Final Assessment for Advanced Control Systems (EEET 2100 and 1368) Question One Multiple Choices (30 marks) 1. The transfer function of a system is given by G(s) = 1 s(s+ 1)(s+ 2) For a unit step input signal, the steady-state value of the step response is (a) 0.5; (b) 0; (c) ∞; (d) −1; (e) none of the above. 2. The transfer function of a system is given by G(s) = 1 (s+2)2 and the pro- portional controller is K = 1. The complementary sensitivity function T (s) is calculated as (a) T (s) = 4 s2+4s+4 ; (b) T (s) = 1 s2+4s+5 ; (c) T (s) = s 2+4s+4 s2+4s+5 ; (d) T (s) = 1 s2+4s+4 ; (e) none of the above. 3. The transfer function of a system is given by G(s) = 1 s+1 . The PI con- troller is chosen to have the structure C(s) = Kc s+1 s where proportional control gain was chosen to be Kc = 10. In order to reduce the effect of disturbance, you would choose (a) Kc = 3; (b) Kc = 10; (c) Kc = 30; (d) Kc = 6; (e) all the above. 4. The transfer function of a double integrating system is given by G(s) = 10 s2−1 . To design a proportional plus derivative controller (PD) without filter, we could select the desired closed-loop poles as (a) −10, −12, −13, −14; (b) 10, 12, 13, 14; (c) −10, −10,−100; (d) −10, −10. (e) none of the above. 5. The transfer function of a system is given by G(s) = 1 s . The input disturbance for this system is known to be a combination of a constant and a sinusoidal signal with frequency ω0 = 1 Hz. To reject this dis- turbance without steady-state error, we choose the resonant controller to have the structure: (a) C(s) = c3s 3+c2s2+c1s+c0 s(s2+1) ; (b) C(s) = c2s 2+c1s+c0 (s2−1) ; (c) C(s) = c2s 2+c1s+c0 (s2+2pi) ; (d) C(s) = c3s 3+c2s2+c1s+c0 s(s2+2pi) ; (e) none of the above. Question Two (20 marks) The transfer function of a process unit in a resource company is described by: G(s) = 1 (s+ 1)(s+ 8)2 e−6s Design a PID controller with filter to control this time delay system. 1. (10 marks) You may use the MATLAB program pidplace.m to find the PID controller parameters or you may choose to find the PID con- troller parameters analytically. Present the PID controller parameters for the three cases where all the desired closed-loop poles are at −1, −0.2 and −0.1, respectively. 2. (10 marks) Assuming that the reference signal to the system is 1, sim- ulate the closed-loop responses for all the three cases. In the simulation of the PID control system, the derivative control is implemented on the output only. Present the control signals and output signals graphically. State your sampling interval and simulation time. Question Three (30 marks) A complex system is controlled by the cascade PID control system as illus- trated in the following figure, where the inner-loop controller is a proportional controller with gain of K to stabilize the unstable system and the outer-loop is controlled by a PID controller. 1. (10 marks) Choose the inner-loop controller gain K such that the inner-loop system has a closed-loop pole at −10 with kp = 1 and find the controller parameters c2, c1, c0 in the outer-loop PID controller C(s), where C(s) = c2s 2 + c1s+ c0 s ; G2(s) = 2 (s+ 1)(s+ 8) All desired closed-loop poles for the outer-loop system are chosen to be −1. (Hint: you may use pole-zero cancelation technique to simplify the computation) 2. (10 marks) The parameter kp is varying and depending on the op- erating conditions. In order to guarantee the closed-loop stability for the variation of kp, the maximum and minimum values of kp need to be determined. Use Routh-Hurwitz stability criterion to determine the maximum and minimum values of kp allowed for the closed-loop control system. 3. (10 marks) With a step reference signal r = 1, simulate the cascade control system for the nominal case where kp = 1, and the cases with the maximum and minimum values of kp determined by the Routh-Hurwitz stability criterion. You need to choose an appropriate sampling interval ∆t and simulation time Ts. You may include a derivative filter for the outer-loop PID controller implementation. – m 6 -C(s) – m- K – kps−10 -G2(s) – 6 + + – – R(s) U(s)∗ Y (s)U(s) Question Four (20 marks) Use the disturbance-observer based approach to design a PID controller for the following system: G(s) = 2 s2 1. (5 marks) Choose the desired closed-loop characteristic polynomial for the proportional plus derivative controller as s2+2ξwns+w 2 n where ξ = 0.707 and wn = 3, while the pole for the estimator is −4. What are the values of K1, K2 and K3? 2. (8 marks) Simulate the closed-loop step response and input distur- bance rejection where the derivative filter time constant τf = 0.1τD. In the simulation, the reference signal r = 1 and the input disturbance has an amplitude of −3 entering the simulation at half of the simula- tion time. The sampling interval ∆t = 0.001 and the simulation time Tsim = 3. What are the maximum and minimum values of the control signal? 3. (7 marks)Evaluate the effect of constraints on the control signal where the constraint parameters umax and umin are chosen to be 90 percent of the control signal’s maximum and minimum amplitude from the previous step. Submit your MATLAB/Simulink programs together with your solutions, simulation results (control signal and output plots) and discussions.
