# H ∞ Robust Tracking Control of Stochastic T-S Fuzzy Systems with Poisson Jumps

Authors: Xiangyun Lin; Weihai Zhang; Bor-Sen Chen
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

A robust adaptiveH∞ tracking control design for nonlinear stochastic systems with both Brownian motion and Poisson jumps is proposed, which is based on Takagi–Sugeno (T-S) type fuzzy techniques. Because the state of to-be-controlled systems cannot be known exactly, to overcome this difficulty, the state estimation systems and error estimation systems are introduced to obtain an augmented system. By using the fuzzy systems to approximate the nonlinear systems, an adaptive fuzzy control is employed to achieve the desired H∞ tracking performance for stochastic systems with exogenous disturbance. A simulation example is presented to illustrate the tracking performance of the proposed design method.


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## Body

## 1. Introduction

Adaptive control theory is a powerful methodology which has been widely applied to design feedback control for systems with parametric uncertainties or for plants with unknown structure or changing operating conditions [1, 2]. Because there are uncertainties in the systems, the laws of adaptive control design schemes no more depend on the fixed description of input-output relationships but are related to the estimation of the system’s state or output and those estimating errors [3–6].The robustH∞ tracking control theory is a powerful methodology to design feedback controllers where the system parametric uncertainties are seen as the exogenous disturbance [7–9]. This methodology is widely applied in networked control systems [10], mobile robots [11], etc. The objective of the robust H∞ tracking control design is to construct an adaptive controller which guarantees the H∞ tracking control performance. Based on the system’s structure, the robust H∞ tracking control designing methods include linear H∞ control and nonlinear H∞ control. Linear H∞ control design problems are based on solving a kind of algebra Riccati inequalities which can be solved by the LMI’s method. And the nonlinear H∞ control design involves solving a nonlinear Hamilton-Jacobi inequality(HJI) [12]. However, it is very difficult to solve the Hamilton-Jacobi equalities or inequalities [13, 14]. In practice, to overcome this difficulty, the fuzzy methods have been applied to the H∞ robust control design of nonlinear systems where the Takagi–Sugeno (T-S) type fuzzy is widely used [15–17].Stochastic systems are applied in economics [18], biology [19], and natural science to describe the randomness in models [20–23]. Based on the distribution properties of the inserted random variables, stochastic systems include Itô-type systems driven by Brownian motion [24, 25], systems driven by Markovian jumps [26], systems driven by Poisson jumps or Lévy process [27], and their compound forms [28–30]. The linear stochastic H∞ theory has been developed since 1990s via the linear matrix inequalities (LMI) approach [31]. The nonlinear stochastic H∞ problems are solved by means of Hamilton-Jacobi equations [25]. In practice, there exist sudden shifts in the systems. In order to describe such phenomenon, Poisson jumps are inserted in the model, so the stochastic systems are driven not only by Brownian motion but also by Poisson jumps or Lévy process [27, 28]. The main complication in the H∞ tracking control design problem studied here is due to the presence of both deterministic, stochastic perturbations and Poisson jumps terms in the system. Nonlinear H∞ tracking theory and fuzzy control design are combined together to construct the adaptive fuzzy-based controller which guarantees the H∞ tracking control performance.This paper is organized as follows: In Section2, some lemmas about stochastic differential equations with Poisson jumps are reviewed, which will be used in the latter theoretical analysis and deductions. In Section 3, the theories of H∞ tracking control are extended to the case of linear stochastic systems with Poisson jumps. In Section 4, the H∞ tracking control design methods based on the Takagi–Sugeno type fuzzy techniques are applied to the nonlinear stochastic systems with Poisson jumps, and the H∞ tracking controller is obtained. In Section 5, the air-to-air missile pursuit systems are presented to show the effectiveness of the proposed method.Notation. For convenience, we adopt the following notations: Rm×n: the set of all real m×n matrices. Rn: the set of all n-dimensional real vectors. A>0(A≥0): the positive definite(semidefinite) matrix A. AT: the transpose of matrix A. I: the identity matrix with proper order. E[ξ]: the expectation of random variable ξ. |x|: the Euclidean norm of vector x∈Rnx.


## 2. Preliminaries

Let(Ω,F,F,Pr) be a complete probability space where F={Ft:t≥0} is a filtration generated by Brownian motion (Wiener process) {W(t),t≥0} and Poisson process {N(t),t≥0} which are two mutually independent stochastic processes:(i)
W(t) is a d-dimensional standard Brownian motion with E[W(t)]=0 and E[W(t)W(t)T]=Id;(ii)
N(t) is a Poisson jump process with rate λ>0 and E[N(t)]=λt.Let(1)Ft=σNs:s≤t∨σWs:s≤t∨N,where N denotes the totality of P-null sets. Then the filtration F=Ftt≥0. We firstly review some basic theories of stochastic differential equations driven by both Brownian motion and Poisson jumps:(2)dxt=ft,xtdt+σt,xtdWt+γt,xtdNt,x0=x0∈Rn.Lemma 1.
Letf(t,x),σ(t,x),γ(t,x) be Rn-valued functions on [0,T]×Rn and, for some positive constant K, satisfy the Lipschitz condition(3)ft,x-ft,y+σt,x-σt,y+γt,x-γt,y≤Kx-y,and linear growth condition(4)ft,x+σt,x+γt,x≤Kxfor all x,y∈Rn, t∈[0,T]. Then the stochastic differential equation (2) has a unique adapted solution with right continuation and left limitation.Under the conditions of Lemma1, we then review the Itô’s formula of (2) (see [32], Chapter 4, Rule 4.24).Lemma 2.
LetV(t,x) be twice continuously differentiable in x and once in t; x(t) is the solution of stochastic differential equation (2). Then(5)dVt,xt=Vtt,xt+VxTt,xtft,xt+12σTt,xtVxxt,xtσt,xtdt+Vxt,xtσt,xtdWt+Vt,xt+γt,xt-Vt,xtdNt.The following lemma is a kind of martingale inequality; see Proposition 7.15 in [33].Lemma 3.
For a finiteTf>0, let {Xt}0≤t≤Tf be a submartingale (or martingale, or supermartingale) with respect to filtrations F={Ft}0≤t≤Tf. Then for any r>0, there exists (6)Prsup0≤t≤Tf⁡Xt≥r≤3rsup0≤t≤Tf⁡EXt.Remark 4.
Because Brownian motionW(t) is a martingale with respect to filtration F, and Poisson process N(t) is a submartingale, by Lemma 3, for every finite Tf>0, there exist (7)Prsup0≤t≤Tf⁡Wt≥r≤3rsup0≤t≤Tf⁡EWt=3rsup0≤t≤Tf⁡2tπ=3r2Tfπ,and (8)Prsup0≤t≤Tf⁡Nt≥r≤3rsup0≤t≤Tf⁡ENt=3rsup0≤t≤Tf⁡λt=3λTfr,or equivalently, (9)Prsup0≤t≤Tf⁡Wt<r≥1-3r2Tfπ,and (10)Prsup0≤t≤Tf⁡Nt<r≥1-3λTfr.Therefore, when r is large enough, |W(t)| and N(t) have upper bound with probability p1(r)=1-3/r2Tf/π and p2(r)=1-3λTf/r, respectively. Furthermore, p1(r)→1 and p2(r)→1 when r→+∞.


## 3.H∞ Robust Tracking Control of Linear Systems

Consider the linear control system with the following forms:(11)dxt=Axt+But+Ewtdt+A1xt+E1wtdW1t+A2xt+E2wtdN1t,yt=Cxt+v,where A∈Rn×n, B∈Rn×nu, E,E1,E2∈Rn×nw, C∈Rny×n, and D∈Rny×nv are the system’s coefficient matrices; x(t)∈Rn is the state, y(t)∈Rny is the measurement, w(t)∈Rnw and v(t)∈Rnv are the exogenous disturbances; and W1(t) is the standard 1-dimensional Wiener process and N1(t) is the Poisson process with Poisson intensity λ>0.Now, we review the basic theory of stochastic differential equations driven by both martingale and Poisson jumps. In order to design the tracking control of system (11), a reference model is suggested as follows:(12)dxrt=Arxrt+rtdt,where xr(t)∈Rn is the desired reference state to be tracked by x(t), Ar∈Rn×n is a specified asymptotically stable matrix, and r(t) is a bounded reference input at the steady state, xr(t)=-Ar-1r(t). In practice, r(t) and Ar are given by user or designer to specify the transient and the steady state of reference signal xr(t) to be tracked.Denote the tracking errors as(13)ert=xt-xrt.For the tracking error er(t), we consider the H∞ robust tracking performance as follows:(14)∫0tfEerTtQertdt∫0tfEw~Ttw~tdt≤ρ2.Here Q is a positive definite matrix, tf>0 is the terminal time of control, and w~(t)=[v(t)Tw(t)Tr(t)T]T. The physical meaning of (14) is that the effect of any w~ on tracking error er(t) must be attenuated below a desired level ρ>0 from the viewpoint of energy. The following observer is proposed to deal with the state estimation of linear system (11):(15)dx^t=Ax^t+But+Lyt-y^tdt,y^t=Cx^t.Denote the estimation errors as(16)et=xt-x^t.Combining (11) and (15), we get(17)det=A-LCet-Lvtdt+A1xt+wtdW1t+A2xt+wtdN1t.Suppose the controller u(t) has the form as(18)ut=Kx^t-xrt.The performance (14) considering control effort is revised as(19)∫0tfEerTtQert+uTtRutdt∫0tfEw~Ttw~tdt≤ρ2,where R is a positive definite matrix. Let x~=[eT,xT,xrT]T and substitute (18) into (11); then system (11) and (17) can be augmented as the following form(20)dx~t=A~x~t+E~w~tdt+A~1x~t+E~1w~tdW1t+A~2x~t+E~2w~tdN1t,and the H∞ performance of (19) equals (21)∫0tfx~TtQ~x~tdt≤ρ2∫0tfw~Ttw~tdtwhere (22)A~=A-LC00-BKA+BK-BK00Ar,A~1=0A100A10000,A~2=0A200A20000,E~=-LE00E000I,E~1=0E100E10000,E~2=0E200E20000,Q~=KTRK-KTRKKTRK-KTRKKTRK+Q-KTRK-QKTRK-KTRK-QKTRK+Q.In order to achieve the H∞ tracking performance (19) with a prescribed attenuation level ρ, an auxiliary symmetric positive definite matrix P~ is suggested. The matrices of A~ and E~ include the to-be-designed matrices K and L. If the initial state x(0), xr(0) and the estimation error e(0) are also considered, the performance of (19) can be described as follows.(23)∫0tfEerTtQert+uTtRutdt≤x~0TP~x~0+ρ2∫0tfEw~Ttw~tdtTheorem 5.
SupposeP~∈S+3n(R),K∈Rnu×n,L∈Rn×ny constitute the solution of the following Riccati matrix inequality:(24)H~P~≤0,(25)ρ2I-E~1TP~E~1-λE~2TP~E~2>0where (26)H~P~=A~TP~+P~A~+A~1TP~A~1+λA~2TP~A~2+Q~+P~E~+A~1TP~E~1+λA~2TP~E~2×ρ2I-E~1TP~E~1-λE~2TP~E~2-1P~E~+A~1TP~E~1+λA~2TP~E~2T.Then the H∞ tracking control performance in (24) is guaranteed for a prescribed ρ2.Proof.
LetV~(x~)=x~TP~x~, x~∈R3n. By Itô formula, we get (27)V~x~tf-V~x~0=∫0tfx~TtA~TP~+P~A~+A~1TP~A~1+λA~2TP~A~2x~t+2x~TtP~E~+A~1TP~E~1+λA~2TP~E~2w~t+w~TE~1TP~E~1+λE~2TP~E~2w~dt+2∫0tfx~TtP~A~1x~t+E~1w~tdW1t+2∫0tfx~TtP~A~2x~t+E~2w~tdN~1t.Taking expectation on the both sides, we obtain (28)EV~x~tf=V~x~0+∫0tfEx~TtA~TP~+P~A~+A~1TP~A~1+λA~2TP~A~2x~t+2x~TtP~E~+A~1TP~E~1+λA~2TP~E~2w~t+w~TE~1TP~E~1+λE~2TP~E~2w~dt.Thus, we have (29)∫0tfEx~TtQ~x~tdt=V~x~0-EV~x~tf+∫0tfEx~TtA~TP~+P~A~+A~1TP~A~1+λA~2TP~A~2+Q~x~t+2x~TtP~E~+A~1TP~E~1+λA~2TP~E~2w~t-w~Tρ2I-E~1TP~E~1-λE~2TP~E~2w~+ρ2w~Ttw~tdt.Completing the square for w~, we have (30)∫0tfEx~TtQ~x~tdt=V~x~0+ρ2∫0tfEw~Ttw~tdt+∫0tfEx~TtH~P~x~tdt-EV~x~tf-∫0tfEw~t-M~-1M~1Tx~tR~2dtwhere M~=ρ2I-E~1TP~E~1-λE~2TP~E~2 and M~1=P~E~+A~1TP~E~1+λA~2TP~E~2. Together with (24), inequality (24) is proved. This ends the proof.In order to find a solution for matrix inequality (24), suppose P~ has the following block form(31)P~=P11000P22000P33,and then(32)H~P~=H~11H~12H~13∗H~22H~23∗∗H~33,where (33)H~11=ATP11+P11A-CTLTP11-P11LC+KTRK+1ρ2P11LLTP11+P11EM-1ETP11,H~12=-KTBTP22+P11EM-1ETP22+E1TP11+P22A1+λE2TP11+P22A2-KTRK,H~13=KTRK,H~22=AT+KTBTP22+P22A+BK+Q+KTRK+A1TP11+P22A1+λA2TP11+P22A2+P22E+A1TP11+P22E1+λA2TP11+P22E2M-1ETP22+E1TP11+P22A1+λE2TP11+P22A2,H~23=-P22BK-KTRK-Q,H~33=ArTP33+P33Ar+Q+KTRK+1ρ2P33P33,and (34)M=ρ2I-E1TP11+P22E1-λE2TP11+P22E2.Because (35)H~11=ATP11+P11A+KTRK+P11EM-1ETP11-ρ2CTC+1ρ2P11L-ρ2CTP11L-ρ2CTT,and (36)H~22=ATP22+P22A-P22BR-1BTP22+Q+A1TP11+P22A1+λA2TP11+P22A2+P22E+A1TP11+P22E1+λA2TP11+P22E2M-1ETP22+E1TP11+P22A1+λE2TP11+P22A2+K+R-1BTP22TRK+R-1BTP22,we take L=ρ2P11-1CT and K=-R-1BTP22. Then(37)H~11=ATP11+P11A+P22BR-1BTP22-ρ2CTC+P11EM-1ETP11,H~12=P11EM-1ETP22+E1TP11+P22A1+λE2TP11+P22A2,H~13=P22BR-1BTP22,H~22=ATP22+P22A-P22BR-1BTP22+Q+A1TP11+P22A1+λA2TP11+P22A2+P22E+A1TP11+P22E1+λA2TP11+P22E2M-1×ETP22+E1TP11+P22A1+λE2TP11+P22A2,H~23=-Q,H~33=ArTP33+P33Ar+Q+P22BR-1BTP22+1ρ2P33P33.Proposition 6.
SupposeP11∈S+(R), P22∈S+(R), and P33∈S+(R) are the solution of the following block matrix inequality:(38)H≔H1100P11E0P22B∗H22-QH2400∗∗H330P330∗∗∗-M00∗∗∗∗-ρ2I0∗∗∗∗∗-R≤0,and(39)M<0,where (40)H11=ATP11+P11A-ρ2CTC,H22=ATP22+P22A-P22BR-1BTP22+Q,H23=-Q,H24=P22E+A1TP11+P22E1+λA2TP11+P22E2,H33=ArTP33+P33Ar+Q.Then P~ with form of (31), L=ρ2P11-1CT and K=-R-1BTP22 are the solutions of (24) and (25). Moreover, the H∞ tracking control performance in (24) is guaranteed for a prescribed ρ2.Proof.
Applying the well-known Schur definiteness criterion to symmetric block matrixH, together with L=ρ2P11-1CT and K=-R-1BTP22, it is easy to see that (38) is equivalent to (24). As far as the H∞ tracking control performance in (24) is guaranteed, it can be directly obtained by Theorem 5.Remark 7.
In equation (32) and (37), if there exists P11>0 such that H~11≤0, this implies (41)ATP11+P11A-CTLTP11-P11LC≤0,i.e., (42)A-LCTP11+P11A-LC≤0.Thus, the following system(43)de¯t=A-LCe¯tdtis stable. Furthermore, if there exists P11>0 such that H~11<0, this implies (44)A-LCTP11+P11A-LC<0.Therefore, system (43) is asymptotically stable; i.e., A-LC is stable. In this paper, the estimation error system (17) can be seen as the extension of (43) where (17) presents the differences between x^(t) and x(t). Furthermore, in (38), H11<0; i.e., (45)ATP11+P11A-ρ2CTC<0,and if C=0, then A is stable. However, in general, C≠0; the matrix A is not necessarily stable.


## 4. Nonlinear Systems and Corresponding Fuzzy Systems

For nonlinear system as in the following forms(46)dxt=fxt+gxtut+Exwtdt+f1xt+E1xwtdW1t+f2xt+E2xwtdN1t,yt=mxt+v,i=1,2,…,r,the reference model is suggested as follows:(47)dxrt=frxrt+rtdt.The observer is proposed to deal with the state estimation of system (46)(48)dx^t=fx^t+gx^tut+Lx~yt-y^tdt,y^t=mx^t.Let (49)et=xt-x^t,and denote x~=[eT,xT,xrT]T, w~=[vT,wT,rT]T. Design the control u with form of(50)ut=Kx~tx^t-xrt,and then(51)dx~t=f~x~t+E~x~tw~tdt+f~1x~t+E~1x~tw~tdW1t+f~2x~t+E~2x~tw~tdN1t,where (52)f~x~=fx-fx^+gx-gx^Kx~x^-xr-Lx~mx-mx^T,fx+gxKx~x^-xrT,frxrTT,f~1x~=f1xT,f1xT,0TT,f~2x~=f2xT,f2xT,0TT,E~x~=-Lx~Ex00Ex000I,E~1x~=0E1x00E1x0000,E~2x~=0E2x00E2x0000.For a positive function l~(x~), an auxiliary positive function V~ is suggested. The H∞ tracking performance considering the initial condition is given by(53)∫0tfEl~x~tdt≤V~x~0+ρ2∫0tfEw~Ttw~tdt.Theorem 8.
SupposeV~:R3n→R+, K(x~), Ł(x~) constitute the solution of the following Hamilton-Jacobi inequality:(54)H~1V~x~≤0,(55)ρ2I-12E~1TV~x~x~E~1-λ2E~2TSE~2>0,where (variable x~ is omitted) (56)H~1V~x~≔V~x~Tf~+λf~2+12f~1TV~x~x~f~1+λ2f~2TSf~2+l~+14V~x~TE~+f~1TV~x~x~E~1+λV~x~TE~2+λf~2TSE~2×ρ2I-12E~1TV~x~x~E~1-λ2E~2TSE~2-1V~x~TE~+f~1TV~x~x~E~1+λV~x~TE~2+λf~2TSE~2T,and S∈S+(R) is a fixed positive definite matrix with(57)V~x~x~x~≤S.Then the H∞ tracking control performance in (53) is guaranteed for a prescribed ρ2>0.Proof.
Applying Itô’s formula toV~(x~(t)), there exist (58)∫0tfEl~x~tdt=V~x~0-EV~x~tf+ρ2∫0tfEw~Ttw~tdt+∫0tfEH~1V~x~t-w~t-ξtΞ2dt,where (59)Ξ=ρ2I-12E~1Tx~tV~x~x~x~tE~1x~t-λ2E~2Tx~tSE~2x~tξt=12Ξ-1V~x~Tx~tE~x~t+f~1Tx~tV~x~x~x~tE~1x~t+λV~x~Tx~tE~2x~t+λf~2Tx~tSE~2x~tT.Keeping in mind that V~ is positive and applying (54), we prove that (60)∫0tfEl~x~tdt≤ρ2∫0tfEw~Ttw~tdt.This ends the proof.The nonlinear system (46), (47), (48), (50), and (51) can be described by T-S fuzzy model as follows.Model Rule i 
IF  ζ(t)is  Mi1, and⋯and  ζp(t)is  Mip,THEN(61)dxt=Aixt+Biut+wtdt+A1ixt+wtdW1t+A2ixt+wtdN1t,yt=Cixt+v,i=1,2,…,r.Corresponding reference model is suggested as follows:(62)dxrt=Arixrt+rtdt,where ζ(t)=(ζ(t),…,ζp(t)) are the premise variables, Mij is the fuzzy set, and r is the number of model rules.The following observer is proposed to deal with the state estimation of fuzzy system (61)(63)dx^t=Aix^t+Biut+Liyt-y^tdt,y^t=Cix^t.Then the controller u(t) has the form as(64)ut=Kix^t-xrt.LetM~ij(ζj(t)) be the grade of membership of ζj(t) in Mij; then the membership function of ζ(t) for the i’th rule is(65)μiζt=M~i1M~i2⋯M~ip≥0,with ∑i=1r‍μi(ζ(t))>0. The standard membership function is defined by(66)hiζt=μiζt∑i=1rμiζt.Then the fuzzy system is inferred as(67)dxt=∑i=1rhiζtAixt+Biut+wtdt+A1ixt+wtdW1t+A2ixt+wtdN1t,yt=∑i=1rhiζtCixt+v,and(68)dx^t=∑i=1rhiζtAix^t+Biut+Liyt-y^tdt,y^t=∑i=1rhiζtCix^t,with estimation errors(69)det=∑i=1rhiζtAiet-Liyt-y^t+wtdt+A1ixt+wtdW1t+A2ixt+wtdN1t.Then the controller u(t) has the form as(70)ut=∑i=1rhiζtKix^t-xrt.and(71)lert,ut=∑i=1rhiζterTtQiert+uTtRiut.


## 5. Simulation Examples

As the application of tracking system, the air-to-air missile pursuit systems are discussed, which are used to shot down escaping target such as aircraft, fighter, or missile [34, 35]. The relative dynamic motion between a homing-missile and the escaping target is described by the following system driven by both Wiener process and Poisson process.(72)dx1t=x2tdt+c1x2tdWt,dx2t=x32tx1t+u1t+v1tdt+c2x32tx1t+v1tdWt,dx3t=-x2tx3tx1t+u2t+v2tdt+c3x2tx3tx1t+v2tdWt+c4x3tx1t+v2tdNt,yt=x1tx3t,x0=x0∈D,t∈0,tf.Here x(t)=(x1(t),x2(t),x3(t))T is the system state which describes the trajectory of missile, concretely; x1(t) is the relative distance between the missile and target; see Figure 1; x2(t) is the radial relative velocity; x3(t) is the tangential relative velocity; see Figure 2; and u(t)=(u1(t),u2(t))T is the input control with u1 and u2 being the missiles acceleration components along the x1 and x2 axes. W(t) denotes the continuous noise of the system such as the air friction caused by air resistance and the inner thermal noise caused by missile operating system. Poisson process N(t) denotes the sudden discrete noise of the system such as directional trimming, or deceptive behavior of the missile to avoid being shot down by shooting out gas from two side jets. y(t) is the measurement and tf is the fighting time which denotes the missile’s effective working time, and C=[101]T.Figure 1
Pursuit guidance course at timet0<t1<t2<tf.Figure 2
The geometric relationship between the target and missile.Taketf=25(sec), x1(0)=7875(m), x2(0)=-343(m/s), x3(0)=-10(rad/s), c1=c2=c3=0.11, c4=0.3, Q=diag(0,1,1)×10-2, R=I2, and the density of Poisson jump N(t) is 0.02; i.e., λ=0.02, ρ2=0.49, and (73)E=E1=001001×10-3,E2=000001×10-3.Let r(t) the trajectory of the target’s motion which is embedded in a reference model as follows:(74)dxr=Arxr+rtdt,xr0=x0,where Ar=-I3 which implies the steady state xr(t)=r(t). Let r(t) be the reference trajectories with (75)rt=t2-362.6t+7987.61.3t2-24.1t-274.4-0.9t2+0.0149t-0.0509,t∈0,25,(76)dx^1t=x^2t+L1yt-y^tdt,dx^2t=x^32tx^1tdt,dx^3t=-x^2tx^3tx^1tdt,y^t=x^1t,t∈0,tf.Choose the fuzzy variable ζ=x3/x1 and the fuzzy rules as follows:Rule 1 (ζ about -0.45).
(77) A 1 = 0 1 0 0 0 - 0.45 0 0.45 0 , A 1 1 = 0 0.3 0 0 0 - 0.135 0 0.135 0 A 2 1 = 0 0 0 0 0 0 0 0.135 0 , P 1 1 = 0.3564 - 1.584 - 0.792 - 1.584 7.4571 1.76 - 0.792 1.76 9.2171 , P 2 1 = 0 0 0 0 0.43 0 0 0 0.43 , P 3 1 = 0.49 0 0 0 0.49 0 0 0 0.49Rule 2 (ζ about -0.3).
(78) A 2 = 0 1 0 0 0 - 0.3 0 0.3 0 , A 1 2 = 0 0.3 0 0 0 - 0.09 0 0.09 0 A 2 2 = 0 0 0 0 0 0 0 - 0.09 0 , P 1 2 = 0.2376 - 1.584 - 0.792 - 1.584 11.1857 2.64 - 0.792 2.64 13.8257 , P 2 2 = 0 0 0 0 0.35 0 0 0 0.35 , P 3 2 = 0.49 0 0 0 0.49 0 0 0 0.49Rule 3 (ζ about -0.15).
(79) A 3 = 0 1 0 0 0 - 0.15 0 0.15 0 , A 1 3 = 0 0.3 0 0 0 - 0.045 0 0.045 0 A 2 3 = 0 0 0 0 0 0 0 - 0.045 0 , P 1 3 = 0.1188 - 1.584 - 0.792 - 1.584 22.3714 5.28 - 0.792 5.28 27.6514 , P 2 3 = 0 0 0 0 0.25 0 0 0 0.25 , P 3 3 = 0.49 0 0 0 0.49 0 0 0 0.49Rule 4 (ζ about -0.07).
(80) A 4 = 0 1 0 0 0 - 0.07 0 0.07 0 , A 1 4 = 0 0.3 0 0 0 - 0.021 0 0.021 0 A 2 4 = 0 0 0 0 0 0 0 - 0.021 0 , P 1 4 = 0.055468 - 1.5848 - 0.7924 - 1.5848 47.9628 11.32 - 0.7924 11.32 59.2828 , P 2 4 = 0 0 0 0 0.17 0 0 0 0.17 , P 3 4 = 0.49 0 0 0 0.49 0 0 0 0.49Rule 5 (ζ about -0.04).
(81) A 5 = 0 1 0 0 0 - 0.04 0 0.04 0 , A 1 5 = 0 0.3 0 0 0 - 0.012 0 0.012 0 A 2 5 = 0 0 0 0 0 0 0 - 0.012 0 P 1 5 = 0.031712 - 1.5856 - 0.7928 - 1.5856 83.9773 19.82 - 0.7928 19.82 103.7973 , P 2 5 = 0 0 0 0 0.13 0 0 0 0.13 , P 3 5 = 0.49 0 0 0 0.49 0 0 0 0.49Rule 6 (ζ about -0.012).
(82) A 6 = 0 1 0 0 0 - 0.012 0 0.012 0 A 1 6 = 0 0.3 0 0 0 - 0.0036 0 0.0036 0 , A 2 6 = 0 0 0 0 0 0 0 - 0.0036 0 P 1 6 = 0.0095126 - 1.5854 - 0.79272 - 1.5854 279.8962 66.06 - 0.79272 66.06 345.9562 , P 2 6 = 0 0 0 0 0.07 0 0 0 0.07 , P 3 6 = 0.49 0 0 0 0.49 0 0 0 0.49Rule 7 (ζ about -0.008).
(83) A 7 = 0 1 0 0 0 - 0.008 0 0.008 0 A 1 7 = 0 0.3 0 0 0 - 0.0024 0 0.0024 0 , A 2 7 = 0 0 0 0 0 0 0 0.0024 0 P 1 7 = 0.0063424 - 1.5856 - 0.7928 - 1.5856 419.8867 99.1 - 0.7928 99.1 518.9867 , P 2 7 = 0 0 0 0 0.06 0 0 0 0.06 , P 3 7 = 0.49 0 0 0 0.49 0 0 0 0.49 .Remark 9.
We find that, in the fuzzy rules, the coefficients ofA(i) have the form with (84)A=01000-b0b0,where b>0. For every rule i, P11, P22, and P33 are solved by the following steps:Step 1.
SolveP11∗>0 such that (85)P11∗=arg⁡maxP11>0⁡αsubject to(86)ATP11+P11A-ρ2CTC≤-αI,(87)α>0.The above optimal P11∗ could be solved by increasing α until P11 cannot be found in LMIs (86) and (87).Step 2.
FixP11=P11∗ solved by Step 1. Suppose P22 has the form of (88)P22=0000pq0qp,where p>0,q≥0. Solve p,q such that(89)H~2≔H~11H~12∗H~22≤0,where H~ij is determined by (37), i,j=1,2. Under the conditions of p>0, p2>q2, for a fixed q, P22 can be solved by increasing p until matrix inequality (89) can be true.Step 3.
FixP11 and P22 which are determined by Steps 1 and 2. The positive definite matrix P33>0 can be obtained by solving the LMIs (38) in which P11 and P22 are fixed matrices.The weighted functionhi(ζ) is given by (90)
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swhere ζ1=-0.45, ζ2=-0.07, ζ3=-0.012, ζ4=-0.008, i=2,3,4,5,6. Figure 3 shows the profiles of hi(ζ), i=1,2,…,7.Figure 3
Fuzzy rules based on weighted functionshi(ζ) defined by (90)–(92).Figure4 shows the sample trajectories of the original system (72) and the corresponding fuzzy system (67) with u=0 and v=0. This illustrates that the T-S fuzzy method can be used to approximate the stochastic nonlinear systems via the fuzzy systems with proper coefficients given by Rule i, i=1,2,…,7.Figure 4
The sample trajectories of the fuzzy system (67) with coefficient matrices provided by Rules 1–7 and the original system (72).By (70), the H∞ tracking fuzzy control u(t)=[u1(t),u2(t)]T is obtained by(93)ut=∑i=17hiζtKix^t-xrt.Here K(i)=-P22(i), i=1,2,…,7. Figure 5 illustrates the sample trajectories of fuzzy control u(t) for system (72) with Poisson process. It shows that the values of control change violently when Poisson jumps occur.Figure 5
The sample trajectories of fuzzy controlu1(t), u2(t) for fuzzy system (67) with coefficient matrices provided by Rules 1–7 corresponding to nonlinear system (72) under Poisson process.Figures6, 7, and 8 show the sample trajectories of the components of the state x(t), the reference xr(t), and the estimation state x^(t). These show that, under the proposed H∞ tracking fuzzy control (93), the state x(t) and its estimation x^(t) of fuzzy system (67) with coefficient matrices provided by Rules 1–7 corresponding to (72) take values around the reference trajectory xr(t). Figure 9 shows the estimation error |e(t)|2=|x(t)-x^(t)|2 which illustrates the performance difference between the state estimation x^(t) and the state x(t). The four figures confirm that the control defined by (93) can achieve the desired performance (53) with l~(x~(t))=l(er(t),u(t)), where l(er(t),u(t)) is defined by (71).Figure 6
The sample trajectories ofx1(t) and xr1(t) of fuzzy system (67) with coefficient matrices provided by Rules 1–7 under fuzzy control (93) corresponding to nonlinear system (72).Figure 7
Trajectories ofx2(t) and xr2(t) of fuzzy system (67) with coefficient matrices provided by Rules 1–7 under fuzzy control (93) corresponding to nonlinear system (72).Figure 8
Trajectories ofx3(t) and xr3(t) of fuzzy system (67) with coefficient matrices provided by Rules 1–7 under fuzzy control (93) corresponding to nonlinear system (72).Figure 9
The estimation errorset2 of fuzzy system (67) with coefficient matrices provided by Rules 1–7 under fuzzy control (93) corresponding to nonlinear system (72).Now, we compare our results with that in [35]. In [35], the guidance law uses the information for the target acceleration bound, and the guidance command is derived based on the nonlinear planar engagement kinematics. In our guidance law design, we consider the nonlinear model of the system with fuzzy method. Some comparisons are given as follows:( 1 ) In the control algorithm of [35], the sliding mode control scheme is used to obtain the guidance law. In our control design, an adaptive fuzzy scheme is employed to cancel the exogenous disturbance v(t) in the stochastic system with Brownian motion and Poisson jumps.( 2 ) The controller obtained in [35] is based on solving a nonlinear function V(x). In our paper, the robust tracking controller is given with fuzzy forms in (70) which is only based on the locally linear approximation.( 3 ) The controller in [35] depends on the sliding surface and the state. Our controller u(t) in (70) depends on the difference between the state x(t) and the reference trajectory xr(t) and the estimation errors e(t).


## 6. Conclusion

TheH∞ tracking control technique combining with adaptive control algorithm and T-S fuzzy method has been studied to design the H∞ tracking control to achieve the desired performance for nonlinear stochastic systems driven by both Brownian motion and Poisson jumps. By using adaptive fuzzy control algorithms, the corresponding adaptive fuzzy control laws are derived, which have been employed to treat the H∞ design problem. A real-world example is used to show the effectiveness of the proposed method. Simulation results illustrate that the proposed adaptive T-S fuzzy control algorithms are practically useful to achieve the H∞ tracking performance.



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*Source: 5257972-2018-10-02.xml*