Definition for finite and infinite zeros of a matrix pencil

Question

I'm currently studying generalized linear systems of the form

$Ey^{\prime}(t) = Ay(t) + Bu(t)$

with controllability pencil pencil $\left[ sE-A \quad B \right]$. In here it is said that $(E,A,B)$ is controllable if the controllability pencil has no finite or infinite zeros. But I miss a definition for infinite and finite zeros and I couldn't find a fiting one.

Answer 1

For some reason, this is not defined in the paper and it is also not defined in many of the papers I have looked at.

In any way, a finite zero for the pencil $\left[ sE-A \quad B \right]$ is a value for $s\in\mathbb{C}$ for which $\left[ sE-A \quad B \right]$ has a rank drop. On the other hand, an infinite zero is when $\left[E \quad B \right]$ has a rank drop.

To formulate a more self-contained answer, let us define the descriptor system $(E,A,B)$ as

$$ E\dot{x}=Ax+Bu\qquad\qquad(1) $$

The system $(E,A)$ is said to be regular if $\det(sE-A)$ is not identically zero.

Therefore the result you state can be reformulated as: The regular system $(E,A,B)$ is completely controllable if and only if

• $\mathrm{rank}\left[sE-A \quad B \right]=n$ for all $s\in\mathbb{C}$, and
• $\mathrm{rank}\left[E \quad B \right]=n$.

The first condition states that the slow subsystem of $(E,A,B)$ is controllable, whereas the second one states that the fast subsystem is controllable.

We may ask the question of stabilizability. The system $(E,A,B)$ is said to be stabilizable if there exists a control law $u=Kx$ such that the system $(E,A+BK)$ is stable.

A necessary and sufficient condition for the regular system $(E,A,B)$ to be stabilizable is that

$$ \mathrm{rank}[sE-A\quad B]=n\ \mathrm{for\ all\ }\Re[s]\ge0. $$