This paper is concerned with the H∞ filtering problem
for networked systems with bounded measurement missing.
A switched linear system model is proposed to describe the
considered filtering error system. A sufficient condition is
derived for the filtering error system to be exponentially stable
and achieve a prescribed H∞ filtering performance level.
The obtained condition establishes quantitative relations among
the H∞ performance level and two parameters
characterizing the measurement missing, namely, the measurement
missing rate bound and the maximal number of consecutive
measurement missing. A convex optimization problem is presented to
design the linear H∞ filters. Finally, an illustrative
example is given to show the effectiveness of the proposed
results.
1. Introduction
A sensor network consists of spatially distributed autonomous sensors to cooperatively monitor physical or environmental conditions. The purpose of a sensor network is to provide users with the information of interest from data gathered by spatially distributed sensors, and it has found applications in a variety of areas, such as moving target localization and tracking, environment monitoring, intelligent transportation systems, and sensor-actuator network-based control (see [1, 2] and the literature therein). Thus, it is not surprising that signal estimation has been one of the most fundamental collaborative information processing problems in sensor networks [3–10]. In the environment of sensor networks, the measurements may be unavailable to the estimators intermittently. The corresponding estimation problem is usually termed estimation with missing, incomplete, or intermittent measurements and has received increasing research attention; see, for example, [11–17] and the references therein.
Measurement missing degrades the performance of the filtering system, and one may be concerned with the problem about how much the filtering performance declines for a certain amount of measurement missing, what is the maximal number of consecutive measurement missing that the filtering system can tolerant to guarantee the desired filtering performance, and so forth. This gives rise to the idea of revealing the relations, especially the quantitative relations, among the filtering performance level and some parameters characterizing the measurement missing, such as the missing rate/probability bound and the maximal number of consecutive measurement missing. Such relations may provide useful guidelines for designing the filtering systems with measurement missing. For example, in the sensor-network-based filtering system, one may purposely suspend some sensor nodes intermittently to save node power without destroying the stability and desired performance level of the filtering systems; in some network-based fast sampling filtering systems, one may consider dropping certain amount of data packets to ensure that the set of the network-based filtering systems is schedulable and meanwhile to guarantee that the overall network-based filtering systems are stable and achieve prescribed filtering performance level and so forth. In these applications, establishing the relations among the filtering performance level and the parameters characterizing the measurement missing is of prior importance. Existing results on this topic can be generally classified into two frameworks, namely, the stochastic framework [18–22] and the deterministic framework [23].
A general structure of the filtering system with measurement missing is shown in Figure 1, where y is the measurement and y^ is the filter input. In the stochastic framework, the measurement missing is usually described by a Bernoulli sequence or a Markov chain. For those results using the Bernoulli sequence, such as [14, 15], the filter input is usually described as y^(k)=γ(k)y(k)+(1-γ(k))y^(k-1) or as y^(k)=γ(k)y(k) by introducing the binary random variable γ(k), where Prob{γ(k)=1}=γ, and γ is usually called the measurement arrival probability while 1-γ is called the measurement missing probability. For those results using Markov chain technique, such as [12, 21], the measurement missing and the measurement arrival are defined as two modes of the Markov chain, and the mode transition probabilities are assumed to be known. In these ways, the stochastic parameters (either the Bernoulli sequence or the Markov chain) are incorporated into the filtering system models, and the relation between the filtering performance level and the measurement missing probability is implicitly established. Note that the maximal number of consecutive measurement missing is an important parameter that characterizes the measurement missing, and it also plays an important role in affecting the filtering performance. However, the relation between the filtering performance level and the maximal number of consecutive measurement missing is not established in all the aforementioned results using stochastic framework. Being different from the stochastic framework, the effects of the measurement missing on the filtering system are treated as time delays in the filter input in the deterministic framework, and the filtering error system with bounded measurement missing is usually described as a deterministic system with bounded time-varying delays. In this way, the relation between the filtering performance level and the maximal number of consecutive measurement missing is established. Some results on this topic can be found, for example, in [7, 23], where the H∞ filtering was investigated for linear continuous-time systems with bounded measurement missing and delays. However, the relation between the filtering performance level and the measurement missing rate/probability is not established in the existing results using deterministic framework.
Diagram of the filtering problem with measurement missing.
By a closer inspection, it is found that the relations among the filtering performance level and the parameters characterizing the measurement missing revealed in the existing results using either the stochastic framework or the deterministic framework are quite implicit. Actually, besides of determining the maximal allowable measurement missing rate/probability bound, one may be interested more in the quantitative results about how much the filtering performance declines for a certain amount of measurement missing and for a certain extent of increase on the maximal number of consecutive measurement missing, or in other words, the quantitative relations among the performance level and the parameters characterizing the measurement missing. Specifically, this gives rise to the problem of how to express the filtering performance level as a function of the parameters characterizing the measurement missing. To the best of the authors’ knowledge, such quantitative relations for the filtering systems with bounded measurement missing have not yet been established in the existing results, which motivates the presented research.
In this paper, the H∞ filtering problem is investigated for networked systems with bounded measurement missing. The filtering error system is described as a switched linear system by using the augmentation technique and by including the numbers of consecutive measurement missing as switching parameters. Based on the obtained switched system model, a sufficient condition is derived for the filtering system to be exponentially stable and achieve a prescribed H∞ performance level. A convex optimization problem is also presented to design the linear H∞ filters which guarantee that the considered filtering system achieves a suboptimal H∞ performance level. The main contributions of the paper are summarized as follows. (1) A switched system model with finite number of subsystems is established to describe the networked filtering system with bounded missing measurements. (2) A quantitative relation is established between the filtering performance and parameters characterizing the measurement missing. Specifically, the exponential decay rate of the filtering error system is given as a monotonic increasing function of the measurement missing rate bound, while the H∞ performance level is given as a monotonic increasing function of both the measurement missing rate bound and the maximal number of consecutive measurement missing. (3) The established relation gives quantitative results about how much the H∞ filtering performance level declines for a certain amount of measurement missing and for a certain extent of increase on the maximal number of consecutive measurement missing. They theoretically reveal that measurement missing degrades the H∞ filtering performance.
2. Modelling of the Filtering System
The considered filtering problem is shown in Figure 1, where the physical plant is described by
(1)x(k+1)=Apx(k)+Bpw(k),y(k)=Cpx(k)+Dpw(k),z(k)=Lpx(k),
where x(k)∈ℜn is the system state, y(k)∈ℜp is the measured output, z(k)∈ℜq is the signal to be estimated, and w(k)∈ℜm is the noise signal which belongs to l2[0,∞). Ap, Bp, Cp, Dp, and Lp are constant matrices with appropriate dimensions, and Ap is assumed to be stable. The full-order linear filter used to estimate the signal z(k) is given by
(2)xf(k+1)=Afxf(k)+Bfy^(k),zf(k)=Cfxf(k),
where xf(k)∈ℜn is the filter state, y^(k)∈ℜp is the filter input, and zf(k)∈ℜq is the estimated signal. Af, Bf, and Cf are filter matrices to be determined.
Since the measurements may be missed during the transmission from the sensor to the filter, the filter input y^ may not be equal to the measured output y. Suppose that the number of consecutive measurement missing i is upper bounded by i≤d and the filter input holds at its last available value if a measurement is missed, where d is a known constant. Then, we have
(3)y^(k)=y(k-i)
if there are i consecutive measurement losses before the time step k+1, where i∈Z0={0,1,…,d}. It can be seen from (3) that the system model of the filter varies over different sampling intervals since the number of consecutive measurement missing is time varying over different sampling periods. By defining the filtering error signal as e(k)=z(k)-zf(k) and denoting
(4)ξ(k)=[xT(k)xT(k-1)⋯xT(k-d)xfT(k)]T,υ(k)=[wT(k)wT(k-1)⋯wT(k-d)]T,
we obtain the following filtering error system model:
(5)Sσ(k):{ξ(k+1)=Aσ(k)ξ(k)+Bσ(k)υ(k)e(k)=Cξ(k),
where σ(k)∈Z0, and
(6)Aσ(k)=[[ApΛaΠ][0Oa]BfCpIaσ(k)Af],Bσ(k)=[[BpΛbOb]BfDpIbσ(k)],C=[[LpOc]-Cf],Λa=[I1OaT],Λa=[I3OdT],Π=[I2Oa],Oa=0∈Rdn×n,Ob=0∈Rdn×(d+1)n,Oc=0∈Rp×dn,Od=0∈Rdm×m,I1=I∈Rn×n,I2=I∈Rdn×dn,I3=I∈Rm×m,Iaσ(k)=[Δa0Δa1⋯Δad],Ibσ(k)=[Δb0Δb1⋯Δbd],Δai={I1,σ(k)=i0,σ(k)≠i,Δbi={I3,σ(k)=i0,σ(k)≠i.
By applying the augmentation technique, the filtering error system is finally described as a nondelayed switched system by including the numbers of consecutive measurement missing as switching parameters. σ(k) serves as the switching signal of system (5) which contains d+1 subsystems. The switching of the subsystems is determined by the measurement missing status. Specifically, for i=0,1,…,d, we have σ(k)=i and system (5) resides in the subsystem Si during the interval [k,k+1) if there is i consecutive measurement missing before time step k+1. This is illustrated in Figure 2, where • represents the measurement that is successfully received by the filter, while ∘ stands for the one that is missed. At time step k-1, the measurement y(k-1) is successfully received by the filter. Then, y^(k-1)=y(k-1), and Sσ(k) resides in S0. At time step k, the measurement y(k) is missed and the filter input holds at its previous value; that is, y^(k) takes y(k-1), and Sσ(k) resides in S1. Then, at time step k+1, the measurement y(k+1) is missed again, and the filter input remains at y(k-1); that is, y^(k+1)=y(k-1), and Sσ(k) resides in S2. The rest can be deduced by analogy. It is seen from the above analysis that the filtering error system Sσ(k) takes on the following characteristics.
For i=1,…,d-1, subsystem Si+1 always appears after the subsystem Si, and subsystem S0 always appears before S1 or after Sd.
Switching between the subsystems of Sσ(k) occurs only when the subsystems Sj (j=1,…,d) appear, and the number of switches of Sσ(k) is determined by the total activation times of Sj. Moreover, at most two switches may be involved when Si appears one time.
An example of measurement missing.
For i=0,1,…,d and 0≤k1≤k≤k2, let Nσ[k1,k2] denote the number of switches of σ(k) and ni[k1,k2] the times that the subsystem Si is activated on the interval [k1,k2]. Then, it follows from the characteristics of Sσ(k) that the following relation holds:
(7)Nσ[k1,k2]≤2∑i=1dni[k1,k2].
Definition 1.
For any switching signal σ(k) and 0≤k1≤k≤k2, let Nσ[k1,k2] denote the number of switches of σ(k) on the interval [k1,k2]. Then, θ[k1,k2]=Nσ[k1,k2]/(k2-k1) is called the switching frequency of σ(k) on the interval [k1,k2].
Definition 2.
For i∈Z0 and 0≤k1<k2, let αi[k1,k2]=ni[k1,k2]/(k2-k1). Then αi[k1,k2] is called the occurrence rate of the subsystem Si on the interval [k1,k2].
The subsystem Sj, j∈{1,…,d} is activated when a measurement missing occurs, so the measurement missing rate on any interval [k1,k2] can be defined as α[k1,k2]=∑i=1dαi[k1,k2]. Then, we have by relation (7) and Definitions 1 and 2 that
(8)θ[k1,k2]≤2α[k1,k2].
Moreover, we have by the characteristics of Sσ(k) that there exists a constant α<1 such that the following relations are true for any t≥0 and l≥0:
(9)α[t,t+l]≤1,l≤dα[t,t+l]≤α<1,l>d.
Remark 3.
The considered filtering system with bounded measurement missing is finally described as a switched linear system Sσ(k) with d+1 subsystems, where S0 describes the filtering system without measurement missing, while Sj (j=1,…,d) describes those with measurement missing. It is reasonable to see that the filtering system will achieve a better H∞ performance level if S0 appears with higher frequency (or, in other words, the measurement missing rate is lower). This is the motivation of the proposed switched system model of the filtering error system, which may enable us to establish the quantitative relation between the H∞ filtering performance and the measurement missing rate bound. Note that the augmentation technique is used to obtain the switched system model, which increases computational cost. From the energy-efficient perspective, the proposed method is applicable to the case where the number of consecutive packet losses is bounded and not too large.
The following definition and assumptions are needed in the derivation of the main results.
Definition 4.
System (5) is said to be exponentially stable with decay rate λ<1 if, for any finite initial state ξ(0)∈ℜ(d+2)n, there exist a constant c>0 such that ∥ξ(k)∥≤cλk∥ξ(0)∥ holds.
Assumption 5.
For any t≥0 and l>d, α[t,t+l] is upper bounded by a constant measurement missing rate bound α; that is, α[t,t+l]≤α, where 0≤α<1.
Assumption 6.
ξ(k)=0 and w(k)=0, for all k≤0.
The H∞ filtering problem to be addressed in this paper is expressed as follows.
Problem HFBMM (H∞ Filtering with Bounded Measurement Missing). For the filtering problem in Figure 1 and a given system (1), determine the matrices Af, Bf, and Cf in filter (2), such that the system (5) with υ(k)=0 is exponentially stable, and the estimation error e satisfies a prescribed H∞ performance level γ (i.e., ∥e∥2≤γ∥υ∥2) under zero initial conditions (i.e., ξ(k)=0,forallk≤0) for all admissible bounded measurement missing.
An H∞ performance condition for the filtering error system (5) is presented in the following theorem.
Theorem 7.
For given scalars μ>1 and 0<λ<1 satisfying μλ<1, if there exist matrices Pi>0, i=0,1,…,d, and a scalar γ0>0, such that the following inequalities hold:
(10)Ωi=[-λ2Pi+CTC00-γ02I]+[AiTBiT]Pi[AiBi]<0,(11)Pα≤μPβ,α,β∈Z0,
then system (5) is exponentially stable with decay rate δ(α)=λμα and achieves a prescribed H∞ performance level γ(α,d)=γ0((1-λ2)(1-(μλ)2(d+1))/(1-(μλ)2)+(1-λ2)(μαλ)2(d+1)/(1-(μαλ)2))1/2, where α is the measurement missing rate bound.
Proof.
Choose the Lyapunov function Vσ(k)(k)=ξT(k)Pσ(k)ξ(k) and define J(k)=∥e(k)∥2-γ02∥υ(k)∥2. For i∈Z0, it follows from (5) and (10) that
(12)Vi(k+1)-λVi(k)+J(k)=ηT(k)Ωiη(k)<0,
where η(k)=[ξT(k)υT(k)]T. For any given integer k≥1, let k1<⋯<ki, i≥1 denotes the switching instants of the switching signal σ(k) on the interval [0,k]. Then, we have by (11) that
(13)Vσ(ki)(ki)=ξT(ki)Pσ(ki)ξ(ki)≤μξT(ki)Pσ(ki-1)ξ(ki)=μVσ(ki-1)(ki),∀i≥1.
It follows from (12) and (13) that
(14)Vσ(k)(k)≤λ2(k-ki)Vσ(ki)(ki)-∑t=kik-1λ2(k-1-t)J(t)≤λ2(k-ki)·μ·Vσ(ki-1)(ki)-∑t=kik-1λ2(k-1-t)J(t)≤λ2(k-ki)·μ·[∑t=ki-1ki-1λ2(ki-ki-1)Vσ(ki-1)(ki-1)ssssssssssssdd-∑t=ki-1ki-1λ2(ki-1-t)J(t)]-∑t=kik-1λ2(k-1-t)J(t)a⋮≤μNσ[0,k-1]λ2(k-ki)λ2(ki-ki-1)⋯λ2k1Vσ(0)(0)-Γ(J(t)),
where
(15)Γ(J(t))=μNσ[0,k-1]λ2(k-ki)λ2(ki-ki-1)⋯λ2(k2-k1)×∑t=0k1-1λ2(k1-1-t)J(t)+μNσ[0,k-1]-1λ2(k-ki)λ2(ki-ki-1)⋯λ2(k3-k2)×∑t=k1k2-1λ2(k2-1-t)J(t)⋮+μ0∑t=kik-1λ2(k-1-t)J(t)=Γ(∥e(t)∥2)-γ02Γ(∥υ(t)∥2).
First, we consider the exponential stability of system (5) for υ(k)=0. It follows from (8), (14), υ(k)=0, and Nσ[0,k-1]=(k-1)·θ[0,k-1]≤k·θ[0,k-1] that
(16)Vσ(k)(k)≤μNσ[0,k-1]λ2(k-ki)λ2(ki-ki-1)⋯λ2k1Vσ(0)(0)≤μ2k·(θ[0,k-1]/2)λ2kVσ(0)(0)≤(μα[0,k-1]λ)2kVσ(0)(0).
It is reasonable in practice that the time horizon k is larger than d. So, it can be further obtained from (16) by assumption 1 that
(17)Vσ(k)(k)≤(μαλ)2kVσ(0)(0)=δ2kVσ(0)(0)
which yields ∥ξ(k)∥≤πM/πmδk∥ξ(0)∥, where πM=maxi∈Z0{λmax(Pi)} and πm=mini∈Z0{λmin(Pi)}. λmax(⋆) and λmin(⋆) represent the maximum and the minimum eigenvalues of the matrix ⋆, respectively. On the other hand, α<1 and μλ<1 guarantee that δ<1. Thus, it is concluded by Definition 4 that system (5) is exponentially stable with decay rate δ.
Next, to prove the H∞ performance, we consider υ(k)≠0. Then, under zero initial condition, we have, by (14) and the fact that Vσ(k)(k)≥0, that Γ(J(t))=Γ(∥e(t)∥2)-γ02Γ(∥υ(t)∥2)≤0, and thus
(18)∑t=0k-1μNσ[t,k-1]λ2(k-1-t)∥e(t)∥2=Γ(∥e(t)∥2)≤γ02Γ(∥υ(t)∥2)=γ02∑t=0k-1μNσ[t,k-1]λ2(k-1-t)∥υ(t)∥2.
It follows from Definition 1 that Nσ[t,k-1]=(k-1-t)·θ[t,k-1]≤2(k-1-t)α[t,k-1], which leads to
(19)1≤μNσ[t,k-1]≤μ2(k-1-t)α[t,k-1].
We have by (18) and (19) that
(20)∑t=0k-1λ2(k-1-t)∥e(t)∥2≤γ02∑t=0k-1(μα[t,k-1]λ)2(k-1-t)∥υ(t)∥2.
Summing both sides of (20) from k=1 to k=+∞ and changing the order of the summation, taking (9) into account, we obtain
(21)(1-λ2)-1∑t=0+∞∥e(t)∥2=∑t=0+∞∥e(t)∥2·(∑k=t+1+∞λ2(k-1-t))=∑k=1+∞∑t=0k-1λ2(k-1-t)∥e(t)∥2≤γ02∑k=1+∞∑t=0k-1(μα[t,k-1]λ)2(k-1-t)∥υ(t)∥2=γ02∑t=0+∞∥υ(t)∥2·(∑k=t+1+∞(μα[t,k-1]λ)2(k-1-t))=γ02∑t=0+∞∥υ(t)∥2·(∑i=0d(μα[t,t+i]λ)2idddddddddddd+∑i=d+1+∞(μα[t,t+i]λ)2i∑i=0d)≤γ02∑t=0+∞∥υ(t)∥2·(∑i=0d(μλ)2i+∑i=d+1+∞(μαλ)2i)=γ02(1-(μλ)2(d+1)1-(μλ)2+(μαλ)2(d+1)1-(μαλ)2)×∑t=0+∞∥υ(t)∥2.
It follows from (21) that ∥e∥2≤γ(α,d)∥υ∥2, which implies that the filtering error system (5) achieves the H∞ performance level γ(α,d). The proof is completed.
Remark 8.
Theorem 7 establishes the quantitative relations among the H∞ filtering performance level, the measurement missing rate bound, and the maximal number of consecutive measurement missing. Specifically, the H∞ filtering performance level γ(α,d) is given as a monotonic increasing function of both the measurement missing rate bound α and the maximal number of consecutive measurement missing d, which implies that some smaller measurement missing rate and maximal number of consecutive measurement missing lead to a better H∞ filtering performance.
Remark 9.
It is obtained from (12) that
(22)Vi(k)<λkV(0)-∑t=0k-1λ(k-1-t)J(t).
It follows from (22) that Vi(k)<λkV(0) for υ(k)=0, which implies that the subsystem Si (i∈Z0) is exponentially stable with decay rate λ. On the other hand, we have by (22) that
(23)∑t=0k-1λ(k-1-t)J(t)<0
under zero initial condition. Summing both sides of (23) from k=1 to k=+∞, we obtain (1-λ2)-1∑t=0+∞J(t)<0, which yields ∥e∥2<γ0∥υ∥2. It implies that the subsystem Si achieves the H∞ performance level γ0. In summary, if (10) is true for 0<λ<1, then the subsystem S0 is exponentially stable with decay rate λ and achieves the H∞ performance level γ0. Note that if there is no measurement missing, then Sσ(k) resides in S0, and we have α=0, d=0 and μ=1. In this case, δ(α) and γ(α,d) in Theorem 7 are, respectively, reduced to λ and γ0, which are just the decay rate and the H∞ performance level of S0, respectively. So, Theorem 7 theoretically reveals that measurement missing degrades the H∞ filtering performance and presents the quantitative result about the effect of the measurement missing on the H∞ filtering performance.
Consider the filtering problem in Figure 1. For given scalars μ>1 and 0<λ<1 satisfying μλ<1, if there exist matrices Qi11>0, Qi22>0, Qi12, i=0,1,…,d, U1, U2, U3, A^f, B^f, and C^f of appropriate dimensions and a scalar γ0>0, such that the following linear matrix inequalities hold:(24)[λ2Qi11λ2Qi120ΔaTU1+IaiTCpTB^fTETΔaTU3+IaiTCpTB^fT[LpOc]T*λ2Qi220A^fTETA^fT-C^fT**γ02IΔbTU1+IbiTDpTB^fTETΔbTU3+IbiTDpTB^fT0***U1+U1T-Qi11U3+EU2-Qi120****U2+U2T-Qi220*****I]>0,(25)Qα≤μQβ,α,β∈Z0,then the filters of form (2) guarantee that the filtering error system (5) is exponentially stable with decay rate δ(α) and achieves a prescribed H∞ performance level γ(α,d), and the filter matrices are given by
(26)Af=U2-1A^f,Bf=U2-1B^f,Cf=C^f,
where δ(α) and γ(α,d) are given in Theorem 7, and
(27)Δa=[ApΛaΠ],Δb=[BpΛbOb],Qi=[Qi11Qi12Qi12TQi22],E=[In×n0dn×dn].
Proof.
By Lemma 1 in [24], (10) is true if and only if there exists a matrix G such that the following inequality holds:
(28)[[λi2Pi-CTC00γ2I][AiTBiT]G*G+GT-Pi]>0,
On the other hand, (24) indicates that U2 is nonsingular, and thus we can always find square and nonsingular matrices M∈Rn×n and G22∈Rn×n satisfying U2=MTG22-1M. Now, introduce the variables U1∈R(d+1)n×(d+1)n and G12∈R(d+1)n×n, and let
(29)G=[U1G12METG22],U3=G12G22-1M,J1=[I00G22-1M],[AfBfCf0]=[M-T00I][A^fB^fC^f0][M-1G2200I].
Then, pre- and postmultiplying (28) and (11) by diag{J1,I,J1} and J1, respectively, following some routine matrix manipulations, yield (24) and (25). Furthermore, denote the filter transfer function from y^(k) to zf(k) by Tf(z)=Cf(zI-Af)-1Bf. Substituting the filter matrices with (29) and taking U2=MTG22-1M into account, we obtain Tf(z)=C^f(zI-U2-1A^f)-1U2-1B^f. So, the filter matrices are given by (26). Then, it follows from Theorem 7 that Theorem 10 is true. The proof is completed.
The condition in Theorem 10 is convex in γ02. So, the convex optimization problem
(30)minρs.t.kk(24)and(25)withρ=γ02
can be formulated to design the H∞ filters. If ρ* is the solution of the optimization problem (30), then the designed filters guarantee that system (5) achieves the following H∞ performance level:
(31)γ*(α,d)=(ρ*(1-λ2)(1-(μλ)2(d+1))1-(μλ)2ffgggggg+ρ*(1-λ2)(μαλ)2(d+1)1-(μαλ)2)1/2.
Remark 11.
Note that γ*(α,d) is monotonic increasing on μ and λ. Therefore, μ and λ should be chosen as small as possible to obtain a better H∞ filtering performance and meanwhile to make the condition μλ<1 easier to be satisfied. Nevertheless, it can be seen from (24) and (25) that some smaller μ and λ will result in a larger value of γ0 and ultimately yield a larger value of γ*(α,d). The following algorithm provides a method on determining the parameters μ and λ that will result in a suboptimal H∞ performance level γ*(α,d).
Algorithm 12.
Step 1. Choose some sufficiently large initial μ and λ such that μλ<1 holds. Set μ0=μ and λ0=λ. Solve the optimization problem (30) and calculate γ*(α,d,μ0,λ0) for given α and d.
Step 2. Decrease λ0 by a certain step length; say, Δλ; that is, set λ0=λ0-Δλ. Then, solve the optimization problem (30) and calculate γ*(α,d,μ0,λ0) for given α and d.
Step 3. If γ*(α,d,μ0,λ0)<γ*(α,d,μ0,λ0+Δλ), then return to Step 2. Otherwise, exit and set λ*=λ0+Δλ.
Step 4. Decrease μ0 by a certain step length; say, Δμ; that is, set μ0=μ0-Δμ. Then, solve the optimization problem (30) and calculate γ*(α,d,μ0,λ*) for the obtained λ* and given α and d.
Step 5. If γ*(α,d,μ0,λ*)<γ*(α,d,μ0+Δμ,λ*), then return to Step 4. Otherwise, exit and set μ*=μ0+Δμ. Then, μ* and λ* are the desired values of the parameters μ and λ, respectively, and γ*(α,d,μ*,λ*) is the desired suboptimal H∞ performance level for given α and d.
5. Illustrative ExamplesExample 5.
Consider a mechanical system with two masses and two springs as that studied in [7]. Its state-space model is given by
(32)x˙(t)=[00100001-k1+k2m1k2m1-cm10k2m2-k2m20-cm2]x(t)+[001m10]w(t),
where x1 and x2 are the positions of masses m1 and m2, respectively. Choose m1=1, m2=0.5, k1=k2=1, and c=0.5, and set T=200ms; then we obtain the discrete-time system model (1) with
(33)Ap=[0.96170.01910.18780.0012280.036970.96290.0024560.1789-0.37320.18530.86780.017870.3528-0.35530.035740.784],Bp=[0.019220.0001250.18780.002456].
Suppose that x1 is measured by a device with noise w(k) and x2 is to be estimated by using the H∞ filters; then we have Cp=[1000] and Lp=[0100]. Furthermore, we choose Dp=0.1.
Choose μ=1.01 and λ=0.96. Suppose that d=2 and α=50%. Then, by solving the optimization problem (23), we obtain the H∞ performance level γ*=0.8001 with the following filter gain matrices:
(34)Af=[-0.0691-0.06660.3304-0.0409-0.52880.91800.07930.1596-1.18960.16871.0783-0.00350.8709-0.3034-0.02190.8090],Bf=[-1.0655-0.5858-0.79880.5298],Cf=[0.0003-1.0002-0.0002-0.0001].
In the simulation, the noise signal w(k) is assumed to be uniformly distributed with [-0.1,0.1] for the interval [0,50]. The measurement missing is generated randomly and is shown in Figure 3(b), where the measurement missing rate is 50%. The trajectories of z(k) and zf(k) are depicted in Figure 3(a). By calculation, we obtain from the results in Figure 3(a) that γa=∥e∥2/∥υ∥2=0.0460<γ*=0.8001, showing the effectiveness of the H∞ filter design. In what follows, we will show the relations among the H∞ performance level, the measurement missing rate bound, and the maximal number of consecutive measurement missing. For μ=1.01, λ=0.96, and d=2, the relation between the H∞ performance level and the measurement missing rate bound is given in Table 1 by solving the optimization problem (30). For μ=1.01, λ=0.96, and α=50%, the relation between the H∞ performance level and the maximal number of consecutive measurement missing is presented in Table 2 by solving the optimization problem (30). In calculating the actual noise attenuation level γa, the noise signal w(k) in Figure 3 is adopted, and the measurement missing is generated randomly under the constraint of the corresponding missing rate bound. It can be seen from Table 1 that the larger the missing rate bound, the worse the H∞ filtering performance. Similarly, Table 2 shows that the larger the maximal number of consecutive measurement missing, the worse the H∞ filtering performance. These verify the statements in Theorem 7. Moreover, γa is always smaller than γ* for different values of α in Table 1, and it is always smaller than γ* for different values of d in Table 2, showing the effectiveness of the H∞ filter design. On the other hand, it can also be seen from Tables 1 and 2 that γ* is much larger than γa for different values of α and d, which indicates that there exists certain extent of conservatism in the H∞ filter design and the estimation of the H∞ performance level γ*.
The relation between the H∞ performance level and the measurement missing rate bound for d=2.
α
0%
20%
50%
80%
γ*
0.7525
0.7704
0.8001
0.8339
γa
0.0470
0.0475
0.0565
0.0926
The relation between the H∞ performance level and the maximal number of consecutive measurement missing for α=50%.
d
1
2
3
γ*
0.5529
0.8001
1.0027
γa
0.0218
0.0424
0.0506
Trajectories of z(k) and zf(k), the noise signal w(k), and the measurement missing status.
6. Conclusions
In this paper, the H∞ filtering problem was investigated for networked systems with bounded measurement missing. A switched system model was proposed to describe the considered system, which helped establish the quantitative relation between the H∞ performance level and two parameters, namely, the measurement missing rate bound and the maximal number of consecutive measurement missing. It has been shown by the example that there exists certain extent of conservatism in the proposed H∞ filter design, and the conservatism is partly introduced by the bounding on the switching frequency of the switched filtering error system given in (8). A tighter bounding on the switching frequency may help reduce the conservatism, which requires more detailed online information about the measurement missing status.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The work is partially supported by Zhejiang Provincial Natural Science Foundation under Grants nos. LQ12F03014 and Y13F010013.
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