soluzione300616

Pagina 1

Control Systems (Prof. Casella) Written Exam – June 30th, 2016 Answers sheet Question 1 Define the external stability and asymptotic stability of the equilibria of a single-input, single- output dynamical system. Let ¯u , ¯y be the equilibrium values of the input and output of the system. Apply an impulse perturbation to the input of amplitude e, u=¯u+ϵδ(t ) . If, for any chosen A > 0 there exist a value d > 0 such that for each |ϵ|<δ it follows that |y(t)−¯y|< A , then the equilibrium is stable. If this condition holds and limt→∞ y(t )=¯y , then the equilibrium is asymptotically stable. Question 2 With reference to a single-input, single-output linear time invariant system, discuss the relationship between the eigenvalues of the A-matrix and the poles of the transfer function between the input and the output. If there are no cancellations, i.e., if the number of the poles is the same as the order of the system, then the poles are the eigenvalues of matrix A. In case of cancellations, the poles are a subset of the eigenvalues of matrix A. Question 3 Consider the turbine system in a large Rankine cycle power plant with reheat. The system is described by these equations, where wHP and wLP are the high and low pressure turbine flow rates, Y are the turbine admittances, P are the pressures, r(P) is the steam density, V is the constant reheater volume, P are the produced mechanical powers, and h are the specific enthalpies of steam at the turbine inlets and outlets . Assume that PHP and YLP are constants and that the changes in the specific enthalpy differences are much smaller than the changes in the flow rates, so that h1, h2, h3, and h4 can also be considered as constants. 3.1 Write down the state and output equations in standard state-space form, considering the high…