2019 06 19 exam answers

Page 1

Control Systems (Prof. Casella) Written Exam – June 19th, 2019 ANSWER SHEET Question 1 State the superposition principle for linear, time-invariant dynamical systems. Explain why it makes the analysis of the system response to inputs easier than in the general case of nonlinear systems. Given a linear, time-invariant system with input u(t), initial state x(0) and output y(t) 1. The motion of the output y(t) is the sum of the free motion, obtained by starting from the given initial state and applying a zero input, and of the forced motion, obtained by starting from zero initial conditions and applying the given input 2. Regarding the free motion, if the application of a certain input u1(t) results in the output y1(t), and the application of a certain input u2(t) results in the output y2(t), then the application of a linear combination of the two inputs au1(t) + bu2(t) results in the same linear combination of the two outputs ay1(t) + by2(t). The superposition principle makes the analysis of the system response much easier because multiplying the input by a certain factor doesn’t change the shape of the output, only its scale, and by the same factor. In the case of a step response, for example, it is only necessary to compute the response to a unit step – the response to step of any other amplitude are simply obtained by multiplying the unit step response by the step amplitude. Also, the superposition principle allows to decompose the input into a sum of fundamental components (e.g., sinusoids, steps), and compute the output as the corresponding combination of the responses to the fundamental components. Question 2 Draw the block diagram of a cascaded feedback control system. Explain in which cases it can be advantageous over a standard feedback controller and what are…