# WeightedH∞ Filtering for a Class of Switched Linear Systems with Additive Time-Varying Delays

Authors: Li-li Li; Guangling Zhang; Xin Ge; Cui-li Jin
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper is concerned with the problem of weightedH∞ filtering for a class of switched linear systems with two additive time-varying delays, which represent a general class of switched time-delay systems with strong practical background. Combining average dwell time (ADT) technique with piecewise Lyapunov functionals, sufficient conditions are established to guarantee the exponential stability and weighted H∞ performance for the filtering error systems. The parameters of the designed switched filters are obtained by solving linear matrix inequalities (LMIs). A modification of Jensen integral inequality is exploited to derive results with less theoretical conservatism and computational complexity. Finally, two examples are given to demonstrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Switched systems, as an important class of hybrid systems, have been extensively investigated during the last decades [1–4]. This is mainly motivated by the hybrid nature of many physical plants and the better performance under a controller switching strategy. Several methods, which are viewed as variations or generalizations of the Lyapunov stability theory, have been employed to cope with the performance analysis and control synthesis problems for switched systems, such as the common Lyapunov function method, the piecewise Lyapunov function method, the average dwell time (ADT) function method, the multiple Lyapunov function method, and the switched Lyapunov function method. Basic problems and recent progress are referred to [1, 2] and the references therein.Switched systems with time delays in system states, control inputs, or switching signals can be modeled as switched delay systems. Due to the coexistence and interaction of the mode switchings and inherent time delays, the features of switched delay systems are very complicated which may even lead to instability and poor performance. Switched delay systems are also with strong practical background such as temperature control system, networked control systems (NCSs), and power systems [5–8]. That is why switched delay systems have gained growing popularity; following in the wake of this, much important progress has been made on the analysis and synthesis of switched delay systems [9–11], where the ADT approach has been proved to be a flexible and powerful tool for switched delay systems. When external noise signals appear in system models, state variables might not be available accurately which need to be estimated. One of the most effective methods for the state estimation problem is to design a filter, which has been concerned for switched delay systems with different performance indexes such as Kalman filtering, L2–L∞ filtering, H2 filtering, and H∞ filtering. Compared with others, H∞ filtering [12, 13] permits the exogenous noises to be arbitrary with bounded energy or average power and without known precise statistics, and it is also more robust to the uncertainties in the external noise signals and system models. In the H∞ filtering setting, a state estimator is designed to guarantee that the filter error system is stable in the absence of the external noise signals and its H∞ performance from the external noise signals to the estimation error is below a prescribed level of noise attenuation. For some representative works on H∞ filtering for switched delay systems, to name a few, [14] investigated an exponential H∞ filtering for continuous-time switched systems with interval time-varying delay to assure the exponential stability with a weighted H∞ performance for the filtering error system via the free-weighting matrix (FWM) technique. A weighted H∞ filter design procedure was developed by using the FWM technique for continuous-time switched time-varying delay systems to achieve the exponential stability with a weighted H∞ performance for the filtering error system in [15]. Reference [16] designed a full-order switched H∞ filter for a class of uncertain switched neutral systems subject to stochastic disturbance and time-varying delays, which guaranteed the robust mean-square exponential stability with a prescribed weighted H∞ performance. On the basis of a filter with Luenberger observer type and a new integral inequality, [17] dealt with the H∞ filtering problem for a class of switched linear neutral systems with time-varying delays. The ADT approach associated with the piecewise Lyapunov functional technology was employed in all of the above four results, and fast convergence and desirable accuracy were guaranteed by the exponential H∞ filters in terms of reasonable error covariance of the filtering process.On the other hand, among the great number of literatures concerning time delay, it is worth pointing out that [18, 19] proposed the additive time-varying delay as a new type of delay for nonswitched continuous-time systems. After that, additive time delay has attracted much attention in the past few years [20–24]. The significance of the additive delays lies in three aspects. First, the additive delay components may describe delays with sharply different properties in system modeling which might not be suitable to combine them together. As mentioned in [18], delays from sensor to controller and from controller to actuator in NCSs are of the case, where delays may not have identical properties. Second, regarding the sum of all additive delay components as one traditional single delay will be very conservative, it is not reasonable and necessary to treat the sum of maximum of single delays as the maximum of the sum of all delays at the same time. Finally, systems with additive delays are of strong application backgrounds such as remote control and NCSs. For instance, [19] utilized the continuous systems with two additive time-varying delays in state to investigate the sampled-data NCSs with network induced delays and data packet dropouts. Recently, by constructing new Lyapunov functionals to avoid some overly boundings, [23, 24] presented stability criteria with fewer matrix variables and less conservatism for systems with additive time-varying delays in control input and system state, respectively. Nevertheless, switched additive time delays systems have been rarely investigated due to the complexity of mode switching.Inspired by the aforementioned discussion, this paper focuses on the weightedH∞ filtering problem for a class of switched systems with additive time-varying delays. Delays under consideration are additive time-varying in the states. New criteria are presented to guarantee delay-dependent exponential stability and a weighted H∞ performance for the filtering error system under ADT switching signals. Then, by solving the corresponding LMIs, the parameters of the designed switched H∞ filters are obtained for all additive time-varying delays. Combining with a modification of Jensen integral inequality [24] instead of FWM technology, the derived conditions are with less theoretical conservatism and computational complexity. Two examples are provided to illustrate the effectiveness of the proposed method. To the best of our knowledge, little work has been addressed concerning stability analysis for switched systems with additive time-varying delays, not to mention the weighted H∞ filtering problem.The remainder of this paper is organized as follows. The weightedH∞ filtering problem for switched systems with additive time-varying is formulated in Section 2. Section 3 presents delay-dependent sufficient conditions on the existence of weighted H∞ filtering which guarantees exponential stability and H∞ performance of the filtering error system, and a corresponding filter is designed. Section 4 gives two examples. Section 5 concludes this paper.Notations. Some standard notations are used in this paper. Rn denotes the n dimensional Euclidean space; P<0  (≤, >, ≥) represents a real negative (negative-semidefinite, positive, positive-semidefinite) definite matrix P;L2[0,+∞) denotes the space of square integrable vector functions on [0,+∞); the superscript T stands for matrix transposition; λmax⁡(P) presents the maximum eigenvalue of P;I is the identity matrix with compatible dimensions; asterisk * denotes symmetric terms in symmetric term in a symmetric matrix.


## 2. Problem Formulation

Consider the following switched linear systems with two additive time-varying delays:(1)x˙(t)=Aσx(t)+Adσxt-d1t-d2t+Bσω(t),y(t)=Cσx(t)+Cdσxt-d1t-d2t+Dσω(t),z(t)=Eσx(t)+Edσxt-d1t-d2t,x(θ)=φθ,θ∈[-d,0],
where x(t)∈Rn is the state vector, y(t)∈Rm is the measured output, z(t)∈Rp is the signal to be estimated, ω(t)∈Rq is the disturbance input which belongs to L2[0,+∞), and φ(θ) is a differentiable vector-valued initial function. The piecewise constant function σ(t):[0,∞)→ψN≜{1,2,…,N} denotes a switching signal to be specified; corresponding switching sequence {x(t0);(i0,t0),(i1,t1),…,(ik,tk),…∣ik∈ψN,k=0,1,…} means that the ikth subsystem is activated when t∈[tk,tk+1). Ai, Adi, Bi, Ci, Cdi, Di, Ei, Edi are known constant matrices with appropriate dimensions; d1(t) and d2(t) represent two time-varying delay components satisfying
(2)0≤d1(t)≤d1,0≤d2(t)≤d2,d˙1(t)≤h1,d˙2(t)≤h2,
where d1, d2, h1, h2 are constants. Set d=d1+d2, h=h1+h2, d(t)=d1(t)+d2(t).Our objective in this paper is to construct a full-order switched filter in the form of(3)x˙f(t)=Afσxf(t)+Bfσy(t),zf(t)=Cfσxf(t)+Dfσy(t),
where xf(t)∈Rn is the filter state, zf(t)∈Rp is the filter output estimating z(t), the matrices Afi, Bfi, Cfi, and Dfi, i∈ψN are the filter parameters to be determined later. Filter (3) is assumed to switch synchronously according to the switching signal σ in system (1).Set the state augmentationξTt=[xT(t)xfT(t)] and the estimation error ϵ(t)=z(t)-zf(t), and thus augmenting system (1) to involve switched linear filter (3) gives the following filter error dynamic system:
(4)ξ˙(t)=A¯σξt+A¯dσξt-dt+B¯σωt,ϵ(t)=C¯σξt+C¯dσξt-dt+D¯σωt,
where
(5)A¯i=Ai0BfiCfiAfi,A¯di=Adi0BfiCdi0,B¯i=BiBfiDi,C¯i=Ei-DfiCi-Cfi,C¯di=Edi-DfiCdi,D¯i=-DfiDi.The following definitions are addressed to derive the desired results.Definition 1 (see [1]).
The equilibriumξ*=0 of the filtering error system (4) is exponentially stable under σ(t), if the solution ξ(t) of system (4) with ω(t)=0 satisfies ξ(t)≤kξ(t0)de-λ(t-t0) for all t≥t0 for constants k≥1 and λ≥0, where · denotes the Euclidean norm, and ξ(t)d=sup⁡-d≤θ≤0{ξ(t+θ),ξ˙(t+θ)}.Definition 2 (see [1]).
For anyT2>T1≥0, let Nσ(T1,T2) denote the number of switchings of σ(t) over (T1,T2). If Nσ(T1,T2)≤N0+(T2-T1)/Ta holds for Ta>0, N0≥0, then Ta is called the average dwell time. As commonly used in literature, choose N0=0.Definition 3.
Forα>0 and γ>0, the weighted H∞ filtering problem of system (1) is solvable if there exists a suitable filter of the form (3) and admissible ADT switching signals such that the filter error system (4) is exponentially stable when ω(t)=0; meanwhile, for any nonzero ω(t)∈L2[0,∞), system (4) has a weighted H∞ performance (α,γ) under zero initial condition φ(θ)=0, θ∈[-d,0]; that is,
(6)∫0∞‍e-αsϵT(s)ϵ(s)ds≤γ2∫0∞‍ωT(s)ω(s)ds.The following modification of Jensen integral inequality is essential to develop the main results with less conservatism.Lemma 4 (see [24]).
For any matrixZ>0, scalars r, rj (j=1,2,3,4) satisfying r>0, 0≤r2-r1≤r, and r4=r-(r2-r1)+r3, there exists a vector function ν such that the integrations concerned are well defined; then the following inequality is true:
(7)∫r1r2‍νsTZν(s)ds+∫r3r4‍νsTZν(s)ds≥1r∫r1r2‍νsds∫r3r4‍νsdsTZM*Z∫r1r2‍ν(s)ds∫r3r4‍ν(s)ds,
if there exists a matrix M such that
(8)ZM*Z≥0.


## 3. Main Results

### 3.1. Exponential Stability and WeightedH∞ Performance Analysis

First, we analyse the exponential stability for filtering error system (4) with ω(t)=0; that is,
(9)ξ˙(t)=A¯σξ(t)+A¯dσξ(t-d(t)),ϵ(t)=C¯σξ(t)+C¯dσξ(t-d(t)).Theorem 5.
Given constantα>0, if there exist matrices Pi>0, Qki>0, Rli>0 and appropriately dimensioned matrices Mli such that
(10)Φi=Φ11iΦ12iΦ13iΦ14iΦ15i0*Φ22iΦ23i0Φ25iΦ26i**Φ33iΦ34iΦ35iΦ36i***Φ44iΦ45iΦ46i****Φ55iΦ56i*****Φ66i<0,(11)RliMli*Rli≥0;
then the system (9) with (2) is exponentially stable under any ADT switching signal satisfying
(12)Ta>Ta*=ln⁡⁡μα.
Moreover, an estimate of state decay is given by
(13)ξ(t)≤bae-λ(t-t0)ξ(t0)d,
where μ≥1, k=1,2,3,4, l=1,2,3 satisfy
(14)Pi≤μPj,Qki≤μQkj,Rli≤μRlj,kkkkkkkkkkkkkkkkkk∀i,j∈ψN,i≠j,(15)Φ11i=PiA¯i+A¯iTPi+Q1i+Q2i+Q3i+αPi+A¯iTRiA¯i-e-αd1d1R1i-e-αdd1R3i,Φ12i=PiA¯di+A¯iTRiA¯di-e-αdd1M3i,Φ13i=e-αd1d1R1i-M1i+e-αdd1R3i,Φ14i=e-αd1d1M1i,Φ15i=e-αdd1M3i,Φ22i=A¯diTRiA¯di-e-αd1-hQ1i+1d1+1d2R3i,Φ23i=e-αdd1M3iT+e-αdd2R3i,Φ25i=e-αdd1R3i+e-αdd2M3iT,Φ26i=-e-αdd2M3iT,Φ33i=-1-h1e-αd1Q2i-2e-αd1d1R1i+e-αd1d1M1iT+M1i-e-αdd1R3i-e-αdd2R3i,Φ34i=e-αd1d1R1i-M1i,Φ35i=-e-αdd1M3i-e-αdd2M3iT,Φ36i=e-αdd2M3iT,Φ44i=-e-αd1Q3i-Q4i-e-αd1d1R1i-e-αd2d2R2i,Φ45i=e-αd2d2R2i-M2i,Φ46i=e-αd2d2M2i,Φ55i=e-αd2d2M2i+M2iT-2R2i-1d1+1d2e-αdR3i,Φ56i=e-αd2d2R2i-M2i+e-αdd2R3i,Φ66i=-e-αdQ4i+1d2R3i-e-αd2d2R2i,λ=12α-ln⁡μTa,Ri=d1R1i+d2R2i+dR3i,a=min⁡∀i∈ψNλmin⁡Pi,b=max⁡∀i∈ψNλmax⁡Pi+32d2max⁡∀i∈ψNλmax⁡R1i+R2i+R3i+4dmax⁡∀i∈ψNλmax⁡Q1i+Q2i+Q3i+Q4i.Proof.
Define the piecewise Lyapunov-Krasoviskii functional candidate of the form(16)Vσξt=V1σξt+V2σξt+V3σξt,
where
(17)V1σξt=ξTtPσξt+∫t-dtt‍eαs-tξTsQ1σξsds+∫t-d1tt‍eαs-tξTsQ2σξsds+∫t-d1t‍eα(s-t)ξT(s)Q3σξ(s)ds,V2σξt=∫t-dt-d1‍eα(s-t)ξT(s)Q4σξ(s)ds,V3σξt=∫-d10‍∫t+θt‍eαs-tξ˙sTR1σξ˙sdsdθ+∫-d-d1‍∫t+θt‍eαs-tξ˙sTR2σξ˙sdsdθ+∫-d0‍∫t+θt‍eαs-tξ˙sTR3σξ˙sdsdθ.
Then taking the time derivative of Vi(ξ(t)) along the trajectory of system (9) yields
(18)V˙1iξt≤2ξTtPiA¯iξt+A¯diξt-dt-α∫t-dtt‍eαs-tξTsQ1iξsds-1-he-αdξTt-dtQ1iξt-dt+ξTtQ1i+Q2i+Q3iξt-α∫t-d1(t)t‍eα(s-t)ξT(s)Q2iξ(s)ds-1-h1e-αd1ξTt-d1tQ2iξt-d1t-α∫t-d1t‍eα(s-t)ξT(s)Q3iξ(s)ds-e-αd1ξTt-d1Q3iξt-d1,V˙2iξt=-α∫t-dt-d1‍eαs-tξTsQ4iξsds+e-αd1ξTt-d1Q4iξt-d1-e-αdξTt-dQ4iξt-d,V˙3iξt≤-α∫-d10‍∫t+θt‍eαs-tξ˙sTR1iξ˙sdsdθ-α∫-d-d1‍∫t+θt‍eα(s-t)ξ˙sTR2iξ˙(s)dsdθ-α∫-d0‍∫t+θt‍eα(s-t)ξ˙sTR3iξ˙(s)dsdθ-∫t-dt‍e-αdξ˙sTR3iξ˙sds-∫t-d1t‍e-αd1ξ˙sTR1iξ˙sds-∫t-dt-d1‍e-αd2ξ˙sTR2iξ˙(s)ds+ξ˙tTRiξ˙(t).
Lemma4 and (11) give the following inequalities:
(19)-∫t-d1t‍e-αd1ξ˙sTR1iξ˙(s)ds=-e-αd1∫t-d1(t)t‍ξ˙T(s)R1iξ˙(s)ds-e-αd1∫t-d1t-d1(t)‍ξ˙T(s)R1iξ˙(s)ds≤-e-αd1d1ϕ1(t)ϕ2(t)TR1iM1i*R1iϕ1(t)ϕ2(t),-∫t-dt-d1‍e-αd2ξ˙sTR2iξ˙(s)ds=e-αd2∫t-d1-d2(t)t-d1‍ξ˙T(s)R2iξ˙(s)ds-e-αd2∫t-dt-d1-d2(t)‍ξ˙T(s)R2iξ˙(s)ds≤-e-αd2d2ϕ3(t)ϕ4(t)TR2iM2i*R2iϕ3(t)ϕ4(t),-∫t-dt‍e-αdξ˙sTR3iξ˙(s)ds=e-αd∫t-d1(t)t‍ξ˙T(s)R3iξ˙(s)ds-e-αd∫t-d(t)t-d1(t)‍ξ˙T(s)R3iξ˙(s)ds-e-αd∫t-d1-d2(t)t-d(t)‍ξ˙T(s)R3iξ˙(s)ds-e-αd∫t-dt-d1-d2t‍ξ˙TsR3iξ˙sds≤-e-αdd1ϕ1(t)ϕ5(t)TR3iM3i*R3iϕ1(t)ϕ5(t)-e-αdd2ϕ6(t)ϕ4(t)TR3iM3i*R3iϕ6(t)ϕ4(t),
where
(20)ϕ1t=ξt-ξt-d1t,ϕ2t=ξt-d1t-ξt-d1,ϕ3t=ξt-d1-ξt-d1-d2t,ϕ4t=ξt-d1-d2t-ξt-d,ϕ5(t)=ξt-d1t-ξt-d1-d2t,ϕ6(t)=ξt-d1t-ξt-dt.
Combining (10) and (16)–(19) leads to
(21)V˙iξt+αViξt≤ηTtΦiηt<0
with ηT(t)=[ξT(t)ξT(t-d(t))ξT(t-d1(t))ξT(t-d1)ξT(t-d1-d2(t))ξT(t-d)].
Whent∈[tk,tk+1), integrating the above inequality from tk to t yields
(22)Vσtξt≤e-αt-tkVσtkξtk.
Using (14) and (16), at switching instant tk, we have
(23)Vσtkξtk≤μVσtk-ξtk-.
Therefore, it follows from (22), (23), and the relation Nσ(t0,t)≤(t-t0)/Ta that
(24)Vσξt≤e-(α-ln⁡μ/Ta)(t-t0)Vσ(t0)ξ(t0).
Noticing that aξ(t)2≤Vσ(t)(ξ(t)) and Vσ(t0)(ξ(t0))≤bξ(t0)d2, we have
(25)ξ(t)2≤1aVσtξt≤bae-(α-ln⁡μ/Ta)(t-t0)ξ(t0)d2,
which completes the proof.Remark 6.
Whenμ=1, we have Ta*=0, which means that all subsystems employ a common Lyapunov functional, and the switching among them can be arbitrary. Meanwhile, let α=0 in (10); it gives asymptotical stability for (9) under arbitrary switching.Remark 7.
Due to the application of the modified Jensen integral inequality, the upper bound ofV3i(ξ(t)) is estimated more tightly, and fewer matrix variables are involved. Therefore, the theoretical conservatism and computational complexity are reduced for the derived conditions.Next, we establish the following weightedH∞ performance criteria for the filter error system (4).Theorem 8.
For scalarsα>0 and γ>0, there exist matrices Pi>0, Qki>0, Rli>0 and appropriately dimensioned matrices Mli such that
(26)Πi=Π11iΠ12i*-I<0;(11) and (14) hold, where μ≥1, k=1,2,3,4, l=1,2,3,
(27)Π11i=ΦiΦ17i*Φ77i,Φ17i=PiB¯i+AiTRiBiAdiTRiBi0000T,Π12i=C¯iTC¯diT0000D¯iTT,Φ77i=-γ2I+BiTRiBi,Γ(t)=ϵT(t)ϵ(t)-γ2ωT(t)ω(t); others are defined in Theorem 5. Then the filtering error system (4) with (2) is exponentially stable with weighted H∞ performance (α,γ) under any ADT switching signal (12).Proof.
From Theorem5, the filter error system (4) is exponentially stable with ω(t)=0. To show the weighted H∞ performance, we choose the Lyapunov-Krosoviskii functional (16). According to Schur complement, (26) is equivalent to Π11i+Π12i(Π12i)T<0 which means
(28)V˙iξt≤-αViξt-Γt.
For any t∈[tk,tk+1), combining (23) and (28) yields
(29)Vσξt≤e-αt-tkVσξtk-∫tkt‍e-α(t-s)Γ(s)ds≤μe-αt-tkVσξtk--∫tkt‍e-α(t-s)Γ(s)ds≤μNσ0,te-αtVσξ0-μNσ(0,t)×∫0t1‍e-α(t-s)Γ(s)ds-⋯-∫tkt‍e-α(t-s)Γ(s)ds≤e-αt+Nσ0,tln⁡μVσξ0-∫0t‍e-α(t-s)+Nσ(s,t)ln⁡μΓ(s)ds.
Under the zero initial condition, multiplying both sides of the above inequality by e-Nσ(0,t)ln⁡μ leads to
(30)0≤-∫0t‍e-α(t-s)-Nσ(0,s)ln⁡μΓ(s)ds,
which is equivalent to
(31)∫0t‍e-αt-s-Nσ0,sln⁡μϵTsϵsds≤γ2∫0t‍e-α(t-s)-Nσ(0,s)ln⁡μωT(s)ω(s)ds.
From (12), Nσ(0,s)ln⁡μ≤αs holds; then one gets
(32)∫0t‍e-α(t-s)-αsϵT(s)ϵ(s)ds≤γ2∫0t‍e-α(t-s)ωT(s)ω(s)ds.
Integrating the above inequality from t=0 to ∞ and exchanging the integration order lead to (6). This completes the proof.


### 3.2. WeightedH∞ Filter Design

Next, parameters of switched filter (3) can be determined by the following theorem.Theorem 9.
For scalarsα>0, γ>0, there exist matrices P1i>0, P3i>0, Qki>0, Rli>0 and appropriately dimensioned matrices P2i, Ni, Mli, A¯fi, B¯fi, C¯fi, D¯fi such that
(33)Π^i=Π^11iΠ^12i*-I<0,(34)Pi≜P1iP2i*P3i>0;(11) and (14) hold for μ≥1, k=1,2,3,4, l=1,2,3 with
(35)Π^11i=Φ^11iΦ^12iΦ13iΦ14iΦ15i0Φ^17i*Φ22iΦ23i0Φ25iΦ26i0**Φ33iΦ34iΦ35iΦ36iΦ^37i***Φ44iΦ45iΦ46i0****Φ55iΦ56i0*****Φ66i0******Φ^77i,Π^12i=(Ei-D¯fiCi)TC¯fiT0000D¯iTT,Φ^11i=ΞiA¯fiT+AiTNi+CiTB¯fi+αNi*A¯fi+A¯fiT+αNi,Φ^12i=P1iAdi+B¯fiCdi+AiTRiAdiNiAdi+B¯fiCdi,Φ^17i=P1iBi+B¯fiDi+AiTRiBiNiBi+B¯fiDi,Φ^37i=AdiTRiBi,Φ^77i=-γ2I,Ξi=P1iAi+AiTP1i+B¯fiCi+CiTB¯fiT+αP1i+AiTRiAi+Q1i+Q2i+Q3i;
other symbols are mentioned earlier. Then there exists a filter of the form (3) such that the weighted H∞ filtering problem of system (1) with (2) is solvable under any ADT switching signal (12). The filter matrices are constructed by
(36)Afi=Ni-1A¯fi,Bfi=Ni-1B¯fi,Cfi=C¯fi,Dfi=D¯fi,
or
(37)Afi=A¯fiNi-1,Bfi=B¯fi,Cfi=C¯fiNi-1,Dfi=D¯fi.Proof.
Set(38)Ni=P2iP3i-1P2iT,A¯fi=P2iAfiP3i-1P2iT,B¯fi=P2iBfi,C¯fi=P2iP3i-TCfiT,L=diag⁡{I,P2iP3i-1,I,I,I,I,I,I},D¯fi=Dfi.
Pre- and postmultiplying (26) by L and LT, respectively, one can get that (33) is equivalent to (10). Thus, the filter error system (4) is exponentially stable with weighted H∞ performance. Furthermore, (34) is equivalent to 0<P1i-P2iP3i-1P2iT=P1i-Ni; then Ni>0 and P2i is nonsingular. Thus we have
(39)AfiBfiCfiDfi=P2i-100IA¯fiB¯fiC¯fiD¯fiP2i-TP3i00I.
From the filter transfer function
(40)Tis=CfisI-Afi-1Bfi+Dfi=C¯fiP2iTP3isI-P2i-1A¯fiP2i-TP3i-1P2i-1B¯fi+D¯fi=C¯fisNi-A¯fi-1B¯fi+D¯fi=C¯fisI-Ni-1A¯fi-1Ni-1B¯fi+D¯fi=C¯fiNi-1sI-A¯fiNi-1-1B¯fi+D¯fi,
the filter matrices are readily established by (36) or (37). This completes the proof.Remark 10.
For convenience, we only consider switched systems with two additive time-varying delays, but the proposed results in this paper can be extended to switched systems with multiple additive time-varying delay components.Remark 11.
Whend1=0 or d2=0, (1) reduces to switched system with a single time-varying delay. Suppose that d2=0 without loss of generality; by setting Q2i=Q4i=0 and R2i=R3i=0 in (16), criterions of stability with weighted H∞ performance for switched systems with a single delay can be obtained similarly.The following algorithm is helpful to derive filter matrices for the weightedH∞ filtering problem under consideration.Design AlgorithmStep 1.
Scalarsα>0, μ≥1 are selected close to 1. γ>0 is also prescribed. Then, LMIs in Theorem 9 are solved for given α,μ, and γ by LMI toolbox in MATLAB.Step 2.
If these LMIs have no solution, then there are two cases. Case 1:α and μ will be increased at a big step size; go back to Step 1. Case 2: the previous α and μ are proper, and go to Step 4.Step 3.
If these LMIs have feasible solutions,μ will be decreased at a small step size, and go to Step 1.Step 4.
α and μ are fixed. Decrease γ similarly until γ is the optimized value. According to (36) or (37), the filter matrices are established. Exit.


## 3.1. Exponential Stability and WeightedH∞ Performance Analysis

First, we analyse the exponential stability for filtering error system (4) with ω(t)=0; that is,
(9)ξ˙(t)=A¯σξ(t)+A¯dσξ(t-d(t)),ϵ(t)=C¯σξ(t)+C¯dσξ(t-d(t)).Theorem 5.
Given constantα>0, if there exist matrices Pi>0, Qki>0, Rli>0 and appropriately dimensioned matrices Mli such that
(10)Φi=Φ11iΦ12iΦ13iΦ14iΦ15i0*Φ22iΦ23i0Φ25iΦ26i**Φ33iΦ34iΦ35iΦ36i***Φ44iΦ45iΦ46i****Φ55iΦ56i*****Φ66i<0,(11)RliMli*Rli≥0;
then the system (9) with (2) is exponentially stable under any ADT switching signal satisfying
(12)Ta>Ta*=ln⁡⁡μα.
Moreover, an estimate of state decay is given by
(13)ξ(t)≤bae-λ(t-t0)ξ(t0)d,
where μ≥1, k=1,2,3,4, l=1,2,3 satisfy
(14)Pi≤μPj,Qki≤μQkj,Rli≤μRlj,kkkkkkkkkkkkkkkkkk∀i,j∈ψN,i≠j,(15)Φ11i=PiA¯i+A¯iTPi+Q1i+Q2i+Q3i+αPi+A¯iTRiA¯i-e-αd1d1R1i-e-αdd1R3i,Φ12i=PiA¯di+A¯iTRiA¯di-e-αdd1M3i,Φ13i=e-αd1d1R1i-M1i+e-αdd1R3i,Φ14i=e-αd1d1M1i,Φ15i=e-αdd1M3i,Φ22i=A¯diTRiA¯di-e-αd1-hQ1i+1d1+1d2R3i,Φ23i=e-αdd1M3iT+e-αdd2R3i,Φ25i=e-αdd1R3i+e-αdd2M3iT,Φ26i=-e-αdd2M3iT,Φ33i=-1-h1e-αd1Q2i-2e-αd1d1R1i+e-αd1d1M1iT+M1i-e-αdd1R3i-e-αdd2R3i,Φ34i=e-αd1d1R1i-M1i,Φ35i=-e-αdd1M3i-e-αdd2M3iT,Φ36i=e-αdd2M3iT,Φ44i=-e-αd1Q3i-Q4i-e-αd1d1R1i-e-αd2d2R2i,Φ45i=e-αd2d2R2i-M2i,Φ46i=e-αd2d2M2i,Φ55i=e-αd2d2M2i+M2iT-2R2i-1d1+1d2e-αdR3i,Φ56i=e-αd2d2R2i-M2i+e-αdd2R3i,Φ66i=-e-αdQ4i+1d2R3i-e-αd2d2R2i,λ=12α-ln⁡μTa,Ri=d1R1i+d2R2i+dR3i,a=min⁡∀i∈ψNλmin⁡Pi,b=max⁡∀i∈ψNλmax⁡Pi+32d2max⁡∀i∈ψNλmax⁡R1i+R2i+R3i+4dmax⁡∀i∈ψNλmax⁡Q1i+Q2i+Q3i+Q4i.Proof.
Define the piecewise Lyapunov-Krasoviskii functional candidate of the form(16)Vσξt=V1σξt+V2σξt+V3σξt,
where
(17)V1σξt=ξTtPσξt+∫t-dtt‍eαs-tξTsQ1σξsds+∫t-d1tt‍eαs-tξTsQ2σξsds+∫t-d1t‍eα(s-t)ξT(s)Q3σξ(s)ds,V2σξt=∫t-dt-d1‍eα(s-t)ξT(s)Q4σξ(s)ds,V3σξt=∫-d10‍∫t+θt‍eαs-tξ˙sTR1σξ˙sdsdθ+∫-d-d1‍∫t+θt‍eαs-tξ˙sTR2σξ˙sdsdθ+∫-d0‍∫t+θt‍eαs-tξ˙sTR3σξ˙sdsdθ.
Then taking the time derivative of Vi(ξ(t)) along the trajectory of system (9) yields
(18)V˙1iξt≤2ξTtPiA¯iξt+A¯diξt-dt-α∫t-dtt‍eαs-tξTsQ1iξsds-1-he-αdξTt-dtQ1iξt-dt+ξTtQ1i+Q2i+Q3iξt-α∫t-d1(t)t‍eα(s-t)ξT(s)Q2iξ(s)ds-1-h1e-αd1ξTt-d1tQ2iξt-d1t-α∫t-d1t‍eα(s-t)ξT(s)Q3iξ(s)ds-e-αd1ξTt-d1Q3iξt-d1,V˙2iξt=-α∫t-dt-d1‍eαs-tξTsQ4iξsds+e-αd1ξTt-d1Q4iξt-d1-e-αdξTt-dQ4iξt-d,V˙3iξt≤-α∫-d10‍∫t+θt‍eαs-tξ˙sTR1iξ˙sdsdθ-α∫-d-d1‍∫t+θt‍eα(s-t)ξ˙sTR2iξ˙(s)dsdθ-α∫-d0‍∫t+θt‍eα(s-t)ξ˙sTR3iξ˙(s)dsdθ-∫t-dt‍e-αdξ˙sTR3iξ˙sds-∫t-d1t‍e-αd1ξ˙sTR1iξ˙sds-∫t-dt-d1‍e-αd2ξ˙sTR2iξ˙(s)ds+ξ˙tTRiξ˙(t).
Lemma4 and (11) give the following inequalities:
(19)-∫t-d1t‍e-αd1ξ˙sTR1iξ˙(s)ds=-e-αd1∫t-d1(t)t‍ξ˙T(s)R1iξ˙(s)ds-e-αd1∫t-d1t-d1(t)‍ξ˙T(s)R1iξ˙(s)ds≤-e-αd1d1ϕ1(t)ϕ2(t)TR1iM1i*R1iϕ1(t)ϕ2(t),-∫t-dt-d1‍e-αd2ξ˙sTR2iξ˙(s)ds=e-αd2∫t-d1-d2(t)t-d1‍ξ˙T(s)R2iξ˙(s)ds-e-αd2∫t-dt-d1-d2(t)‍ξ˙T(s)R2iξ˙(s)ds≤-e-αd2d2ϕ3(t)ϕ4(t)TR2iM2i*R2iϕ3(t)ϕ4(t),-∫t-dt‍e-αdξ˙sTR3iξ˙(s)ds=e-αd∫t-d1(t)t‍ξ˙T(s)R3iξ˙(s)ds-e-αd∫t-d(t)t-d1(t)‍ξ˙T(s)R3iξ˙(s)ds-e-αd∫t-d1-d2(t)t-d(t)‍ξ˙T(s)R3iξ˙(s)ds-e-αd∫t-dt-d1-d2t‍ξ˙TsR3iξ˙sds≤-e-αdd1ϕ1(t)ϕ5(t)TR3iM3i*R3iϕ1(t)ϕ5(t)-e-αdd2ϕ6(t)ϕ4(t)TR3iM3i*R3iϕ6(t)ϕ4(t),
where
(20)ϕ1t=ξt-ξt-d1t,ϕ2t=ξt-d1t-ξt-d1,ϕ3t=ξt-d1-ξt-d1-d2t,ϕ4t=ξt-d1-d2t-ξt-d,ϕ5(t)=ξt-d1t-ξt-d1-d2t,ϕ6(t)=ξt-d1t-ξt-dt.
Combining (10) and (16)–(19) leads to
(21)V˙iξt+αViξt≤ηTtΦiηt<0
with ηT(t)=[ξT(t)ξT(t-d(t))ξT(t-d1(t))ξT(t-d1)ξT(t-d1-d2(t))ξT(t-d)].
Whent∈[tk,tk+1), integrating the above inequality from tk to t yields
(22)Vσtξt≤e-αt-tkVσtkξtk.
Using (14) and (16), at switching instant tk, we have
(23)Vσtkξtk≤μVσtk-ξtk-.
Therefore, it follows from (22), (23), and the relation Nσ(t0,t)≤(t-t0)/Ta that
(24)Vσξt≤e-(α-ln⁡μ/Ta)(t-t0)Vσ(t0)ξ(t0).
Noticing that aξ(t)2≤Vσ(t)(ξ(t)) and Vσ(t0)(ξ(t0))≤bξ(t0)d2, we have
(25)ξ(t)2≤1aVσtξt≤bae-(α-ln⁡μ/Ta)(t-t0)ξ(t0)d2,
which completes the proof.Remark 6.
Whenμ=1, we have Ta*=0, which means that all subsystems employ a common Lyapunov functional, and the switching among them can be arbitrary. Meanwhile, let α=0 in (10); it gives asymptotical stability for (9) under arbitrary switching.Remark 7.
Due to the application of the modified Jensen integral inequality, the upper bound ofV3i(ξ(t)) is estimated more tightly, and fewer matrix variables are involved. Therefore, the theoretical conservatism and computational complexity are reduced for the derived conditions.Next, we establish the following weightedH∞ performance criteria for the filter error system (4).Theorem 8.
For scalarsα>0 and γ>0, there exist matrices Pi>0, Qki>0, Rli>0 and appropriately dimensioned matrices Mli such that
(26)Πi=Π11iΠ12i*-I<0;(11) and (14) hold, where μ≥1, k=1,2,3,4, l=1,2,3,
(27)Π11i=ΦiΦ17i*Φ77i,Φ17i=PiB¯i+AiTRiBiAdiTRiBi0000T,Π12i=C¯iTC¯diT0000D¯iTT,Φ77i=-γ2I+BiTRiBi,Γ(t)=ϵT(t)ϵ(t)-γ2ωT(t)ω(t); others are defined in Theorem 5. Then the filtering error system (4) with (2) is exponentially stable with weighted H∞ performance (α,γ) under any ADT switching signal (12).Proof.
From Theorem5, the filter error system (4) is exponentially stable with ω(t)=0. To show the weighted H∞ performance, we choose the Lyapunov-Krosoviskii functional (16). According to Schur complement, (26) is equivalent to Π11i+Π12i(Π12i)T<0 which means
(28)V˙iξt≤-αViξt-Γt.
For any t∈[tk,tk+1), combining (23) and (28) yields
(29)Vσξt≤e-αt-tkVσξtk-∫tkt‍e-α(t-s)Γ(s)ds≤μe-αt-tkVσξtk--∫tkt‍e-α(t-s)Γ(s)ds≤μNσ0,te-αtVσξ0-μNσ(0,t)×∫0t1‍e-α(t-s)Γ(s)ds-⋯-∫tkt‍e-α(t-s)Γ(s)ds≤e-αt+Nσ0,tln⁡μVσξ0-∫0t‍e-α(t-s)+Nσ(s,t)ln⁡μΓ(s)ds.
Under the zero initial condition, multiplying both sides of the above inequality by e-Nσ(0,t)ln⁡μ leads to
(30)0≤-∫0t‍e-α(t-s)-Nσ(0,s)ln⁡μΓ(s)ds,
which is equivalent to
(31)∫0t‍e-αt-s-Nσ0,sln⁡μϵTsϵsds≤γ2∫0t‍e-α(t-s)-Nσ(0,s)ln⁡μωT(s)ω(s)ds.
From (12), Nσ(0,s)ln⁡μ≤αs holds; then one gets
(32)∫0t‍e-α(t-s)-αsϵT(s)ϵ(s)ds≤γ2∫0t‍e-α(t-s)ωT(s)ω(s)ds.
Integrating the above inequality from t=0 to ∞ and exchanging the integration order lead to (6). This completes the proof.


## 3.2. WeightedH∞ Filter Design

Next, parameters of switched filter (3) can be determined by the following theorem.Theorem 9.
For scalarsα>0, γ>0, there exist matrices P1i>0, P3i>0, Qki>0, Rli>0 and appropriately dimensioned matrices P2i, Ni, Mli, A¯fi, B¯fi, C¯fi, D¯fi such that
(33)Π^i=Π^11iΠ^12i*-I<0,(34)Pi≜P1iP2i*P3i>0;(11) and (14) hold for μ≥1, k=1,2,3,4, l=1,2,3 with
(35)Π^11i=Φ^11iΦ^12iΦ13iΦ14iΦ15i0Φ^17i*Φ22iΦ23i0Φ25iΦ26i0**Φ33iΦ34iΦ35iΦ36iΦ^37i***Φ44iΦ45iΦ46i0****Φ55iΦ56i0*****Φ66i0******Φ^77i,Π^12i=(Ei-D¯fiCi)TC¯fiT0000D¯iTT,Φ^11i=ΞiA¯fiT+AiTNi+CiTB¯fi+αNi*A¯fi+A¯fiT+αNi,Φ^12i=P1iAdi+B¯fiCdi+AiTRiAdiNiAdi+B¯fiCdi,Φ^17i=P1iBi+B¯fiDi+AiTRiBiNiBi+B¯fiDi,Φ^37i=AdiTRiBi,Φ^77i=-γ2I,Ξi=P1iAi+AiTP1i+B¯fiCi+CiTB¯fiT+αP1i+AiTRiAi+Q1i+Q2i+Q3i;
other symbols are mentioned earlier. Then there exists a filter of the form (3) such that the weighted H∞ filtering problem of system (1) with (2) is solvable under any ADT switching signal (12). The filter matrices are constructed by
(36)Afi=Ni-1A¯fi,Bfi=Ni-1B¯fi,Cfi=C¯fi,Dfi=D¯fi,
or
(37)Afi=A¯fiNi-1,Bfi=B¯fi,Cfi=C¯fiNi-1,Dfi=D¯fi.Proof.
Set(38)Ni=P2iP3i-1P2iT,A¯fi=P2iAfiP3i-1P2iT,B¯fi=P2iBfi,C¯fi=P2iP3i-TCfiT,L=diag⁡{I,P2iP3i-1,I,I,I,I,I,I},D¯fi=Dfi.
Pre- and postmultiplying (26) by L and LT, respectively, one can get that (33) is equivalent to (10). Thus, the filter error system (4) is exponentially stable with weighted H∞ performance. Furthermore, (34) is equivalent to 0<P1i-P2iP3i-1P2iT=P1i-Ni; then Ni>0 and P2i is nonsingular. Thus we have
(39)AfiBfiCfiDfi=P2i-100IA¯fiB¯fiC¯fiD¯fiP2i-TP3i00I.
From the filter transfer function
(40)Tis=CfisI-Afi-1Bfi+Dfi=C¯fiP2iTP3isI-P2i-1A¯fiP2i-TP3i-1P2i-1B¯fi+D¯fi=C¯fisNi-A¯fi-1B¯fi+D¯fi=C¯fisI-Ni-1A¯fi-1Ni-1B¯fi+D¯fi=C¯fiNi-1sI-A¯fiNi-1-1B¯fi+D¯fi,
the filter matrices are readily established by (36) or (37). This completes the proof.Remark 10.
For convenience, we only consider switched systems with two additive time-varying delays, but the proposed results in this paper can be extended to switched systems with multiple additive time-varying delay components.Remark 11.
Whend1=0 or d2=0, (1) reduces to switched system with a single time-varying delay. Suppose that d2=0 without loss of generality; by setting Q2i=Q4i=0 and R2i=R3i=0 in (16), criterions of stability with weighted H∞ performance for switched systems with a single delay can be obtained similarly.The following algorithm is helpful to derive filter matrices for the weightedH∞ filtering problem under consideration.Design AlgorithmStep 1.
Scalarsα>0, μ≥1 are selected close to 1. γ>0 is also prescribed. Then, LMIs in Theorem 9 are solved for given α,μ, and γ by LMI toolbox in MATLAB.Step 2.
If these LMIs have no solution, then there are two cases. Case 1:α and μ will be increased at a big step size; go back to Step 1. Case 2: the previous α and μ are proper, and go to Step 4.Step 3.
If these LMIs have feasible solutions,μ will be decreased at a small step size, and go to Step 1.Step 4.
α and μ are fixed. Decrease γ similarly until γ is the optimized value. According to (36) or (37), the filter matrices are established. Exit.


## 4. Example

Two examples are presented in this section to illustrate the proposed results.Example 1.
Consider switched system (1) with (2) consisting of two subsystems where subsystem 1 is described by
(41)A1=-20-0.50.2,Ad1=-100-0.2,B1=0-0.5,C1=-10.3,Cd1=0.9-0.5,D1=0.4,E1=0.5-1.3,Ed1=-0.90.1,
and subsystem 2 is described by
(42)A2=-0.90-0.80,Ad2=00.500.4,B2=0.5-0.7,C2=0.5-0.9,Cd2=0.3-1,D2=0.2,E2=0.2-1,Ed2=-0.8-0.5.
Let d1(t)=0.3+0.2sin⁡t, d2(t)=0.9-0.3cos⁡⁡t, α=0.5,  γ=0.9, Ta=0.8046,  μ=1.3>1. It can be easily checked that d1=0.5, d2=0.6, h1=0.2, h2=0.3, Ta*=ln⁡μ/α=0.4807. Solving LMIs (11), (14), (33), and (34) gets the designed filter with the following parameterized matrices:
(43)Af1=-0.5792-3.5696-1.24000.3236,Bf1=1.01240.1474,Cf1=-0.03710.3802,Df1=0.1262,Af2=0.37440.82321.0259-2.0503,Bf2=-0.23001.1451,Cf2=-0.03720.1669,Df2=-0.0282.
Under the initial condition ξ0=-10.700.1T and ω(t)=0.2sin(0.1t) which belongs to L2[0,∞), the state responses of the filter error system (4) are shown in Figure 1, and the estimation error is displayed in Figure 2. Figure 3 shows the corresponding ADT switching signal satisfying (12). The simulation results imply that the desired goal is well achieved. The admissible minimum feasible γ for different cases is shown in Table 1 to ensure the exponential stability for filtering error system.Table 1
Minimum feasibleγ for different cases.
d
1
0.6
0.6
0.6
0.7
0.7
0.7
d
2
0.3
0.5
0.8
0.3
0.5
0.8
γ
min
⁡
1.2700
0.9397
0.0746
0.8918
0.2219
0.0026Figure 1
The state responses of the filtering error system.Figure 2
Filtering error.Figure 3
Switching signal.Example 2.
Consider a satellite yaw angles control system with noise perturbation, which is given by the following state-space representation [13]:
(44)1000010000J10000J2θ˙1tθ˙2tδ˙1tδ˙2t=00100001-kk-ffk-kf-fθ1tθ2tδ1tδ2t+000001-10u(t)+0101ω(t),y(t)=1101θ1(t)θ2(t)δ1(t)δ2(t)T.
The satellite yaw angles system consists of two rigid bodies joined by a flexible link. This link is modelled as a spring with torque k and viscous damping f. Respectively, θ1 and θ2 denote the yaw angles for the main body and the instrumentation module of the satellite, δ1=θ˙1, δ2=θ˙2. u(t) is the control torque, while J1 and J2 are the moments of inertia of two bodies. Here we choose J1=J2=1, k=0.3, f=0.004, and x(t)=θ1(t)θ2(t)δ1(t)δ2(t)T. According to the method proposed for this model in [19], the gain matrix for the state-feedback controller u(t)=Kx(t-d1-d2(t)) is given by
(45)K=0.0087-1.00000.0610-0.5800-0.0046-0.004780.00181.0000.
Thus, the system is described by
(46)x˙(t)=Ax(t)+Adx(t-d1-d2(t))+Bω(t),y(t)=Cx(t)+Dω(t),zt=Ext,
where
(47)A=00100001-0.30.3-0.0040.0040.3-0.30.004-0.004,Ad=00000000-0.0046-0.00480.00181.0000-0.00871.0000-0.06100.5800,B=0101,C=1101,D=0,E=0000.
We design a packet-loss-dependent filter of the following form [7]:
(48)x˙f(t)=Afixf(t)+Bfiy^(t),zf(t)=Cfix(t),i∈1,2.
When i=1, it indicates that a packet is successfully received by the filter, and y^(t)=y(t); then the filter model is described by
(49)x˙f(t)=Af1xf(t)+Bf1Cx(t)+Bf1Dω(t),zf(t)=Cf1x(t).
When i=2, it means that a packet is lost, and y^(t)=0; then the filter model is given by
(50)x˙f(t)=Af2xf(t),zf(t)=Cf2x(t).
Setting ξTt=[xT(t)xfT(t)], ϵ(t)=z(t)-zf(t), we obtain the following filtering error system:
(51)ξ˙(t)=A¯σξ(t)+A¯dσξ(t-d1-d2(t))+B¯σω(t),ϵ(t)=C¯σξ(t),
where
(52)A¯1=A0Bf1CAf1,A¯2=A00Af2,A¯d1=A¯d2=Ad000,B¯1=BBf1D,B¯2=B0,C¯1=E-Cf10,C¯2=E-Cf20,σ=1,2.
Choose d1=0.1, d2=0.4, h1=0, h2=0.2, α=0.7, μ=1.3, γ=0.9. By using Theorem 9, we obtain the following gain matrices of the packet-less-dependent filter:
(53)Af1=-0.5106-0.0411-0.0437-0.01951.3969-1.35940.4521-2.31960.8576-0.0883-2.7691-0.69751.04261.1355-1.2184-4.6745,Bf1=-0.0946-1.7884-1.0630-2.9718,Af2=-0.5452-0.0923-0.10140.0485-0.3012-0.51210.3895-1.32740.5511-0.2017-2.0697-0.2610-0.97641.3459-1.6402-2.2371,Bf2=0000,Cf1=0.1103-0.7832-0.10770,Cf2=0.1729-1.8773-0.33910.
Under the initial condition ξ0=-10.70-0.5-101-1T and ω(t)=0.1sin⁡(πt/2) belonging to L2[0,∞), the state responses of the filter error system (51) are shown in Figure 4, and the estimation error is displayed in Figure 5. Figure 6 shows the corresponding ADT switching signal satisfying (12).Figure 4
The state responses of the filtering error dynamic system.Figure 5
Filtering error.Figure 6
Switching signal.


## 5. Conclusion

The weightedH∞ filtering problem for a class of switched linear system with two additive time-varying delays is investigated. First, new criterions are proposed to ensure exponential stability and a weighted H∞ performance for the filtering error system under ADT switching signals. Then filter matrices of the desired switched filters are obtained. Less conservative and computational complex conditions are developed by applying a modification of Jensen integral inequality. Finally, two examples are given to show the effectiveness of the proposed method.



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