300617answers

Page 1

Control Systems (Prof. Casella) Written Exam – June 30th, 2017 Answer Sheet Question 1 State how to determine the stability of a single-input single-output linear time-invariant dynamical system described by its transfer function G(s). Then, explain why in this case stability is an attribute of the system and not of its individual equilibria. A system is asymptotically stable if all the poles p of G(s) have Re(p) < 0 A system is stable if all the poles of all the poles p of G(s) have Re(p)  0 and the poles with Re(p) = 0 have multiplicity one. Otherwise the system is unstable. Due to the superposition principle, the response of the output to a step perturbation of the input has the same shape no matter what the equilibrium is; therefore, all equilibria of the given linear system with transfer function G(s) have the same stability properties, which allows to improperly refer the stability to the system as a whole. Question 2 Consider the control system represented by the following block diagram. There are two sets A and B of properties of this system. All properties in set A can be achieved simultaneously, as well as all properties in set B, while there is a trade-off between each property in set A and each property in set B. Tag the following properties as belonging to set A or B. A Fast response of controlled output y to changes of set point y° A Good rejection of disturbance d B Good rejection of disturbance n A Large value of crossover frequency wc A Wide bandwidth of transfer function between the set point y° and the controlled output y B Small sensitivity of the control variable u to changes of the set point y° B Small sensitivity of the control variable u to changes of the disturbance d B Small sensitivity of the control variable u to changes of the disturbance…