New stability and robust stability results are given based on weaker conservative assumptions. First, new boundary condition is designed. It is less conservative and has broader application range than that has been given. Then, we derive the results which have the same form, but under a weaker conservative assumption. Meanwhile, the process of the proofs has been simplified. Finally, an example is given to illustrate our results. Our results can be extended to the fields of stabilization, filtering and state estimation, and so forth.
Over the past decades, Fornasini-Marchesini (FM) model has been applied in many practical problems, for example, the control of sheet-forming processes [
The boundary condition of (
The main goal of the present paper is to find stability and robust stability criteria for two dimensional stochastic systems based on weaker conservative assumptions. First, new boundary condition is designed. It is less conservative and has broader application range than Assumption
The following notation is used in this paper. For an
First, we rewrite the 2D stochastic system model as follows:
The boundary condition is independent of
Assumption
Assumption
For example, we assume the boundary state of system (
Similar to [
The two-dimensional discrete stochastic system (
Definition
In this section, we discuss mean-square asymptotic stability for 2D discrete stochastic systems (
Given constant matrices
The 2D discrete stochastic system (
Let
Let
From Assumption
From Definition
Let
Let
Now, we prove that
Clearly, let
Continue this procedure, and we can obtain that
Theorem
Theorem
Before proceeding further, we give the following lemma which will be used in the following proofs frequently.
Given appropriately dimensioned matrices
Next, we present the robust stability result for system (
The main task of this subsection is to establish the robust mean-square asymptotic stability for two-dimensional stochastic system (
First, we give the following assumptions.
Assume that the matrices
Now, we have the robust stability result for system (
The 2D discrete stochastic system (
With the result of Theorem
It can be written as (
Theorem
We can get similar results correspond to [
We can get similar results about robust
In this section, we illustrate our results for 2D discrete stochastic system (
Consider two-dimensional stochastic system (
First, we assume that the system matrices are perfectly known, that is,
Figures
State variable
State variable
Now, we assume that the uncertain parameter
In this paper, new stability and robust stability results are given based on weaker conservative assumptions. A new boundary condition is designed. It is less conservative and has broader application range than that has been given. Then, we derive the results which have the same form, but under a weaker conservative assumption. Meanwhile, the process of the proofs has been simplified. Our results can be extended to the fields of stabilization, filtering and state estimation, and so forth.
This work was partially supported by National Natural Science Foundation of China (10571036) and the Key Discipline Development Program of Beijing Municipal Commission (XK100080537).