Stability and level of passivity

Question

Given two interconnected systems $G_1$, $G_2$ where $G_1$ is strictly passive and $G_2$ is passive. Is the feedback interconnection of $G_1$ and $G_2$ asymptotically stable or only stable?

And is the answer the same if one (or both) are linear systems?

Answer 1

I am assuming you have a system of the form: $$ \begin{aligned} \dot x_i & = f_i (x_i,e_i) \\ y_i & = g_i(x_i,e_i) \end{aligned} $$ Then the origin of the closed loop system formed by the feedback interconnection of two time invariant dynamical systems is asymptotically stable if

• Both feedback components are strictly passive
• Both feedback components are output strictly passie and zero-state observable or
• One component is strictly passive and the other one is output strictly passive and zero state observable. If the storage function of each component is radially unbounded, the origin is asymptotically stable.

See for example theorems in Chapter 6 of Nonlinear Systems by Hassan Khalil.