School of Electrical Engineering &
Telecommunications
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:
This lab is also conducted remotely through Microsoft Teams for online students. The lab exercise
needs to be done on the lab computer as you will implement your control system on the actual water
tank system in the lab. Please note that you will lose access to the lab computer 15 minutes
before the lab end time. Make sure to copy all your codes, the auto-generated data files, and
screenshots of your results to your own computer before the last 15 minutes.
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.

G (s) =
U(s)
= K +
Ki , ⎯
Z⎯OH→ G (z)= U(z) =K + Kih ,
CT E(s) p s DT E(z) p z−1
(1)
2


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.


yref (k)






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


e(k) u(k) y(k)
PID Controller
Process
G(z) Gc(z)
Solution:
3

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 Kp and 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 Kp and 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 Appendix for 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.
4

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
to Fig. 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.
5

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 and to 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?
6

Appendix
Many practical control problems are solved by PID-controllers. The well-known version of the PID-
controller can be described by the following equation [1],

u(t)=K (e(t)+
1

t
e()d +T
de(t)
)
p Ti 0
d dt
(A1)
where u is the control input, e is the difference between the reference signal yref (the set-point signal)
and process output y (see Fig. 1), Kp is the proportional gain of the controller, Ti is the integration
time or reset time, and Td is the derivative time. This version can also be described in terms of
controller gains as below

t de(t)

K =
Kp

u(t)=Kpe(t)+Ki  e()d +Kd → i Ti 0 dt K =K T d p d

(A2)
where Ki and Kd are integration gain and derivative gain, respectively. The PID-controller was
originally implemented using analog technology that went through several development stages, that
is, pneumatic valves, relays and motors, transistors and Op-Amps, and integrated circuits. In this
section, we will discuss the digital PID-controller in some detail.
Pure derivative cannot, and should not, be implemented, because it amplifies measurement noise
in the control loop. The gain of the derivative must be limited and therefore, the PID control law is
implemented in Laplace domain as follows,

de(t)
⎯La⎯plac⎯e T⎯rans⎯form→ sE(s)
s
E(s)=
Ns
E(s),
dt
1+
s N +s
N

(A3)
where N is the gain of derivative at high frequencies. Thus, the transfer function for the PID controller
is given as follows,

GCT (s)=
U(s)
=Kp(1+
1
+
Tds ),
E(s) Tis 1+
s
N

(A4)

Discretization of PID Controller
There are multiple methods for discretization of PID controller equation given in Eq. (A4), for instance
Zero-Order-Hold (ZOH) method you used in pre-Lab exercise (not an easy method for discretization
of the derivative part). The integral part can be discretised by using Forward Euler method, Backward
Euler method, or trapezoidal (Tustin or bipolar transformation) method. The derivative part can be
discretised by Forward Difference method or Backward Difference method. However, given the
special form of the approximate derivate part in Eq. (A3), we can use Forward Euler method for both
integrator and differentiator to discretize 1/s term since

s
=
N
.
1+
s
N
1
+1
N s

(A5)
In forward Euler method, the integrator part, I(t), is approximated as below,
7


I(t)=
1

t
e()d

I(kh)=I(kh−h)+
h
e(kh−h)
 Ti 0 ⎯Fo⎯rwa⎯rd E⎯ule⎯r, ⎯t =k→h
 Ti ,  1  h
I(s)=
T s
E(s) I(z) =
T (z −1)
E(z)
 i  i

(A6)
and, the derivative part, D(t), (or the term 1/s) in (A5) can also be discretised as follows,

D(s)=Td
N
E(s) ⎯Fo⎯rwa⎯rd E⎯ule⎯r, ⎯t=k→h D(z)=Td
N
E(z).
N
1
+1 N
h
+1
s z −1

(A7)
Therefore, The transfer function of the PID controller is given as the following,

GDT (z)=
U(z)
=Kp(1+
h
+Td
N
).
E(z) Ti (z −1) N
h
+1

z −1

(A8)



Tunning a PID Controller with the Ziegler-Nichols methods
A PID-controller has parameters Kp, Ti, Td, and N, that must be chosen. There is another parameter,
Tl, known as tracking-time constant or expected-time constant of the closed-loop system. The
primary parameters are Kp, Ti, and Td. Parameter N can often be given a fixed default value, for
example, N =100. Some special methods have, however, been developed to tune the PID
parameters experimentally. The behavior of the discrete-time PID-controller is very close to the
analog PID-controller if the sampling interval is short. The traditional tuning rules for continuous-
time controllers can thus be used. There are two classical heuristic rules developed by Ziegler and
Nichols (1942) that can be used to determine the controller parameters: the step- response method
and the ultimate-sensitivity method.
1. Step-response method
In this method, the unit step response of the open-loop process is determined experimentally.
The technique can be applied to processes whose step response is monotone or essentially
monotone apart from an initial non-minimum phase characteristic. To use the method the
tangent to the step response that has the steepest slope is drawn and the intersections of the
tangent with the axes are determined as shown in Fig. A1. The controller parameters are then
obtained from Table A1. The Ziegler-Nichols rule was designed to give good response to load
disturbances. It does, however, give low damping of the dominant poles.
8

t
Fig. A1. Unit step response of a system with parameters needed for Ziegler-Nichols step-response method.


Table A1: PID parameters obtained from the Ziegler-Nichols step-response method, a = RL.

Controller Type Kp Ti Td Tl
P 1/a 4L
PI 0.9/a 3L 5.7L
PID 1.2/a 2L 0.5L 3.4L
2. Ultimate-sensitivity method
In this method, the key idea is to determine the point where the Nyquist curve of the open-loop
system intersects the negative real axis. This is done by connecting the controller to the process
and setting the parameters so that pure proportional control is obtained. The gain of the
controller is then increased until the closed-loop system reaches the stability limit (self-
oscillation). The gain critical gain, Ku, when this occurs and the period of the oscillation, Tu,
are determined. These parameters are called ultimate gain and ultimate period. The controller
parameters are then given by Table A2.

Table A2: PID parameters obtained from the Ziegler-Nichols ultimate-sensitivity method.

Controller Type Kp Ti Td Tl
P 0.5Ku
Tu
PI 0.45Ku Tu /1.2
1.4Tu
PID 0.6Ku Tu /2 Tu /8 0.85Tu
y(t) Unit step response
Steepest
Slope R
9
References
[1] K. J. Astrom and B. Wittenmark, Adaptive Control, Chapter 8. 2nd ed., Upper Saddle River, NJ: Prentice Hall, 1995.
V1 by Dr. Hailong Huang, July 2018.
V0 by Dr. Arash KHATAMIANFAR, July 2017.