FACULTY OF AUTOMATION AND COMPUTER SCIENCE

Abstract of the PhD Thesis

Contributions to the Modelling, Simulation and Advanced Control of Nonlinear Systems with Fast Dynamics, Applied in

Electromagnetic Levitation

PhD Student: eng. Adrian-Vasile DUKA

Thesis advisor: Prof.dr.eng. Mihail ABRUDEAN

Contents:

1. Introduction

2. Current State in the Field of Fuzzy Control

3. Modelling of the Electromagnetic Levitation Process

4. Modelling and Fuzzy-PD Control of the Electromagnetic Levitation System

5. Modelling and Fuzzy Model Reference Adaptive Control of the Electromagnetic

Levitation System

6. General Conclusions and Personal Contributions

Thesis outline:

The thesis contains the author’s results of the fundamental and applied research in

the field of automatic control, regarding the modeling, simulation and control, based on

advanced algorithms, of nonlinear systems with fast dynamics. The studies are based on an

electromagnetic levitation plant with fast dynamics, which displays an unstable and

nonlinear character, affected by uncertainties.

The three directions followed by the author on the course of this thesis were: the

theoretical and analytical study, system modeling and simulation, as well as the

experimental study. They allowed the fundamentation of some theoretical and applied

aspects regarding: the integration of neural networks in system modeling, the design of

linear fuzzy controllers equivalent to conventional controllers, the tunning of fuzzy

parameters (scaling gains, universes of discourse, number of membership functions) and

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their influence on the system’s response, the way the linear fuzzy controller is made nonlinear

and the way it could be included in a fuzzy model reference adaptive control system.

Chapter 1 represents an introduction in the subject of the thesis and a short

description of each chapter is provided.

Chapter 2 starts with a short review of some chronological events in the history of

fuzzy systems and presents a series of achievements in this field. The theoretical aspects of

fuzzy logic for understanding fuzzy control are presented synthetically. Fuzzy sets are

introduced, as an extension of the classical notion of set, and the most widely used

membership functions are presented together with the fuzzy logic operations. The notion of

linguistic variable and the theory of approximate reasoning, which provides a framework for

reasoning in the face of imprecise and uncertain information, are also presented.

In the section dedicated to fuzzy controllers, the principles of this type of control

action are presented, and the components of a fuzzy controller are described, as well as

different ways of their implementation. The structure of the fuzzy controller used in the

follwing chapters is emphasized.

Chapter 3 is focused on the development of a mathematical model for a one degree of

freedom, attraction type, electromagnetic levitation system used for suspending in midair a

ferromagnetic object at predetermined distances. A short history and some of the ways to

achieve levitation are introduced, before starting the mathematical modeling.

Figure 1 shows the principle of electromagnetic levitation.

Figure 1. Principle of electromagnetic levitation

The dynamical model of the magnetic levitation system is described by the following

nonlinear equation:

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( ) ( ) ( ) ( )( )

2

2

2,, ,,, ⎟⎟

⎠

⎞⎜⎜⎝

⎛=−=

txtiCtxiftxifmg

dttxdm ee (1)

where fe(i,x,t) is the electromagnetic force that counteracts the weight of the levitated ferro-

magnetic object (a steel ball in this case), x is the distance between the electromagnet and the

steel ball, i is the current through the coil, C is a nonlinear electromagnetic parameter, m is the

mass of the levitated ball, g is the gravitational constant.

The equilibrium situation, when levitation is achieved, is described by equation (2). 2

0

0⎟⎟⎠

⎞⎜⎜⎝

⎛=

XI

Cmg (2)

This equation indicates the existence of several pairs of values (X0, I0) which define

different equilibrium points. The experiments, together with this equation, indicate a

nonlinear behavior of the electromagnetic parameter C.

The difficulty of fully understanding and modeling the phenomenons encountered in

the case of the electromagnetic levitation system, have led the author to the idea of

introducing a neural network, which approximated the electromagnetic parameter C, in the

model of the plant. A two-input, one-output feed-forward neural network was used for this

task, having 8 neurons in the hidden layer. The network was trained using experimental

values for X0, I0 and C, which were determined when equilibrium was achieved. The

Levenberg-Marquardt training algorithm was used, which assured fast convergence of the

training error.

A Simulink model was finally developed which indicated a nonlinear dynamic plant

that was open-loop unstable and affected by parametrical uncertainties. As a result feedback

fuzzy control was proposed to stabilize the plant.

Chapter 4 introduces a design methodology for fuzzy controllers which is based on

the aspects of conventional controller design, namely on PID-type controllers.

Since fuzzy controllers are nonlinear, setting the controller parameters (gains, membership

functions etc.) can be often quite difficult and most of the time is done in an ad-hoc manner.

A fuzzy controller is nothing more than a nonlinear controller, having one or more inputs and

outputs. The shape of its nonlinear characteristic can be modeled into various shapes by

adequately choosing the different parameters in the structure of the fuzzy controller.

This is a consequence of the universal approximation theorem for fuzzy systems,

which creates the premises for the existence of a fuzzy system, which under certain

assumptions, is capable of approximating any control characteristic produced by all the

known conventional controllers, including the PID.

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In order to determine a fuzzy controller of this type, Taylor expansion was used to

linearize the electromagnetic force fe(i,x,t) and a transfer function was determined for the

plant, which helped in the design of a conventional phase-lead controller. This controller

provided the starting point in the development of the fuzzy controller. A digital version of the

phase-lead controller (equation 3) was used and it’s PD component was replaced by a linear

fuzzy system according to the following equations.

( ) ( ) ( ) ( )11 210 −++−= keqkeqkuqku (3)

To show the PD component, equation (3) was rewritten as follows:

( ) ( ) ( )

⎟⎠⎞

⎜⎝⎛ −−

+=Δ

Δ+−=

TkekeTkeKku

kukuqku

dp)1()()()(

10

(4)

and the PD component ∆u(k) was replaced by a linear equivalent fuzzy system producing the

next function:

( ) ( )( ) ⎟⎠⎞

⎜⎝⎛ −−

+=⋅⋅⋅=ΔT

kekeTkeKkcgkegfgku dpceu)1()()(,)( (5)

where ge, gc, gu are the fuzzy system’s scaling gains which respect the following relation:

de

c

peu

Tgg

Kgg

=

=⋅

(6)

Apart from equations (6), for the equivalence in (5) to hold, the following design

choices for the fuzzy system are requiered:

- triangular input sets that cross at 50%;

- complete rule base;

- algebric product for the AND connective in the premisis;

- use output singletons, positions determined by the sum of the peak positions of the

input sets;

- center-of-gravity defuzzification.

With these design choices the control surface degenerates to a diagonal plane.

Since there are no specifications regarding the number of fuzzy sets requiered, the size

of the universe of discourse or the way the scalling gains are to be chosen, except equation

(6), a model of the closed-loop system controlled by the Fuzzy-PD linear controller was

developed in order to simulate the behavior of the system for various choices of these

parameters.

The experiments and the results provided by these simulations led to a linear structure

of the Fuzzy-PD controller, which represented the starting point for a nonlinear fuzzy

controller for the electromagnetic levitation device.

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In the end, by introducing heuristic knowledge, about the way a human expert would

control the process, and considering the linear structure of the controller developed earlier by

analytical means, a nonlinear Fuzzy-PD controller was developed.

Its structure is presented in Figure 2.

Figure 2. The nonlinear Fuzzy-PD controller

The success with the direct fuzzy controller, applied to the magnetic levitation plant,

presented in chapter 3 is used in the design of a learning Fuzzy-PD controller. This approach

is based on the Fuzzy Model Reference Learning Control structure and is presented in

Chapter 5. The learning algorithm is based on the on-line adaptive tuning of the centers of

the output membership functions of the Fuzzy-PD system in the controller presented earlier.

Figure 3 shows the general Learning Fuzzy-PD Control structure as applied for the

positioning system based on electromagnetic levitation.

Figure 3. Learning Fuzzy-PD control structure

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The control structure consists of four main parts: the process (plant, sensor, execution

element), the reference model, the direct Fuzzy-PD controller, and the learning mechanism

(inverse fuzzy model, rule base modifier).

The learning mechanism tries to adjust the controller parameters so that the closed

loop system (expressed through r(kT) and y(kT)) behaves as the reference model (expressed

by r(kT) and ym(kT)). This way, two loops are used to control the plant: the control loop

(lower) in which the controller acts by modifying the command u(kT) so that the output y(kT)

follows the reference r(kT) and the adaptation loop (upper) which makes the output of the

plant y(kT) follow the output of the reference model ym(kT) by adjusting the fuzzy controller’s

parameters.

The reference model is chosen to generate the desired trajectory, ym, for the plant

output y to follow. In this case, to allow simplified computations, the output of the reference

model was considered identical to the reference.

( ) ( )kTrkTym = (7)

An additional fuzzy system was developed called “fuzzy inverse model” which adjusts

the centers of the output membership functions of the Fuzzy-PD system developed earlier,

used to control the process.

The output of the inverse fuzzy model is an adaptation factor p(kT) which is used by

the rule base modifier to adjust the centers of the output membership functions of the Fuzzy-

PD system in the controller. The adaptation is stopped when p(kT) gets very small and the

changes made to the rule base are no longer significant.

The implementation of the Fuzzy Model Reference Adaptive Control system for

electromagnetic levitation is presented in Figure 4.

Figure 4. The adaptive Fuzzy-PD control system

Next some experimental results are presented. A comparative response between the

three control strategies used throughout this thesis is shown in Figure 5. For the three cases,

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the set-point of the closed loop system corresponded to the initial linearization point (I0, X0).

Discrete phase-lead control, direct Fuzzy-PD control and Learning Fuzzy-PD control were

considered.

The changes made to the initial Fuzzy-PD linear controller (which behaved identical to

the phase lead system), by making it nonlinear, show just a little improvement in the response.

Even though, the nonlinear transformation did not reduce sufficiently enough the stationary

error, specific to PD control strategies, a dynamical change in the structure of the Fuzzy-PD

controller, introduced by the adaptive system, shows the reduction of the stationary error and

a more accurate tracking of the reference.

Figure 5. Compartive response: phase lead, direct Fuzzy-PD, adaptive Fuzzy-PD

The tracking capabilities of the system are shown in Figure 6, where a square and a

sine trajectory are considered for exemplification. The graphics show the system’s ability to

position very accurately the levitated steel ball along the required trajectories, with very fast

convergence.

Figure 6. Tracking of square (a) and sine (b) trajectories

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Chapter 5 ends with the presentation of the way the fuzzy adaptive system was

implemented by software and the programming techniques adopted in order to optimize the

execution time of the fuzzy algorithms.

Chapter 6 discusses the general conclusions of this thesis and presents the personal

contributions of the author and possible future studies. The main personal contributions

presented can be summarized as follows:

- the determination of the mathematical models for the electromagnetic levitation

system and the development of a Simulink model which uses a neural network to

approximate the electromagnetic parameter C;

- the design and implementation of a phase-lead controller for the plant, in both

continuous and discrete form;

- the design and implementation of a nonlinear Fuzzy-PD controller which uses a

nonlinear fuzzy system to replace the PD component of the conventional controller.

For this purpose the final nonlinear controller resulted as a combination of an

analytical design method for linear fuzzy controllers with heuristical methods;

- the study of the influence of the fuzzy parameters (scaling gains, universe of

discourse, number of fuzzy setes) on the response of the closed loop system

controlled by the linear Fuzzy-PD controller;

- the use of the nonlinear Fuzzy-PD controller in an adaptive control scheme;

- the modification of the FMRLC scheme, by using a reference model having the

same output as the reference, and modifying the rule base periodically at fixed

time intervals;

- a comparative study using three types of controllers for the plant;

- the development of Simulink models for all the components of the systems

introduced in this paper and their implementation in an experimental device;

- a very accurate positioning system based on electromagnetic levitation.