Consider the following unity feedback system where Cs K Gs

Consider the following unity feedback system, where C(s) = K. G(s) = 1/s(s+1)(s+5). (a) Sketch the squareroot locus. (b) Use the refinement of the squareroot locus to find the value of the gain K, for which the closed-loop system will be critically damped. (c) Find the value of the squareroot locus at its intersection with the imaginary axis, and the corresponding gain K. (d) Use the Matlab command \"rlocus\" to do Part (c), and compare the result with the answer you got in Part (c).

Solution

We will begin by examining a simple control system. We will use a controller that is just a gain - a proportional contoller - and some simple feedback to control a first order system. That system is shown in the diagram below. It has an adjustable gain, K, and a single pole at s = -a.

        The closed loop transfer function is:

                    CLTF(s) = K/(s + a + K).