Energy Engineering - Control Systems

Full exam

Control Systems (Prof. Casella) Written Exam – Sep 7th , 2021 Surname:.............................................. Name: ................................................... Reg. Number:............................... Signature:...................................... Notices: This booklet is comprised of 7 sheets – Check that it is complete and fill in the cover. Write your answers in the blank spaces with short arguments, including only the main steps in the derivation of the results. You are not allowed to leave the classroom unless you hand in the exam paper or withdraw from the exam. You are not allowed to consult books or lecture notes of any kind. Please hand in only this booklet at the end of the exam – no loose sheets. The clarity and order of your answers will influence how your exam is graded. Question 1 Define the various types of stability of equilibria in a dynamical system. Then, explain why all the equilibria in a given LTI dynamical system have the same stability property. Question 2 The step response of a LTI system approaches zero asymptotically as time approaches infinity. What can you say about its transfer function? Please motivate your answer clearly. Question 3 Consider a pipe carrying a liquid with constant density r, starting from a source at constant relative pressure P i, ending with a valve with a variable flow coefficient A v discharging a mass flow rate w into the atmosphere. Assuming incompressible flow and neglecting friction, the equations of the system are given on the right, where L is the constant pipe length, A is the constant pipe cross-section, P o is the pipe outlet relative pressure. 3.1 Write down the state and output equations of the system in standard state-space form, considering the valve coefficient A v as input, the mass flow rate w as a state, and the mass flow rate w and outlet pressure P o as outputs. 3.2 Compute the equilibrium conditions of the system 3.3 Write down the linearized state-space equations of the system around the equilibrium computed at point 3.2 3.4 Compute the transfer functions of the system between the deviation of the input DA v and the corresponding deviations of the outputs Dw and DP o around the generic equilibrium found at point 3.2.L Adw dt+P o−P i=0 w=A v√ρP oP iP ow 3.5 Draw the unit step response plots of the two transfer functions computed at point 3.4. 3.6 (Water hammer effect) Consider a scenario where the system is initially at equilibrium with certain non-zero values of the mass flow rate and outlet pressure. At time t = 0, the valve flow coefficient is suddenly reduced by a certain amount DA v. Based on the previously found result, draw a plot of the pipe outlet pressure P o, and comment on the consequence that this behaviour may have on the mechanical integrity of the pipe. Question 4 Consider the following block diagramA(s)= 1 s B(s)= 2 1+2s 4.1 Compute the transfer functions between the inputs u and v, and the output y. 4.2 Determine for which values of the parameter K the system is asymptotically stable.u -Ky A(s) C(s)- v B(s) C(s)= 10 1+4s Question 5 Consider the following control system, where the unit of time constants is the second: 5.1 Design a PI controller with a bandwidth of 0.08 rad/s and at least 60° phase margin. Please draw the Bode plot of the loop transfer function explicitly. 5.2 Compute the asymptotic amplitude of the error corresponding to n = sin(0.01t).H (s)=− 1 (1+2s)2 G (s)=0.3 1 (1+2s)4d H(s) C(s)y° -G(s)yu n 5.3 Suppose that the gain of G(s) becomes 0.6, but the control system remains the same. What happens to the performance of the closed loop system? 5.4 Design (if possible) a disturbance compensator which rejects the effects of the disturbance d in the bandwidth 0-0.3 rad/s. 5.5Is it possible to increase the bandwidth of the feedback loop to 1 rad/s by means of a PID controller? Motivate your answer.