UNIVERSITY COLLEGE DUBLIN SCHOOL OF ELECTRICAL & ELECTRONIC ENG. EEEN40010 CONTROL THEORY Laboratory CT_1 DOMINANT POLE PLACEMENT VIA ROOT LOCUS 1. Objective: To practice the control design method of dominant pole placement via root locus. Before starting this laboratory generate some personalised digits. Let a, b, c be the last three digits of your student number. For example, if your student number is 19306227 then a = 2, b = 2 and c = 7. If any of these digits is 0 then add 1. For example, if your student number is 19306207 then a = 2, b = 1 and c = 7. In this laboratory description document illustrative MATLAB commands are shown in purple. The problems which you are to solve are shown in blue. You must produce a report for this laboratory. This report will be graded and that grade will carry a weight of 28% when calculating your overall module grade. This is a slight reduction on previous years in light of student feedback. The number of grade steps which can be earned for each of the problems is given in the problem statement and a coarse grading scheme follows each problem statement. In designing using root loci one generally sketches a few root loci to see the expected shape for choices of various parameters (i.e. various introduced poles and zeros). To achieve a high grade you must do the same. All root locus design finally comes down to applying the rlocfind command and picking a gain. You must show on the sketched root locus/loci where you clicked to determine the associated gain, explaining why in accompanying text. If sketched by hand root loci should be subsequently scanned/photographed and included as an image in the report. This is the single exception to the general rule regarding acceptability of handwritten material. It is generally not acceptable, but here the requirement is for you to reveal your understanding of the root locus drawing rules by applying them and assessors cannot judge that understanding if you have the MATLAB command rlocus draw those root loci for you. You may alternatively, if you wish use sketching software to produce these images of “sketched” root loci for your report. To reiterate, what is sought here is evidence that you understand and can apply the root locus drawing rules, because this skill is useful in controller design using the root locus method. The following comments apply to all design laboratories so take note. These comments are extremely important. Traditionally the main loss of grades is due to a failure to properly comprehend the stipulations below. Consider general reporting requirements discussed in previous laboratory. The comments here are particular to the issue of reporting on a design. In reporting a design, report in detail all of the steps taken in the design and the reasons for taking them. The majority of grades are assigned for this careful description of your design process. Pay very close attention to this requirement. Just simply reporting controller gains without any accompanying explanation of how they were obtained and why you have some expectation of good resulting performance will result in few if any grades being earned. This stipulation almost certainly negates the value of using more advanced MATLAB tools such as syms, Sisotool or Control System Designer. These will do too much of the design for you in a rather ad hoc manner and they amount to little more than fast methods for randomly selecting controller gains resulting in rapid convergence to designs that work but for which there is no accompanying reason why they would be expected to work. We often as a result obtain designs that work “in-the-box”, i.e. in the ideal simulation environment, but fail when any of the host of assumptions which have been made do not hold. The rather course grading schemes which follow each problem statement also very slightly nudge you towards a proper layout of the report of your designs. 2. Controller Design: Problem 1: (9 grade steps) A plant has transfer function = 6(+3.9+) (+3.1)(+6.2+3+) (1) where a, b, c are your personal digits. By employing dominant pole placement via the root locus design a PID controller such that: (i) the closed-loop system is stable (ii) there is zero steady state error to a step input (iii) the 2% settling time does not exceed 85% of the settling time of the plant, but should not be less than 150% of the 10%-90% rise time, with the latter restriction taking priority in the event that these two specifications are in conflict. (iv) the PO% does not exceed 50% of the percentage overshoot of the plant or 20%, whichever is larger. Your design should employ controller gains which are small where possible, as unnecessarily large gains yield poorer performance in general. Upon completion of your design simulate the closed-loop system to determine whether design specifications have been achieved, at least in the ideal simulation environment. A control design should be robust, meaning that it should continue to work in spite of inaccuracies such as incorrect modelling. Investigate the robustness of your control design by considering the performance of the closed-loop system in the event where your modelling of the plant zero has been incorrect and the true zero differs by up to 10% How far from its nominal value can your modelled plant zero vary before the closed-loop performance is completely compromised, i.e. not even stable? Grading Scheme: Conversion from specifications to desired dominant pole(s) and requirement or otherwise of I-term (2 grade steps), controller design using root locus (4 grade steps), numerical closed- loop performance and comparison with specifications (1 grade step), investigate robustness of performance to poorly modelled plant zero (2 grade steps). Problem 2: (12 grade steps) A SISO plant with input F and output y is described by the following ODE (the damped, driven pendulum equation): 2 2 + + () = 1 () (2) Let m = a+2b,  = 3√ and l = 0.5(a+2b) where a, b, c are your personal digits. As usual g denotes standard gravity. The nominal deflection y and the nominal input force F are both taken to be 0. The desired operating point is: = 0, ̇ = 0. Obtain a linearised model of the system in the locality of this operating point. Determine the transfer function of this local LTI model. Find the steady-state error to a step input, the 2% settling time and the percentage overshoot for this linearised system. By employing dominant pole placement via root locus design a practical PID controller (i.e. a 4- term controller where the D-term, if included, is not an ideal differentiator but rather incorporates the filter 1/(TD s+1) considered in lectures) such that: (i) the closed-loop system is stable (ii) there is zero steady state error to a step input (iii) the 2% settling time does not exceed 50% of the settling time of the plant itself but should not be less than 150% of the 10%-90% rise time, with the latter restriction taking priority in the event that these two specifications are in conflict. (iv) the PO% does not exceed 40% of the overshoot of the plant itself or 20% whichever is larger. Determine the step response of the resulting closed loop system and present data to show that all specifications have been met. If the circuitry realising the various controller gains has two failure mechanisms, one where the realised gain fails low (to 1% of nominal) and one where it fails high (to 500% of nominal) is the closed-loop system failsafe? Determine the equations of motion of the closed-loop system incorporating the PID controller and the nonlinear global model of the plant given in Eq. (2). Hint: unless a P-term only is present in setting up these equations of motion you will have to model both the plant and the controller, and subsequently the whole closed-loop system using the state-space models which were very briefly discussed through a single example in CT_0 and which will be discussed in far more detail later in lectures. Using for example ode45 find the step response of the resulting closed-loop system for a number of different choices of size of the step. If you have the necessary knowledge then you may avoid use of ode45 and alternatively employ Simulink in this task. Confirm that the linearised closed-loop system and the actual closed-loop system have very similar step responses provided the size of the step input is very small. How are the steady-state error, settling time and overshoot affected by increasing the size of the step input? Grading Scheme: Obtain linearised local model, determine transfer function, numerically evaluate step response characteristics (1.5 grade steps), conversion from specifications to desired dominant pole(s) and requirement or otherwise of I-term (2 grade steps), controller design using root locus (4 grade steps), numerical closed-loop performance and comparison with specifications (1 grade step), failsafe investigation (1 grade step), obtain global model of closed-loop system (1 grade step), numerically determine step response for several values of size of step input, confirm agreement of local and global model for small step input (0.75 grade step), determine effect of size of step input on performance characteristics (0.75 grade step).