The estimation problem is investigated for a class of stochastic nonlinear systems with distributed time-varying delays and missing measurements. The considered distributed time-varying delays, stochastic nonlinearities, and missing measurements are modeled in random ways governed by Bernoulli stochastic variables. The discussed nonlinearities are expressed by the statistical means. By using the linear matrix inequality method, a sufficient condition is established to guarantee the mean-square stability of the estimation error, and then the estimator parameters are characterized by the solution to a set of LMIs. Finally, a simulation example is exploited to show the effectiveness of the proposed design procedures.
1. Introduction
In the past decades, estimation techniques have been extensively investigated in many complex dynamical processes of networks such as target tracking [1], advanced aircrafts, and manufacturing processes. A number of estimation methods have been proposed in the literature, most of them are under the assumption that the measurements always contain true signals with the disturbances and the noises, see for example, [2–9]. But, in practical applications, the measurements may contain missing measurements due to many reasons such as the sensor temporal failures, network congestion, multipath fading, and high maneuverability of the tracked targets. Because of the clear engineering signification, the estimation problems with missing measurements have received attention, see for example [10–22].
Recently, with the rapid development of networks, novel methods and flexible models have been devoted, but the research of missing measurements is still a challenge, and the Bernoulli-based distributed model has still been a hot approach to modeling the missing observation cases. For example, in [10], the missing probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution over the interval [01]. Packet dropouts and communication delays are considered simultaneously in [12]. The variance-constrained dissipative control problem for a class of stochastic nonlinear systems with multiple degraded measurements in [13], where the degraded probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution over a given interval. The H∞ filtering problem has been addressed in [20] for a class of nonlinear systems with randomly occurring incomplete information, where the considered incomplete information includes both the sensor saturations and the missing measurements, a regional sensor model has been designed to account for both the randomly occurring sensor saturation and missing measurement in a unified representation, based on this sensor model, a newfangled H∞ filter with a certain ellipsoid constraint has been researched such that the filtering error dynamics is locally mean-square asymptotically stable and the H∞-norm requirement is satisfied.
On the other hand, time delays are frequently encountered in real-world application such as communications, engineering, and biological systems. The occurrence of time delays may induce instability, oscillation, and poor performance. Consequently, research on time-delay systems has been a topic of recurring interest over the past decades. Current efforts can be classified into several categories, for example, simple delay and multiple delays [12], delay-independence [23, 24] and delay-dependence [5, 8, 25–30], time-varying delays [31, 32] and constant delays, retarded-type delay and neutral-type delay [30, 33], and mixed delays [34, 35]. However, in some applications, such as these systems connected over a wireless networks/or neural networks, as pointed out in [36], networks usually have a spatial extent due of the presence of a multitude of parallel pathways with a variety of axon sizes and lengths, and therefore the propagation delays can be distributed over a period of time, so it is essential to describe the distributed time delay under the probability framework as possible as. In this paper, the probability distribution of the time-vary delays are described for Itô type discrete-time stochastic distribution by a binary switching sequence satisfying the Bernoulli-distributed model.
Motivated by the aforementioned discussions, in this paper, we model the stochastic nonlinearities, the missing measurements, and the distributed time-vary delays by Bernoulli distributed white sequence with known conditional probability distribution. We aim at designing a estimator such that, for all possible measurements missing and distributed time-vary delays to obtain the estimation error system mean-square stable. The solvability of the addressed estimation problem can be expressed as the feasibility of a set of LMIs. Finally, a numerical simulation example is exploited to show the effectiveness of the results derived. The main contributions of this paper are summarized as the following: (1) a new estimation problem is studied for the stochastic nonlinear systems with both distributed time-vary delays and measurements missing phenomenon; (2) a mean-square stable performance is taken into consideration for the addressed stochastic nonlinear systems with distributed time-vary delays and missing measurements.
The rest of this paper is organized as follows. Section 2 briefly introduces the problem under consideration. In Section 3, a sufficient condition is established such that, for the missing measurements, the randomly distributed time-varying delays and nonlinearities, the estimation error system is the mean-square stability. A numerical example is given in Section 4. This paper is concluded in Section 5.
Notations.
The notation used here is fairly standard except where otherwise stated. ℝn, ℝn×m, and 𝕀+ denote, respectively, the n-dimension Euclidean space, the set of all n×m real matrices, and the set of nonnegative integers. (Ω,ℱ,{ℱk}k∈𝕀+,ℙ) is complete filtered probability space, Ω is the sample space, ℱ is the σ-algebra of subsets of the sample space, and ℙ is the probability measure on ℱ. 𝔼{x} stands for the expectation of the stochastic variable x. Prob{·} is used for the occurrence probability of the event “·”. The superscript “T” stands for matrix transposition. P>0(P≥0) means that matrix P is real symmetric and positive definite (positive semi-definite). λmin(·) denotes the minimum eigenvalue of a matrix. I and 0 represent the identity matrix and the zero matrix with appropriate dimensions, respectively. diag{X1,X2,…,Xn} stands for a block-diagonal matrix with matrices X1,X2,…,Xn on the diagonal. In symmetric block matrices or long matrix expressions, we use “*” to represent a term, that is, induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.
2. Problem Formulation and Preliminaries
Consider the following class of stochastic nonlinear system with distributed time-varying delays:
(2.1)x(k+1)=Ax(k)+κ1(k)B∑m=-τ(k)-1x(k+m)+κ2(k)f(x(k))+E1x(k)w(k),y(k)=κ3(k)Cx(k)+E2x(k)w(k),z(k)=H1x(k),
where x(k)∈ℝn is the state vector, y(k)∈ℝm is the measured output vector, z(k)∈ℝq is the signal to be estimated, w(k) is a one-dimensional, zero-mean, Gaussian white noise sequence on a probability space (Ω,ℱ,{ℱk}k∈𝕀+,ℙ) with 𝔼{ω2(k)}=1, A, B, C, E1, E2, and H1 are known real constant matrices with appropriate dimensions, τ(k)denoting time-varying delays are positive integers and bounded, namely, 0<τl≤τ(k)≤τu, the stochastic variables κ1(k)∈ℝ, κ2(k)∈ℝ, and κ3(k)∈ℝ are Bernoulli distributed white sequence taking the values of 0 and 1 with
(2.2)Prob{κ1(k)=1}=𝔼{κ1(k)}:=α1,(2.3)Prob{κ1(k)=0}:=1-α1,(2.4)Prob{κ2(k)=1}=𝔼{κ2(k)}:=α2,(2.5)Prob{κ2(k)=0}:=1-α2,(2.6)Prob{κ3(k)=1}=𝔼{κ3(k)}:=α3,(2.7)Prob{κ3(k)=0}:=1-α3,
where α1∈[01], α2∈[01], and α3∈[01] are known positive scalars.
Remark 2.1.
The nonlinear stochastic f(x(k)) is assumed to have the following for all x(k):
(2.8)𝔼{f(x(k))∣x(k)}=0,𝔼{f(x(k))fT(x(k))∣x(k)}=0,k≠j,𝔼{f(x(k))fT(x(k))∣x(k)}≤∑i=1qΠixT(k)Φix(k),
where q is a known nonnegative integer, Πi=Π-iΠ-iT, Πi, Π-i, and Φi(i=1,…,q) are known matrices with appropriate dimensions. For convenience, one assumes that f(x(k)) is unrelated with κ1(k), κ2(k), κ3(k), and ω(k).
In this paper, we aim at designing a linear estimator of the following structure:
(2.9)xf(k+1)=Afxf(k)+Aky(k),z^(k)=H2xf(k),z^(0)=0,
where xf∈ℝn is the state estimate, z^(k) is the estimate output, H2 is a known real constant matrix with appropriate dimension, and Af and Ak are estimator parameters to be determined.
By defining x^(k)=[xT(k)xfT(k)]T, we have the following augmented system:
(2.10)x^(k+1)=𝒜x^(k)+𝒜-x^(k)+ℬ∑m=-τ(k)-1x^(k+m)+ℬ-∑m=-τ(k)-1x^(k+m)+𝒩h(k)+𝒩-h(k)+ℰx^(k)w(k),
where
(2.11)𝒜=[A0α3AkCAf],𝒜-=[00(κ3(k)-α3)AkC0],ℬ=[α1B000],ℬ-=[(κ1(k)-α1)B000],x^(k+i)=[x(k+i)xf(k+i)],h(k)=[f(x(k))0],ℰ=[E10AkE20],𝒩^=[I000],𝒩=α2𝒩^,𝒩-=(κ2(k)-α2)𝒩^.
Observe the system (2.10) and let x^(k;φ) denote the state trajectory from the initial data x^(s)=φ(s) on -ξM≤s≤-ξm. Obviously, x^(k;0)≡0 is the trivial solution of system (2.10) corresponding to the initial data φ=0.
In what follows, we aim to design a linear estimator of the form (2.9) for system (2.1) such that, for all admissible randomly occurring distributed time-varying delays, missing measurements, stochastic nonlinearities, and estimation error system (2.10) is mean-square stable.
3. Main Results
The following lemmas are essential in establishing our main results.
Lemma 3.1 (Schur Complement).
There are constant matrices Υ1, Υ2, and Υ3 where Υ1=Υ1T and Υ2=Υ2T>0, then Υ1+Υ3TΥ2-1Υ3<0 if and only if [Υ1Υ3TΥ3-Υ2]<0.
Lemma 3.2.
Let 𝒲∈ℝn×n be a positive semidefinite matrix, xi∈ℝn be a vector, and ai≥0(i=1,2,…) be scalars. If the series concerned are convergent, then the following inequality holds [35]
(3.1)(∑i=1+∞aixi)T𝒲(∑i=1+∞aixi)≤(∑i=1+∞ai)∑i=1+∞aixiT𝒲xi.
In the following theorem, Lyapunov stability theorem and a LMI-based method are combined together to deal with the stability analysis issue for the estimator design of the discrete-time stochastic nonlinear system with distributed time-varying delays and missing measurements. A sufficient condition is derived that ensures the solvability of the estimation problem.
Theorem 3.3.
Given the estimator parameters Af and Ak consider the estimation error system (2.10). If there exist positive definite matrices P=PT>0,Q=QT>0, and positive scalars ϖi>0(i=1,2,…,q) such that the following matrix inequalities,
(3.2)[-P*******0-Q******Pℰ0-P*****β1Q00-Q****P𝒜Pℬ00-P***β3P0000-P**β4Φ^00000-Ξ*0β2Pℱ00000-P]<0,(3.3)Δ=[-ϖiI*P𝒩^Π-i-P]<0,i=1,2,…,q,
hold, where
(3.4)β1=(τu+12(τu-τl)(τu+τl-1))1/2,β2=(α1(1-α1))1/2,β3=(α3(1-α3))1/2,β4=(α2)1/2,Φ^=[ϖ1Φ-11/2,…,ϖqΦ-q1/2]T,Φ-i=[Φi000],Π-i=[π^iπ^i],Ξ=diag{ϖ1I,…,ϖqI},ℱ=[B000],
then the estimation error system (2.10) is mean-square stable.
Proof.
Define the following Lyapunov functional candidate for system (2.10):
(3.5)V(x^(k),k)=x^T(k)Px^(k)+∑i=-τ(k)-1∑j=k+ik-1x^T(j)Qx^(j)+∑i=-τu-τl-1∑j=i+1-1∑n=k+jk-1x^T(n)Qx^(n).
By calculating the difference of the Lyapunov functional (3.5), based on Lemma 3.2, one has,
(3.6)𝔼{▵V(x^(k),k)}=𝔼{V(x^(k+1),k+1)∣x^(k)}-V{(x^(k),k)}=[𝒜x^(k)+ℬ∑m=-τ(k)-1x^(k+m)]TP[𝒜x^(k)+ℬ∑m=-τ(k)-1x^(k+m)]+𝔼{[ℬ-∑m=-τ(k)-1x^(k+m)]TP[ℬ-∑m=-τ(k)-1x^(k+m)]}+α3(1-α3)x^T(k)Px^(k)+x^T(k)ℰTPℰx^(k)+α2𝔼{hT(k)𝒩^TP𝒩^h(k)}-x^T(k)Px^(k)+∑i=-τ(k+1)-1∑j=k+i+1kx^T(j)Qx^(j)-∑i=-τ(k)-1∑j=k+ik-1x^T(j)Qx^(j)+∑i=-τu-τl-1∑j=i+1-1[∑n=k+j+1k-∑n=k+jk-1]x^T(n)Qx^(n)=[𝒜x^(k)+ℬ∑m=-τ(k)-1x^(k+m)]TP[𝒜x^(k)+ℬ∑m=-τ(k)-1x^(k+m)]+𝔼{[ℬ-∑m=-τ(k)-1x^(k+m)]TP[ℬ-∑m=-τ(k)-1x^(k+m)]}+α3(1-α3)x^T(k)Px^(k)+x^T(k)ℰTPℰx^(k)+α2𝔼{hT(k)𝒩^TP𝒩^h(k)}-x^T(k)Px^(k)+∑i=-τ(k+1)-1[∑j=k+i+1k-1x^T(j)Qx^(j)+x^T(k)Qx^(k)]-∑i=-τ(k)-1[∑j=k+i+1k-1x^T(j)Qx^(j)+x^T(k+i)Qx^(k+i)]+∑i=-τu-τl-1∑j=i+1-1[x^T(k)Qx^(k)-x^T(k+j)Qx^(k+j)]≤[𝒜x^(k)+ℬ∑m=-τ(k)-1x^(k+m)]TP[𝒜x^(k)+ℬ∑m=-τ(k)-1x^(k+m)]+𝔼{[ℬ-∑m=-τ(k)-1x^(k+m)]T[ℬ-∑m=-τ(k)-1x^(k+m)]}+α3(1-α3)x^T(k)Px^(k)+x^T(k)ℰTPℰx^(k)+α2𝔼{hT(k)𝒩^TP𝒩^h(k)}-x^T(k)Px^(k)+∑i=-τu-τl-1∑j=k+i+1k-1x^T(j)Qx^(j)+∑i=-τl-1∑j=k+i+1k-1x^T(j)Qx^(j)+τux^T(k)Qx^(k)-∑i=-τl-1∑j=k+i+1k-1x^T(j)Qx^(j)-∑i=-τ(k)-1x^T(k+i)Qx^(k+j)+12(τu-τl)(τu+τl-1)x^T(k)Qx^(k)-∑i=-τu-τl-1∑j=k+i+1k-1x^T(j)Qx^(j)=[𝒜x^(k)+ℬ∑m=-τ(k)-1x^(k+m)]TP[𝒜x^(k)+ℬ∑m=-τ(k)-1x^(k+m)]+𝔼{[ℬ-∑m=-τ(k)-1x^(k+m)]TP[ℬ-∑m=-τ(k)-1x^(k+m)]}+α3(1-α3)x^T(k)Px^(k)+x^T(k)ℰTPℰx^(k)+α2𝔼{hT(k)𝒩^TP𝒩^h(k)}-x^T(k)Px^(k)+τux^T(k)Qx^(k)-∑i=-τ(k)-1x^T(k+i)Qx^(k+i)+12(τu-τl)(τu+τl-1)x^T(k)Qx^(k)≤[𝒜x^(k)+ℬ∑m=-τ(k)-1x^(k+m)]TP[𝒜x^(k)+ℬ∑m=-τ(k)-1x^(k+m)]+𝔼{[ℬ-∑m=-τ(k)-1x^(k+m)]TP[ℬ-∑m=-τ(k)-1x^(k+m)]}+α3(1-α3)x^T(k)Px^(k)+x^T(k)ℰTPℰx^(k)+α2𝔼{hT(k)𝒩^TP𝒩^h(k)}-x^T(k)Px^(k)+τux^T(k)Qx^(k)+12(τu-τl)(τu+τl-1)x^T(k)Qx^(k)-∑i=-τ(k)-1x^T(k+i)Qx^(k+i)≤[𝒜x^(k)+ℬ∑m=-τ(k)-1x^(k+m)]TP[𝒜x^(k)+ℬ∑m=-τ(k)-1x^(k+m)]+𝔼{[ℬ-∑m=-τ(k)-1x^(k+m)]TP[ℬ-∑m=-τ(k)-1x^(k+m)]}+α3(1-α3)x^T(k)Px^(k)+x^T(k)ℰTPℰx^(k)+α2𝔼{hT(k)𝒩^TP𝒩^h(k)}-x^T(k)Px^(k)+τux^T(k)Qx^(k)+12(τu-τl)(τu+τl-1)x^T(k)Qx^(k)-τu-1(∑i=-τ(k)-1x^T(k+i))Q(∑i=-τ(k)-1x^(k+i)).
From (2.8), it can be seen that
(3.7)𝔼{hT(k)𝒩^TP𝒩^h(k)}≤∑i=1q[x^T(k)Φ-ix^(k)]tr(𝒩^Πi𝒩^TP),
where Πi:=Π-iΠ-iT with Φ-i and Π-i defined in (3.4).
Furthermore,
(3.8)𝔼{[ℬ∑m=-τ(k)-1x^(k+m)]TP[ℬ∑m=-τ(k)-1x^(k+m)]}≤β22∑m=-τ(k)-1x^T(k+m)ℱTPℱ∑m=-τ(k)-1x^(k+m),
where β2 is defined in (3.4).
From (3.6)–(3.8), one has
(3.9)𝔼{▵V(x^(k),k)}≤𝔼{ηT(k)Θη(k)},
where η(k)=[x^T(k),∑i=-τ(k)-1x^T(k+i)]T and
(3.10)Θ=[Θ1𝒜Pℬ*Θ2],
where Θ1=-P + ℰTPℰ + (τu+(1/2)(τu-τl)(τu+τl-1))Q + 𝒜TP𝒜 + β42∑i=1qΦ-itr(𝒩^Πi𝒩^TP) + β32P,Θ2 = -(1/τu)Q + ℬTPℬ + β22ℱTPℱ, β3, β4, ℱ are defined in (3.4).
From Lemma 3.1, (3.10) holds if and only if tr(𝒩^Πi𝒩^TP). Furthermore, by Lemma 3.1, one can obtain from (3.2), (3.3) that Θ<0 and, subsequently,
(3.11)𝔼{▵V(x^(k),k)}<-λmin(Θ)|x^(k)|2.
Thus, the augmented estimation system (2.10) is mean-square stable.
The following theorem is focused on the design of the desired estimation parameters Af and Ak by using the results in Theorem 3.3.
Theorem 3.4.
Consider the augmented estimation system (2.10) with given estimator parameters. If there exist positive-definite matrices S=ST>0,R=RT>0,Q=QT>0, matrices A~f,A~k, and positive scalars ϖi>0,(i=1,2,…,q) such that the following linear matrix inequalities holds
(3.12)Γ=[-S************-S-R***********00-Q**********SE1E10-S*********ϕ1ϕ20-S-R********β1Qβ1Q000-Q*******SASAα1SB000-S******ϕ3ϕ4α1RB000-S-R*****β3Sβ3S000000-S****β3Sβ3R000000-S-R***Φ~Φ~00000000-Ξ**00β2SB00000000-R*00β2RB00000000-S-R]<0,(3.13)[-ϖiI**Sπ^i-S*Rπ^i+A~fπ^i-S-R]<0,i=1,2,…,q,(3.14)S-R<0
hold, where α1 is defined in (2.2), β1,β2,β3, and β4 are defined in (3.4),
(3.15)Φ~T=[β4[ϖ1Φ11/2]T,…,β4[ϖqΦq1/2]T],ϕ1=RE1+A~fE2,ϕ2=RE1+A~fE2,ϕ3=RA+α2A~fC+A~k,ϕ4=RA+α2A~fC,
then the estimator parameters are designed as
(3.16)Ak=X12-1A~f,Af=X12-1A~kS-1(Y12T)-1,
where X12,Y12 are any square and nonsingular matrices satisfying X12Y12T=I-RS-1<0, then the estimation error system (2.10) is mean-square stable.
Proof.
Recall that our goal is to derive the expression of the estimator parameters from (2.9). To do this, we partition P and P-1 as
(3.17)P=[RX12X12TX22],P-1=[S-1Y12Y12TY22],
where the partitioning of P and P-1 is compatible with that of 𝒜 defined in (2.11), that is, R∈Rn×n,X12∈Rn×n,X22∈Rn×n,S∈Rn×n,Y12∈Rn×n, and Y22∈Rn×n. Define
(3.18)T1=[S-1IY12T0],T2=[IR0X12T]
which imply that PT1=T2 and T1TPT1=T1TT2.
By applying the congruence transformations diag{T1,I,T1,I,T1,T1,I,…,I,T1} and the congruence transformations diag{S,I,I,S,I,I,S,I,S,I,I,…,I,S,I} to (3.2), we have (3.12).
Again, performing the congruence transformation diag{I,T1} to (3.3) lead to (3.19)
(3.19)[-ϖiI**π^i-S-1*Rπ^i+X12Akπ^i-I-R]<0,i=1,2,…,q.
Then, one uses congruence transformation diag{I,S,I} to (3.19) and we have
(3.20)[-ϖiI**Sπ^i-S*Rπ^i+X12Akπ^i-S-R]<0,i=1,2,…,q.
Furthermore, if (3.12) is feasible, we have [-S-S-S-R]<0 or [-S-1IIR]>0.
It follows directly from XX-1=I that I-RS-1=X12Y12T<0. Hence, one can always find square and nonsingular X12 and Y12 [37]. Therefore, this completes the proof.
4. Numerical Example
In this section, an example is presented to illustrate the usefulness and flexibility of the estimator design method developed in this paper. The system data of (2.1)–(2.9) are the following:
(4.1)A=[0.1500.20.1],B=[0.09000.09],C=[1.2001.2],E1=[0.12000.12],E2=[0.12000.12],H1=H2=[0.6000.6],
where n=q=2,τ(k)=1+(1+(-1)k),τl=1,τu=3.
f(x(k)) describes the stochastic nonlinear function of the states in (2.1), which is bounded as follows:
(4.2)𝔼{f(x(k))f(x(k))T∣x(k)}=[0.220.22][0.220.22]TxT(k)[0.11000.11]x(k).
Let α1=0.2,α2=0.3, and α3=0.9. Using Matlab LMI Toolbox to solve the LMIs in (3.12)–(3.14), one has
(4.3)S=[0.6726-0.0035-0.00350.6563],R=[1.8796-0.0041-0.00411.8411],Q=[0.0668-0.0013-0.00130.0693],ϖ1=1.0776,ϖ2=1.3335.
Thus, we can calculate the estimator parameters as follows:
(4.4)Af=[0.13250.04860.0462-0.1465],Ak=[-0.9144-0.1677-0.1684-0.8160].
Remark 4.1.
Seldom of the estimation literature explicitly introduce the effects of the estimators by the digits in the graphs, for example [18]. In this paper, some digits are marked in Figures 1–4. Figures 1–2 show the actual measurements and ideal measurements. Figures 3–4 plot the estimation errors. From these digits in the graphs, it can be seen that the designed estimator performs well.
Actual Measurements y1(1,k) and ideal Measurements y2(1,k).
Actual Measurements y1(2,k) and ideal Measurements y2(2,k).
Estimation Errors z~(1,k).
Estimation Errors z~(2,k).
5. Conclusions
In this paper, we research the estimation problem for a class of stochastic nonlinear systems with both the probabilistic distributed time-varying delays and missing measurements. The distributed time-varying delays and missing measurements are assumed to occur in random ways, and the occurring probabilities are governed by Bernoulli stochastic variables. A linear estimator is designed such that, for the admissible random distributed delays, the stochastic disturbances, and the stochastic nonlinearities, the error dynamics of the estimation process is mean-square stable. At last, an illustrative example has been exploited to show the effectiveness of the proposed approach. In the future, we plan to consider the estimation problem with Markovian switching is in the finite-horizon case, and the nonlinearities are in more general forms.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grant nos. 60974030 and 61104125, the Science and Technology Project of Education Department in Fujian Province under Grant no. JA11211, and the Science and Technology Innovation Platform “CAD/CAM Engineering Research Center of Universities of Fujian Province” which is set up in Putian University, Putian, Fujian Province, China.
BugalloM. F.XuS.Performance comparison of EKF and particle filtering methods for maneuvering targets2007174774786ChenC.LiuH.GuanX.H∞ filtering of time-delay T-S fuzzy systems based on piecewise Lyapunov-Krasovskii functional2009891019982005CostaO.Stationary filter for linear minimum means quare error estimator of discretetime markovian jump systems200248813511356GaoH.LamJ.XieL.WangC.New approach to mixed H2/H∞ filtering for polytopic discrete-time systems20055383183319210.1109/TSP.2005.8511162169659GaoH.WangC.A delay-dependent approach to robust H∞ filtering for uncertain discrete-time state-delayed systems20045261631164010.1109/TSP.2004.8271882068994GaoH.LamJ.WangC.XuS.Robust H∞ filtering for 2D stochastic systems200423647950510.1007/s00034-004-1121-02107096GaoH.LamJ.WangC.Robust energy-to-peak filter design for stochastic time-delay systems200655210111110.1016/j.sysconle.2005.05.0052187838ZBL1129.93538ZhouW.SuH.ChuJ.Delay-dependent H1 filtering for singular Markovian jump time-delay systems201090618151824SubramanianA.SayedA. H.Multiobjective filter design for uncertain stochastic time-delay systems200449114915410.1109/TAC.2003.8214222028557WeiG.WangZ.ShuH.Robust filtering with stochastic nonlinearities and multiple missing measurements200945383684110.1016/j.automatica.2008.10.0282527273ZBL1168.93407ShenB.WangZ.ShuH.WeiG.On nonlinear H∞ filtering for discrete-time stochastic systems with missing measurements20085392170218010.1109/TAC.2008.9301992459593WeiG.WangZ.HeX.ShuH.Filtering for networked stochastic time-delay systems with sector nonlinearity20095617175WangZ.LamJ.MaL.BoY.GuoZ.Variance-constrained dissipative observer-based control for a class of nonlinear stochastic systems with degraded measurements2011377264565810.1016/j.jmaa.2010.11.0382769164ZBL1214.93104WangZ.YangF.HoD. W. C.LiuX.Robust H∞ filtering for stochstic time-delay systems with missing measurements200654725792587ShenB.WangZ.HungY. S.Distributed H∞-consensus filtering in sensor networks with multiple missing measurements: the finite-horizon case201046101682168810.1016/j.automatica.2010.06.0252877323LingQ.LemmonM. D.Optimal dropout compensation in networked control systemsProceedings of the IEEE Conference on Decosion and Control2003Honolulu, Hawaii, USA670675SinopoliB.SchenatoL.FranceschettiM.PoollaK.JordanM. I.SastryS. S.Kalman filtering with intermittent observations20044991453146410.1109/TAC.2004.8341212086911CheY.ShuH.WangZ.Nonlinear systems filtering with missing measurements and randomly distributed delays20111663641WangZ.ShenB.ShuH.WeiG.Quantized H-infinity control for nonlinear stochastic time-delay systems with missing measurements201257614311444WangZ.ShenB.LiuX.H∞ filtering with randomly occurring sensor saturations and missing measurements201248355656210.1016/j.automatica.2012.01.0082889455ZBL1244.93162ShenB.WangZ.LiangJ.LiuY.Recent advances on filtering and control for nonlinear stochastic complex systems with incomplete information: a survey201220121653075910.1155/2012/530759DongH.WangZ.GaoH.Distributed filtering for a class of time-varying systems over sensor networks with quantization errors and successive packet dropouts201260631643173KapilaV.HaddadW. M.Memoryless H1 controllers for discrete-time systems with time delay199834811411144KimJ. H.ParkH. B.H∞ state feedback control for generalized continuous/discrete time-delay system19993581443145110.1016/S0005-1098(99)00038-21831484ZBL0954.93011de SouzaC. E.LiX.Delay-dependent robust H∞ control of uncertain linear state-delayed systems19993571313132110.1016/S0005-1098(99)00025-41829975FridmanE.ShakedU.A descriptor system approach to H∞ control of linear time-delay systems200247225327010.1109/9.9833531881892FridmanE.ShakedU.An improved stabilization method for linear time-delay systems200247111931193710.1109/TAC.2002.8044621937712MoonY. S.ParkP.KwonW. H.LeeY. S.Delay-dependent robust stabilization of uncertain state-delayed systems200174141447145510.1080/002071701100671161857590ZBL1023.93055ZhangY.XuS.ZouY.LuJ.Delay-dependent robust stabilization for uncertain discrete-time fuzzy Markovian jump systems with mode-dependent time delays20111641668110.1016/j.fss.2010.09.0152741096ZBL1217.93156LiuZ.-W.ZhangH.-G.Delay-dependent stability for systems with fast-varying neutral-type delays via a PTVD compensation201036114715210.3724/SP.J.1004.2010.001472676563YueD.LiH.Synchronization stability of continuous/discrete complex dynamical networks with interval time-varying delays20107348096819LiuY.WangZ.LiuX.Robust H∞ filtering for discrete nonlinear stochastic systems with time-varying delay2008341131833610.1016/j.jmaa.2007.10.0192394087ZBL1245.93131XuS.LamJ.YangC.H∞ and positive-real control for linear neutral delay systems20014681321132610.1109/9.9409431847343ZBL1008.93033RakkiyappanR.BalasubramaniamP.Delay-probability-distribution-dependent stability of uncertain stochastic genetic regulatory networks with mixed time-varying delays: an LMI approach20104360060710.1016/j.nahs.2010.03.0072645875ZBL1200.93119LiuY.WangZ.LiuX.State estimation for discrete-time Markovian jumping neural networks with mixed mode-dependent delays2008372487147715510.1016/j.physleta.2008.10.0452474610ZBL1227.92002ShuH.WangZ.LüZ.Global asymptotic stability of uncertain stochastic bi-directional associative memory networks with discrete and distributed delays200980349050510.1016/j.matcom.2008.07.0072576440ZBL1195.34125SchererC.GahinetP.ChilaliM.Multiobjective output-feedback control via LMI optimization199742789691110.1109/9.5999691469832ZBL0883.93024