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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 non- exclusive 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.
The author retains ownashp, of the copyright in this thesis. Neither the thesis oor substantial extracts fiom it may be printed or otherwise reproâuced without the author's permission.
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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 mul-
tivariable 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.
1 am in debt to my farnilyt who have done ao much to make my dreams corne true.
Without their love and encouragement, L would achieve Little.
Finaily, 1 have 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
. . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background Knowledge 2
. . . . . . . . . . . . . . . . . . 1.2 The Rotoflex Web Handling Machine 3
. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline of the Thesis 4
2 Modeling For W e b Tension Control 6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction 6
. . . . . . . . . . . . . . . 2.2 Modeling for the Web Mechanicd System 7
. . . . . . . . . . . . . . . . . . . . 2.2.1 The Mode1 of a Fkee Web 7
. . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 ModelS of Roiiers 10
. . . . . . . . . . . . . . . 2.2.3 Models of Web Mechanical Systems 12
3 The Servomechanism Control Problem 25
. . . . . . . . . . . . . . . . . . . 3.1 Robust Servomechanism Problems 25
. . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Servo-cornpensator 27
. . . . . . . . . . . . . . . . . . . . . 3.2.1 The ControlIer Structure 27
. . . . . . . . . . . . . . . . . . . 3.2.2 StabiIizing Controller Design 30
4 The Perfect Control Problem 31
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction 31
. . . . . . . . . . . . . . . . . . . . . . 4.2 The Perfect Contsol Problem 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
7.2.2 Another Model for the Web System . . . . . . . . . . . . . . . 7.3 Cheap controller design . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1 Experimental Results-Controller #1 . . . . . . . . . . . . . . 7.3.2 Discussion of the Experimental h l t s Obtained Fkom Con-
troller #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Ekperimentd Redts-Controller #2 . . . . . . . . . . . . . .
7.4 High Gain Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 The Properties of the Web System
7.4.2 Design of The Sattuation Controller . . . . . . . . . . . . . . . 7.4.3 Combining The High Gain Control Design With The Saturation
Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 Simulation and Experimental Results . . . . . . . . . . . . . .
CONTENTS vi
7.5 nining Reguiator Design . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.5.1 Preliminary Ekperiment . . . . . . . . . . . . . . . . . . . . . 83
7.5.2 Results Obtained Rom Experiments . . . . . . . . . . . . . . 88
8 Conclusions Dl
8.1 Main Contributions of This Thesis . . . . . . . . . . . . . . . . . . . 91
8.2 E'urtherResearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
A Properties of the Saturation Contmller 96
A.1 Structure of The Saturation ControUer . . . . . . . . . . . . . . . . . 96
A.2 The Response of the Saturation Controlier . . . . . . . . . . . . . . . 97
A.2.1 Case 1-Direct State Feedback . . . . . . . . . . . . . . 97
A.2.2 Using an observer . . . . . . . . . . . . . . . . . . . . . . . . . 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
B.2.1 Cheap Control Servomechanism Controller . . . . . . . . . . . 107
B.2.2 High Gain Servomechanism Conttoller . . . . . . . . . . . . . 110
C Scripts For Controlier Designe 113
C.1 Matlab Codes For Controller Design . . . . . . . . . . . . . . . . . . 113
C.l.l The Script To Obtain the University of Toronto Web Machine
State Space Mode1 . . . . . . . . . . . . . . . . . . . . . . . . 113
C.1.2 The Script To Normaiize The Linear System And Add Filters 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-
troller and to Export the ControUer As A C Head File . . . . 124
C.2 C Codes For Controllet Imp1enientations . . . . . . . . . . . . . . . . 127
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 1
. . . . . . . . . . . . . . . . . . . . . . . 7.1 List of Identified Parameters 52
. . . . . . . . . . . . . . . . . . . . . . 7.2 Steady State Value Difference 83
List of Figures
. . . . . . . . . . . . . . . 1.1 Overview of University of Toronto Machine
2.1 The overall electrical/rnechanical system of a web handling machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A Fkee Span Web
. . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 A Single Tension Span
. . . . . . . . . . . . . . . . . . . . . . . . 2.4 ARollerinaWebSystem
. . . . . . . . . . . . . . . . . . . . . . . 2.5 AWinderinaWebSystem
. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 An+lspansystem
. . . . . . . . . . . . 2.7 Calculation of the inertia and radius of a winder
. . . . . . . . . . . . . . 2.8 Reduced System Is a First Order RC circuit
. . . . . . . . . . . . . . . . . . . . 2.9 A High Frequency Model Module
. . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 High F'requency Model
. . . . . . . . . . . . . . 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~%on of The Response of the BIack Box Model and The &ai
MachineUnwinder Torque Step . . . . . . . . . . . . . . . . . . . . 7.7 The Web System Structure Mer S d n g . . . . . . . . . . . . . . . . 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 (Con-
troller #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
7.20 Large Input Overshoot of the High Gain Controller . . . . . . . . . . 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 . . . . . . . .
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
7.28 Open Loop Response to Torque Step Input 6U = [0.6 O 0IT . . . . . 84
7.29 Open Loop Response to Torqne Step Input 6U = [O 0.6 O]= . . . . . 85
7.30 Open Loop Response to Torque Step Input CW = [O O 90.31~ . . . . . 86
7.31 Tuning Reguiator Ekperiment: Responses to Fa step input . . . . . . 88
7.32 Tuning Regulator Experiment: Responses to Fb step input . . . . . 89
7.33 W g Regulator Expeciment: Responses to velocity step input . . . 90
A.1 The Overd Structure of A High Gain Design With Saturation Con-
troller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
A.2 Simplified Structure of figureA . 1 . . . . . . . . . . . . . . . . . . . . . 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 nom 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 w of 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 cek (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 imple-
mented 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 ob-
tained. 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 7
2.2 Modeling for the Web 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 direc-
tions 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 (mass per 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
apprdate iy 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 F0 and 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, Fi to F,+* denote the friction torques on the rollers, Ui to 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+l Di + @, . fin+* denote the steady state operating dues .
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 (approxi-
mately 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 - , n + 1, we have:
which can be reWIrtten as:
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 9 by & and add ail of the resulting equations
together; we thence obtain:
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 cm see that the (2n + 3)th order system has been sirnplified
fist order RC circuit as given in fig 2.8: where
r-------- 1 v 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 Web 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 negligi-
ble, 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-resistor circuit of
figure 2.9.
i X(t) To(t) - i (os cwnnt)
I - ,--------A (M voltage)
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 Web 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 -
S Y C
a
- - - I
(BI (Cl
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.ng equation (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
outputs, g , ,~ E 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.
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 cded 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 ~uficzent conditions that there ezists a so-
lution 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, - -- , q.
5. y i s contained in y,,,, i e . , the outputa to be rtgulated are meosumble.
3.2 The Servo-compensator
3.2.1 The Controller Structure
Given the design lrequencies &, i = 1,2, - - - , p, define a matrix t$ E PX'
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 con-
troller 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
tracking and mb& control. The role of the etabilizing controiler is to stabilize the
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).
Figure 3.1: The general structure of a sewomechanism controller
Y *=f ;= Sem0 Ym . Cornpen- KO Plant + sator J
B. T ,
D
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 solu-
tion 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 stabilizing controiler 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
Pro blem with Perfect Control (RSPPC) [DS87].
4.2 The Perfect Control Problem
4.2.1 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.
Mead 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 pa-
rameter 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), i.e. V6 > O, VA > O and dl initial conditions =(O), €(O), ~ ( 0 ) ~ fr(0), c . (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 [KT + 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:
(a) K:(s) = 1, i.e., perfect decoupled tracking occurs.
(b) liwda K:(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 servomechanism problem
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 prob-
lem 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 the RSP with
peqfed control if and only if the follomng wndàtons dl hot&
2. (C, A, B, D) is minimal phase and m 2 r.
3. y is contained in y,,, .
Pdect Controuer Design U~ing Cheap 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)+ Bu
where Ka is the observer gain found so that A - K.C, is stable.
O ther 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 systemts infinite transmission zeros (12) will be intmduced.
Chapter 5
A High Gain St abilizing 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 Mat hematical Preliminaries
5.2.1 Transmission Zeros
Let G(s) be a rational transfer hinction matrix; then it can be trandormed to the
Smit h-McMillan form.
where 4 ( s ) , &(a) are polynomiais with the property that for aîi i = 1,2, , r, E ~ - ~ (s)
àivides ~ ( s ) and &-&) divides &(a).
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 wil l 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 cded 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-
dues , 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
sum of open loop pales)-(sum of open loop zeros) h = i 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 ordm 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 i th order
zeros, then for every ith order 12, Say, j th 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 neces-
sarily 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 non-
square 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 non-
singular matrices P, Q € Ilmxm such 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 IZ s t ~ c t u r e can 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(4 me 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 IZ while 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, , h, and consader
the high-gain cornpewotor (5.13) toith (5.16) and (5.1 y), where ri 2 ai - 1. Choose
po = 1 , p j = 2 , j=1,2,-•,h, G1 =QG, 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
where a (s) 9 daag(t& (s) , ci&), - - - , b'(s) ), with bi (s ) a monac ml-coeficient Hurunruntz
polynomid of degwe Ji - 1, i = 1,2, -- , m, and where Ï'(s) is an upper tdangular
polynomzal matriz with an integer 1 on 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 f i; 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*, - - , & are
the set of the zeros of the Hurwitz polyn0rnial6~ (8) , 6~ (3) , ,if,,, (s) respectively.
The above property of the system C (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- , wi i l 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, - , rm are 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, co~ ider the 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 dedbed 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 STABILI '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 Regulat or Cont roller
As discussed before, to solve the robust servomechanism problem using the serve-
compensetor, 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 GAi is 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 K2 can 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, 30 that on
opplying u = &q, where K = (D - CA-%)+ 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 Implement at ion
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 Univer-
sity 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.
7.2 Models for Controller Implementat ions
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 1 Unmodeled Part (Active Rollers) 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 YA denote 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
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
d u e 0.0415m 0.0415m 0.0415m 0.0175kgm' 0.0322 kgm2
3
Parameter 1 meaning roa 1 Radius of the bare unwinder Tb
Tob
JO= Jb
Radius of transport roUer (Nip) Radius of bare rewinder Inertia of bare unwinder Inertia of transport roller
JO= b~ bb
4 e
Inertia of bare rewinder 1 0.0234kgm2 Damping fiction coefficient of unwinder Damping fnction coefficient of transport roller Damping fiction coefncient of rewinder Web thickness
1/254Nms 1/165Nms 1/258Nmo O.O015/(2xn) m
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 oper-
ating 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
for the response of the real machine. This is about 5 times faster than the of tixm 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 (YA or 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 multivarïable state 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 IdenWied Model and The Real Systern 30
1 - Real I 1 I 1 I I I ! i.. .* - - . ........ ........ ....... -1
t5 I 1 t 1 I I
O 2 4 6 8 10 12 14 Tendon Fb Response of the Identifiecl Madel and The Real System
Figure 7.4: Cornparison of The Response of the Black Box Model and The Real Machine-Rewinder Torque Step
13l I I I I I I I
O 2 4 6 8 10 12 14 Speed Response of the Identifid Model and The Red System
1 . & 1 I ! i I I 1
0.8 .......... . . . . . . . . . . . . . . . . . . . . . :. ........................... .:-. . . . . . . . . . . . .; . . , - - - Real - - Identifid
Tension Fa Response of the ldenaiffed Model and The Real System
1 1 I 1 I I I O 2 4 6 8 10 12 14
Tension Fb Response of the Identifieci Model and The Real Systern
Figure 7.5: Cornparison of The Response of the Black Box Mode1 and The Real MachineNip Torque S tep
1 1 1 I 1 1 I O 2 4 6 8 I O 12 14
Speed Response of the ldentified Model and The Red System 1 1 1 I 1 I I
0.8
. . . . . . . . . . . . . . . . . . . . . . . . . . < . . . . *
. . . . . . . . . . . . . . . . . . . . . L . . . . . . . . . . -
1 1
O 2 4 6 8 10 12 14
- Real - - ldentified - .......... :.. .......................................... i . . . . . . . . . . . . . . . . . . . . . . -
-
Tension Fa Response of the ldentMied Model and The ReaI Systern
2 4 6 8 10 $2 14 Tension Fb Response of the ldentified Model and The Real System
121 1 1 I 1 1 1 1 O 2 4 6 8 10 12 14
Speed Response of the Identifid Model and The Real Systern 0.4 . 1 1 I I I 1 - Real
.......... ; ............. .:. ............ .; . . . . . . . . . . . . . * . . . . . . . . . . . . . . . . . . . . . . . -
Figure 7.6: Cornparison of The Response of the Black Box Model and The Real Machine-Unwinder Torque Step
In the foilowing sections, controllers will be designed based on both modeis and
the results obtained WU be compami.
In thh section, three design methodologies: cheap control design, high-gain controL
design and tuning regdatoc design wiii be camed out.
7.3 Cheap controller design
Choosing the Right ScaIing
It cm be observed that both mode1 #1 and #2 of the web system form a controllable
and observable system. However, we note that the outputs of these models c m have
very Werent scale ranges, i.e., the tensions can vary from ON to over IOON, whereas
the machine speed varies only within f 5mls for normal operation. To avoid numerical
problems in the controller design, we therefote need to nomalize the outputs.
The following transformation is made:
This normalizes the magnitude of the output to lie in the interval [O II (see figure
Design of the Servocompensator
The reference sipals to the system are assumed to be constant corresponding to
the tension and velocity set points which are to be tracked. There are several main
sources of disturbance to the web system:
1. Constant type o&t signais. These signais include sources such as the static
fiction torque.
2. Periodic type disturbance signala. These signals inchde disturbances associ-
ated with the mechanical vibration in the web system such as the periodical
disturbance arising from unbalanceci roiiers.
3. Stochastic type disturbance signals. These signals include Bignals such as the
wound-in tension Tw(t) from the unwinder, which according to (2.15), has very
Iittle influence on the system d y n d c s .
Since the fkequency information of the periodic type disturbance and the spec-
tnim information of the stochastic type disturbance are unknown, the assumption of
sssurning constant disturbances d only be made.
Thus, the servocompensator can be simply chosen to be:
where y is the plant output, yref is the reference signal and 6 is the servocompensator
state,
The b a l closed loop system has the structure as given in figure 7.8.
where T = diag(&, &, &) and K is found by solving the cheap control problem
described in chapter 4.
7.3.1 Exparimental Results-Controiier #1
A controiier is first designeci based on the mode1 #1, and the resulting controller is
called controlier #1. The foiiowing plots @ven in figure 7.9 and figure 7.11 give the
Figure 7.8: The Structure of the Overall Controller
closed loop response for the above controller when the cheap control gain c is chosen
to be l e 4 and the observer gain eo is chosen to be l e 4 The complete mode1 of this
controlier can be found in Appendix B.1.
Cheap Conttol Servo Controller experiments, Fa response to Fa step
36.5 37 37.5 38 38.5 39 39.5 40 40.5 41 41.5 Cheap Control Servo Controller experirnents, Fb response to Fa step
50 1 1 I ! 1 I i I 1 I - Response - - Reference
4 - ........................ ;. ...... .;. ...... -; ....... .:. ....... .:. . . . . . . . ; . . . . . . . . . . . . . . . . . . . . .
6 . 37 37.5 38 38.5 39 39.5 40 40.5 41 41.5 Cheap Control Servo Controller experirnents, Vb response to Fa step
1 1 1 1 I 1 I I I I I
. . . - Response ...... : . . . . . . . . : . . . . . . . ; . . . . . . . .:. ..... .:. . . . . : . . . . . . . . . . . , . . . : - - - Reference
Figure 7.9: Output Responses to Tension Fa Step Reference Input (controller #L)
Cheap Control SWO Controller experimnts, Fa response to Fb step
05.5 46 46.5 47 47.5 48 48.5 49 49.5 50 50.5 Cheap Control Servo Controller expen'rnents, Fb response to Fb step
45.5 46 46.5 47 47.5 48 48.5 49 49.5 50 50.5 Cheap Control Sewo Controller expen'ments, Vb response to Fb step
Figure 7.10: Output Responses to Tension Fb Step Reference Input (controller #1)
Cheap Controt Servo Controtler expetirnents, Fa msponse to Vb step
L W
13.5 14 14.5 15 15.5 16 16.5 17 17.5 18 18.5 Cheap Control Sewo Controller experiments, Fb response to Vb step
1 - - Reference / j 40 -. . .., .......................... .; ........ ;. ...... .;. . . . . . . .- . . . . . . . ..:. . . . . . . ..:.. . . . . . * : . . . . . . . . ; . . .
30- : - - .
Cheap Control Sewo Controller experirnents, Vb response to Vb step
V
13.5 14 14.5 15 15.5 16 16.5 17 17.5 18 18.5
Figure 7.11: Output Responses to velocity Step Reference Input (controller #1)
1 I I b I 1 I I I I I
. - Response .: ; ; : .: , ; ..,. ................. . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . .
- - Reference - L , .. . . :. . . . . . . . .:. . . . . . . . i . . . . . . . . . . . . . . . . . . . .: . . . . . .: . . . . . . . . . . . : . . . . . . . .
7.3.2 Discussion of the Experimental Results Obtained Rom
Controiier #l
(a) Transient m o n s e and Interactions
A very important criteria for controlIer design is the transient response time, As we
can see fiom figure 7.9 to 7.11, the teneion response has a settling time of approxi-
mately 0.3s and a speed response of less than ls, which is much faster than the SIS0
PID controllers commarciaiiy used in web system control. Another criteria for con-
troller performance ie the interaction between outputs, i.e., it is desired that tension
set point change in one span should not cause variation in the response of the tension
of the other span or the machine speed. A h the speed set point change should cause
as little variation in the tension responses. As can be seen in figures 7.9, 7.10 and
7.11, the largest interaction occurs when the speed set point is changed. However, in
this case the variation of tension is approxhately 1N (corresponding to less than 5%
of the tension set points), wbich is negligible by the industrial criteria of 25% of the
tension set point values.
(b) Steady Stata Variations
The maximum variation of tension at steady state is approximately 1 to 2N (corre-
sponding to 2% - 5% of the tension reference values), which is much lower than the
required industrial criteria of 10% of the set point d u e .
(c) Cornparison With Theoretic Rasults
It is interesting to compare the experimental rmlts of the real machine with the
resuits obtained from the simulation of the identified model #1 to determine the
effect of unmodeIed uncertainties on the overall closed loop system.
It is obsewed fiom figure 7.12 that the experimental results and the results o b
tained fkom the simdation on the model #1 are very similar, They have almost the
same response speede. It is interesting to note that the large uncextainnty of system
dynarnics causeci by the idem as discussed in section 7.2.1 has very little effect on the
CHAPTER 7. CONTROLLER WLEMENTAZ7ON 67
Comparison of SImulated and Experimental Rqsuits: Tension Fa Step Reference 1 I 1 ! I
.............. .:. ..
Comparfson of Simulated and Experimental Results: Tension Fb Step Reference
...... 35 - . . . . . . . . . . . . . . . . . . . . - . . a . .
: - Actual Response 30 - ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ; - - Reference -
: . - - Sfrnulated Response 2s I i 1
45 46 47 48 49 50 5 1 Comparison of Simulated and Experimental Results: Velocity Step Reference
0.8 1 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;. . . . . . . . . . . . . . . 2 . 4
. .-
: - ActuaI Response . . . . . . . : . . . . - - Refetence - . . .
: . - - Simulated Response 0.4 I 1 I 1 1
13 14 15 t6 17 18 19
Figure 7.12: Comparison of the closed loop responses of the experimental results and the theoretical results of the mode1 #1 (controller #1)
closed loop system. This is because that the large uncertainties in system dynamic
matrix A are very s m d compared to the feedback tems.
(d) mirthet Discussion-Control of Nodinear System
To use linear control design for noalineu systems, the general practice is to treat
the nonlinear system as a family of LTI systems at several operating points, which is
indexed by some state variables and possibly some exogenous parameters, and then
design a family of h e m controllers for each of the fixed operating points, and thence
use gain-scheduling. In the web system, the dominate nonlinear aspects of the web
CHAPTm 7. CONTROLLER XlMPLEMENTATION
machine c m be approurimated by 8 M y of LTI systems:
which is parameterized by the radius ru. A new LTI approximation of the web system
(corresponding to a new value of ta) and a new controller is then designed when the
desird perlomance of the LTI controiler can not be maintainecl for the closed loop
noniinear system. Figure 7.13 and 7.14 give the response of the nonlinear web system
(7.10) at different operating points, using the previous designed LTI cheap control
servocompensator controller #l. It is interesting to note that for dl of the operating
ranges, the LTI controiler psoduces similar responses. Thus, the gain scheduhg
procedure described above is not necessary.
Figure 7.13: The Simulated Closed Loop Response of Tension Fb for Difterent Oper- ating Points (controuer #1 applied to mode1 #1)
Figure 7.14: The Closed Loop Experimental Response of Tension Fb for DiEerent Operating Points (controller #1)
Rom the modehg discussion of chapter 2, the scheduling variable ro has the
dynsmic equation:
and so the variation speed of ra is detennined by the web speed and the web thickness
e. As a fhrther check, it is shown in figure 7.15, that under the same Bingle LTI con-
trouer #1, the clased loop nonlinear system remains stable and does not sigdicantly
deteriorate in response for web thickness up to 5mm for the same web material.
Figure 7.15: The Closed Loop Simulated Response of Tension Fb for Different Web Thickness (controiler #1 applied on model #1)
7.3.3 Experimental Results-Controller #2
A cheap servornechanism controller was also designed based on the 6th order model
#2 with c = IO-^ and eo = 104. This controlier is called controller #2. Figure 7.16
to 7.18 show the experimentd results obtained when this controller is applied to the
web machine. A complete description of this controller is given in Appendur B.1.
Cheap Control Sewo Controller experirnents, Fa response to Fa step i k 1 1 I
1 - Response 1 ! I !
1 - - Reference 1 '. ..................... .". ........... . . . . . . > . . .
. . . . . . . . . . . . , - . : . . . . . . . . . . . . . : . . . . . . . . . . . . . "
- - -
22.5 23 23.5 24 24.5 25 25.5 26 Cheap Control Sewo Controller experfments, Fb response to Fa step
50 1 1 I I I I I I - Response - - Reference 40 - . ............................. .:. - . . . . .- . . . .,;. ........... ;. . - . . . . . . . . . .:. . . . . . . . . . . . ,:. . . . . . . . . :. -
.. .;. . . . . . . . . . . . . . . . . . . . . . . :... . . . . . . ; . . . . . . . . . . . . : * . . . . . . ..............................,,
201 ' I 1 1 t I 1 I
22.5 23 23.5 24 24.5 25 25.5 26 Cheap Control Senro Controller experiments, Vb response to Fa step
1 1 I 1 L l I I , - - Response . . . . . . . . . . . .: . . . . . . . . . m . . i ............ ; . . . . . . . . . . . . ;. ........... .:. . . . . . . . . . . . : - - - Reference
Figure 7.16: Experimentd Results of Response To Tension Fa Step Input (Controller #a
L W -- -- - - -
925 93 93.5 94 94.5 95 95.5 Cheap Control S m Controller experiments, fb response to Fb step
Cheap Control Servo Cantroller experiments, Fa response to Fb step 50 - r I 1 I I I t
50 r I I I I 1 I
- Response
40
"' 92.5 I I I I 1 1 I
93 93.5 94 94.5 95 95.5 Cheap Control Servo Controller experiments, Vb response to Fb step
Figure 7.17: Experimental Results of Response To Tension Fb Step Input (Controiler #2)
- Response - - Reference
: ; . . . . . . . -*. ............... .*. ............... .:- ............ ; ............ ; . . . . . - . . . . . . .;. . . . . . . . . . . . .:. -
Cheap Control Sewa Controller expwiments, Fa response to Vb step
201 ' I 1 I I I 1 I J
71 71.5 72 725 73 73.5 74 74.5 Cheap Contrat S e m Controller experiments, Fb response to Vb step
50 j 1 L I I L I 1 1 - Response - - Reference a-: .............,...'......................;.................................................. . . . ; .
Cheap Control Servo Controlfer experiments, Vb response to Vb step J 1 I I 1 r I 1
- Response . . . . . . . . . : . . . . . . . . . . . ; . . . . . . . . . . . . .:. . . . . . . . . . . :. . . . . . . . . . . ; . . . . . . . . - - - Reference
Figure 7.18: Experirnentd Results of Response To Velocity Step Input (Contmiler 12)
We can observe in this case that the controller #2 based on the n = 6 model
#2 provides excellent tracking for the closed loop system. Interestingly, however, the
response is not better than the controller #1 designed based on the n = 3 model #1.
7.4 High Gain Controller Design
7.4.1 The Pmperties of the W e b System
The high gain controlier is designed based on the reduced order low frequency model,
which is a minimum phase system. Hence the high gain controiler design methodology
introduced in chapter 5 can be applied.
CHAPTER 7. CONTROLLER IMPtEMENTAïïON 74
As a furthet observation, we observe that the web system to be controlied (mode1
#1) has a fidl rank CB matrix, which means that it has no finite transmission zeros
and has only 3 fust order infinite transmission zeros, with no higher order inhite
transmission zeros.
7.4.2 Design of The Saturation Controller
A large obstacle for hi& gain controUer implementation is the input saturation prob-
lem, Le., as the loop gain becornes larger, the plant inputs become larger than which
the teal physicd plant can provide. Given now the high gain controller design intro-
duced in Chirpter 5, assume that the high gain controller is chosen to be the form
as introduced in (5.23) of section 5.4.2. This implies that when the web system to be
controiled is started from rest (y@) = O), the control input amplitude will h~comr
unbounded as é + 0.
The input saturation problem is a common issue for dl controller designs. However
it is of particular importance in the high gain control design because unlike other
design methodologies such as the cheap control design, decreasing the loop gain may
result in an unstable closeci loop system for the high gain design. Clamping the
control input signal also may not solve the problem because this will not guarantee
overall stability or asymptotic tracking of the system.
To resolve this problem, we will design a cascaded saturation controller and high
gain controller structure as indicated in figure 7.19.
The saturation controller is chosen to be the following[DJ91]:
where 5 is the obeerved state of the composite system consisting of the high gain con- - -
trollet and the plant, and A, B, C and D are the system matrices of the same system
t r r o r x - Y D + cœladk Plant '
Figure 7.19: The Overall Structure of a High Gain Design - Saturation Controller
(see figure 7.19). z is the saturation controller output, and cl > O is the saturation
controller gain. It can be noted that in this saturation controller design, a servocom-
pensator is included, and hence cloaed loop asymptotic tracking is guaranteed.
Properties of the Saturation Controller
To illustrate the properties of the saturation controller, it is necessary to examine the
plant input under a high gain controller to see what elements should be constrained
in the p l a t input.
Figure 7.20: Large Input Overshoot of the High Gain Controller
Figure 7.20 shows a representative plant input response of the closed loop system
composeci of the web system (mode1 #1) and a high gain controlIer under a step
tension reference set point change. It is observed f h n figure 7.20 that at the reference
change instant, the error of the output and the reference signal becomes very large,
which causes the large overshoot shown in figure 7.20.
The saturation controller (7.13) h a the property that by on-line tuning the satu-
ration controller gain Q, we can reduce the overshoot of the plant input to dowable
values. The smder the gain y, the smaller the overshoot. However, reducing Q wiil
&O slow down the plant output response.
The detaiied analysis of the properties of the saturation controiler is presented in
appendix A.
7.4.3 Combining The High Gain Control Design With The
Saturation Controller Design
The design is c d e d out in two stages. First we design the high gain controlier
to stabilize the web systern and achieve a "best" response, and then we apply a
saturation controller to constrain the input amplitude.
The two stage saturation and high gain design methodology provides the free-
dom to achieve desirable control properties and satisfy the control input constraint.
The following section gives some representative simulation and experimentd r d t s
obtained using the above high gain-saturation controller design on the Rotoff ex ma-
chine.
7.4.4 Simulation and Experimental Results - Cornparison of Simulation Results
The high gain controller is implemented using the decomposition method introduced
in section 5.4.1 (the complete design scripts can be found ui appendiv C.1.4). The
high gain controllet gain is chosen to be 4e - 1 and the saturation controller gain is
CHAPTER 7. CONTROLLER WLEMENTATION 78
chosen to be 30 in the find design (see appendix B.1 for the compIete mode1 for this
controller).
The resulting controller displays very good control properties.
Simulation rmuits show that the high gain controller ha9 Little interaction between
the output channels (almost achieving decoupled control), under the sarne control
magnitude input constraint as useci in the cheap control design. As shown in figure
7.21, we see that a 0.3m/s speed set point change approximately causes only 0.05%
of variation in tensions which is some 10 t h e s better than the cheap control case (see
figure 7.22).
Figure 7.21: Simulated Output of the High Gain Controller
Figure 7-22: SimuIated Output of the Cheap Control Perfect Controller
Figures 7.23 and 7.24 give the corresponding control inputs for the high gain-
saturation controiier, and the cheap controiler design. It is interesthg to note that,
with ahnoet the same contml sort (the input amplitude), the high gain-saturation
controiier produces much better control results than the cheap gain control design.
Figure 7.23: The Input of the High Gain Controller
Figure 7.24: The Input of the Cheap Control Perfect Controiier
Figures 7.25 to 7.27 give the experimental closed loop responses obtained on the web
machine 6 t h the above high gain-saturation controIler applied. It is obsenred that
the hi& gain-saturation controuer provides a response speed similar to the cheap
control perfecf controllet.
C W T E R 7. CONTROLLER IMPLEMENTATION
It is also seen fiom figures 7.25 to 7.27, that the clased loop system under high
gain-saturation control shows no interaction when a refarence set point changes, which
is an improvement compared to the cheap control experimental results of figures 7.9
to 7.11.
High W n Controller experiments, Fa response to Fa step 50 1 1 I I - Response - - Reference v-
40 -.. .......................... :...., ...... ....*..........;....,.....,.*....:,...............
20 t I I I 1 1 15 16 17 18 19 20
High Gain Controller experiments, Fb responae to Fa step 50 - 1 I 1 i I
Figure 7.25: Output Responses to Tension Fa Step Reference Input
40
High Gain Controller experirnents, Vb response to Fa step
- Response - - Refemnce - .. . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
, 1 I 1 1 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 r - - Response - - Reference
High Qain Controîier expefiments, Fa response to Fb step
Hlgh Gain Corrtroller e~perimnts, Fb response to Fb step
I I I I I I - Response - - Reference
*O' 2'5 I 1 t 1 1 I
26 27 28 29 30 High Gain Controllet experirnents, Vb response to Fb step
: i
50
40
30
-...,,..............., .............................................................................
Figure 7.26: Output Responses to Tension Fb Step Reference Input
1 r ! t I I 1
I 7 1 I I I
- Response - - Reference : :
... i . . S . . . . . . . . . . . . i .............. .i.. . . . . . . . . . . . . . :. , . . . . . . . . . . . . . : . . . , . . . - i
, .
- . . . . . . . . . . . . . . . . ; ... *
- .................... ; w
- Respon- - - Reference
CHAPTER 7. CONTROLLER IMPLEMENTATION
High Gain Contmtler experiments, Fa response to Vb step
20 t t 1 1 t t t 1 I 1 1 1 33 33.5 S 34.5 35 35.5 36 3ô.5 37 37.5 38
Hiqh Gain Controller experiments, Fb response to Vb step
20 t t 1 1 1 t I 1 I 1 1 1 33 33.5 34 34.5 35 35.5 36 36.5 37 37.5 38
High Gain Controller experiments, Vb response to Vb step I r . I I 1 l I 1 I I 1
1 - Response ... ; .*..... .: . . . . . . . :. . . . . . . . ; ... . . . . .: . . . . . . . : . . . . . . . . . . . . . . . .:. . . . . ;. . . . - - Reference ; - - - - - - . - . F
I
Figure 7.27: Output Responses to velocity Step Reference Input
7.5 Tuning Regulator Design
From the previous controller implementation discussion, we recognize that a "good
identified model" is very important in controller design. However, this identification
process is a very tedious procedure, and a reliable model is always very ~ M c u i t to
obtain. Hence a diaerent methodology which bypasses the plant model identification
procedure of controuer design is appealing. If the plant is assumed to be LTI and
open loop stable, then it is shown in [Dav76a] that a tuning regulator design approach
can be used which requires no mathematical model of the system.
In web control, the main control objective is to track constant tension and constant
velocity set points, and reject any constant unmeastuable disturbance in the system.
Rom the r d f s in chapter 6, the eolvabiüty verscation condition and controuer
design for this problem can be easily done by carrying out a set of open loop step
response experiments.
On using the approach of pav76a], it is assumed that the plant to be controlled can
be described by a LTI model. In the case when this annimption can not be satisfied,
then the approach of [Dav76a] can be extendeci to nonlinear plants by applying gain-
scheduling techniques, e.g., see [SD86].
In the case of the web system, the analysis canied out in the modehg section
of chapter 2 indicates that the web system is in fact a nonlinear system. However
this analysis also indicates that the web system cm be approximately described by
a LTI rnodel, and so this assmption wiil also be made here (as was done using the
previous controlier design approaches). The analysis of chapter 2 also indicates that
the web system c m be described by a model which is open loop stable, as has been
demonstrateci by experiment, and so the assumptions required by the tuning regulator
approach hold.
A set of open loop step torque input experiments was then carried out on the
University of Toronto web machine for the case when ro = 0.0732m and rc = 0.0853m,
and figures 7.28 to 7.30 give the response of the outputs Ta, v and Tb.
The correspondhg steady state values obtained under these step inputs are listed
in table 7.2.
Table 7.2: Steady State Value Difference
1 Steady State Value Difference ' $
0.3667 0.3778 -0.1453
t5Fr 0.7962
1
0.7796 5.2389
1
7.7637 -1.5849 1.3173
Torque Step Input bui = Torque Step Input dU' = TorqueStepInputbU3=0,
>.6, 0, OIT g, 0.6, OIT
O, -O.3lT
Open bop Tension a Response to Toque Ua Step lnput
Figure 7.28: Open Loop Response to Torque Step Input t5U = [0.6 O O]=
N 1 I
35 - 1
- - steady state value
=O I I 1 1 t 1
2 4 6 8 10 12 14 Open Loop Velodty Response to Toque Ua Step Input
1 I I J 1 I 1
0.8 - - 2 -
0.4 - - - - - - - - - - -
- actual response - - steady state value I - ) I
O 2 4 6 8 10 12 14 Open Loop Tension b Response to Torque Ua Step lnput
24 t 1 I I 1 1 1 O 2 4 6 8 10 12 14
Open toop Velodty Response to Toque Ub Step Input 1 - 1
A 1 1 I l t - actual response - - steady state vaiue
0.6 - - 0.4 - 3
0.2 I 1
O 2 4 6 8 10 12 14 Open Loop Tension b Response to Toque Ub Step Input
- actual response - - ~teady $tût9 ~ 1 ~ 9 - 21 t 3
20 - II
Figure 7.29: Open Loop Response to Torque Step Input 6U = [O 0.6 O]*
CHAPTER 7. CONTROLLER IMPLEMENTATION 86
Opecr Loop Tawion a Response to Toque Uc Step lnput
32 -
- actuaf response - - steady state value
=0 I I I t I I
2 4 6 8 10 12 Open Loop Velocity Response to Torque Uc Step lnput
0.4 I I I I I I - actual response
o:, 1 I t I I I I 2 4 6 8 10 12 14
Open Loop Tension b Response to Toque Uc Step lnput 301 I I I I I 1 I
20- - - - - . z - - - - - - - - - C - - - - - - - actual response
- - steady state value 15 I I t I I I
0 2 4 6 8 10 12 14
Figure 7.30: Open Loop Response to Torque Step Input bU = [0 0 -0.31~
According to the results of chapter 6, when the control objective is tracking/rejecting
constant signals, the information obtained in table 7.2 is only required in order to
solve the servomechanism problem for the web system using the tuning regulator
controller approach.
Determination of the Steady State Gain Matrix G
Definition 7.1 Given a LTI system:
the steady-steady gain matriz of the system (7.14) for constant reference input signab
is given byr
It than foiiows from chapter 6 that there exista a servomechanism contmller for the
web system for constant reference/disturbance signals if and only if rank(G) = dim(y)
the aumber of inputs of the web system is at least equal to the number of outputs
and the gain mat* G has a rank not smaller than the number of outputs.
For the web system used in this experiment, the gain matrix G cm be determined
from table 7.2:
and there exisfs a solution to the servomechanism problem since:
rank(G) = 3 = number of outputs
The tuning Reguiator is given by:
where Tm=[, v, and Tw are reference signals for tension a, velocity and tension b
respectively, and Tu, u, and Tb denote the actual measured tension a, velocity and
CHAPTER 7. CONTROCLER WLEMENTATTON 88
tension b. ~p > O and q > O are scaiers found by "on-line" tuning to achieve the
best control effect.
7.5.2 Results O btained R o m Experiments
After applying the controller 7.18 to the real plant, the resdtant closed loop step
responses obtained are given in 7.31 to 7.33.
TuninaReg experiments, Fa response ta Fa step 50 . 1 I I ! I t - Response - - Reference . ..........,..... .'* ...... ....................................... :.
20 I i I I 1 I 1
29 30 31 32 33 34 35 TuningJeg experintents, 1% response to Fa step
20 I I I 1 I I I
29 30 31 32 33 34 35 TuningJeg experirnents, Fô tesportse to f a step
50 1 r I 1 1 I I
40
Figure 7.31: 'Iiining Reguiator Experiment: Responses to Fa step input
- Response - - Reference - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : .
L I r ï I 1 I , . - Responci. - - Reference ....... :. a . . . . . . . . . . . :. ............ .:* . . . . . . . . . . . . . :* . . . . . . . . . . . . .: . . . . . . . . . . . . . : -
TuningJeg expertments, Fa response to Fb step - - -
50 . 1 t 1 1 t I - Response : - - Reference f
qg ;.. ................................... ;- . . a . . . . .......; ............... 1.. ........... ...:. . . . . . . . . . -
"' 52 I t I I I 1
53 54 55 56 57 TuningJeg expsriments, Fb response to Fb step
Figure 7.32: M g Reguiator Experiment: Responses to Fb step input
*O' 52 1 1 I 1 1 I
53 54 55 56 57 TuninaReg experfments, Vb response to Fb step
, 0.5' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I I I 1 I I
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... : : . . . . . . . . . . . . . . .; . . . . . : - i
, - Response , - - Reference
CHAPTER 7. CONTROLLER IMPLEMENTATlON
TunimReg enperimenls, Fa response to Velocity step
13 14 15 16 17 18 19 Tunin~~Reg expetiments, Fb response to Velocity step
201 ' I 1 I 1 I 1 1
13 t4 15 16 17 18 19 Tunin~Reg experiments, Vb response to Veloclty step
- 13 14 15 16 17 18 19
Figure 7.33: nining Regulator Experiment: Responses to velocity step input
Behavior of the Tuning Reguiator
Rom figure 7.31 to 7.33, it can be observed that the tension responses display satis-
factory behavior with respect to both transient state and steady state, with the time
constant less than 1 second. However, we aiso observe that the speed input response
is somewhat oscillatory. This could be improved by adding a rate feedback term in
the controller 7-18.
Chapter 8
Conclusions
8.1 Main Contributions of This Thesis
The web handiing system has been treated as a multivariable servomechanism control
problem in this thesis, and three different types of servomechanism controllen have
been designed and implemented on the University of Toronto web machine.
1. A new modeling method has been presented. Using this method, a high fie-
quency model and a reduced order low fiequency model have been developed.
2. A 6th order LTI model of the web system was initially obtained by applying
"black box" identification techniques, which appeaxed to have a better fit to the
input/output behavior of the web system compared to the nominal 3rd order
LTI benchmark model used in the web system.
3. Using both the n = 6 and n = 3 models of the web system, a "cheap control
smrnedianiwi controiIer" was designed and implemented on the University
of Toronto web handling machine. The r d t i n g closed Ioop system displayed
excellent tension and speed respoose, and the output response was very fast
with very üttle interaction occuring between output channeIs.
4. A new type of controiier-a " high gain-saturation servomechanism controiler" ,
was introduced and implemented on the same web system.
This confroIIer design produced excellent tension and speed response, with zero
interaction occuring between the output channels.
5. Finally, under the assumption that a mode1 of the web system is not avail-
able, on carrying out a set of steady state experiments, a Utunhg regulator
servomechanism controller" was designed and implemented on the University
of Toronto web machine. This controller also achieved excellent response.
8.2 Further Research
A desirable goal of semmechanism contmller design is to design a controiler that
achieves the required control objectives with as little a priori knowledge of the un-
derlying plant as possible. The tuning regulator is an example of a very good design
methodology which achieves this goal. However, given that the web system param-
eters may change over time due to aging, mechanicd Wear, etc., it may be desirable
to have a controller which "adapts" to such changes. Hence some type of adaptive or
self-tuning modification of this controlier is needed i n future implementation of the
tuning regulator design methodology to industrial web systems.
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Appendix A
Properties of the Saturat ion
Cont roller
In the high gain controlier design, due to nature of the high loop gain, when the
error is large (Le. at the instant of reference set point change), the input to the plant
may become too large for the physicai plant to provide. In this AppendUr, a modified
type of saturation controiler wiiI be introduced. In this saturation controiler design,
given a plant and a high gain controller, by applying on-line tuning of the saturation
contmîler gain, it c m always be ensured that the maximum value of the plant input
can be reduced to acceptable levals.
A.l Structure of The Saturation Controiier
The sole purpose of the saturation controiler is to limit the plant input amplitudes.
Hence we will nrst design our high gain controller (or other type of controiler), and
then apply the saturation controller to the resuiting closed loop system consieting of
the high gain controller and the plant. This type of controller structure is illustrated
in figure A.1.
APPENDDC A. PROPERlïES OF THE SATURATTON CONTROLLER 97
Figure A.1: The Overdl Structure of A High Gain Design With Saturation Controller
The saturation controller is chosen to be the following[D J911:
where 5 is the can be the exact state feedback or the estirnateci state of the closed loop
system consisting of the high gain controller and the plant. We denote this composite
system to be (c, A, B, 6). Here z is the saturation controller output, and €1 > O is the
saturation controller gain. It can be noted that in this saturation controlier design, a
servocompensator is included; hence closed loop asymptotic tracking is guaranteed.
A.2 The Response of the Saturation Controller
A.2.1 Case 1-Direct State Feedback
On camying out a simple coordinate transformation:
we can solve for z(t) in figure A.1, aemuning that the reference input yrcf and the
disturbance w are constant.
A P P m I X A. PROPERTES OF THE SATURATION CONTROLLER 98
This response ha9 the Laplace trandorm of:
which means that figure A.1 c m be transformed to a simpier structure in figure A.2
HiIi- . u Y
Qmdla Plant
Figure A.2: Simpiified Structure of figureA.1
where TI is a mat& of filtele of the fom:
where r is the number of outputs of the plant.
Denote the transfer function matrices of the high gain controller and the plant to
be Gh(s) and Gp(s) , then the inputs u(s) to the plant can be expressed as:
which can be expressed as:
APPENDIX A. PROPERTES OF TtIE SATURATION CONTROLLER 99
where Go is a constant matrix and G8(s) is a proper rational matrix of s.
The decomposition of (A.6) exiets and is unique since we can write G as:
where, due to the fact that both the high gain controller and the plant transfer
hnction matrices are proper, every element g&), i = 1, - , m; j = 1, - * , T of G
is proper. Hence we cm decompose g,(s) into a constant part and a part which is a
multiple of s:
where nfj, d>j, and 95 are constants, nfj(s) and G j ( s ) are polynomial function of s,
and g,rj(s) is a proper rational function of S.
Then the result (A.6) arises directly fiom (A.9).
Thus we c m decompose u(s) as u(s) = q (s) + us (9):
where:
and
(A. 10)
APPENDlX A. PR0PELM:IEhS OF THE SATURATON CONTROLtER 100
The Relationship Between uo(t) And The Saturation Control Gain 4
The correspondhg time domain response of uo(s) is:
We can eady conclude from A.12 that the response of uo(t) has the follwoing
properties:
1. The steady state response of ~ ( t ) is Gaz,, .
2. These is no overshoot in the response of uo(t).
3. The rate of the change of uo(t):
at any instant t can be made arbitradly srnall by simply decreasing €1, Le., the
input can be made as slow as possible.
We can see kom the above properties, that the input uO(t) provides the necessary
input for asymptotic tracking. Hence we can cal1 it the tracking input.
The Relationship Between u,(t) And The Saturation Control Gain cl
On expanding us($), we obtain:
APPENDIX A. PROPERTES OF THE SATURATION CONTROLLER 101
We can now solve for u&) by canying out an inverse Laplace transform of
u$(s), i = 1, - * - ,m. We wiil do this by carrying out the inverse Laplace transform
of each single component of u;(s):
(A. 15)
where a = 1,- ,m andj = 1,- , r .
Rom equation (A.9), we lmow that set of the poles of G,(s) lies in the set of the
poles of G(s); hence we can express g&) ae:
where Ckl, k = 1,- ,&, k = 1 , e - ,6& and Cd, a = 1,- ,&, k = 1,- ,da are
constant coefficients. Here -Ak are the real stable poles of G(s), and -ok f jwk are the
complex conjucate pairs of stable poles of the closed loop composite system consisting
of the plant and the high gain controlier. Jk is the multiplicity of the kth real pole
and 6, is the multiplicity of the compler conjugate poles.
C1(u; j (s) ) can be obtained by solving for each
(A. 17)
and
(A. 18)
For simplicity, we wiil just discw the case of (A.17). Similar results c m be
obtained for the case of (A.18).
APPENDIX A. PROPERTES OF SATURATlON CONTROCLER 102
Now kom complex analysis, we lmow that:
&* = (* + z=i (,+*,y " -.) w here:
And so fiom the resdt of (A.20), we obtain:
' ti-le-Art Coe-qt + C ci - (i - l)!
A(<) "1 ,
On examining the structure of uk(t) given in A.21 and A.20, the following prop
erties of u,(t) now directly follows:
1. The steady state response of u,(t) is O.
2. The maximum amptitude of every element u';(t), i = 1 , + , r of u,(t) can be
made arbitrarily s m d by simply reducing €1.
Conclusions
By decomposing the plant input into two parts: the input uo(t) and the input u,(t),
it is easy to understand how the saturation controlIer works, and how the gain q
influences the maximum amplitude of the plant input. Reducing the gain q causes
the input ua(t) to approach 0, and hence the plant input approaches the tracking input
~ ( t ) , which has no input ovemhoot at all. However, reducing wil l also reduce the
speed of the input ~ ( t ) , and hence will also slow dom the plant input u(t) and the
system output response.
APPENDLX A. PROPEICTIES OF THE Si4TURATION CONTROLLER 103
A.2.2 Usingan observer
According to [DJ91], when an estimate of the state 2 in (A.1) is obtained from an
observer:
the corresponding response of t(s) is given by:
We observe fiom (A.23) that the response of z(s) consists of two parts PL(s) and
&(s). PL($) is exactly the same as for the direct state feedback case, and P2(s) is an
extra term which results when the obsemed state is used.
In this case, when the reference and the disturbance signais are constant, the same
conclusions made in section A.2J still hold*
Appendix B
Models of the University of
Toronto Web Machine and
Cont rollers
B. 1 Models of the University of Toronto Machine
Models me expressed in the following state space representation:
where u = [ma mb m,] and y = [Fa v Fc], with:
ma the torque output to the rewinder drive (signal M28 in MTSC32 manual) mb the torcpe output to the nip drive (signai M29 in MTSC32 manual) m, the torque output to the unwinder drive (signal M30 in MTSC32 manual) Fa the tension of the rewinding zone v the velocity of the nip Fb the tension of the unwinding zone
APPENDIX B, MO= OF THE UNIVERSITY OF TORONTO WEB M A C ' AND C
AU models include flters of the fonn:
connecting to each tension output.
B.1.1 Model #1: A Reduced Order Low F'requency Model
B.1.2 Model #2: A Black Box6th Order Model
APPENDtXB. MODELS OF THE UNNERSITYOF TORONTO WEB MACHINE AND C
B .2 Controllers
Controllers are given by the following model:
where
the reference signal FW of tension Fa (tension of the rewinding zone) the reference signal v,f of nip velocity v the reference signal FW of tension Fb (tension of the unwinding zone) &ef - Fa Ur,/ - v &efœF,
and U is the torque input vector to the drives as described in section B.1.
APPENDIX B. MODELS OF TIiE UNIVERSITY OF TORONTO FVE. iblACm\rE AVD C
B.2.1 Cheap Control Servomechanism Controller
Controlier #1: Controller Design Based On The Reduced Order Low Re-
quency Mode1
For e = le - 4 and eo = le - 4, the controller is given by:
APPErVDIX B. MODELS OF THE: UNIVE:RSIW OF TORONTO WEB MACHINE .4N. C
Controk #2: Controller Design Based On The 6th Order Black Box
Mode1
Columas 1 through 7
Columas 8 through 9
APPENDIXB. MODELS OF TEE UNZVERSITYOF TORONTO WEB MACHINE AND C
Columne 1 through 7
Columns 8 through 9
APPENDLX B. MODELS OF THE UNlVERSITY OF TORONTO WEI3 MACHXNE AND C
B .2.2 High Gain Servomechanism Controller
This controuer is a cascade connection of a high gain controller and saturation con-
trouer. When the high gain controUer gain is 4e - 1 and saturation controller gain is
30, the overall controuer is given by:
Ac =
Colums 1 through 6
Columns 7 through 12
APPENDLX B. MODELS OF THE UNIVERSITY OF TORONTO WEB MACHINE AND Cf
Cc =
Columns 1 through 6
Colurmts f through 12
APPENDIX B, MODELS OF THE UWVERSITY OF TORONTO WEB MACEIüW AND C
Appendix C
Scripts For Controller Designs
C.l MatIab Codes For Controller Design
C.1.1 The Script To Obtain the University of Toronto Web
Machine State Space Mode1
This function is used to obtain the reduced order low fiequency model of the Uni-
versity of Toronto web machine model. It uses the modeling method described in
chapter 2.
Furiction: [iinsys] = get lin-eystem(into);
This function i e used to get a linear system.
1 nput : inf O (Plant information vector)
irifo(1) to iafo(3) : inertia of the three rollers (from
rewinder t o unvinder) , info(4) t o infoC6) : radius of the three rollers (from
rewinder to unwinder) , info(7) t o info(9): frict ion coefficients of the three rollers
(fioom rewinder t o unwinder) . Output: lin-sye (the linear reduced order s tate space rnodel obtained)
APPEMIIX C. SCRIPTS FOR CONTROLLER DESiGNS
f unction [lin-sys] = get-lin-system(iafo)
Ja = M o (1) ; Job = iafo (2) ; Jc = info (3) ;
ra = i n f o (4) ; rbrinfo (5) ; rc - inf O (6) ;
bA = in fo (7) ; bB = inf O (8) ; bC = inf O (9) ;
Cl= Cb+Cc; CS= Cc; Y13 Yb+Yc; Y2= Yc;
C-i=YI-(Y*Ci)/C; C,2=Y2-(Y*C2)/C;
C.1.2 The Script To Normalize The Linear System And Add
Filters
X Iaput: Base-oax i s the vector of the maximum vaïue of the outputil
-4PPENDLX C. SCRIPTS FOR CONTROLLEX DESIGNS 115
X Output: my,sys i s the normalized system with filters connected t o
x the tension measurements.
function my,ryr - syrteqrelative (Base-mad
X web properties
rho = 1100 ; U = 0.168 ; e =O .00015/(2*pi) ; prU2 = pi*rho*W/2 ;
% bare shaft inertias
JOa = 0.01746 ; Job = 0.03223 ; JOC = 0.023363;
R m i n = 0.0415 ; Rmax = 0.15 ;
ra = Rmin ;
rc = Rmax;
rb = rO ;
X Shaft inertiaa (range) a i t h web on
Ja = JOa + prW2*(ra. "4 - r0-4) ;
Jc = JOc + prW2*(rcba4 - r0-4) ;
X Damping factors for the ihaftri, denoted ma,mb,mc
ma = 1/254.4878 ; mb = 1/16S .O867 ; mcW258 -3219 ;
APPENDIX C. S W T S FOR CONTROLLER DESIGNS
Ia = Ja; Ib = Jb; Icdc;
X This part si11 calculate the system after filtering
C . 1.3 Scripts for Cheap Control Servomechanism Cont roller
Design
Cheap Control Design 8
(Modified from E.J. Davison's routine "senrob.mn)
fwction [k8koJ=ee~obd(a8b8c,d,e,eo)
'/r This fimction i r uaed t o find the cheap controller gain and the obser
g SERVOBD
x k,ko]=semob(a,b,c,d,e,eo)
APPENDZX C. SCRIFTS FOR CONTROLLER DESIGNS
% u-k2*neta +ki*(observed etate) , where k=kl k2]
X Observer Design
ko = observe(eo ,a,b, c) ; % observer gain
Obtaining the State Space Representation of the Cheap Control Servomech-
anism Controuer
Function: [coatlis] = se~ode(system,e,eo)
X This function obtains the rtate space model of the controllet from
X the given plant state space model uring the function I1servobdw.
X The input t o the obtained controller is [yref error] ', the output of
X the controller is the input u to the plant. i . 8 . . the controller has
X the f ollowing st ate space representation :
X dot (Xc) = cont,ri. a*Xc + cont-as . b* Cyref error] >
X 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 the given syitern
r = rc; X niimbet of output8 of the given system
m = mb; % nupiber of inputs of the given system
% Now use the servob routine t o compute the control ler
% Now construct t he controller Btate space representation
X The control ler is of the following structure
% u = ki*X + K2* ke th i
% ühere X is the observed state from a f u l l order linear state observer:
% Dot(hat (x)) = A*(hat (x) ) + B*u + ko * (y - hat (y))
% hat(y) = C * hat(x) + D*u
% and dot (kethi) = y - yref - error
% For the control ler 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 he input t o the plant
Acont = CAO- Ko*CO + (BO-Ko*DO) * K I (BO-Ko*DO) *K2 ; teros Cr. n) zero8 Cr . r) ;
Bcoat = Mo -Ko;zeros (r,d -eyeCr. t) 1 ;
Ccont = [KI K 2 f ;
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 Con-
trouer 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 De-
sign
X Specify the 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 tabi l i ty . Iteol1 should be chosen from a large value as le4, t h i s
X tesul 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, the resultant closed loop system ma. be slow, the value is the
X t o achieve desired reirponee.
% first get the plant system, t h i s system export the
X relat ive output, y,% = y./y,max and the state are
X relat ive States x,# = 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 + BcJ*(y-v)
% u = Cc,H*Xh + Dc J*(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 hg,ss.a] ;
comp-b = [zero8 (length(p1ant. a) , r) ;-hg,ss . 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*yref I<=l
X for this C a m , the output8 are alrrady relative outputr.
APPEN.DE C. SCRIPTS FOR CONTROLLER DESIGNS
% which means the rystem has already been scaïed
tou = inv(comp-88 .d-comp,ar .c*inv(comp-sr. a) *comp,ss. b) ;
theta = given-theta ;
observer design
ko .= obrerve (eo , compeil . a. conp,rs. b , comp,rs, c) ; X observer gain
X now get the saturation controller for the composite system of the form:
% dot Ceta) = y-yref
% u = theta*tou*contp,ss . c*ino(comp,ss. a) *hat (x) -theta*tau*eta
X where h a t W i r the observed state from the following full order observe
% dot (hat ( x ) ) = comp-SB. a*hat (x) +camp-8s. b*u+ko (y-comp-sr . c*hat (x) -camp-ss . d
X This controller cm then be expressed by:
% dot ( X l ) = all-Cont,ss. a*X1 + all,Cont,ss. b* [yref error] >
% v = all_Cont_ss . c*X1 + ail-Cont-ss. d* Cyref errorJ >
all-Ccont = ~theta*tou*comp,ss .c*inv(comp-8s .a) (-theta) * tod ;
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 to 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 S. a*X1 + all-Cont-as. b* [yref error]
% v = all,Cont,ss. 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:
X dot (Xc) = owraïl,Cont_ss. a*Xc + overal1,Cont-as. b* [yref errod
% 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 zero8 Cr, n-all-Cont) 3 ;
overail-D = zeros Cm, 2-1 ;
% Because the controller is designed t o control the relat ive 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, > . . \progran\high-gain-cont . ha ) ;
C.1.5 The Sc~ipt to 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
f unction C] = Output-Controller (Cont , f i le)
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 pro ject 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 in the C program.
f printf (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 ta t ic f loat xRi]=<' .n) ;
for i = 1:n-1
fprintf (out, 'O, >) ;
end f printf (out, O) ; \r\n ) ;
f priiitf (out, if (State ! -6) \r\n<\r\n ) ;
for i = l:n
fprintf (out. k[XiJ=O;\r\nS ,i-1) ;
end
for i = 1:m
f printf (out, 'Cont,inputs [Xi] =O ; \r\nJ , i-1) ;
end
for i -= 1:r
fprintf (out, 'Plant-inputs [Xi] =O ; \r\n , i-1) ;
end
APPENDIX C. SCRIPTS FOR CONTROLLER DESIGN$
f printf (out, 9\r\n8 ) ;
f printf (out, \r\n ;
fprintf (out, %\nB ;
for i = 1:r
fprintf (out,~Plaat,iap~tr~~~~~~,i-i) ;
for j = 1:n
fprintf(out,>(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 J ' 0 ' ) 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.2 C Codes For Controller Implement at ions
C.2.1 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) ; --callad by the MTSC32 during i n i t phase
// void UofT-Interrupt(void); --called by the MTSC32 at every sampling inst
/********* NECmSARY HEADm INCtUDES *******+***/ ainclude csysdef.h>
#inchde cec32. h>
#include <kernel32.h>
#include <bina2 + h>
#inchde <ana32,11.h>
#inchde <enc32,29+h>
Binclude <math.h>
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
#def ine su-7.8
#def ine su-7.9
#def ine sw-7-10
#def ine sw,7,1i
#def ine su,7,12
#define su-7-13
#def ine su,?, 14
Xdef ine sw-7-15
#def ine sw-7-16
Xdefine su-7-17
#define su-7-18
#define su-7-19
#def ine sw-7-20
#define sw-7-21
#def ine sw-7-22
#def ine su,7,23
#def ine sw-7-24
#def ine su-7.25
#define swJ,26
#def ine su-7-27
#def ine su-7-28
awitches C71 &
switches [7] &
switches [7] O
switches 173 &
switches 171 &
switches 171 &
switches 171 &
switches C71 &
switches 171 &
switches [7] &
switches [?] &
switches [7] &
switches 171 &
switches [7] &
ewitches [7] C
switchesD] &
switches L?] &
suitches [7] &
switches 171 &
switches 171 t
switches 173 &
switches [Il Ir
APPENDIX C. SCRIPTS FOR CONTROLLER DESIGNS
#def ine 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 s ignals-ex [24)
Wdef ine mfb,ref signal8,ex [Z6]
// define e r ro 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 the modified tensions t o be monitored
#def ine mFa s ignaïs-ex 1303
Wef ine mFb signals,ex[31]
// define the plant output values
#der ine Fa-modi (signal8 Cl31 +O .31795) /l. 4457
// l inear i ty between the measurement and the real tension. In future work,
// these linearity valuas should be modified from the corresponding set up
// values.
#def ine Vb-act signal8 C23 *(3.1415926/30) *O. 0415
#der ine Fb-modi (signal8 CI41 -1.7522) /1.3314 //linearity between measu/rea
// def ine the ref erence vaïues
APPENDIX C. SCRIPTS FOR CONTRQIILER DESIGN"
Mef ine Ref ,Fa signala D2J
Mefine Ref ,Vb 8 ignalr [21]
Xdef in8 Ref Jb s ignals [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 to shaft C
/* Setup Values */ #define offilet
/S*WSS******S**S* INIT mCTION ***************/ void
Uof T-Init (void) (
version-string [O] 10 ;
strcpy (version-string , ~ServoCompensator\O ) ;
// This string should be changed to name of the controller t o be
// implemented.
APPENDIX C. SCRlPTS FOR CONTROLLER DESIGNS
f loat Cont-inputs 161 ;
f l o a t Plant-inputs 131 ;
Cont-input s [O] = Ref ,Fa;
Cont-inputs W = Ref Jb;
~ont-inputsBJ = Ref-Fb;
ont-inputs [3J = Ref S a - F a ~ o d i ;
ont-inputs M = Ref -Vb - Vb-act ;
Cont-inputs CS] = Ref Jb - 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 (State=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 * rc, rtart up behavior, rollers are static
1
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~esponding set 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 ;
