Consider the following descriptor system: $\{\bf A, \bf B, \bf C, \bf D, \bf E\}$
With $\bf A \in \mathbb{C}^{\it p \times n}$, $\bf B \in \mathbb{C}^{\it p \times m}$, $\bf C \in \mathbb{C}^{\it l \times n}$, $\bf D \in \mathbb{C}^{\it l \times m}$ and $\bf E \in \mathbb{C}^{\it p \times n}$
if $\bf A$ and $\bf E$ are square matrices, ie. if $p=n$, then the $s$-domain transfer function of this descriptor system is:
$$\textbf{H}(s) = \textbf{C}(s\textbf{E}-\textbf{A})^{-1}\textbf{B}+\textbf{D}$$
My question is:
- if $p \ne n$ does that imply that the transfer function does not exist / is "ill defined"? or is there an alternative definition of the transfer function which can be used in that situation?.
- If the transfer function cannot be defined in this situation then: what can be said about the system? ie. what is the (intuitive) reason for the lack of a well-defined transfer function?.
Attempting to answer my own question:
We have: $$\textbf E\dot x(t)=\textbf Ax(t)+\textbf Bu(t)$$ $$y(t)=\textbf Cx(t)+\textbf Du(t)$$ Laplace transform: $$s\textbf Ex(s)=\textbf Ax(s)+\textbf Bu(s)$$ $$y(s)=\textbf Cx(s)+\textbf Du(s)$$ Solving for $x(s)$: $$(s\textbf E-\textbf A)x(s)=\textbf Bu(s)$$ $$x(s) \approx (s\textbf E-\textbf A)^{\dagger}\textbf Bu(s)+\textbf F \textbf Zu(s)$$ Where the columns of $\bf F$ are orthogonal vectors that span the null-space of $s\textbf E-\textbf A$, and $\textbf Z$ is arbitrary.
We then have: $$y(s)\approx (\textbf C(s\textbf E-\textbf A)^{\dagger}\textbf B+\textbf C\textbf F\textbf Z+\textbf D)u(s)$$ $$\textbf H(s)\approx \textbf C(s\textbf E-\textbf A)^{\dagger}\textbf B+\textbf C\textbf F\textbf Z+\textbf D$$
If the system is fully or under -determined then all the equations hold with an equality sign. If on the other hand the system is fully or over -determined then $\bf F = 0$.