Stability and Robust Stability of 2D Discrete Stochastic Systems

Abstract

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.

1. Introduction

Over the past decades, Fornasini-Marchesini (FM) model has been applied in many practical problems, for example, the control of sheet-forming processes [1], circuits, signal processing, and discrimination of some partial differential equations with initial-boundary conditions [2–6]. Asymptotic stability for 2D deterministic systems based on FM models also has been developed quite successfully. Several methods have been proposed, for example, using Lyapunov function [7, 8], using LMI technique [9], and using the nonnegative matrices theory [10–12]. For linear 2D model in general case, stability has been discussed in [13]. However, very few effort has been made toward the analysis and synthesis of 2D stochastic systems (2DSSs) with stochastic system matrices. The mean-square stability of 2DSS has been discussed in [14, 15]. The state estimation problem has been discussed in [16]. The 𝐻_∞ filtering problem has been discussed in [17]. But all of them are based on the conservative assumption which is listed as follows.

Assumption 1.1. The boundary condition of (2.1) is independent of 𝑣(𝑖,𝑗) and is assumed to satisfy lim_𝑁→∞𝔼^𝑁_𝑘=1‖𝑥(0,𝑘)‖²+‖𝑥(𝑘,0)‖²<∞. (1.1)

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 1.1. 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. At last, an example is given to illustrate our results.

The following notation is used in this paper. For an 𝑛-dimensional vector of real elements 𝑥∈ℝ^𝑛, ‖𝑥‖=(𝑥^𝑇𝑥)^1/2 denotes the 2-norm, where the superscript 𝑇 stands for matrix transposition. 𝔼{𝑥} denotes the expected value of 𝑥. In symmetric block matrices or long matrix expressions, we use an asterisk (*) 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. 2D Stochastic System Model

First, we rewrite the 2D stochastic system model as follows:𝑥𝑖+1,𝑗+1=𝐴₁+𝑀₁𝑣𝑖,𝑗𝑥𝑖,𝑗+1+𝐴₂+𝑀₂𝑣𝑖,𝑗𝑥𝑖+1,𝑗, (2.1) where 𝐴₁, 𝐴₂ are system matrices with compatible dimensions, 𝑀₁, 𝑀₂ are appropriately dimensioned matrices, and 𝑣(𝑖,𝑗) is a standard random scalar signal satisfying 𝔼{𝑣(𝑖,𝑗)}=0 and 𝔼𝑣𝑣(=1=(),0≠().𝑖,𝑗𝑚,𝑛)𝑖,𝑗𝑚,𝑛𝑖,𝑗𝑚,𝑛 (2.2) We make the following assumption on the boundary condition which is less conservative.

Assumption 2.1. The boundary condition is independent of 𝑣(𝑖,𝑗) and is assumed to satisfy lim_𝑘→∞𝔼‖𝑥(0,𝑘)‖²=0,lim_𝑘→∞𝔼‖𝑥(𝑘,0)‖²𝔼𝑥(=0,‖𝑘,0)‖²<∞,𝐸‖𝑥(𝑘,0)‖²<∞forany𝑘≥1. (2.3)

Remark 2.2. Assumption 2.1 does not yield any loss of generality since the 2DDS (2.1) is not asymptotically stable if the initial states do not satisfy Assumption 2.1. We can see that Assumption 2.1 is necessary if the 2DDS (2.1) is asymptotically stable.

Remark 2.3. Assumption 2.1 is less conservative and has broader application range than Assumption 1.1.

For example, we assume the boundary state of system (2.1) satisfies 𝔼{‖𝑥(0,𝑘)‖²}=𝔼{‖𝑥(𝑘,0)‖²}=1/𝑘, where 𝑘=1,2,…,∞. So, we havelim_𝑘→∞𝔼‖𝑥(0,𝑘)‖²=0,lim_𝑘→∞𝔼‖𝑥(𝑘,0)‖²=0. (2.4) Clearly, the above boundary state does not meet Assumption 1.1, so the results in [14–17] can not be used on system (2.1) with Assumption 1.1. Although the above-boundary state meets Assumption 2.1, we can use our conclusions on system (2.1) with Assumption 2.1.

Similar to [14], we give the following definition which will be used throughout the paper.

Definition 2.4. The two-dimensional discrete stochastic system (2.1) with Assumption 2.1 is said to be mean-square asymptotically stable if under the zero input and for every initial condition 𝔼{‖𝑥(0,0)‖²}<∞, lim_{𝑖+𝑗→∞}𝔼𝑥𝑖,𝑗²=0. (2.5)

Remark 2.5. Definition 2.4 is more general and has broader application range than Definition 2 given in [14].

3. Asymptotic Stability and Robust Stability

In this section, we discuss mean-square asymptotic stability for 2D discrete stochastic systems (2.1). Then, we extend the result into the fields of robust stability.

Lemma 3.1 (Schur’s Complement [18]). Given constant matrices 𝐶, 𝐿, and 𝐷 of appropriate dimensions where 𝐶 and 𝐷 are symmetric and 𝐷>0, then the inequality 𝐶+𝐿^𝑇𝐷𝐿<0 holds if and only if 𝐶𝐿^𝑇𝐿−𝐷⁻¹<0or−𝐷⁻¹𝐿𝐿^𝑇𝐶<0. (3.1)

3.1. Asymptotic Stability

Theorem 3.2. The 2D discrete stochastic system (2.1) is mean-square asymptotically stable if there exist two positive-definite matrices 𝑃₁ and 𝑃₂ satisfying −𝑃₁0𝐴^𝑇₁𝑃𝑀^𝑇₁𝑃∗−𝑃₂𝐴^𝑇₂𝑃𝑀^𝑇₂𝑃∗∗−𝑃0∗∗∗−𝑃<0, (3.2) where 𝑃=𝑃₁+𝑃₂.

Proof. Let 𝐶=−𝑃₁00−𝑃₂,𝐿^𝑇=𝐴^𝑇₁𝑃𝑀^𝑇₁𝑃𝑀^𝑇₂𝑃𝑀^𝑇₂𝑃𝑃,𝐷=⁻¹00𝑃⁻¹, (3.3) by Lemma 3.1, and LMI (3.2) is equivalent to 𝐴Ψ=^𝑇1𝑃𝐴₁+𝑀^𝑇₁𝑃𝑀₁−𝑃₁𝐴^𝑇₁𝑃𝐴₂+𝑀^𝑇₁𝑃𝑀₂∗𝐴^𝑇₂𝑃𝐴₂+𝑀^𝑇₂𝑃𝑀₂−𝑃₂<0. (3.4)
Let Δ𝑉𝑖+1,𝑗+1=𝑉𝑖+1,𝑗+1−𝑉₁𝑖,𝑗+1−𝑉₂𝑖+1,𝑗, (3.5) where 𝑉₁𝑖,𝑗=𝔼𝑥^𝑇𝑖,𝑗𝑃₁𝑥,𝑉𝑖,𝑗₂𝑖,𝑗=𝔼𝑥^𝑇𝑖,𝑗𝑃₂𝑥𝑖,𝑗. (3.6) Substitute (2.1) into (3.5) and let 𝑥∶=^{𝑥(𝑖,𝑗+1)}_{𝑥(𝑖+1,𝑗)}, then we have Δ𝑉𝑖+1,𝑗+1=𝔼𝑥^𝑇𝑖+1,𝑗+1𝑃₁𝑥𝑖+1,𝑗+1+𝔼𝑥^𝑇𝑖+1,𝑗+1𝑃₂𝑥𝑥𝑖+1,𝑗+1−𝔼^𝑇𝑖,𝑗+1𝑃₁𝑥𝑖,𝑗+1−𝔼𝑥^𝑇𝑖+1,𝑗𝑃₂𝑥𝑥𝑖+1,𝑗=𝔼^𝑇𝑖+1,𝑗+1𝑃₁+𝑃₂𝑥𝑥𝑖+1,𝑗+1−𝔼^𝑇𝑖,𝑗+1𝑃₁𝑥𝑖,𝑗+1−𝔼𝑥^𝑇𝑖+1,𝑗𝑃₂𝑥𝐴𝑖+1,𝑗=𝔼¹+𝑀₁𝑣𝑖,𝑗𝑥𝑖,𝑗+1+𝐴₂+𝑀₂𝑣𝑖,𝑗𝑥𝑖+1,𝑗^𝑇𝑃𝐴₁+𝑀₁𝑣𝑖,𝑗𝑥𝑖,𝑗+1+𝐴₂+𝑀₂𝑣𝑥𝑥𝑖,𝑗𝑖+1,𝑗−𝔼^𝑇𝑖,𝑗+1𝑃₁𝑥𝑖,𝑗+1+𝔼𝑥^𝑇𝑖+1,𝑗𝑃₂𝑥=𝐴𝑖+1,𝑗₁𝑥𝑖,𝑗+1+𝐴₂𝑥𝑖+1,𝑗^𝑇𝑃𝐴₁𝑥𝑖,𝑗+1+𝐴₂𝑥+𝑀𝑖+1,𝑗₁𝑥𝑖,𝑗+1+𝑀₂𝑥𝑖+1,𝑗^𝑇𝑃𝑀₁𝑥𝑖,𝑗+1+𝑀₂𝑥𝑥𝑖+1,𝑗−𝔼^𝑇𝑖,𝑗+1𝑃₁x𝑖,𝑗+1+𝔼𝑥^𝑇𝑖+1,𝑗𝑃₂𝑥=𝑥𝑖+1,𝑗_𝑇𝐴^𝑇₁𝐴^𝑇₂𝑃𝐴₁𝐴₂𝑥+𝑥^𝑇𝑀^𝑇₁𝑀^𝑇₂𝑃𝑀₁𝑀₂𝑥−𝑥^𝑇𝑃₁00𝑃₂𝑥=𝔼𝑥^𝑇Ψ𝑥. (3.7) Hence, for any 𝑥≠0, we have Δ𝑉𝑖+1,𝑗+1=𝔼𝑥^𝑇Ψ𝑥<0. (3.8)
From Assumption 2.1, we have 𝔼{‖𝑥(𝑖,0)‖²}→0 and 𝔼{‖𝑥(0,𝑖)‖²}→0 as 𝑖→∞.
From Definition 2.4, we can see that, to show mean-square asymptotically stable, we only need to prove that lim_{𝑖+𝑗→∞}𝔼{‖𝑥(𝑖,𝑗)‖²}=0, that is, to prove that 𝔼{‖𝑥(𝑖,0)‖²}→0, 𝔼{‖𝑥(𝑖,1)‖²}→0,…,𝔼{‖𝑥(𝑖,∞)‖²}→0, and 𝔼{‖𝑥(0,𝑖)‖²}→0, 𝔼{‖𝑥(1,𝑖)‖²}→0,…,𝔼{‖𝑥(∞,𝑖)‖²}→0 as 𝑖→∞. So, we only need to prove that for any natural number 𝑘≥1, 𝔼{‖𝑥(𝑖,𝑘)‖²}→0 and 𝔼{‖𝑥(𝑘,𝑖)‖²}→0 as 𝑖→∞. To prove this, we need the following two steps.
Step 1. Let 𝑘=1, to prove 𝔼{‖𝑥(𝑖,1)‖}→0 and 𝔼{‖𝑥(1,𝑖)‖}→0 as 𝑖→∞.
Step 2. Let 𝑘=2,…,∞, to prove 𝔼{‖𝑥(𝑖,𝑘)‖}→0 and 𝔼{‖𝑥(𝑘,𝑖)‖}→0 as 𝑖→∞.
Now, we prove that 𝔼{‖𝑥(𝑖,1)‖}→0 as 𝑖→∞. From (3.5) and (3.8), we have 𝑉₁𝑖+1,𝑗+1+𝑉₂𝑖+1,𝑗+1−𝑉₁𝑖,𝑗+1−𝑉₂𝑖+1,𝑗<0, (3.9) which implies 𝑉₁𝑖+1,𝑗+1<𝑉₁𝑖,𝑗+1or𝑉₂𝑖+1,𝑗+1<𝑉₂𝑖+1,𝑗if𝔼{‖𝑥(𝑖,1)‖}>0. (3.10)
Clearly, let 𝑖→∞ and 𝑗=0, and substitute them into (3.10), then we get 𝑉₁(∞,1)<𝑉₁(∞,1)or𝑉₂(∞,1)<𝑉₂(∞,0)=0if𝔼{‖𝑥(∞,1)‖}>0. (3.11) Since both 𝑉₁(∞,1)<𝑉₁(∞,1) and 𝑉₂(∞,1)<0 are false, (3.11) is false. Hence, the condition 𝔼{‖𝑥(∞,1)‖}>0 in (3.11) is false. Thus, it yields 𝔼{‖𝑥(∞,1)‖}=0. Similarly, we can get that 𝔼{‖𝑥(1,∞)‖}=0.
Continue this procedure, and we can obtain that 𝔼{‖𝑥(𝑖,𝑘)‖²}→0 and 𝔼{‖𝑥(𝑘,𝑖)‖²}→0 as 𝑖→∞ for any natural number 𝑘≥2. It implies that lim_{𝑖+𝑗→∞}𝔼{‖𝑥(𝑖,𝑗)‖²}=0. Therefore, from Definition 2.4, the system (2.1) is mean-square asymptotically stable. The proof is completed.

Remark 3.3. Theorem 3.2 gives a sufficient condition for the mean-square asymptotical stability of system (2.1). It is equivalent to [14, Theorem 2] in the form. However, it has broader application range because the assumption is weaker.

Remark 3.4. Theorem 3.2 is also equivalent to in [14, Theorem 1] from [14, Theorem 3] in the form. However, it has broader application range.

Before proceeding further, we give the following lemma which will be used in the following proofs frequently.

Lemma 3.5 (see [15]). Given appropriately dimensioned matrices 𝑅₁, 𝑅₂, 𝑅₃, with 𝑅^𝑇₁=𝑅₁, then 𝑅₁+𝑅₃𝑊_𝑖,𝑗𝑅₂+𝑅^𝑇₂𝑊^𝑇_𝑖,𝑗𝑅^𝑇₃<0 (3.12) holds for all 𝑊_𝑖,𝑗 satisfying 𝑊^𝑇_𝑖,𝑗𝑊_𝑖,𝑗≤𝐼 if and only if, for some 𝛿>0, 𝑅₁+𝛿⁻¹𝑅₃𝑅^𝑇₃+𝛿𝑅^𝑇₂𝑅₂<0. (3.13)

Next, we present the robust stability result for system (2.1) with norm-bounded uncertain matrices.

3.2. Robust Stability

The main task of this subsection is to establish the robust mean-square asymptotic stability for two-dimensional stochastic system (2.1) with uncertain matrix data.

First, we give the following assumptions.

Assumption 3.6. Assume that the matrices 𝐴₁, 𝐴₂, 𝑀₁, 𝑀₂ of system (2.1) have the following form: 𝐴₁=𝐴₁₀+Δ𝐴₁,𝑀₁=𝑀₁₀+Δ𝑀₁,𝐴₂=𝐴₂₀+Δ𝐴₂,𝑀₂=𝑀₂₀+Δ𝑀₂, (3.14) where 𝐴₁₀, 𝐴₂₀, 𝑀₁₀, 𝑀₂₀ are known constant matrices with appropriate dimensions. Δ𝐴₁, Δ𝐴₂, Δ𝑀₁, Δ𝑀₂ are real-valued time-varying matrix functions representing norm-bounded parameter uncertainties satisfying Δ𝐴₁Δ𝐴₂Δ𝑀₁Δ𝑀₂=𝐺Δ_𝑖,𝑗𝐻₁𝐻₂𝐻₃𝐻₄, (3.15) where Δ_𝑖,𝑗 is a real uncertain matrix function with Lebesgue measurable elements satisfying Δ^𝑇_𝑖,𝑗Δ_𝑖,𝑗≤𝐼, (3.16) and 𝐺, 𝐻₁, 𝐻₂, 𝐻₃, 𝐻₄ are known real constant matrices with appropriate dimensions. These matrices specify how the uncertain parameters in Δ_𝑖,𝑗 enter the nominal matrices 𝐴₁₀, 𝐴₂₀, 𝑀₁₀, 𝑀₂₀.

Now, we have the robust stability result for system (2.1) with norm-bounded uncertain matrices.

Theorem 3.7. The 2D discrete stochastic system (2.1) with Assumption 3.6 is robustly mean-square asymptotically stable if there exist two positive-definite matrices 𝑃₁, 𝑃₂ and a scalar 𝛿>0 satisfying −𝑃₁+𝛿𝐻^𝑇₁𝐻₁+𝐻^𝑇₃𝐻₃𝛿𝐻^𝑇₁𝐻₂+𝐻^𝑇₃𝐻₄𝐴^𝑇₁₀𝑃𝑀^𝑇₁₀𝑃00∗−𝑃₂𝛿𝐻^𝑇₂𝐻₂+𝐻^𝑇₄𝐻₄𝐴^𝑇₂₀𝑃𝑀^𝑇₂₀𝑃00∗∗−𝑃0𝑃𝐺0∗∗∗−𝑃0𝑃𝐺∗∗∗∗−𝛿𝐼0∗∗∗∗∗−𝛿𝐼<0, (3.17) where 𝑃=𝑃₁+𝑃₂.

Proof. With the result of Theorem 3.2, substituting the norm-bounded uncertain matrices 𝐴₁, 𝐴₂, 𝑀₁, 𝑀₂ defined in (3.14) into (3.2), we have −𝑃₁0𝐴₁₀+Δ𝐴₁^𝑇𝑃𝑀₁₀+Δ𝑀₁^𝑇𝑃∗−𝑃₂𝐴₂₀+Δ𝐴₂^𝑇𝑃𝑀₂₀+Δ𝑀₂^𝑇𝑃∗∗−𝑃0∗∗∗−𝑃<0, (3.18) where 𝑃=𝑃₁+𝑃₂.
It can be written as (3.12) with 𝑅₁=−𝑃₁0𝐴^𝑇₁₀𝑃𝑀^𝑇₁₀𝑃∗−𝑃₂𝐴^𝑇₂₀𝑃𝑀^𝑇₂₀𝑃∗∗−𝑃0∗∗∗−𝑃,𝑅₂=𝐻₁𝐻₂𝐻00₃𝐻₄00,𝑅₃=,𝑊0000𝑃𝐺00𝑃𝐺_𝑖,𝑗=Δ_𝑖,𝑗00Δ_𝑖,𝑗. (3.19) By Lemma 3.5, we get −𝑃₁+𝛿𝐻^𝑇₁𝐻₁+𝐻^𝑇₃𝐻₃𝛿𝐻^𝑇₁𝐻₂+𝐻^𝑇₃𝐻₄𝐴^𝑇₁₀𝑃𝑀^𝑇₁₀𝑃∗−𝑃₂+𝛿𝐻^𝑇₂𝐻₂+𝐻^𝑇₄𝐻₄𝐴^𝑇₂₀𝑃𝑀^𝑇₂₀𝑃∗∗−𝑃+𝛿⁻¹𝑃𝐺(𝑃𝐺)^𝑇0∗∗∗−𝑃+𝛿⁻¹𝑃𝐺(𝑃𝐺)^𝑇<0. (3.20) It can be rewritten as 𝐶+𝐿^𝑇𝐷𝐿<0, (3.21) with 𝐶=−𝑃₁+𝛿𝐻^𝑇₁𝐻₁+𝐻^𝑇₃𝐻₃𝛿𝐻^𝑇₁𝐻₂+𝐻^𝑇₃𝐻₄𝐴^𝑇₁₀𝑃𝑀^𝑇₁₀𝑃∗−𝑃₂+𝛿𝐻^𝑇₂𝐻₂+𝐻^𝑇₄𝐻₄𝐴^𝑇₂₀𝑃𝑀^𝑇₂₀𝑃,𝐿∗∗−𝑃0∗∗∗−𝑃_𝑇=𝛿0000𝑃𝐺00𝑃𝐺,𝐷=⁻¹𝐼00𝛿⁻¹𝐼. (3.22) Using Lemma 3.1 (Schur’s Complement), we get 𝐶𝐿^𝑇𝐿−𝐷⁻¹<0, (3.23) which is (3.17). The proof is completed.

Remark 3.8. Theorem 3.7 is equivalent to [14, Theorem 4] in the form. However, it has broader application range.

Remark 3.9. We can get similar results correspond to [14, Theorems 5–8].

Remark 3.10. We can get similar results about robust 𝐻_∞ filtering correspond to [17, Theorems 1–5].

4. Example

In this section, we illustrate our results for 2D discrete stochastic system (2.1) through an example. All computations in this section are carried out by Matlab 7.8.0.347.

Consider two-dimensional stochastic system (2.1) with two state variables 𝑥₁, 𝑥₂, and the following system matrices: 𝐴₁=0.100.50.15+0.2𝜌,𝑀₁=,𝐴0.0600+0.1𝜌0.05+0.2𝜌₂=0.20+0.1𝜌0.30.2,𝑀₂=,𝔼𝑣𝔼𝑣𝑣(==(),≠().00.050.05+0.15𝜌0𝑖,𝑗=0,𝑖,𝑗𝑚,𝑛)1for𝑖,𝑗𝑚,𝑛0for𝑖,𝑗𝑚,𝑛 (4.1) The boundary condition is assumed to satisfy Assumption 3.6 but does not satisfy Assumption 2.1. For example,‖𝑥(0,𝑘)‖²=‖𝑥(𝑘,0)‖²=1𝑘. (4.2)

First, we assume that the system matrices are perfectly known, that is, 𝜌=0. We can not determine the stability of the system using the conclusions of [14] because the boundary condition does not satisfy the assumption given in [14]. However, we can determine the stability of the system by Theorem 3.2 because the boundary condition satisfies our assumption. Using Matlab to solve the inequality (3.2), we get that there exist two positive matrices 𝑃₁=𝑃₂=^{0.8612−0.0497}_{−0.04970.7026}, such that inequality (3.2) is true. So the system is stable.

Figures 1 and 2 show the two state variables of the above system. It can be seen that the system is asymptotically stable too.

Figure 1 

State variable 𝑥1 of system  (2.1).

Figure 2 

State variable 𝑥2 of system  (2.1).

Now, we assume that the uncertain parameter 𝜌, satisfying |𝜌|≤1. we have the matrices in Assumption 3.6 as follows: 𝐴₁₀=0.100.50.15,𝑀₁₀=0.06000.05,𝐴₂₀=0.200.30.2,𝑀₂₀=,𝐻00.050.050₁=0000.2,𝐻₂=00.100,𝐻₃=000.10.2,𝐻₄=,000.150𝐺=1001,Δ_𝑖𝑗=,𝔼𝑣𝔼𝑣𝑣(==(),≠().𝜌00𝜌𝑖,𝑗=0,𝑖,𝑗𝑚,𝑛)1for𝑖,𝑗𝑚,𝑛0for𝑖,𝑗𝑚,𝑛 (4.3) Similar to the case of stability, we can not determine the robust stability of the system using the conclusions of [14]. However, we can determine the stability of the system by Theorem 3.7 because the boundary condition satisfies our assumption. Using Matlab to solve the inequality (3.17), we get that there exist two positive matrices𝑃₁=2.05920.07170.07170.9343,𝑃₂=2.2844−0.1540−0.15400.5537, (4.4) and a scalar 𝛿=6.5538>0, such that inequality (3.17) is true. So the system is robustly stable.

5. Conclusions

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.

Acknowledgments

This work was partially supported by National Natural Science Foundation of China (10571036) and the Key Discipline Development Program of Beijing Municipal Commission (XK100080537).