Stability Problems in 2-D Systems

Abstract

A 2-D system is defined by the following set of equations [1,2]:

$$\begin{gathered} {\text{x(h + 1,}}\,{\text{k + 1)}}\,{\text{ = }}\,{\text{A}}_{\text{1}} \,{\text{x}}({\text{h + 1,}}\,{\text{k}})\, + \,{\text{A}}_{\text{2}} \,{\text{x(h,k + 1)}}\,{\text{ + }}\,{\text{B}}_{\text{1}} \,{\text{u(h + 1,k)}}\,{\text{ + }}\,{\text{B}}_{\text{2}} \,{\text{u(h,k + 1)}}\, \hfill \\ \end{gathered}$$

((1))

where u(h,k), the input value at (h,k), and y(h,k), the output value at (h,k), are in ℝ and h,k ∈ ℤ. The vector x ∈ X =ℝⁿ is called the local state space and A_i∈ ℤ^n×n, B_i∈ ℤ^n×1, C∈ ℤ^1×n, i = 1,2.