Absolute stability for System with nonlinear output function?

Question

Given a nonlinear system like

$$ \begin{split} \dot{x} &= Ax - B\phi(y)\\ y &= C^T x. \end{split} \tag{1} $$

If the nonlinear function $\phi$ fulfills the well known sector conditons on absolute stability, we can easily check for stability of the nonlinear system (at least we get sufficient conditions).

However, given a system like

$$ \begin{split} \dot{x} &= Ax - B\phi(y) \\ y &= g(x) \end{split} \tag{2} $$

with the output $y$ being a nonlinear function of the state vector, $y = g(x)$.

Question: Is it still possible to analyse (absolute) stability of $(2)$ using the same (or similar) techniques as for system $(1)$? Only for certain kinds of $g$? Or are all bets off?

Answer 1

A system in the form of $(2)$ can also be written in the form of $(1)$ using

$$ \left\{ \begin{align} \dot{x} =\,& A\,x - B\,\phi(g(y)) \\ y =\,& x \end{align} \right. \tag{3} $$

So $C$ is the identity matrix and the resulting nonlinear function is $\phi(g(y))$. After this you can use the same techniques for analyzing stability as for $(1)$.

Namely for stability analysis it does not matter how you define the output $y$, however this will affect the observability of the system.