Meaning of non-degenerate function

Question

I am reading some control theory literature. One of the assumptions made about some transfer (rational) functions is that they should be "non-degenerate". What does that exactly mean?

Answer 1

I recommend you reading the paper:

Ferreira, P. G. "On degenerate systems." International Journal of Control 24.4 (1976): 585-588

(and references therein). I take the definition from this paper:

Let $x\in R^n,u\in R^m,y\in R^q$, $$ \begin{align} \dot{x}=Ax+Bu\\ y=Cx+Du \end{align} $$

and $P(s)=\begin{pmatrix}sI-A&B\\-C&D\end{pmatrix}$ the Rosenbrock matrix.

Then the system is degenerate if and only if

$$ \textrm{rank} \{ P(s) \} < n + \min(m,q) $$

for all complex $s$ and the rank should be taken over the field of complex numbers.

• For SISO systems, degenerate is equivalent to the transfer function being identical to zero (for all $s$).
• For MIMO systems the situation is a bit more complicated (as usually). If the MIMO transfer function is identical to zero (for all $s$) the system is degenerate, but the converse is generally not the case.