Energy Engineering - Control Systems

Full exam

Control Systems (Prof. Casella) Written Exam – July 10th , 2023 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 Consider a generic linear time-invariant SISO dynamical system, described by its transfer function.. State under which conditions the unit step response asymptotically approaches a constant value. Then, state under which additional conditions that value is actually zero. Question 2 Explain why controlling systems that have an open-loop step response showing an undershoot is in general difficult. Question 3 The equations shown on the right describe the temperature dynamics in an insulated pipe modelled by three finite volumes, assuming perfect insulation, negligible wall heat capacity, and a liquid fluid with constant density. w is the fluid mass flow rate, h j are the specific enthalpy values at each volume boundary, c is the constant specific heat capacity of the fluid, and M is the constant total mass of fluid in the pipe. 3.1 Write down the system equations in standard state-space form, considering T 0 and w as inputs and T 3 as output. 3.2 Compute the equilibrium conditions for generic constant values of T 0 and w, including w = 0. 3.3 Write down the system's linearized equations around equilibria with w  0 (Hint: do use the equilibrium conditions to simplify the equations as much as possible).dE 1 dt=w(h 0−h 1) dE 2 dt=w(h 1−h 2) dE 3 dt=w(h 2−h 3) E j=Mc 3T j,j=1,...,3 h j=cT j,j=0,...,3 3.4 Compute the transfer functions between the deviations of the inputs DT 0, Dw and the deviation of the output D T 3 . and write them down in gain/time constant form. 3.5 Draw the diagrams of the unit step response of the transfer function found at point 3.4. Question 4 Consider the following block diagram 4.1 Compute the transfer functions between the inputs u and v and the output y. 4.2 Determine for which values of K the transfer functions are asymptotically stable and show an undershoot in their step response.A(s)= 2 1+4s B(s)= 1 1+s C(s)= 1+4s su B(s)-KyA(s) C(s)D(s) v- D (s)= 1 1+2s Question 5 Consider the following system, where the unit of time constants is the second: 5.1 Design a standard single-loop feedback PI or PID controller with a bandwidth of 0.02 rad/s and at least 60° phase margin. Draw the block diagram of the overall control system and the Bode plot of |L(jw)|. 5.2 Design a cascaded PI or PID controller with the same settling time of the output y(t) to a step- change of the set point y°(t), which is also able to reject the disturbance d(t) in the frequency range 0-1 rad/s. Draw the block diagram of the overall system and the Bode plot of |L(jw)|.G 2(s)=101 (1+100s)(1+10s)G 1(s)=50 (1+s)(1+0.1s)d G 1(s) G 2(s)y v nu 5.3 Suppose now that the sensor measuring v(t) in the cascaded controller fails, its output getting stuck to the lowest value in the measurement range. Explain how the whole control system will react and how the system output will change as a consequence of that. 5.4 Assume now that you wanted to obtain the same disturbance rejection performance of the cascaded controller of point 5.2 (i.e., reject disturbances in the frequency range 0-1 rad/s) by means of a single-loop feedback controller. Compare the performance of this system with the performance of the system designed at point 5.2, regarding the response of the manipulated variable u(t) to high-frequency measurement noise n(t).