Transfer function is $0$?

Question

Given the continuous time state space model:

$\dot{x}(t)=Ax(t)+Bu(t)$, $\quad y(t)=Cx(t), \quad t\in R^{+}$ with: $\left[ \begin{array}{c|c} A & B \\ \hline C & \\ \end{array} \right]$ = $ \left[ \begin{array}{ccc|c} 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ \hline 0 & 1 & 0 \\ \end{array} \right]$

Using: $C(sI-A)^{-1}B$ yields the following transfer function:

$0$.

I'm used to seeing $s$-terms in the denominator, for instance: $\frac{1}{s+7}$. Which then provides the pole location(s) and thus the stability.

What does this zero say about stability?

Answer 1

Your equations give: $$y = x_2\\sx_2=x_3\\sx_3=0$$ Which implies that: $$s^2y=0$$ Thus the transfer function is indeed zero.

Such systems are called "finite-memory" (particularly for discrete-time systems) and their matrices are nilpotent: $$\exists n | A^n = \mathbb{0} $$ Finite-memory systems have null output in a finite amount of time (as opposed to the usual, asymptotic behaviour of stable systems) when input is also null.