Question

A second-order process is described by its transfer function G(s) = (s+1)(843) and a PI controller by Consider feedback control with unit feedback gain as shown in Figure 1 A disturbance D(s) exists, and to achieve zero steady-state error, a small integral component is applied. Technical limitations restrict the controller gain kp to values of 0.2 or less. The goal is to examine the influence of the controller parameter k on the dynamic response. D(s) Controller Process X(s) Y(s) Figure 1: Block diagram of a second-order process with PI-control. 1. Are there any stability issues? Specifically, are there any range re- strictions for kp and ki that have to be considered for the closed-loop (5 points) system to remain stable? 2. Determine the characteristic polynomial of the closed-loop system in a form suitable for root-locus analysis, that is, obtain the "root locus form" of the closed-loop system and extract the open-loop transfer function F(s): Note that the closed-loop denominator is the same for the setpoint X(s) and for the disturbance D(s). For the purpose of drawing a root locus diagram, a distinction between the independent variables is not (5 points) necessary 3. We now choose2. Plot the pole-zero diagram of F(o), then draw the asymptotes of the root locus diagram in the s-plane. Deter- mine the number of branches, their angle with respect to the positive real axis, ard the intersection point of the asymptotes ơA. Next, de- termine the branchoff point ơB and finally sketch the complete root (10 points) locus curve. 4. Determine the value of kr for (a) the critically damped case (double pole) and for the underdamped case with no undershoot (poles on a (5 points) 45 line in the complex plane Note: You need to perform these steps manually. In particular, you need to show how you determined ki in Part 4. An optional template for your sketch is provided in Figure 2. You are encouraged, however, to verify your results with Scilab

Answer 1

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5 ニF(s) .