Stable compensators in parallel to stabilize arbitrary proper rational SISO plants

Abstract

We consider stabilization of linear time-invariant (LTI) and single input single output (SISO) plants in the frequency domain from a fresh perspective. Compensators that are themselves stable are sometimes preferred because they make starting the system easier. Such starting remains easy if there is a stable compensator in parallel with the plant rather than in a feedback loop. In such an arrangement, we explain why it is possible to stabilize all plants whose transfer functions are proper rational functions of the Laplace variable $s$. In our proposed architecture we have (i) an optional compensator $C_s(s)$ in series with the plant $P(s)$, (ii) a necessary compensator $C_p(s)$ in parallel with $C_s(s)P(s)$, along with (iii) a feedback gain $K$ for the combined new plant $C_s(s)P(s)+C_p(s)$. We show that stabilization with stable $C_s(s)$ and $C_p(s)$ is always possible. In our proposed solution the closed-loop plant is biproper and has all its zeros in the left half plane, so there is a $K_0$ such that the plant is stable for $K>K_0$. We are not aware of prior use of parallel compensators with such a goal. Our proposed architecture works even for plants that are impossible to stabilize with stable compensators in the usual single-loop feedback architecture. Several examples are provided.