Multivariable Servomechanism Controller Design of Web ... - TSpace

Multivariable Servomechanism Controller Design of Web Handling Systems

Weixuan Liu

A thesis submitted in conformity with the requirements for the Degree of Master of Applied Science

Graduate Department of Electncal and Computer Engineering University of Toronto

@Copyright by Weixuan Liu, 2000

The author has grrmted a nonexclusive licence ailowing the National Library of Canada to reproduce, loan, distn'bute or sel1 copies of this thesis in microform, paper or eiectronic forma~s.

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Multivariable Servomechanism Controuer Design of Web Handliag Systems

M.A.Sc., 2000 Weixuan Liu Graduate Department of Electrical and Cornputer Engineering University of Toronto Abstract

In traditional web handling processes, web tension and speed are controiled assuming that the web system consists of a number of single input and single output systems.

This assumption often r d t s in large interactions occurring in the closed loop system between the control loops, and hence results in high quality control being difficult to achieve.

In this thesis, the control of the web handiing processes is treated as a multivariable servomechanism problem. Three types of controller designs-the "cheap control senromechanism controlie?' , the "high gain servomechanism controller" and the "tuning regulator" are studied and implemented on the University of Toronto web machine.

The experimental results obtained show that these controllers provide

excellent tension and speed response.

I am trdy gratefbi to my supervisor, Profeesor Edward J. Davison for his patient guidance and financid support throughout my study and research in the System Control Group. 1am in debt to my farnilytwho have done ao much to make my dreams corne true.

Without their love and encouragement, L would achieve Little.

Finaily, 1have to thank my colleagues in the System Control Group for their help

and friendships. Special thanks go to Chuan Ma, Gaby Saad and Ken Pu, who have treated me as a family member and made my life enjoyable.

Contents 1 Introduction

2

Background Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2 The Rotoflex Web Handling Machine

3

1.3 Outline of the Thesis

.................. ...........................

4

1.1

2 Modeling For W e b Tension Control

2.1 2.2

................................ Modeling for the Web Mechanicd System . . . . . . . . . . . . . . . 2.2.1 The Mode1 of a Fkee Web . . . . . . . . . . . . . . . . . . . . 2.2.2 ModelS of Roiiers . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Models of Web Mechanical Systems . . . . . . . . . . . . . . . Introduction

3 The Servomechanism Control Problem 3.1 3.2

................... The Servo-cornpensator . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The ControlIer Structure . . . . . . . . . . . . . . . . . . . . . 3.2.2 StabiIizing Controller Design . . . . . . . . . . . . . . . . . . . Robust Servomechanism Problems

4 The Perfect Control Problem

4.1 4.2

6

6

7 7

10 12 25

25

27 27 30

31

................................ The Perfect Contsol Problem . . . . . . . . . . . . . . . . . . . . . .

31

................... ....................

Introduction

31

4.2.1

Development of the Problam

31

4.2.2

Design for Perféct Control

33

CONTENTS 5 A High Gain Stabwing controller

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2 Mathematical Preliminaries 5.2.1 lkansrnission Zeros 5.2.2

....................... ........................ .............. .....................

'hmnission Zero at Infinity[Mac89]

5.2.3 Minimum Phase System

5.2.4 Hurwitz Polynomial . . . . . . . . . . . . . . . . . . . . . . . .

....................... ...........................

5.3 The mdtivsriable root locus

5.4 Controller Synthesis

5.4.1 Approach 1-Decomposition

5.4.2

...................

Approach 2-Generalized Differential Interactor . . . . . . . . .

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Tuning Regulator Controller

7 Controller Impiementation

7.1 Introduction

................................

7.2 Models for Controller hplementations 7.2.1 A Reduced Order Model

.................

.....................

............... Cheap controller design . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Experimental Results-Controller #1 . . . . . . . . . . . . . . 7.2.2 Another Model for the Web System

7.3

7.3.2 Discussion of the Experimental h l t s Obtained Fkom Con-

............................. 7.3.3 Ekperimentd Redts-Controller #2 . . . . . . . . . . . . . . High Gain Controller Design . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 The Properties of the Web System . . . . . . . . . . . . . . . troller #1

7.4

7.4.2 Design of The Sattuation Controller . . . . . . . . . . . . . . . 7.4.3 Combining The High Gain Control Design With The Saturation

......................... Simulation and Experimental Results . . . . . . . . . . . . . .

Controller Design 7.4.4

CONTENTS 7.5

vi

nining Reguiator Design

82

7.5.1

......................... Preliminary Ekperiment . . . . . . . . . . . . . . . . . . . . .

83

7.5.2

Results Obtained Rom Experiments

..............

8 Conclusions

8.1

Dl

................... .............................

Main Contributions of This Thesis

8.2 E'urtherResearch

88

A Properties of the Saturation Contmller

91 92 96

................. The Response of the Saturation Controlier . . . . . . . . . . . . . . . A.2.1 Case 1-Direct State Feedback . . . . . . . . . . . . . . A.2.2 Using an observer . . . . . . . . . . . . . . . . . . . . . . . . .

A.1 Structure of The Saturation ControUer

96

A.2

97 97

103

B Models of the University of Toronto W e b Machine and Controllers 104 B.l Models of the University of Toronto Machine . . . . . . . . . . . . . . 104 B.l.l Mode1 #1: A Reduced Order Low Frequency Mode1 .

..... ...........

105

B.1.2 Mode1 #2: A Black Box 6th Order Mode1

105

5.2 ControUers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

...........

107

.............

110

B.2.1 Cheap Control Servomechanism Controller B.2.2 High Gain Servomechanism Conttoller

C Scripts For Controlier Designe C.1 Matlab Codes For Controller Design

113

..................

113

C.l.l The Script To Obtain the University of Toronto Web Machine

State Space Mode1

........................

C.1.2 The Script To Normaiize The Linear System And Add Filters

113 114

C.1.3 Scripts for Cheap Control Servomechanism Controller Design . 116 C.1.4 Scripts for High Gain Servomechanism Controller Design

...

120

C.1.5 The Script to Discretize the continuous t h e state space con-

....

124

................

127

troller and to Export the ControUer As A C Head File C.2 C Codes For Controllet Imp1enientations

CONTENTS

vii

C.2.i C Codes For The Implementation of the High Gain and Cheap

Control Servomechanism Controllers

. . . . . . .. . . . . . .

127

List of Tables 1

List of Notation

..............................

....................... Steady State Value Difference . . . . . . . . . . . . . . . . . . . . . .

1

7.1 List of Identified Parameters

52

7.2

83

List of Figures 1.1 Overview of University of Toronto Machine

...............

2.1 The overall electrical/rnechanical system of a web handling machine

.

............................. A Single Tension Span . . . . . . . . . . . . . . . . . . . . . . . . . . ARollerinaWebSystem . . . . . . . . . . . . . . . . . . . . . . . . AWinderinaWebSystem . . . . . . . . . . . . . . . . . . . . . . . An+lspansystem . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 A Fkee Span Web 2.3

2.4 2.5

2.6

2.7 Calculation of the inertia and radius of a winder

............

.............. A High Frequency Model Module . . . . . . . . . . . . . . . . . . . . High F'requency Model . . . . . . . . . . . . . . . . . . . . . . . . . .

2.8 Reduced System Is a First Order RC circuit 2.9

2.10

2.11 Block Diagram of the High Fkequency Model

..............

2.12 The derivation of the mode1 for the reduced order system 3.1 The general structure of a servomechanisrn controIIer 5.1 A Generic Multivariable Root Loci

......

.........

...................

7.1 The Illustrative Diagram of the Rotoflex Machine

...........

7.2 The Reduced Low Ftequency Model of the Rotoflex Machine

....

.

7.3 The w o n s e of the Open Ioop Real System vs the ldentified Model

7.4 Cornparison of The Response of the Black Box Model and The Real

Machindtewinder Torque Step

.....................

LIST

FIGURES

7.5 Cornparison of The Regponse of the Black Box Model and The Red

Maihine-Nip Torque Step

. .. . . . . . .. . .. ... . . .. .. . .

7.6 Compa~%onof The Response of the BIack Box Model and The &ai

. . . . . . . . .. . . . . . . . . .. The Web System Structure Mer S d n g . . . . . . . . . . . . . . . . MachineUnwinder Torque Step

7.7

7.8 The Structure of the Overall Controller

.. ... . . . . ... .. . .

7.9 Output Responses to Tension Fa Step Reference Input (controller #1)

7.10 Output Responses to Tension Fb Step Reference Input (controller #1)

7.11 Output Responses to velocity Step Reference Input (controller #1)

.

7.12 Cornparison of the closed loop responses of the experimentai results and the theoretical results of the model #1 (controller #1)

.....

7.13 The Simulated Closed Loop Response of Tension Fb for DifFerent O p erating Points (controller #l applied to model #1)

. ..... . . ..

7.14 The Closed Loop Expehental Response of Tension Fb for DiEerent

Operating Points (controller #1)

... . . . . .. . . . . . . . . . . .

7.15 The Closed Loop Simulated Response of Tension Fb for DifTerent Web

Thickness (controller #1 applied on model #1)

............

7.16 Experimentd Ebuits of Response To Tension Fa Step Input (Controller #2)

..... .... .. .. ... .. .......... ....

7.17 Experimental R.esults of Response To Tension Fb Step Input (Con-

trouer #2)

. . . . . . . . . . . . . . . . .. . ... . . . . . . . . . .

7.18 Experimentd Resdts of Response To Velocity Step Input (Controller #2)

.....*..............................

7.19 The Overd Structure of a High Gain Design - Saturation ControUer

........ .. 7.21 Simulated Output of the High Gain Controuer . . . . . . . . . . . . 7.22 Simulated Output of the Cheap Control Perfect Controller . . . . . . 7.23 The Input of the High Gain Controller . . . . . . . . . . . . . . . . . 7.24 The Input of the Cheap Contml Pedect Controller . . . . . . . . . . 7.25 Output Responses to Tension Fa Step Reference Input . . . . . . . . 7.20 Large Input Overshoot of the High Gain Controller

LIST OF FIGURES

xi

7.26 Output Responses to Tension Fb Step Reference Input

81

7.27 Output Responses to velocity Step Reference Input

........ ..........

82

.....

84

..... 7.30 Open Loop Response to Torque Step Input CW= [O O 90.31~ . . . . . 7.31 Tuning Reguiator Ekperiment: Responses to Fa step input . . . . . .

85

.....

89

...

90

7.28 Open Loop Response to Torque Step Input 6U = [0.6 O 0IT

7.29 Open Loop Response to Torqne Step Input 6U = [O 0.6 O]=

7.32 Tuning Regulator Experiment: Responses to Fb step input

7.33 W

g Regulator Expeciment: Responses to velocity step input

86 88

A.1 The Overd Structure of A High Gain Design With Saturation Con-

................................... Simplified Structure of figureA .1 . . . . . . . . . . . . . . . . . . . . .

troller

A.2

97 98

ml'OF FIGURES The following notation and expressions wiil be used throughout the thesis. Table 1: List of Notation

The set of real numbers. The set of positive real numbers. The set of complex nurnbers. The set of the n-dimensional real vector space. The set of the n-dimensional cornplex vector space. The set of the m x n real matrices. The set of the m x n complex matrices. The set of complex numbers with strictly negative real parts. The set of complex numbers lying on the closed right hand part of the complex pli The set of the eigenvalues of the square rnatrix M The field of rational functions with real coefficients The commutative ring of polynornials with real coefficients The field of rational function with complex coefncients The commutative ring of polynomials with complex coefficients The Euclidean n o m of a matrix

Chapter 1 Introduction 1.1

Background Knowledge

In industry, paper, plastic and other elastic thin materials are offen used in the manufacturing of commercial products by using a continuous process. In this case, the paper or other material is typicdy unrolled fiom a large roll using a series of rollers and a rewinder, forming what ia called a web.

To produce an end product fkom a raw web material, such as kom a paper machine or a film extruder, two Ends of processes are involved: Web Converthg and Web

Handlllig. Web converthg includes al1 those processes which are requiied to modify the physical properties of the web materid such as Coating, Slitting, Metalking, Drying and Embossing, etc. The web handling processes, on the other hand, consists of those processes which are associated with the transportation aspects of

the web. The main purpose of the web handling process is to transport web with

maximum throughput (speed)and with minimm damage[Roi98]. To achieve this, web tension contml is crucial because of the following reasons:

1. Web tension affects the geometry of the web, such as the apparent length and

width of the web. 2. High web tension prevents the loss of traction on the rollers; however too high

web tension will cause a web break to occur.

3. Web tension control helps to reduce wrinlüng. In paxticular, high process tension wili help decrease the wrinkling caused by a misaligrunent of rollers; however, excessively high tension will cause more wridding to occur on very thin materiais. Hence, appmpriate web tension control is very important. 4. Web tension affecfs the wound-in tension and the shape of the final product

roll, and hence the FOU quality.

For these reasons, it is essentiai in web handling to contrai the web tension at a desired value as closely as possible. Nomally, web tension should be set at 1045% of the

web's yield strength and should be kept within 10% of this value during the system's steady ninning state and 25% of this value at speed set-point changes [Roi98]. Almost ail the components of a web machine influence the web tension. We will now study the properties of the web machine which was useà in the experimental part

of this thesis at the University of Toronto, Department of Electrical and Cornputer Engineering, and is manufactureci by Rotoflex International Inc..

1.2 The Rotoflex Web Handling Machine As a typical small size web handling machine, the University of Toronto web machine

has al1 the necessary components associated with a web handling process. Rollers are essential parts of a web handling machine. In any machine, there are two type of rollers: (i) externdy torque dnven rollers such as the unwinder, rewinder and the nip roller and (ii) the web driven roilers (idem). These devices are &O called "transport rollers" in industry because they are not intended to change the physicai

properties of the web. The traditionai role of a "nipped rouer" is to step the tension up or down between sections of processes, and hence create different tension zones for

different processes. In designing a controlier for a web systern, the nipped rouer torque input and the wound rouer torque inputs (rewinder and unwinder) provide multiple

inputs for mdtivariable tension/speed control. These torque inputs are regulated by

PWM drives. The toque outputs fkom the PWM

drives can be either positive or

CHAPTER I . INTRODUCTION

Figure 1.1: O v e ~ e wof University of Toronto Machine negative, and hence can either act as "drives" or "brakes" in web control. Besides

the nip and winders, there are also some web driven roilem (idem), which provide

additional inertia to the web system. The tensions of the web system are measured by load c e k (which provide better

precision than the dancer system). To provide real-time monitoring of time varying information such as inertia of the unwinder and rewinder, there are two diameter sensors which measure the changing diameters of the unwinder and rewinder, in the

U of T web machine.

1.3 Outline of the Thesis The r a t of the thesis deab with the control of web handling systems. This thesis starts fkom the modeling of web handling systems; in ptvticuiar, in chapter 2, a reduced order Iow frequency linear model and a high fkequency model are developed for web system control design.

In chapters 3, the robust servomechanism is introduced, and the structure of the uservocompensator" is presented.

W T E R 1, INTRODUCTION

5

Chapters 4, 5 and 6 introduce the design of three s t a b ' i g controllers. Two different types of penect control design-a cheap control design and a high gain con-

troiler design are given in chapter 4 and 5 respectively, and in chapter 6, a "tuning regulator" controiier, which does not require a mathematical mode1 of the system is

presented.

AU three types of servomechanism controller design methodologies are implemented on the University of Toronto web handling machine, and the results are presented in chapter 7. Chapter 8 concludes the work in this thesis and presents some suggestion for future

reseaxch.

Chapter 2 Modeling For Web Tension Control Introduction To achieve satisfactory control of the web tension and velocity in a web machine, an approxiniate mathematical mode1 describing the physical web system will be obtained. The complete electrical/mechanicaI components of the web handling machine influences the control of the web tension; however the web mechanic part of the sys-

tem is the core part. It is also to be noted that the tension seneors introduce high frequency measmement noise to the whole web system, whiie the drive block places restrictions on the control signals (i.e. amplitude and rate thresholds of the output signais nom the control system.). We will concentrate on the modeling of the most important part of the web system: the web mechanical system.

Figure 2.1: The overaii electricai/mechanical system of a web handling machine

CHAPTER 2. MODE&ING FOR WEB TENSION CONTROL

2.2

7

Modeling for the W e b Mechanical System

The most influentid components of a web system are the roilem and web. We will begin by developing mathematicai models of these two prominent elements and then

deveiop a mode1 of the overd system.

2.2.1

The Mode1 of a Fkee W e b Front View

Left View

Figure 2.2: A Free Span Web

A strip of web under longitudinal stretch will experience strains in dl three directions MD, CD and ZD as shown in fig2.2:

where the subscript s represents the state of being stretched and the subscript O

represents the original unstretched state. The subscripts x, w , h represent the MD,

CD and ZD directions respectively. In the following paragraphs, the subscription x for the MD direction will be omitteà.

It follows, according to ( 2 4 , that the maap per unit length of the stretched web m,, and the mass per unit length of the unstretched web mo have the following

CHAPTER 2. MODEWG FOR WEB TENSION CONTROL relstionship:

where p and A denote the density (massper unit length) and the cross section area of the web span respectively. Here r denotes strain in the MD direction as describeci

in (2.1).

Figure 2.3: A Single Tension Span Now consider a one span web system. On applying the mass conservation law on

the web span in fig 2.3, i.e, the rate of the mass uicrease in the web span equals the rate of mass entering the web span niinus the rate of leaving this span, we obtain:

Under the assumption that the strain in the web is UfUformly distributecl, the strain in the web span in figure 2.3 is given by ci(%,t ) = q ( t ) ,which implies that p(x, t ) = h ( t )

and A(x, t ) = Ai(t) must also be m e .

CHIAPTER 2. MODELING FOR WFJ3 TENSION CONTROL This implies that:

On applying the result of (2.2) and (2.4) to (2.3), we then 0btai.n the foiiowing dynamic equation for the web

Aseuming the strain e is very s m d (< l),then:

On appiying (2.6) to (2.5), we obtain the dynamic equation of the web for the s m d strain mode1

However, our real interest in modeling is in the tension in the web span rather

than the strain. In th5 case, fiom mechanics, it is known that tension and strain are

a p p r d a t e i y related by Hooke's Iaw:

where T is the tension deveioped in the web, e is the strain of the web from the

unstretched state, A is the crosssection area of the web in the unstretched state, and

E, a constant, is the Young% modulus of the web. Assume now that the cross-section area of the web, when at unstretched state, does not change dong the web, and apply (2.8) to (2.7); we then obtain the dynamic

2.2.2

Models of Rollers

The roller is another very important cornponant in a web handling machine. The winders and the web itself provide the necessaxy tools to produce the Tension, Nip

and Torque (TNT) to produce a "goodn roll, which is the ultimate goal of web handling processing.

Figure 2.4: A Rouer in a Web System A roller in a web system is driven by the web tension and conesponding extemal

motor torque inputs. In a web machine, the overall system is designed for the web to interact with the roller in a state of either floating, sliding or tracking with a

d e r . Here we will only consider the case when the machine hm been designed to work in the tracking mode.

In this case, the speed of a roller in a web system such as represented in figure 2.4 should satipSr the following dynamic equation:

CHAPTER 2. MODEIIING FOR WEB TENSION CONTROL

11

where U is the external torque appiied by the motor and 3 is the fiction of the shaft,

Figure 2.5: A Winder in a Web System

The dynamic behavior of a winder (rewinder or unwinder) is somewhat more

complex because of the changing inertia (J) and radius (r) of the d e r . However, since (2.10) must still hold tme, the dynamic equation for a winder can be described by:

On simplifying?the dynamic equation for v ( t ) then becomes:

In the derivation of (2.12), the relationship:

where w ( t ) is the anguiar speed and e is the thickness of the web is used. The fiction torque 3 is generaily a noniixtear function of the angular velocity. As obserwd in the identification study of the Rotoflex machine [Bor99], the fiction force

can be chaxacterized by a quaciratic function or a function of the fom: 3 = (?)=+d, where a, e, c, d, are constant coefficients and w is the angular veiocity of the shaR.

However for practicai use, the foilowing Iinearized mode1 for the friction force is often

CHAPTER 2. MODEL1ZVG FOR WEB TENSION CONTROL enough:

where F0and b are constants.

2.2.3

Models of Web Mechanical Systems

Figure 2.6: A n + 1 span system

An overd model of the mechanical behavior of a web system, consisting of a unwinder, a rewinder and n non-winder rollers, can be directly obtained fiom the previous sub-component models obtained in sections 2.2.1 and 2.2.2 as following:

Here, if the first r o k (roller number 1) is the rewinder, then the last roller (rolier number n + 2) is the unwinder, and they have varying radii and inertia, given by the

CHAPTER 2. MODELRVG FOR 'WEB TENSION CONTROL foilowing equations:

where (see fig 2.7), the radius r and inertia J with a subscript O denotes the radius and inertia of the ah&, rl denotes the radius of the rewinder, Ji denotes the inertia of the rewinder, r,+l denotes the overall radius of the unwinder and JM denotes the overall inertia of the unwinder. Here VI to vn+l denote the iinear speeds of the n

+2

rollers, Fito F,+* denote the friction torques on the rollers, Uito Un+? denote the extemal motor torque applied to the rollers, Ti to Tn+I denote the tensions in the

n + 1 web spans and Tw is the "wound in tension" in the unwinder, which is stored in the roll by the previous processes.

Figure 2.7: Calculation of the inertia and radius of a winder

This system is nonlinear. In order

to apply linear multivariable control

design

methods to find a controller for thie system , we must linearize (2.15) and (2.16). However, direct linearization for a single equilibrium point is not feasible for this system because the states ri (t) and r,+r(t) do not have steady state operating state. Hence, we will carry out a partial lineaxbation, i.e. m will linearize the system on

CWAPTER 2. MODELING MIR WEB TENSION CONTROL the equiiibrium space defineà as:

where

- Ti -

pn+lDi

+

@,

. fin+* denote the steady state operating d u e s .

In this case, the hemlized mode1 becomes:

where the state variables vi, i = 1,

,n + 2 and Tj,i = 1,

,n + 1 now denote

the difFerence of the actuai correspondhg system state and the steady state operating values.

On examinhg the dynamic equation (2.16) of the r d , it is quite obvious that i i ( t ) ,i = 1, n + 2 is very s m d because the thicicness e of the web is small (approximately 10-5m). Hence the system (2.16) r d t s in a slow varying system with respect

to the radii ri(t), r = 1,n

+ 2.

Another feature we can observe fiom the dynamic equations (2.15) is the multiple t h e scale structure. In particder, from (2.15), for j = 1,2,

O

which can be reWIrtten as:

-

,n + 1, we have:

CWAPTER 2. MODELING FOR WEB TENSION CONTROL

where é

6. Then gince the web normdy has a high modulus, G is very large, which

implies that c is very small. Thus the web system hm a two time scale structure,

the very fast dynamics of the web tension components cansed by the high value of the web modulus and the relatively very slow dynamics of the velocities components.

This irnplies that further simplification can be obtained using singuiar perturbation methods.

The Reduced Slow Dynamic Mode1 Consider the singuler perturbed system

where x E R" and z E IF;then in the caee when c + 0, the reduced order system

can be found by solving g(t, z,h(x),O) = O for h(x).

Consider now the web system (2.15):

In this case, it can be eady o b s e d that if c = O, then the solution to (2.23) is given

CHAPTER 2. MODELING FOR W;EZB TENSION CONTROL by:

This result reflects the general fact that when the stifFness of a web material is extremely high, the web material can be treated as xigid body, and hence al1 the web spens move with the same speed.

In this case, the dynamic equations for the docities of the system are reduced to a single dynamic equation, Le.,

Q

= v , i= 1,2,

-

,n +2 and the dynamic equations

of the tensions are represented as only aigebraic relationships.

A state space representation for the reduced system can be easily obtained in this case.

h(2.15), denote

then on letting vi = v , i= 1,2,

. ,n + 2, we obtein:

For every equation in (2.26), divide together; we thence obtain:

9 by & and add ail of the resulting equations

CELWTER 2. MODEZLING FOR WEB TENSION CONTROL

The output equation can be found by setting the right hand sides to be equal for all of the equations in (2.26). However we wilI end up with a very mesgr expression with

no indication of the physicd rneaning. To overcorne this problem, this equation be derived later as a simplification of the high order model.

Rom (2.27), we c m see that the (2n +3)th order system has been sirnplified fist order RC circuit as given in fig 2.8: where

v

r--------1

1

I

Figure 2.8: Reduced System 1s a First Order RC circuit

with Y being the conductance, the reciprocal of the resistance.

The reduced system (2.27) ha9 a very simple fom. However, it does not capture the high fiequency information in the system. To completely undetstand the stability properties of the system, we should include the dominant hÏgh frequency efects of

the web system.To do this, we need to look at a mote complicated model containhg

CHAPTER 2. MODELING FOR WEB TENSION CONTROL high fkequency information.

High Frequency Modal of the W e b System To derive a high fiequency mode1 of the web system, consider the noniineat dynadc equation for tensions in the web given by (2.20):

1 where B = O.

Unlike the singular perturbation andysis approach, we now want to keep the detailed dynamic behavior of the tension

ternia.

It is observed that because of the

very s m d value of c, the nonlineax components of the tension €Nj(t) and dVn+l (j), j = 1,2,- , n + 1 of (2.29) is very smd, i.e.

Hence as c + O, we can &op the nonlinear terms fiom the tension equations, and in this case, the tension equation (2.29) equations become linear equations:

This is just the dynarnic equation for an ordinary spring. This hplies that when nonlùiear factors caused by chmges of the density throughout web spcrns are negligible, a web span acts exactly as a @ring.

CHAPTER 2. MODELING FOR WEB TENSION CONTROL

19

On combining the linearized mode1 derived before in (2.18)with (2.31), we now obtain a complete hi& fkequency linear model:

Let

US

now associate {vi(t), x(t)),i = 1,2,

,n + 1 and

Tw)as corne-

sponding to single module tems Mi:

We can describe these

two equations by a capacitor-inductor-resistorcircuit of

figure 2.9.

-

To(t)

i

i

I

,--------A

X(t) (os cwnnt)

-

(Mvoltage)

Figure 2.9: A High Fkequency Mode1 Module

CHAPTER 2. MODELING FOR WEB TENSION CONTROL where:

From the diagram of the module of a single model (figure 2.9), we can see that web tension is an output from one module which becornes an input to the next module, which agrees with the weli hown fact that W e b tension is traferted dong the

web spans. Every single module is a second order LCR Ladder circuit.

The complete web system then is a series connection of these modules, starting from the roller 1 to roller n

+ 2 as given in figure 2.10.

Figure 2.10: High Fkequency Mode1 Alternately, it can be represented the block diagram 2.11. Let us now return to the reduced system. It is obvious that the reduced slow system is only a simpIification when the modulus G is set to oo (c

+ O),

in which

case the inductance Ii is O. In this case, the resistors and capacitors form a simple

parallel connection, and hence they can be lumped together forming the C and Y as given in figure 2.8. We are now ready to derive the output equation for the reduced slow model.

CKAPTER 2. MODEIJNG FOR WEB TENSION CONTROL;

Figure 2.11: Block Diagram of the High Requency Mode1

-

v

Ti

a

Y

S

C a

--

-

(BI

(Cl

I

Figure 2.12: The derivation of the mode1 for the reduced order system

CHAPTER 2. MODELRVG FOR WEB TENSION CONTROL

22

As shown in part C of figure 2.12, the output tension Ti can be expresseci as:

where CiAl, YidL and SiAt are the lumped capacitance, conductance and curent

sources fiom the module i

+ 1 to module n + 2 respectively, hence:

On carrying out a Laplace transform on (2.35), we obtain:

Petform the same procedure, we can also get that:

where C,Y and S are the overd Iumped capacitance, conductance and curent source

Erom the module 1 to module n + 2 respectively, given by:

CHAPTER 2. MODELING FOR WEI3 TENSION CONTROL On appLyi.ngequation (2.38) to (2.37), we obtain:

n+2-r'tmm

On writing the output equation described by (2.40) for the reduced slow system

in matrix form, and on applying the rdationship Si = +Uj(t), we obtain:

where

As a summary, on combining the output equation (2.41) and the dynamic equation (2.27), we then obtain the complete state space modei of the low frequency mode1 of

C W T E R 2. MODELING FOR WEB TENSION CONTROL the web system:

where T,CT and DT are given in 2.41.

The modeling method proposed here provides physicd insight into the system, and provides a simple systematic numerical algonthm to determine the approximate space representation of a web system when the number of web spans is large (2 3).

Chapter 3

The Servomechanism Control Problem 3.1

Robust Servomechanism Problems

Let a linear tirne invariant system be described by the state-space model:

where x E R" are the States, u E lP are the inputs, w E

are the unmeasurable

disturbances, y E IF are the outputa to be regulated, y, E Rrm are the meamrable

RI are the reference inputs, and e E R are the error signals. It is assumed that A, B, C,D,Cm,Dm are known and E,F may or may not be known. outputs, g , , ~ E

Consider the class of referencefdisturbance signals which are described by the

CEIAPTER 3. TEE SERVOIMECHANISM CONTROL PROBLEM

where the pairs (C,,&) and (&Ad) are observable, and where zank(C,) = r and

Let &(s) and A&)

denote the minimal polynomiale of A, and Ad respectively.

Defme Ai, i = 1,2,...,p to be the zeros (multiplicities included) of the least common multiple polynomial A(a) of A&) and A&).

These zeros are a h called the design

frequencies of the robust servomechanism problem.

Remark 3.1 It is tu be noted that eornmon types of 8ignak such os constant, mmp and sinwoid signaii al1 satish (3.2).

It is now desired to find a robust controiler which will achieve the following control

objectives (Dav76bl: 1. The closed-loop system is stable. 2. Asymptotic regulation occurs (Le., e(t) -+ O as t -+ oo) for al1 reference inputs gr,,

and disturbances w describeci by (U), and for aU plant and controller initial

States.

3. Asymptotic regulation occm for any perturbations in the plant parameters

(C, A, B, D,Cm,Dm)of (3.1), which do not de-stabilize the resuitant perturbed system. Such a control problem is c d e d a robust servomechanism conttol problem. Any dynamk linear controUer nhich soIves the robust servomechanism problem

C U T E R 3. THE SERVOMECHANLSM CONTROL PROBLEM is assumed to have the foliowing structure:

where xc E E is the controller state, and u E ii"L is the output of the controller to

the input of system (3.1). The following existence conditions are obtained to solve this problem:

Lemma 3.1 [h76b] The necessory and ~uficzentconditions that there ezists a solution to the mbwt seruomechanh problern for (9.1) and (3.2) are that the follouàng conditiow should al1 hold: 1. (A,B ) i s stabilizable.

2. (C,A) is dectectable. 3. m z r .

.

The tmnsrnission zeros of (C, A, B, D) do not coàncide with A, i = 1,2,

5. y i s contained in y,, i e . , the outputa to be rtgulated are meosumble.

3.2 3.2.1

The Servo-compensator The Controller Structure

Given the design lrequencies &, i = 1,2,

- - - ,p, define a matrix

t$

E PX'

- -- ,q.

CKAPTER 3. THE SERVO2MECivrSM CONTROL PROBLEM where &, &,

- ,6, are the coefficients of the following polynomial:

The servo-compensator for (3.1) then has the following structure:

where in the above equation

where

Assuming that a solution to the servomechsnism probiem for (3.1) acists, a controller which solves the servomechaism problern for (3.1) is then given by:

The state O of (3.9) is the output of a stabiliaing controller which has the following general structure

where us P [&

& F uqT

The servo-compensator serves to achieve the control objectives of asymptotic

CWAPTER 3. THE SERVOIMECHANISM CONTROL PROBLElM

29

control. The role of the etabilizing controiler is to stabilize the

tracking and mb&

resultant overail augmentecl system, obtained when the servo-compensator (3.6) is applied to (3.1) to produce:

where, if the conditions of lemma 3.1 hold, the triple

is stabilizable and detectable, and hes the same fixecl modes as the plant (C,A, B).

*=f=; +

Sem0

.

Cornpensator

Ym

KO

Plant

B.

,

T

Y D

J

Figure 3.1: The general structure of a sewomechanism controller

3.2.2

Stabilizing Controller Design

DifEerent methods can be used to design the stabilizing controller (3.10) for the ser-

vomechanism controk. In this thesis, three methods will be used: a cheap control approach, a high gain stabiliaing controllar approach and a tuning regulator controller spproach.

Chapter 4

The Perfect Control Problem 4.1 Introduction Rom the past chapter, it is clear that given a class of modeled tracking/disturbance signals (design signals) and the plant mode1 satiafying Lemma 3.1, there exists a solution to the servomechanism problem so that asymptotic reference tracking/disturbance

rejection occurs for any arbitrary plant perturbations which do not produce instsbility for the overd closed-loop system. The controller muet contain a servocompenaator

and a stabiliPng controller, where the design of the semocompensator is unique and the design of stabilizing controller is not. It is now desired to solve the foilowing type of pro blem: given a class of unmodeled signals which lie outside of the design signals,

find a stabilizingcontroiler so that arbitrary good approximate error regulation occurs and arbitrarily good transient response occurs for ali the bounded initial conditions of the plant and controiler. The above problem is cded the Robust Servomechanisrn

Problem with Perfect Control (RSPPC)[DS87].

4.2 4.2.1

The Perfect Control Problem Development of the Problem

A description of the RSPPC will be dehed here.

CHAPTER 4. THE PERFECT CONTROC PROBLEM

32

Given a plant describeci by (3. l), consider the class of unmodeled refetencefdistmbance sigpals which are linear combinations of signais of the following type:

where t , are nonnegative integers, S E C+ and A,,

\, CC+ occur in conjugate pairs,

and fjW E E,iit E Rn.

M e a d of considering a single feedback controller (3.9) and a single stabilizing controller (3.10), consider now a family of controliers parameterizeà by a single parameter c, Le.

where is the output the servo-compensator (3.6) and 5 is the output of a stabihing controller:

Assume now that the conditions of Lemma 3.1 are satisfied and that for each fixed c > O, a servomechanism controiler describeci by (4.2) to (4.3)

has been found to solve

the robust servomechanism problem for the design fkequencies describeci by (3.1). Assume now that the unmodeled reference/disturbance signais (4.1) are applied

to the resultant closed loop system; it is desireà now that for any fked X E Ci in the signals (4.1) the closed-loop system should possess the foliowing properties [DS87]: 1. Achieving ArbatmRly Good Approxàmate Emr R&ation(AGAER), O, VA

> O and dl initial conditions =(O),

i.e. V6 >

c.

€(O), ~ ( 0fr )(~ 0), (O) locateà on their

respective unit spheres, there exists a t~ > O and T > O such that the resultant steady-state e w r , which is denoted by e,, (t),has the pmperties that Ilerr(t) 11 < 6, V t E [ K T

+ A].

CHAPTER 4. THE PERFECT CONTROL PROBLEM 2. Achieving Pedect Control,

Let the output of the r d t a n t closed loop sytem be denoted by:

where K:(s),K$(s), Hg(s),H ' ( s ) ,H$(s)are the corresponding transfer funcc tions obtained after applying the sei.vomechanism controller (4.2) and (4.3) to the plant (3.1). It is then deeired that, in the point-wiee convergence seose:

K:(s)= 1,i.e., perfect decoupled tracking occurs.

(a)

(b) liwdaK:(s) = O, i.e., perfect disturbance rejection occurs.

(c) perfect initial condition response occurs, i.e.

i. ïim,+o H;(s)= O

ii.

Hi (s) = O

iü. l i ~ H;(s) + ~ =O

When lime+oH{(s) = O does not hold, the problem is called a robwt remomechanism problem uith perfect control, otherwise it is calleci a robust servomechanismproblem with complete perfect control.

4.2.2

Design for Perfect Control

It is shown in [DS8?] that there exists no solution to the mbwt servomechanism problem uith complete perfect

controL However, there exista a solution to the mbwt

servomechanism pmbtem with pedbct control provided the following conditions al1

hold:

Lemma 4.1 [DS6v Given the plant ( X I ) , there

Qists a solution to

peqfed control if and only if the follomng wndàtons dl hot&

the RSP with

2.

(C, A, B, D)is minimal phase and m 2 r.

3. y is contained in y,,

.

Pdect Controuer Design U~ingCheap Control Assume now that the above conditions hold for a given plant. Coneider the system in which the plant (3.1) is augmentecl with the servocompenaator (3.6):

where CU

(1,O,

-

,O), and

T

P

Define z E lF to be

Then we form the system (6,Â, Ê), where:

CWAPTER 4. TWE PERFECT CONTROL PROBLEM A pedect controiier is now found by finding a feedback controller

to rninimize the following performance index

J, = JOOD(Iz + eufu)dt where e > O is a scalar.

It is well known that a solution to the above control problem is given by:

where P,is the unique positive semidefinite solution of the Algebraic Riccati equation

The perfect controller is then implemented by using an observer, Le.,

where:

f

= ( A - koCm)i+k&,-D,u)+

where Kais the observer gain found so that A - K.C,

Bu is stable.

Other Perfect Controllers

The Design of the perfect controller is not unique. In the next chapter, a new type of high gain controiIer design [ZD94a] for minimm phase plants ,which requires ody

information regarding the systemtsinfinite transmission zeros (12) will be intmduced.

Chapter 5

A High Gain Stabilizing controller 5.1

Introduction

High gain controliers have been widely usad as stabilizing controllers because of their

ability to compensate for nonlinearity and uncertainties in the system.Also, because

the amplitude of the sensitivity funcfion can be signiscantly reduced, disturbance rejection of the closed loop system can be greatly improved.

5.2 5.2.1

Mathematical Preliminaries Transmission Zeros

Let G(s) be a rational transfer hinction matrix; then it can be trandormed to the

Smith-McMillan form.

where 4 ( s ) , &(a) are polynomiais with the property that for aîi i= 1,2, àivides ~ ( s and ) &-&) divides &(a).

,r , E ~ (s) - ~

CHAPTER 5. A HIGH GAIN STABICXZING CONTROLLER The transmisaion aeros are defined as the zesos of poiynomial Z(s) = ci (a) €* (s)

€&?)

(5.2)

If the transmimion zero has multiplicity 6, it is said to be a trtmsdmion zero of order 6. In the foHowing chapters, a trammission zero will be simply cailed a zero.

5.2.2

Transmission Zero at Infinity[Mac89]

Let H(X)

G(h); then the infinite transmission zeros of G(s) of order o at s = a,

are defineci to be the zeros of H(B)of order o at s = O. In the following chapters, an

infinite transmission zero wili be denoted by 12. #

5.2.3

Minimum Phase System

A system is c d e d minimum phase if none of its transmisson zeros lies in the cloaed right hand plant C+.

5.2.4

Hurwita Polynomial

A nth order real coefacient polynomial on X f (A)

= a#

+a#-'

+-+

(5.3)

is c d e d a Hurwitz polynomial if all of its zeros have negative real parts.

5.3

The multivariable root locus

In the SIS0 case, the eigenvaiues of a closed loop system are very eesy to analyze for high gain feedback. However, for a multivariable system, due to the multi-input

and multi-output geometrid structure, the behavior of the closed loop system eigen-

d u e s , as the feedbdc gain inmeases, is much more compiicated than for the SIS0

CRAPTER 5. A HIGK GAIN STABILIZXNG CONTROLLER

38

To study the behavior of multivmiable systems under high gain, it is convenient to

study the multi-variable root locus of the system. It is weil known that for a SIS0 strictly proper system with n poles and m zeros, the root loci begin at the open-loop poles, and m branches of the root loci terminate

at the finite zeros of the system,with the other n - m branches going to infinity dong asymptotes with angles of:

which intersect at the point h = i sum of

open loop pales)-(sum of open loop zeros) n-m

(5-5)

However when one studies the multivanable case, the root locus plot is not this straight forwatd. The following is a plot of the root locus of a strictly proper

square multivariable system,whose cornplete description of this system is described in [KS76].

Figure 5.1: A Generic Mdtivariable Root Loci

In the above plot, rn observe that a multivariable system can have 12s of multiple

CHAPTER 5. A NIGH GAIN S T A B E I ' G COn'TROttER

39

orders. The root loci approach zeros of Merent o r d m with dinerent rates in a

Butterworth pattern. For systems with W t e zems of order larger than 1, there may be frimilies of root loci approachixtg such infinite zeros. For example, if a system has v ith order zeros, then for every ith order 12, Say, jth ith order zero, the root loci approach the this infinite zero (12) dong asymptotes satisfying the following equation defining a

Butterworth Pattern

where X is some complex constant.

In this case, there are i branches of root loci. They approach infinity with a rate of I(Ak)! 1, and have angles with the positive axis of

The center (called a pivot in [KE79]) of the above rays of radiation does not necessarily lie on the real axis and may corne in complex conjugate pairs[KE79], which is

very different fkom the SIS0 case.

It is clear that whenever Butterworth Patterns of order higher than 2 occurs in a system, this system is unstable at high gain feedback. Hence the main design

objective in high gain controller design is to eliminate the occurrence of high order Butterworth Patterns. We went the closed loop system root loci to approach the h i t e zeros of the minimal phase system or/and go

off to infinity in the open lefi

hand complex plane, Le., the closeà loop system can have only IZ of order 1.

5.4

Controller Synthesis

The controiler will be designeci for square syetem, i.e., the number of inputs m of a system is equal to the number of the outputa r. Recogniaing the fact that a nonsquare invertible and minimal phase system can be squared down to a square minimal

CHAPTER 5. A HIGH GAIN S T A B K 1 . G CONTROLLR

40

phase system via dynamic feedback, such as introduced in [SS88], the results here can

be very easily extendeci to nonecpare systems.

5.4.1

Approach 1-Decomposition

The structure of the 12s of a given system is the most important information in high gain controller design. This etmcture can be constmcted by the spectrum

decomposition or the singuiar value decompoaition. Decompdtion of the Tramfer mindion Matrix Theorem 5.1 [ZD94b] Gàuen an inuertible systern ( C , A , B ) , the= Qist two nonsingular matrices P, Q € Ilmxmsuch that:

where ci, i= 1,2,

- ,h are the orders of the 12s of this system.

Then fkom FS761, the decompoeed systems (Ci, Mi, Bi) has oniy 12s with orders higher than oi, i = 1,2,

,h.

CHAPTER 5. A HIGE GAIN SWILIZING CONTROLLR

41

The detsils ofusing this method to find the IZs t ~ c t u r ecan be found in [KS76]and [KE79]. Another method to determine the IZ structure can be found bom Kdath's '

work in (Kai8O], where a bilinear tranformation is used.

Controller Structure With these decomposition resuits, we can design a controiier on the following system:

Denote

and

Consider a controller structure of the foiiowing:

where Gi( s ) ,G2(4me constant matrices. Let

Gi= QG and G2= P,where G is an arbitary matrix which will be chosen

later in the controlier design.

Note the fact that the characteristic polynomial P(s) of the closed loop system on applying controller (5.13) to the original system (5.12) ie given by:

P ( s ) = det(1,

- GiC(s,€))Oz H ( s ) )

= det(1,

- GC(s,c)R*(s))

We can now see that appiyhg the controller (5.13) to the original system is quivalent

CHAPTER 5. A 131GH G r n STABILIrnG CONTROLLER to applying a controller

to the decomposai system (5.10). Choose now:

Ci(s, c) will be designed to compensate for the open loop IZwhile maintainhg a stable

proper controiier and without introducing new high order IZ as c

+ O.

Define the

foiiowing structure for Ci(s, E )

where 4 is chosen to be Hurwita polynomial of order Oi - 1 and

Ti

2 4 - 1.

This high gain controiler has the foilowing property: Theorem 5.2 (ZD94bI Consider a minimum phase square system (5.12) and the class of monic Hurwitz polyomiol &(s) of d e g m q - 1, i = 1,2,

the high-gain cornpewotor (5.13) toith (5.16) and (5.1 y), po = 1 , p j = 2 , j=1,2,-•,h,

,h, and consader where ri 2 ai - 1. Choose

G1= Q G , G 2 = P , andletG=diag(G1,C&,- ,Ch)

such that sp(ÇiGi) C C-;then as a

+ O,

the closed loop system is asymptotically

stable, and 2ts eigenvalues are gàven by (multiplicities excluded):

where ho denotea the fintte tmnsmksion zeros of the open loop sytem (C,A, B), and 1\, denotes the zems of &(s).

5.4.2

Approach 2-Generalized DifFerentiaI Interactor

CHAPTER 5. A HIGH GAIN STABXLIZING CONTROLLER

Instead of decornpothg the system and tweaking the system not to have high order

IZ,a simple substitution is to fmd a dynamic invariant feedback to make the system have ody 1st order 12. To do this, let us introduce the concept of a generalized diffetential interector[ZD94a]. Deflnition 5.1 A na x na polynomM1 matriz &(s)

ib called the

(Mt)generaüsad

dinerential interactor of H(e) if P hos the following structure

a

where (s) 9daag(t& (s),ci&), polynomid of degwe Ji

- - - ,b'(s)), with bi

( s ) a monac ml-coeficient Hurunruntz

- 1, i = 1,2, -- ,m, and where Ï'(s)

is an upper tdangular

polynomzal matriz with an integer 1 o n the diagonal, such that

w h m j(H iS O non-sinplor constant matriz. A detailed way of determining the interactor &(s)

has been given in [ZD94a]

The dennition 5.1 implies that the following holds: Theorem 5.3 Given a non-singular m x rn atrictly pmper hnsfer funetion matriz H ( s ) , then there olwuys exists a genmlàzed differenttal intemetor ZH(s)such that

The fact that

CB is bill rank states that this system can have only rn 1st order

12 and no other higher order IZ.Denote the overd order of the sgstem (C, A, 8) to

CHAPTER 5. A HIGH GAIN STABILtlZING CONTROLLER be fi; then it has R - m finite zeros, and they are given by

where

denotes the finite zeros of the system (C, A, B ) and Ai, A*,

the set of the zeros of the Hurwitz polyn0rnial6~( 8 ), 6~(3) ,

The above property of the system C

- - , & are

,if,, (s) respectively.

(C, A, B) maLes the high gain controller

design for É: very easy. A static controller

where ii is the input to

and G is chosen such that A(CÉG)E C- , wiil suffice.

This controller has the following property: Lemma 5.1 [ZD94a]Giuen o minimum phase square qstem (C, A, B ) , consader the tmnsfumed system (C,A, 8 ) defined in theorem 5.3; then aftet tapplying the contmller (5.29), where A(CBG)E Cm,the closed Iwp system

has the property thut as c

+ O,

it is osynptotically stable, and its eigenudues are

given by

where O(€) denotes the order of the small saler c.

The result (5.25) results fiom singular perturbation theory [Kok86]. Using the same theory, Zhang and Davison in [ZD94a] have proposed a dynamic high gain

CHAPTER 5. A HGH GAXN S T A B W G CONTROLLER feedback controller of the form:

where ri, r2,

- ,rmare positive integers. This controlier r d t s in a asymptoticaliy

stable closed loop system with a time scale of 3.

Lemma 5.2 Given a minimum phme square ~ystem,c o ~ i d e rthe system (C,A, B) as defined in (5.21), and opply the contmller (5.261, whem X(CBG) E Co;then os c

+ 0, the closed loop system 6s asynptoticully stable and the closed bop eagmudues

ate gàven by

w h m the ET' Xq matriz A, is defined as the

5.5

Conclusion

The previous controllers d e d b e d in chapters 4 and 5 have the property that they require either: (a)a "satisfactory" complete mode1 deacRbing the plant, or (b) a

knowledge of some aspects of the plant, e.g. that the plant be minimum phase and a knowledge of the Markov parameters of the plant. It is oRen difncdt in industrial

control problems to determine such a prion knowledge, and the question arises: can one stili design a controller to solve the RSP when this knowledge is not available? In

CHAPTER 5. A HIGH GAIN S T A B I L I ' G CONTROLLER

46

[Dav76b], Davison proposecl a type of mukari8ble tunhg regulator which can solve the RSP provided certain mild conditions hold.

Chapter 6

Tuning Regulator Controller As discussed before, to solve the robust servomechanism problem using the servecompensetor, two design steps should be carried out: the servocompensator design and the stabilizing controller design. The servocompensator design is based only on the signals to be tracked/rejected, and hence is independent of the plant information.

Plant information is only needed for stabilizing the overall system. Let us examine

what information is required to accompibh this. Given a plant:

where y,

= y, then from Lemma 3.1, the conditions for a solution to exist to solve

the robust servomechanism problem are:

1. (A,B) is stabilizable. 2. (C,A) is dectectabie.

4. The tr-on

zeros of (C,A, B, D)do not coincide with &, i = 1,2,

--O

,p.

where Xi, A2,

-

+

-

,& are the

roots for the characteristic polynomial for the refer-

ence/disturbance signais:

On making the assumption that the plant is open loop stable, we can see that the first two conditions are automatically satisfied. The third and forth conditions are equivalent to the r d condition of the system rnatrix:

which is equivalent to the condition:

where GAiis the steady date gain of the plant at the frequency A; this condition cm be experimentally determined [Dav76a].

Thus the eristence of a solution to the sarvomechanism problem can be checked experimentdy by determining the steady state gain matrices (6.4). Assuming that such a solution exists, it is then pointed out in Pav76aj that a controller which solves the servomechanism problem is given by:

where q is the output of the servo-compensator aseociated with Al, A=,

-*

,Xp, end

where K2can be determined experimentaliy in conjunction with some one dimensional

"online tuning"; moreover, when the control objective is to track/reject only constant signais, there is a very simple way to determine K2.

Lemma 6.1 pav76bI Given the stable system (6.1), where the number of inputs nt is ut l&

aqud to the number of outpufs r, let the semocompenaafor i j = y

-

be apptied; then y mnk(D - CA-lB) = r, there ensb un r* > O, opplying u = &q, where K = (D- CA-%)+

30

that on

a (LI - CAalB)'{(D- CA-%)(D

-

CA-iB)')-l, the fesuttant closeà Ioop sgstem ia stable Ve E (O, Z], and protides asymptotic tmckàng/re~~ection for corutont signab.

Chapter 7

Controller Implementation

7.1 Introduction In this chapter, three controller designs: cheap controller design, high gain controller design and the tuning regulator controller design wili be implemented on the University of Toronto web machine.

The Univerity of Toronto web machine is a two span machine as iliustrated in fig

Nip

Figure 7.1: The Iilusfrative Diagram of the Rotoflex Machine

The major components of this system consist of an unWinder, a winder, a aip and the web conneethg them. ki between the rewinder and the nip, and betnreen

the nip and the unwinder, there are a number of rollers as shown in figure 1.1. ki

system identification and controk design, these idlers are ignored, whkh inhoduces uncertainty to the system. This uncertainty is asswned to be insignincant.

Models for Controller Implementations

7.2 7.2.1

A Reduced Order Model

The web material used in the axperiments camed out in the thesis is paper, which

has a very high elasticity moddus, and hence can be considered as a sti!T material.

The web model is aseumecl to be the reduced order low frequency model. It can be verified fiom the modeling of chapter 2, that the web system hm the structure as shown in figure 7.2.

Modeled Part (Active Rollers)

1

Unmodeled Part i (Idlers) I

Dynamic Equation Figure 7.2: The Reduced Low F'requency Model of the Rotoflex Machine

C W T E R 7. CONTROLLER IMPLEMENTArnON where

where the subscriptions r, n and u denote the rewinder, aip and unwinàer respectively and the subscripts 1,

,n represent the n idlem. Here C and Y denote the lumped

effect of the rewinder, nip and winder, and Ca and YAdenote the lumped effect of

al1 idlers on the system. Hence we can see that the idlers add a perturbation term to the system parameters for the reduced order model.

Table 7.1: List of Identifid Parameters Parameter 1 meaning due 1 Radius of the bare unwinder 0.0415m roa Radius of transport roUer (Nip) 0.0415m Tb Radius of bare rewinder 0.0415m Tob Inertia of bare unwinder 0.0175kgm' JO= Inertia of transport roller 0.0322kgm2 Jb Inertia of bare rewinder 1 0.0234kgm2 JO= Damping fiction coefficient of unwinder 1/254Nms b~ Damping fnction coefficient of transport roller 1/165Nms bb Damping fiction coefncient of rewinder 1/258Nmo 4 e Web thickness O.O015/(2xn) m

3

Using the parameters in table 7.1 obtained from the identification results in [Bor99], we are able to obtain a reduced order model (7.2) by ignoring the idlem

CHAPTER 7. C O N T R O WLEMENTATION ~ and fixing the radius ru at 0.0833m. 6

= -0.167961~+ [0.479610.84649 O.4I162][ma mb m,IT

Since the real system has filters of the structure:

connected to each of the tension outputs, such an addition of filters was applied to system (7.2) to obtain the foiiowing n = 3 model of the system:

The response of the system (7.4) is compared to the response of the real plant by carrying out a set of open loop rewinder torque step input experiments at the operating point ra = 0.0833m. The plot 7.3 shows the difference between the responses of the system (7.4) and the response of the actual system for the case of a step input

in the rewinder torque.

We observe fkom figure 7 3 that the ignored idiers have little eaect on the the tension responses. A step input on mwinder torque causes ahost the same amplitudes

of tension increase and tension responses to occur for both the model (7.4) and the

Figure 7.3: The Response of the Open loop Real System vs. the Identified Mode1 real system. However, on the other hand, the speed responses of the identified model

aad the real system m e r significantly. A step input in rewinder torque causes approximately 5 times grester change in speed response for the model (7.4) than for the real system. This is because the real

system has more fiction than the model (7.4). From figure 7.2, we know that that under a step increase bSi on the rewinder torque input, the amplitude of the resulting speed change bv at steady state is determined by the overall friction of the system:

With no significant difFerence between the friction coeeicients of the external driven rollers (rewinder, nip and unwinder) and the idlers, the idlers play a sig-

nifiant role in (7.5) because of their large number, which resuits in a speed response amplitude of the model (7.4) significantly larger than the red response.

The speed response of the model (7.4) is also significantly slower than the red system. The time constant of the speed response of the real system is determined by

the overail inertia (C

+ Ca)and the ovetali damping (Y +Y*):

The trnmodeled idlers will result in an increase in both the overad inertia and the overd damping. To determine which one has the dominant influence, we wül

compare the time constant of the model (7.4) with the real plant. By fitting a first order dynamic equation to the response of the real system we obtain a time constant

of tixm for the response of the real machine. This is about 5 times faster than the time constant & of the model (7.4), which is determined by:

Rom previous observation, we know that the steady state change of the model 7.4 is about 5 times larger than the real system, which means that Y + Yn is about 5 times larger than Y. This implies that the C + Ca is alrnost equal to C. Hence, the dominant influence of the idlen is the damping (YAor the idler frictions) which they introduce to the system, rather than the inertia. This conclusion agrees with our assumption that the idlem inertia is insignificant. Overd speaking, model (7.4) has a very good match in tension responses with

the red plant. For speed response, the main difference of the real system and model (7.4) is that the real system has more damping.

The final identified system used for controller design in this thesis is obtained by

fixing ru = 0.0415m (smdest radius) and rc = 0.15 (largest radius). The n = 3 system obtained in this case is called the identified system #1, which is described in

Appendix B.1.

7.2.2

Another Mode1 for the W e b System

Ftom the open loop step experiments, we c m also apply a multivarïablestate space

identification method as desaibed in [Dav99) to obtain a LTI model, by treating the web system as a black bac. In this case,a 6th order syatem is identined which gives

a better speed response mat& than model #l. T b model is called identifid model #2. (Complete information of the 6th order model can be found in Appendix B.1.)

Figure 7.4 to 7.6 show the step responaes of the identifieci 6th order system and the real system, which have exdent agreement as compaced to the case of model #i*

CHWTER 7. CONTROLLER llMP1;EMENTATIT)N Tension Fa Response of the IdenWiedModel and The Real Systern 30

1 - Real

1

I

I

1

I

I

. ! ........i. ........ *.

I

t5 O

2

13l O

2

1

1

I

1

- -....... t

1

I

I

I

I

I

6 8 10 Speed Response of the IdentifidModel and The Red System I

!

i

I

14

I

I

12

14

- Real 0.8 - - - Identifid .......... :. ........................... .:-.. . . . . . . . . . . .;.. . . . . . . . . . . . . .&

-1

I

4 6 8 10 12 Tendon Fb Response of the Identifiecl Madel and The Real System

4

I

I

1

,

.........

-

Figure 7.4: Cornparison of The Response of the Black Box Model and The Real Machine-Rewinder Torque Step

Tension Fa Response of the ldenaiffedModel and The Real System

1

1

I

1

I

I

I

4 6 8 10 12 Tension Fb Response of the IdentifieciModeland The Real Systern

O

2

1

1

1

I

1

1

O

2

4

6

8

IO

12

14

I

14

Speed Response of the ldentified Model and The Red System 1

0.8

1

1

I

1

I

I

i............

.........

- Real - - ldentified-..........:............................................

.-

-

..........................

u

= cont-sr .c*Xc + cont-si .d* Cyref error] '

APPENDIX C. S C ' T S FOR CORTTROLLER DESIGNS

Cnb,mb} = site(B0);

Crc ,ncl = size(C0) ; n = nb;

% nuiber of atates o f t h e given syitern

r = rc;

X niimbet of output8 of the given system

m = mb;

% nupiber of inputs of t h e given system

% Now use t h e servob routine t o compute the controller

% Now construct t h e controller Btate space representation

X The controller is of the following structure % u = ki*X + K2* k e t h i %

ühere X is t h e observed state from a f u l l order linear state observer: Dot(hat (x)) = A*(hat (x) ) + B*u + ko

%

hat(y) = C

% %

and

dot (kethi) = y

* hat(x)

-

yref

*

(y

- hat (y))

+ D*u

- error

% For the controller s t a t e space representation, the input t o t h e Yi controlier is t h e output from t h e plant, and t h e output of t h e

X controller is t h e input t o t h e plant

.

Acont = CAO- Ko*CO + (BO-Ko*DO) *KI (BO-Ko*DO) *K2;teros Cr.n) zero8 Cr r) ; Bcoat = Mo -Ko;zeros (r,d -eyeCr. t)1; Ccont = [KI K2f ; Dcont = zero8(m,r*2) ;

APPENDLX C. SCRETS FOR CONTROLLER DESIGN$

cont-ss = se (Acont ,Bcont ,Ccont ,Dcont);

The Overall Design Script Pot The Cheap Control Servomechanism Controuer Design

X F i r s t nonualize the obtained linear rystem and serially comect each tensio

-

% output8 to a sensor as the real syatem. F a ~ a x= 130; Fb-max

130; Vbsax = 5;

Base-max = [Fa-max Vb-max Fb,maxJ ; plant = syatem_relative (Base-max) ;

X here get the controller [cont ,as]

= servo,de (plant) ;

% Nov the actual designed controller vil1 take the

X relative reference and error as the inputs, so it % is necessary to modify the B. D matrice. here assume % the inputa to the controller are references and errors.

X Now export this controller t o C

APPENDIX C. SCRIPTS FOR CONTROLLER DESIGNS

C.1.4

Scripts for High Gain Servomechanism Controller Design

X Specify t h e high gain coatcoller gain % p ~ i , h ~ ~the . weighting scaler f o r t h X observer design %O" and the saturation controller gain ltgiven,thetaM. The Y. high gain controller gain wepsi-h" rhould s t a r t from a very small valus aa X t o ensure s t a b i l i t y .

Iteol1 should be chosen from a large value as le4, t h i s

X t e s u l t in a slow response rpeed. then it should be tuned t o smaller values X satiafactory response speed is achieved. The saturation i s chosen from a va X large value, t h e resultant closed loop system ma. be slow, t h e value is the X t o achieve desired reirponee.

% first get the plant system, t h i s system export the

X relative output, y,% = y./y,max X relative States x,#

and the state are

= x./x-mu;

X This function is iimilar t o l~system~relative.m", the difference is t h a t it % also nornalize the plant s t a t e al80 t o be in [O 11 . plant = syrtem,state,relative

(Base-nax)

;

APPENDE C. SCRIPTS FOR CONTROLLER DESIGNS

X High gain contcoller design

X The state spaca representation of the high gain controller X The high gain controller is expressed in the following fonn: X %

dot(Xh) = Ac,H*Xh

u

= Cc,H*Xh

+ BcJ*(y-v)

+ DcJ*(y-v)

X The composite syrtem of the plant and the high gain controller. X Becauae hg-8s.d is O, we obtain a simplet expression as following: comp-a = Lp1ant.a plaiit.b*hg-ss.c;hg,ss.b*plant.c comp-b = [zero8 (length(p1ant. a) ,r) ;-hg,ss

hg,ss.a]

.b] ;

comp-c = [plant. c zero8 (length(hg-sr. a) 11;

comp-d = zero8(r) ;

camp-ss = sa (comp-a,camp-b. comp-c ,camp-d) ;

X Now design the input limitiDg controller X first needto rcale the output8 to make lyrefa*yrefI

%

%

v

.

= all_Cont_ss c*X1 + ail-Cont-ss. d* Cyref errorJ >

all-Ccont = ~theta*tou*comp,ss .c*inv(comp-8s .a) (-theta) * t o d ; al1,Dcont = zeros (m,2*r) ;

all-Acont = [(camp-8s .a-ko*comp,rs

.c)

zeros (n-xhat ,r) ;

zeros Cr,n-xhat) zeros Cr)1 +[((comp,ss

.b-ko*comp,ss

.dl *aïl,Ccont) ;zero8 Cr,n_xhat+r)3 ;

al1,Bcont = [ko -ko ;zeros (r) -eye (r)1;

APPENDIX C. SCRIPTS FOR CONTROLLER DESIGNS

123

X Now it is needed t o transform the controller t o the form that can be applie X t o the input of the real plant. rather than the input of the raturation X controller. %

This is done by augmenthg the controller:

dot (X1) = aïl,Cont,r

%

v

= all,Cont,ss.

S . a*X1

+ all-Cont-as. b* [yref error]

c*X1 + aïl,Cont,re

.d* [yref error]

X and t h e high pin controller: %

dot (%hl= Ac-H*Xh + Bc,H*(y-v)

u = Cc,H*Xh X t o get a new rystem: %

dot (Xc) = owraïl,Cont_ss. a*Xc + overal1,Cont-as. b* [yref errod

X %

u

= overall-Cent-8s. c*Xc + overall-Cont-ss .d* Cyref error] >

overal1,A = [hg-sa .a (-hg-se .b) *all,Cont-as. c; zeroa (n-all-cont ,n-hg) aU_Cont_ss.d ;

overall-B = [-hg-as. b*aiï,Cont,ss. d+ [hg-ss .b (-hg,s

S.

b) 3 ;ail-Cont -ss .b] ;

overall-C = [hg-ss .c zero8Cr,n-all-Cont) 3 ; overail-D = zeros Cm,2-1 ; % Because the controller is designed t o control the relative outputs, % scaiing on e J yref ahould be made, L e . . transformation should be made to 0

% and Dcont.

APPENDIX C. SCRXPTS FOR CONTROLLER DESIGNS

Output,Controller (overall-Contœ8s, >

C.1.5

..\progran\high-gain-cont .ha) ;

The Sc~iptto Discretize the continuous time state space controller and to Export the Controller As A

C Head File Fùnction: [ ] = Output-Controller(Cont ,fi)

X Input: Cont : State Space Representation of the Controller

x

file: Name of the output C header f i l e for the controller function C] = Output-Controller (Cont , f i l e )

Acont = Cont .a; Bcont = Cont .b; Ccont = Cont .c ; Dcont = C0nt.d; % The continoue model is discretized by a bilinear transformation. This w i l l

.

X project the open l e f t hand side coordinate plane into the mit circle hen X the discretized model will not change the stability property of the contin % t h e model. [A,

B ,C ,Dl = bilinear (Acont ,Bcont.Ccont ,Dcont ,400)

;

APPENDXX C. SCRIPTS FOR CONTROLCER DESIGNS

X NOW get the dimension informations of the aystem

out = f opencf i l e ,'wb ) ;

X Here the name ~Breroocomprn8atorBt ii the name of the coritrol function t o be

X called i n the C program. fprintf (out. 'void servocompensator(double *Cent-inputs, double

*Plant-inputs) \r\nl ) ; fprintf (out, >C\r\n\r\n1 ) ;

f printf (out, int i ;\r\a ) ; fprintf (out. int j ;\r\n\r\nJ

fprintf (out, ' s t a t i c float xRi]= ); end f printf (out, O) ;\r\n ) ; f priiitf (out, if (State!-6) \r\n(W)*xCXiI+',C(i,j) ,j-1) ; end

fprintf (out, '0') for j = 1:m fprintf (out, J+(Yd) *Cent-inputs [%il '

J) ,J-1) ;

end f printf (out, > ;\r\n8 )

end

fprintf (out, > \r\ns) ;

for i = 1:n fprintf (out, 'xd[Xi] =' ,i-1) ;

for j = l:n fprintf(~ut,~(Xf)*xmi]+>,A(i,j),j-1);

end f p r b t f (out '0 ') J

for j = 1:n fprintf (out,>+(Xf)*Cont,iiiputsCXiJ ,Mi,j) ,j-1) ;

end fprintf (out, ' ;\r\nS ;

end

fprintf (out, \r\n\r\n8 ; f printf (out, f or Ci-O ;i < X i ; i++)\W ,n) ; f p t h t f (out. x Cil = xd Cil ;\r\a\r\n

;

f printf (out, '\r\n>\r\n ; f close (out) ;

C

C.2 C.2.1

Codes For Controller Implementat ions

C Codes For The Implementation of the High Gain and Cheap Control Servomechanism Controllers

//UofT. c

-- 1 October 1997,

mp Web Technology, FA

//To be implemented at University of Toronto //The f ollowing f unctiona murt be included: //

void Uof T-Init Cvoid) ;

//

void UofT-Interrupt(void); --called by the MTSC32 at every sampling inst

/*********

--callad by the MTSC32 during i n i t phase

NECmSARY HEADm INCtUDES

*******+***/

ainclude csysdef.h> #inchde cec32.h> #include #include +

#inchde #inchde Binclude Llinclude t8serpocompensator.h~ // this name should be chaaged to

// appropriate head file of the

APPENDIX C. SCRIPTS FOR CONTROLLER DESIGNS // controller t o be impleneated #def ine State signais C3SJ

/***********************************************/

/*

Operation Svitches */

#define sw-7,7

awitches C71 &

#def ine su-7.8

switches [7] &

#def ine su-7.9

switches [7] O

#def ine sw-7-10

switches 173 &

#def ine sw,7,1i

switches 171 &

#def ine su,7,12

switches 171 &

#define su-7-13

switches 171 &

#def ine su,?, 14

switches C71 &

Xdef ine sw-7-15

switches 171 &

#def ine sw-7-16

switches [7] &

Xdefine su-7-17

switches [?] &

#define su-7-18

switches [7] &

#define su-7-19

switches 171 &

#def ine sw-7-20

switches [7] &

#define sw-7-21

ewitches [7] C

#def ine sw-7-22

switchesD] &

#def ine su,7,23

switches L?]

#def ine sw-7-24

suitches [7] &

#def ine su-7.25

switches 171 &

#define swJ,26

switches 171 t

#def ine su-7-27

switches 173 &

#def ine su-7-28

switches[Il Ir

&

APPENDIX C. SCRIPTS FOR CONTROLLER DESIGNS #define su-7-29

switches[7] &

0x10000000

#define sw-7-30

suitches C7J &

0x20000000

#define sw-7-31

switchesn] &

0x40000000

#def ine sw-7-32

switches[?J k

0x80000000

/* Honitor Signais */ // define the plant references t o be monitored

Xdef ine Sa-ref signals-ex [23] Xdef ine mVb,ref

signals-ex [24)

Wdef ine mfb,ref

signal8,ex [Z6]

// define e r r o r signal t o be monitored

tdef ine Fa-err eignals,ex 1261 #def ine Vb-err eipala-ex [27] Mef ine Fb-err signala-ex 1281 // define t h e modified tensions t o be monitored

#def ine mFa signaïs-ex 1303 Wef ine mFb signals,ex[31]

// define the plant output values

#derine Fa-modi (signal8Cl31+O .31795) /l.4457 // l i n e a r i t y between the measurement and t h e real tension. In future work, // these linearity valuas should be modified from t h e corresponding set up // values.

#def ine Vb-act

signal8 C23 *(3.1415926/30) *O. 0415

#derine Fb-modi (signal8 CI41-1.7522) /1.3314

// def ine the reference vaïues

//linearity between measu/rea

APPENDIX C. SCRIPTS FOR CONTRQIILERDESIGN" Mef ine Ref ,Fa signala D2J

Mefine Ref ,Vb

8 ignalr [21]

Xdef in8 Ref Jb signals[33]

/*

Reference values */

// def ine the control signalil Mefine ma deshed-vvalus CS41 // Torque t o shaft A #def ine ab desired-values[SS]

/ / Torque t o shaft B

#define mc desired,vaiuea[56J

/ / Torque t o shaft C

/* Setup Values */ #define offilet

/S*WSS******S**S*

INIT mCTION

***************/

void

UofT-Init (void) (

version-string [O] 10 ;

strcpy (version-string ,~ServoCompensator\O) ; // This string should be changed t o name of the controller t o be

// implemented.

APPENDIX C. SCRlPTS FOR CONTROLLER DESIGNS float Cont-inputs 161 ; f l o a t Plant-inputs 131 ;

Cont-input s [O] = Ref,Fa; Cont-inputs W = Ref J b ; ~ont-inputsBJ = Ref-Fb; ont-inputs [3J = Ref S a ont-inputs M = Ref-Vb

Cont-inputs CS] = Ref Jb

- F a ~ o d;i

- Vb-act ;

- Fbaodi ;

servocoapensator(Cont,inputr ,Plant-inputs) ;

//This function should be changed to the controller function name in //the appropriate head file of the controller to be implemented. /* have to adjust the outpots here */

output-ma = Plant-inputs [O] ; output-rab = Plant-inputs Cl] ; output-mc = Plant-inputs [23 ; if ( S t a t e = 4 )

C Plant-inpute CO] = Ref,Fa*aignals CS41 ; //ref a

* ra,

a t a r t up behaviot, rolleri, are static

Plant-input s c21 = -Ref,Fb*signaïs [liU ; //refb

1

*

rc, rtart up behavior, rollers are static

APPENDLX C. SCRIPTS FOR CONTROCLER DESIGNS

132

// These coefficient values are used t o correct for the linearity of th8 // input rigiiali.

They 8hould ba modified by the co~espondingset up

// values in the future work.

@a-ref = Cont-input s [O] ; mVb-ref = Cont,input 8 C l ] ;

mFb,ref

= Cont,input s C23 ;