Fuzzy Supervised PID controller.pdf

DESIGN OF FUZZY-SUPERVISED PID CONTROLLER FOR MAGLEV SYSTEMS

B.Tech. Project By MIRZA ABDUL WARIS BEGH (10289) AAKASH AGRAWAL (10288) GOPAL BHARADWAJ (10265) MOHAN LAL (09223)

DEPARTMENT OF ELECTRICAL ENGINEERING NATIONAL INSTITUTE OF TECHNOLOGY, HAMIRPUR - 177005 (INDIA) May, 2014

DESIGN OF FUZZY-SUPERVISED PID CONTROLLER FOR MAGLEV SYSTEMS A PROJECT Submitted in partial fulfilment of the requirements for the award for the degree of

BACHELOR OF TECHNOLOGY by MIRZA ABDUL WARIS BEGH (10289) AAKASH AGRAWAL (10288) GOPAL BHARADWAJ (10265) MOHAN LAL (09223) Under the Guidance of Dr. Bharat Bhushan Sharma

DEPARTMENT OF ELECTRICAL ENGINEERING NATIONAL INSTITUTE OF TECHNOLOGY, HAMIRPUR - 177005 (INDIA) May, 2014 II

Copyright © NIT HAMIRPUR, 2014

III

CANDIDATE’S DECLARARTION We hereby certify that the work which is being presented in the project report entitled “DESIGN OF FUZZY-SUPERVISED PID CONTROLLER FOR MAGLEV SYSTEMS,” in partial fulfillment of the requirements for the award of degree of the Bachelor of Technology and submitted in the Department of Electrical Engineering, National Institute of Technology, Hamirpur H.P. is an authentic record of our own work carried out during a period from January 2014 to May 2014 under the supervision of Dr. Bharat Bhushan Sharma, Assistant Professor, Department of Electrical Engineering, N.I.T. Hamirpur. The matter presented in this project report has not been submitted by us for the award of any other degree of this or any other university/institute. Sd/MIRZA ABDUL WARIS BEGH (10289) AAKASH AGRAWAL (10288) GOPAL BHARADWAJ (10265) MOHAN LAL (09223)

This is to certify that above statement made by the candidate is correct to the best of my knowledge.

Date:

Sd/Dr. Bharat Bhushan Sharma Assistant Professor, EED

The project Viva-Voce Examination of the Candidates Mirza Abdul Waris Begh (10289), Aakash Agrawal (10288), Gopal Bharadwaj (10265), Mohan Lal (09223) has been held on____________________.

Dr. Bharat Bhushan Sharma Project Supervisor Electrical Engg. Dept.

---------------------------------External Examiner

IV

ACKNOWLEDGEMENT First things first we find it hard to express our gratefulness to Almighty GOD in words for bestowing upon us His deepest blessings and providing us with the most wonderful opportunity in the form of life of a human being and for the warmth and kindness he has showered upon us. We feel great pleasure in acknowledging our deepest gratitude to our revered guide and mentor, Dr. Bharat Bhushan Sharma, Assistant Professor, Electrical Engineering Department, National Institute of Technology Hamirpur, under whose firm guidance, motivation and vigilant supervision we succeeded in completing our work. He infused into us the enthusiasm to work on this topic. His tolerant nature accepted our shortcomings and he synergized his impeccable knowledge with our curiosity to learn into this fruitful result. We would sincerely thank Prof. Y. R. Sood, HOD, Electrical Engineering Department who suggested many related points and is always very helpful and constructive. Words are inadequate to express our heartfelt gratitude to our affectionate parents who have shown so much confidence in us and by whose efforts and blessings we have reached here. We would also like to thank all the faculty members of Department of Electrical Engineering for their continuous moral support and encouragement. Last but not the least we wish to express heartiest thanks to our friends and colleagues for their support, love and inspiration.

Date: MIRZA ABDUL WARIS BEGH (10289) AAKASH AGRAWAL (10288) GOPAL BHARADWAJ (10265) MOHAN LAL (09223)

V

Abstract The magnetic levitation system is a mechatronic system already acknowledged and accepted by the field experts. For such a system it is desired to propose a suitable controller for positioning a metal sphere in air space by the help of an electromagnetic force. In the ideal situation, the magnetic force produced by the current from an electromagnet counteracts the weight of the metal sphere. Nevertheless, the electromagnetic force is very sensitive, and presence of noise induces accelerating forces on the metal sphere, causing the sphere to move into the unbalanced region. Fuzzy logic controller (FLC) is an attractive alternative to existing classical or modern controllers for designing the challenging Non-linear control systems. Fuzzy rules are very easy to learn and use, even by non-experts. It typically takes only a few rules to describe systems that may require several lines of conventional software code, which reduces the design complexity. By considering these advantages, this project presents the design and analysis of a Fuzzy logic based supervision controller for the magnetic levitation system. Additionally, a classical PID controller is also designed to compare the performance of both types of controllers. Results reveal that Fuzzy supervised PID controller is found to give better transient and steady state results compared to the classical PID. Keywords: Fuzzy Logic; PID controller; Fuzzy supervised PID controller; PID tuning; Maglev; Magnetic Levitation;

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Contents

Page No.

List of Figures………………………………………………………………... IX List of Abbreviations…………………………………………………............ X List of Tables…………………………………………………………….....

X

List of symbols…………………………………………………………….....

XI

1 1.1 1.2 1.3 1.4 1.5 1.6

Introduction……………………………………………………….... Overview……………………………………….................................. Magnetic Levitation………………………………………................. Maglev………………………………………...................................... Objective……………………………………….................................. Motivation………………………………………................................ Organisation of the report………………………………………........

1 1 2 3 4 5 5

2 2.1 2.2 2.3

Magnetic Levitation System……………………………………….. Maglev Systems………………………………………....................... Principle………………………………………................................... System Model Description……………………………………….......

6 6 7 7

3 3.1 3.1.1 3.2 3.2.1 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.4 3.5

Fuzzy Logic………………………………………............................. Fuzzy Sets………………………………………................................ Linguistic variables……………………………………….................. Fuzzy control………………………………………............................ Fuzzy control system design……………………………………….... Fuzzy reasoning………………………………………....................... Fuzzy rules………………………………………............................... Fuzzy inference system………………………………………............ Fuzzification………………………………………............................. Inference………………………………………................................... Defuzzification………………………………………......................... Different types of Fuzzy Logic controllers………………………….. Discussion………………………………………................................

12 12 12 13 13 14 14 15 15 16 17 18 19

4 4.1 4.2 4.3 4.4 4.4.1 4.4.2

Fuzzy Controller………………………………………..................... Fuzzy supervisory control………………………………………........ Supervision of conventional controllers……………………………... Fuzzy tuning of PID controllers……………………………………... Fuzzy gain scheduling……………………………………….............. Construction of a heuristic schedule gain…………………………… Construction of a schedule gain by fuzzy identification……………..

20 20 21 21 22 23 23 VII

4.4.3

Construction of a gain schedule using the PDC method…………….. 23

5 5.1 5.1.1 5.2 5.3 5.4

Tuned PID Controller………………………………………............ Overview……………………………………….................................. Tuning and its Purpose………………………………………............ Trial and error method………………………………………............. Pole placement method………………………………………............ Ziegler Nichols method………………………………………............

25 25 25 25 26 27

6 6.1 6.2 6.3 6.4 6.5 6.6

Fuzzy Supervised PID-Controller………………………………… Fuzzification………………………………………............................. Inference engine………………………………………....................... Rule Base………………………………………................................. Defuzzification………………………………………......................... Adjusting fuzzy membership functions and rules…………………… Results and Discussion……………………………………….............

29 29 30 31 33 34 35

7

Conclusion........................................................................................... 38

References........................................................................................................

39

VIII

LIST OF FIGURES Fig (1.1) Fig (1.2)

Trans-rapid 09 at the Emsland test facility in Germany. SC-Maglev in Yamanashi Prefecture, Japan.

Fig (2.1) Fig (2.2) Fig (2.3) Fig (2.4) Fig (2.5)

Principle of Magnetic Levitation. Close view of the levitating steel ball. Schematic Diagram of the Magnetic Levitation Unit. A magnetic ball bearing system. Block diagram of Magnetic levitation system.

Fig (3.1) Fig (3.2) Fig (3.3) Fig (3.4) Fig (3.5) Fig (3.6) Fig (3.7) Fig (3.8)

Conventional sets and fuzzy sets. Terms of fuzzy logic. Different shapes of membership functions. Fuzzy controller diagram. The structure of the fuzzy logic inference system. Fuzzification of a crisp input and a fuzzy input The fuzzy inference using the Min-inference. Defuzzification methods

Fig (4.1) Fig (4.2) Fig (4.3) Fig (4.4) Fig (4.5)

Fuzzy Supervisory controller Fuzzy PID auto-tuner Conventional fuzzy gain scheduler Tank PDC concept

Fig (5.1) Fig (5.2) Fig (5.3) Fig (5.4)

Unit step response of the system G(s) tuned with trial and error method. Response of a system tuned with Pole Placement Method. Control Scheme for Ziegler Nichols Method. Unit step response of the system G(s) tuned with ZL method

Fig (6.1) Fig (6.2) Fig (6.3) Fig (6.4) Fig (6.5) Fig (6.6) Fig (6.7) Fig (6.8) Fig (6.9) Fig (6.10) Fig (6.11) Fig (6.12) Fig (6.13) Fig (6.14) Fig (6.15)

Triangular Membership functions of input variable „error‟. Triangular Membership functions of input variable „error-rate‟. Triangular Membership functions of output variable „KP‟ Triangular Membership functions of output variable „KI‟. Triangular Membership functions of output variable „KD‟. Fuzzy Logic Rule-Base in SIMULINK®. Surface plot of Fuzzy Logic Rule-Base for variable KP. Surface plot of Fuzzy Logic Rule-Base for variable KI. Surface plot of Fuzzy Logic Rule-Base for variable KD. Structure of fuzzy logic controller. Block diagram of self-tuning Fuzzy Supervised PID controller. SIMULINK® model of self-tuning Fuzzy Supervised PID controller. Error rate (de) v/s time response of Fuzzy tuned PID controlled system. Output v/s time response of Fuzzy tuned PID controlled system. Error (e) v/s time response of Fuzzy tuned PID controlled system.

IX

LIST OF ABBREVIATIONS DC PID PI PLC MagLev MJ mph CO2 FIS Max Min COA WA PDC ZN FL NL NM NS ZE PS PM PL SSE FLC

Direct Current Proportional, Integral and Derivative Proportional & Integral Programmable Logic Controller Magnetic Levitation Mega Joules Miles per hour Carbon Dioxide Fuzzy inference system Maximum Minimum Center of Area Weighted average Parallel Distributed Compensation Ziegler Nichols Method Fuzzy Logic Negative Large Negative Medium Negative Small Zero Positive Small Positive Medium Positive Large Steady-State Error Fuzzy Logic Controller

LIST OF TABLES Table 1 Table 2 Table 3 Table 4

Parameters of the Magnetic Levitation system Rule base Parameters for KP. Rule base Parameters for KI. Rule base Parameters for KD.

X

LIST OF SYMBOLS

ω Fm I X Vs Tr Mp H0 M R L1 β I0 Ks K1 K2 u e de Kp Kd Ki ζ *fuz

Frequency (rad/sec) Magnetic force Electromagnetic current Air gap length Sensor output Rise time Maximum overshoot Equilibrium height of ball (m) Mass of ball bearing (kg) Resistance (Ω) Inductor (H) Constant related to magnetic force (Nm2/A2) Equilibrium current of the coil (A) Sensor gain factor (V/m) constant (N/A) constant (N/m) The control signal provided by the PID controller to the plant Error Differential of error Proportional gain Derivative gain. Integral gain Relative damping coefficient Output fuzzy variable

XI

Chapter-1

Introduction 1.1

Overview

In the control of plants with good performances, engineers are often faced to design controllers in order to improve static and dynamic behavior of plants. Usually the improvement of performances is observed on the system responses. For illustration, an example of a DC machine is chosen. Two cases of study are presented: I. In open loop, the velocity response depends on the mechanical time constant of the DC machine (time response) and the value of the power supply. Indeed, for each value of power supply, a velocity value is reached in steady state. Therefore the DC machine can reach any value of velocity which depends only of the power supply. In this case no possibility to improve performances. II. For a specific need, the open loop control is not sufficient. Engineers are faced to a problem of control in order to reach a desired velocity response according to defined specifications such as disturbance rejection, insensitivity to the variation of the plant parameters, stability for any operation point, fast rise-time, minimum Settling time, minimum overshoot and a steady state error null. Also, the designed control is related to other constraints such as the cost, computation complexity, manufacturability, reliability, adaptability, understand-ability and politics [1]. In general the design of the control needs to identify the dynamic behavior of the system. Therefore a dynamic model of the plant is developed in order to reproduce the real response in open loop. Developing a model for a plant is a complex task which needs time and an intuitive understanding of the plant‘s dynamics. Usually, on the basis of some assumptions to choose, a simplified model is developed and the physical parameters of the established model are identified using some experimental responses. If the model is nonlinear, we need to linearize the model around a steady state point in order to get a simplified linear model. Therefore a linear controller is designed with techniques from classical control such as pole placement or frequency domain methods. Using the mathematical model and the designed controller, a simulation in closed loop is carried out in order to study and to analyse its performances. This step of study consists to adjust controller parameters until performances are reached for a given set point. In the last step, the designed controller is implemented via, for example, a microprocessor, and evaluating the performance of the closed-loop system (again, possibly leading to redesign). In industry most of the time engineers are interested in linear controllers such as proportional- integral-derivative (PID) control or state controllers. Over 90% of the controllers in operation today are PID controllers (or at least some form of PID controller like a P or PI controller). This approach is often viewed as simple, reliable, and easy to understand and to implement on PLCs. Also performances of the plant are on-line improved by adjusting only gains. In spite of the advantages of PID controllers, the process performances are never reached. This is due mainly to the accuracy of the model used to design controller and not properly to the controller. 1

Introduction

In the development of analytical models, variation of physical parameters, operation conditions and disturbances are not taken into account. This is due to the difficulty to identify all the physical phenomena in the process and to find an appropriate model for each. Therefore the closed loop specifications of the process are not maintained. Engineers are however in the need to adjust permanently PID controllers even it‘s a heavy task. For these different reasons, a new approach of control is proposed taking into account the constraints related to the process, the parameter variation and disturbances. This type of control, named fuzzy control, is designed from the operator experience on the process over many years. Under different operation conditions, linguistic rules are established taking into account constraints and environment process. In this case, modelling the process is not necessary and the designed fuzzy controller is sufficient to ensure the desired performances. Currently, Fuzzy control has been used in a wide variety of applications in engineering, science, business, medicine, psychology, and other fields [1]. For instance, in engineering some potential application areas include the following: 1. 2. 3. 4. 5. 6. 7. 8.

1.2

Aircraft/spacecraft: Flight control, engine control, avionics systems, failure diagnosis, navigation, and satellite attitude control. Automated highway systems: Automatic steering, braking, and throttle control for vehicles. Automobiles: Brakes, transmission, suspension, and engine control. Autonomous vehicles: Ground and underwater. Manufacturing systems: Scheduling and deposition process control. Power industry: Motor control, power control/distribution, and load estimation. Process control: Temperature, pressure, and level control, failure diagnosis, distillation column control, and desalination processes. Robotics: Position control and path planning.

Magnetic Levitation

Magnetic levitation, maglev, or magnetic suspension is a method by which an object is suspended with no support other than magnetic fields. Magnetic force is used to counteract the effects of the gravitational and any other accelerations. The two primary issues involved in magnetic levitation are lifting force: providing an upward force sufficient to counteract gravity, and stability: insuring that the system does not spontaneously slide or flip into a configuration where the lift is neutralized. Magnetic levitation is used for maglev trains, magnetic bearings and for product display purposes. The Magnetic Levitation System (Maglev) serves as a simple model of devices that have become more popular in recent years such as Maglev trains and magnetic bearings. Maglev trains have been recently tested and some lines are already available for example in Shanghai. Magnetic bearings are used in turbines for the same reason as Maglev trains are being built, which is low friction in the bearing itself. Already many turbines are in commercial used where the rotating shaft is levitated with magnetic force. Some other magnetic bearing applications include pumps, fans and other rotating machines.

2

Introduction

1.3

Maglev

Maglev is a completely new mode of transport that will join the ship, the wheel, and the airplane as a mainstay in moving people and goods throughout the world. Maglev has unique advantages over these earlier modes of transport and will radically transform society and the world economy in the 21st Century. Compared to ships and wheeled vehicles—autos, trucks, and trains—it moves passengers and freight at much higher speed and lower cost, using less energy. Compared to airplanes, which travel at similar speeds, Maglev moves passengers and freight at much lower cost, and in much greater volume. In addition to its enormous impact on transport, Maglev will allow millions of human beings to travel into space, and can move vast amounts of water over long distances to eliminate droughts. In Maglev—which is short for MAGnetic LEVitation—high speed vehicles are lifted by magnetic repulsion, and propelled along an elevated guide-way by powerful magnets attached to the vehicle. The vehicles do not physically contact the guide-way, do not need engines, and do not burn fuel. Instead, they are magnetically propelled by electric power fed to coils located on the guide-way. There are four basic reasons why Maglev is important: 







Maglev is a much better way to move people and freight than by existing modes. It is cheaper, faster, not congested, and has a much longer service life. A Maglev guide-way can transport tens of thousands of passengers per day along with thousands of piggyback trucks and automobiles. Maglev guideways will last for 50 years or more with minimal maintenance, because there is no mechanical contact and wear, and because the vehicle loads are uniformly distributed, rather than concentrated at wheels. Similarly, Maglev vehicles will have much longer lifetimes than autos, trucks, and airplanes. Maglev is very energy efficient. Unlike autos, trucks, and airplanes, Maglev does not burn oil, but instead consumes electricity, which can be produced by coal-fired, nuclear, hydro, fusion, wind, or solar power plants. At 300 miles per hour in the open atmosphere, Maglev consumes only 0.4 MJ per passenger mile, compared to 4 MJ per passenger mile of oil fuel for a 20-miles-pergallon auto that carries 1.8 people (the national average) at 60 miles per hour (mph). At 150 mph in the atmosphere, Maglev consumes only 0.1 of a MJ per passenger mile, which is just 2 % of the energy consumption of a typical 60mph auto. In low-pressure tunnels or tubes, like those proposed for Switzerland‘s Metro system, energy consumption per passenger mile will shrink to the equivalent of 10,000 miles per gallon. Maglev vehicles emit no pollution. When they consume electricity, no carbon dioxide is emitted. Even if they use electricity from coal- or natural-gas-fired power plants, the resulting CO2 emission is much less than that from autos, trucks, and airplanes, because of Maglev‘s very high energy efficiency. Maglev has further environmental benefits. Maglev vehicles are much quieter than autos, trucks, and airplanes, which is particularly important for urban and suburban areas. Moreover, because Maglev uses unobtrusive narrow-beam elevated guide-ways, its footprint on the land is much smaller than that of highways, airports, and railroad tracks. Maglev has major safety advantages over highway vehicles, trains, and airplanes. The distance between Maglev vehicles on a guide-way, and the 3

Introduction

speed of the vehicles, are automatically controlled and maintained by the frequency of the electric power fed to the guide-way. There is no possibility of collisions between vehicles on the guide-way. Moreover, since the guide-ways are elevated, there is no possibility of collisions with autos or trucks at grade crossings.

Fig. 1.1: Trans-rapid 09 at the Emsland test facility in Germany

Fig. 1.2: SC-Maglev in Yamanashi Prefecture, Japan

1.4

Objective

If the model of a system is known with a good accuracy, industrialists prefer to implement a fuzzy PID supervisor to their existing PID controllers where gains are on-line adjusted taking into account different operation conditions, variation of plant parameters and disturbances. Supervisory control is a type of adaptive control since it seeks to observe the current behavior of the control system and modify the controller to improve the performance. It is a multilayer (hierarchical) controller with the supervisory at the highest level; the supervisor controller can use any available data 4

Introduction

from the control system to characterize the systems current behavior and generate outputs that are not direct command inputs to the plant. Rather, they dictate changes to another controller that generates these command inputs [2]. Because PID controllers are often not properly tuned (e.g., due to plant parameter variations or operating condition changes), there is a significant need to develop methods for the automatic tuning of PID controllers. The supervisor is trying to recognize when the controller is not properly tuned and then seeks to adjust the PID gains to obtain improved performance. When there is heuristic knowledge available on how to tune PID controllers while in operation, there is the opportunity to utilize fuzzy control methods as the supervisor that tunes or coordinates the application of conventional controllers, this approach shouldn‘t be confused with Fuzzy-PID controllers, which are PID controllers realized by fuzzy control methods [3]. Overall, fuzzy PID auto-tuners tend to be very application dependent and it is difficult to present a general approach to on-line fuzzy PID auto-tuning that will work for a wide variety of applications [2]. There are different configurations that incorporating fuzzy controllers with PID controllers, examples are: replacing PID with fuzzy controller, using fuzzy controller to adjust PID parameters, and using fuzzy controller to add to PID output [4 - 9]. In the method presented here, we use a PID controller to create a stable equilibrium point of the position of a magnetically levitated rotor, and a fuzzy controller to adjust gains of the PID controller based on the operating conditions to improve the performance of the system. The controller is simulated using SIMULINK®, and the performance of the PID controller alone is compared to the performance of the hybrid controller.

1.5

Motivation

Design of controllers is an area of research that has been explored since the advent of human intellect. In the initial stage of development of control theory most of the effort was concentrated into algorithms and methods that constitute the classical control. It did not incorporate real time or real life problems. In the last century a large effort was put for designing controllers that can act like a human operator and imitate decisions like humans. Thus the neural algorithms and fuzzy based approaches were discovered. The key motivation behind this project has been the interesting field of Fuzzy based controllers. Fuzzy controllers are designed to make decisions like a human but unlike neural networks; fuzzy systems tend to be more domain specific.

1.6

Organisation of the Report

This work is organised as follows. In Chapter 2 the Magnetic levitation System and its system model is presented. Chapter 3 discusses the Fuzzy Logic and a brief description of fuzzy control is given. In Chapter 4 extensive description of how to design a fuzzy controller is given. In Chapter 5 tuning of PID controller using conventional methods is analysed. Chapter 6 presents a detailed discussion on Fuzzy supervised PID controller and analysis of system using SIMULINK® model. Finally conclusion and future scope is discussed in Chapter 7.

5

Chapter-2

Magnetic Levitation System 2.1

Maglev Systems

Magnetic levitation systems are systems in which a rotor or a stationary object is suspended in magnetic field. Magnetic levitation systems have practical importance in many engineering systems such as in high-speed maglev passenger trains, frictionless bearings, levitation of wind tunnel models, vibration isolation of sensitive machinery, levitation of molten metal in induction furnaces, and levitation of metal slabs during manufacturing. The maglev systems can be classified as attractive systems or repulsive systems based on the source of levitation forces. Magnetic levitation of a rotating disk typically incorporates four or more electromagnets to levitate a ferromagnetic disk without contact with the surroundings, where levitation is accomplished through automatic control of the electromagnet coils currents. Position sensors are required to sense the position of the disk, and a controller uses position sensor outputs to apply stiffness and damping forces to the rotor to achieve a desired dynamic response. Active magnetic levitation systems are being increasingly used in industrial applications where minimum friction is desired or in harsh environments where traditional bearings and their associated lubrication systems are considered unacceptable, as discussed in [10, 11]. Such systems are inherently open-loop unstable, and require means of control to stabilize their operation; this is generally done by creating a closed loop system using feedback control. The requirement of controllers introduces flexibility into the dynamic response of the systems, which can also be designed to compensate for noises and vibrations that would affect the operation. Also, these systems are highly nonlinear, and in order to obtain a transfer function to describe them, number of approximations have to be made; hence, the design of linear controllers can produce the desired dynamic response only for the region in which the linear model was created. Many non-linear control algorithms were introduced in earlier research [12, 13] and a comparison between using linear and non-linear methods of controlling magnetic levitation systems was discussed in [14]. In recent years, a lot of works have been reported in the literature for controlling magnetic levitation systems. The feedback linearization technique has been used to design control laws for magnetic levitation systems [15, 16]. The input-output, inputstate, and exact linearization techniques have been used to develop nonlinear controllers [17, 18]. Other types of nonlinear controllers based on nonlinear methods have been reported in the literature [19]. Robust linear controller methods such as H∞ optimal control, μ-synthesis, and Q-parameterization have also been applied to control magnetic levitation systems [20]. Control laws based on phase space, linear controller design, the gain scheduling approach, and neural network techniques [21] have also been used to control magnetic levitation systems. In this project design and simulation of a new supervisory control strategy for magnetic levitation systems that incorporates a fuzzy controller to tune the gains of a discrete PID controller is studied. 6

Magnetic Levitation System

2.2

Principle

The basic principle of a simple electromagnetic suspension system is shown in Fig. 2.1. The magnetic force applied by the electromagnet is opposite to gravity and maintains the suspended steel ball in a levitated position. The magnetic force Fm depends on the electromagnet current I, electro-magnet characteristics, and the air gap X between the steel ball and the electromagnet.

Fig. 2.1: Principle of Magnetic Levitation

2.3

System Model Description

Fig. 2.2: Close view of the levitating steel ball The Magnetic levitation system as shown Fig. 2.2 consists of a magnetic sphere suspension system. The objective of the system is to control the vertical position of the ball by adjusting the current in the electromagnet through the input voltage. The metal sphere is suspended in air by the electromagnetic force generated by an electromagnet. The Magnetic levitation system consists of an electromagnet, a metal sphere and an infra-red sphere position sensor. The magnetic ball suspension system can be categorized into two systems: a mechanical system and an electrical system. The sphere position in the mechanical system can be controlled by adjusting the current through the electromagnet where the current through the electromagnet in the electrical system can be controlled by applying controlled voltage across the electromagnet terminals. 7


Fig. 2.3: Schematic Diagram of the Magnetic Levitation Unit From Ampere‘s circuit law and faraday‘s inductive law, the magnitude of the force f(h,i) exerted across an air gap h by an electromagnet through which current i flows can be described as: f (h,i ) = −

i 2 dL(h) 2 dh

(1)

The total inductance L is a function of the distance and given by L(h) = L1 +

L0 H 0 h

(2)

Where L1 is the inductance of the electromagnetic (coil) in the absence of the levitated object, L0 is the additional inductance contributed by its presence, and X0 is the equilibrium position. The parameters are determined by the geometry and construction of the electromagnet, and can be determined experimentally. Substituting equation (2) into (1) yields: 2

LX ⎛i⎞ ⎛i⎞ f = 0 0⎜ ⎟ =β⎜ ⎟ 2 ⎝h⎠ ⎝h⎠

2

(3)

8


Fig. 2.4: A magnetic ball bearing system

β=

L0 H 0 2

(4)

⎛ ∂f ⎞ ⎛ ∂f ⎞ ⎛ ∂f ⎞ ⎛ ∂f ⎞ f = f0 + ⎜ ⎟ ΔL0 + ⎜ ⎟ ΔH 0 + ⎜ ⎟ Δi + ⎜ ⎟ Δ h ⎝ ∂i ⎠ ⎝ ∂h ⎠ ⎝ ∂L0 ⎠ ⎝ ∂H 0 ⎠

(5)

Eliminating higher order terms give ⎛ ∂f ⎞ ⎛ ∂f ⎞ f = f 0 + ⎜ ⎟ Δi + ⎜ ⎟ Δh ⎝ ∂i ⎠ ⎝ ∂h ⎠ Evaluating equation (6) using (4) and (5) yields ⎛ I ⎞ ⎛ 2β I 0 ⎞ ⎛ 2β I 0 2 ⎞ f = β ⎜ 0 ⎟+⎜ Δ i − Δh ⎜ 2 ⎟ 3 ⎟ ⎝ H0 ⎠ ⎝ H0 ⎠ ⎝ H0 ⎠

(6)

(7)

Where, I0 is the equilibrium value. At equilibrium, the weight of the object is suspended by the electromagnet force, f0. The force required to maintain equilibrium, f1, is f1 = f − f 0 (8) Combining equations (7) and (8) gives ⎛ 2β I 0 ⎞ ⎛ 2β I 02 ⎞ Δh f1 = ⎜ Δ i − ⎟ ⎜ 2 3 ⎟ ⎝ H0 ⎠ ⎝ H0 ⎠ The voltage equation of the electromagnetic coil is given in equation 1.

V = iR + L(h)

di dt

(9)

(10)

Assuming the suspended object remains close to its equilibrium position, h=h0, and therefore

L(h) = L1 + L0

(11)

Also assuming that L1 >> L0, equation (10) can be simplified as

9


V = iR + L1

di dt

(12)

The principal equation for the suspended object comes by applying Newton‘s second law of motion. For this one degree of freedom system, a force balance taken at the centre of gravity of the object yields d 2h = − f1 dt 2 The sensor can be modelled as a gain element, M

Vs = K s h

(13)

(14)

Where Vs is the sensor output voltage and Ks is an experimental gain between the object‘s position and the output voltage. −M

d 2 h ⎛ 2β I 0 ⎞ ⎛ 2β I 0 2 ⎞ =⎜ ⎟i − ⎜ ⎟h dt 2 ⎝ H 0 2 ⎠ ⎝ H 03 ⎠ −M

Where,

K1 =

(15)

d 2h = K1i − K 2 h dt 2

2β I 0 H 02

and

K2 =

(16)

2β I 0 2 H 03

The Laplace transform of above equation obtained as:

(M s2 − K2 )H(s) = −K1 I(s)

(17)

The Laplace transform of equation (28) is I (s) =

V (s) Ls+R

(18)

The overall transfer function of the Maglev system is obtained as: K s K1 V (s) ML1 G (s) = s = V ( s) ⎛ R ⎞ ⎛ 2 K2 ⎞ ⎜ s + ⎟⎜ s − ⎟ L1 ⎠ ⎝ M ⎠ ⎝ −

(19)

Table 1 summarizes the variables and parameters use in this problem. Here the problem is to maintain the ball at its operating point (position) of 0.03 meters from the coil.

10


Table1. Parameters of the Magnetic Levitation system Parameters

Description

Values

H0 M R L1 β I0 Ks K1 K2

Equilibrium height of ball (m) Mass of ball bearing (kg) Resistance (Ω) Inductor (H) Constant related to magnetic force (Nm2/A2) Equilibrium current of the coil (A) Sensor gain factor (V/m) constant (N/A) constant (N/m)

0.03 0.225 2.48 0.18 7.93×10-5 5 200 0.882 147

Fig. 2.5: Block diagram of Magnetic levitation system

11

Chapter-3

Fuzzy Logic Fuzzy logic is a logical system providing a mathematical framework to capture the uncertainties associated with human cognitive systems such as thinking and reasoning. Simply, it simulates human thinking which operates more likely on symbols than exact values. In fact, our daily thoughts and communication are full of these symbols or fuzzy expressions. This chapter gives a brief introduction to the main concepts of fuzzy logic.

3.1

Fuzzy sets

In conventional set theory, an element either belongs to the set or not. Fuzzy logic is a generalization of the conventional logic. In fuzzy set theory, the element can belong to the set partially with a certain degree. The difference between conventional crisp and fuzzy sets is illustrated in Fig. 3.1. Let us consider tree example sets the poors, the averages and the richs in the universe of discourse ‗wealth‘. In conventional logic, persons are divided into these three groups crisply. The fuzzy sets have no crisp boundaries, but a person can simultaneously be a member of several groups with different degrees. For example, a person can be rich with degree of 0.1 and average with 0.7.

Fig. 3.1: Conventional sets and fuzzy sets. 3.1.1 Linguistic variables The main advantage of fuzzy logic is that words or sentences can be used as expressions instead of numeric values. The associative expressions are called linguistic variables. They are common in our daily life. Let us consider the fuzzy variable velocity. It can be, for example, divided into three linguistic variables: slow, medium, and fast which are fuzzy sets, as show in Fig. 3.2. Each linguistic variable is represented by membership function in the universe of discourse. For example, the membership function of the linguistic variable slow could be defined by

12

Fuzzy Logic

1,   µ slow (v) =   v-----–-----3---5-   1 –  25  ,

v ∈ [0, 35] , v ∈ [35, 60] .

(20)

The membership function values can vary between zero and unity and they can have many shapes, as shown in Fig. 3.3. The selection of the shape for a fuzzy set is subjective and particular rules do not exist, but the singleton type of membership function (fuzzy unit set) is usually employed only for the output variables of the fuzzy reasoning.

Fig. 3.2: Terms of fuzzy logic.

Fig. 3.3: Different shapes of membership functions. (a) Z-shaped, (b) trapezoidal, (c) bell-shaped, (d) triangular and (e) singleton.

3.2

Fuzzy control

Fuzzy control is useful in some cases where the control processes are too complex to analyse by conventional quantitative techniques. Fuzzy control design is very interesting for industrial processes where modelling is not easy to make or conception of nonlinear controllers for industrial processes with models. The available sources of information of a process are interpreted qualitatively, inexactly or uncertainly. The main advantages of fuzzy logic control remains in [1]: • Parallel or distributed multiple fuzzy rules –complex nonlinear • Linguistic control, linguistic terms –human knowledge • Robust control 3.2.1 Fuzzy control system design Fig. 3.4 gives the fuzzy controller block diagram, where we show a fuzzy controller embedded in a closed-loop control system. The plant outputs are denoted by y(t), its 13

Fuzzy Logic

inputs are denoted by u(t), and the reference input to the fuzzy controller is denoted by r(t). The design of fuzzy logic controller is based on four main components [1]: 1. 2. 3. 4.

The fuzzification interface which transforms input crisp values to fuzzy values The knowledge base which contains knowledge of the application domain and the control objectives The decision-making logic which performs inference for fuzzy control actions The defuzzification interface which provides the control signal to the process.

Fig. 3.4: Fuzzy controller diagram

3.3

Fuzzy reasoning

3.3.1

Fuzzy rules

Fuzzy reasoning is usually performed using if -then rules. The fuzzy rules define the connection between input and output fuzzy (linguistic) variables. The rule consists of two parts: an antecedent and a consequence part. The Inference block is used to link the input variables to the output variable denoted XR and considered as a linguistic variable given by a set of rules:

XR = (IF (condition 1), THEN (consequence 1) IF (condition 2), THEN (consequence 2) ………………………………………… ………………………………………… ………………………………………… OR IF (condition n), THEN (consequence n).

OR OR

n corresponds to the product of the number of membership functions of each input variable of the fuzzy logic controller. In these rules, the fuzzy operators AND, OR link the input variables in the ―condition‖ while the fuzzy operator OR links the different rules. The choice of these operators for inference depends obviously on the static and dynamic behaviours of the 14

Fuzzy Logic

system to control. The numerical processing of the inference is carried out by three methods [22]: 1. 2. 3.

max-prod inference method max-min inference method sum-prod inference method

3.3.2 Fuzzy inference system The fuzzy inference system (FIS) performs fuzzy reasoning. The basic FIS is composed of five functional blocks, as depicted in Fig. 3.5. The knowledge base consists of the data base and the rule base. The fuzzy sets are defined in the data base and fuzzy rules in the rule base. The decision-making unit executes fuzzy reasoning rules taking fuzzified inputs of FIS as inputs and delivering the fuzzy result to the defuzzifier, which produces the output of the FIS.

Fig. 3.5: The structure of the fuzzy logic inference system. The operation of the FIS is illustrated in Fig. 3.7. First, crisp inputs x and y are fed into a FIS. In the second stage, they are fuzzified. After that, the fuzzified inputs are combined according to the fuzzy rules in the knowledge base. Finally, the results of all rules are combined and defuzzified. In the following, each stage is described in more detail. 3.3.3 Fuzzification In the fuzzification, the crisp input values are transformed to fuzzy values. If the input has a crisp value, the matching against the membership function of linguistic variable is shown in Fig. 3.6(a). If the input contains noise, it can be modeled by using a fuzzy input value. In this case the fuzzy output is the intersection of fuzzy input and the linguistic variable membership functions as shown in Fig. 3.6(b). However, the crisp input value fuzzification is mostly used because of its simplicity. The fuzzification block contains generally preliminary data which are obtained from: • •

Conversion of measured variables with analog/digital converters. Preprocessing of the measured variables in order to get the state, error, state error derivation and state error integral of the variables to control (output variables or other state variables).

15

Fuzzy Logic

• Choice of membership functions for the input and output variables namely the shape, the number and distribution. Usually three to five triangular or Gaussian membership functions are used with a uniform distribution presenting 50% of overlapping. More than seven membership functions, the algorithm processing becomes long and presents a drawback for fast industrial processes.

Fig. 3.6: Fuzzification of a crisp input (on the left) and a fuzzy input (on the right). 3.3.4 Inference The decision making unit performs the inference operations on the fuzzy rules. The fuzzy values within a fuzzy rule are aggregated with connective operators like intersection (AND), union (OR) and complement (NOT). Due to the use of the multivalued logic, the connective operators for fuzzy logic differ from the ones used in the Boolean logic. The operators can be defined in several ways, but the following are the best-established ones [23]:

Fig. 3.7: The fuzzy inference using the Min-inference. Intersection AND (µ A, µ B ) = min {µ A, µ B} ,

(21)

AND (µ A, µ B ) = µ A µ B ,

(22) 16

Fuzzy Logic

Union OR (µ A, µ B ) = max {µ A, µ B } ,

(23)

Complement NOT (µ A ) = 1 – µ A ,

(24)

Where µA and µB are membership values which are combined by operators. The firing strengths of the fuzzy rules are computed by employing above operators. The operation of the intersection is shown in Fig. 3.7. The final output fuzzy sets are obtained either scaling (Max-Dot method) or cutting (Max-Min) according to the firing strength of the fuzzy rules. If the output fuzzy sets are singletons, they are not handled by the firing strengths in this stage. 3.3.5

Defuzzification

In the defuzzification stage, the outputs of the fuzzy rules are combined to a crisp output value. Several defuzzification strategies have been suggested [24]. The most common method is the center of area (COA) defuzzification strategy, illustrated in Fig. 3.8. Assuming a discrete universe of discount, the crisp output Z is produced by searching the center of gravity of consequence fuzzy sets according to m

∑ µC (zi )

zi

Z = i----=----m0------------------------- ,

(25)

∑ µi (zi )

i=0

where m is the number of quantization levels of the output, zi is the amount of output at the quantization level i, and µi(zi) represents its membership value in C. If only singletons are used as the consequences of fuzzy rules, the natural defuzzification method is the weighted average (WA). It can be considered as a special case of COA defuzzification method is the weighted average (WA). The WA method combines the consequences of the fuzzy rules to the output of the inference system z according to n

∑ µi zi =0 Z = i-------------------n

(26)

∑ µi

i=0

where n is number of fuzzy rules, µi is the firing strength of the rule, and is the output value of the ith singleton.

17

Fuzzy Logic

Fig. 3.8: Defuzzification methods On the left, three fuzzy rules which have singleton output fire. The output is computed by using weighted average strategy. On the right, two fuzzy rules fires. The crisp output is the centre of the area.

3.4

Different types of Fuzzy Logic controllers

On the basis of the consequence of rules given above, different types of fuzzy logic controllers are presented. 1. If the consequence is a membership function or a fuzzy set, the fuzzy controller is Mamdani type. In this case, the processing of inference uses often the max-min or max- prod inference method while for defuzzification, the center of gravity method is often used and in some cases we use the maximum value method if fast control is needed. 2. If the consequence is a linear combination of the input variables of the fuzzy logic controller. Indeed each rule corresponds to a local linear controller around a steady state. Consequently, the set of the established rules correspond to a nonlinear controller. In this case, we use max-min or max-prod inference method and for defuzzification, we often use the weight average method. Also, there exist other types of fuzzy logic controllers such as Larsen or Tsukamoto [25]. Most of time Mamdani and TSK controllers are used in the design of controllers for nonlinear systems with or without models [22]. The advantages of the design of a fuzzy logic controller using Mamdani type are an intuitive method, used at a big scale and well suited for translation of human experience on linguistic rules. On the other hand, the advantages of a fuzzy logic controller using a Takagi-Sugeno type are: • • •

Good operation with linear techniques (the consequence of a rule is linear) Good operation with optimization techniques and parameters adaptation of a controller Continuous transfer characteristics very suited for systems with a model fast processing of information

18

Fuzzy Logic

3.5

Discussion

The different steps followed in the processing of the input variables of the fuzzy logic controller namely fuzzification, inference and defuzzification, allows obtaining a nonlinear characteristic. Indeed it‘s an advantage when compared to the classical control. The nonlinearity of this characteristic depends on some parameters. For example the number, the type and the distribution of membership functions. Also, other parameters can be considered such as the number of rules and inference methods. Finally, the nonlinearity can be more or less pronounced depending on all these parameters. In this case we consider the fuzzy logic controller as a nonlinear controller. Another possibility to get a nonlinear controller is to design and to add a fuzzy supervision to a PID controller. Industrialists are motivated to keep PID controllers which are well known and to add a fuzzy supervisor which modifies on-line PID parameters in order to reach and to maintain high performances whatever the parameters change and operations conditions maybe. In the design of the fuzzy supervision, the outputs are the PID parameters to provide on-line to the PID controller.

19

Chapter-4

Fuzzy Controller 4.1

Fuzzy supervisory control

Fuzzy Supervisory controller is a multilayer (hierarchical) controller with the supervisor at the highest level, as shown in Figure 4.1. The fuzzy supervisor can use any available data from the control system to characterize the system‘s current behavior so that it knows how to change the controller and ultimately achieve the desired specifications. In addition, the supervisor can be used to integrate other information into the control decision-making process.

Fig. 4.1: Fuzzy Supervisory controller Conceptually, the design of the supervisory controller can then proceed in the same manner as it did for direct fuzzy controllers (fuzzification, inference and defuzzification): either via the gathering of heuristic control knowledge or via training data that we gather from an experiment. The form of the knowledge or data is, however, somewhat different than in the simple fuzzy control problem. For instance, the type of heuristic knowledge that is used in a supervisor may take one of the following two forms: 1. Information from a human control system operator who observes the behavior of an existing control system (often a conventional control system) and knows how this controller should be tuned under various operating conditions. 2. Information gathered by a control engineer who knows that under different operating conditions controller parameters should be tuned according to certain rules. Fuzzy supervisor is characterized by: 1.

2.

The outputs which are not control signals to provide to the control system but they are parameters to provide to the controller in order to compute the appropriate control. Fuzzy supervision associated to the controller can be considered as an adaptive controller 20

Fuzzy Controller

3.

Fuzzy supervisor can integrate different types of information to resolve problems of control.

4.2

Supervision of conventional controllers

Most controllers in operation today have been developed using conventional control methods. There are, however, many situations where these controllers are not properly tuned and there is heuristic knowledge available on how to tune them while they are in operation. There is then the opportunity to utilize fuzzy control methods as the supervisor that tunes or coordinates the application of conventional controllers. In this part, supervision of conventional controllers concerns only PID controllers and how the supervisor can act as a gain scheduler.

4.3

Fuzzy tuning of PID controllers

Over 90% of the controllers in operation today are PID controllers. This is because PID controllers are easy to understand, easy to explain to others, and easy to implement. Moreover, they are often available at little extra cost since they are often incorporated into the programmable logic controllers (PLCs) that are used to control many industrial processes. Unfortunately, many of the PID loops that are in operation today are in continual need of monitoring and adjustment since they can easily become improperly tuned. Because PID controllers are often not properly tuned (e.g., due to plant parameter variations or operating condition changes), there is a significant need to develop methods for the automatic tuning of PID controllers for nonlinear systems where the model is not well known. In this method, the fuzzy supervisor knows, from a response time, when the controller is not well tuned and acts by adjusting the controller gains in order to improve system performances. The principle scheme of the fuzzy PID auto tuner [1] is given by Fig. 4.2.

Fig. 4.2: Fuzzy PID auto-tuner

21

Fuzzy Controller

The block ―Behavior Recognition‖ is used to characterize and analyze the current response of the system and provides information to the ―PID Designer‖ in order to determine the new parameters of the PID controllers and to improve performances. The basic form of a PID controller is given by: u(t)=KP e(t)+KI e(τ) dτ+KD d e(t) dt 0 t

(27)

Where u is the control signal provided by the PID controller to the plant. e is the error deuced from the reference input r and the plant output y. Kp is the proportional gain, Ki is the integral gain, and Kd is the derivative gain. In this case, the adjustment of PID parameters is carried out by some candidate rules as follows • If steady-state error is large Then increase the proportional gain. • If the response is oscillatory Then increase the derivative gain. • If the response is sluggish Then increase the proportional gain. • If the steady-state error is too big Then adjust the integral gain. • If the overshoot is too big Then decrease the proportional gain. In these rules conditions are deal with the block ―Behavior Recognition" and consequences are evaluated by the block ―PID Designer‖ of the fuzzy supervisor. In some applications controller gains are quantified according to different types of responses a priori identified from experiments on the real process and implemented on the block ―Behavior Recognition‖ [1].

4.4

Fuzzy gain scheduling

Conventional gain scheduling involves using extra information from the plant, environment, or users to tune (via ―schedules‖) the gains of a controller. The overall scheme is shown in Figure 4.3. A gain schedule is simply an interpolator that takes as inputs the operating condition and provides values of the gains as its outputs. One way to construct this interpolator is to view the data associations between operating conditions and controller gains.

Fig. 4.3: Conventional fuzzy gain scheduler

22

Fuzzy Controller

The controller gains are established on the basis of information collected from the plant to control, the operator or the environment. Three approaches are proposed for the construction of the fuzzy gain scheduling [1]: • • •

Heuristic Gain Schedule Construction Construction of gain schedule by fuzzy identification Construction of gain schedule using the PDC Distributed Compensation method)

method

(Parallel

4.4.1 Construction of a heuristic schedule gain This method is applied for plants with specific particularities not involved in the design of classical controllers. The PID parameters are deduced intuitively and the rules used for the adjustment of parameters are heuristic. This is for example the case of a tank with an oval shape (Fig. 4.4). In the heuristic rules, the condition corresponds to the water levels and the consequence corresponds to the values of the controller gain [1]. Each rule covers a set of water levels taking into account the tank section. For low levels, the gain is higher in order to get high flow rates and for high water levels, the gain is small in order to get small flow rates. This approach is very useful for systems without models.

Fig. 4.4: Tank 4.4.2 Construction of a schedule gain by fuzzy identification This approach is useful for plants where we know a priori how to adjust the controller gains under different operation conditions [1]. For example if a control engineer knows how to adjust gain controller according to certain rules, he can represent this data by a fuzzy model of Mamdani or TSK type. Indeed it‘s the equivalent of a set of controllers which are active in terms of the operation points. Also, the gain controllers are deduced on-line by the inference mechanism between controllers for any operation point. Indeed it‘s a soft transition from controller to another one. 4.4.3 Construction of a gain schedule using the PDC method This approach is applied particularly for processes that can be modelled. Most of time, the established models are nonlinear. In this case, the nonlinear model is replaced by a sum of linearized models around different operation points [1]&[26]. For each 23

Fuzzy Controller

linearized model, a linear controller is designed (Fig. 4.5). These linear controllers could be PI, PD or PID state controllers. The set of the designed controllers is finally a nonlinear controller which is a fuzzy controller. In this approach the n linearized models and the n corresponding controllers are rules which are active simultaneously two by two since the condition is similar for both, thus the name of the method "Parallel Distributed Compensation‖. In all the approaches presented above, performances are not used directly when designing controllers. Also non linearity, disturbances and variation parameters of the plant are not taken into account in the systems with models. In some cases stability is not ensured particularly when a change set point occurs or when a disturbance is present. In the case of the PDC approach, local and global stabilities are checked using Lyapunov theory [1]. For the other approaches, stability is checked when implementing fuzzy supervision for classical controllers.

Fig. 4.5: PDC concept Usually specifications and performances in closed loop are a priori defined. Therefore, it‘s more interesting to use them and to design controllers ensuring stability and same performances in closed loop whatever the operation conditions maybe.

24

Chapter-5

Tuned PID Controller 5.1

Overview

Tuning of PID controllers has always been an area of active interest in the process control industry. Ziegler Nichols Method (ZN) is one of the best conventional methods of tuning. Though ZN tunes systems very optimally, a better performance is needed for very fine response and this is obtained by using Fuzzy Logic (FL) methodology which is highly effective. 5.1.1

Tuning and its Purpose

A PID may have to be tuned when:  Careful consideration was not given to the units of gains and other parameters.  The process dynamics were not well-understood when the gains were first set, or the dynamics have (for any reason) changed.  Some characteristics of the control system are direction-dependent (e.g. actuator piston area, heat-up/cool-down of powerful heaters).  You (as designer or operator) think the controllers can perform better. Always remember to check the hardware first because there are many conditions under which the PID may not have to be tuned. These conditions are when:  A control valve sticks. Valves must be able to respond to commands.  A control valve is stripped out from high-pressure flow where the valve‘s response to a command must have some effect on the system.  Measurement taps are plugged, or sensors are disconnected. Bad measurements may have you correcting for errors that don‘t exist. Once fix these hardware problems then depending on the responses we obtain an appropriate decision can be taken whether or not to tune a PID controller. [27]

5.2

Trial and error method

This process is a very time consuming process as a lot of permutations and combinations are involved. Though much iteration is performed the final result is not satisfactory. A balance is not obtained between the rise time and % overshoot even though a lot of possible combinations of the gains are incorporated. Continuous cycling may be objectionable because the process is pushed to the stability limit. Consequently, if external disturbances or a change in the process occurs during controller tuning, it results in unstable operation. The tuning process is not applicable to processes that are open loop unstable because such processes typically are unstable at high and low values of Kc but are stable at an intermediate range of values. It can be observed in Fig. 5.1 that large overshoot is obtained as the program is written for faster rise time hence compromising with overshoot. All the time response specifications cannot be balanced using trial and error method. 25

Tuned PID Controller

Fig. 5.1: Unit step response of the system G(s) tuned with trial and error method.

5.3

Pole placement method

If the process is described by a low-order transfer function, a complete pole placement design can be performed. Consider for example the process described by the secondorder model.

G (s ) =

KP (1 + sT1 )(1 + sT2 )

(28)

This model has three parameters. By using a PID controller, which also has three parameters, it is possible to arbitrarily place the three poles of the closed loop system. The transfer function of the PID controller in parallel form can be written as GC (s) =

(

K 1 + sTI + s 2TI TD sTI

)

(29)

The characteristic equation of the closed loop system becomes ⎛ 1 1 K KT ⎞ ⎛ 1 K K⎞ K K s3 +s2⎜⎜ + + P D ⎟⎟+s⎜⎜ + P ⎟⎟+ P =0 1 2 ⎠ ⎝TT 1 2 TT 1 2 ⎠ TTT I 1 2 ⎝T1 T2 TT

(30)

A suitable closed-loop characteristic equation of a third-order system is

(s +αω )(s2 + 2ζωs +ω 2 ) = 0

(31)

which contains two dominant poles with relative damping (ζ) and frequency (ω), and a real pole at –αω. Identifying the coefficients in these two characteristic equations determines the PID parameters K, TI and TD. The solution is T1T2ω 2 (1 + 2ζα ) − 1 K= KP

(32)

2

26


T1T2ω (1 + 2ζα ) − 1 T1T2αω 3 T T ω (α + 2ζ ) − T1 − T2 TD = 1 2 T1T2ω 2 (1 + 2ζα ) − 1 TI =

(33) (34)

Fig. 5.2: Response of a system tuned with Pole Placement Method.

5.4

Ziegler Nichols method t ⎤ ⎡ de (t) 1 + ∫ e(τ )dτ ⎥ u(t) = Kp ⎢e(t)+Td dt Ti 0 ⎦ ⎣

(35)

Ziegler Nichols formula ensures good load disturbance attenuation, but it generally provides a poor phase margin and therefore it produces a large overshoot and settling time in the step-response. The overall control scheme for Ziegler Nichols Method is shown in Fig 5.3.

Fig. 5.3: Control Scheme for Ziegler Nichols Method.

27


Fig. 5.4: Unit step response of the system G(s) tuned with ZL method.

28

Chapter-6

Fuzzy Supervised PID-Controller The fuzzy logic controller for the given Maglev system is designed in the following sections. The various blocks related to this controller and rules governing the operation of the controller are also detailed in the subsequent sections.

6.1

Fuzzification

The success of this work, and the like, depends on how good this stage is conducted. In this stage, the crisp variables „e‟ and „de‟ are converted in to fuzzy variables ‗e’ and ‗de’ respectively. The membership functions associated to the control variables have been chosen with triangular shapes as shown in Fig. 6.1 - Fig. 6.5. The universe of discourse of all the input and output Variables are established as the suitable scaling factors are chosen to brought the input and output variables to this universe of discourse. Each universe of discourse is divided into seven overlapping fuzzy sets: NL (Negative Large), NM (Negative Medium), NS (Negative Small), ZE (Zero), PS (Positive Small), PM (positive Medium), and PL (Positive Large). Each fuzzy variable is a member of the subsets with a degree of membership μ varying between 0 (nonmember) and 1 (full-member). All the membership functions have asymmetrical shape with more crowding near the origin (steady state). This permits higher precision at Steady state [27].

Fig. 6.1: Triangular Membership functions of input variable ‗error‘.

Fig. 6.2: Triangular Membership functions of input variable ‗error-rate‘. 29

Fuzzy Supervised PID-Controller

6.2

Inference engine

Knowledge base involves defining the rules represented as IF-THEN statements governing the relationship between input and output variables in terms of membership functions. In this stage, the variables ‗e’ and ‗de’ are processed by an inference engine that executes 25 rules (5x5) as shown in Table 2. These rules are established using the knowledge of the system behavior and the experience of the control engineers. Each rule is expressed in the form as in the following example: IF {e is Negative Large} AND {de is Positive Large} THEN {*fuz is Zero}. Different inference engines can be used to produce the fuzzy set values for the output fuzzy variable ‗*fuz’. In this work, the Max-product inference method [27] has been used. Table 2: Rule base Parameters for KP. de

NL

NS

ZE

PS

PL

PVL PML PMS PML PVL

PVL PML PS PML PVL

PVL PML PVS PML PVL

PVL PL PMS PL PVL

PVL PVL PMS PVL PVL

e NL NS ZE PS PL

Fig. 6.3: Triangular Membership functions of output variable ‗KP‘. Table 3: Rule base Parameters for KI. de

NL

NS

ZE

PS

PL

PM PMS PS PMS PM

PM PMS PS PMS PM

PM PMS PVS PMS PM

PM PMS PS PMS PM

PM PMS PS PMS PM

e NL NS ZE PS PL

30


Fig. 6.4: Triangular Membership functions of output variable ‗KI‘. Table 4: Rule base Parameters for KD De

NL

NS

ZE

PS

PL

PVS PMS PS PML PVL

PMS PML PMS PVL PVL

PM PL PVS PVL PVL

PL PVL PL PVL PVL

PVL PVL PVL PVL PVL

e NL NS ZE PS PL

Fig. 6.5: Triangular Membership functions of output variable ‗KD‘.

6.3

Rule Base

A decision making logic which is stimulating a human decision process, infers fuzzy control action from the knowledge of control rules and linguistic variable definitions. The rules are “If-Then” format and formally the If side is called the condition and the Then side is called conclusion. The computer is able to execute the rules and compute the control signals depending on the measured input error (e) and change in error (de). In a rule based controller the control strategy is stored in more or less natural language. A rule base controller is easy to understand and easy to maintain for a nonspecialist end user and equivalent controller could be implemented using control techniques.

31


Fig. 6.6: Fuzzy Logic Rule-Base in SIMULINK®.

Fig. 6.7: Surface plot of Fuzzy Logic Rule-Base for variable KP.

32


Fig. 6.8: Surface plot of Fuzzy Logic Rule-Base for variable KI.

Fig. 6.9: Surface plot of Fuzzy Logic Rule-Base for variable KD.

6.4

Defuzzification

The reverse of fuzzification is called as defuzzification. The use of Fuzzy Logic Controller (FLC) produces output in linguistic variables (Fuzzy number). According to real world requirements, the linguistic variables have to be transformed to crisp output.

νCOGS =

P i µC (xi ).xi P i µG (xi )

(36)

In order to define fuzzy membership function, designers choose many different shapes based on their preference and experience. There are generally four types of membership functions used: 33


   

Trapezoidal MF Triangular MF Gaussian MF Generalized bell MF

Fig. 6.10: Structure of fuzzy logic controller. Implementation of an FLC requires the choice of four key factors: I. Number of fuzzy sets that constitute linguistic variables. II. Mapping of the measurements onto the support sets. III. Control protocol that determines the controller behavior. IV. Shape of membership functions. PID parameters fuzzy self-tuning is to find the fuzzy relationship between the three parameters of PID and "e" and "de", and according to the principle of fuzzy control, to modify the three parameters in order to meet different requirements for control parameters when "e" and "de" are different, and to make the control object a good dynamic and static performance [29].

6.5

Adjusting fuzzy membership functions and rules

In order to improve the performance of FLC, the rules and membership functions are adjusted. The membership functions are adjusted by making the area of membership functions near ZE region narrower to produce finer control resolution. On the other hand, making the area far from ZE region wider gives faster control response. Also the performance can be improved by changing the severity of rules. An experiment to study the effect of rise time (Tr), maximum overshoot (Mp) and steady-state error (SSE) when varying KP, KI and KD was conducted. The results of the experiment were used to develop 25-rules for the FLC of KP, KI and KD are the output variables and from error and change of error are the input variables. Triangular membership functions are selected.

34


Fig. 6.11: Block diagram of self-tuning Fuzzy Supervised PID controller.

6.6

Results and Discussion

Fig. 6.12: SIMULINK® model of self-tuning Fuzzy Supervised PID controller. Fig.6.12 shows the SIMULINK® model of the Maglev system in addition to the Fuzzy tuned PID controller. The results shows that the controlled electromagnet current can stabilize the disturbances that otherwise, would cause the ball to either fall or attach itself to the electromagnet. From the simulation results shown in Fig. 6.14, it can be observed that the fuzzy controller has better transient response than the classical controller. The overshoot of the FLC controller is 6% compared to 18% in the classical case. Furthermore, FLC has a faster transient response; it reaches to steady state in 0.9 sec to that of 2 sec in classical PID. In comparison to the steady state value, both controllers satisfactorily attain the steady state value. According to the different responses, it‘s clear that performances are better with the fuzzy control (Fuzzy supervised PID controller) in comparison to the classical control. Indeed, the designed Fuzzy PID supervisor provides on-line the PID parameters KP, KI, and KD allowing to reach the desired position with good performances and we consider the performances better because the PID parameters vary in order to control the ball position very quickly and without overshoot.

35


Fig. 6.13: Error rate (de) v/s time response of Fuzzy tuned PID controlled Maglev System model.

Fig. 6.14: Output v/s time response of Fuzzy tuned PID controlled Maglev System model.

36


Fig. 6.15: Error (e) v/s time response of Fuzzy tuned PID controlled Maglev System model.

37

Chapter-7

Conclusion In the control of a nonlinear process, classical control is robust but not optimal for the complete range of operation conditions. Indeed, the design of one controller is not sufficient to ensure good performances and stability for all the operation set points. Also, the variation of physical parameters of a process over time affects the performances. Therefore a continuous adjustment of controller gains is necessary to improve and eventually to maintain performances. On the basis of the proposed approach, performances are used a priori in the design of the fuzzy PID supervision taking into account the variation of parameters and operation conditions. Indeed in terms of both, the designed fuzzy PID supervision provides on-line the appropriate gains to the PID controllers ensuring the same performances whatever the operation conditions maybe. From the simulation results, it has been shown that the fuzzy controller can stabilize the system efficiently. Also the performance during the transient period of the fuzzy system is better in the sense that less overshoot was obtained. Moreover, the fuzzy controller provides a zero steady state error. Based on the simulation results it is concluded that the Fuzzy Logic supervision based controller can stabilize the system efficiently and accurately compared to a classical PID controller. Further work in this direction could be the analysis of the Maglev system and design of a controller using more advanced analysis techniques like GA, Ant Colony algorithm, Neuro-Fuzzy algorithm and ultimately practical implementation of the designed controller onto the magnetic levitation system.

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