Energy Engineering - Control Systems

Full exam

Control Systems (Prof. Casella) Written Exam – July 23rd , 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 following control system. The controlled variable y is measured by a very fast sensor, that has negligible dynamics, but is affected by a lot of high- frequency noise n. The low-pass filter F(s) is thus introduced to reduce the effect of such sensor noise on the system performance. Discuss all the effects that the filter can have on the various aspects of the control system performance, particularly regarding the dynamic performance and the control sensitivity to the sensor noise. Assuming that you need to obtain a specified bandwidth and phase margin of the system, discuss possible design criteria for the filter F(s), highlighting potential trade-offs in the design process. Question 2 Assume that you need to control an asymptotically stable LTI system. Using transfer functions and block diagrams, explain why stability is a potential concern when using feedback control, while it is not when feed-forward control is used. Try to be as precise and rigorous as possible in your explanation. Question 3 Consider the motion of a skydiver, descending towards the ground suspended to a parachute with a vertical descent velocity v. The skydiver is subject to the force of gravity F g and to the drag of the parachute F d. The drag is proportional to the square of the vertical velocity and to a drag coefficient k d(u), which depends on the shape of the parachute that can be changed by pulling the control strings by a certain lenght u, and is thus a function of that variable. Assume that function is smooth and monotonously increasing. The equations describing the system are the following, where M is the skydiver's mass, g the acceleration of gravity and a is the skydiver's acceleration towards the ground. 3.1 Write down the state and output equations in standard state-space form, considering u as input and v, a as outputs 3.2 Compute the equilibrium conditions (i.e., constant velocity) for the system 3.3 Write down the system's linearized equations around a generic equilibrium 3.4 Compute the transfer functions of the system between small deviations of the input Du and the corresponding deviations of the outputs Dv and Da around the equilibrium point computed at the previous step; write them down in gain/time constant form. 3.5 Draw the step response plots of the two transfer functions computed at point 3.4.nd C(s)y° -G(s)yu F(s) Ma=Fg−F d F g=Mg F d=k d(u)v 2 a= ˙v Question 4 Considering the following block diagram A(s)=10 s B(s)=1−s 1+10s 4.1 Compute the transfer functions between the input u and the outputs y and v. 4.2 Determine for which values of the parameter K the system is asymptotically stable and shows an undershoot in the step response of the output y. 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 0.005 rad/s and at least 50° phase margin. Please draw the Bode plot of the loop transfer function. 5.2 Design a PI or PID controller with a bandwidth of 0.005 rad/s and at least 60° phase margin. Please draw the Bode plot of the loop transfer function. 5.3 Plot the qualitative diagrams of the response of the controlled output y to a step change in the set point y° for the two controllers designed at points 5.1 and 5.2. 5.4Consider adding a static disturbance compensator to the control diagram. Assuming the transfer functions are known with very good approximation, compare the dynamic performance you can get from this controller and from the controller designed at point 5.2, in terms of disturbance rejectionC(s)=51+10s 1+su K-A(s) C(s)B(s) -y v H(s)=2e−100 s (1+200s)G (s )=10e −100s ( 1 +200s )(1 +30s )d H(s) C(s)y° -G(s)yu