5 Controller Design Consider a plant y-3y = u with y(0) =-1. In this problem, we want to design a proportional controller: The objective is for y(t) to track the set-point r (a) Is the plant stable? (b) Find the differential equation describing the closed-loop feedback system (c) For what values of ki is the closed-loop system stable? (d) Find the DC gain of the closed-loop system from r to y in terms of ki and k2. What should this DC gain be for the tracking objective? (e) Choose ki and k2 to satisfy the following requirements y(t) r in steady state. lu(t)l 6 for all 12 0, when r = 2 The 95% settling time (three times the time constant) as small as possible. This must be smaller than 6 sec (f) Plot y versus t and u versus t to verify that your controller satisfies the requirements 5 Controller Design Consider a plant y-3y = u with y(0) =-1. In this problem, we want to design a proportional controller: The objective is for y(t) to track the set-point r (a) Is the plant stable? (b) Find the differential equation describing the closed-loop feedback system (c) For what values of ki is the closed-loop system stable? (d) Find the DC gain of the closed-loop system from r to y in terms of ki and k2. What should this DC gain be for the tracking objective? (e) Choose ki and k2 to satisfy the following requirements y(t) r in steady state. lu(t)l 6 for all 12 0, when r = 2 The 95% settling time (three times the time constant) as small as possible. This must be smaller than 6 sec (f) Plot y versus t and u versus t to verify that your controller satisfies the requirements