# ResilientL2-L∞ Filtering of Uncertain Markovian Jumping Systems within the Finite-Time Interval

Authors: Shuping He
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper studies the resilientL2-L∞ filtering problem for a class of uncertain Markovian jumping systems within the finite-time interval. The objective is to design such a resilient filter that the finite-time L2-L∞ gain from the unknown input to an estimation error is minimized or guaranteed to be less than or equal to a prescribed value. Based on the selected Lyapunov-Krasovskii functional, sufficient conditions are obtained for the existence of the desired resilient L2-L∞ filter which also guarantees the stochastic finite-time boundedness of the filtering error dynamic systems. In terms of linear matrix inequalities (LMIs) techniques, the sufficient condition on the existence of finite-time resilient L2-L∞ filter is presented and proved. The filter matrices can be solved directly by using the existing LMIs optimization techniques. A numerical example is given at last to illustrate the effectiveness of the proposed approach.


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## Body

## 1. Introduction

More recently, the finite-time stability and control problems have received great attention in the literature; see [1–6]. Compared with the Lyapunov stable dynamical systems, a finite-time stable dynamical system does not require the steady-state behavior of control dynamics over an infinite-time interval and the asymptotic pattern of system trajectories. The main attention may be related to the transient characteristics of the dynamical systems over a fixed finite-time interval, for instance, keeping the acceptable values in a prescribed bound in the presence of saturations. However, more details are related to the stability and control problems of various dynamic systems, and very few reports in the literature consider the filtering problems.Since the Kalman filtering theory [7] has been introduced in the early 1960s, the filtering problem has been extensively investigated. In the filtering scheme, its objective is to estimate the unavailable state variables (or a linear combination of the states) of a given system. During the past decades, many filtering schemes have been developed, such as Kalman filtering [8], H∞ filtering [9], reduced-order H∞ filtering [10], and L2-L∞ filtering [11]. Then, extension of this effort to the problem of resilient Kalman filtering with respect to estimator gain perturbations was considered in [12]. And the resilient H∞ filtering [13] was also raised. Among the filtering schemes, the resilient L2-L∞ filtering was not considered. In practical engineering applications, the peak values of filtering error should always be considered. Compared with the H∞ filtering scheme, the external disturbances are both assumed to be energy bounded; but L2-L∞ filtering setting requires the mapping from the external disturbances to the filtering error is minimized or no larger than some prescribed level in terms of the L2-L∞ performance norm.In this paper, we have studied the resilient finite-timeL2-L∞ filtering problem for uncertain Markouvian Jumping Systems (MJSs). Firstly, the augmented filtering error dynamic system is constructed based on the state estimated filter with resilient filtering parameters. Secondly, a sufficient condition is established on the existence of the robust finite-time filter such that the filtering error dynamic MJSs are finite-time bounded and satisfy a prescribed level of L2-L∞ disturbance attenuation with the finite-time interval. And the design criterion is presented by means of LMIs techniques. Subsequently, the robust finite-time L2-L∞ filter matrices can be solved directly by using the existing LMIs optimization algorithms. In order to illustrate the proposed result, a numerical example is given at last.Let us introduce some notations. The symbolsℜn and ℜn×m stand for an n-dimensional Euclidean space and the set of all n×m real matrices, respectively, AT and A-1 denote the matrix transpose and matrix inverse, diag{AB} represents the block-diagonal matrix of A and B, σmax(A) and σmin(A), respectively, denote the maximal and minimal eigenvalues of a real matrix A, ∥*∥ denotes the Euclidean norm of vectors, E{*} denotes the mathematics statistical expectation of the stochastic process or vector, P>0 stands for a positive-definite matrix, I is the unit matrix with appropriate dimensions, 0 is the zero matrix with appropriate dimensions, and * means the symmetric terms in a symmetric matrix.


## 2. Problem Formulation

Given a probability space(Ω,F,Pr) where Ω is the sample space, F is the algebra of events, and Pr is the probability measure defined on F. Let us consider a class of linear uncertain MJSs defined in the probability space (Ω,F,Pr) and described by the following differential equations:
(1)x˙(t)=[A(rt)+ΔA(rt)]x(t)+B(rt)w(t),y(t)=[C(rt)+ΔC(rt)]x(t)+D(rt)w(t),z(t)=E(rt)x(t),x(t)=x0,rt=r0,t=0,
where x(t)∈ℜn is the state, y(t)∈ℜl is the measured output, w(t)∈L2m[0+∞) is the unknown input, z(t)∈ℜq is the controlled output, and x0 and r0 are, respectively, the initial states and mode. A(rt), B(rt), C(rt), D(rt), and E(rt) are known mode-dependent constant matrices with appropriate dimensions. The jump parameter rt represents a continuous-time discrete-state Markov stochastic process taking values on a finite set M={1,2,…,N} with transition rate matrix Π={πij}, i,j∈M, and has the following transition probability from mode i at time t to mode j at time t+Δt as
(2)Pij=Pr{rt+Δt=j∣rt=i}={πijΔt+o(Δt),i≠j,1+πiiΔt+o(Δt),i=j,
where Δt>0 and limΔt→0(o(Δt)/Δt)→0.In this relation,πij≥0 is the transition probability rates from mode i at time t to mode j(i≠j) at time t+Δt, and
(3)∑j=1,j≠iNπij=-πiifori,j∈M,i≠j.For presentation convenience, we denoteA(rt), B(rt), C(rt), D(rt), E(rt), ΔA(rt), and ΔC(rt) as Ai, Bi, Ci, Di, Ei, ΔAi, and ΔCi, respectively.And the matrices with the symbolΔ(*) are considered as the uncertain matrices satisfying the following conditions:
(4)[ΔAiΔCi]=[M1iM2i]Γi(t)N1i,
where M1i, M2i, and N1i are known mode-dependent matrices with appropriate dimensions, and Γi(t) is the time-varying unknown matrix function with Lebesgue norm measurable elements satisfying ΓiT(t)Γi(t)≤I.Remark 1.
It is always impossible to obtain the exact mathematical model of practical dynamics due to the complexity process, the environmental noises, and the difficulties of measuring various and uncertain parameters, and so forth; thus, the model of practical dynamics to be controlled almost contains some types of uncertainties. In general, the uncertaintiesΔ(*) in (1) satisfying the restraining conditions (4) and ΓiT(t)Γi(t)≤I are said to be admissible. The unknown mode-dependent matrix Γi(t) can also be allowed to be state dependent; that is, Γi(t)=Γi(t,x(t)), as long as ∥Γi(t,x(t))∥≤1 is satisfied.We now consider the following resilient filter:(5)x^˙(t)=(Afi+ΔAfi)x^(t)+Bfiy(t),z^(t)=(Cfi+ΔCfi)x^(t),x^(t)=x^0,rt=r0,t=0,
where x^(t)∈ℜn is the filter state, z^(t)∈ℜq is the filer output, x^0 is the initial estimation states, and the mode-dependent matrices Afi, Bfi, and Cfi are unknown filter parameters to be designed for each value i∈M. ΔAfi and ΔCfi are uncertain filter parameter matrices satisfying the following conditions:
(6)[ΔAfiΔCfi]=[M3iM4i]Γfi(t)N2i,
where M3i, M4i, N2i, and Γfi(t) are defined similarly as (4).The objective of this paper is to design the resilientL2-L∞ filter of uncertain MJSs in (1) and obtain an estimate z^(t) of the signal z(t) such that the defined guaranteed performance criteria are minimized in an L2-L∞ estimation error sense. Define e(t)=x(t)-x^(t) and r(t)=z(t)-z^(t), such that the filtering error dynamic MJSs are given by
(7)e˙(t)=(Afi+ΔAfi)e(t)+[Ai+ΔAi-(Afi+ΔAfi)-Bfi(Ci+ΔCi)]x(t)+(Bi-BfiDi)w(t),r(t)=(Cfi+ΔCfi)e(t)+[Ei-(Cfi+ΔCfi)]x(t).Letξ(t)=[x(t)e(t)], and we have
(8)ξ˙(t)=A-iξ(t)+B-iw(t),r(t)=C-iξ(t),
where
(9)A-i=[Ai+ΔAi0Ai+ΔAi-(Afi+ΔAfi)-Bfi(Ci+ΔCi)Afi+ΔAfi],B-i=[BiBi-BfiDi],C-i=[Ei-(Cfi+ΔCfi)Cfi+ΔCfi].The external disturbancew(t) is varying and satisfies the constraint condition with respect to the finite-time interval [0T] as follows:
(10)∫0TwT(t)w(t)dt≤W,
where W is a positive scalar.Definition 2.
Given a time-constantT>0, the filtering error dynamic MJSs (8) (setting w(t)=0) are said to be stochastically finite-time stable (FTS) with respect to (c1c2TR~i), if
(11)E{ξT(0)R~iξ(0)}≤c1⟹E{ξT(t)R~iξ(t)}<c2,t∈[0T],
where c1>0, c2>0, R~i=diag[RiRi]>0, and Ri is the weight coefficient matrix.Definition 3.
Given a time-constantT>0, the filtering error dynamic MJSs (8) are stochastically finite-time bounded (FTB) with respect to (c1c2TR~iW), wherein c1>0, c2>0, R~i>0, if condition (10) holds.Definition 4.
For the filtering error dynamic MJSs (8), if there exist filter parameters Afi, Bfi, and Cfi, as well as a positive scalar γ, such that the filtering error dynamic MJSs (8) are stochastically FTB and under the zero-valued initial condition, the system output error satisfies the following cost function inequality for T>0 with attenuation γ>0 and for all admissible w(t) with the constraint condition (10):
(12)J=∥r(t)∥E∞T-γ∥w(t)∥2T<0,
where ∥r(t)∥E∞T = supt∈[0T]E[∥r(t)∥], ∥w(t)∥2T = ∫0TwT(t)w(t)dt.
Then, the resilient filter (5) is called the stochastic finite-time L2-L∞ filter of the uncertain dynamic MJSs (1) with γ-disturbance attenuation.


## 3. Main Results

In this section, we will study the robust stochastic finite-time resilient filtering problem for the filtering error dynamic MJSs (8) in an L2-L∞ estimation error sense. Before proceeding with the study, the following lemma is needed.Lemma 5 (see [14]).
LetT, M, and N be real matrices with appropriate dimensions. Then, for all time-varying unknown matrix function F(t) satisfying FT(t)F(t)≤I, the following relation holds:
(13)T+MF(t)N+NTFT(t)MT>0,
if and only if there exists a positive scalar α>0, such that
(14)T+α-1MMTαNTN<0.Theorem 6.
For givenT>0, η>0, c1>0, and R~i>0, the filtering error dynamic MJSs (8) are stochastically FTB with respect to (c1c2TR~iW) and have a L2-L∞ performance level γ>0, if there exist positive constants, γ>0, c2>0, and mode-dependent symmetric positive-definite matrices P~i, such that
(15)[P~iA-i+A-iTP~i+∑j=1NπijP~j-ηP~iP~iB-iBiTP~i-e-ηTI]<0,(16)C-iTC-i<γ2P~i,(17)eηTc1σ-P^+Wη(1-e-ηT)<c2σ_P^,
where P^i=R~i-1/2P~iR~i-1/2, σ-P^=maxi∈Mσmax(P^i), and σ_P^=mini∈Mσmin(P^i).Proof.
Let the mode at timet be i; that is, rt=r∈M. Take the stochastic Lyapunov-Krasovskii functional V(ξ(t),rt,t>0):ℜn×M×ℜ+→ℜ+ as
(18)V(ξ(t),i)=ξT(t)P~iξ(t).
Then, we introduce a weak infinitesimal generatorℑ[*] (see [15, 16]), acting on V(ξ(t),i), for all i∈M, which is defined as
(19)ℑV(ξ(t),i)=limΔt→01Δt[E{V(ξ(t+Δt),rt+Δt,t+Δt)∣ξ(t),rt=i(ξ(t+Δt),rt+Δt,t+Δt)}-V(ξ(t),i,t)].
The time derivative ofV(ξ(t),i) along the trajectories of the filtering error dynamic MJSs (8) is given by
(20)ℑV(ξ(t),i)=2ξT(t)P~iξ˙(t)+ξT(t)∑j=1NπijP~jξ(t)=ξT(t)(P~iA-i+A-iTP~i+∑j=1NπijP~j)ξ(t)+2ξT(t)P~iB-iw(t).
Considering theL2-L∞ filtering performance for the dynamic filtering error system (8), we introduce the following cost function by Definition 4 with t≥0:
(21)J(t)=E[ℑV(ξ(t),i)]-ηE[V(ξ(t),i)]-e-ηTwT(t)w(t).
It follows from relation (16) that J(t)<0; that is,
(22)E[ℑV(ξ(t),i)]<ηE[V(ξ(t),i)]+e-ηTwT(t)w(t).
Then, multiplying the previous inequality bye-ηt, we have
(23)E{ℑ[e-ηtV(ξ(t),i)]}<e-η(T+t)wT(t)w(t).
Integrating the above inequality from0 to T, we have
(24)e-ηTE{V(ξ(T),i)}-E{V(ξ(0),r0)}<e-ηT∫0Te-ηswT(s)w(s)ds.
ConsideringV(ξ(0),r0)≥0, as well as the zero initial condition; that is, ξ(0)=0, for t>0, then it follows that
(25)E{V(ξ(T),i)}<∫0Te-ηswT(s)w(s)ds.
Then, it can be verified from the defined Lyapunov-Krasovskii functional that(26)E{ξT(T)P~iξ(T)}=E{V(ξ(T),i)}<∫0Te-ηswT(s)w(s)ds.
By (15) and within the finite-time interval [0T], we can also get
(27)E{rT(T)r(T)}=E{ξT(T)C-iTC-iξ(T)}<γ2E{ξT(T)P~iξ(T)}=γ2E{V(ξ(T),i)}<γ2∫0Te-ηswT(s)w(s)ds<γ2∫0TwT(s)w(s)ds.
Since the previous inequality is always true for anyT>0, the following relation:
(28)supt∈[0T]E[∥r(t)∥]<γ∫0TwT(s)w(s)ds.
It is easy to get the following result:(29)∥r(t)∥E∞T=γ∑i=1Nπi{supt∈[0T]E[∥r(t)∥]}<γ∑i=1Nπi∫0TwT(s)w(s)ds.=γ∫0TwT(s)w(s)ds=γ∥w(t)∥2T.
Therefore, the cost function inequality (10) can be guaranteed, which implies J=∥r(t)∥E∞T-γ∥w(t)∥2T<0.
Denote thatP^i=R~i-1/2P~iR~i-1/2, σ-P^=maxi∈Mσmax(P^i), and σ_P^=mini∈Mσmin(P^i). From equality (24), we have
(30)E{ξT(t)P~iξ(t)}=E{V(ξ(t),i)}<∫0te-ηswT(s)w(s)ds+eηtE{V(ξ(0),r0)}<eηtE{V(ξ(0),r0)}+W∫0te-ηsds<eηtc1σ-P^+Wη(1-e-ηt)≤eηTc1σ-P^+Wη(1-e-ηT).
On the other hand, it results from the stochastic Lyapunov-Krasovskii functional that(31)E{ξT(t)P~iξ(t)}≥σ_P^E{ξT(t)R~iξ(t)}.
Then, we can get(32)E{ξT(t)R~iξ(t)}<eηTc1σ-P^+(W/η)(1-e-ηT)σ_P^.
It implies that for allt∈[0T], we have E{ξT(t)R~iξ(t)}<c2 by condition (17). This completes the proof.Theorem 7.
For givenT>0, η>0, c1>0, and Ri>0, the filtering error dynamic MJSs (8) are stochastically FTB with respect to (c1c2TRiW) with Ri∈ℜn×n>0 and have a prescribed L2-L∞ performance level γ>0, if there exist a set of mode-dependent symmetric positive-definite matrices Pi, a set of mode-dependent matrices Xi, Yi, and a positive scalar σ1 and mode-dependent sequences αi,βi,λi, satisfying the following matrix inequalities for all i∈M:
(33)[Λ1i*PiBiPiM1i0Λ2iΛ3iPiBi-YiDiPiM1i-YiM2iPiM3i**-e-ηTI00***-αiI0****-βiI]<0,(34)[-Pi0-EiT+CfiTN2iT0-Pi-CfiT-N2iT**-γ2I+λiM4iM4iT0***-λiI]<0,(35)Ri<Pi<σ1Ri,(36)eηTc1σ1+Wη(1-e-ηT)<c2,
where Λ1i = PiAi+AiTPi + ∑j=1NπijPj-ηPi + αiN1iTN1i+βiN2iTN2i, Λ2i=PiAi − Xi-YiCi-βiN2iTN2i, Λ2i=Xi+XiT + ∑j=1NπijPj − ηPi + βiN2iTN2i.
Moreover, the suitable filter parameters can be given as(37)Afi=Pi-1Xi,Bfi=Pi-1Yi,Cfi=Cfi.Proof.
For convenience, we setP~i=diag{Pi,Pi}. Then, we can get the following relations according to matrix inequalities (15) and (16):
(38)Πi+ΔΠ1i+ΔΠ2i<0,(39)Σi+ΔΣi<0,
where (40)Πi=[PiAi+AiTPi+∑j=1NπijPj-ηPi*PiBiPiAi-PiAfi-PiBfiCiPiAfi+AfiTPi+∑j=1NπijPj-ηPiPiBi-PiBfiDi**-e-ηTI],Σi=[-Pi0-EiT+CfiT0-Pi-CfiT**-γ2I],ΔΠ1i=[PiΔAi+ΔAiTPi*0PiΔAi-PiBfiΔCi00**0],ΔΠ2i=[0*0-PiΔAfiPiΔAfi+ΔAfiTPi0**0],ΔΣi=[00ΔCfiT00-ΔCfiT**0].
Referring to Lemma5, ΔΠ1i and ΔΠ2i can be presented as the following form:
(41)ΔΠ1i=L1iΓi(t)L2i+L2iTΓiT(t)L1iT<αi-1L1iL1iT+αiL2iTL2i,ΔΠ2i=L3iΓfi(t)L4i+L4iTΓfiT(t)L3iT<βi-1L3iL3iT+βiL4iTL4i,ΔΣi=L5iΓfi(t)L6i+L6iTΓfiT(t)L5iT<λiL5iL5iT+λi-1L6iTL6i,
where L1i=col[PiM1iPiM1i-PiBfiM2i0], L2i=[N1i00],  L3i=col[0PiM3i0],  L4i=[-N2iN2i0],  L5i = col[00M4i], and  L6i=[N2i-N2i0].
Then, inequalities (15) and (16) are equivalent to LMIs (38) and (39) by letting Xi=PiAfi and Yi=PiBfi.
On the other hand, we setR~i=diag[RiRi], and LMI (35) implies that
(42)1<σ_P^=mini∈Mσmin(P^i),σ-P^=maxi∈Mσmax(P~i)<σ1.
Then, recalling condition (17), we can get LMI (36). This completes the proof.To obtain an optimal finite-timeL2-L∞ filtering performance against unknown inputs, uncertainties, and model errors, the attenuation lever γ2 can be reduced to the minimum possible value such that LMIs (33)–(36) are satisfied. The optimization problem can be described as follows:
(43)minPi,Xi,Yi,Cfi,Dfi,αi,βi,λi,ρρs.t.LMIs(33)–(36)withρ=γ2.Remark 8.
In Theorems6 and 7, if c2 is a variable to be solved, then (17) and (36) can be always satisfied as long as c2 is sufficiently large. For these, we can also fix γ and look for the optimal admissible c1 or c2 guaranteeing the stochastic finite-time boundedness of desired filtering error dynamic properties.


## 4. Numeral Examples

Example 9.
Consider a class of MJSs with two operation modes described as follows.
Mode 1:(44)A1=[-21-1-1],B1=[-0.20.1],C1=[10.5],D1=0.1,E1=[0.10.2],M11=[00.2],M21=-0.1,M31=[0.1-0.1],M41=0.2,N11=[0.20.1],N21=[0.1-0.2];
Mode 2:(45)A2=[03-1-2],B2=[0.1-0.2],C2=[11],D2=-0.2,E2=[0.2-0.1],M12=[-0.10.1],M22=0.2,M32=[0.10.2],M42=-0.2,N12=[-0.10.1],N22=[0.10.1].
The mode switching is governed by a Markov chain that has the following transition rate matrix:
(46)Π=[-0.30.30.5-0.5].In this paper, we choose the initial values forW=2, T=4, η=0.25, and Ri=I2. Then, we fix γ=0.8 and look for the optimal admissible c2 with different c1 which can guarantee the stochastic finite-time boundedness of desired filtering error dynamic properties. Figure 1 gives the optimal minimal admissible c2 with different initial upper bound c1.Figure 1
The optimal minimal upper boundc2 with different initial c1.Forc1=1, we solve LMIs (33)–(36) by Theorem 7 and the optimization algorithm (43) and get the following optimal resilient finite-time L2-L∞ filter:
(47)Af1=[-7.7193-3.3359-6.9662-6.6546],Bf1=[3.01896.7316],Cf1=[0.050.1];Af2=[-1.30040.3608-3.0043-9.2166],Bf2=[1.14152.0586],Cf2=[0.1-0.05].
And then, we can also get the attenuation lever as γ=0.0777 and the relevant upper bound c2=592.45.To show the effectiveness of the designed optimal resilient finite-timeL2-L∞ filter, we assume that the unknown inputs are unknown white noise with noise power 0.05 over a finite-time interval t∈[04]. For the selected initial conditions x(0)=[0.5-0.5] and r0=2, the simulation results of the jumping modes and the response of system states are shown in Figures 2, 3, and 4. It is clear from the simulation figures that the estimated states can track the real states smoothly.Figure 2
System response with statex1(t).Figure 3
System response with statex2(t).Figure 4
System response with output errorr(t).


## 5. Conclusion

The resilient finite-timeL2-L∞ filtering problems for a class of stochastic MJSs with uncertain parameters have been studied. By using the Lyapunov-Krasovskii functional approach and LMIs optimization techniques, a sufficient condition is derived such that the filtering dynamic error MJSs are finite-time bounded and satisfy a prescribed level of L2-L∞ disturbance attenuation in a finite time-interval. The robust resilient filter gains can be solved directly by using the existing LMIs. Simulation results illustrate the effectiveness of the proposed a pproach.



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