Estimation for Stochastic Nonlinear Systems with Randomly Distributed Time-Varying Delays and Missing Measurements

Abstract

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. ℝ^𝑛, ℝ^𝑛×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,{ℱ_𝑘}_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). 𝜆_min(⋅) 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,…,𝑋_𝑛} stands for a block-diagonal matrix with matrices 𝑋₁,𝑋₂,…,𝑋_𝑛 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: 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝜅₁(𝑘)𝐵⁻¹_{(𝑘)𝑚=−𝜏}𝑥(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), (2.1) where 𝑥(𝑘)∈ℝ^𝑛 is the state vector, 𝑦(𝑘)∈ℝ^𝑚 is the measured output vector, 𝑧(𝑘)∈ℝ^𝑞 is the signal to be estimated, 𝑤(𝑘) is a one-dimensional, zero-mean, Gaussian white noise sequence on a probability space (Ω,ℱ,{ℱ_𝑘}_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<𝜏_𝑙≤𝜏(𝑘)≤𝜏_𝑢, 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 Prob𝜅₁(𝑘)=1=𝔼𝜅₁(𝑘)∶=𝛼₁,Prob𝜅₁(𝑘)=0∶=1−𝛼₁,Prob𝜅₂(𝑘)=1=𝔼𝜅₂(𝑘)∶=𝛼₂,Prob𝜅₂(𝑘)=0∶=1−𝛼₂,Prob𝜅₃(𝑘)=1=𝔼𝜅₃(𝑘)∶=𝛼₃,Prob𝜅₃(𝑘)=0∶=1−𝛼₃, (2.2)(2.3)(2.4)(2.5)(2.6)(2.7) 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): 𝔼𝑓(𝑥(𝑘(𝑘𝔼𝑓(𝑥(𝑘𝑓))∣𝑥)=0,))^𝑇(𝑥(𝑘(𝑘)𝔼𝑓(𝑥(𝑘𝑓))∣𝑥=0,𝑘≠𝑗,))^𝑇(𝑥(𝑘))∣𝑥(𝑘)≤^𝑞_𝑖=1Π_𝑖𝑥^𝑇(𝑘)Φ_𝑖𝑥(𝑘), (2.8) where 𝑞 is a known nonnegative integer, Π_𝑖=Π_𝑖Π^𝑇_𝑖, Π_𝑖, Π_𝑖, and Φ_𝑖(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: 𝑥_𝑓(𝑘+1)=𝐴_𝑓𝑥_𝑓(𝑘)+𝐴_𝑘𝑦(𝑘),𝑧(𝑘)=𝐻₂𝑥_𝑓(𝑘),𝑧(0)=0, (2.9) where 𝑥_𝑓∈ℝ^𝑛 is the state estimate, 𝑧(𝑘) is the estimate output, 𝐻₂ is a known real constant matrix with appropriate dimension, and 𝐴_𝑓 and 𝐴_𝑘 are estimator parameters to be determined.

By defining 𝑥(𝑘)=[𝑥^𝑇(𝑘)𝑥^𝑇_𝑓(𝑘)]^𝑇, we have the following augmented system: 𝑥(𝑘+1)=𝒜𝑥(𝑘)+𝒜𝑥(𝑘)+ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)+ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥()(𝑘)+𝑘+𝑚+𝒩ℎ𝒩ℎ(𝑘)+ℰ𝑥(𝑘)𝑤(𝑘), (2.10) where 𝛼𝒜=𝐴0₃𝐴_𝑘𝐶𝐴_𝑓,𝜅𝒜=00₃(𝑘)−𝛼₃𝐴_𝑘𝛼𝐶0,ℬ=¹,𝐵000𝜅ℬ=¹(𝑘)−𝛼₁,𝑥()=𝑥()𝑥𝐵000𝑘+𝑖𝑘+𝑖_𝑓()(𝑘)=𝑓(𝑥(𝑘0,𝐸𝑘+𝑖,ℎ))ℰ=¹0𝐴_𝑘𝐸₂0,𝒩=𝐼000,𝒩=𝛼₂𝒩,𝒩=𝜅₂(𝑘)−𝛼₂𝒩. (2.11)

Observe the system (2.10) and let 𝑥(𝑘;𝜑) denote the state trajectory from the initial data 𝑥(𝑠)=𝜑(𝑠) on −𝜉_𝑀≤𝑠≤−𝜉_𝑚. 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=Υ^𝑇₁ and Υ₂=Υ^𝑇₂>0, then Υ₁+Υ^𝑇₃Υ₂⁻¹Υ₃<0 if and only if ^Υ1Υ𝑇3_Υ3−Υ2<0.

Lemma 3.2. Let 𝒲∈ℝ^𝑛×𝑛 be a positive semidefinite matrix, 𝑥_𝑖∈ℝ^𝑛 be a vector, and 𝑎_𝑖≥0(𝑖=1,2,…) be scalars. If the series concerned are convergent, then the following inequality holds [35] ^+∞_𝑖=1𝑎_𝑖𝑥_𝑖^𝑇𝒲^+∞_𝑖=1𝑎_𝑖𝑥_𝑖≤^+∞_𝑖=1𝑎_𝑖^+∞_𝑖=1𝑎_𝑖𝑥^𝑇_𝑖𝒲𝑥_𝑖. (3.1)

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 𝐴_𝑓 and 𝐴_𝑘 consider the estimation error system (2.10). If there exist positive definite matrices 𝑃=𝑃^𝑇>0,𝑄=𝑄^𝑇>0, and positive scalars 𝜛_𝑖>0(𝑖=1,2,…,𝑞) such that the following matrix inequalities, 𝛽−𝑃∗∗∗∗∗∗∗0−𝑄∗∗∗∗∗∗𝑃ℰ0−𝑃∗∗∗∗∗₁𝛽𝑄00−𝑄∗∗∗∗𝑃𝒜𝑃ℬ00−𝑃∗∗∗₃𝛽𝑃0000−𝑃∗∗₄Φ00000−Ξ∗0𝛽₂𝑃ℱ00000−𝑃<0,Δ=−𝜛_𝑖𝑃𝒩𝐼∗Π_𝑖−𝑃<0,𝑖=1,2,…,𝑞, (3.2)(3.3) hold, where 𝛽₁=𝜏_𝑢+12𝜏_𝑢−𝜏_𝑙𝜏_𝑢+𝜏_𝑙−1^1/2,𝛽₂=𝛼₁1−𝛼₁^1/2,𝛽₃=𝛼₃1−𝛼₃^1/2,𝛽₄=𝛼₂^1/2,Φ=𝜛₁Φ₁^1/2,…,𝜛_𝑞Φ_𝑞^1/2𝑇,Φ_𝑖=Φ_𝑖0,00Π_𝑖=𝜋_𝑖𝜋_𝑖,Ξ=diag𝜛₁𝐼,…,𝜛_𝑞𝐼,ℱ=,𝐵000 (3.4) then the estimation error system (2.10) is mean-square stable.

Proof. Define the following Lyapunov functional candidate for system (2.10): 𝑉(𝑥(𝑘),𝑘)=𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+⁻¹_{(𝑘)𝑖=−𝜏}^𝑘−1_{𝑗=𝑘+𝑖}𝑥^𝑇𝑗𝑄𝑥𝑗+^{−𝜏𝑙−1}_{𝑖=−𝜏𝑢}⁻¹_𝑗=𝑖+1^𝑘−1_{𝑛=𝑘+𝑗}𝑥^𝑇(𝑛)𝑄𝑥(𝑛). (3.5)
By calculating the difference of the Lyapunov functional (3.5), based on Lemma 3.2, one has, 𝔼{(𝑥(𝑘){𝑉(𝑥())∣𝑥(𝑘𝑥(𝑘)=𝒜𝑥(𝑘)▵𝑉,𝑘)}=𝔼𝑘+1,𝑘+1)}−𝑉{(,𝑘)}+ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)^𝑇𝑃𝒜𝑥(𝑘)+ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)+𝔼ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)^𝑇𝑃ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)+𝛼₃(1−𝛼3)𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+𝑥^𝑇(𝑘)ℰ^𝑇𝑃ℰ𝑥(𝑘)+𝛼₂𝔼ℎ^𝑇(𝑘)𝒩^𝑇𝑃𝒩ℎ(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+⁻¹₍₎^𝑘_{𝑖=−𝜏𝑘+1𝑗=𝑘+𝑖+1}𝑥^𝑇𝑗𝑄𝑥𝑗−⁻¹_{(𝑘)𝑖=−𝜏}^𝑘−1_{𝑗=𝑘+𝑖}𝑥^𝑇𝑗𝑄𝑥𝑗+^{−𝜏𝑙−1}_{𝑖=−𝜏𝑢}⁻¹_𝑗=𝑖+1^𝑘_{𝑛=𝑘+𝑗+1}−^𝑘−1_{𝑛=𝑘+𝑗}𝑥^𝑇(𝑛)𝑄𝑥(𝑛)=𝒜𝑥(𝑘)+ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)^𝑇𝑃𝒜𝑥(𝑘)+ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)+𝔼ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)^𝑇𝑃ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)+𝛼₃(1−𝛼3)𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+𝑥^𝑇(𝑘)ℰ^𝑇𝑃ℰ𝑥(𝑘)+𝛼₂𝔼ℎ^𝑇(𝑘)𝒩^𝑇𝑃𝒩ℎ(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+⁻¹_{()𝑖=−𝜏𝑘+1}^𝑘−1_{𝑗=𝑘+𝑖+1}𝑥^𝑇𝑗𝑄𝑥𝑗+𝑥^𝑇(𝑘)𝑄𝑥(𝑘)−⁻¹_{(𝑘)𝑖=−𝜏}^𝑘−1_{𝑗=𝑘+𝑖+1}𝑥^𝑇𝑗𝑄𝑥𝑗+𝑥^𝑇()𝑄𝑥()+𝑘+𝑖𝑘+𝑖_{−𝜏𝑙−1𝑖=−𝜏𝑢−1𝑗=𝑖+1}𝑥^𝑇(𝑘)𝑄𝑥(𝑘)−𝑥^𝑇𝑄𝑥≤𝒜𝑥(𝑘)𝑘+𝑗𝑘+𝑗+ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)^𝑇𝑃𝒜𝑥(𝑘)+ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)+𝔼ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)^𝑇ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)+𝛼₃(1−𝛼3)𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+𝑥^𝑇(𝑘)ℰ^𝑇𝑃ℰ𝑥(𝑘)+𝛼₂𝔼ℎ^𝑇(𝑘)𝒩^𝑇𝑃𝒩ℎ(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+^{−𝜏𝑙−1}_{𝑖=−𝜏𝑢}^𝑘−1_{𝑗=𝑘+𝑖+1}𝑥^𝑇𝑗𝑄𝑥𝑗+⁻¹_{𝑖=−𝜏𝑙}^𝑘−1_{𝑗=𝑘+𝑖+1}𝑥^𝑇𝑗𝑄𝑥𝑗+𝜏_𝑢𝑥^𝑇(𝑘)𝑄𝑥(𝑘)−⁻¹_{𝑖=−𝜏𝑙}^𝑘−1_{𝑗=𝑘+𝑖+1}𝑥^𝑇𝑗𝑄𝑥𝑗−⁻¹_{(𝑘)𝑖=−𝜏}𝑥^𝑇()𝑄𝑥+1𝑘+𝑖𝑘+𝑗2𝜏_𝑢−𝜏_𝑙𝜏_𝑢+𝜏_𝑙−1𝑥^𝑇(𝑘)𝑄𝑥(𝑘)−^{−𝜏𝑙−1}_{𝑖=−𝜏𝑢}^𝑘−1_{𝑗=𝑘+𝑖+1}𝑥^𝑇𝑗𝑄𝑥𝑗=𝒜𝑥(𝑘)+ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)^𝑇𝑃𝒜𝑥(𝑘)+ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)+𝔼ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)^𝑇𝑃ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)+𝛼₃(1−𝛼3)𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+𝑥^𝑇(𝑘)ℰ^𝑇𝑃ℰ𝑥(𝑘)+𝛼₂𝔼ℎ^𝑇(𝑘)𝒩^𝑇𝑃𝒩ℎ(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+𝜏_𝑢𝑥^𝑇(𝑘)𝑄𝑥(𝑘)−⁻¹_{(𝑘)𝑖=−𝜏}𝑥^𝑇()𝑄𝑥()+1𝑘+𝑖𝑘+𝑖2𝜏_𝑢−𝜏_𝑙𝜏_𝑢+𝜏_𝑙−1𝑥^𝑇(𝑘)𝑄𝑥(𝑘)≤𝒜𝑥(𝑘)+ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)^𝑇𝑃𝒜𝑥(𝑘)+ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)+𝔼ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)^𝑇𝑃ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)+𝛼₃(1−𝛼3)𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+𝑥^𝑇(𝑘)ℰ^𝑇𝑃ℰ𝑥(𝑘)+𝛼₂𝔼ℎ^𝑇(𝑘)𝒩^𝑇𝑃𝒩ℎ(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+𝜏_𝑢𝑥^𝑇(𝑘)𝑄𝑥(𝑘)+12𝜏_𝑢−𝜏_𝑙𝜏_𝑢+𝜏_𝑙−1𝑥^𝑇(𝑘)𝑄𝑥(𝑘)−⁻¹_{(𝑘)𝑖=−𝜏}𝑥^𝑇()𝑄𝑥()≤𝒜𝑥(𝑘)𝑘+𝑖𝑘+𝑖+ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)^𝑇𝑃𝒜𝑥(𝑘)+ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)+𝔼ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)^𝑇𝑃ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)+𝛼₃(1−𝛼3)𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+𝑥^𝑇(𝑘)ℰ^𝑇𝑃ℰ𝑥(𝑘)+𝛼₂𝔼ℎ^𝑇(𝑘)𝒩^𝑇𝑃𝒩ℎ(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+𝜏_𝑢𝑥^𝑇(𝑘)𝑄𝑥(𝑘)+12𝜏_𝑢−𝜏_𝑙𝜏_𝑢+𝜏_𝑙−1𝑥^𝑇(𝑘)𝑄𝑥(𝑘)−𝜏_𝑢⁻¹⁻¹_{(𝑘)𝑖=−𝜏}𝑥^𝑇(𝑘+𝑖)𝑄⁻¹_{(𝑘)𝑖=−𝜏}𝑥(𝑘+𝑖). (3.6)
From (2.8), it can be seen that 𝔼ℎ^𝑇(𝑘)𝒩^𝑇𝑃𝒩ℎ(𝑘)≤^𝑞_𝑖=1𝑥^𝑇(𝑘)Φ_𝑖𝑥(𝑘)tr𝒩Π_𝑖𝒩^𝑇𝑃, (3.7) where Π_𝑖∶=Π_𝑖Π^𝑇_𝑖 with Φ_𝑖 and Π_𝑖 defined in (3.4).
Furthermore, 𝔼ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)^𝑇𝑃ℬ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚)≤𝛽²₂⁻¹_{(𝑘)𝑚=−𝜏}𝑥^𝑇(𝑘+𝑚)ℱ^𝑇𝑃ℱ⁻¹_{(𝑘)𝑚=−𝜏}𝑥(𝑘+𝑚), (3.8) where 𝛽₂ is defined in (3.4).
From (3.6)–(3.8), one has 𝔼{▵𝑉(𝑥(𝑘),𝑘)}≤𝔼𝜂^𝑇(𝑘)Θ𝜂(𝑘), (3.9) where 𝜂(𝑘)=[𝑥^𝑇(𝑘),∑⁻¹_{𝑖=−𝜏(𝑘)}𝑥^𝑇(𝑘+𝑖)]^𝑇 and ΘΘ=¹𝒜𝑃ℬ∗Θ₂, (3.10) where Θ₁=−𝑃 + ℰ^𝑇𝑃ℰ + (𝜏_𝑢+(1/2)(𝜏_𝑢−𝜏_𝑙)(𝜏_𝑢+𝜏_𝑙−1))𝑄 + 𝒜^𝑇𝑃𝒜 + 𝛽²₄∑^𝑞_𝑖=1Φ_𝑖tr(𝒩Π_𝑖𝒩^𝑇𝑃) + 𝛽²₃𝑃,Θ₂ = −(1/𝜏_𝑢)𝑄 + ℬ^𝑇𝑃ℬ + 𝛽²₂ℱ^𝑇𝑃ℱ, 𝛽₃, 𝛽₄, ℱ are defined in (3.4).
From Lemma 3.1, (3.10) holds if and only if tr(𝒩Π_𝑖𝒩^𝑇𝑃). Furthermore, by Lemma 3.1, one can obtain from (3.2), (3.3) that Θ<0 and, subsequently, 𝔼{▵𝑉(𝑥(𝑘),𝑘)}<−𝜆_min(Θ)|𝑥(𝑘)|². (3.11) Thus, the augmented estimation system (2.10) is mean-square stable.

The following theorem is focused on the design of the desired estimation parameters 𝐴_𝑓 and 𝐴_𝑘 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=𝑆^𝑇>0,𝑅=𝑅^𝑇>0,𝑄=𝑄^𝑇>0, matrices 𝐴_𝑓,𝐴_𝑘, and positive scalars 𝜛_𝑖>0,(𝑖=1,2,…,𝑞) such that the following linear matrix inequalities holds Γ=−𝑆∗∗∗∗∗∗∗∗∗∗∗∗−𝑆−𝑅∗∗∗∗∗∗∗∗∗∗∗00−𝑄∗∗∗∗∗∗∗∗∗∗𝑆𝐸₁𝐸₁𝜙0−𝑆∗∗∗∗∗∗∗∗∗₁𝜙₂𝛽0−𝑆−𝑅∗∗∗∗∗∗∗∗₁𝑄𝛽₁𝑄000−𝑄∗∗∗∗∗∗∗𝑆𝐴𝑆𝐴𝛼₁𝜙𝑆𝐵000−𝑆∗∗∗∗∗∗₃𝜙₄𝛼₁𝛽𝑅𝐵000−𝑆−𝑅∗∗∗∗∗₃𝑆𝛽₃𝛽𝑆000000−𝑆∗∗∗∗₃𝑆𝛽₃Φ𝑅000000−𝑆−𝑅∗∗∗Φ00000000−Ξ∗∗00𝛽₂𝑆𝐵00000000−𝑅∗00𝛽₂𝑅𝐵00000000−𝑆−𝑅<0,−𝜛_𝑖𝑆𝜋𝐼∗∗_𝑖𝑅𝜋−𝑆∗_𝑖+𝐴_𝑓𝜋_𝑖−𝑆−𝑅<0,𝑖=1,2,…,𝑞,𝑆−𝑅<0 (3.12)(3.13)(3.14) hold, where 𝛼₁ is defined in (2.2), 𝛽₁,𝛽₂,𝛽₃, and 𝛽₄ are defined in (3.4), Φ^𝑇=𝛽₄𝜛₁Φ₁^1/2𝑇,…,𝛽₄𝜛_𝑞Φ_𝑞^1/2𝑇,𝜙₁=𝑅𝐸₁+𝐴_𝑓𝐸₂,𝜙₂=𝑅𝐸₁+𝐴_𝑓𝐸₂,𝜙₃=𝑅𝐴+𝛼₂𝐴_𝑓𝐶+𝐴_𝑘,𝜙₄=𝑅𝐴+𝛼₂𝐴_𝑓𝐶, (3.15) then the estimator parameters are designed as 𝐴_𝑘=𝑋⁻¹₁₂𝐴_𝑓,𝐴_𝑓=𝑋⁻¹₁₂𝐴_𝑘𝑆⁻¹𝑌^𝑇₁₂⁻¹, (3.16) where 𝑋₁₂,𝑌₁₂ are any square and nonsingular matrices satisfying 𝑋₁₂𝑌^𝑇₁₂=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 𝑃=𝑅𝑋₁₂𝑋^𝑇₁₂𝑋₂₂,𝑃⁻¹=𝑆⁻¹𝑌₁₂𝑌^𝑇₁₂𝑌₂₂, (3.17) where the partitioning of 𝑃 and 𝑃⁻¹ is compatible with that of 𝒜 defined in (2.11), that is, 𝑅∈𝑅^𝑛×𝑛,𝑋₁₂∈𝑅^𝑛×𝑛,𝑋₂₂∈𝑅^𝑛×𝑛,𝑆∈𝑅^𝑛×𝑛,𝑌₁₂∈𝑅^𝑛×𝑛, and   𝑌₂₂∈𝑅^𝑛×𝑛. Define 𝑇₁=𝑆⁻¹𝐼𝑌^𝑇₁₂0,𝑇₂=𝐼𝑅0𝑋^𝑇₁₂ (3.18) which imply that 𝑃𝑇₁=𝑇₂ and 𝑇^𝑇₁𝑃𝑇₁=𝑇^𝑇₁𝑇₂.
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) −𝜛_𝑖𝜋𝐼∗∗_𝑖−𝑆⁻¹∗𝑅𝜋_𝑖+𝑋₁₂𝐴_𝑘𝜋_𝑖−𝐼−𝑅<0,𝑖=1,2,…,𝑞. (3.19)
Then, one uses congruence transformation diag{𝐼,𝑆,𝐼} to (3.19) and we have −𝜛_𝑖𝑆𝜋𝐼∗∗_𝑖𝑅𝜋−𝑆∗_𝑖+𝑋₁₂𝐴_𝑘𝜋_𝑖−𝑆−𝑅<0,𝑖=1,2,…,𝑞. (3.20)
Furthermore, if (3.12) is feasible, we have ^{−𝑆−𝑆}_{−𝑆−𝑅}<0 or ^{−𝑆−1𝐼}_𝐼𝑅>0.
It follows directly from 𝑋𝑋⁻¹=𝐼 that 𝐼−𝑅𝑆⁻¹=𝑋₁₂𝑌^𝑇₁₂<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: ,𝐸𝐴=0.1500.20.1,𝐵=0.09000.09,𝐶=1.2001.2₁=0.12000.12,𝐸₂=0.12000.12,𝐻₁=𝐻₂=,0.6000.6 (4.1) where 𝑛=𝑞=2,𝜏(𝑘)=1+(1+(−1)^𝑘),𝜏_𝑙=1,𝜏_𝑢=3.

𝑓(𝑥(𝑘)) describes the stochastic nonlinear function of the states in (2.1), which is bounded as follows: 𝔼𝑓(𝑥(𝑘))𝑓(𝑥(𝑘))^𝑇∣𝑥(𝑘)=0.220.220.220.22^𝑇𝑥^𝑇(𝑘)𝑥(𝑘).0.11000.11 (4.2) 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 𝑆=,0.6726−0.0035−0.00350.6563,𝑅=1.8796−0.0041−0.00411.8411𝑄=0.0668−0.0013−0.00130.0693,𝜛₁=1.0776,𝜛₂=1.3335. (4.3)

Thus, we can calculate the estimator parameters as follows: 𝐴_𝑓=0.13250.04860.0462−0.1465,𝐴_𝑘=−0.9144−0.1677−0.1684−0.8160. (4.4)

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.

Figure 1 

Actual Measurements 𝑦1(1,𝑘) and ideal Measurements 𝑦2(1,𝑘).

Figure 2 

Actual Measurements 𝑦1(2,𝑘) and ideal Measurements 𝑦2(2,𝑘).

Figure 3 

Estimation Errors 𝑧(1,𝑘).

Figure 4 

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.