stability of linear system

Question

I have a discrete system of the form

$x(k+k_o) = Ax(k+k_o-1) + Bx(k)$

where $A$ and $B$ are $n\times n$ matrices ($k_o>0$). I want to know about the stability of the system when

1. both A and B have eigen values inside the unit circle
2. Eigen values of B are inside the unit circle while largest eigen value of A lies exactly on the unit circle.

Answer 1

In general, it is difficult to conclude on the stability of the system. In other words, the eigenvalues of the matrices A and B does not provide more information on the stability of the system.

For example, we take $k_0 = 1$ and $A = B = 3/4$. In this case, the matrices A and B have eigen values inside the unit circle but the system is unstable.

Similary, if take $A = B = 1/3$, the system is stability. In the same way, we can take some examples in case 2.

Answer 2

By defining a new state as

$$ z(k) = \begin{bmatrix} x(k) \\ x(k-1) \\ \vdots \\ x(k-k_o+1) \end{bmatrix}, $$

then the difference equation can be written as

$$ z(k+1) = \bar{A}\,z(k), $$

with

$$ \bar{A} = \begin{bmatrix} A & 0 & \cdots & 0 & B \\ I & 0 & \cdots & 0 & 0 \\ 0 & I & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & 0 \\ 0 & \cdots & 0 & I & 0 \end{bmatrix}. $$

So stability can be checked by evaluating the eigenvalues of $\bar{A}$ (which must be Schur).