Project Report

CHAPTER 1

INTRODUCTION

1

1.1 Motivation

Modern surface mount machines, used in the placement of electronic circuit

components onto circuit boards, use servo motors to rapidly position the machine

axes. AC induction motors are commonly utilized in the surface mount machine

due to their properties. AC induction motors have several characteristics superior

to DC motor. Such as a maintenance free structure, relatively lower cost than

equivalent size DC motors, and greater power output ranging from a fraction of a

horsepower to 10,000hp. AC induction motor, indeed, are the workhorses of

today's industry.

Currently, the Proportional, Integral, and Derivative (PID) controller structure is the

most common in use in industry, mainly due to the fact that it is relatively easy to

design and implement. However, despite of its widespread use, the PID controller

does have a number of limitations. One of the main drawbacks of PID controller is

the task of tuning gains to achieve a set of desired dosed-loop performance

specifications. Since performance specifications generally conflict with each other,

the task of tuning gains to meet several closed-loop performance specifications

simultaneously requires considerable time and experience. For the case of low

performance specifications, tuning PID gains is not difficult. However, as

performance specifications become more stringent, i.e. higher performance, the

task of tuning gains becomes increasingly difficult due to the fact that multiple

simultaneous specifications typically conflict with each other. Therefore, there

exists a need for a reliable control design rnethod to systematically design closed

loop controls to meet all specifications simultaneously. This, in fact, is the main

goal of this thesis.

In order to overcome some problems that faced by PID controller, the other type of

control methods can be developed such as Linear-Quadratic Regulator (LQR)

optimal control. LQR is a control scheme that gives the best possible performance

2

with respect to some given measure of performance. The performance measure is

a quadratic function composed of state vector and control input.

1.2 Background of the Project:

Control engineering is one subject which is perceived as being the most theoretical

and most difficult to understand. In industries, application of motor control system

is important to operate some processes. An average home in some developed

countries uses a dozen or more electric motors. In some application the Induction

motor is required to maintain its desired speed when load is applied or

disturbances occur. This kind of system can be controlled using PID, Fuzzy, LQR

and other more.

In this project, Linear Quadratic Regulator (LQR) controller is introduced in order to

control the INDUCTION MOTOR performance as we required. MATLAB/

SIMULINK is used to design and tune the LQR controller and be simulated to

mathematical model of the Induction Motor. From the simulation the LQR controller

in MATLAB/SIMULINK can be interfaced with the actual Induction Motor using

appropriate data acquisition card.

The Linear Quadratic Regulator (LQR) controller is a new method of controlling the

motor. Linear Quadratic Regulator (LQR) is theory of optimal control concerned

with operating a dynamic system at minimum cost.

3

1.3 Proposed research

The nature of this research is to develop a control method for the optimization of

performance of an AC Induction motor. The control method used in this project is a

robust controller which is well known as ‘’Linear Quadratic Regulator’’ or just LQR.

The dynamic model of AC induction motor is developed in this thesis on which this

control method is to be applied to develop a controller of our concern.

1.4 Objective of the Project

The main objective of this project is to build control system for controlling the

speed of AC induction motor. The design procedure is as follows:

To Design and produce the simulation of the LQR controller.

To Implement LQR controller onto the dynamic mathematical model AC induction motor.

To implement another type of controller which is a well known PID controller onto the same dynamic model of AC induction motor.

To compare the simulation results of LQR with the results of PID in order to show that the proposed LQR controller model is better than the other controller model.

4

Chapter 2

Background Theory and

Literature Review

5

2.1 Structure of the AC Induction Motor

2.1.1 Internal Structure of the AC Induction Motor

A motor basically consists of two parts. i.e. the stator and the rotor. The internal

structure of AC induction motor is depicted in Figure 2.1. The stator is a stationary

part that sets the magnetic field. The stator is a permanent magnet in brush type

DC motors or a set of wire windings in AC induction motors. The rotor is a rotating

part holding a shaft that couples mechanical load. A typical rotor of the squirrel-

cage type found in AC induction motor has aluminium bars connected to the rings

that short the ends together.

Figure 2.1: Internal Structure of Squirrel-cage Type AC Induction Motor

2.1.2 Rotating Magnetic Field

Since the power source of the AC induction motor is three-phase alternating

current (AC), the stator has three pairs of windings. These three pairs of windings

create a set of magnetic poles (Figure 2.2). Each phase of current establish

rotating field in the stator. In the squirrel cage rotor, the current is induced due to

6

the rotating field. Since the ends of the bars are shortened, the induced current

creates a new magnetic field in the rotor and is attracted by the revolving field

produced by stator currents. Consequently, as the magnetic field rotates, the rotor

rotates.

2.2 Dynamic Modeling of AC Induction Motor

In AC machines, all signals exhibit a sinusoidal wave form. In other words, in

three-phase AC machines, the space vector, such as the flux linkage vector, the

voltage vector and the current vector are sinusoidal wave forms. Unfortunately,

alternating properties are not convenient for control analysis purposes. This

problem can be solved by introducing rotating transformed d-q coordinates (D-Q

coordinates) with arbitrary speed. This results in signals which are time-varying DC

signals, and hence are easier to analyze and manipulate in control schemes than

AC signals. Based on this coordinates transformation, the flux vector control

method is developed and has been applied to AC induction motors

(Trzynadlowski, 1994).

In this section, a coordinate transformation will be discussed first. Dynamic

modeling of the AC induction motor will then be addressed.

7

Figure 2.2: Current Phasors and Space Vectors at ωt = 0 ° and a = 60°

2.2.1 d-q Coordinate Transformation

In AC machines, the space vectors can be described as follows:

Figure 2.3: Space Vector Components in the d-q Axes

8

Q axis circuit

D axis circuit

Fig. 2.4 D-Q Equivalent Circuit on a Synchronous Frame

F ss =Fase

jws+¿ Fbs ej (wt− 2

3 п) + F cs e

j (wt− 43 п

)…….. (2.1)

Where :

F = vector representing space vector superscript s represents stationary reference frame

9

subscript s represents stator

Since the three components of the space vector, F ss, constitute one vector, it can

be projected in the complex plane which has a real component and an imaginary

component (Figure 2.3). Therefore if we define the real axis as the d-axis and the

imaginary axis as the q-axis then the transformation of "abc" phases into "dq" axis

is possible.

For instance, three-phase currents that have magnitudeia,ib and ic of phases a, b,

and c, respectively, can be expressed as follows:

I ss= I a e jws + I b e

j (wt− 23 п

)+ I b ej(wt− 4

3 п)=I as

+ I bs + I cs………….(2.2)

Figure-2.5. d-q - axis superimposed into a three-phase induction motor.

10

The transformation abc to d-q is performed as:

In these equations, the axis reference frame has been transformed. Therefore, the

stator axis as well as the rotor axis is transformed to a reference axis rotating with

an arbitrary angular velocity. In this way, the sinusoidal coupling between the rotor

and the stator circuit with the rotor position can be eliminated, if the stator and the

rotor both refer to the same reference frame. Here, we assume that the referred

frame of reference is rotating with the synchronous speed.

Figure 2.6: Currents Transformation

11

2.2.2 Electromechanical Dynamic Modeling of AC Induction Motor

AC induction motors are described by nonlinear dynamic equations, which can be

expressed as a set of differential equations representing both the electrical system

dynamics and the mechanical system dynamics. The dynamic behaviour of a

balanced three phase induction motor in the electromechanical model consists of

five ordinary differential equations given below, expressed in stator fixed d-q

coordinates (Atkinson et al., 1991).

A two phase d-q model of an Induction machine rotating at the synchronous speed

is introduced which will help to carry out the decoupled control concept to the

induction machine. This model can be summarized by the following equations

...............................(2.3)

..........................(2.4)

The stator and rotor fluxes are given by the following relations:

.......................................(2.5)

.......................................(2.6)

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In equations 1 to 4, the voltages, currents and fluxes space vectors are function of

the corresponding three-phase variables. As an example, the stator current space

vector is linked to the corresponding three phase currents by the following relation:

...................(2.7)

Where a =e j 2 п /3. The produced electromagnetic torque is given by

....................................(2.8)

Figure 2.7: Reference frames and space vector representation

Using the d-q coordinate system, as illustrated in Figure 2.7, and separating the

machine variables state vectors into their real and imaginary parts, the well-known

Induction motor model expressed in terms of the state variables is obtained from

equations 2.3 to 2.8 and is given by:

13

..........(2.9)

In equation (2.9 coefficient of dispersion Ϭ is given by:

......................................(2.10)

As shown in Figure 2.7, the d-axis is aligned with the rotor flux space vector.

Under this condition we have; ψqr = 0 and ψdr = ψr. Consequently, the induction

motor model established in the rotor flux field coordinate is then given by the

equations 2.11 to 2.14.

14

…...(2.11)

.....................................(2.12)

..................................(2.13)

...................................(2.14)

In ordinary use, only stator voltages, currents and rotor speed are available for

measurement. In this case, the d-q stator voltages and currents are obtained from

the corresponding α−β stationary reference frame variables through an

appropriate transformation involving rotor flux space vector angle θe, as shown in

Figure 2.7. This transformation is given by:

........................(2.15)

15

In equation 2.15, "x" is a voltage, a current or a flux. As mentioned before, θe is the

rotor flux space vector angle. In direct vector control, the rotor flux is available for

measurement or is estimated from measured stator voltages and currents. The

rotor flux angle is then given by:

..................................(2.16)

The rotor flux amplitude is obtained by solving equation 2.13, and its spatial

position is given by:

............................(2.17)

The Indirect vector control strategy can now satisfactorily be achieved since both

amplitude of rotor flux vector and its spatial position are known. As in DC

machines, the torque and the flux are controlled independently: The

electromagnetic torque Te is controlled by Iqs (torque producing current), and the

flux is controlled by Ids (flux producing current).

2.3 Literature Review of AC Induction Motor Control

In the AC induction motor control area, many control methods have already been

developed. Induction motor scalar control with variable frequency has been

investigated by some authors (Mixon, 1984). However, it has been shown that the

steady-state torque under scalar control can be controlled accurately, but the

transient torque response is unsatisfactory. To obtain high performance of AC

16

induction motors, fast torque response is required. Thus, for fast response of

torque, the Flux Vector Control Method, originally introduced by Blaschke (1971),

has been employed by many authors to control AC induction motors (Ho and Sen.

1988). It has been shown that the flux vector controlled AC induction motors have

better performance than scalar controlled motors by comparing the behaviour of

an induction motor driven by flux vector control with a motor driven by scalar

control (Finch et al., 1998). Recently, Direct Torque Control Method is developed

by Tiitinen et al. (1994) for better torque response and it is an ongoing research

area.

For position control of AC induction motors, many authors have conducted

experiments of AC induction motor position control with flux vector control theory

(Vukosavic and Stojic, 1993). Some of authors have employed a fuzzy sliding

control scheme for AC induction motor position control (Chen and Hsu, 1994).

Generally, Proportional Integral Derivative (PID) controller is widely used in

industry for position control. However, PID controller has a number of

disadvantages. The main disadvantage of PID-type control is its gain-tuning

nature. To tune the gains properly to achieve better performance requires

extensive time and experience. In fact, it is tedious and time consuming to tune the

PID gains so that all desired performance specifications are met simultaneously

(Liu, 1998 Control theory has been a classical conceptualisation of feedback and

control of the physical system, from an engineering point of view. This theory

needs a rigorous definition of many mathematically complex tools. However,

different classical and robust control algorithms have been proposed for the

optimisation control law computing.

The most common ones are LQR, LQG, H2, H ∝ control and eigen structure

assignment [Chiappa (1998), Kubika and Livet (1994), Kucera (1986), Kucera

(1992), Sobel and Shapiro (1985), Tsui (2001)]. But recently, it has been verified

that the approaches issued from the soft computing such as Fuzzy Logic, Genetic

Algorithms, Neural Network and Ant System are simple, practical, adaptable and

computationally efficient to solve several practical optimisation problems including

17

hard industrial objectives. Moreover, the infinite horizon linear quadratic regulator

(LQR) and the eigen structure assignment (EA) are among the most popular

controller design techniques for MIMO (Multi Input Multi Output) systems. The

advantage of EA is that if the specifications are given in terms of system eigen

structure, the eigen structure can be achieved exactly for the desired stability and

dynamic performance. However, the EA suffers from some limitations as that the

system performance may not be optimised in some practical cases, such as

minimum control effort, and that the system requirements are often not easily

specified in terms of eigen values/vectors. Furthermore, the LQR could be used to

optimise the controller design by minimizing a quadratic cost function of system

response and control energy.

For deterministic system, the LQR-based control design generally guarantees the

closed loop stability and certain degrees of robustness, but may not easily achieve

specific system performances due to the difficulty in the selection of the synthesis

matrices Q and R. Hence, via the EA techniques, the choice of the LQR design

matrices Q and R presents a nice problem to be studied by applying the efficient

heuristics and methods inspired from Nature’s Laws, especially the ant system

optimisation metaheuristic. Initially, submitted to application by Dorigo et al

[Colorni et al. (1992), Davis and Clark (1995), Stützle and Dorigo (2002)], the ant

system optimisation presents a class of general algorithms of optimisation. The

main underlying idea, essentially inspired from the behaviour of real ants,

represents a parallel search of several constructive computational solutions. These

latter are based on the problem characteristic data and on a dynamic structure

memory containing information on the quality of previous solutions. Moreover, the

ant system metaheuristic has been successfully applied to a variety of

combinatorial optimization problems such as the travelling salesman problem and

different variants of the scheduling problem. Convergence proofs for the Ant

Colony Optimization algorithms can be found in [Gutjahr (2003), Stützle and

Dorigo (2002)].

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CHAPTER 3

LINEAR QUADRATIC REGULATOR

19

Linear Quadratic Regulator

3.1 Introduction

The theory of optimal control is concerned with operating a dynamic system at

minimum cost. The case where the system dynamics are described by a set of

linear differential equations and the cost is described by a quadratic functional is

called the LQ problem. One of the main results in the theory is that the solution is

provided by the linear-quadratic regulator (LQR), a feedback controller whose

equations are given below. The LQR is an important part of the solution to the

LQG problem. Like the LQR problem itself, the LQG problem is one of the most

fundamental problems in control theory.

In layman's terms this means that the settings of a (regulating) controller governing

either a machine or process (like an airplane or chemical reactor) are found by

using a mathematical algorithm that minimizes a cost function with weighting

factors supplied by a human (engineer). The "cost" (function) is often defined as a

sum of the deviations of key measurements from their desired values. In effect this

algorithm therefore finds those controller settings that minimize the undesired

deviations, like deviations from desired altitude or process temperature. Often the

magnitude of the control action itself is included in this sum so as to keep the

energy expended by the control action itself limited.

In effect, the LQR algorithm takes care of the tedious work done by the control

systems engineer in optimizing the controller. However, the engineer still needs to

specify the weighting factors and compare the results with the specified design

goals. Often this means that controller synthesis will still be an iterative process

where the engineer judges the produced "optimal" controllers through simulation

and then adjusts the weighting factors to get a controller more in line with the

specified design goals.

The LQR algorithm is, at its core, just an automated way of finding an appropriate

state-feedback controller. And as such it is not uncommon to find that control

20

engineers prefer alternative methods like full state feedback (also known as pole

placement) to find a controller over the use of the LQR algorithm. With these the

engineer has a much clearer linkage between adjusted parameters and the

resulting changes in controller behaviour. Difficulty in finding the right weighting

factors limits the application of the LQR based controller synthesis.

This chapter will explain the literature study that is related to the project task. The

information gets from several sources such as websites, journals, books,

magazines, handout and others.

3.2 Working Principle of Linear Quadratic Regulator (LQR):

The linear quadratic regulator (LQR) is a well-known design technique that

provides practical feedback gains. For the derivation of the linear quadratic

regulator, assume that the plant to be written in state-space form as:

.........................................(3.1)

And that all of the n states x are available for the controller. The feedback gain is a

matrix K of the optimal control vector

....................................................(3.2)

so as to minimize the performance index

.......................................(3.3)

21

Where Q is a positive-definite (or positive-semi definite) Hermitian or real

symmetric matrix and R is a positive-definite Hermitian or real symmetric matrix.

Note that the second term on the right-hand side of the Equation 3 accounts for

the expenditure of the energy of the control signals. The matrices Q and R

determine the relative importance of the error and the expenditure of this energy.

In this problem, assume that the control vector u(t) is unconstrained.

As will be seen later, the linear control law given by equation (2) is the optimal

Therefore, if the unknown elements of the matrix K are determined so as to

minimize performance index, then is optimal for any initial state

x(0). The block diagram showing the optimal configuration is shown in Figure

below :

Figure 3.1: Optimal Regulator System

Now let solve the optimization problem. Substituting Equation 3.2 into Equation 3.1

............................(3.4)

In the following derivations, assume that the matrix A-BK is stable, or that the

eigen values of A-BK have negative real parts.

22

Substituting Equation 2 into Equation 3 yields:

..................................(3.5)

Let set,

..................................(3.6)

Where P is a positive definite Hermitian or real symmetric matrix. Then obtain

..(3.7)

Comparing both sides of this last equation and noting that this equation must hold

true for any x, it require that

...................... (3.8)

It can be proved that if A-BK is a stable matrix, there exists a positive-definite

matrix P that satisfies Equation 8. Hence the next procedure is to determine the

elements of P from Equation 8 and see if it is positive definite. (Note that more

than one matrix P may satisfy this equation. If the system is stable, there always

exists one-positive matrix P to satisfy this equation. This means that, if to solve this

equation and find one positive-definite matrix P, the system is stable. Other P

matrices that satisfy this equation are not positive definite and must be discarded.

23

The performance index J can be evaluated as

.........................(3.9)

Since all eigen values of A-BK are assumed to have negative real parts, it have

x(∞) 0. Therefore, J can be obtain

..............................................(3.10)

Thus, the performance index J can be obtained in terms of the initial condition x(0)

and P. To obtain the solution to the quadratic optimal control problem, proceed as

follows: Since R has been assumed to be a positive-definite Hermitian or real

symmetric matrix, it can be written

R=T.T ........................................................(3.11)

Where T is a non-singular matrix. Then Equation 3.8 can be written as

.................... (3.12)

This can be rewritten as:

..

(3.13)

The minimization of J with respect to K requires the minimization of

...................(3.14)

24

with respect to K. Since this last expression is non negative, the minimum occurs

when it is zero, or when

...........................................(3.15)

Hence

.................................(3.16)

Equation 3.16 gives optimal matrix K. Thus, the optimal control law to the

quadratic optimal control problem when the performance index is given by

Equation 3.3 is linear and is given by

................................. (3.17)

5equation:

............................... (3.18)

Equation 3.18 is called the reduced-matrix Riccati equation. The design steps

may be stated as follows:

1. Solve Equation 3.18, the reduced-matrix Riccati equation, for the matrix P.

2. [If a positive-definite matrix P exists (certain systems may not have a positive-

definite matrix P), the system is stable, or matrix A-BK is stable.]

3. Substitute this matrix P into Equation 3.16. The resulting matrix K is the optimal

matrix.

25

Note that if the matrix A-BK is stable, the present method always gives the correct

result. Finally, note that if the performance index is given in terms of the output

vector rather than the state vector, that is

....................................(3.19)

Then the index can be modified by using the output equation

Y=CX....................................................(3.20)

To

................................(3.21)

and the design steps presented in this part can be applied to obtain optimal matrix

K.

26

CHAPTER 4

INDUCTION MOTOR

27

4.1 INTRODUCTION

Induction motors are widely used in industry, for their simple and robust structure,

higher torque-to-weight ratio, higher reliability and ability to operate in hazardous

environment. However, unlike DC motor, their dynamic response is sluggish and

control is a challenging task, because of the inherent coupling between rotor

current and air gap flux, responsible for the torque production. The control of IM in

field coordinates using vector control (also known as field oriented control), leads

to decoupling between the flux and torque, thus, resulting in improved dynamic

torque and speed responses. Significant advances have been made in vector

Control of induction motors since its inception.

A disadvantage of the conventional field-oriented controller is, the method

assumes that, the magnitude of the rotor flux is regulated to a constant value.

Though good dynamic current (or torque) and speed responses are obtained with

vector control, the torque is only asymptotically decoupled from the flux, i.e.,

decoupling is obtained only in steady state, when the flux amplitude is constant.

Coupling is still present, when flux is weakened in order to operate the motor at

higher speed within the input voltage saturation limits, or when flux is adjusted in

order to maximize power efficiency. This has led to further research on application

of differential geometry, to develop the control techniques for linearization and

decoupling control.

After the theory was proposed, it has drawn attention of many researchers for

further development and implementation. These techniques have resulted in

solutions to several problems, including feedback linearization, input-output

linearization and decoupling control. Reference achieved decoupling of torque and

flux by a static multivariable state-feedback controller. Decoupling is also obtained

in by a static state-feedback controller using the amplitude and frequency of the

supply voltage as inputs. A voltage command input-output linearization controller is

28

developed and a current command input- output linearization controller is reported.

Feedback linearizing control technique is used in to design a controller for

switched reluctance motor.

4.2 Mathematical Modeling of Induction Motor

4.2.1 STATE SPACE MODEL

From the voltage equations of the induction motor in the synchronously rotating d-

q axes reference frame, the state space model with stator current and rotor flux

components as state variables is:

…………………………………..(4.1)

The stator current which is measurable is taken as the output, which is expressed

as

is = [ 1 0 ] [ is

Ψ r] ……………………………………(4.2)

Where,

,

and

A11 = - a1 I - ωcJ,

29

A12 = a2 I – P a3 ωr J,

A21 = a5 I,

A22 = -a4 I – (ωe - P ωr) J,

B1 c I.

and

C = Lr / ( Ls Lr - Lm2 ),

a1 = c R s + c Rr Lm2 / Lr

2,

a2 = c Rr Lm / Lr2,

a3 = c Lm / Lr,

a4 = Rr / Lr,

a5 = Rr Lm / Lr,

R s , Rr , Ls , Lr , Lm : Motor parameters ( given in appendix )

P : Number of pole pairs,

ωr : Mechanical rotor angular velocity,

ωe : Synchronous electrical angular velocity,

V ds, V qs : d-q axis stator phase voltages,

ids , iqs : d-q axis stator phase currents,

Ψ dr , Ψ qr : d-q axis rotor fluxes.

The torque developed by the motor is:

30

T e= K t (Ψ dr iqs - Ψ qr ids )…………………………………………..(4.3)

Where, K t = 3PLm / 2Lr.

4.2.2 Controller Design

4.2.2.1 Linearizing Control

The conditions required for vector control [1] are:

Ψ qr = 0 and

From (1),

…………………………(4.4)

Indirect vector control is obtained, when

a5 iqs = (ωe – P ωr ) Ψ dr……………………………………..……….(4.5)

Or,

ωe = P ωr + a5 iqs/Ψ dr……………………………………….……….(4.6)

When the above equation is satisfied, the dynamic behaviour of the induction

motor is:

………………………………….(4.7)

……………………………(4.8)

31

……………………………….…………………(4.9)

T e = K t Ψ dr iqs……………………………………………………..………(4.10)

The concept behind field oriented control is that rotor flux can be controlled

according to (4.9), with ids acting as the control input. The q-axis current

component iqs serves as an input in order to control the torque (4.10) as a product

of Ψ dr and iqs. Even the field oriented induction motor model described by (4.7-

4.10) has nonlinearity and interaction. The speed emf term ((Or Idr) appearing in

(4.8) makes the current dynamics nonlinear and speed dependent. Equations

(4.7) and (4.8) show that interaction between current components exists, in the

rotating reference frame. The transition from field oriented voltage components,

vds and vqs to current components as in (4.7) and (4.8) involves leakage time

constants and interactions. During the flux transient period (4.9), coupling of flux

and torque is apparent from (4.7) to (4.10). The interaction between current

components and nonlinearity in the overall system is eliminated by using the

linearization control approach. This approach consists of change of coordinates

and use of nonlinear inputs to linearize the system equations. The developed

torque, Te is considered as a state variable, replacing iqs in the induction motor

model. Differentiating (4.10) and simplifying with substitution of(4.6), (4.8), (4.9):

………….(4.11)

The nonlinearities in (4.7) and (4.11) are put together and then replaced by

nonlinear functions of the form u1 and u2 respectively. With these linearizing

inputs u1 and u2, (4.7) and (4.11) are then modified to (4.12) and (4.14)

respectively. The Induction motor system is now transformed into two linear and

decoupled subsystems: electrical and mechanical. Electrical subsystem is

represented by the state equation:

32

……………………………..……………………(4.12)

………………………….……………………………(4.13)

Mechanical subsystem is represented by the state equation:

……………………………………………………….…..(4.14)

…………………………………………………………..(4.15)

The stator input voltage components V qs and V ds in terms of u1 and u2 are:

V ds = ( - ωe iqs + u1 )/ c……………………………………………………………(4.16)

V qs = 1c

¿………………………………………….(4.17)

The transformed model given above is valid only for Ψ dr≠ 0. Since the induction

motor system described by (4.12)-(4.15) is linear and decoupled, the developed

torque (or the speed) and the rotor flux are independently controlled. LQR

Controller is designed to improve the performance of induction motor.

33

CHAPTER 5

CONTROLLER DESIGN AND ANALYSES

34

5.1 Mechanical subsystem of induction motor:

As I have derived the equations for the mechanical subsystem of induction motor

in the last chapter4, are given as:

ddt

Te = -( a1 + a4 ) T e + u2………………………………………………………….(5.1)

ddt

ωr = (T e - T l – βωr )/ J…………………………………..……………………...(5.2)

These above written equations can be shown in state space equation form like

shown below:

ddt [T e

ωr] = [−(a1+a4) 0

1 /J −β /J ] [T e

ωr] + [10] u2 …………………………………….(5.3)

Comparing the above equation with the standard state space equation:

ẋ = A x + B u

we get,

A = [−(a1+a4) 01 /J −β /J ]

B = [10]For the electrical subsystem:

u=u2,

y=ωr,

35

Thus for output equation:

Y = c x

ωr = [ 0 1 ] [T e

ωr]

Thus we get,

C = [ 0 1 ]

5.2 Rating & Parameters of the motor:

Let us take following ratings for our use,

3-Ф , 50 hz, 0.75 kw, 220 v, 3A , 1440 rpm

Stator and Rotor Resistances: Rs = .6 ohms, Rr = .6 ohms

Self inductance: Ls = Lr = 0.26 H

Mutual Inductances: Lm = 0.24 H

Moment of inertia of motor and load : J = 0.0088kg.m2

Viscous friction coefficient β = 0.003 N.m.s/rad.

Now substituting these values in the parameters:

C = 26,

36

a1 = 28.89,

a4 = 2.3075,

1/J = 113.6363,

β/J = 0.3409,

Thus matrices become:

A = [−31.1976 0113.6363 −0.3409]

B = [10]C = [ 0 1 ]

5.3 Algebraic Riccati Equation (ARE)

The Algebraic Riccati Equation is:

ATP+PA+Q-PBR−1 BTP=0………………….(5.4)

There are three kinds of symmetric solutions to the ARE. The stabilizing solution

P+, the anti-stabilizing solution P_ , and the mixed solutions PӨ with both negative

and positive eigen value real parts. All three solution types are needed to know the

complete phase portrait of the Riccati differential equations. Given a symmetric

solution for the ARE, Bucy in [4] calls a solution structurally stable if it is

continuously deformed and keeps the same closed-loop stability properties

(inertia) under data perturbation. It can easily be shown that P+ and P_ are

structurally stable. However, P_ may or may not be structurally stable depending

on how the RHP and LHP eigen values of the corresponding Hamiltonian are

37

combined. Bucy's main result in [4] relates the invertibility of Jacobian matrix J to

the structural stability of the solutions to ARE.

5.3.1: Theorem 13 (Bucy)

A solution P of the ARE is structurally stable if and only if J is invertible.

Structurally unstable solutions are good candidates for bifurcation. Depending on

the Eigen structure of the associated Hamiltonian matrix, a mixed solution may be

structurally unstable. In fact from Bucy's work, it can be shown that repeated eigen

values of the Hamiltonian matrix is a necessary condition for the mixed solution of

the ARE to be structurally unstable. Therefore, for these solutions we can define a

Riccati map such that the solution is a critical point of the Riccati map and we can

further analyze the solution to decide its infinitesimal V -stability. That criterion will

determine whether that particular mixed solution bifurcates under data

perturbation. Mixed solutions, PӨ’s, have an application in smoothing problems

when we need to estimate the state of the system using both past and future

measurements. The potential instability of the smoothing solution can be justified

from the fact that smoothing requires a choice (of past and future observations)

and that it might not be possible to make that choice continuously under data

perturbation.

5.4 Derivation of transfer function of induction motor:

The closed loop transfer function of the induction motor can be derived using

these two equations (5.1-5.2):

ddt

Te = -( a1 + a4 ) T e + u2………….(5.1)

ddt

ωr = (T e - T l – βωr )/ J…………..(5.2)

38

Taking laplace transform in both equations, we get

sT e(s) = -( a1 + a4 ) T e(s) + u2(s),

¿ = u2(s),

Or, u2(s) = ¿………………….(5.5)

And,

sωr = {T e(s) – βωr (s)}/ J,

Jsωr = T e(s) – βωr (s),

(Js+β)ωr (s) = T e(s),

ωr (s) = T e(s)/(Js+β), ……………………………..(5.6)

Dividing equation (5.6) by (5.5), we get

ωr (s )u2(s)

=1/ (Js+β)(s+a1+a4)

ωr (s )u2(s)

= 1(s+a1+a4)(Js+β) …………………………………(5.7)

Above is the transfer function of induction motor. Now after putting the values of

parameters, we get

113.6

T(s) = --------------------------------- ………………………….(5.8)

s^2 + 31.54 s + 124.3

Now as we have the transfer function of the system this is nothing but the

induction motor, we will concentrate on solving the Algebraic Riccati Equation

(ARE):

39

ATP+PA+Q-PBR−1 BTP=0………………….(5.4)

As we know that in the ARE we have to choose the values of Q and R by hit and

trial method or with the help of some advanced techniques like ANT and GA

techniques. But we have the limitations of using these techniques so we will go

with the hit and trial method.

After many trials I have come with these values of Q and R:

Q= [0.25 0; 0 .30] andR= [.3]

Usually Q is a function of C and is related as:

Q = CT * C

Where C is [0 1], so Q should have been:

Q = [0 0; 0 1]

But different systems respond differently with different values, so I had to

manipulate the value of Q near about this calculated value to become Q = [0.25 0;

0 0.30].

Using the function built in MATLAB for solving continuous time ARE which is:

[KK,S,e] =lqr(A,B,Q,R)Or,

[KK,S,e] =care(A,B,Q,R)

Here S = Arbitrary matrix P,And KK = system gain matrix K

A = [−31.1976 0113.6363 −0.3409]

40

B = [10]C = [ 0 1 ]

This gives the values as:

S =

0.9415 0.2704 0.2704 0.0825 e = -30.9973 -3.6820

As the value of K gives U as,

U = - K.x

= - R−1 BTP.x ………………………………………………………………….(5.9)

Thus K is given by,

K = R−1 BTP…………………….………………………………………………(5.10)

Putting the values in (5.10) gives,

K = [0.3]−1[10]T

[0.9415 0.27040.2704 0.0825 ]

= [3.1384 0.9014]

= [k1 k2]

41

input outputInduction

Motor

model

Figure 5.1 Expected Induction Motor Model

In this model, the induction motor is represented in its state space model form i.e.

X=Ax+Bu

And Y=Cx

And K is also in matrix form.

After applying the step signal, the system response which is also known as step re

is given by as follows:

Fig.5.2: Time response of the closed-loop system with proposed LQR controller

42

K

5.5.1. LQR response specifications:

Eigenvalue Damping Frequency

-3.68e+000 1.00e+000 3.68e+000

-3.10e+001 1.00e+000 3.10e+001

(Frequencies expressed in rad/seconds)

Rise Time: 0.8124

Settling Time: 1.0969

Settling Min: 0.9461

Settling Max: 0.9949

Overshoot: 0

Undershoot: 0

Peak: 0.9949

Peak Time: 1.9776

After seeing the response of induction motor using LQR technique, it is obvious

that the response of the system is critically damped. This means that the ξ i.e.

damping ratio is unity which is its maximum value and it shows fastest response to

reach unity which is its desired final value.

5.5: Induction motor model with unity feedback

43

outputInduction

Motor

model

input

Figure 5.3 Induction motor model with unity feedback

Step response of closed loop unity feedback system is shown below:

Fig.5.4: Time response of the closed-loop system without proposed LQR controller

44

5.5.1. Closed loop response specifications without using LQR:

Rise Time: 0.6564

Settling Time: 0.8884

Settling Min: 0.8696

Settling Max: 0.9138

Overshoot: 0

Undershoot: 0

Peak: 0.9138

Peak Time: 1.6162

From the above response curve it clear that this response is a critically damped

response but there is steady state error which can be calculated as:

ess = lim ¿t → ∞

¿ [ r(t)-c(t) ]

Where, r(t)=1,

And c(t)= 0.9138

Thus ess = 0.0862.

%ess = 0.0862* 100

= 8.62 %

45

CHAPTER 6

CONTROLLER DESIGN WITH PID

46

6.1: PID Controller

A proportional–integral–derivative controller (PID controller) is a generic

control loop feedback mechanism (controller) widely used in industrial control

systems – a PID is the most commonly used feedback controller. A PID controller

calculates an "error" value as the difference between a measured process variable

and a desired set point. The controller attempts to minimize the error by adjusting

the process control inputs.

The PID controller calculation (algorithm) involves three separate constant

parameters, and is accordingly sometimes called three-term control: the

proportional, the integral and derivative values, denoted P, I, and D. Heuristically,

these values can be interpreted in terms of time: P depends on the present error, I

on the accumulation of past errors, and D is a prediction of future errors, based on

current rate of change.[1] The weighted sum of these three actions is used to adjust

the process via a control element such as the position of a control valve, or the

power supplied to a heating element.

In the absence of knowledge of the underlying process, a PID controller has

historically been considered to be the best controller.[2] By tuning the three

parameters in the PID controller algorithm, the controller can provide control action

47

designed for specific process requirements. The response of the controller can be

described in terms of the responsiveness of the controller to an error, the degree

to which the controller overshoots the set point and the degree of system

oscillation. Note that the use of the PID algorithm for control does not guarantee

optimal control of the system or system stability.

Some applications may require using only one or two actions to provide the

appropriate system control. This is achieved by setting the other parameters to

zero. A PID controller will be called a PI, PD, P or I controller in the absence of the

respective control actions. PI controllers are fairly common, since derivative action

is sensitive to measurement noise, whereas the absence of an integral term may

prevent the system from reaching its target value due to the control action.

Figure 6.1: PID controller

Proportional term:

The proportional term produces an output value that is proportional to the current

error value. The proportional response can be adjusted by multiplying the error by

a constant Kp, called the proportional gain constant.

48

The proportional term is given by:

A high proportional gain results in a large change in the output for a given change

in the error. If the proportional gain is too high, the system can become unstable

(see the section on loop tuning). In contrast, a small gain results in a small output

response to a large input error, and a less responsive or less sensitive controller. If

the proportional gain is too low, the control action may be too small when

responding to system disturbances. Tuning theory and industrial practice indicate

that the proportional term should contribute the bulk of the output change.

Droop

Because a non-zero error is required to drive the controller, a pure proportional

controller generally operates with a steady-state error, referred to as droop.[note 1]

Droop is proportional to the process gain and inversely proportional to proportional

gain. Droop may be mitigated by adding a compensating bias term to the setpoint

or output, or corrected by adding an integral term.

Integral term:

The contribution from the integral term is proportional to both the magnitude of the

error and the duration of the error. The integral in a PID controller is the sum of the

instantaneous error over time and gives the accumulated offset that should have

been corrected previously. The accumulated error is then multiplied by the integral

gain ( ) and added to the controller output.

The integral term is given by:

49

The integral term accelerates the movement of the process towards setpoint and

eliminates the residual steady-state error that occurs with a pure proportional

controller. However, since the integral term responds to accumulated errors from

the past, it can cause the present value to overshoot the setpoint value (see the

section on loop tuning).

Derivative term:

The derivative of the process error is calculated by determining the slope of the

error over time and multiplying this rate of change by the derivative gain . The

magnitude of the contribution of the derivative term to the overall control action is

termed the derivative gain, .

The derivative term is given by:

The derivative term slows the rate of change of the controller output. Derivative

control is used to reduce the magnitude of the overshoot produced by the integral

component and improve the combined controller-process stability. However, the

derivative term slows the transient response of the controller. Also, differentiation

of a signal amplifies noise and thus this term in the controller is highly sensitive to

noise in the error term, and can cause a process to become unstable if the noise

and the derivative gain are sufficiently large. Hence an approximation to a

differentiator with a limited bandwidth is more commonly used. Such a circuit is

known as a phase-lead compensator.

6.2: Step response of induction motor when applied with PID

controller:

50

For calculating the step response we need to tune PID parameters that is setting

P,I and D gains. According to our requirements, I have chosen some specifications

as follows:

Kp = 5Ki = 150Kd = .5

Fig.6.2: Time response of the closed-loop system with PID controller

Rise Time: 0.0991

Settling Time: 2.5936

Settling Min: 0.6993

51

Settling Max: 1.4184

Overshoot: 41.8368

Undershoot: 0

Peak: 1.4184

Peak Time: 0.2303

CHAPTER 7

COMPARISON RESULTS AND CONCLUSION

52

7.1: Comparing graphs of all the three responses:

53

Fig.7.1: Time response comparison between closed loop, proposed LQR

and PID controller

Fig.7.2: Impulse response comparison between proposed LQR

and PID controller

54

7.2: Comparison table:

Parameters LQR PID

Rise Time(sec) 0.8124 0.0991

Settling time(sec) 1.0969 2.5936

Settling Min(sec) 0.9461 0.6993

Settling Max(sec) 0.9949 1.4184

Overshoot 0 41.8368

Undershoot 0 0

Peak 0.9949 1.4184

Peak time(sec) 1.9776 0.2303

7.3: Result:

From the above graphs and tables it is clear that the Fig.5.4 demonstrates the

stable but the error prone behaviour of the closed loop induction motor. The time

response of the closed loop system with the simulated PID controller and LQR

controller are shown in Fig.5.2 and Fig.6.2 respectively. In order to investigate and

evaluation the performance of theses controllers easily, the step response and the

impulse response of closed-loop system with both controllers are shown in Fig.7.1

and 7.2 respectively. From figures 5.2 and 6.2 it can be realized that both of these

controllers are suitable to utilize to control the induction motor due to both can give

almost zero steady-state error, fast response and no overshoots at the transient

response. However, the results has proven that the LQR method acts better than

the PID controller in terms of its faster response.

7.4: Conclusion

55

In this study, based on the mathematical model of induction motor, two controllers,

PID and LQR are designed and compared to investigate a more appropriate

control method. The simulation results demonstrate that both of these controllers

are effective and suitable for improving the time domain characteristics of system

response, such as settling time and overshoots. According to the results, LQR

method gives the better performance compared to PID controller. However, as a

method, the determination of PID parameters are easier to obtain using LQR

method.

REFERENCES

56

[1] Dawson, D.M., Hu, J., Burg, T.C., Nonlinear Control of Eiectric Machinery, Marcel Dekker, Inc., New York, 1998

[2] Ho, E.Y.Y, Sen, P.C., "Decouplhg Control of Induction Motor Drives", IEEE Transactions on Industrial Electronics, vol. 3 5, no. 2, pp. 25 3 - 262, 1988.

[3] W. Leonhard, Control of Electrical Drives, Springer Verag Berlin, 1990.

[4] D. I. Kim, I. J. Ha and M. S. Ko, "Control of induction motors via feedback linearization with input-output decoupling," Int. Journal of Control, 51 (4), 1990, pp. 863-883.

[5] A. Isidori, A. J. Krener, C. Gori-Giorgi, and S. Monaco, "Nonlinear decoupling via feedback: A differential-geometric approach," IEEE Trans. on Automatic Control, vol. 26, 1981, pp. 331-345.

[6] T. J. Tarn, A. K. Bejczy, A. Isidori and Y. Chen, "Nonlinear feedback in robot arm control," Proceedings of 23 Conference on Decision and Control, December 1984, pp.736-751.

[7] Z. Krzeminski, "Nonlinear control of induction motor," IFAC 10t World Congress on Automatic Control, vol. 3, Munich, 1987, pp. 349-354.

[8] A. De Luca, and G. Ulivi, "Design of exact nonlinear controller for induction motors," IEEE Trans. on Automatic Control, 34 (12), Dec. 1989, pp 1304-1307.

[9] R. Marino, S. Peresada, and P. Valigi, "Adaptive input-output linearizing control of induction motors," IEEE Trans. on Automatic Control, vol. 38, no. 2, 1993, pp. 208-221.

57

[10] M. Illic'-Spong, R. Marino, S. M. Peresada and D. G. Taylor, "Feedback linearizing control of switched reluctance motors," IEEE Trans. on Automatic Control, vol. 32, no. 5, May 1987, pp. 371-379.

[11] W. H Wonham, "On pole assignment in multi input controllable linear systems," IEEE Trans. on Automatic Control, vol. 12, December 1967, pp. 660-665.

[I2] H. W. Smith and E. J. Davison, "Design of industrial regulators", Proc. IEE,vol. 119, no. 8, August 1972, pp. 1210-1216.

[13]. Fatima Gurbuz, Eyup Akpinar, “Stability Analysis of a closed-loop control for a Pulse Width Modulated DC Motor Drive", Turkish Journal of Electrical Engineering, vol. 10, No.3, 2002.

[14]. Bimal K. Bose," Modern Power Electronics & ACDrives”, Pearson Education, 2002.

[15]. Katsuhiko Ogata, “Modern Control Engineering”, Prentice-Hall of India Pvt. Ltd, 2000.

[16]. Design and Analysis of Control Systems, Arthur G.O. Mutambara, CRC Press, London, 1999.

[17]. Linear Control Systems Engineering, Morris Driels, McGraw Hill international edition, 1995.

[18]. Modern Control Design with Matlab and Simulink, Ashish Tewari, John Wiley and Sons Ltd., 2002.

[19]. ‘’The Induction Motor’’ Herbert Vicker

[20]. ‘’Generalized Theory of AC Machinery’’ P.S.Bimbhara

58

APPENDICES

59

Appendix A.

1: abc to dq0 transformation

Vd= 2/3 (Va*sinwt + Vb*sin(wt-2pi/3) + Vc*sin(wt+2pi/3)

Vq= 2/3 (Va*coswt + Vb*cos(wt-2pi/3) + Vc*cos(wt+2pi/3)

V0= 1/3 (Va + Vb + Vc )

60

Figure (i.1): Simulation Model for abc to dq0 transformation

2. dq0 to abc transformation

Va= Vd*sinwt + Vq*cos(wt) + Vo

Vb= Vd*sin(wt-2pi/3) + Vq*cos(wt-2pi/3) + Vo

Vc= Vd*sin(wt+2pi/3) + Vq*cos(wt+2pi/3) + Vo

61

Figure (i.2): Simulation Model for dq0 to abc transformation

Appendix B.

NOMENCLATURE

62

63