Necessary stability condition for a second order discrete time system $x(k+2) = Ax(k+1) + Bx(k)$

Question

Let $x(k) \in \mathbb{R}^n$, $A \in \mathbb{R}^{n\times n}$, $B \in \mathbb{R}^{n\times n}$. Consider the following discrete time system:

$$x(k+2) = Ax(k+1) + Bx(k)$$ where $x(1) = Ax(0)$ and $x(0) \neq 0$. What is the stability condition for this system? All I can find is the $\textbf{necessary}$ stability condition for first order system where $B=0$ which is $|\lambda(A)| < 1$.

Thanks a lot!

Answer 1

You can rewrite this system as $$\begin{bmatrix}x(k+1)\\x(k+2)\end{bmatrix}=\begin{bmatrix}0&I\\B&A\end{bmatrix}\begin{bmatrix}x(k)\\x(k+1)\end{bmatrix}$$ Now this system is stable if and only if all eigenvalues of $\begin{bmatrix}0&I\\B&A\end{bmatrix}$ are in the unit circle.