1. Module 5570 Control Systems and Design
ELEC 5570 Control Systems and Design
Benjamin Chong – Room G62
School of EEE, University of Leeds

2. Module 5570 Control Systems and Design
Recommended Text
Modern Control Systems (7th Edition),
Dorf and Bishop, Prentice Hall, 2008
Relevant Chapters
Chapter 2: Mathematical models of systems
Chapter 5: Performance of feedback control systems
Chapter 7: Root locus method
Chapter 10: The design of feedback control systems
Chapter 13; Digital control systems
School of EEE, University of Leeds

3. Chapter 1 – System Modelling
Module 5570 Control Systems and Design
Chapter 1 – System Modelling
A simple position control system
Power Amp. K R2 R1 Ra La Va
Pot.
Kt + R
r(t)
(t)
Before we start discussing the design of controllers, let us have a look of a simple control system. The figure above shows a position control system. The control system consists of two major elements:- the plant (in this case the motor and its power supply) and the controller (the op-amp and the resistors). The dc motor is driven by a power amplifier, and a sensor is used to measure the angle of the motor shaft. An analogue control circuit is used to control the motor shaft to follow the input demand r(t). The task of the control design is to select the values of the two resistors (R1 and R2) such that the difference between the input demand and motor output angle is kept small. This example is perhaps the simplest form of control. Many practical controllers normally have more complex structures than the proportional control in order to meet stability and performance specifications, and therefore the control design will have to deal with a number of design issues.
So what issues are involved ?
Plant
Controller
Controller output
= Plant input
School of EEE, University of Leeds

4. Chapter 1 – System Modelling
Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Development of a Control System
Representation of plant
Requirements/specifications
Design techniques
Verification/validation
Implementation and testing
Typically, control system design involves the following steps. First, a representation of the plant must be found by using modelling and/or testing methods. The next task is normally to specify the basic requirements for the control system, i.e. what is expected from the system. Then, appropriate design techniques are chosen and controller is designed. The controller is then verified so ensure that it meets those requirements before it is implemented and tested.
This module is mainly focussed on the study of controller design, and we shall be discussing “classical control” – designing controllers in frequency domain (s-plane for continuous-time systems and z-plane for discrete-time systems). Therefore, it is important to revise Laplace transforms and system transfer functions (which you should have studied before).
School of EEE, University of Leeds

5. Chapter 1 – System Modelling
Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Plant (The system to be controlled)
Electrical, Mechanical, Aeronautical, Chemical (or combination)
Mathematical Modelling (Differential equations, transfer functions)
Model simplification (Identification of parts which can be neglected)
Different plants will pose different problems for the design, and it is therefore essential to understand the characteristics of a plant and the difficulties and constraints the plant poses in order to design a satisfactory controller. In almost all cases, the plant must be represented using mathematical equations (i.e. the model), and the process of deriving the equations is known as modelling. For some systems (e.g. high order and/or non-linear systems), the full mathematical model might be too complex to be used for control design. Normally, a model reduction and/or linearisation process is carried out to derive a simpler model that represents key characteristics of the plant, whilst neglecting less significant elements.
Because control is used in a wide range of applications, the modelling of the plants can often only be dealt with by people working in the relevant areas. This chapter will only give a few examples to show mathematics can be used to derive some simple models.
School of EEE, University of Leeds

6. Chapter 1 – System Modelling
Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Example:- A two-mass mechanical system mb mw Ks Cs Kp
Inertial reference
F:- vector sum of all forces (in all directions)
a:- vector acceleration of each body
m:- mass of the body
The figure above shows a two-mass system, which can be used as the model of a quarter of a car (i.e. a quarter of the car body m2, one wheel m1 and the suspension components Ks, Cs and Kp).
The mathematical model of the mechanical system, i.e. the equations of motion, can be derived by applying Newton’s law F = ma, where ‘F’ is the vector sum of all forces applied to each body in a system, ‘a’ is the vector acceleration of each body with respect to an inertial reference frame, and m is the mass of the body.
School of EEE, University of Leeds

7. Chapter 1 – System Modelling
Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Example:- A two-mass mechanical system
Forces on the wheel and suspension mb mw Ks Cs Kp xb xw xg
Inertial reference Fs Fp
Equations of motion
Forces caused by the suspension can be expressed as Fp (in the tyre) and Fs (between the body and the wheel), and the equations of motios for the body and the wheel are obtained using Newton’s law.
School of EEE, University of Leeds

8. Chapter 1 – System Modelling
Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Example:- A two-mass mechanical system
Equations of Motion
In the s-domain – take Laplace transforms
The equations of motion can be expressed in both the time-domain and frequency domain. From the frequency domain representation, transfer functions can be readily derived. If the input is xg and output is xb, the transfer function (in s-plane) H(s) = Xb(s)/Xg(s) is given above.
Once the transfer function is obtained, the properties of the system can be examined using MATLAB (a program is given below).
% Define the system parameters
mb = 250; mw = 50; Ks = 10000; Cs = 125; Kp = ;
% Define the numerator and denominator coefficients of the transfer function
num = [Kp*Cs, Kp*Ks];
den = [mw*mb, Cs*(mb+mw), (mw*Ks+mb*Ks+mb*Kp), Kp*Cs, Kp*Ks];
sys = tf(num,den); % sys is the transfer function –a MATLAB “object”
% Look at the frequency response using Bode plots (angular frequency range -
% 100 points between 10-1 and 103 rads/s)
w0 = logspace(-1, 3, 100); figure(1); bode(sys,w0); grid;
% Look at a step response from 0 to 10 seconds
[y,t] = step(sys, 10); figure(2); plot(t,y); grid;
Transfer function from xg to xb
School of EEE, University of Leeds

9. Chapter 1 – System Modelling
Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Example:- An electric circuit
Kirchhoff’s laws (voltage and current)
Relations between voltage and current for R, L, C components L R C va
The figure above shows an R-L-C circuit. The input is the voltage supply and the output is either voltage across or current through the resistive load.
Deriving the mathematical model of the electrical system replies on the Kirchhoff’s current and voltage laws, and the relations between the voltage and current of R, L, C components.
Reminder:-
Kirchhoff’s current law:- the algebraic sum of currents leaving a junction or node equals the algebraic sum of currents entering that node.
Kirchhoff’s voltage law:- the algebraic sum of all voltages taken around a closed path in a circuit is zero.
School of EEE, University of Leeds

10. Chapter 1 – System Modelling
Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Example:- An electric circuit L R C va
Applying the Kirchhoff’s current and voltage laws gives:
1) iL = iR + iC
2) vL + vR –va = 0
For the inductor, we have
vL = L d(iL)/dt
Using the V-I relation for the capacitor:
–vR + va = L d(iR + iC)/dt = L d(iR)/dt + L d[C d(vC)/dt]/dt
As vC = vR, we have
–vR + va = L d(iR)/dt + LC d2(vR)/dt
School of EEE, University of Leeds

11. Chapter 1 – System Modelling
Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Example:- An electric circuit L R C va
Hence:
In the s-domain:
As vR = R iR, we have
LC d2(vR)/dt2 + L/R d(vR)/dt + vR = va or
LCR d2(iR)/dt2 + L d(iR)/dt + R iR = va
The equations can be derived in the frequency domain directly. Try this.
Try to obtain the transfer functions Va(s)/VR(s) and Va(s)/IR(s)
School of EEE, University of Leeds

12. Chapter 1 – System Modelling
Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Example:- A position control system
Power Amp. K R2 R1 Ra La Va
Pot.
kt + R
(t) ia
r(t)
The next example is the position control system that was looked at previously, and this example will be used extensively throughout the course.
The power amplifier can be considered to be a constant gain. The dc motor may be divided into electrical and mechanical parts, and their mathematical equations are given above, where
va -- armature voltage
ia -- armature current
La -- armature inductance
Ra -- armature resistance
K -- voltage gain of power amplifier.
kt -- back emf constant = torque constant
 -- the motor speed
J -- moment of inertia of the motor armature and its load
B – viscous friction constant
Electrical part
Mechanical part
School of EEE, University of Leeds

13. Chapter 1 – System Modelling
Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Example:- A position control system
In the s-domain:-
The equations can be expressed in the s-domain, which results in a transfer function between the angular the input armature voltage and (output) velocity of the motor.
The approximations neglects the armature inductance, La. This is an example of model simplification.
School of EEE, University of Leeds

14. Chapter 1 – System Modelling
Module 5570 Control Systems and Design
Chapter 1 – System Modelling
The transfer function between angular armature voltage
and output velocity is therefore given by
If we now include the voltage gain of the amplifier,
the transfer function of the forward path (in green) is
G(s) is therefore the “model” of the “plant”.
School of EEE, University of Leeds

15. Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Transfer Function – Input/Output Relationship
Derived by simultaneously manipulating
multiple differential equations
- This approach is straightforward for systems of 2 order or less
Another approach which employs state-space
modelling technique can be used.
A set of (differential) equations describing a system can be either solved simultaneously or combined into a single, larger order differential equations by taking derivatives and substituting one into the other. The standard methods of solving differential equations are then applied.
However, such a process can be tedious, and the methods employed vary in complexity as the order of the differential equation increases. Instead, state equations are used to describe the dynamics of a physical system.
- This approach can also be applied to higher order systems
- Can be used when there are multiple input/outputs
School of EEE, University of Leeds

16. Module 5570 Control Systems and Design
Chapter 1 – System Modelling
State-space modelling technique
-a method for describing a system in terms of a set of first order linear differential equations.
In the general form they can be expressed as
State-space modelling technique enables a system to be described in a unified approach.
It is simply a collection of first-order differential equations that together represent exactly the same as the original larger differential equation.
The collection of state variables at any given time is known as the state of the system, and the set of all values that can be taken on by the state is known as the state space.
In normal case, there is one or more input(s) and one or more output(s). Therefore, state-space modelling technique can be used not just for single-input-single-output system but also for single-input-multiple-output, multiple-input-single-output, multiple-input-multiple-output systems.
School of EEE, University of Leeds

17. Module 5570 Control Systems and Design
Chapter 1 – System Modelling
State-space modelling technique
where
is an n  1 vector of state variables
is a p  1 vector
of inputs
is an m  1 vector
of outputs
represents the set of state variables
represents the input set
represents the output set
A is state matrix, B is the input matrix, C is the output matrix, and D is the feedthrough matrix
A is an n  n matrix, B is an n  p matrix, C is an m  n matrix, D is an m  p matrix.
School of EEE, University of Leeds

18. Module 5570 Control Systems and Design
Chapter 1 – System Modelling
State-space modelling technique
(single-input, single-output (SISO) system)
We have
p = m =1
We can write B = b (nx1 column vector), C = c which is a 1  n row vector and d is scalar, i.e.
In the previous slide, the state space equation is in general form and it is for multiple-input-multiple-output (MIMO) .
If the input and/or output is one dimensional (i.e. single), the associated matrix is written in lower case; b, c or d. This is to indicate that they are simple columns (b), rows (c) or scalars (d), respectively.
School of EEE, University of Leeds

19. Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Relation between State Space and Transfer Function Matrix Descriptions In
take Laplace transforms with X(0) = 0
It may be reasonably asked at this point how the state space equation which is in time domain, relates to the transfer function representation of systems.
The most fundamental to answer to this question is that transfer functions are defined for systems with zero initial conditions only. State variable descriptions are therefore more appropriate when transients due to non-zero initial conditions are important.
Also a transfer functions is mainly dealing with SISO.
In this module, we will analyse how we can derive the system transfer function from a set of state space equations.
School of EEE, University of Leeds

20. Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Hence
and
Hence
Hence, we have a description between input and output (in s-domain) expressed as
School of EEE, University of Leeds

21. Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Application of state-space modelling technique for
transfer function derivation
State variable
Input/output
Note: Input is independent
School of EEE, University of Leeds

22. Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Application of state-space modelling technique for
transfer function derivation
School of EEE, University of Leeds

23. Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Application of state-space modelling technique for
transfer function derivation
This
School of EEE, University of Leeds

24. Module 5570 Control Systems and Design
Chapter 1 – System Modelling
Application of state-space modelling technique for
transfer function derivation
School of EEE, University of Leeds