Energy Engineering - Control Systems

Full exam

Control Systems (Prof. Casella) Written Exam – June 17th , 2020 Notices: This PDF file has 3 pages. You don't need to print it out, you can read it directly on your PC screen. Write your answers preferably on white sheets of paper and with a blue or black pen, to maximize readability of the scans. Please try as much as possible to write in a clear and ordely manner. Please do not hand in your minutes, only clean and well-written answers to the questions. It is not necessary to report all the mathematical derivations, though the main ones could be helpful to understand the process you followed to solve the problems. You are not allowed to leave the virtual classroom unless you upload the exam paper or withdraw from the exam. You are not allowed to consult books or lecture notes of any kind, nor to communicate with other people with any means during the exam. When you are finished, make a single-PDF scan of your answer sheets named with your Person Code (e.g. 10035506.pdf) and upload it via Forms. Please make sure the PDF file has the answers in the same order as the questions, to expedite correction. In case of problems with the Forms submission process, you can send the pdf file to [email protected]. Please only consider this as a last-resort backup solution. The clarity and order of your answers will influence how your exam is graded. Question 1 Consider the problem of controlling the room temperature in a building during winter. The room is heated with hot water radiators. Consider these three solutions: 1.The supply temperature of the boiler is determined on the basis of the outside temperature, measured by an outdoor sensor or obtained from a meteo web server, by means of a suitably calibrated curve. The radiator valves are always open. 2.The supply temperature of the boiler is fixed. The valves on the radiators are equipped with thermostatic heads, that open or close the valve based on the sensed room temperature 3.The supply temperature of the boiler is determined on the basis of the outside temperature, measured by an outdoor sensor or obtained from a meteo web server, by means of a suitably calibrated curve. The valves on the radiators are equipped with thermostatic heads, that open or close the valve based on the sensed room temperature Classify the three solutions as open-loop or closed loop, giving motivations for your choice. Then, discuss the pros and cons of the three solutions in terms of performance. Question 2 Explain what is the role of Integral action and Derivative action in a PID controller. In particular, point out in which cases each of the two actions is mandatory or at least beneficial. Question 3 Consider the model of a pandemic outbreak, where S is the number of susceptible individuals, E the number of exposed individuals, who are incubating the disease but are not contagious, I is the number of infectious individuals, I e is daily rate of incoming infectious immigrants, and D is rate of new infectious cases per unit time, i.e., the daily number of reported new cases. The parameter b > 0 indicates the infectiousness rate, the parameter e > 0 is the inverse of the average incubation time and the parameter g > 0 is the inverse of the average infectiousness time. 3.1 Assume that the outbreak is at the beginning, i.e. S  N. Under this assumption, write down a simplified second-order, LTI model in state-space form, considering I e as input and D as output. 3.2 Compute the equilibrium conditions for the system derived at point 3.1 3.3 Compute the transfer function from the number of infectious immigrants I e to the daily rate of new infectious case D and write it down in pole-zero form. 3.4 Determine the stability of the equilbria found at point 3.2. Discuss the stability of the equilibria as a function of the reproduction number R 0 = b/g, which is the average number of subjects that are infected by one infectious subject. 3.5 Consider now the full third-order dynamical system, still under the simplifying assumption that S  N. Does the transfer function from I e to D involve pole-zero cancellations? Motivate your answer appropriately.dS dt=−βSI N dE dt=βSI N−ϵE dI dt=ϵE−γI+I e D=ϵE Question 4 Consider the following block diagram 4.1 Compute the transfer functions between the inputs u and v and the output y. 4.2Using Bode's stability criterion for feedback systems, determine for which non-zero values of K the system is asymptotically stable 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 0.025 rad/s and at least 60° phase margin. Please draw the Bode plot of the loop transfer function and briefly comment how you carried out the design. 5.2 Design a PI or PID controller with a bandwidth of 0.14 rad/s and at least 40° phase margin. Please draw the Bode plot of the loop transfer function and briefly comment how you carried out the design. 5.3 Plot the step response of the controlled output y to a unit step change of the set point y°. 5.4 Compare the response of the manipulated variable u to high-frequency feedback noise n in the two cases of 5.1 and 5.2. What is the price to pay for the increased bandwidth? 5.5 Assume that a sensor with a time delay of 3 s is used for the control of this system. Discuss the impact of this additional delay on the performance of the two controllers designed at points 5.1 and 5.2H(s )=−20 ( 1 +100s )(1 +15s )d C(s)y° -G(s)yu nu C(s)A(s)y B(s) - -v G (s )=4 ( 1 +100s )(1 +15s )(1 +5s )A(s)=K s B(s)=10 1+s C(s)=1+0.2s 1+0.1s