Energy Engineering - Control Systems

Full exam

Control Systems (Prof. Casella) Written Exam – June 18th , 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 Explain how to assess the stability of a given equilibrium of a nonlinear system described by means of state-space equations. Discuss the similarities and differences with the stability criterion for LTI systems described by transfer functions. Question 2 Discuss how LTI systems can be used as filters in signal processing, what are the main types of filters and what are their possible applications. Question 3 Consider the system represented in the figure, which is part of a recuperated closed-cycle oxy-combustion power generation system. The two compressors on the left pump an oxygen flow w 1 and a carbon dioxide flow w 2 into a recuperator, where they are preheated. The two flows w 3 and w 4 leaving the recuperator react with a fuel flow rate w 5 in a burner and are expanded in a gas turbine with flow rate w 6. Assuming ideal gas and nearly constant temperatures in the recuperator, neglecting the pressure losses in the circuit, and assuming choked flow in the first stage of the gas turbine, the system can be described by the following equations, where C is a constant compressibility coefficient, assumed equal for both the O 2 and CO 2 channels, k is a constant turbine flow coefficient, and p is the gas pressure in those channels and in the combustor. Please note that the system is linear, so there is no need to linearize it. 3.1 Write down the state and output equations of the system in standard state-space form, considering the pressure p as a state, the mass flow rates w 1, w 2, and w 5 as inputs, and the mass flow rates w 3, w 4, and w 6 as outputs. Make sure to write the state-space equations in fully explicit form, with right-hand sides as a function of states and inputs only. (Hint: sum the first two equations together) 3.2 Compute the equilibrium conditions. 3.4 Compute the transfer functions of the system between the input w 1 and the outputs w 3, w 4, w 6.Cdp dt=w 1−w 3 Cdp dt=w 2−w 4 w 6=w 3+w 4+w 5 w 6=kp 3.5 Draw the unit step response plots of the three transfer functions computed at point 3.4. 3.6 Suppose to start from an equilibrium condition where w 1 and w 2 are the same, and that w 1 is suddenly increased by an amount A at t = 0. Plot the time history of the two flow rates w 3 and w 4 on the same diagram. Question 4 Considering the following block diagramA(s)= 10 1+s B(s)=1 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. C(s)= 10 1+2s 1+5su K-A(s) C(s)B(s)yv Question 5 Consider the following control system, where the unit of time constants is the second: 5.1 Design a PI or PID controller with a bandwidth of 0.01 rad/s and at least 60° phase margin. Try to minimize the settling time of the response of y to a step change of d. + 5.2 Compute the static error corresponding to a unit step input applied to the disturbance d.H(s)=2 (1+10s)2G (s)= 4 (1+10 s)3d H(s) C(s)y° -G(s)yu n 5.3 Suppose that the employed sensor is affected by strong measurement noise n in the bandwidth 1-10 rad. Propose an improvement of the control system that guarantees the same rejection of disturbance d, while substantially reducing the effect of the sensor noise on the manipulated variable u. 5.4 Assume that the actual gain of G(s) is in fact 2, instead of 4. How does this affect the performance of the two controllers designed at points 5.1 and 5.3? 5.5Is it possible to bring the bandwidth of the control system to 1 rad/s by means of a PID controller? Motivate your answer.