Module 16-5157 Сontrol and Instrumentation Dynamical System Analysis & Control Assignment 2016/17 Date Set: 17.10.16 Last Date for Handing In: 10.03.17 Please hand in to CANTOR Reception, before 15:30 hrs. Background The system to be studied involves the positioning of a mechanical load characterised by mass and frictional damping. The load is moved by a hydraulic actuator which responds to a control voltage, and the coupling arrangement between the actuator and load has to be able to cope with a certain amount of flexing - so it is "springy". Hydraulic Actuator Coupling and load Control voltage u (V) Actuator position q (mm) Load position y (mm) Figure 1 - The Open-Loop System The Laplace transfer function model of the hydraulic actuator is: 8 8 ( ) ( ) + = U s s Q s The LTF model of the coupling and load is: Q s s s P Y s + + + = 4 20 14 5. ( ) ( ) 2 , where P is the last digit of your student registration number. Your task is to investigate the open-loop behaviour of this system and design and test closed-loop controllers to change the behaviour, in an effort to meet the specification given below. Specification The test input to the system (both in open- and closed-loop) is to be a step of 5V. The indicators to be used in evaluating the performance of the system are the usual time-domain ones: • The steady-state error between y and the applied step input must be zero. Thus, since e = r - y, a step input of h Volts at r must cause an ultimate output movement of h mm at y in the closed-loop system. • The percentage overshoot in the step response must be less than 5% (of the final value). • The rise time from 10% to 90% of the final value must be less than 400ms. • The settling time to within ±1% of the final value must be less than 2s. The assignment questions follow. Note that questions 1 to 4 are to be done completely "by hand", with no computer assistance. Question 5 is to be done using MATLAB. ALL THE WORK WHICH YOU SUBMIT MUST BE YOUR OWNTask 1. For the time being, ignore the hydraulic actuator, so that the system model becomes Q s s s P Y s + + + = 4 20 14 5. ( ) ( ) 2 . By analysing the steady-state behaviour of this system, and by sketching its step response to a 5 mm step input at q (which you can do by comparison with the standard second-order curves), discuss the open-loop performance of this model when compared with the specification above. Your sketch of the response need not be particularly accurate, but it must at least show the values you have used in your discussions. [20 Marks] Task 2. Remembering that the model used in Question 1 was actually an approximation to a third-order model, now imagine that the hydraulic actuator LTF is now reinstated to give the full third-order model of Fig.1, with a 5V input step now applied at u. No numerical results are expected here, but you should now explain why you would expect each of the performance indicators to get better, or worse, or be unaffected by this change, compared with your discussion in Question 1. [10 Marks] Task 3. In an attempt to improve its performance, the full model is now to be included in a closed-loop unity-negative-feedback (UNF) arrangement, as shown in Fig. 2. The test input (still a step change of 5V) is now applied to the closed-loop at r. Figure 2 - Closed-loop arrangement If the "Controller" is initially a simple proportional gain, Kp , what values of Kp can be used without the system becoming unstable? [25 Marks] Task 4. For the closed-loop arrangement of Fig. 2, and with the "Controller" still as a simple gain, Kp , derive a general expression for the steady-state error of the system, following a step change of 5V at the input r. From this expression, what do you conclude about the usefulness of a simple gain as the controller for this system? [20 Marks]Task 5. Use MATLAB / SIMULINK to carry out the following exercise (SIMULINK is the easiest approach, but it can all be done with MATLAB commands if you wish): Set up the system of Fig. 2 with a PID controller (if you use SIMULINK, you may use the "PID Controller" block or you can build your own PID controller in SIMULINK using an integrator, a differentiator, a summer and three gains). What controller settings meet the specification? In tuning the controller, you might start with the Ziegler-Nichols settings, but they will probably not meet this specification, so will need further tuning. Try to achieve a "conventional" response, so that the magnitude of the first overshoot is greater than that of the first undershoot, etc. (at least up to the settling time). Present plots, properly labelled, of the output y and of the control signal u. Comment on the performance. [25 Marks] TOTAL: 100 MARKS