx¯+x¯TC¯TPC¯x¯dt+x¯T0Px¯0≤∑i=1L∑j=1LhimthjmtE∫0Tx¯T3P2+A¯ijTP+PA¯ij+C¯TPC¯+Q2+k˜jTR1k˜j+ΦTΦ+ΩjTΩj+2ΞiTΞix¯dt+E∫0TvT∑i=1LhimtB¯iT∑i=1LhimtB¯ivdt+x¯T0Px¯0≤x¯T0Px¯0+E∫0TvT∑i=1LhimtB¯iT∑i=1LhimtB¯ivdt+∑i=1L∑j=1LhimthjmtE∫0Tx¯T4P2+A¯ijTP+PA¯ij+C¯TPC¯+Q2+k˜jTR2k˜j+ΦTΦ+ΩjTΩj+2ΞiTΞix¯dt≤x¯02trP+tr∑i=1LhimtB¯iR0∑i=1LhimtB¯iT,which shows that (b) holds. The proof of this theorem is concluded.
According to Theorem1, suboptimal H2/H∞ control design has been transformed into solving the optimization problem minP>0trP under the constraint of (16) and (17). However, because P22, Li, and W11 are coupled in some components, the optimization problem is not convex. So, we need to convert it into convex optimization problems.
ExpressP, Q1, and Q2 as follows:(21)P=P1100P22,Q1=Q11100Q221,Q2=Q11200Q222.
Plugging these representations into (16) and (17), one gets(22)A11A12A21A22<0,(23)B11B12B21B22<0,where(24)A11=3P112+P11Ai+Bikj+Ai+BikjTP11+CTP22C+γ−2P11LiLiTP11+Q111+ΔATΔA+kjTΔBTΔBkj+2ΔDTLiTLiΔD,A12=P11LiDi+CTP22C−γ−2P11LiLiTP22+ΔATΔA+2ΔDTLiTLiΔD,A21=DiTLiTP11+CTP22C−γ−2P22LiLiTP11+ΔATΔA+2ΔDTLiTLiΔD,A22=3P222+P22Ai−LiDi+Ai−LiDiTP22+CTP22C+γ−2P22LiLiTP22+Q221+ΔATΔA+2ΔDTLiTLiΔD,B11=4P112+P11Ai+Bikj+Ai+BikjTP11+CTP22C+Q112+kjTR˜2kj+ΔATΔA+kjTΔBTΔBkj+2ΔDTLiTLiΔD,B12=P11LiDi+CTP22C+ΔATΔA+2ΔDTLiTLiΔD,B21=DiTLiTP11+CTP22C+ΔATΔA+2ΔDTLiTLiΔD,B22=4P222+P22Ai−LiDi+Ai−LiDiTP22+CTP22C+Q222+ΔATΔA+2ΔDTLiTLiΔD.
Next, letW=diagW11,I with W11=P11−1. Multiplying both sides of (22) and (23) by W and setting Zi=P22Li, Yj=kjW11, we have(25)C11C12C21C22<0,(26)D11D12D21D22<0,where(27)C11=3I+AiW11+W11AiT+BiYj+YjTBiT+W11CTP22CW11+γ−2LiLiT+W11Q111W11+W11ΔATΔAW11+YjTΔBTΔBYj+2W11ΔDTLiTLiΔDW11,C12=LiDi+W11CTP22C−γ−2LiZiT+W11ΔATΔA+2W11ΔDTLiTLiΔD,C21=DiTLiT+CTP22CW11−γ−2ZiLiT+ΔATΔAW11+2ΔDTLiTLiΔDW11,C22=3I+P22Ai+AiTP22−ZiDi−DiTZiT+CTP22C+γ−2ZiZiT+Q221+ΔATΔA+2ΔDTLiTLiΔD,D11=4I+AiW11+W11AiT+BiYj+YjTBiT+W11CTP22CW11+W11Q112W11+YjTR˜2Yj+YjTΔBTΔBYj+W11ΔATΔAW11+2W11ΔDTLiTLiΔDW11,D12=LiDi+W11CTP22C+W11ΔATΔA+2W11ΔDTLiTLiΔD,D21=DiTLiT+CTP22CW11+ΔATΔAW11+2ΔDTLiTLiΔDW11,D22=4P222+P22Ai+AiTP22−ZiDi−DiTZiT+CTP22C+Q222+ΔATΔA+2ΔDTLiTLiΔD.
By Schur’s complement, (25) and (26) can be rewritten as(28)M11W11YjTM140W11M22000YjT0M3300M14T00M44Li000LiT−γ2I<0,N11W11YjTN14W11N2200Yj0N330N14T00N44<0,where(29)M11=3I+AiW11+W11AiT+BiYj+YjTBiT+YjTΔBTΔBYj,M14=LiDi+W11TCTP22C−γ−2LiZiT+W11ΔATΔA+2W11ΔDTLiTLiΔD,M22=−Q111+ΔATΔA+2ΔDTLiTLiΔD−1,M33=−ΔBTΔB−1,M44=3P222+P22Ai+AiTP22−ZiDi−DiTZiT+CTP22C+γ−2ZiZiT+Q221+ΔATΔA+2ΔDTLiTLiΔD,N11=4I+AiW11+W11AiT+BiYj+YjTBiT,N14=LiDi+W11CTP22C+W11ΔATΔA+2W11ΔDTLiTLiΔD,N22=−Q112+ΔATΔA+2ΔDTLiTLiΔD−1,N33=−R1+ΔBTΔB−1,N44=4P222+P22Ai+AiTP22−ZiDi−DiTZiT+CTP22C+Q222+ΔATΔA+2ΔDTLiTLiΔD.
Noticing that if (16) and (17) are true, then M44<0 and N44<0, the following two inequalities hold:(30)3P222+P22Ai+AiTP22−ZiDi−DiTZiT+CTP22C+γ−2ZiZiT+Q221+ΔATΔA+2ΔDTLiTLiΔD<0,4P222+P22Ai+AiTP22−ZiDi−DiTZiT+CTP22C+Q222+ΔATΔA+2ΔDTLiTLiΔD<0,which can be written as the following LMIs:(31)G1P22ZiP22−13I0ZiT0−γ2I<0,(32)G2P22P22−14I<0,where(33)G1=P22Ai+AiTP22−ZiDi−DiTZiT+CTP22C+Q221+ΔATΔA+2ΔDTLiTLiΔD,G2=P22Ai+AiTP22−ZiDi−DiTZiT+CTP22C+Q222+ΔATΔA+2ΔDTLiTLiΔD.
Therefore, the observer-based suboptimal stochasticH2/H∞ control design can be transformed into solving a two-step convex optimization problem.
The first step: under the constraint of (31) and (32), solve the convex optimization problem:(34)minP22>0trP22.It can be obtained that P22, Zi and Li=P22−1Zi.
The second step: under the constraint of (25) and (26), solve the following convex optimization problem:(35)minW11>0trW11.
We can getP11=W11−1 and the feedback gain kj. A suboptimal solution P=diagP11,P22 and compensator (7) are achieved.
To sum up, we state the following main result.Theorem 2.
If the above convex optimization problems (34) and (35) have solutions, then Li=P22−1Zi and kj=YjW11−1. Moreover, we have u∗t=YjW11−1x^t, and J∗x¯,u∗=trW11−1+P22x¯02+tr∑i=1hhimtB¯iR0∑i=1LhimtB¯iT.


## 4. A Numerical Example

For system (5), we define the fuzzy number as “big and small” and assume its coefficient matrices are(36)A1=0.23.22.80.5,A2=0.62.63.40.3,B1=0−0.3,B2=0−0.2,C=2.532.82,D1=D2=11.Give(37)ΔA=0.40.60.60.2,ΔB=00.1.We haveΔl=0, Δh=0 and ΔC=0, ΔD=0.Chooseγ2=0.9. Using LMI toolbox in MATLAB, we get(38)P22=0.3140−0.3253−0.32530.3761,W11=4.9206−4.9454−4.94544.9958,Z1=1.09711.1117,Z2=1.07521.1196,Y1=0.942234.3978,Y2=−1.150829.9927.According toLi=P22−1Zi, kj=YjW11−1, the parameters of observer and controller are(39)L1=63.096657.5249,L2=62.637057.1486,k1=1.40251.3952×103,k2=1.13741.1319×103.Taking the controlleru∗t into account, the simulation results are shown in Figure 1. It is shown that the system can achieve the desired control effects under the fuzzy controller.Figure 1
The trajectories of statesx1 and x2.


## 5. Conclusions

In this paper, the mixedH2/H∞ output-feedback control problem for stochastic nonlinear systems in a finite horizon has been studied. Firstly, the nonlinear system is transformed into a linear fuzzy model by T-S rules, and the error between the original system and the fuzzy one has been considered. A fuzzy observer-based two-step convex optimization method has been proposed to treat the suboptimal H2/H∞ problem. The method is simple and effective. The closed-loop system can guarantee the robustness and minimize the energy output. Since time delays exist widely in practical systems, how to generalize the obtained H2/H∞ output-feedback controller design method to stochastic nonlinear systems with delays is one of the directions of future research.



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