The delay-dependent exponential
Time delays are quite often encountered in various practical engineering systems, and they are regarded as one of the main sources causing instability and degrading performance of control systems [
Since certain unavoidable stochastic perturbations are widely existing in many engineering systems, stochastic systems have gained considerable research attention over the past few years [
In the field of stochastic dynamic system with time delays, the filtering problem, which is to estimate the unavailable state of variables of a given control system, is also an important issue. Kalman filtering scheme is a well-known effective way to deal with the filtering problem. However, it has some limitations in practical applications due to the fact that it assumes that the system and its disturbances are exactly known, that is, stationary Gaussian noised with known statistics. Under this view, recently,
This paper focuses on the problems of delay-dependent
Consider the following stochastic systems with mixed delays and nonlinear perturbations:
For system (
Define
The objective of this paper is to design full-order the filtering error system ( under the zero initial condition, the filtering error system (
with
Before presenting the main results of this paper, we introduce the following lemmas, which will be essential to our derivation.
For a given symmetrical matrix
For any positive symmetric matrix
For any positive symmetric matrix
In this section, a new delay-dependent condition of the
Consider the stochastic time-delay system (
First, show the asymptotic stability of system (
Next, denote
Notice the fact of (
Thus,
By Schur complement lemma, it is easy to show that
By Dynkin’s formula, there exists
On the other hand, from (
Now, we will establish the
Define
Moreover, by Schur complement, (
Therefore, if (
In system (
Consider the stochastic time-delay system (
In this section, we will focus on the design of
Consider the stochastic time-delay system (
In this case, the parameters of a desired filter in the form of (
From (
Set
Define
Substitute
On the other hand, (
Therefore, by Theorem
When deriving the results in Theorem
Following the similar method in Theorem
Consider the stochastic time-delay system (
The results presented in Theorem
Consider the stochastic time-delay system (
Moreover, for the nonlinear functions, we let
In the case of
The upper bound of
Methods |
|
---|---|
Theorem |
7.481 |
Theorem |
8.190 |
Theorem |
8.317 |
Theorem |
8.379 |
Consider the stochastic time-delay system (
Given
From Corollary
The upper bound of
Methods |
|
---|---|
[ |
1.725 |
Corollary |
3.755 |
Corollary |
5.054 |
Corollary |
5.111 |
Corollary |
5.688 |
Corollary |
5.920 |
It can be seen from the results that the conservatism can be reduced with the increase of partition integers. However, it is necessary to point out that the less conservatism is at the cost of a higher computational complexity.
In this paper, a new approach has been developed to investigate the problems of delay-dependent
The project is supported by Nature Science Foundation of China (NSFC, Grant no. 51277022). The authors would also like to thank the editor and the anonymous reviewers for their valuable comments and suggestions which helped to improve the quality of this paper.