ELEC4632 Lab 5

Design and Real-time implementation of PI control system for set-point control

In this lab, you are going to design, simulate, and implement a discrete-time proportional-integral (PI) control system on an actual W-T system in real-time. The PI controller is from the proportional- integral-derivative (PID) controller category without its derivative part. Similar to the situation you faced in Lab 4, our identified model from Lab 4 may no longer be valid for your specific W-T setup since some valves positions may have been changed. Therefore, for the purpose of this lab, you must repeat system identification, as what you did in Lab 2 and Lab 4, to be able to design and simulate your PID control system and validate it by implementing your controller on the actual W-T system. However, in real world, having a valid model for PID control system design is not essential since the structure of PID controller only depends on the reference signal, yref, and the measured output, y.  In  our  case,  to  be  able  to  simulate  the  control  system  first   before  its  real-time implementation, we must have a working model of the W-T process. More details on PID controller structure in both continuous-time and its equivalent discrete-time are provided in the Appendix, which you are highly encouraged to read.

Note:

Pre-lab Exercise

a.   If the PI transfer function in continuous-time is given as GCT (s), in Eq.(1)on the left, show that its equivalent discrete-time form in Z-domain is obtained as GDT (z), Eq.(1)on the right, using Zero-Order-Hold (ZOH) method. Note that in Eq. (1), Kp is the proportional gain, Ki is the integral gain, and h is the sampling time. Write your answer in the provided box.

b.   Use inverse Z-transform. for GDT (z) in Eq.(1) to find the difference equation relating the input (error signal e(k) = yref (k) - y(k)) to the output (control input u(k)) of the discrete-time PI controller as shown in Fig. 1. Write your answer in the provided box.

c.   Using the  block diagram of given in Fig. 1, find the closed-loop transfer function Gcl(z) = Y(z)/Yref(z), and the transfer function from reference signal to control input, i.e., U(z)/Yref(z). Write your answer in the provided box.

Fig. 1. Block diagram of a feedback control system with PID controller.

Lab Exercise (2 marks)

Make sure that you have your lecture notes on the topic of “Digital Control  System Characteristics”, and previous lab notes and your personal notes with yourself. You can always access lecture and lab notes via Moodle as well. Now, follow the steps below for this lab exercise.

1. System Identification (0.5 marks, checked after 50 minutes)

a.   Repeat system identification on the W-T system as you did before in Lab 2. Show your results to the demonstrator to receive the mark for this part of the lab exercise by plotting the figures that you were asked before in Labs 1 and 2.

2. PI control design (0.5 marks, checked after 1 hour 30 minutes)

a.   Once a good model is obtained, use either MATLAB or Simulink to design and simulate a PI controller for your identified system (check the remarks below). You can assume initial conditions are zero. Use the same reference signal for the control system to  track as fast as possible without exceeding the control input limits, i.e., yref (k) = {0, 0.7, -0.2, 0.5, 0} with each level period to be 140×0.75 = 105sec. For a proper design, you need to find good values of PI controller gains Kpand Ki such that the overshoot remains less than 2 percent. Also, make sure your choice of PI controller gains results in control input remaining within its limits.

b.  There are several methods for selecting proper PI gains to achieve the desired control performance. The most trivial method is using the trial-and-error approach. We know that from many resources that increasing Kp reduces the rise time (increasing response speed) and adding Ki removes steady state error as well as slowing down the response while it increases overshoot. You can begin by choosing Kp = 0.5 and Kp = 0.01, run the simulation and examine the output following the reference signal and control input. To improve the control system performance, change Kp by 0.05 increments/decrements and Ki by 0.005 increments/decrements. Other famous empirical methods to find suitable Kpand Kp is the so- called Ziegler-Nichols Step-Response method and Ziegler-Nichols Ultimate- Sensitivity method, which is explained in Appendix. MATLAB and Simulink also provide automatic tuning for PID, particularly if you use built-in PID Controller block in Simulink.

Remarks:

-     Note that we are designing just a PI controller here. The reason for not using the full PID controller with derivative action is that the measurements are noisy and adding derivative of the error signal to the controller would aggravate the noise in the closed-loop system. Refer to Appendixfor more discussion on reducing the side effects of derivative action in PID controllers.

-    To design the PI control system in MATLAB, you can define the closed-loop transfer function Gcl(z) = Y(z)/Yref(z) (obtained in your prelab) and then use “lsim” function to simulate the control system. Note that you can only get the output when using lsim with Gcl.Therefore, you also need to use lsim a second time with the transfer function  from reference signal to control input U(z)/Yref(z) and the simulated output obtained to find the control input values separately. You may find the MATLAB functions “feedback” and “pid” useful. You can also use other methods in MATLAB.

-    To design the PI control system  in Simulink, You can build the block diagram using transfer functions similar to Fig. 2. Another possible way is to build the system using state space representation (i.e. using G, H and C obtained from system identification) and using the built- in PID Controller block in Simulink.

3. Real-time implementation of the control system (1 marks, checked after 2 hours)

a.  When you are satisfied with simulation results, download the pre-built Simulink file named WaterTankSysControlPID.slx from Moodle. Set your model and controller parameter in the real-time model as shown in Fig. 3. The parameters are PI Controller block parameters Kp and Ki, and input and output offsets, u_offset and y_offset, respectively. You just need to assign them with their values in  MATLAB Workspace as Simulink can read them from there. Moreover, you should change the default values in Saturation TANK#1 block to Vmax and Vmin of your W-T setup.

b.  After making sure all the proper settings are in place, run the Simulink model. The running time is set to 5×140×0.75 = 525sec or 8 minutes and 45 seconds, and the program will stop after this time. The data will be recorded in MATLAB Workspace as PIDLogData in Structure   format,    and    it    is    saved    in    the    directory    “Documents/MATLAB”    as PIDControlData_0.mat. Similar to what was explained in Lab 2 for data recording, if you repeat the experiment, the new data will be save under the same name with an increment of one unit, so you would never lose any test data.

c.   Copy the auto-generated data files (found in Documents/MATLAB) for both system identification (i.e. all SysIdenData_x.mat) and control system implementation (i.e. all PIDControlData_x.mat) in addition to your codes and/or Simulink model  to your own computer. You will need these for the final report.

d.   Finally, extract the data as shown below and plot them against your simulated results similar toFig. 4.

treal = PIDLogData.time;

yref = PIDLogData.signals(1).values(:,1);

yreal = PIDLogData.signals(1).values(:,2);

ureal = PIDLogData.signals(2).values;

Fig. 3. Pre-built Simulink model for real-time output state feedback control with observer.

This figure  illustrates the  actual  PID  control  results  obtained from  one  of the W-T systems. It compares the real-time results with the simulated ones. The PI controller gains were chosen as Kp = 0.68 and Kp = 0.03 for this test with zero initial conditions. As you can see, the practical results are quite similar to the simulation ones for output signal in Fig.4(a) and control input  signal in Fig. 4(b). Both simulated output and control input signals are shifted up by output offset and input offset, respectively. This confirms that the identified model was accurate enough to represent the process andto be used for the controller design, as well as validity of the PI control system design.

Fig. 4. Comparison between simulated and actual PI control of W-T system in set-point tracking, (a) Output for different water levels, (b) Control input.

a. Optional as Bonus: Can you explain the behaviour of the real-time control operation in the first period (initial transient behavior) shown in Fig. 4? Why is the control input in Fig. 4(b) saturated at the beginning?