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

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.


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 .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{X 1 , X 2 , . . ., X n } stands for a block-diagonal matrix with matrices X 1 , X 2 , . . ., 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.

Problem Formulation and Preliminaries
Consider the following class of stochastic nonlinear system with distributed time-varying delays: 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{ω 2 k } 1, A, B, C, E 1 , E 2 , and H 1 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 κ 1 k ∈ R, κ 2 k ∈ R, and κ 3 k ∈ R are Bernoulli distributed white sequence taking the values of 0 and 1 with where α 1 ∈ 0 1 , α 2 ∈ 0 1 , and α 3 ∈ 0 1 are known positive scalars.

Mathematical Problems in Engineering
Remark 2.1.The nonlinear stochastic f x k is assumed to have the following for all x k : where q is a known nonnegative integer, Π i Π i Π 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 κ 1 k , κ 2 k , κ 3 k , and ω k .
In this paper, we aim at designing a linear estimator of the following structure: where x f ∈ R n is the state estimate, z k is the estimate output, H 2 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:

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.

Main Results
The following lemmas are essential in establishing our main results.Lemma 3.1 Schur Complement .There are constant matrices Υ 1 , Υ 2 , and Υ 3 where 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 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 i > 0 i 1, 2, . . ., q such that the following matrix inequalities, hold, where then the estimation error system 2.10 is mean-square stable.

Mathematical Problems in Engineering
Proof.Define the following Lyapunov functional candidate for system 2.10 :

3.5
By calculating the difference of the Lyapunov functional 3.5 , based on Lemma 3.2, one has,

3.6
From 2.8 , it can be seen that where where β 2 is defined in 3.4 .
From 3.6 -3.8 , one has where , F are defined in 3.4 .From Lemma 3.1, 3.10 holds if and only if tr NΠ i N T P .Furthermore, by Lemma 3.1, one can obtain from 3.2 , 3.3 that Θ < 0 and, subsequently, 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 i > 0, i 1, 2, . . ., q such that the following linear matrix inequalities holds hold, where α 1 is defined in 2.2 , β 1 , β 2 , β 3 , and β 4 are defined in 3.4 ,

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: 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.

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.