# Nonfragile Quantized Dissipative Filter for Nonlinear Networked Systems

Authors: Kewang Huang; Jianfeng Wang; Feng Pan
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

To address the problem of filter parameter perturbation in nonlinear networked systems, a nonfragile quantized dissipative filter is designed by considering the coexistence of random one-step time delay, multipacket losses, and quantization error. We acquired the sufficient conditions for the existence of filter by choosing appropriate Lyapunov function as well as utilizing linear matrix inequality. Furthermore, we obtained the parameter expressions of the designed filter. The designed filter could meet the performance requirements of stability and dissipativity for the filter error system under the condition of allowed time delays, packet loss probability, and quantization density. The effectiveness of the designed filter is verified by numerical simulation.


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## Body

## 1. Introduction

The word “filter” is derived from electrical engineering, where filters are used to transform electrical signals from one form to another, especially to eliminate various frequencies in a signal. This means that the main role of the filter is to keep frequency contents in the desired band and to eliminate other parts when the external input signal passes the filter. Nowadays, the filter plays an important role in networked control systems. When information is transmitted in an infinite network with limited bandwidth, such problems as overload, packet losses, and so on will occur if the data is too large. Therefore, the signal needs to be quantized in advance. Nevertheless, signal quantization will cause quantization error unavoidably. This will influence system performance and may lead to system instability [1–6].The modern networked control systems are developing rapidly by virtue of low cost, easy installation, high reliability/flexibility, strong fault tolerance/diagnosis ability, and convenient remote manipulation/control. In network environments, network-induced delays and packet dropouts caused by limited network bandwidth and/or congested network traffic should be taken into account [7]. The occurrence of time delays can lead to poor performance or even destroy the stability of networks [8, 9]. Due to these problems, the signals delivered to filters may be incomplete in real implementation. Consequently, the traditional techniques for designing filters may lead to unsatisfactory performance. Many scholars investigated H∞ filtering and passive filtering [5, 6, 10]. H∞ filtering only extracts the gain of information, whereas passive filtering only considers phase information. The theory of dissipative systems includes some basic tools, such as passive theorem, bounded real lemma, and Kalman–Yakubovich lemma and provides an effective framework for stability analysis and design of control systems. Thus, in the past several decades, dissipativity is applied to a wide range of fields, such as systems, circuits, and complex networks [11]. However, the design of filter based on dissipative theory can consider both gain and phase information. Therefore, investigating dissipative filtering is of great significance [12–19].Ma et al. [12] investigated the problem of robust dissipative filter for discrete systems with nonlinear factors and constant time-delay, and they obtained the sufficient conditions for the existence of the filter by constructing appropriate Lyapunov function. Lin, Yang, and Wang [14] addressed the problem of nonlinear fuzzy filtering with multipacket losses, whereas they obtained the sufficient conditions for the existence of the filter based on linear matrix inequality (LMI) using T-S fuzzy model. Wang, Zhang, and Han [15] solved the problem of dissipative filter by considering the existence of time-varying delay during transmission for neutral systems, and an event-triggered communication scheme was introduced to design a suitable filter. Similarly, they obtained the parameter expressions of the corresponding filter by solving LMI. Tao et al. [16] discussed the problem of dissipative filter for Markov-jump systems, and slack matrix variables were introduced to eliminate the cross-coupling between the Lyapunov matrix and system matrices. Furthermore, they also obtained the sufficient conditions for the existence of the filter by solving LMI. Yang, Yang, and Liu [18] settled the problem of stability and dissipativity for nonlinear systems with filter parameter perturbation. However, they adopted a more general type of filter gain and obtained the parameter expressions of the corresponding filter by choosing suitable Lyapunov function as well as utilizing LMI. Xia et al. [19] focused on the problem of the coexistence of mixed delays and unknown transition rates for Markov-jump systems, and they obtained the sufficient conditions for the existence of the designed filter by constructing a parameter-dependent Lyapunov–Krasovskii function based on delay-partitioning approach.Unfortunately, time delays, packet losses, and quantization error were independently considered in the above-mentioned research. However, they may coexist in real network transmission. Moreover, the performance of filter may be influenced by the change of its own parameters. Therefore, time delays, packet losses, and quantization error were simultaneously considered in this paper. We designed the corresponding nonfragile quantized dissipative filter and acquired the sufficient condition for the existence of the filter based on Lyapunov stability theory and linear matrix inequality. Furthermore, we derived the parameter expression of the filter.The rest of this paper is organized as follows. Section2 outlines the problem formulation. Section 3 details the design of nonfragile quantized dissipative filter. Section 4 demonstrates numerical simulation results and discussions. Section 5 concludes this paper.


## 2. Problem Formulation

Consider the following nonlinear discrete system:(1)xk+1=Axk+Bwk+fk,xkyk=Cxk+Dwkzk=L1xk+L2wkwhere x(k)∈Rn is the state vector, z(k)∈Rq is the signal to be estimated, y(k)∈RP is measurement output, w(k)∈Rm is the disturbance, and w(k)∈L2[0,∞); A∈Rn×n, B∈Rn×m, C∈Rn×n, D∈Rn×m, L1∈Rq×n, and L2∈Rq×m are constant matrices, respectively; f(k,x(k))∈Rn is a nonlinear vector that meets Lipschitz condition; that is, f(k,x(k))≤W2x(k); W2 is a constant matrix.Considering the limited bandwidth, the logarithmic quantizer is utilized to relieve the tension on the network channel [1, 2]. We adopt the definition of logarithmic quantizer given in [2], and the quantization level is denoted as(2)U=±ui,ui=ρiu0,i=±1,±2,…∪±u0∪0,0<ρ<1,u0>0.The corresponding logarithmic quantizerq(v) is defined as(3)qv=uiif11+δui<v<11-δui,v>00ifv=0-q-vifv<0where ρ is quantization density, δ=1-ρ/1+ρ represents the upper bound of quantization error, and v is the signal sampled before quantization.The measurement output after quantization can be described as(4)y~=qyUtilizing the sector boundary method, the quantization error of measurement output can be defined as(5)ek≜qyk-yk≜Δkyk,Δk∈-δ,δThen, (4) can be expressed as(6)y~=1+Δky.Considering that there may be a random one-step delay or multipacket losses in real received measurement output, the corresponding mathematical model can be expressed as(7)y¯k=αky~k+1-αk1-αk-1βky~k-1+1-αk1-1-αk-1βky_k-1where y¯(k)∈RP is the real received measurement output.Substituting (6) into (7), we have(8)y¯k=αk1+Δkyk+1-αk1-αk-1βk1+Δkyk-1+1-αk1-1-αk-1βky_k-1where α(k) and β(k) are uncorrelated random sequences with Bernoulli distribution.Respectively, they meet the following statistical probabilities:(9)probαk=1=α-,probαk=0=1-α-probβk=1=β-,probβk=0=1-β-where prob{·} represents event probability.From (8), we have the following:P α ( k ) = 0 , α ( k - 1 ) = 0 , β ( k ) = 1 = ( 1 - α - ) 2 β - is the probability of one-step time delay; Pα(k)=0,α(k-1)=0,β(k)=0+Pα(k)=0,α(k-1)=1=(1-α-)2(1-β-)+(1-α-)α- is the probability of the packet loss; Pα(k)=1=α- is the probability of timely received data.For convenience later, we define the following variable:(10)y^k=1-αkβk+11+Δkyk+1-1-αkβk+1y_kCombining (8) and (10), we have(11)y-k=αk1+Δkyk+1-αky^k-1Further, combining (1), (10), and (11), we have the following augmented system:(12)x-k+1=A-x-k+B-wk+Gfk,xky-k=C-x-k+D-wkzk=L_x_k+L2wkwhere(13)x_k+1=xk+1Ty∧kTy_kTT,x_k∈Rn+2p,A_=A0+αkA1+1-αkβk+1A2,B_=B0+αkB1+1-αkβk+1B2,C_=C0+αkC1,D-=αkD1,L-=L100,A0=A000I00I0,B0=B00,C0=0I0,A1=0001+ΔkC-I01+ΔkC-I0,B1=01+ΔkD1+ΔkDC1=1+ΔkC-I0,A2=0001+ΔkC-I0000,B2=01+ΔkD0,G=I0,D1=1+ΔkD.


## 3. Nonfragile Dissipative Filter Design

Consider designing the following filter:(14)x^k+1=Afdx^k+Bfdy-kzfk=Cfdx^k+Dfdy-kwhere x∧k∈Rn+p is the state estimation of the filter; zf(k)∈Rq is the estimation of z(k), and Af∈R(n+2p)×(n+2p), Bf∈R(n+2p)×p, Cf∈Rq×(n+2p), and Df∈Rq×p are parameter matrices of the filter, respectively. Afd, Bfd, Cfd, and Dfd can be updated as follows:(15)Afd=Af+ΔAf,Bfd=Bf+ΔBfCfd=Cf+ΔCf,Dfd=Df+ΔDfwhere Af,Bf,Cf, and Df are given parameter matrices and their corresponding perturbation can be calculated as follows:(16)ΔAf=H1F1kE1,ΔBf=H2F2kE2ΔCf=H3F3kE3,ΔDf=H4F4kE4where H1∈R(n+2p)×r, H2∈R(n+2p)×r, H3∈Rn×r, H4∈Rq×r, E1∈Rr×(n+2p), E2∈Rr×(n+2p), E3∈Rr×(n+2p), E4∈Rr×p, and Fi(k) meet:(17)FikTFik≤I,i=1,2,3,4Combining (12) and (14), we can obtain the following filter error system:(18)x~k+1=A~x~k+B~wk+G~fk,xkek=C~x~k+D~wkwhere x~(k)=x-(k)Tx^(k)TT and e(k)=z(k)-zf(k) is the estimated error:(19)A~=A_0BfdC_Afd=A~0+αkA~1+1-αkβk+1A~2,D~=L2-DfdDαkB~=B_BfdD_=B~0+αkB~1+1-αkβk+1B~2C~=L_-DfdC_-Cfd=C~0+αkC~1A~0=A00BfdC0Afd,A~1=A10BfdC10,A~2=A2000,B~0=B00,B~1=B1BfdDB~2=B20,C~0=L_-DfdC0-Cfd,C~1=-DfdC10,D~=L2-DfdD1akDefiningA∗=EA~, B∗=EB~, C∗=EC~, D∗=ED~, Ad=EA¯, Bd=EB¯, Cd=EC¯, and Dd=ED¯ where E represents expectation and letting x,H,yt=E[∑0tx(k)THy(k)], we select the following secondary energy supply function for (18):(20)E-w,e,t=e,Q,et+2e,S,wt+w,R,wtwhere Q=QT, S, and R=RT are matrices of appropriate dimensions, respectively.The aim of this paper is to design a filter by considering the coexistence of random one-step time delay, multipacket losses, and parameter perturbation. The designed filter should be able to estimate the system state and the signal to be estimated accurately, and it should meet the following two performance requirements:① Whenw(k)=0, if there exists a scalar χ>0 such that the inequality E∑k=0∞x~(k)2<χsup-τ≤i≤0Eϕ(i)2 holds, then system (18) is stochastically stable.② For anyt>0 and nonzero w(k)∈L2[0,∞), if there exists a scalar a>0 such that the inequality E[a∑0tw(k)Tw(k)]<E-(w,e,t) holds under zero initial conditions, then system (18) is strictly (Q,S,R) dissipative.


### 3.1. Dissipativity Analysis

Lemma 1 (see [20]).
Given matricesN=NT, H and E are of appropriate dimensions, and matrix F meets FTF≤I such that N+HFE+ETFTHT<0. If there exists a positive ε-, then the sufficient and necessary condition of N+HFE+ETFTHT<0 changes into N+ε-HHT+ε--1ETE<0.
The detailed proof of Lemma1 can be found in [20]; it is omitted here.Theorem 2.
For given matricesQ=QT<0, R=RT, and S, assuming that Afd,BfdCfd,Dfd are known, if there exists symmetric positive definite matrix P and constant τ>0 such that (21) holds, then system (18) is stochastically stable and strictly (Q,S,R) dissipative.(21)-P+H0∗∗∗∗∗∗∗0-τI∗∗∗∗∗∗∗-STC∗0-D∗TS-STD∗-R∗∗∗∗∗∗A∗G~B∗-P-1∗∗∗∗∗σ1A~10σ1B~10-P-1∗∗∗∗σ2A~20σ2B~200-P-1∗∗∗σ3A~1,20σ3B~1,2000-P-1∗∗C~∗0D~∗0000Q-1∗σ4C~10-σ4DfD100000Q-1<0where A~1,2=A~1-A~2, σ1=α-(1-α-)(1-β_),σ2=(1-α-)2(1-β_)β_, σ3=α-β_(1-α-), and σ4=α-(1-α-).Proof.
We partition the proof into two parts.
Firstly, when w(k)=0, we prove that system (18) is stochastically stable.
Consider constructing the following Lyapunov function:(22)Vx~k=x~kTPx~k
Define(23)EΔVx~k+τxkTW2TW2xk-τfkTfk=EVx~k+1∣x~k-Vx~k+τxkTW2TW2xk-τfkTfk=ζkTΦζkwhere(24)ζk=x~kTfkTT,Φ=-P+H00-τI-Γ1TΓ2TP~-1Γ1Γ2Γ1=A∗Tβ1A~1Tβ2A~2Tσ3A~1,2TT,Γ2=G~T000T,G~=G0H=τW1TW1000,W1=W20,P~=P-10000P-10000P-10000P-1,A~1,2=A~1-A~2
According to Schur complement lemma,Φ<0 is equivalent to the following equation:(25)-P+H∗∗∗∗∗0-τI∗∗∗∗A∗G~-P-1∗∗∗σ1A~100-P-1∗∗σ2A~2000-P-1∗σ3A~1,20000-P-1<0
Considering thatf(k)Tf(k)≤x(k)TW2TW2x(k), if there exists τ>0, then the inequalities τx(k)TW2TW2x(k)-τf(k)Tf(k)≥0 hold.
Further, when (25) holds, we have(26)EΔVx~k=EVx~k+1∣x~k-Vx~k≤ζkTΦζk<0
Therefore, there exists a scalarc such that the inequalities E[V(x~(k+1)-V(x~(k))]≤cx~(k)2 hold. According to the method proposed in [18], we can also prove that there exists a constant χ>0 such that the inequalities E∑k=0∞x~(k)2<χsup-τ≤i≤0Eϕ(i)2 hold. This means that system (18) is stochastically stable.
Secondly, when w(k)≠0, we prove that system (18) is strictly (Q,S,R) dissipative.
Under zero initial state conditions, we define(27)EΔVx~k-EekTQek+2ekTSwk+wkTR-aIwk+τxkTW2TW2xk-τfkTfk=ηkTΛ+adiag⁡0,Iηkwhere(28)ηk=x~kTfkTwkTT,Λ=-P+H∗∗0-τI∗-STC~∗0-STD∗-D∗TS-R-G1TG2TG3TP~-1G1G2G3,P~=diag⁡-P-1,-P-1,-P-1,-P-1,Q-1,Q-1,G1=A∗Tσ1A~1Tσ2A~2Tσ3A~1-A~2TC∗Tσ4C~1TTG3=B∗Tσ1B~1Tσ2B~1Tσ3B~1-B~2TD∗Tσ4D~TT,G2=G~T0000T.
According to Schur complement lemma,Λ<0 is equivalent to (21).WhenΛ<0, there must exist a sufficiently small scalar a>0 such that the inequalities (Λ+adiag⁡(0,I))<0 hold. This means that (27) is less than zero.Similar to the proof in the first part, we have(29)EΔVx~k-EekTQek+2ekTSwk+wkTR-aIwk<0Further, summingk in (29) from 0 to t, we can obtain(30)EV(x~T+1-EV(x~0+Ea∑0twkTwk<E-w,e,t.Considering that system (18) is stochastically stable and V(x~(0))=0 and V(x~(T+1))≥0 under zero initial state conditions, we haveE[a∑0tw(k)Tw(k)]<E-(w,e,t). This means that system (18) is strictly (Q,S,R) dissipative.This completes the proof.


### 3.2. Solution of Filter Parameters

Theorem 3.
For given correlated perturbation parameter matricesH^1∈R(n+2p)×r,H^2∈R(n+2p)×r,H3∈Rn×r,H4∈Rq×r,E^1∈Rr×(n+2p),E2∈Rr×(n+2p),E^3∈Rr×(n+2p),E4∈Rr×p and matrices Q=QT<0, R=RT, and S, if there exist constants ε>0,εi>0(i=1,2,3,4,5) and τ>0 and matrices X∈R(n+2p)×(n+2p), Z∈R(n+2p)×(n+2p), A∧f∈R(n+2p)×(n+2p), B∧f∈R(n+2p)×p, C∧f∈Rq×(n+2p), and D∧f∈Rq×p, the following LMI holds:(31)φ1φ2Tφ2φ3<0Then system (18) is stochastically stable and strictly (Q,S,R) dissipative, where(32)φ1=Ξ1Ξ2TΞ2Ξ3,φ2=O1O2O3O4φ3=diag⁡-ε1I,-ε1I,-ε1I,-ε1I,-ε2I,-ε2I,-ε3I,-ε3I,-ε4I,-ε4I,-ε5I,-ε5IΞ1=Ξ11Ξ12TΞ12Ξ13,Ξ11=-X+τW1TW1+εδ2I-Z+τW1TW1+εδ2I0-Z+τW1TW1+εδ2I-Z+τW1TW1+εδ2I000-τIΞ13=M1∗0M2∗M3∗Ξ12=-STL-+STDfC--STL-+STDfC-+STC^f0XA-d+B^fC-dXA-d+B^fC-d+A^fXGZA-dZA-dZGσ1XA-1+B^fC-1σ1XA-1+B^fC-10σ1ZA-1σ1ZA-10σ2XA-2σ2XA-20σ2ZA-2σ2ZA-20σ3XA-1,2+B^fC-1σ3XA-1,2+B^fC-10σ3ZA-1,2σ3ZA-1,20L_-DfC-dL_-DfC-d-C^f0-σ4DfC-1-σ4DfC-10M1∗=-D~d∗TS-STD~d∗T-R+εδ2,D~d∗=L2-a-D^fDM2∗T=M22TZB-dTσ1M23Tσ1ZB-1Tσ2XB-2Tσ2XB-2Tσ3M24Tσ3ZB-1,2TL2-DfDa_T-σ4DfDTM22=XB-d+B^fDa_,M23=XB-1+B^fD,M24=XB-1,2+B^fDM3∗=diag⁡M11,M11,M11,M11,M12M11=-X-Z-Z-Z,M12=Q-100Q-1Ξ2=Ξ21Ξ22Ξ21=000UTDfdTSα-M1TX+α-M11TB^fT+δ-M2TXα-M1TZ+δ-M2TZσ1M1TX+σ1M11TB^fTσ1M1TZ0000000000000000000α-DTDfdTSα-N1TX+a-DTB^fT+δ-N2TXα-N1TZσ1N1TX+σ1DTB^fTσ1N1TZΞ22=σ2M2TXσ2M2TZσ3M1TX+σ3M11TB^fT-σ3M2TXσ3M1TZ-σ3M2TZ-α-UTDfT-σ4UTDfT000000000000σ2N2TX0σ3N1TX+σ3DTB^fT-σ3N2TXσ3N1TZ-α-DTDfT-σ4DTDfTM1=000C00C00,M2=000C00000,N1=0DD,N2=0D0,U=C00Ξ3=-εI0000-εI0000-εI0000-εI,A-1=000C-I0C-I0,B-1=0DD,B-2=0D0,C-1=C-I0,A-2=000C-I0000,A-1,2=A-1-A-2,B-1,2=B-1-B-2A_d=A0+α-A-1+δ-A-2,Bd_=B0+α-B-1+δ-B-2,C_d=C0+α-C-1,δ-=1-α-β_O1=0000H^2T0000ε1E2C-dε1E2C-d0ε1·α-E2D00000000000σ1H^2T00ε1E2C-1ε1E2C-10ε1E2D000000000H^1T00000ε2E^10000000,O2=00000000000000ε1·α-E2M1100ε1·α-E2D0σ3H^2T000000000000ε1E2M1100ε1E2D000000000000000000O3=000H^3TS000000ε3E^30000000000H4TS00000ε4E4C-dε4E4C-d0ε4E4Da-00000000000000ε5E4C-1ε5E4C-10ε5E4D00000,O4=000-H^3T00000000000000000-H4T0000000000ε4E4U00ε4E4D0000-σ4H4T000000000ε5E4U00ε5E4DIf (31) has a solution, then the parameters of the nonfragile dissipative filter can be calculated as follows:(33)Af=W-1A∧fZ-1V-T,Bf=W-1B∧f,Cf=C∧fZ-1V-T,Df=DfH1=W-1H∧1,E1=E∧1Z-1V-T,H2=W-1H∧2,E3=E∧3Z-1V-Twhere W and V are nonsingular constant matrices that meet WVT=I-XZ-1.Proof.
Define(34)P=XWWT-WTZ-1V-T,Ω1=M10BfdM110,Ω2=M2000,Γ1=N1BfdD,Γ2=N20Γ3=-DfdU0,Γ4=-DfdD,and then (21) is equivalent to the following LMI:(35)Σ+0∗∗∗∗∗∗∗∗00∗∗∗∗∗∗∗-STΓ3ΔkI0-α-Γ4TΔkI-α-Γ4ΔkI∗∗∗∗∗∗α-Ω1+δ-Ω2ΔkI0α-Γ1+δ-Γ2ΔkI0∗∗∗∗∗σ1Ω1ΔkI0σ1Γ1ΔkI00∗∗∗∗σ2Ω2ΔkI0σ2Γ2ΔkI000∗∗∗σ3Ω1-Ω2ΔkI0σ3Γ1-Γ2ΔkI0000∗∗α-Γ30α-Γ400000∗σ4Γ30σ4Γ4000000<0where(36)Σ=Σ11Σ12TΣ12Σ22,Σ11=-P+τHTH+εδ2∗∗0-τI∗-STC~d∗0-D~d∗TS-STD~d∗-R+εδ2Σ22=diag⁡-P-1,-P-1,-P-1,-P-1,Q-1,Q-1Σ12T=A~d∗Tσ1A~1dTσ2A~2dTσ3A~1,2dTC~d∗Tσ4C~1dTG~T00000B~d∗Tσ1B~1dTσ2B~2dTσ3B~1,2dTD~d∗T-σ4DfD1dTA~d∗=Ad_0BfdCd_Afd,B~d∗=B-d0BfdDd_,C~d∗=L_-DfdCd_-Cfd,A~2d=A-2000,A~1d=A-10BfdC-10,B~1d=B-1BfdD,C~1d=-DfdC-10,A~1,2d=A~1d-A~2dB~1,2d=B~1d-B~2d,B~2d=B-20.
Equation (35) can be simplified as Σ+HM+MTHT<0, where(37)H=000000-STΓ30-a-Γ4TI-a-Γ4a-Ω1+δ-Ω20a-Γ1+δ-Γ2σ1Ω10σ1Γ1σ2Ω20σ2Γ2σ3Ω1-Ω20σ3Γ1-Γ2a-Γ30a-Γ4σ4Γ30σ4Γ4,M=ΔkI000000000ΔkI000000000ΔkI000000
Considering thatΔk2<δ2, based on Schur complement Lemma, we have(38)ψ1ψ2Tψ2ψ3<0where(39)ψ1=∏1∏3T∏3∏2,∏1=-P+τHTH+εδ2∗∗0-τI∗-STC~d∗0-D~d∗TS-STD~d∗-R+εδ2,∏2=diag⁡-P-1,-P-1,-P-1,-P-1,Q-1,Q-1,∏3T=A~d∗Tσ1A~1dTσ2A~2dTσ3A~1,2dTC~d∗Tσ4C~1dTG~T00000B~d∗Tσ1B~1dTσ2B~2dTσ3B~1,2dTD~d∗T-σ4DfD1dTψ2=00-Γ3TSα-Ω1T+δ-Ω2Tσ1Ω1Tσ2Ω2Tσ3Ω1-Ω2α-Γ3Tσ4Γ3T00000000000-α-Γ4TSα-Γ1T+δ-Γ2Tσ1Γ1Tσ2Γ2Tσ3Γ1-Γ2α-Γ4Tσ4Γ4T,ψ3=-εI000-εI000-εI
LetΣ1=XIVT0 and Σ2=IZ-10VT; (38) can be simplified as (31) with the following steps:
Step 1.Premultiply (38) with diag{Σ2T,I,I,Σ1T,Σ1T,Σ1T,Σ1T,I,I,I,I}.
Step 2.Postmultiply diag{Σ2,I,I,Σ1,Σ1,Σ1,Σ1,I,I,I,I} on the basis of Step 1.
Step 3.Premultiply diag{I,Z,I,I,I,Z,I,Z,I,Z,I,Z,I,I,I,I,I,I} on the basis of Step 2.
Step 4.Postmultiply diag{I,Z,I,I,I,Z,I,Z,I,Z,I,Z,I,I,I,I,I,I} on the basis of Step 3.
Step 5.Separate certainty with uncertainty in Step 4, and repeatedly utilize Lemma 1 and Schur complement Lemma.
This completes the proof.Remark 1.
The solution method of linear inequality (31) is general and will bring less conservatism. We can always get the solutions.Remark 2.
According to Theorem3, the filter parameters of the system can be obtained by the feasible solution of linear inequality (31). Random delays and packet losses are assumed to occur simultaneously in the networked systems. Compared with the results of considering only random time delays or packet losses, the derivation process is more complicated due to the increase of random variables in this paper.


## 3.1. Dissipativity Analysis

Lemma 1 (see [20]).
Given matricesN=NT, H and E are of appropriate dimensions, and matrix F meets FTF≤I such that N+HFE+ETFTHT<0. If there exists a positive ε-, then the sufficient and necessary condition of N+HFE+ETFTHT<0 changes into N+ε-HHT+ε--1ETE<0.
The detailed proof of Lemma1 can be found in [20]; it is omitted here.Theorem 2.
For given matricesQ=QT<0, R=RT, and S, assuming that Afd,BfdCfd,Dfd are known, if there exists symmetric positive definite matrix P and constant τ>0 such that (21) holds, then system (18) is stochastically stable and strictly (Q,S,R) dissipative.(21)-P+H0∗∗∗∗∗∗∗0-τI∗∗∗∗∗∗∗-STC∗0-D∗TS-STD∗-R∗∗∗∗∗∗A∗G~B∗-P-1∗∗∗∗∗σ1A~10σ1B~10-P-1∗∗∗∗σ2A~20σ2B~200-P-1∗∗∗σ3A~1,20σ3B~1,2000-P-1∗∗C~∗0D~∗0000Q-1∗σ4C~10-σ4DfD100000Q-1<0where A~1,2=A~1-A~2, σ1=α-(1-α-)(1-β_),σ2=(1-α-)2(1-β_)β_, σ3=α-β_(1-α-), and σ4=α-(1-α-).Proof.
We partition the proof into two parts.
Firstly, when w(k)=0, we prove that system (18) is stochastically stable.
Consider constructing the following Lyapunov function:(22)Vx~k=x~kTPx~k
Define(23)EΔVx~k+τxkTW2TW2xk-τfkTfk=EVx~k+1∣x~k-Vx~k+τxkTW2TW2xk-τfkTfk=ζkTΦζkwhere(24)ζk=x~kTfkTT,Φ=-P+H00-τI-Γ1TΓ2TP~-1Γ1Γ2Γ1=A∗Tβ1A~1Tβ2A~2Tσ3A~1,2TT,Γ2=G~T000T,G~=G0H=τW1TW1000,W1=W20,P~=P-10000P-10000P-10000P-1,A~1,2=A~1-A~2
According to Schur complement lemma,Φ<0 is equivalent to the following equation:(25)-P+H∗∗∗∗∗0-τI∗∗∗∗A∗G~-P-1∗∗∗σ1A~100-P-1∗∗σ2A~2000-P-1∗σ3A~1,20000-P-1<0
Considering thatf(k)Tf(k)≤x(k)TW2TW2x(k), if there exists τ>0, then the inequalities τx(k)TW2TW2x(k)-τf(k)Tf(k)≥0 hold.
Further, when (25) holds, we have(26)EΔVx~k=EVx~k+1∣x~k-Vx~k≤ζkTΦζk<0
Therefore, there exists a scalarc such that the inequalities E[V(x~(k+1)-V(x~(k))]≤cx~(k)2 hold. According to the method proposed in [18], we can also prove that there exists a constant χ>0 such that the inequalities E∑k=0∞x~(k)2<χsup-τ≤i≤0Eϕ(i)2 hold. This means that system (18) is stochastically stable.
Secondly, when w(k)≠0, we prove that system (18) is strictly (Q,S,R) dissipative.
Under zero initial state conditions, we define(27)EΔVx~k-EekTQek+2ekTSwk+wkTR-aIwk+τxkTW2TW2xk-τfkTfk=ηkTΛ+adiag⁡0,Iηkwhere(28)ηk=x~kTfkTwkTT,Λ=-P+H∗∗0-τI∗-STC~∗0-STD∗-D∗TS-R-G1TG2TG3TP~-1G1G2G3,P~=diag⁡-P-1,-P-1,-P-1,-P-1,Q-1,Q-1,G1=A∗Tσ1A~1Tσ2A~2Tσ3A~1-A~2TC∗Tσ4C~1TTG3=B∗Tσ1B~1Tσ2B~1Tσ3B~1-B~2TD∗Tσ4D~TT,G2=G~T0000T.
According to Schur complement lemma,Λ<0 is equivalent to (21).WhenΛ<0, there must exist a sufficiently small scalar a>0 such that the inequalities (Λ+adiag⁡(0,I))<0 hold. This means that (27) is less than zero.Similar to the proof in the first part, we have(29)EΔVx~k-EekTQek+2ekTSwk+wkTR-aIwk<0Further, summingk in (29) from 0 to t, we can obtain(30)EV(x~T+1-EV(x~0+Ea∑0twkTwk<E-w,e,t.Considering that system (18) is stochastically stable and V(x~(0))=0 and V(x~(T+1))≥0 under zero initial state conditions, we haveE[a∑0tw(k)Tw(k)]<E-(w,e,t). This means that system (18) is strictly (Q,S,R) dissipative.This completes the proof.


## 3.2. Solution of Filter Parameters

Theorem 3.
For given correlated perturbation parameter matricesH^1∈R(n+2p)×r,H^2∈R(n+2p)×r,H3∈Rn×r,H4∈Rq×r,E^1∈Rr×(n+2p),E2∈Rr×(n+2p),E^3∈Rr×(n+2p),E4∈Rr×p and matrices Q=QT<0, R=RT, and S, if there exist constants ε>0,εi>0(i=1,2,3,4,5) and τ>0 and matrices X∈R(n+2p)×(n+2p), Z∈R(n+2p)×(n+2p), A∧f∈R(n+2p)×(n+2p), B∧f∈R(n+2p)×p, C∧f∈Rq×(n+2p), and D∧f∈Rq×p, the following LMI holds:(31)φ1φ2Tφ2φ3<0Then system (18) is stochastically stable and strictly (Q,S,R) dissipative, where(32)φ1=Ξ1Ξ2TΞ2Ξ3,φ2=O1O2O3O4φ3=diag⁡-ε1I,-ε1I,-ε1I,-ε1I,-ε2I,-ε2I,-ε3I,-ε3I,-ε4I,-ε4I,-ε5I,-ε5IΞ1=Ξ11Ξ12TΞ12Ξ13,Ξ11=-X+τW1TW1+εδ2I-Z+τW1TW1+εδ2I0-Z+τW1TW1+εδ2I-Z+τW1TW1+εδ2I000-τIΞ13=M1∗0M2∗M3∗Ξ12=-STL-+STDfC--STL-+STDfC-+STC^f0XA-d+B^fC-dXA-d+B^fC-d+A^fXGZA-dZA-dZGσ1XA-1+B^fC-1σ1XA-1+B^fC-10σ1ZA-1σ1ZA-10σ2XA-2σ2XA-20σ2ZA-2σ2ZA-20σ3XA-1,2+B^fC-1σ3XA-1,2+B^fC-10σ3ZA-1,2σ3ZA-1,20L_-DfC-dL_-DfC-d-C^f0-σ4DfC-1-σ4DfC-10M1∗=-D~d∗TS-STD~d∗T-R+εδ2,D~d∗=L2-a-D^fDM2∗T=M22TZB-dTσ1M23Tσ1ZB-1Tσ2XB-2Tσ2XB-2Tσ3M24Tσ3ZB-1,2TL2-DfDa_T-σ4DfDTM22=XB-d+B^fDa_,M23=XB-1+B^fD,M24=XB-1,2+B^fDM3∗=diag⁡M11,M11,M11,M11,M12M11=-X-Z-Z-Z,M12=Q-100Q-1Ξ2=Ξ21Ξ22Ξ21=000UTDfdTSα-M1TX+α-M11TB^fT+δ-M2TXα-M1TZ+δ-M2TZσ1M1TX+σ1M11TB^fTσ1M1TZ0000000000000000000α-DTDfdTSα-N1TX+a-DTB^fT+δ-N2TXα-N1TZσ1N1TX+σ1DTB^fTσ1N1TZΞ22=σ2M2TXσ2M2TZσ3M1TX+σ3M11TB^fT-σ3M2TXσ3M1TZ-σ3M2TZ-α-UTDfT-σ4UTDfT000000000000σ2N2TX0σ3N1TX+σ3DTB^fT-σ3N2TXσ3N1TZ-α-DTDfT-σ4DTDfTM1=000C00C00,M2=000C00000,N1=0DD,N2=0D0,U=C00Ξ3=-εI0000-εI0000-εI0000-εI,A-1=000C-I0C-I0,B-1=0DD,B-2=0D0,C-1=C-I0,A-2=000C-I0000,A-1,2=A-1-A-2,B-1,2=B-1-B-2A_d=A0+α-A-1+δ-A-2,Bd_=B0+α-B-1+δ-B-2,C_d=C0+α-C-1,δ-=1-α-β_O1=0000H^2T0000ε1E2C-dε1E2C-d0ε1·α-E2D00000000000σ1H^2T00ε1E2C-1ε1E2C-10ε1E2D000000000H^1T00000ε2E^10000000,O2=00000000000000ε1·α-E2M1100ε1·α-E2D0σ3H^2T000000000000ε1E2M1100ε1E2D000000000000000000O3=000H^3TS000000ε3E^30000000000H4TS00000ε4E4C-dε4E4C-d0ε4E4Da-00000000000000ε5E4C-1ε5E4C-10ε5E4D00000,O4=000-H^3T00000000000000000-H4T0000000000ε4E4U00ε4E4D0000-σ4H4T000000000ε5E4U00ε5E4DIf (31) has a solution, then the parameters of the nonfragile dissipative filter can be calculated as follows:(33)Af=W-1A∧fZ-1V-T,Bf=W-1B∧f,Cf=C∧fZ-1V-T,Df=DfH1=W-1H∧1,E1=E∧1Z-1V-T,H2=W-1H∧2,E3=E∧3Z-1V-Twhere W and V are nonsingular constant matrices that meet WVT=I-XZ-1.Proof.
Define(34)P=XWWT-WTZ-1V-T,Ω1=M10BfdM110,Ω2=M2000,Γ1=N1BfdD,Γ2=N20Γ3=-DfdU0,Γ4=-DfdD,and then (21) is equivalent to the following LMI:(35)Σ+0∗∗∗∗∗∗∗∗00∗∗∗∗∗∗∗-STΓ3ΔkI0-α-Γ4TΔkI-α-Γ4ΔkI∗∗∗∗∗∗α-Ω1+δ-Ω2ΔkI0α-Γ1+δ-Γ2ΔkI0∗∗∗∗∗σ1Ω1ΔkI0σ1Γ1ΔkI00∗∗∗∗σ2Ω2ΔkI0σ2Γ2ΔkI000∗∗∗σ3Ω1-Ω2ΔkI0σ3Γ1-Γ2ΔkI0000∗∗α-Γ30α-Γ400000∗σ4Γ30σ4Γ4000000<0where(36)Σ=Σ11Σ12TΣ12Σ22,Σ11=-P+τHTH+εδ2∗∗0-τI∗-STC~d∗0-D~d∗TS-STD~d∗-R+εδ2Σ22=diag⁡-P-1,-P-1,-P-1,-P-1,Q-1,Q-1Σ12T=A~d∗Tσ1A~1dTσ2A~2dTσ3A~1,2dTC~d∗Tσ4C~1dTG~T00000B~d∗Tσ1B~1dTσ2B~2dTσ3B~1,2dTD~d∗T-σ4DfD1dTA~d∗=Ad_0BfdCd_Afd,B~d∗=B-d0BfdDd_,C~d∗=L_-DfdCd_-Cfd,A~2d=A-2000,A~1d=A-10BfdC-10,B~1d=B-1BfdD,C~1d=-DfdC-10,A~1,2d=A~1d-A~2dB~1,2d=B~1d-B~2d,B~2d=B-20.
Equation (35) can be simplified as Σ+HM+MTHT<0, where(37)H=000000-STΓ30-a-Γ4TI-a-Γ4a-Ω1+δ-Ω20a-Γ1+δ-Γ2σ1Ω10σ1Γ1σ2Ω20σ2Γ2σ3Ω1-Ω20σ3Γ1-Γ2a-Γ30a-Γ4σ4Γ30σ4Γ4,M=ΔkI000000000ΔkI000000000ΔkI000000
Considering thatΔk2<δ2, based on Schur complement Lemma, we have(38)ψ1ψ2Tψ2ψ3<0where(39)ψ1=∏1∏3T∏3∏2,∏1=-P+τHTH+εδ2∗∗0-τI∗-STC~d∗0-D~d∗TS-STD~d∗-R+εδ2,∏2=diag⁡-P-1,-P-1,-P-1,-P-1,Q-1,Q-1,∏3T=A~d∗Tσ1A~1dTσ2A~2dTσ3A~1,2dTC~d∗Tσ4C~1dTG~T00000B~d∗Tσ1B~1dTσ2B~2dTσ3B~1,2dTD~d∗T-σ4DfD1dTψ2=00-Γ3TSα-Ω1T+δ-Ω2Tσ1Ω1Tσ2Ω2Tσ3Ω1-Ω2α-Γ3Tσ4Γ3T00000000000-α-Γ4TSα-Γ1T+δ-Γ2Tσ1Γ1Tσ2Γ2Tσ3Γ1-Γ2α-Γ4Tσ4Γ4T,ψ3=-εI000-εI000-εI
LetΣ1=XIVT0 and Σ2=IZ-10VT; (38) can be simplified as (31) with the following steps:
Step 1.Premultiply (38) with diag{Σ2T,I,I,Σ1T,Σ1T,Σ1T,Σ1T,I,I,I,I}.
Step 2.Postmultiply diag{Σ2,I,I,Σ1,Σ1,Σ1,Σ1,I,I,I,I} on the basis of Step 1.
Step 3.Premultiply diag{I,Z,I,I,I,Z,I,Z,I,Z,I,Z,I,I,I,I,I,I} on the basis of Step 2.
Step 4.Postmultiply diag{I,Z,I,I,I,Z,I,Z,I,Z,I,Z,I,I,I,I,I,I} on the basis of Step 3.
Step 5.Separate certainty with uncertainty in Step 4, and repeatedly utilize Lemma 1 and Schur complement Lemma.
This completes the proof.Remark 1.
The solution method of linear inequality (31) is general and will bring less conservatism. We can always get the solutions.Remark 2.
According to Theorem3, the filter parameters of the system can be obtained by the feasible solution of linear inequality (31). Random delays and packet losses are assumed to occur simultaneously in the networked systems. Compared with the results of considering only random time delays or packet losses, the derivation process is more complicated due to the increase of random variables in this paper.


## 4. Numerical Simulation and Discussion

To verify the effectiveness of the proposed filter, we adopt the following system parameters given in [18]:(40)A=-0.20.10.40.7-0.10.1-0.3-0.2-0.1,B=-1-1.50.5,D=0.9,C=1-21,L1=-21-2,L2=0H∧1=1000001000001000001000001,E∧1=0.01∗-1000-110001-100101000-1-10100,H∧2=11111,E2=0.02H4=1,E4=0.02H3=10002,E∧3=0.02∗10001-1000-110010-1100010100W2=0.010000.010000.01,fk,xk=0.01sin⁡xk10.01sin⁡xk20.01sin⁡xk3F1k=F3k=sin⁡0.1k00000sin⁡0.1k00000sin⁡0.1k00000sin⁡0.1k00000sin⁡0.1kF2k=F4k=sin⁡0.1kHere, we choose α-=0.7,β-=0.8, and ρ=0.95. Assume that disturbance w(k) is in the form of sin⁡(k)∗e-0.1k. The parameters of dissipative filter are set as Q=-I, S=0.5I, and R=12I. According to Theorem 3, the parameters of traditional quantized dissipative filter without parameter perturbation and the parameters of the nonfragile quantized dissipative filter with parameter perturbation can be obtained by Matlab LMI toolbox. Then, we can obtain the performance metric γ of corresponding dissipative filter based on the formula γ=∑(e(k))/∑(w(k)).The parameters and performance metricγ of the nonfragile quantized dissipative with parameter perturbation filter are listed as follows:(41)Af=-1.3330-1.47330.50600.65550.00090.61610.3589-0.1316-0.3262-0.0005-1.0025-1.77690.3872-0.7236-0.00030.0001-0.10660.07830.16670.0000-0.0388-0.06660.02540.02370.0000,Bf=-0.56880.1140-0.5597-0.2520-131.1128,Cf=-0.1075-0.43790.24800.46290.0002,Df=-0.1801,γ=0.9893.The parameters of traditional quantized dissipative filter without parameter perturbation are listed as follows:(42)Af=-1.0905-1.16450.46710.49630.00130.48570.2565-0.1398-0.1720-0.0008-0.8796-1.38180.2421-0.3477-0.0025-0.0634-0.23730.11960.11120.0000-0.0385-0.06110.02310.0179-0.0000,Bf=-0.55390.1482-0.4579-0.3401-52.8181Cf=-0.0619-0.40090.23500.32710.0005,Df=-0.1726Note that there exists parameter perturbation in real networked systems; that is,ΔAf, ΔBf, ΔCf, and ΔDf are unequal to zero. We can obtain the corresponding performance metric of traditional quantized dissipative filter with parameter perturbation in real networked systems; that is, γ=1.0233. It can be seen that the performance metric of the nonfragile quantized dissipative filter is less than its counterpart in the traditional quantized dissipative filter without parameter perturbation. This means that the nonfragile quantized dissipative filter is of better disturbance rejection capability than the traditional quantized dissipative filter.The curves ofzf(k) estimated by nonfragile quantized dissipative filter and the signal to be estimated z(k) are shown in Figure 1, respectively. We can see that z(k) will converge to zf(k) within finite times k. This means that the designed nonfragile quantized dissipative filter is effective in real networked systems with time delays and packet losses.Figure 1
The curves ofzf(k) and z(k).We setρ=0.95; considering the existence of different probability of time delays and packet losses in networked systems, the performance metric γ of nonfragile quantized dissipative filter under the same quantization density is shown in Table 1. We can see that the nonfragile quantized dissipative filter is of better disturbance rejection capability than traditional quantized dissipative filter under given probability of time delays and packet losses. Furthermore, the performance metric γ of nonfragile quantized dissipative filter will become larger and the estimation result of nonfragile quantized dissipative filter will become worse with the increase of the probability of time delays and packet losses. Therefore, the probability of time delays and packet losses may influence system performance.Table 1
Performance metricγ under different PTD and PPL.
α - β - PTD PPL γ of TQDF γ of NFQDF 0.6 0.7 0.112 0.288 1.1511 1.1101 0.7 0.8 0.072 0.228 1.0233 0.9893 0.8 0.9 0.036 0.164 1.0013 0.9375 Note that PTD, PPL, TQDF, and NFQDF are the abbreviations of probability of time delays, probability of the packet loss, traditional quantized dissipative filter, and nonfragile quantized dissipative filter, respectively.We setα-=0.7,β-=0.8; considering three different quantization densities, the performance metric γ of nonfragile quantized dissipative filter under the same probability of time delays and packet losses is shown in Table 2. We can see that the nonfragile quantized dissipative filter is of better disturbance rejection capability than traditional quantized dissipative filter under given quantization density. Furthermore, the performance metric γ of nonfragile quantized dissipative filter will become larger and the estimation result of nonfragile quantized dissipative filter will become worse with the decrease of quantization density. Therefore, the quantization density may also influence system performance.Table 2
Performance metricγ under different QD.
QD RQE γ of TQDF γ of NFQDF 0.75 [-0.1428,0.1428] 1.3401 1.2639 0.85 [-0.0811,0.0811] 1.1243 1.0623 0.95 [-0.0256,0.0256] 1.0233 0.9893 Note that QD, RQE, TQDF, and NFQDF are the abbreviations of quantization density, range of quantization error, traditional quantized dissipative filter, and nonfragile quantized dissipative filter, respectively.


## 5. Conclusion

In this paper, we investigate the problem of nonfragile quantized dissipative filter with parameter perturbation in nonlinear networked systems. Compared with the designed filter in [12–20], the designed nonfragile quantized dissipative filter in this paper considers the coexistence of time delays, packet losses, and quantization error in the real networked systems. The designed filter decreases the conservation problem by considering filter parameter perturbation. We obtain the sufficient conditions for the existence of nonfragile quantized dissipative filter by constructing appropriate Lyapunov function and utilizing linear matrix inequality method. Furthermore, we obtain the expression of the filter parameters. The simulation results show that the designed nonfragile quantized dissipative filter is better than traditional quantized dissipative filter without parameter perturbation.



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*Source: 5659873-2019-03-03.xml*