The PID controller is not always sufficient for every system. Depending on the system, a PID controller does not always allow you to improve the transient performance arbitrarily. Consider the open-loop transfer function below, which is similar to the hydraulic system open loop transfer function. K KGH(S) = s(s – 1 + i)(s – 1 - i) A. Use RL tool to plot the root locus for this system for three PD controller design cases: a) add a derivative zero to the right half plane s = +1, b) add a derivative zero at s =-0.5, c) add a derivative zero at s=-3. These represent all the possible derivative zero locations. Do any of these cases shift the root locus so that the closed-loop complex poles are moved to the left of the open loop poles, thus making them respond faster? (hint, the answer is no). This does not mean that you cannot find a controller that improves the settling time, however it would be higher order than a PID controller. Print out the 3 root locus plots. B. You would need to add a controller with a set of complex zeros instead of a PID controller. We don't cover this topic in this course but we can use Matlab to try it out. Use RLtool to draw the root locus and simulate the step response using a controller with complex zeros at s = -4 = 4i. Print out the root locus and the step response. Show that this type of controller can indeed improve the settling time. Print out the root locus and the step response. C. There is a problem with the controller in part B. Recall from lecture that we cannot actually build a device that predicts the future, that is, with a higher order numerator than denominator. So, to actually build this controller we would have to add complex poles to the controller. Redo the plots from part B by adding complex poles far to the left. Find a location for the complex poles that does not change the shape of the branches nor the desired pole location you got from part B.