Energy Engineering - Control Systems

Full exam

Control Systems (Prof. Casella) Written Exam – June 19th , 2019 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 State the superposition principle for linear, time-invariant dynamical systems. Explain why it makes the analysis of the system response to inputs easier than in the general case of nonlinear systems. Question 2 Draw the block diagram of a cascaded feedback control system. Explain in which cases it can be advantageous over a standard feedback controller and what are the tuning criteria for its components. Question 3 Consider an open vertical cylindrical tank, where two incoming mass flow rates of water w 1 and w 2 at temperatures T 1 and T 2 are mixed. An outlet mass flow rate w o at temperature T is then drawn from the bottom of the tank. Assuming a constant specific heat capacity c and de = dh = cdT, where e and h are the specific energy and enthalpy of the liquid, the system is described as: 3.1Write the system equations in standard state-space form, considering w 1, w 2, w o, T 1 and T 2 as inputs and l, T as states and outputs. 3.2Compute the equilibrium conditions of the system, assuming non-zero flow rates and level. 3.3Write down the linearized equations for the system, taking into account the equilibrium conditions in order to simplify them as much as possible. dM dt=w1+w 2−w o dE dt=w 1h 1+w 2h 2−w oh o M=ρAl E=McT h 1=cT 1 h 2=cT 2 h o=cT 3.4 Compute the transfer functions from the deviation of the mass flow rate Dw 1 to the deviations of level Dl and of the outlet temperature DT, writing them down in gain/time constant form. 3.5 Draw the diagrams of the unit step responses of the transfer function found at point 3.4. 3.6 Consider the transfer function between Dw 1 and D T. Is it possible to conclude that there is a pole/zero cancellation in the system? If so, can you estimate the value of the cancelled pole? Question 4 Consider the following block diagram A(s)=2 s B(s )=5 1 +10s 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 two transfer functions are both asymptotically stable. C (s )=1 1 +su Ky A(s)C(s) - B(s)v - 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 0.002 rad/s and at least 60° phase margin. 5.2 Design a PID controller with N = 10 and the maximum bandwidth you can get with a phase margin of 60°H(s )=−20 sG (s )=−5 s (1 +100s )(1 +20s )d H(s) C(s)y° -G(s)yu n 5.3 Draw an approximated plot of the step response of the controlled output y to a unit step applied to the disturbance d for both controllers. 5.4With reference to the PID controller designed at point 5.2, discuss the effects of reducing the parameter N on the control system performance