Is a feedback system with an unstable component and the other component being zero internally stable?

Question

So let's consider a system like I described, say looking like such: Where $K, P_{1}$ and $P_{2}$ are all multivariable transfer function matrices. In this case technically it could be presented as interconnection of two systems, $K$ and $(P_{1}-P_{2})$. As far as I know, the internal stability then occurs as long as $||K(P_{1}-P_{2})||_{\infty} < 1$ (please correct me if I'm wrong). What happens then if $K$ is unstable but also $P_{1} = P_{2}$ are stable? Is the internal stability criterion fulfilled then?

Answer 1

Internal stability means that all the closed-loop transfer functions are stable. In the above diagram you have not defined the external inputs (references, disturbances, measurement noise) and outputs. Typically, in the plant output $P_1-P_2$ there will be some measurement noise $\eta$ that will be added as external input into your system. So you want (among others) the transfer function from $\eta$ to $u$ (the controller's $K$ output) to be stable i.e. $$[I+K(P_2-P_1)]^{-1}K$$ to be stable. When $P_1=P_2$ this transfer function is unstable for unstable $K$.