# NonfragileH∞ Filter Design for Nonlinear Continuous-Time System with Interval Time-Varying Delay

Authors: Zhongda Lu; Guoliang Zhang; Yi Sun; Jie Sun; Fangming Jin; Fengxia Xu
Journal: Journal of Control Science and Engineering
 (2018)

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## Abstract

This paper investigates nonfragileH∞ filter design for a class of continuous-time delayed Takagi-Sugeno (T-S) fuzzy systems with interval time-varying delays. Filter parameters occur multiplicative gain variations according to the filter’s implementation, to handle this variations, a nonfragile H∞ filter is presented and a novel filtering error system is established. The nonfragile H∞ filter guarantees the filtering error system to be asymptotically stable and satisfies given H∞ performance index. By constructing a novel Lyapunov-Krasovskii function and using the linear matrix inequality (LMI), delay-dependent conditions are exploited to derive sufficient conditions for nonfragile designing H∞ filter. Using new matrix decoupling method to reduce the computational complexity, the filter parameters can be obtained by solving a set of linear matrix inequalities (LMIs). Finally, numerical examples are given to show the effectiveness of the proposed method.


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## Body

## 1. Introduction

As we all known, in practical control systems, nonlinearity and time delay phenomena are often encountered in various industry and control system, such as networked control system and mechanical drive control system. The control of nonlinear systems has been explored and studied by many scholars in related fields. T-S fuzzy model is a powerful tool to deal with nonlinearity; much effort has been devoted on the networked control system for T-S fuzzy system or time-delayed (see [1–4]). The actuator and sensor faults estimation based on T-S fuzzy model with unmeasurable premise variables were investigated in [5]. The problem of exponential stabilization for sampled-data T-S fuzzy control systems with packet dropouts was investigated in [6]; a switched system approach is proposed to model the data-missing phenomenon. There always exist many kinds of noise interference in the process of transmission among real industrial control system’s signal, causing the error between the obtained signals and the desired signals; in order to obtain the accurate data information about the control signal and eliminate the influence of disturbances on the system, it is essential to be filtering. At present, there are the Kalman filtering, fault detection filter, L2 filtering, H∞ filtering, and so on. Compared with other filtering methods, H∞ filtering does not need the exactly known statistics of the external disturbance [7, 8] and H∞ filtering has excellent robustness against unmodeled dynamics. In recent years, H∞ filtering system based on the Takagi-Sugeno (T-S) model has attracted much attention from the control community [9, 10], many studies have addressed H∞ filtering for T-S fuzzy systems with time-varying delay, and the proposed H∞ filtering technology has been applied to many actual communications system. Authors in literature [11–13] investigated the problem of H∞ filter design for continuous-time via Takagi-Sugeno fuzzy model approach. In literature [14, 15], the problem of H∞ filtering for a class of discrete fuzzy system has been reported. Based on discrete inequality technique and the Lyapunov-Krasovskii functional approach, sufficient conditions for the existence of admissible filters are established in terms of linear matrix inequalities. In literature [16], the event-triggered H∞ filtering for networked control systems with quantization and network-induced delays was investigated; it improved the usage of network resource.However, in practical system, it is difficult for an exactly implemented filter to meet the actual requirements because inaccuracies or uncertainties, which include collection error and component aging, may occur during filter implementation. It often degrades the performance of the control system and even instability; the filter has a higher sensitivity to the parameter uncertainty [17]. Thus, we need to design nonfragile H∞ filter considering the parameter variation and uncertainty. Some achievements have been reported in journal about nonfragile H∞ filtering for T-S fuzzy systems with time-varying delay. In literature [18], design an H∞ filter with the gain variations such that the filtering error system was quadratically D stable and guarantees a prescribed H∞ performance level. Literature [19] is concerned with the problem of nonfragile H∞ filtering for discrete-time nonlinear systems and considered additive interval uncertainty. In literature [20], the designed nonfragile H∞ filter was in standard form and the filter was designed, which have two types of multiplicative gain variations; these models were in standard form. In literature [21], the problem of nonfragile H∞ filter design for linear continuous-time systems was studied; it proposed a notion of structured vertex separator. In literature [22], this paper studied the nonfragile H∞ filtering problem for a class of discrete-time T-S fuzzy systems with both randomly occurring gain variations and channel fading. In literature [23], the problem of nonfragile H∞ filter design for linear continuous-time systems has been studied. The filter has been designed; it included additive gain variations. In literature [24], it studied the nonfragile filtering design for a kind of fuzzy stochastic system with time-varying delay and parameter uncertainties. Sufficient conditions for stochastic input-to-state stability (SISS) of the fuzzy stochastic systems were obtained. Papers proposed the filter design methods with occurring additive gain variations according to the filter’s implementation.Motivated by the aforementioned discussion, in this paper, a nonfragileH∞ filter design method is proposed to enhance the nonfragility of the filter. By considering the multiplicative gain variations and interval time-varying delays according to the filter implementation, a novel filtering error system is established. Different from some existing works, Jensen’s inequality is used to tackle the integral items of the derivative of Lyapunov-Krasovskii; a more relaxed H∞ performance stability criterion is derived. By constructing a novel Lyapunov-Krasovskii function and using the linear matrix inequality (LMI), delay-dependent conditions are exploited to derive sufficient conditions for nonfragile designing H∞ filter. Our objective is to design nonfragile H∞ filter which guarantees the filtering error system to be asymptotically stable and satisfies given H∞ performance index. The filter parameters can be obtained by solving a set of linear matrix inequalities (LMIs).The rest of this paper is organized as follows. The problem formulation is stated in Section2; nonfragile filter scheme and filtering error system are employed to enhance system’s stabilization. Stability analysis and fuzzy filter design are obtained in Section 3; by constructing a Lyapunov-Krasovskii functional, a new stability criterion is proposed to prove being less conservative than the existing ones. An applicable H∞ filter is designed in Section 4, which guarantees stability and a desire performance of the filtering error system. In order to show the effectiveness of the proposed method, simulation results are presented in Section 5.


## 2. Problem Formulation

Consider a nonlinear system with time-varying delay which could be approximated by a class of T-S fuzzy systems with time-varying delays. The T-S fuzzy model with plant rules can be described by the following.Plant Rule i. If θ1(t) is Mi1... and θr(t) is Mip, then(1)x˙t=Aixt+Aτixt-τt+Biωtyt=Cixt+Cτixt-τt+Diωtzt=Eixtx ( t ) = φ ( t ), where φ(t) is the continuous initial vector function defined on -τ2,0, Mij are the fuzzy sets, i=1,2,3,…,r, and j=1,2,3,…,p is the number of IF-THEN rules. θ1(t),θ2(t)…θr(t) are the premise variables, x(t)∈Rn is the state vector, y(t)∈Rm is the measured output, z(t)∈Rp is the signal vector to be estimated, ω(t)∈Rq is the disturbance signal vector which belongs to ω(t)∈l20,∞, and Ai, Aτi, Bi, Bτi, Ci, Cτi, Di, Dτi, Ei are known constant matrices with appropriate dimensions. τ(t) is interval time-varying delay that satisfies the following inequality: τ1≤τ(t)<τ2, τ˙(t)≤d where τ1, τ2, and d are constant scalars.By employing the commonly used center-average defuzzifier, product interference, and singleton fuzzifier, the overall fuzzy model is inferred as follows:(2)x˙t=∑i=1rhiθtAixt+Aτixt-τt+Biωtyt=∑i=1rhiθtCixt+Cτixt-τt+Diωtzt=∑i=1rhiθtEixtwhereθ=θ1,…,θrT, hi(θ(t))=hi(θ(t))/∑irhi(θ(t)), hi(θ(t))=∏j=1pMij(θj(t)), Mij(•) represents the grade of membership for θjt in Mij, 0≤hiθt≤1i=1,2,…,r, and ∑i=1rhi(θ(t))>0. It can be seen that hi(θ(t))≥0(i=1,2,…,r), ∑i=1rhi(θ(t))=1.Consider the nonfragile fuzzy filter with multiplicative gain uncertainties; we design the following fuzzyH∞ filter:(3)x˙fit=Afi+ΔAfitxft+Bfi+ΔBfitytzfit=Cfi+ΔCfitxftConsider the followingH∞ filter form which is analogous to the fuzzy control form through parallel distributed compensation.Plant Rule i. If θ1(t) is Mi1... and θp(t) is Mip, then(4)x˙ft=∑i=1rhiθtAfi+ΔAfitxft+Bfi+ΔBfitytzft=∑i=1rhiθtCfi+ΔCfitxftx f ( 0 ) = x f 0, where xf0 is the continuous initial vector function, xf(t)∈Rn is the filter state vector, the estimated signal vector z(t) is zf(t), and Afi, Bfi, Cfi, i=1,2,3,…,r are the filter parameters. ΔAfi(t), ΔBfi(t), ΔCfi(t) represent the gain variations.The multiplicative gain uncertainties are defined as(5)ΔAfit=AfiM1iKAtN1iΔBfit=BfiM2iKBtN2iΔCfit=CfiM3iKCtN3iwhere M1i, N1i, M2i, N2i, M3i, N3i are constant matrices with appropriate dimensions and KA(t), KB(t), KC(t) are uncertain matrices bounded, such that(6)KATtKAt<I,KBTtKBt<I,KCTtKCt<I.By combining (2) with (4), we can obtain the following filtering error system:(7)ξ˙t=A-tξt+A-τtξt-τt+B-tωtet=zt-zft=E-tξtwhere(8)ξt=xTtxfTtTA-t=A^t+ΔA^t=At0BftCtAft+00ΔBftCtΔAftA-τt=A^τt+ΔA^τt=Aτt0BftCτt0+00ΔBftCτt0B-t=B^t+ΔB^t=BtBftDt+0ΔBftDtE-t=E^t+ΔE^t=Et-Cft+0-ΔCftA^t=∑i=1r∑j=1rhihjAj0BfiCjAfi=At0BftCtAftA^τt=∑i=1r∑j=1rhihjAτj0BfiCτj0=Aτt0BftCτt0B^t=∑i=1r∑j=1rhihjBjBfiDj=BtBftDtE^t=∑i=1r∑j=1rhihjEj-Cfi=Et-CftΔA^t=∑i=1r∑j=1rhihj00ΔBfiCjΔAfi=00ΔBftCtΔAftΔA^τt=∑i=1r∑j=1rhihj00ΔBfiCτj0=00ΔBftCτt0ΔB^t=∑i=1r∑j=1rhihj0ΔBfiDj=0ΔBftDtΔE^t=∑i=1r∑j=1rhihj0-ΔCfi=0-ΔCftIn this paper, our purpose is to design the fuzzyH∞ filter in the form of (3), meanwhile, satisfying the following requirements.( 1 ) The filtering error system (7) with ω(t)=0 is said to be asymptotic stability for any initial condition( 2 ) For a given positive scalar γ>0, the filtering error system (7) is said to be asymptotically stable with guaranteed H∞ performance γ, if it is asymptotic stability and the filtering error e(t) satisfies(9)∫0Let2dt≤γ2∫0Lωt2dtfor all L>0 and nonzero ω(t)∈L20,∞ subject to the zero initial condition.


## 3. Stability andH∞ Filtering Performance Analysis

The purpose of this paper is to design nonfragileH∞ filter such that the filtering error system (7) is asymptotically stable with H∞ performance index. A sufficient condition is presented in the following theorem to guarantee the existence of the filter in form of (3).Lemma 1.
LetX, Y, and K be real matrices with appropriate dimensions and KiT(t)Ki(t)<I. Then, for any scalar δ>0,(10)XKtY+YTKtXT≤εXXT+ε-1YTYLemma 2.
For any vectorsx,y∈Rn and any scalar ε>0, matrices D,F,E are real matrices of appropriate dimensions with FT(t)F(t)<I; then the following inequalities hold:(11)2xDFtEy≤εxTDDTx+ε-1yTEETyLemma 3.
LetV, H, E, Q, and F be real matrices of appropriate dimensions such that Q>0 and FTF≤I. Then, for any scalar ε>0 such that Q-1-εHHT>0, we have(12)V+HFETQV+HFE≤VTQ-1-εHHT-1V+ε-1ETELemma 4.
LetP>0 and Ai,Al be any real matrices of appropriate dimensions. Then(13)AiTPAl+AlTPAi≤AiTPAi+AlTPAlLemma 5 (Jenson’s inequality).
Supposeτ1≤τ(t)≤τ2 and x(t)∈Rn; for any positive matrix R∈Rn×n, the following inequality holds:(14)-τ2-τ1∫t-τ2t-τ1x˙TsRx˙sds≤xt-τ1xt-τ2T-RRR-Rxt-τ1xt-τ2Theorem 6.
For nonlinear systems (1) and the filtering error system (7), the given positive scalar H∞ performance γ and the filtering error system (7) are asymptotically stable with H∞ performance γ if there exist symmetric positive scalars ε2>0, ε6>0, ε10>0, ε29>0, ε15>0, ε25>0, ε20>0, ε21>0, ε22>0, ε11>0, ε12>0 and symmetric positive definite matrices P^>0, Q^1>0, Q^2>0, Q^3>0, R^1>0, R^2>0 such that we have the following inequality.(15)IM~3∗ε20-1>0(16)R^1-1M~2∗ε1k-1>0,k=5,9(17)R^2-1M~2∗ε2k-1>0,k=5,9(18)R^1-1α11M~1α12M~2∗I0∗∗I>0(19)R^2-1α21M~1α22M~2∗I0∗∗I>0(20)Φ=Φ11Φ12R^10Φ15∗Φ22000∗∗Φ3300∗∗∗Φ440∗∗∗∗Φ55<0(21)Φ11=P^A^t+A^TtP^+Q^1+Q^2+Q^3-R^1+E^TtI-ε20M~3M~3T-1E^t+α42P^M~2M~2TP^+α32P^M~1M~1TP^+α52N~1N~1+α62N~2N~2+β12A^TtR^1-1-ε11M~1M~1T-ε12M~2M~2T-1A^t+β22A^TtR^2-1-ε21M~1M~1T-ε22M~2M~2T-1A^t+ε20-1N~3TN~3Φ12=P^A^τt,Φ15=P^B^t,Φ33=-Q^2-R^1+R^2,Φ44=-Q^3-R^2,Φ22=-1-dQ^1+α22N~2τTN~2τ+β12A^τTtR^1-1-ε15M~2M~2T-1A^τt+β22A^τTtR^2-1-ε25M~2M~2T-1A^τt,Φ55=α12N~2DTN~2D-γ2I+β12B^τTtR^1-1-ε19M~2M~2T-1B^τt+β22B^τTtR^2-1-ε29M~2M~2T-1B^τtM~1=0AftM1,M~2=0BftM2,M~3=M3,N~1=0N1,N~2=N2Ct0,N~2τ=N2Cτt0,N~2τ=N2Cτt0,N~2D=N2Dt,N~3=0-N3Cft,β1=3τ12,β2=3τ22-τ12,α1=β12ε19-1+β22ε29-1+ε10-1,α2=β12ε15-1+β22ε25-1+ε6-1,α3=ε21,α4=ε2+ε6+ε10,α5=ε21-1+β12ε11-1+β22ε22-1,α6=β12ε12-1+β22ε22-1Proof.
We construct a novel Lyapunov-Krasovskii function as follows:(22)Vξt=ξTtP^ξt+∫t-τttξTsQ^1ξsds+∫t-τ1tξTsQ^2ξsds+∫t-τ2tξTsQ^3ξsds+τ1∫-τ10∫t+θtξ˙TsR^1ξ˙sdsdθ+τ2-τ1∫-τ2-τ1∫t+θtξ˙TsR^2ξ˙sdsdθ(23)V˙=2ξTtP^A-tξt+2ξTtP^A-τtξt-τt+2ξTtP^B-tωt+ξTtQ^1+Q^2+Q^3ξt-1-τ˙tξTt-τtQ^1ξt-τt-ξTt-τ1Q^2ξt-τ1-ξTt-τ2Q^3ξt-τ2+τ12ξ˙TtR^1ξ˙t+τ2-τ12ξ˙TtR^2ξ˙t-τ1∫t-τ1tξ˙TsR^1ξ˙sds-τ2-τ1∫t-τ1tξ˙TsR^2ξ˙sds(24)V˙≤2ξTtP^A^tξ˙t+2ξTtP^ΔA^tξt+2ξTtP^tA^τtξt-τt+ξTtQ^1+Q^2+Q^3ξt+2ξTtP^ΔB^tωt+2ξTtP^tΔA^τtξt-τt-1-dξTt-τtQ^1ξt-τt-ξTt-τ1Q^2ξt-τ1+2ξTtP^B^tωt-ξTt-τ2Q^3ξt-τ2+τ12ξ˙TtR^1ξ˙t+τ2-τ12ξ˙TtR^2ξ˙t-τ1∫t-τ1tξ˙TsR^1ξ˙sds-τ2-τ1∫t-τ2t-τ1ξ˙TsR^2ξ˙sds
By Lemma2, for ∀ε2>0, ∀ε21>0, we can obtain(25)2ξTtP^ΔA^tξt=2ξTtP^M~2iKitN~2i+M~1iKitN~1iξt≤ξTtε2P^M~2M~2TP^+ε21P^M~1M~1TP^+ε2-1N~2TN~2+ε21-1N~1TN~1ξt
Similar to (25), if there exists ∀ε20>0, we can obtain(26)eTtet=ξTtE-TtE-tξt=ξTtE^t+ΔE^tTE^t+ΔE^tξt≤ξTtE^TtI-ε20M~3M~3T-1E^tξt+ε20-1ξTtN~3TN~3ξt
By Lemma2, if there exists ∀ε6>0, we can obtain(27)2ξTtP^ΔA^τtξt-τt=2ξTtP^M~2KitN~2τξt-τt≤ξTtε6P^M~2M~2TP^ξt+ξTt-τtε6-1N~2τTN~2τξt-τt
Similar to (25), if there exists ∀ε10>0, we can obtain(28)2ξTtP^ΔB^tωt=2ξTtP^M~2KitN~2Dωt≤ξTtε10P^M~2M~2TP^ξt+ωTtε10-1N~2DTN~2Dωt
Combining with inequalities (25), (27), and (28) gives(29)2ξTtP^ΔA^tξt+2ξTtP^ΔA^τtξt-τt+2ξTtP^ΔB^tωt≤ξTε2P^M~2M~2TP^+ε2-1N~2N~2T+ε21P^M~1M~1TP^+ε21-1N~1N~1Tξt+ξTtε10P^M~2M~2TP^ξt+ξTtε6P^M~2M~2TP^ξt+ξTt-τtε6-1N~2τTN~2τξt-τt+ωTtε10-1N~2TN~2ωt
Lemma4 gives(30)τ12ξ˙TtR^1ξ˙t=τ12A-tξt+A-τtξt-τt+B-tωtTR^1A-tξt+A-τtξt-τt+B-tωt≤3τ12ξTtA-TR^1A-tξt+3τ12ξTt-τtA-τTt-τtR^1A-τt-τtξt-τt+3τ12ωTtB-TtR^1B-tωt
where(31)A-TtR^1A-t=A^t+ΔA^tTR^1A^t+ΔA^t=A^t+M~2KitN~2+M~1KitN~1TR^1A^t+M~2KitN~2+M~1KitN~1(32)A-τTtR^1A-τt=A^τt+ΔA^τtTR^1A^τt+ΔA^τt=A^τt+M~2KitN~2τTR^1A^τt+M~2KitN~2τ(33)B-TtP^B-t=B^τt+ΔB^tTR^1B^t+ΔB^t=B^t+M~2KitN~2DTR^1B^τt+M~2KitN~2D
By Lemma3, if there exist ε11>0, ε12>0 such that R^1-1-ε11M~1M~1T-ε12M~2M~2T>0, from (31), it follows that(34)A^t+M~2KitN~2+M~1KitN~1TR^1A^t+M~2KitN~2+M~1KitN~1≤A^Tt+M~2KitN~2TR^1-1-ε11M~1M~1T-1A^t+M~2KitN~2+ε11-1N~1TN~1≤A^TtR^1-1-ε11M~1M~1T-ε12M~2M~2T-1A^t+ε12-1N~2TN~2+ε11-1N~1TN~1
By Lemma3, if there exist ε15>0, such that R^1-1-ε15M~2M~2T>0, from (32), it follows that(35)A^τt+M~2KitN~2τTR^1A^τt+M~2KitN~2τ≤A^τTtR^1-1-ε15M~2M~2T-1A^τt+ε15-1N~2τTN~2τ
By Lemma3, if there exist ε19>0 such that R^1-1-ε19M~2M~2T>0, from (33), we can get(36)B^t+M~2KitN~2DTR^1B^τt+M~2KitN~2D≤B^TtR^1-1-ε19M~2M~2T-1B^t+ε19-1N~2DTN~2D
Combining with formulas (31)~(36), from (30), we have(37)τ12ξ˙TtR^1ξ˙t≤ξTt3τ12A^TtR^1-1-ε11M~1M~1T-ε12M~2M~2T-1A^t+ε12-1N~2TN~2+ε11-1N~1TN~1ξt+ξTt-τt3τ12A^TτtR^1-1-ε15M~2M~2TA^τTt+ε15-1N~2τTN~2τξTt-τt+ωTt3τ12B^TtR^1-1-ε19M~2M~2TB^t+ε19-1N~2DTN~2Dωt
Similarly, for formula (30) we have(38)τ2-τ12ξ˙TtR^2ξ˙t≤ξTt3τ2-τ12A^TtR^2-1-ε21M~1M~1T-ε22M~2M~2T-1A^t+3τ2-τ12ε22-1N~2TN~2+3τ2-τ12ε21-1N~1TN~1ξt+ξTt-τt3τ2-τ12A^TtR^2-1-ε25M~2M~2T-1A^t+3τ2-τ12ε25-1N~1TN~1ξt-τt+ωTt3τ2-τ12B^TtR^2-1-ε29M~2M~2T-1B^t+3τ2-τ12ε29-1N~2DTN~2Dωt
By Lemma5, according to Jensen’s inequality, we have that(39)-τ1∫t-τ1tξ˙TsR^1ξ˙sds≤ξtξt-τ1T-R^1R^1∗-R^1ξtξt-τ1(40)-τ2-τ1∫t-τ2t-τ1ξ˙TsR^2ξ˙sds≤ξt-τ1ξt-τ2T-R^2R^2∗-R^2ξt-τ1ξt-τ2
Combining with formulas (25)~(30), from (23), we have(41)V˙ξt≤ηTtΦ-ηt
where(42)Φ-=Φ-11Φ-12R^10Φ-15∗Φ-22000∗∗Φ-3300∗∗∗Φ-440∗∗∗∗Φ-55<0Φ-11=P^A^t+A^TtP^+Q^1+Q^2+Q^3-R^1+α42P^M~2M~2TP^+α32P^M~1M~1TP^+α52N~1N~1+α62N~2N~2+β12A^TtR^1-1-ε11M~1M~1T-ε12M~2M~2T-1A^t+β22A^TtR^2-1-ε21M~1M~1T-ε22M~2M~2T-1A^tΦ-12=P^A^τt,Φ-15=P^B^t,Φ-33=-Q^2-R^1+R^2,Φ-44=-Q^3-R^2,Φ-22=-1-dQ^1+ε6-1N~2τTN~2τ+α22N~1τTN~1τ+β12A^τTtR^1-1-ε15M~2M~2T-1A^τt+β22A^τTtR^2-1-ε25M~2M~2T-1A^τtΦ-55=α12N~2DTN~2D+β12B^TtR^1-1-ε19M~2M~2T-1B^t+β22B^TtR^2-1-ε29M~2M~2T-1B^t
By Schur complement and formula (26), we have(43)V˙ξt+eTtet-γ2ωTtωt≤ηTtΦηt
where(44)ηTt=ξTtξTt-τtξTt-τ1ξTt-τ2ωTt
Consequently, it follows from inequality (20), V(ξ(t))t=0=0 and V(ξ(t))t=L≥0, and we have ∫0Le(t)2-γ2ω(t)2dt+V(ξ(t))t=L-V(ξ(t))t=0≤0, which implies that (9) holds.
Thus,H∞ performance is verified. In addition, when the zero disturbance input ω(t)=0, by Schur complement, we can obtain that the time derivative of Lyapunov-Krasovskii V˙ξ(t)≤0; that means that the filtering error system (7) with ω(t)=0 is asymptotically stable.


## 4. FuzzyH∞ Filter Design

Theorem6 provides a sufficient condition for H∞ filter design with time delay and satisfied the H∞ performance. However, there exist some coupled matrix variables in the matrix inequality (15); H∞ filter parameter can not be calculated directly. In order to decouple the variables in (15), we will use decoupling technique. Using this method, inequality (15) can be equivalently expressed in another form; hence, we can obtain H∞ filter parameter.Theorem 7.
For given scalarsδ>0,  ε2i>0,  ε6i>0,  ε10i>0, ε19i>0, ε29i>0,  ε15i>0,  ε25i>0,  ε20i>0,  ε21i>0,  ε22i>0,  ε11i>0, and ε12i>0, the filtering error system (6) is asymptotically stable as well as with the H∞ performance level γ, if there exist matrices P^>0,  Q^1>0,  Q^2>0,  Q^3>0,  R^1>0,  R^2>0,  Afi,  Bfi, and Cfi such that the following linear matrix inequalities are satisfied:(45)R^1-1M~2i∗ε1ki-1>0,k=5,9(46)R^2-1M~2i∗ε2ki-1>0,k=5,9(47)R^1-1α11iM~1iα12iM~2i∗I0∗∗I>0(48)R^2-1α21iM~1iα22iM~2i∗I0∗∗I>0(49)Ψij=Ψ1Ψ2∗Ψ4where(50)Ψ1=Ψ111Ψ112Ψ113Ψ1140Ψ1150Ψ118∗-I000000∗∗Ψ13300000∗∗∗Ψ144Ψ145000∗∗∗∗-I000∗∗∗∗∗Ψ166Ψ1670∗∗∗∗∗∗-I0∗∗∗∗∗∗∗-IΨ111=ψ11ψ1200ψ15∗-1-dQ^1000∗∗-Q^2-R^1+R^200∗∗∗-Q^3-R^20∗∗∗∗-γ2Iψ11=P11Aj+B-fiCjA-fiP12TAj+B-fiCjA-fi+P11Aj+B-fiCjA-fiP12TAj+B-fiCjA-fiT+Q^1+Q^2+Q^3-R^1,ψ12=P11Aτj+B-fiCτj0P12TAτj+B-fiCτj0,ψ15=P11Bj+B-fiDjP12TBj+B-fiDj,Ψ112=α4iP^M~2iα3iP^M~1i00,Ψ113=α5iN~1iTα6iN~2iTα20iN~3iTE^jT0000,Ψ114=β2iA^jTP^0,Ψ115=β2iA^jTP^0,Ψ118=0α7iN~2τiT000,Ψ133=-diagI,I,I,I-α20i2M~3iM~3iT,Ψ144=-2δP^+δ2R^1,Ψ145=α11iP^M~1iα12iP^M~2i,Ψ166=-2δP^+δ2R^2,Ψ167=α21iP^M~1iα22iP^M~2i,Ψ2=Ψ2110Ψ2130Ψ215Ψ2160Ψ2180000000000000000000000000000000000000000000000000000000000000000,Ψ211=0β1iA^jTP^000,Ψ213=0β2iA^τjTP^000,Ψ215=0000α1iN~2DiT,Ψ216=0000β1iB^jTP^,Ψ218=0000β2iB^jTP^,Ψ4=Ψ411α15iP^M~2i0000000∗-I0000000∗∗Ψ433α25iP^M~2i00000∗∗∗-I00000∗∗∗∗-I0000∗∗∗∗∗Ψ466α19iP^M~2i00∗∗∗∗∗∗-I00∗∗∗∗∗∗∗Ψ488α29iP^M~2i∗∗∗∗∗∗∗∗-IΨ411=-2δP^+δ2R^1,Ψ433=-2δP^+δ2R^2,Ψ466=-2δP^+δ2R^1,Ψ488=-2δP^+δ2R^2,M~2i=0BfiM2i,M~1i=0AfiM1i,M~3i=M3i,N~1i=0N1i,N~2i=N2iCj0,N~2τi=N2iCτj0,N~2Di=N2iCDj,N~3i=0-N3iCfi
We can obtain the filter parameters as follows:(51)Afi=P12-1A-fi,Bfi=P12-1B-fi,Cfi=C-fiProof.
By Schur complement formula, the matrix inequality conditions (20) in Theorem 6 can be described as the following matrix inequalities:(52)Γ=Γ1Γ2∗Γ4where(53)Γ1=Γ111Γ112Γ113Γ1140Γ1150Γ118∗-I000000∗∗Γ13300000∗∗∗-R^1-1Γ145000∗∗∗∗-I000∗∗∗∗∗-R^2-1Γ1670∗∗∗∗∗∗-I0∗∗∗∗∗∗∗-IΓ111=P^A^t+A^TtP^+Q^1+Q^2+Q^3P^A^τt00P^B^t∗-1-dQ^1000∗∗-Q^200∗∗∗-Q^30∗∗∗∗-γ2IΓ112=α4P^M~2α3P^M~100,Γ114=β1A^TtP^0,Ψ113=α5N~1Tα6N~2Tα20N~3TE^T0000,Γ115=β2A^TtP^0,Γ118=0α7N~2τT000,Γ133=-diagI,I,I,I-α202M~3M~3T,Γ145=α11P^M~1α12P^M~2,Γ167=α21P^M~1α22P^M~2,Γ2=Γ2110Γ2130Γ215Γ2160Γ2180000000000000000000000000000000000000000000000000000000000000000,Γ211=0β1A^TtP^000,Γ213=0β2A^τTtP^000,Γ215=0000α1N~2DT,Γ216=0000β1B^TtP^,Γ218=0000β2B^TtP^,Γ4=Γ411α15P^M~20000000∗-I0000000∗∗Γ433α25P^M~200000∗∗∗-I00000∗∗∗∗-I0000∗∗∗∗∗Γ466α19P^M~200∗∗∗∗∗∗-I00∗∗∗∗∗∗∗Γ488α29P^M~2∗∗∗∗∗∗∗∗-IΓ411=-2δP^+δ2R^1,Γ433=-2δP^+δ2R^2,Γ466=-2δP^+δ2R^1,Γ488=-2δP^+δ2R^2.
LetP^=P11P12∗P12, P^A^(t)=P11A(t)+B-f(t)C(t)A-f(t)P12TA(t)+B-f(t)C(t)A-f(t), P^A^τ(t)=P11Aτ(t)+B-f(t)Cτ(t)0P12TAτ(t)+B-f(t)Cτ(t)0, P^B^(t)=P11B(t)+B-f(t)D(t)P12TB(t)+B-f(t)D(t), β1=3τ12, β2=3(τ22-τ12), α1=β12ε19-1+β22ε29-1+ε10-1, α2=β12ε15-1+β22ε25-1+ε6-1, α3=ε21, α4=ε2+ε6+ε10, α5=ε21-1+β12ε11-1+β22ε22-1, α6=β12ε12-1+β22ε22-1, α7=ε6, α11=ε11, α12=ε12, α15=ε15, α25=ε25, α19=ε19, α20=ε20, α21=ε21, α22=ε22, α29=ε29, Δ(t)=∑i=1rhi(θ(t))Δi(t), where Δ=A,B,D,Aτ, A-f,B-f,C-f.
According toδR^-P^R^-1δR^-P^≥0, we have that -P^R^-1P^≤-2δP^+δ2R^ holds for any scalar δ>0. The filter parameters in (3) can be designed as (51). This completes the proof.


## 5. Simulation Example

Consider the following nonlinear systems with time-varying delays:(54)x˙t=∑i=12hiAixt+Aτixt-τt+Biωtyt=∑i=12hiCixt+Cτixt-τt+Diωtzt=∑i=12hiEixtwhere(55)A1=-2.10.11-2,A2=-1.90-0.2-1.1,Aτ1=-1.10.1-0.8-0.9,Aτ1=-0.90.1-1.1-1.2,B1=1-0.2,B1=0.30.1,D1=0.3,D2=-0.6,C1=10,C2=0.50.6Cτ1=-0.80.6,Cτ2=-0.21,E1=1-0.5,E2=-0.20.3The known matrices in (5) are given by(56)M11=M12=0.50.40.30.2,M21=M22=0.50.3,M31=M32=0.080.02,N11=N12=0.30.60.20.1,N21=N22=0.1-0.2,N31=N32=0.20.3,Klt=sint,wherel=A,B,C.The disturbance signalω(t) is given as follows:(57)ωt=sin⁡t,0<t<150other.We can get the desired nonfragile filter by solving LMIS (45)-(49) in Theorem 7; the nonfragile parameter matrices are given as follows:(58)Af1=-3.13296.22680.7361-7.4593,Af2=-2.75537.6590-0.8303-8.3416,Bf1=-0.4755-1.1658,Bf2=-0.2459-1.2213,Cf1=-0.20310.2124,Cf2=0.2231-0.3065In addition, the new method provides less conservative design result; we can obtain a smallerγ=0.4996. Figures 1 and 2 show the response of system’s states x(t) and filter’s states xf(t), respectively. Figure 3 shows the trajectories of z(t) and its estimates zf(t). The estimation error e(t) is depicted in Figure 4.Figure 1
The response curve of system statesx1,x2.Figure 2
The response curve of system statesxf1, xf2.Figure 3
The response curve of vectorz(t), zf(t).Figure 4
The response curve of vectore(t).The random disturbance signal is given as follows:(59)ωt=αtsint,0<t<150other,where α(t) denotes Bernoulli’s random event. The simulation results are obtained as shown in Figures 5–9.Figure 5
The event occurrence probabilityα(t).Figure 6
The response curve of system statesx1, x2.Figure 7
The response curve of system statesxf1, xf2.Figure 8
The response curve of vectorz(t), zf(t).Figure 9
The response curve of vectore(t).The stochastic variablesα(t) are Bernoulli-distributed white sequences; Figure 5 shows the event occurrence probability. Figures 6 and 7 show the response of system’s states x1(t),x2(t) and filter’s states xf1(t),xf2(t), respectively. Figure 8 shows the trajectories of z(t) and its estimates zf(t). The system error e(t) is depicted in Figure 9.The results show that theH∞ nonfragile filter can make the system have good stability. The system generates small overshoot; this filter can efficiently reduce the influence of external disturbance and uncertainty; it can enhance control precision and dynamic qualities of the system.


## 6. Conclusions

This paper studied the fuzzy nonfragileH∞ filter design problem for a class of nonlinear systems with an interval time-varying delays; meanwhile, the designing H∞ filter with multiplicative gain variations was considered. By constructing a new Lyapunov-Krasovskii functional, we obtained a sufficient condition for designing the nonfragile H∞ filter such that the filtering error system is asymptotically stable and satisfied the given H∞ performance index. This nonfragile filter design method enhances the nonfragility of the filter and reduces some conservatism. A numerical example has shown the effectiveness of the proposed method. Future research includes event-triggered nonfragile H∞ filter design for nonlinear system considering packet dropout and interval time-varying delays. Moreover, type 2 fuzzy filter design for nonlinear system with time-varying delays also can be further considered for the future investigation.



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*Source: 8198603-2018-09-05.xml*