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Control Systems (Prof. Casella) Written Exam – July 2nd, 2015 Answer Sheet Question 1 With reference to feed-back control systems, explain why the input-output dynamic behaviour of the plant to be controlled is crucial for the system performance. Also explain why, on the other hand, a very precise knowledge of that dynamic behaviour is not required, if the controller is designed properly. The input-output dynamics of the plant is a key factor in determining the dynamic behaviour of the closed-loop system, first and foremost stability, then response speed and ability to closely follow the set point and reject disturbances. A controller which is not correctly designed for a certain plant dynamics can make the closed-loop system unstable. On the other hand, if the controller is designed with a good phase margin, significant but yet small changes in the plant dynamics will not alter its stability, and lead to changes of closed-loop dynamic features such as response time, settling time or damping of oscillations that are acceptable in most use cases. Question 2 Briefly explain why the stability of a linear system with transfer function G(s) is completely determined by its poles. Then, state precisely under which conditions such system will be asymptotically stable, simply stable, or unstable. The external stability of a linear system is determined by its impulse response, in particular by the fact that it is bounded or unbounded, and that it does or does not approache zero as time approaches infinity. Now, the impulse response is the inverse Laplace transformation of G(s), which can be decomposed in the summation of simple rational functions whose poles are the poles of G(s), and whose inverse transforms are the components of the impulse response. Therefore, the stability…