Chapter 5.P2, Problem 5P

In the following problem, we ask the reader to some of the details left out of the above discussion, to analyse the closed-loop system for the stability properties, and to conduct a numerical simulation of the nonlinear system.

Use the Routh criterion to show that if the gain factor $K$ is sufficiently large, $H_k(s)$ will have two poles with positive real parts and the corresponding closed-loop system is therefore unstable. Derive an expression for $K_c$, that value of $K$ at which the real parts of a pair of conjugate complex poles of $H_k(s)$ are equal to $0$.