ABSTRACT
This thesis presents the dynamic modelling of a linear quadratic regulator based optimal direct current (DC) motor controller for improved performance. This study investigates the optimal use of a controller in the control of DC motor speed. This work also carried out a comparison of time response specification between conventional Proportional-Integral-Derivatives (PID) controller and Linear Quadratic Regulator (LQR) for a speed control of a separately excited DC motor. The goal is to determine which control strategy delivers better performance with respect to DC motor’s speed. The control method to be implemented in this work is the LQR which is a state space controller and the model of SEDM is presented in state space form. MATLAB/SIMULINK program was developed and used to solve the steady state and transient mathematical models of the machines. The test for both controllability and observability were carried out in the rank of matrix two. When the PID and LQR were compared, it was observed from the plots of speed against time at open loop with unity value of 1Volt and 1Ohm, the SEDM system with the LQR control stabilizes faster at the speed of 1rad/sec and after about 1seconds. This shows the superiority of the LQR to the PID that stabilizes at a speed of 41rad/sec after about 80seconds which is too long. The LQR responses were compared with PID, the LQR shows a rise time of 1.26seconds, settling time of 1.99seconds, overshoot of 0.525%, steady state of unity (1) and a peak amplitude of unity (1). While the PID also give a rise time of 0.0404seconds, settling time of 2.76seconds, overshoot of 84.8%, steady state of 1 and a peak amplitude of 1.85. The PID and LQR were also varied for different values of moment of inertia for J (J = 0.01), 0.5J and 2J. The variation were also carried out for different values of Viscosity of b (b = 0.00003), 0.5b, and 0.2b. The LQR rises and settles in 1secs with a unity steady state value of 1 in almost all the variations. These performance makes the LQR more superior to the other types of conventional controllers. Simulation results shows that the proposed controller (Linear Quadratic Regulator) gives better performance when compared with the PID controller. Performance of these controllers have been verified through simulation using MATLAB/SIMULINK software application package. According to the simulation results, the LQR method gives the better performance, such as settling time, steady state and overshoot compared to conventional PID controller.
TABLE OF CONTENTS
Title page i
Declaration ii
Certification iii
Dedication iv
Acknowledgements v
Table of Content vii
List of Symbols ix
List of Tables x
List of Figures xi
Abstract xii
CHAPTER 1: INTRODUCTION
1.1 Background of the Study 1
1.2 Problem Statement 5
1.3 Aim and Objectives of the Study 5
1.4 Significance of the study 6
1.5 Scope of the Study 6
CHAPTER 2: LITERATURE REVIEW
2.1 Theoretical Background of Direct Current Motor 8
2.2 Types of DC Motor 10
2.2.1 Separately excited DC motors 10
2.2.2 Series DC motors 11
2.2.3 Shunt DC motors 11
2.2.4 Compound DC motors 12
2.2.5 Stepper motors 12
2.3 Application of Dc Motors 13
2.4 Basic Operational Principles of Motors 13
2.5 Characteristics of DC Servomotors 14
2.6 The Concept of Conventional Controllers 16
2.6.1 Proportional (P) control action 16
2.6.2 Proportional plus integral (PI) control action 17
2.6.3 Proportional-integral-derivative (PID) control action 18
2.7 PID Tuning Methods 20
2.7.1 Manual tuning method 21
2.7.2 Zeigler-Nichols tuning specifications (closed-loop) 22
2.8 Introduction of Linear Quadratic Regulator (LQR) 22
2.9 Reviewed Papers 24
2.10 Research Gaps 25
CHAPTER 3: MATERIALS AND METHODS
3.1 Materials 36
3.2 Modelling and Control Design 36
3.2.1 System modelling 36
3.3 Open Loop Simulation 40
3.4 Control Design 42
3.4.1 Test for controllability and observability 43
3.4.2 Selection of Q and R matrices 45
CHAPTER 4: RESULTS AND DISCUSSION
4.1 Simulink Modelling 47
4.2 Simulink Model of Direct Current Motor 47
4.3 Linear Quadratic Regulator (LQR) Simulation Results 58
4.4 Summary of results 62
4.5 Simulation Result for PID and LQR for the Variation of Moment of Inertia 62
4.6 Simulation Result for PID and LQR for the Variation of Viscosity 65
CHAPTER 5: CONCLUSION, RECOMMENDATIONS AND
CONTRIBUTION TO KNOWLEDGE
5.1 Conclusion 67
5.2 Recommendations 68
5.3 Contribution to Knowledge 68
REFERENCES 70
APPENDICES 75
LIST OF SYMBOLS AND ABBREVIATIONS USED
A = n × n constant matrix
B = n × 1 constant matrix
Bm = viscous friction coefficient (kgm2/s)
C = 1 × n constant matrix
D = constant
ea(t) = applied voltage (V)
eb(t) = back emf (V)
ia(t) = armature current (A)
Jm = moment of inertia of rotor (kg.m2)
Kb = back emf constant (V/rad/s) (Kb =Ki)
Ki = torque constant (Nm/A)
La = armature inductance (H)
Ra = armature resistance (Ω)
tf = final time(sec)
TL(t) = load torque (Nm)
Tm (t) = motor torque (Nm)
u = control signal
x = state vector
y = output signal
θm (t) = rotor displacement (rad)
ωm (t) = rotor angular velocity (rad/s)
LIST OF TABLES
2.1: Effects of a step response in changing K parameters for a
Manually-tuned PID Controller 21
2.2: Zeigler-Nichols Tuning specifications (Copeland, 2008) 22
4.1: Parameters for SEDM 57
4.2: Parameters for LQR control simulation 58
4.3: Speed variation and characteristics for PID and LQR 61
4.4: Speed variation for LQR and PID for 0.5J 64
4.5: Speed variation for LQR and PID for J 64
4.6: Speed variation for LQR and PID for 2J 64
4.7: Speed variation for PID and LQR for 0.5b, b and 2b 66
LIST OF FIGURE
1.1: The diagram showing schematic of the DC motor 4
1.1: The diagram showing schematic of the DC motor 4
2.1: Equivalent circuit of the SEDM (Separately excited DC motor) 10
2.2: Equivalent circuit of a series DC motor (Beauvais, 2003) 11
2.3: Shunt DC Motors 11
2.4: Equivalent circuit of a compound DC motor 12
2.5: Basic Operation of Motors 14
2.6: Toque vs Speed Relationship Motor 15
2.7: Torgue vs Speed and Power vs Speed Characteristic of SEDM (Separately Excited DC Motor) 15
2.8: Block diagram of a proportional controller 17
2.9: Block diagram representation of a PID controller 19
3.1: Schematic Diagram of Separately Excited DC Motor 37
3.2: Block diagram of state space model 41
3.3: Open-loop step response of SEDM system 41
3.4: LQR control scheme for SEDM 46
4.1: Resultant Simulink model of SEDM 48
4.2: Armature current of the SEDM against time 49
4.3: Field current of the SEDM against time 50
4.4: Speed of a SEDM against time 51
4.5: Electromechanical Torque of the SEDM against time 52
4.6: Armature current of the SEDM against time 53
4.7: Field current of the SEDM against time 54
4.8: Mechanical rotor speed of the SEDM against time 55
4.9: Electromechanical Torque of the SEDM against time 56
4.10: Speed output with LQR and typical Q and R values 59
4.11: PID step response 60
4.12: PID and LQR step responses 61
4.13: Graph of PID step response for variation of moment of inertia 62
4.14: Graph of LQR step response for variation of moment of inertia 63
4.15: Graph of PID and LQR step response for variation of viscosity 65
CHAPTER 1
INTRODUCTION
1.1 BACKGROUND OF THE STUDY
It is a significant reality to know that the DC (direct current) machines have been seen as the most widespread transponders devices; their distinguished importance/advantage is that the voltage and current (Volt-Ampere) or the characteristics of the speed torque of these machines are so much flexible and adjustable and are subjected to operate in both steady state and dynamic operations. Electrical machines are highly essential in our modern society and our day-day lives (Slemon et al., 1982). The direct current machines are used to produce electrical energy whether it is required in the usable heat form (thermal application), mechanical power form (electrical motor in industrial application), illuminating form (lighting system) or conversion system.
The immediate current machines likewise perform or work as a motor in a direct current form and as a generator. In a few different ways, the exhibition and activity of a DC machine can change starting with one mode or structure then onto the next consequently and alternately or the other path round. It is most advantageous to take a gander at the different working of the DC generator in light of the fact that the enthusiasm of this generator is the terminal voltage with burden and speed variety, while in a DC engine, the distinctive highlights are the connection or association existing apart from everything else of power (torque) and (speed) and the modification or compliance to different types of mechanical burden (Sen et al., 1989).
Another very important or imperative feature of a DC machine is due to the potentiality or capability of developing an extensive number or collection of torque/speed characteristic, its economical speed control (Rizzoni et al., 1996), and forever-multiplying intricacy or enlargement of industrial processes that needs larger flexibility from electrical machines in terms of special characteristics (Ryff et al., 1987).
Direct current motor been long established are Fractional-Kilowatt dc motors that have higher efficiencies and are minute in frame and sizes than their AC competitors. However, the DC motors have some natural disadvantages too concerned with commutation and magnetic phenomena. Traditional or ordinary commutation in a small machine has approximate small sections, giving rise to emf and torque ripples. Brushes are responsible for volt-drop, friction wear and radio interference. Primarily, huge type of this motor can be used as control devices (tacho generators) for speed sensing and servomotor for positioning and tracking, used in machine tools. Printing press textile mills, pumps, hoist, and conveyors fans (Slemon et al, 1982).
During the 1930s, there were three mode controllers with P, I and D (PID) exercises ended up being monetarily open and expanded no matter how you look at it mechanical affirmation. These sorts of controllers are up till now one of the most comprehensively used controllers in strategy adventures. It wins on account of various incredible features of this count, for instance, ease, quality and wide relevance.
Various diverse tuning strategies which have been proposed from 1942 up to now for expanding better and dynamically sufficient control structure in like manner response subject to our appealing control goals, for instance, percent of overshoot, integral of absolute value of the error (IAE), settling time, manipulated variable behaviour etc. Some methods of tunings in this controllers. Many of these tuning systems have contemplated, only one of these objectives as a standard for their tuning and some of them have developed their estimation by pondering more than one of the referenced model. In this write-up, we have broken down the displays of a couple of tuning procedures. For multiplication study first, second and third order structures with dead time have been used and it was normal that the components of system is known.
Real study has been performed for some set core interests. Due to the excellent speed control characteristics of a DC motor, it has been commonly used in industry (such as cars, trucks and aircraft) even though its maintenance costs are higher than the induction motor. As a result, authors have driven their attentions to a method of control called position control of direct current motor and prepared several methods to control speed of such motors. Due to their excellent control characteristics, DC motors have been widely used in industrial applications. Generally, the DC motors have two windings. These are the armature winding and the field winding. When the armature is supplied through brushes and a commutator that switches the direction of the current in the armature winding, this armature is also responsible for producing a unidirectional torque.
To produce different characteristics, the two motor windings are usually connected indifferent configurations, such as, series, shunt or separately excited. In terms of the control theory, series or shunt connection of the two motor windings produces a system with single degree of freedom, as in this case the system will have only a single input, which is the motor supply voltage. Motors with separated armature and field windings have two degrees of freedom as they have two control inputs, which are the motor armature voltage Va and the motor field voltage Vf. These two voltages are usually supplied to the motor by two controlled DC voltage sources. Figure 1.1 shows the diagram of the direct current motor.
Figure 1.1: The diagram showing schematic of the DC motor.
The interaction of the motor armature current and the magnetic field are responsible for the motor current production. The current in the motor field winding, produces the motor torque. When the armature of the DC motor rotates, an emf is formed in the armature winding when the armature of the motor rotates. This voltage is called the back emf voltage, and it is directly alike to that of the motor angular speed. As a multi-input, multi-output (MIMO) system, different approaches is long been proposed in designing the controller of the DC motor. In this thesis, two approaches are used. These are the PID and LQR controller approach (Whalley et al., 2006). Other approaches were also proposed, the Inverse Nyquist Array approach by (Rosenbrock et al., 1936) and the optimal control approach (Kalman R.E., 1960), least effort approach is based on minimizing a performance index. The controller, in this approach, has two loops; the inner loop ensures stable dynamics of the closed loop system, while the outer loop provides some dynamics that help to provide a specified steady stateng (Whalley et al., 2006). In the inverse Nyquist array approach, developed by (Rosenbrock et al., 1936) in1969, and in order to decrease the system output interaction, a diagonally dominant closed loop transfer function has to be found. This will reduce the design process to the design of a set of independent single loops, in which single input single output (SISO) control methods can be used. Gershgorin’s band theorem will be used to test the diagonal dominance of the closed loop transfer function matrix. In the optimal control approach, state feedback is used. The design problem is to find a state feedback matrix that will minimize a quadratic performance index, whilst simultaneously providing acceptable performance conditions.
1.2 PROBLEM STATEMENT
It is significant to control the speed of the DC motor. The DC motor speed control is an important area to be considered in every processes. Due to disturbance present in the surrounding of the DC motor, the speed may change. This changes will sometimes cause the speed not to be maintained as desired. The controller called PID (proportional integral derivatives) is applied widely to control the industrial processes. The PID controllers are desired in the industrial process because of their principle and simple structure. These controllers provide a very good behaviour for controlling various systems. However, the proportional integral derivatives in several cases such as disturbances or parameter variations is not appropriate. For these challenges and problems that are faced by PID controller to be solved, the LQR (linear quadratic regulator) type of control methods can be developed or adopted.
1.3 AIM AND OBJECTIVES OF THE STUDY
This main aim of this work is to study the dynamic modelling of a linear quadratic regulator based optimal direct current motor for improved performance
The specific objectives of this work are:
i) To study and understand LQR technique.
ii) To investigate the current developments in DC motor controllers.
iii) To develop a mathematical model for the DC Motor.
iv) To validate theoretical findings by the use of a computer simulations package.
v) To compare the motor speed performance using PID and LQR
1.4 SIGNIFICANCE OF THE STUDY
The successfully dynamic modelling of the direct current motor will find significance in the following areas:
i) Direct current motor control is very important in the industries for the regulation of production machines, they are used in the operation of sewing machines, the generation of solar uses direct current principles for energy generation.
ii) Direct current motors are also useful in the following areas:
a) For paper machines, diesel-electric propulsion of ships in steel rolling mills.
b) For drives requiring very high starting torque and where adjustable varying speed is satisfactory.
c) For hoists, cranes, trolley cars, conveyors, electric locomotives
d) For shears, conveyors, crushers, bending rolls, ice making machines, air compressor.
e) For experimental and research work.
1.5 SCOPE OF THE STUDY
This work is the dynamic modelling of a LQR based optimal direct current motor controller for improved performance. This research is to carry out control techniques that can be applied in controlling the speed of a direct current with the application of a controller (LQR). Since the performance of a machine is a vital factor for a big production line. This research will therefore, examine the efficiency and performance of the DC motor with the application of a controller and the entire LQR design controller will be done using a software application package called MATLAB/SIMULINK. Please, note here that this entire work and it’s results is based on SIMULATION using MATLAB/SIMULINK as no practical hardware implementation of any form will be done or carried out.
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