eindhoven university of technology master limit cycle ...limit cycles occur, is the pipet of...

Eindhoven University of Technology

MASTER

Limit cycle behaviour of a mechanical servo-system

van de Laar, J.H.A.

Award date:1999

Link to publication

DisclaimerThis document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Studenttheses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the documentas presented in the repository. The required complexity or quality of research of student theses may vary by program, and the requiredminimum study period may vary in duration.

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

https://research.tue.nl/en/studentTheses/306863f5-b1bd-43dd-b5d9-cce5433f50a4

Limit cycle behaviour of a mechanical

SWVQ-SyStern

J.H.A. van de Laar Report No. WFW 99.008

Master’s thesis

Professor : Prof.dr.ir. J.J. Kok Coaches : Dr.ir. M.J.G. van de Molengraft

Dr.ir. M. Steinbuch (Philips CFT Eindhoven)

Eindhoven, April 1999

Eindhoven University of Technology Department of Mechanical Engineering Section Systems and Control

Author: Jacco van de Laar

Title: Limit cycle behaviour of a mechanical servo-system

Samenvatting

Het onderwerp van het onderzoek is het limit cycle gedrag van mechanische servo-systemen en hoe dit gedrag kan worden beinvloed door het toepassen van regeltechnieken. Een van de mechanische servo-systemen waar limit cycles optreden is de pipet van de Advanced Compo- nent Mounter van Philips. Zulke limit cycles zijn periodieke oplossingen van het systeem welke veroorzaakt worden door een combinatie van regelaar en niet-lineariteiten. In een produktie proces zijn limit cycles niet gewenst omdat ze het systeem beschadigen en de nauwkeurigheid van het positioneren verkleinen.

Bij het gebruik van een PD-regelaar in combinatie met een encoder als niet-lineariteit, kunnen met het Shooting Method algoritme meerdere periodieke oplossingen worden gevonden welke naast elkaar bestaan. Bij het berekenen van deze periodieke oplossingen worden de eigenschappen van de regelaar en de niet-lineariteit gelijk gehouden. Het verschil tussen de periodieke oplossingen wordt veroorzaakt door het verschil in begintoestand, i.e., beginposi- tie, van de massatraagheid. Helaas had geen van de gevonden periodieke oplossingen betere eigenschappen dan de limit cycle welke optreedt aan de pipet van de ACM. Tijdens de simu- laties kwam naar voren dat het gebruikte algoritme, in de huidige vorm, tekort schoot. Als het systeem een grote discontinuiteit, zoals een encoder, bevat, wordt het moeilijk het pad van de periodieke oplossing te volgen. Een van de meer recente ontwikkelingen op het gebied van bi- furcaties is de uitbreiding van de theorie naar discontinue systemen. De hieruit voortkomende verbeterde Shooting en Pathfollowing algoritme zal wel in staat zijn om de niet-stabiele periodieke oplossingen van een systeem met een rigorouze discontinuiteit te berekenen. Zo zal het wel mogelijk zijn om een volledig pad van stabiele en instabiele periodieke oplossingen van het systeem te volgen.

Naast het bovenstaande is er ook gekeken naar een systeem met een PID-regelaar en wrijv- ing als niet-lineariteit. Bij deze configuratie is het opvallend om te zien bat de amplitude van de limit cycle sterk afhankelijk is van de bandbreedte van de regelaar. Zo sterk zelfs dat in bepaalde bandbreedte gebieden de limit cycle niet meer optreedt. De gebruikte vorm van de regelaar blijkt van invloed op de plaats waar deze bandbreedte gebieden zich bevinden. Door de vorm van de regelaar gunstig te kiezen, kan er een bandbreedte gebied ontstaan waarbin- nen geen limit cycle meer optreedt en bovendien aan alle andere technische specificaties van het systeem voldaan kan worden.

Keywords: Limit cycle, Shooting Method algorithm, Mechanical servo-system

Author: Jacco van de Laar

Title: Limit cycle behaviour of a mechanical servo-system

Abstract

The topic of the report is the limit cycle behaviour of a mechanical servo-system and how this behaviour can be influenced using control techniques. One of those servo-systems at which limit cycles occur, is the pipet of Philips’ Advanced Component Mounter. Such limit cycles are periodic solutions of the system caused by a combination of controller and nonlinearities. In a production process, limit cycles are unfavourable because they limit the positioning performance of and cause damage to the system.

With the Shooting Method algorithm, co-existing periodic solutions are found if a combination of PD-controller and encoder is used. These different periodic solutions are calculated using the same controller and nonlinearity properties. The difference between these periodic solutions is purely caused by the chosen initial condition, i.e., the initial position, of the inertia. Unfortunately, none of these co-existing periodic solutions has better properties than the one occurring at the pipet. During the simulations, a drawback of the Shooting Method algorithm, in its present form, came to light. If the system contains a rigorous discontinuity such as an encoder, pathfollowing of the periodic solutions faces difficulties. Only stable periodic solutions of the system with an encoder as nonlinearity can be calculated with the algorithm. One of the more recent developments in the field of bifurcations is the extension of the theory to discontinuous systems. The resulting improved Shooting and Pathfollowing algorithm will be able to calculate the unstable periodic solutions of a system with a rigorous discontinuity and thus follow a complete path of stable and unstable periodic solutions of the system.

Next to the above, a look is taken at a system with a PID-controller in combination with friction as nonlinearity. It is interesting to see that the amplitude of the limit cycle is strongly influenced by the bandwidth of the controller. So strong that in some controller bandwidth areas the limit cycle no longer appears. The position of these bandwidth areas depends on the shape of the controller. By carefully choosing the shape of the controller, a bandwidth area can be found in which not only the limit cycle no longer appears, but the other technical specifications of the system might be met as well.

Keywords: Limit cycle, Shooting Method algorithm, Mechanical servo-system

1 Introduction 8

2 Theory 11 2.1 Fundamentals of Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Limit Cycle Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 The Shooting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Simulation setup 19

3.3 Simulation algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 System configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Controller 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Resul t s and Discussion 27 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Encoder and PD-controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.1 Limit cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Limit cycle behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2.3 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Friction and PD-controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 Friction and PID-controlier . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.1 Limit cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.4.2 Limit cycle behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

27

27

38

5 Conclusions and Recommendations 43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Conclusions 43

5.2 Recommendations 44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A Monodromy matrix 46

B Derivation of controller parameters 48 B.l Equation for the gain K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 B.2 Controller parameters of Controller B . . . . . . . . . . . . . . . . . . . . . . 50

4

Eist of Figures

1.1 Pipet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Long-term response of the system for a range of excitation frequencies (schematic 1.2 Example of a limit cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

representation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Model of a nonlinear system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Block-model of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Stribeck curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Phase diagram of limit cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Classification of bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Floquet multipliers in the complex plane . . . . . . . . . . . . . . . . . . . . . 15 Fundamental solution matrix for one of the states of the system . . . . . . . . 16 Example of a periodic solution . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Newton-Rhapson algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Block-model of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Model of the process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 SwitchModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4 Model of a system with PD-controller and friction . . . . . . . . . . . . . . . 23 3.5 Standard shape of the controller . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.6 Sensitivity Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1 Behaviour of the position. velocity and control effort of the system . . . . . . 28 4.2 Influence of the bandwidth on the properties of the limit cycle . . . . . . . . 29

4.4 30 4.5 Amplitude of the limit cycle vs . cut off frequency of the low pass filter . . . . 31 4.6 Floquet multipliers of the stable periodic solutions A (o) and B (+) . . . . . 31 4.7 32

4.9 Behaviour of the position, velocity and control effort of the system . . . . . . 34 4.10 Behaviour of the position, velocity and control effort of the system . . . . . . 35 4.11 Behaviour of the position, velocity and control effort of the system . . . . . . 36 4.12 Behaviour of the position, velocity and control effort of the system . . . . . . 38

40 4.14 Limit cycle behaviour for small (-) and big (- -) difference between friction levels 40

4.3 Bodeplot of three different controllers . . . . . . . . . . . . . . . . . . . . . . 30 Limit cycles for low (-) and high (- -) initial position . . . . . . . . . . . . . .

Indication of the path with unstable periodic solutions . . . . . . . . . . . . . 4.8 End-position of the inertia vs . bandwidth . . . . . . . . . . . . . . . . . . . . 33

4.13 Limit cycle behaviour for low (-) and high (- -) I-action

4.15 Long-term response of the system for ZO = 37r . [rad] . . . . . . . . . . . 41

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

5

LIST OF FIGURES

4.16 Openloop FRFs of controllers A (-) and B (- -) . . . . . . . . . . . . . . . . . 41 4.17 Long-term response of the system for ICO = 37r . [rad] . . . . . . . . . . . 42

A. l System response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6

List of Tables

3.1 Process. encoder and friction parameter values . . . . . . . . . . . . . . . . . 22 3.2 Controller parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

B.l Controller B parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7

Chapter 1

Introduction

Mechanical servo-systems are designed to track a reference motion as accurately and quickly as possible. Many industrial machines such as the Advanced Component Mounter (ACM) from Philips use such a mechanical servo-system to position components on Printed Circuit Boards (PCB). The components can vary from fairly big IC’s to non-standard formed ones. The ACM can be used as a stand-alone machine or can be placed in a so-called Powerline with other machines for assembling complete TCBs. A TTT-robot, which can make an X-Y-2 motion, is equipped with a pipet to pick and place the component. In Figure 1.1 a picture of such a pipet is shown.

Figure 1.1: Pipet

Before being placed on the PCB, the angular position of the component is checked by an optical measurement system and is changed toward the desired position by adjusting the angular position of the pipet. An encoder is used to measure the angular position of the pipet. Besides this encoder, friction at the pipet is another nonlinearity which influences the

8

1. introduction

positioning of the component. If a constant reference signal is fed to the controlled system, one would expect the system to come to rest after a period of time. Due to the nonlinearities and controller, the pipet will not come to rest at its designated angular position, but will instead oscillate around it. Such an undesirable periodic motion is called a limit cycle. In this report a closer look is taken at this phenomenon called limit cycle or limit cycling. Limit cycles are periodic solutions of autonomous nonlinear systems. An example of a limit cycle, as it could occur in a mechanical servo-system, is shown in Figure 1.2. Although the reference signa! eq~a!s zero, the systerri does riot come to rest.

Figure 1.2: Example of a limit cycle

There are two reasons why limit cycles are unfavourable:

0 Limit cycles can cause damage to the controlled system.

o Limit cycling limits the positioning performance of controlled systems. The limit cycle causes a periodic tracking error equal to the size of the limit cycle amplitude, which leads to a poor accuracy.

In [i] the influence which the nonlinearities of the system (friction, encoder) and I-action have on the shape of the limit cycle was investigated. To what extent simple nonlinear models of the system are sufficient to predict limit cycles was another part of the research in [i]. In view of this research the possibility of the co-existence of more than one periodic solution is investigated in this report. Here, the purpose is to see whether next to the stable periodic solution other stable or unstable periodic solutions exist under the same circumstances, ex- pressed in terms of the controller parameters, encoder resolution and friction model. If this is the case and the properties of one of the other periodic solutions are favourable, then the system could be forced into this periodic solution by applying active control. In [LI such a scheme is implemented for a nonlinear dynamic system. Figure 1.3 shows a schematic representation of the response of the system for a range of excitation frequencies. The figure shows that if the system is excitated with a frequency higher than 30 [Hz], two periodic solution co-exist. For an excitation frequency of 40 [Hz], a stable periodic solution with a period of 22' [SI and an amplitude of approximately lop2 [m] co-exists with an unstable periodic solution with a period of T [SI and an amplitude of around 5 . [m]. The latter periodic solution clearly has the better properties. The vibration amplitude of the system can now be reduced

9

1. introduction

considerably by applying active control. The system can then be forced from the stable 4 subharmonic response of high amplitude into the co-existing unstable harmonic response of low amplitude.

O

O 0 O O0

O 0 stable ," : O 0 8 0

O 0 O 0

O 0 O O0

O 0 stable ," : O 0 8 0

O 0 O 0

O O

o@o

O O o n O O., "d O g

%- O

O U

++ ++++ +

unstable O00 o 0

+++++++++++

10-51 10 20 30 40 50 60

frequency [Hz]

Figure 1.3: Long-term response of the system for a range of excitation frequencies (schematic represent ation)

After this introduction, the theory of the limit cycle phenomenon will be addressed in Chapter 2. Some fundamental information on this phenomenon and corresponding terminologies will be given to better understand the remainder of this report. Furthermore, the algorithm used to calculate the periodic solutions of the system will be described. Next, Chapter 3 deals with the model of the system used for simulations. In this chapter, the different configurations of the systems model, the derivation of the controller parameters and the simulation algorithms are addressed. The results of the simulations are given in Chapter 4. Together with the information given in the first part of the report, these results will be discussed to see how the different controller and nonlinearity variables influence the limit cycle behaviour of the system. Finally, in Chapter 5, conclusions are drawn and recommendations for further research are given from this discussion.

10

Chapter 2

Theory

A limit cycle or limit cycling is a phenomenon which only occurs in nonlinear systems. A model of the nonlinear system used in this report is shown in the figure below. The model is a closed-loop system consisting of a controller and a process. The nonlinearity is applied on the output (encoder) or on the process itself (friction).

Y

Figure 2.1: Model of a nonlinear system

In the first part of this chapter, some fundamental information is given about limit cycles. The limit cycle phenomenon will be described together with frequently used terms in this report. The second part of this chapter deals with the Shooting Method algorithm, which is used to calculate the periodic solutions of the system.

2.1 Fundamentals of Limit Cycles

2.1.1 Limit Cycle Phenomenon

Because of the large extend of information about the limit cycle phenomenon, the information given in this paragraph is limited to one of the system configurations discussed in this report, a controller with friction as nonlinearity. In [3], it is said that the combination of controller and friction leads to limit cycling when tracking at low velocities or with velocity reversal. Let the system be modelled as shown in Figure 2.2. With this model a virtual experiment will be done to get a feeling of what happens if the system is limit cycling. We want to move the mass to a reference position gref. To move the mass from its current position to y r e f , a certain amount of force F is needed to overcome the static friction (stick phase). With an ordinary PD-controller the static friction will be overcome but the reference position will not be reached. The mass will remain at a constant deviation from the desired position, depending on the proportional gain of the controller (in case of position feedback). For the

11

2. Theory

e

Figure 2.2: Block-model of the system

controller to be able to build up enough force so the mass will reach the reference position, an I-action is included in the controller. With this I-action, the static friction is overcome and the mass moves towards g r e f , the mass is now in slip phase. Instead of going straight to the reference position and remain there, the mass overshoots this reference position. The reason for this lies in the difference between the friction level in stick phase and slip phase. A model for the friction in the system is shown in Figure 2.3, the Stribeck curve.

Friction

Velocity

Figure 2.3: Stribeck curve

The curve shows a dip in friction force at the transition from stick to slip phase. The moment the mass moves, the friction comes into slip phase. The friction in this phase is less than the one in stick phase. At the transition point, the force generated by the controller is larger than the force needed to move the mass. Because of this excessive force, the mass does reach its designated position g r e f , but overshoots it resulting in a new position error. The above described phenomenon will constantly repeat itself which results in a movement as shown in Figure 1.2, a limit cycle.

12

2. Theory

The shape of the limit cycle is determined by at least two parameters:

The level of I-action determines the frequency of the limit cycle. The more I-action is used, the faster the necessary amount of force to overcome the static friction is build up, and thus the faster the mass moves.

The difference between the friction level in stick phase and the one in slip phase determines the amplitude of the limit cycle. If the difference between the two friction levels is smaiier, the difference between the force generated in stick phase arid the ûne needed in slip phase will also be smaller. Consequently, the overshoot of the mass, i.e., the amplitude of the limit cycle, will be smaller as well.

In [4] a general description of a limit cycle is given. A limit cycle is defined as a closed and isolated trajectory in the state space. A limit cycle is a periodic motion of the state. If the limit cycle starts at state 2, this state will be reached again after T seconds, where T is the period time of the limit cycle. Therefore, a limit cycle is called a closed trajectory. If initial conditions are chosen in the vicinity of the limit cycle, the limit cycle is the limit of the system response in forward or backward time. This goes for initial conditions on either side of the limit cycle, that is why the limit cycle is an isolated trajectory. In Figure 2.4 two types of limit cycles are shown, i.e., stable and unstable ones. If all the trajectories in the vicinity of the limit cycle converge to it for t + 00, the limit cycle is called stable. On the other hand, if all the trajectories in the vicinity of the limit cycle diverge from it for t -f 00, the limit cycle is called unstable.

stable limit cycle '1 unstable limit cycle

Figure 2.4: Phase diagram of limit cycles

2.1.2 Terminology

After explaining what a limit cycle is and when it occurs, other terms are necessary to be addressed for understanding the remainder of this report. A bifurcation is a qualitative change in the systems behaviour [5, 61. During the motion of the system, the system parameters are understood to remain constant. If a system parameter, say r , changes, the behaviour of the system will also change. A small change in T will result in a small quantitative change of the systems behaviour. There is a possibility however that the small change in T causes a

13

2. Theory

qualitative change of the systems behaviour. The parameter value r at which the bifurcation occurs is called the bifurcation value rc. A system can lose its stability at a local bifurcation under increasing values of system parameter r if a fundamental primary path becomes unstable at T = rc. If a bifurcation value is passed, the local stability of a periodic solution, the number of (co-existing) periodic solutions and the type of periodic solutions may change. If the path bifurcates and a secondary path is created, a classification of the bifurcation can be made by looking at the form and stability of this secondary path.

stable (-) unstable (- -)

T C

parameter value r

(a) Example of a flip bifurcation

7 ' C

parameter value r

(b) Example of a fold bifurcation

Figure 2.5: Classification of bifurcations

In Figure 2.5, two examples of a bifurcation are given. In both cases the path bifurcates at r = rc. The bifurcation in Figure 2.5(a) is an example of a flip bifurcation, the continuous fundamental path loses its stability as it intersects a stable secondary path which only exists for T > rc. The bifurcation in Figure 2.5(b) is called a fold bifurcation, the fundamental path loses its stability as it intersects an unstable secondary path which only exists for r < rc. At a bifurcation point, the fundamental path loses its stability.

The local stability of a periodic solution can be determined using Floquet multipliers. Floquet multipliers are the eigenvalues of the fundamental solution matrix (see appendix A). If all Floquet multipliers have a modulus less than unity, the periodic solution is stable. A periodic solution is unstable if at least one of the Floquet multipliers is larger than unity, see Figure 2.6. A loss of stability at a bifurcation point corresponds to a Floquet multiplier passing through the unit cycle. A bifurcation can be classified by looking at the Floquet multipliers belonging to the periodic solutions in the vicinity of the bifurcation. If a real Floquet multiplier passes through -1 we have a flip bifurcation. A flip bifurcation generates a secondary path with periodic solutions with period 2T, as can be seen in Figure 2.5(a). In the case of a single real Floquet multiplier passing through fl, we have a cyclic fold bifurcation, as shown in 2.5(b), where the branch turns around and stability is exchanged.

14

2. Theory

Figure 2.6: Floquet multipliers in the complex plane

2.2 The Shooting Method

In [2], the finite element package DIANA was originally planned to be used to simulate a model of the dynamic system. With DIANA, periodic solutions of nonlinear mechanical systems can be calculated using path following techniques in combination with a finite difference method to solve the two-point boundary problem. At first, DIANA was used to try and find periodic solutions of the systems investigated in this report, but in the process the following problems were

e

e

e

encountered:

To be able to use DIANA, the system must have n x 2 states. If the PD-controller is used in the system, there are only 5 states. Therefore this restriction is not met. This problem could be by-passed by introducing a very small mass, but this causes numerical problems when calculating the periodic solutions and is therefore no option.

When the I-action is added to the system, the system has 6 states. However, DIANA dernands that the system can be translated into a system consisting of only mass-spring- damper systems. It is impossible to do this with the controllers different parts and their inputs and outputs, so this demand is not met.

To draw a bifurcation diagram, periodic solutions are calculated. To do this in DIANA, the system has to be smooth. This means there may not be any jump in the fundamental solution matrix of the system. In Figure 2.7 the fundamental solution matrix for one of the states of the system with PD-controller and encoder as nonlinearity is plotted. It clearly shows a jump, so this restriction is not met either.

Because of these problems DIANA is not suitable and therefore not used for calculating the periodic solutions of the systems investigated in this report. The research described in [7] presents a very good alternative for DIANA. Next to a simple and efficient friction model, a shooting method algorithm for calculating periodic solutions is presented in this paper. In the following this Shooting Method algorithm will be discussed. Consider an nth-order, autonomous, nonlinear system represented by the state equation

x = f(x)

15

2. Theory

time [SI

Figure 2.7: Fundamental solution matrix for one of the states of the system

where Z d~ z is a column with the n state variables of the system, t is time and f is dt ’ a column of nonlinear functions of the components of z. The system is called autonomous because f does not depend on t. In an initial value problem, the initial condition is usually given at t = t o . Because f is independent of t , solutions based at time t o # O, can be translated to t o , therefore the initial condition reads

x ( t = O ) = z o (2.2)

The solution of the state Equation (2.1) with initial condition (2.2) is often written as d t ( z0 ) ,

to explicitly show the dependence on the initial condition. An example of $t(zo) with zo = [zol xo21T is visualised in the figure below, O < tl < t 2 < T .

Figure 2.8: Example of a periodic solution

The Shooting Method algorithm finds periodic solutions of the system by solving a two-point Boundary- Vulue Problem (BVP), in which solutions are sought for

H ( z 0 , T ) = d T ( X 0 ) - zo = o, (2 .3 )

16

2. Theory

where T is the period time of the periodic solution and zo is a state on the limit cycle. $*(zo) is the solution of the state Equation (2.1) with initial condition (2.2) after T seconds. In case of a periodic solution with period T , $T(ZO) will be equal to zo. In order to solve BVPs, Initial Value Methods (IVMs) are used. A BVP can be represented in terms of fundamental solutions which on their part are solutions of associated Initial Vulue Problems (IVPs). 'The simplest IVM for BVPs is the single or simple Shooting Method. The way the solutions for ( 2 . 3 ) are sought with the Shooting Method algorithm shows a clear resemblance with the Newton-Raphson a!gorft,hm for finding zeros, shown in the figure below.

Figure 2.9: Newton-Rhapson algorithm

Given an approximation near the zero a, di+') of the linear function passing through ( ~ ( 2 1 , H ( d i ) ) ) with slope D H ( d i ) ) will be a much better approximation. This linear function is given by

y = H ( x ( ~ ) ) + D H ( Z ( ~ ) ) ( Z - = O

and therefore its zero is

In the case of the Shooting Method algorithm, the approximation di) consists of an approximation of the initial condition which is a state on the limit cycle (zo) and an approximation of the period time T.

SH SH SXO ST

-H = -Azo + -AT

17

2. Theory

@T is called the monodromy matrix. The monodromy matrix is the fundamental solution matrix (see appendix A) after the period time T . @T tells to what extend &T is changed if ZO is changed with an infinitely small perturbation 6x0:

Sq5T = (PTSZO

Because (2.4) is a system of n equations with n+ 1 unknowns (n components of 20 and period T ) , an extra constraint is added to make the system solvable,

fT(zo)TAzo = 0 (2.5)

This constraint restricts the state correction term AZO to be orthogonal to f ~ . From (2.4) and (2.5) the following iterative scheme emerges,

With this iterative scheme is iterated

scheme zeros of H can be found using initial guesses 20' and S('). The until some convergence criterion is met. The same convergence criteria

apply to the Shooting Method algorithm as to the Newton-Raphson algorithm. The returned value for period T could be a multiple of the actual period. Therefore one should check this value to see if it is the minimum period of the solution. The Shooting Method algorithm needs an initial condition to get started. This initial condition must be a state in the neighbourhood of the periodic solution together with a fairly accurate prediction of the period time T . If a 'too-fur-away' initial condition is used, the Shooting Method algorithm will not be able to find a periodic solution. For more information about the Shooting Method algorithm, the reader is referred to [8].

18

Chapter 3

Simulation setup

In [i] an experimental setup of a servo-system was put together to investigate the influence of encoder resolution, friction and integral action on the limit cycle behaviour. From this experimental setup, a model was derived and validated for simulations in Matlab and a controller was designed to stabilise the system. For the Matlab simulations in this report, the same model is used. The controller is basically the same just that a different tuning method is applied. This way, the bandwidth of the controller can be changed without endangering the stability of the system. A block-model of the system is shown in Figure 3.1. It consists of a controller part and a process part with nonlinearities. The different parts of the model will be discussed in this chapter.

Figure 3.1: Block-model of the system

3.1 Process

The process is modelled as a cylinder on which a torque is applied as is shown in Figure 3.2. The state-space representation of the process is:

An amplifier converts the controller output signal u into a current signal, where Ka is the amplifier gain. The motor generates a torque T, proportional to this current with motor

19

3. Simulation setup

constant Km. The system inertia is denoted by J, and the angular position of the cylinder by <p. The transfer function of the model of the process is represented by:

I T,

Figure 3.2: Model of the process

The aim is to control the angular position of the cylinder. The angular position of the cylinder is measured using an encoder. The encoder used in the simulations has a resolution of 10.000 [e]. This means that one encoder increment equals 27r/i0.000 [rad] (w 6.3 . [rad]). The encoder is implemented in the model by rounding the real angular position using the resolution of the encoder.

27T [radl (27r/i0.000) . io.ooo

cp (p = round(

The rounded value (e is compared to the reference position, < p r e f . The difference between the two , perr = pref - (p, is the input for the controller.

In [7] the Swatch Model is proposed as model for the friction in the system. The Switch Model can be seen as an improved version of the Karnopp Model. The classical Karnopp Model generates ordinary differential equations but suffers from numerical instabilities in the stick phase. The advantage of the Switch Model is that its algorithm maintains continuity of the state vector and yields a set of ordinary differential equations that do not suffer from numerical instabilities. The Switch Model, shown in Figure 3.3, divides the friction into three different sets of ordinary differential equations: one for the slip phase ( i ) , a second for the stick to slip transition (ii) and a third for the stick phase (iii). The state vector is inspected at each time-step to determine in which of the three modes the system is. Because an exact value of zero will rarely be computed, a region of small velocity is defined as I+/ < 7. If the velocity of the system lies outside this region, the system is in slip mode. The system is in transition from stick to slip if the velocity lies within the small region defined above and the torque generated by the motor exceeds the static friction.

20

3. simulation setup

The state-space model of the system for the three modes is:

( 9 if l(L?l > 7 then

(ii)

r

r

slip 1 (P - % s i g n ( )

stick-slip transition 1 (L? - 2 sign(u)

(iii) else

r (P 1

To ensure the state remains continuous, the acceleration of the inertia during the stick mode is set to - (L?dF. This way the velocity is forced to zero instead of simply setting the velocity to zero as is done in the Karnopp Model. Experiments in [i] led to the following values for the friction:

o Static friction, T, = 0.037 [Nm]

o Coulomb friction, T, = 0.032 [Nm]

ity

Figure 3.3: Switch Model

The values of the parameters discussed in this paragraph are summarised in Table 3.1.

21

3. Simulation setup

Process Ka = A 2.4 [VI /1 Encoder

Km = 0.315

Jm = 1.956-10-3 [kgm2] 11

R 10.000 incr Irevl

T, = 0.037 [Nm]

T~ = 0.03% [Nm]

Table 3.1: Process, encoder and friction parameter values

3.2 Controller

The controller, designed in [i], is to provide a stabilisation of the linear dynamics of the system. The controller consists of a series connection of a lead-lag compensator, a low pass filter and an I-action. Since the nonlinear influences are not accounted for in the controller design, actual stability properties can be different. With a phase of -180", the Nyquist curve of the process model goes through the point -1. To stabilise the system a lead-lag compensator is used.

Tds + 1 Transfer function of the lead-lag compensator : Cil(s) = K xs +

With X > 1, the lead-lag compensator provides a phase lead for frequencies in the neighbourhood of the point -1, from w = to w = -[--I. The low pass filter is included in the controller to diminish the influence of the unmodelled dynamics and high frequency disturbances on the stability.

i rad x rad r d T d s

Transfer function of the low pass filter : Cl,(s) = s2 + 2 P W l S + w;

where p is the damping-coefficient and wi is the cut-off frequency. To overcome the stick-slip friction, a certain torque is needed. With only a PD-controller, the pipet will not move for small position errors. In the figure below a model of a system with PD-controller and friction as nonlinearity is shown. The controller can be modelled as a spring (P-action) with constant P = K and a damper (D-action) with constant D = Krd. The inertia Jm suffers from a certain friction momentum Tfric. The inertia is to be moved to its reference position vTef. From this, the equation of motion can be derived

Perr = Pre f -

At t = O, the velocity and acceleration of the inertia and the reference signal are chosen equal to zero. Together with a constant friction level, i.e., Tfric = T,, this leads to

22

3. Simulation setuD

Figure 3.4: Model of a system with PD-controller and friction

From this equation it is clear that for a small position error, a large P-action is required to overcome the static friction and move the pipet. Instead of using a large P-action, an I-action is added to the controller. The I-action enables the controller to generate enough effort for the motor to create the torque needed to overcome the static friction.

ris + 1 Transfer function of the I-action : Ci = Ti s

The filter provides integral action up to the frequency fi 7 & [Hz]. Bad tuning of the controller parameters can lead to poor disturbance attenuation. Therefore, as is custom at Philips, a different tuning method is used for the simulation. The controller is changed by altering the bandwidth of the controller. The bandwidth fb is the frequency at which the open loop gain of the controller and process together equals 1,

IHC(S>llHP(S)l = 1 ( 3 4

A schematic picture of the more or less standard shape of the controller is shown in Figure 3.5.

pass filter

Figure 3.5: Standard shape of the controller

23

3. Simulation setup

For the controller to hold its shape, the parameters f i , f d and fw, all depend on the controller bandwidth f b [Hz]. Naturally, the P-, I- and D-action change for different values of the bandwidth, but the shape of the controller remains the same. By tuning the controller in this way, the controllers behaviour will not deteriorate. Figure 3.6 shows a sensitivity function of a system where the controller parameters are badly tuned. In the figure, a peak is clearly visible. Such a peak is an indication for poor disturbance attenuation. This means that if a disturbance is applied to the closed-loop system with a frequency of, in this case, around 15 [EZ], this disturbaxe will Se amplified instead af being suppressed. The higher the pe&, the poorer the disturbance attenuation. By applying the different tuning as mentioned above, a sensitivity function as the one in Figure 3.6 and thus poor disturbance attenuation of the system is prevented.

1 O'

E 10.'

10-2 1 oo 1 o'

frequency [Hz] 1 o2

Figure 3.6: Sensitivity Function

In the standard shape of the controller, as it is used at Philips, the control parameters ri, r d and w1 depend on the bandwidth f b . The parameters ,b' and X have constant values and the gain K is calculated by solving the equation lHc(s)llHp(s)l = 1. In appendix B.l, the equation for the gain K is derived, resulting in

2n2 f ; J r n d F J144p2 + 352

3 2 / 3 7 K a K m K = (3.3)

The parameters p, A, Ka, K, and J, are all known. The gain K can now be calculated by implementing the desired controller bandwidth f b [Hz] into Equation 3.3. In Table 3.2 a summary is given of the values of the control parameters discussed in this paragraph.

24

3. simulation setm

Controller

0.5 í-1

Table 3.2: Controller parameter values

3.3 Simulation algorithm

The Shooting Method algorithm [7] and the standard Matlab [9] function ode45 are used for the simulations in this report. The standard ode45 function is a commonly used Matlab function for solving the systems differential equations. The ode45 function is used to monitor the behaviour of the system as a function of time and to find a suitable starting point for the Shooting Method algorithm. The Shooting Method algorithm is build around the standard Matlab function ode45 and is used to find periodic solutions of the system.

3.4 System configuration

In this paragraph, the different configurations of the systems used in the simulations are introduced. In most cases the controller, process and friction parameters have values as presented earlier in this chapter. For the few simulations which differ from these values, the proper values are given in the next chapter where the results of the simulations are given. The block-models of the different system configurations are shown below. Regrettably, a system configuration of PID-controller and encoder and friction both as nonlinearities is not used in simulations due to time limits.

o Encoder and PD controller process

25

3. Simulation setup

o Friction and PD controller process

friction j 1~ o F'riction and PID controller

process

friction j

i t

26

Chapter 4

Results and Discussion

4.1 Introduction

In this chapter the different models discussed in the previous chapter are simulated numeri- cally. Firstly, the results of the simulation with the encoder as only nonlinearity in combination with a PD-controller are discussed. It will be shown that if one uses this configuration, a bifurcation can occur under certain circumstances. The periodic solutions in the neighbourhood of this bifurcation will be checked to be able to say more about the stability of the solution and to determine the type of bifurcation. Next, a look is taken at the system with a PD-controller and friction as only nonlinearity. No limit cycle occurs with this system configuration, but the inertia comes to rest at a certain static offset from the reference position. Finally, a system with a PID-controller in combination with friction as only nonlinearity is used. This configuration results in a friction induced limit cycle. The influence of controller and friction parameters will be discussed together with the possibilities to eliminate the friction induced limit cycle. For each of the configurations mentioned above, the shape of the limit cycle and the reasons for the occurrence of a limit cycle will be explained using the results of the simulations.

4.2 Encoder and PD-contro!!er

The system consisting of a PD controller and encoder as nonlinearity results in an encoder induced limit cycle. In [i] can be read that the encoder resolution determines the amplitude of the limit cycle.

4.2.1 Limit cycle

To get more insight of the limit cycle phenomenon, a look is taken at the behaviour of the position, velocity and control effort. It is sufficient to look at half a period of the limit cycle because the system configuration at hand causes a symmetric limit cycle. Therefore the second half of the period shows the same behaviour just in the opposite direction. In Figure 4.1 the encoder induced limit cycle is plotted with a reference position equal to O [rad]. Four different transitions are determined.

27

4. Results and Discussion

-0 02 ' -4 -3 -a -1 o 1 a 3

position [rad] x 1 ~ 4

(a) limit cycle

time [SI

(c) velocity vs. time

I I 4 O 0.1 0.2 O3 0.4 O 5 0.6 O7

time [SI

(b) position vs. time

(d) control effort vs. time

Figure 4.1: Behaviour of the position, velocity and control effort of the system

o Part I: point O to 1 In this part, the real position is rounded to O increments. The position error equals zero, therefore no control effort is generated. Because no control effort is generated, the velocity of the inertia remains constant and the position error increases.

o Part 11: point 1 to 2 The real value of the position exceeds +i increments. Therefore the encoder rounds the position to +1 increments. A position error is fed to the controller and a control effort is generated. Because of this control effort, the velocity of the inertia decreases. At point 2, the control effort and position error reach their maximum as the velocity equals zero.

28


0 Part 111: point 2 to 4 In this part the position error decreases. At point 3, the real value of the position equals +$ increments. After this point, the position is rounded by the encoder to O increments. Next to the position error, the control effort decreases as well. The absolute value of the velocity increases in this part and reaches a maximum at point 4.

o Part IV: point 4 to 6 The real valiie nf the pnsitiori is roiinded tc zeTc by the encoder in this part. The centre! effort is not set to zero immediately, but dims to zero. The velocity of the inertia shows the same behaviour. At point 5, the velocity nearly reaches zero. Therefore a plateau is visible at this point in the plot of position vs. time.

4.2.2 Limit cycle behaviour

The main features of the limit cycle are its amplitude and its frequency. First a look will be taken at the influence of the bandwidth of the controller on these two features. The controller is tuned with the method mentioned in the previous chapter. The bandwidth range is 5 [Hz] to 80 [Hz].

(a) amplitnde vs. time (b) period time vs. time

Figure 4.2: Influence of the bandwidth on the properties of the h i t cycle

In the figures above can be seen that the increase of bandwidth has no effect on the amplitude of the periodic solution. Whatever initial position is chosen, the resulting amplitude of the periodic solution equals half an encoder increment, 3.2 . lop4 [rad]. This in contrast to the period time of the periodic solution. By increasing the bandwidth of the controller, the controller is made more stiff. Therefore the controller will react faster to any position error which results in a smaller period time.

4.2.3 Bifurcation

The situation in which a change of controller parameters causes a qualitative change in the systems behaviour is called a bifurcation. With the encoder as nonlinearity a bifurcation will occur when the position of the inertia does not only oscillate between -1 and +1 increments

29


but shoots through this interval and oscillates between - 2 and + 2 increments, depending on the initial condition. For this phenomenon to occur the damping in the system must be reduced. To do this, the shape of the controller has to be altered. The damping of the system can be reduced By reducing the D-action of the controller, the damping of the system becomes smaller. This can be done by increasing Td or decreasing wl.

' -1 00 1 o' 10' frequency [Hr]

Figure 4.3: Bodeplot of three different controllers

In Figure 4.3 is visible how the controllers openloop Frequency Response Functions (FRFs) change if the values for T d or wl are altered. In both cases a reduction of the D-action is the result. Here, the choice is made to vary the cut-off frequency of the low pass filter, wl, to see if a bifurcation occurs. It turns out that for low values of wl more than one periodic solution exists, depending on the initial position of the inertia. In Figure 4.4 the position of the inertia vs. time is plotted for two different initial positions. It clearly shows two periodic solutions with different amplitudes under the same system configuration and controller parameter values.

time [SI

Figure 4.4: Limit cycles for low (-) and high (- -) initial position

By increasing q, and thus increasing the damping of the system, different initial positions will ultimately result in the same periodic solution. The point at which the different periodic solutions, which exist for low values of wl, merge into just one periodic solution, is called a bifurcation point. To find out where the bifurcation point is situated in the bifurcation diagram, the value for wl is increased from 100 [Hz] to 125 [Hz].

30


o 5 104 106 108 110 1 4

Figure 4.5: Amplitude of the limit cycle vs. cut off frequency of the low pass filter

In Figure 4.5, the amplitudes of the stable periodic solution are plotted versus the cut off frequency of the low pass filter, wz. The bifurcationdiagram clearly shows three branches with stable periodic solutions of the system. For low values of wl (wl < 113) where the damping in the system is small, more than one stable periodic solution exist. The middle branch is calculated with an initial position equal to 2 . [rad]. If the value for wz is increased, the periodic solution of this branch come to a hold at wz = 113 [%$I. After this point the initial position will result in a stable periodic solution on the bottom branch of the bifurcationdiagram. Therefore point B, at wz = 113 [%$I, is called a bifurcation point. Unfortunately the bifurcation diagram is incomplete. Apparently the Shooting Method algorithm is unable to find the unstable periodic solutions of the system at hand. To be able to say something about how the periodic solutions behave between two stable branches of the bifurcationdiagram, a look is taken at the Floquet multipliers in the vicinity of the bifurcationp oint .

Figure 4.6: Floquet multipliers of the stable periodic solutions A (o) and B (+)

In Figure 4.6, the Floquet multipliers of the periodic solutions of point A (o) and B (+) in Figure 4.5 are plotted. Because the system is autonomous, one of the Floquet Multipliers equals +l. To monitor their behaviour, the Floquet Multipliers of periodic solution A are

31

4. Results and Discussion ~

paired to the corresponding Floquet Multipliers of periodic solution B. The closer the periodic solutions are to the bifurcation point, the closer one of the corresponding Floquet multipliers is to the point +1, which implements a cyclic-fold bifurcation as discussed in Chapter 2. So for low values of wl a cyclic-fold bzfurcation can occur. Although the behaviour of the Floquet Multipliers indicate the occurrence of a cyclic-fold bifurcation, the exact path of the unstable periodic solutions can not be determined. Therefore, the only information which can be added to Figure 4.5 is an indication of the area where the unstable periodic solutions exist, as can be seen in t h e figwe below.

o 0 0

I 118 4

Figure 4.9: Indication of the path with unstable periodic solutions

Let’s presume the working area of the controller is the one where the bifurcation can occur. In this area, for ’weak’ controller properties, the amplitude of the periodic solution depends on the initial position of the inertia. The more the initial position of the inertia divers from the reference position, the greater the amplitude of the periodic solution. For instance, if the difference between the initial position of the inertia and the reference position equals 4 increments, the amplitude of the resulting periodic solution will be around the same value. At the same time, another periodic solution exists with an amplitude of around 1 increment. By applying active control it should be possible, according to the research in [2], to push the periodic solution with amplitude of 4 increments into the periodic solution with amplitude of 1 increment. In the system at hand, this only applies to the ’weak’ controller case. If the more or less standard values for the parameters of the controller are used (q = 300 [rad/s]), the damping in the system is too large for more than one stable periodic solution to exist. In that case, the inertia will ultimately reach this periodic solution whatever initial position is chosen.

32


6-

4.3 Friction and PD-controller

O

O

With the combination of PD-controller and friction as system configuration, no limit cycle occurs but the reference position is not reached either. The controller properties, i.e., the controllers bandwidth, do have an influence on the end-position of the inertia. This can be seen in Figure 4.8 where the end-position of the inertia is shown versus the bandwidth of the controller.

ä L

8 4 - 8 a

2-

IO-' 10,

O

O

O

O

O

o O

o o O o c , oOOG-.-

1 O0

O 0 @O

bandwidth [Hr] O

Figure 4.8: End-position of the inertia vs. bandwidth

This figure can be divided into three sections. In the following, a look is taken at the behaviour of the position, velocity and control effort for one situation of each of these sections. The friction level applied on the system will also be addressed. For this, the reader is referred to Figure 3.3.

Section A

In this section, the end-position of the inertia moves closer to its reference position, i.e., O [rad], for increasing bandwidth value. In the figures, three transitions are determined.

o Part I: point O to 1 In Chapter 3, the friction model used in the simulation is discussed. In this model a small stick-band is defined around the velocity value O [%l. For velocities within this stick-band, the system is in stick phase. This is the case in part I, the velocity lies in the stick band and the friction momentum equals T,.

o Part 11: point 1 to 2 At point 1, the velocity leaves the stick band and the inertia moves towards its reference position. The system is in slip phase now and the friction momentum equals T,. The control effort consist of a proportional part and a differential part. Until point 2 is reached, the decrease of the absolute value of the proportional part, i.e., IPyl, is smaller than the increase of the absolute value of the differential part, i.e., lD+l. Therefore, the resulting absolute value of the control effort increases in this part. After point 2 this is no longer the case and the absolute value of the control effort decreases.

33


(a) position vs. time

0.05

8 4 1

O 0 42

~

I i i

Pme [s]

(b) velocity vs. time

LA 0.004 0.W6 0.008 C 0.012 0.014 O

bme [s]

(c) control effort vs. time


o Part 111: point 2 to 3 In poir;t 3 , the control effort is too small to compensate the friction momentum T,. The inertia comes to a halt and the system switches to stick phase. The friction momentum now equals T,. The controller was not able to generate enough effort to overcome the friction in the slip phase. It will therefore be impossible for the controller to generate the effort needed to overcome the friction momentum in stick phase because T, > T,. The inertia will therefore remain at its present position.

34


Section B

In this section, the system shows an overshoot. The inertia overshoots its reference position and the movement has to be reversed. Because the controller is to 'weak', this overshoot can not be compensated and the inertia is not moved significant closer to its reference position.

(a) position vs. time (b) velocity vs. time



o Part I: point O to 1 Because of the higher bandwidth value, a larger P-action is applied. The velocity will therefore exceed the earlier mentioned stick band much faster. The system switches to slip phase and the friction momentum changes from T, to T,. The control effort consist of a proportional part and a differential part. Until point 1 is reached, the decrease of the absolute value of the proportional part, i.e., IPyl, is smaller than the increase of the absolute value of the differential part, i.e., ID+l. Therefore, the resulting absolute value of the control effort increases in this part. After point 1 this is no longer the case and the absolute value of the control effort decreases.

e Part 11: point 1 to 3 The inertia reaches its reference position at point 2 but can not hold it. From point 2,

35


the inertia moves away from its reference position. Therefore, the movement has to be reversed. Before the movement can be reversed, the inertia has to come to a halt. This happens at point 3.

o Part 111: point 3 to 4 In this part, the overshoot is insignificantly compensated. At point 4, the control effort becomes too small to overcome the friction in the slip phase and the system switches to

the one in slip phase, the controller will not be able to generate the effort needed to move the inertia. The inertia therefore remains at this position.

stick phase. The fïictiûn leve! iiûw eqUals T,. BecaUse this frictim !eve! is higher than

Section C

In this section, the system still shows an overshoot. Now the controller is strong enough to compensate this overshoot significantly and the inertia is moved closer to its reference position.

.4 o 1_ o w 2

-

4

L ".u1 O012 0.014 O.ui0 0.018

[SI

(a) position vs. time

2

3 9 ,

L , o w 4 ,

- I L O 0.002 0 . ~ 4 0 . u d 0.Wö ú.ui ii.uir 0.014 C.vis L u i d 0.02

time [SI

E O01 O012 O014 time [SI

(b) velocity vs. time


6 0018 O02


36


o Part I: point O to 1 Because of the higher bandwidth value, a larger P-action is applied. The velocity will therefore exceed the earlier mentioned stick band even faster than in the previous section. The system switches to slip phase and the friction momentum changes from T' to T,. The control effort consist of a proportional part and a differential part. Until point 1 is reached, the decrease of the absolute value of the proportional part, i.e., IPyl, is smaller than the increase of the absolute value of the differential part, i.e., lD<c.l. Therefare, the resulting 2bsohte vahe ef the contrd efbrt , increases in this part. After point 1 this is no longer the case and the absolute value of the control effort decreases.

o Part 11: point 1 to 3 The position of the inertia reaches the reference position at point '2 and overshoots it. The inertia now moves away from its reference position. The movement has to be reversed and at point 3 the inertia comes to a halt.

o Part 111: point 3 to 4 At point 3, the velocity is zero and thus the system switches to stick phase. At this point, the control effort, i.e., Py, is large enough to overcome the static friction momentum and the inertia ones again moves towards its reference position. From that moment on, the control effort decreases and at point 4 it becomes smaller than the friction momentum, i.e., T,. The inertia comes to a standstill and the system switches to the stick phase. The friction momentum in this phase is larger than the one in slip phase. The controller was not able to generate enough effort to overcome the friction in the slip phase. It will therefore not be able to overcome the static friction and thus the inertia will remain at its present position.

37


-002

-003

-004

-0 o5

4.4 Friction and PID-controller

-

-

-

In the previous section, the PD-controller was not able to get the inertia to its reference position. Therefore an I-action is added to the controller in this section.

4.4.1 Limit cycle

In the following a look is taken at the behaviour of the position, velocity and control effort during half a period of the limit cycle. Half a period is sufficient because the system configuration at hand causes a symmetric limit cycle. Therefore the second half of the period shows the same behaviour just in the opposite direction. In Figure 4.12 the friction induced limit cycle is plotted with a reference position equal to O [rad]. Again, four different transitions are determined.

(a) limit cycle

(c) velocity vs. time

I , 2 3

time Is]

(b) position vs. time

(d) control effort vs. time

6


38


Part I: point O to 1 In this part, a position error is determined. The inertia stands still, hence the velocity is zero. The friction is in stick phase. The control effort increases to move the inertia towards the reference position. At point 1 the stick phase is overwon and the inertia starts to move.

Part 11: point 1 to 2

in the stick phase. There is less effort needed to keep the inertia moving thus the control effort decreases. The position error decreases and passes the reference position at point 2. At that point, the velocity reaches its maximum.

Part 111: point 2 to 3 The velocity of the inertia decreases and reaches zero at point 3. Because the inertia still has velocity when reaching the reference position, the reference position can not be held. The inertia overshoots the reference position and increases to the other side. The control effort decreases more in this part, so much that the friction returns to the stick phase. The figure with the control effort shows a irregularity at the point where the slip phase changes to stick phase. In point 3, the stick phase is reached again.

Part IV: point 3 to 4 The friction in the stick phase is larger the momentum generated by the motor. The velocity of the inertia remains zero and the position error remains constant. Again a control effort is build up (in the opposite direction) to overcome the friction and move the inertia.

rnL- f-:-L:-- . .- “1:- ,L,”, I Ilt: lL lLII lu l l is I I U W ;u B l l p p l l a B G . The friction in the dip phase is !ower tha:: the frictim

4.4.2 Limit cycle behaviour

Integral action

In [i], the influence of the integral action on the limit cycle behaviour was checked. It turned out that the integral action only influenced the limit cycles frequency and not significantly its amplitude. In Figure 4.13 two situations are plotted one with a low integral action and one with a larger integral action. The plot with the lower integral action shows a lower frequency of the limit cycle as the other shows a larger frequency. The amplitude doesn’t differ very much between the two plots. The increase of the frequency of the limit cycle for the situation with more integral action can be explained as follows. The integral action of the controller enables the controller to build up the effort needed to overcome the static friction. The more integral action is applied, the faster the build up of control effort will be and thus the faster enough momentum is generated by the motor to overcome the static friction. If the amplitude of the limit cycle is not a problem for the proper functioning of the process at hand, a lower I-action will lower the frequency of the limit cycle. This might be useful in practice.

fiiction coefficients

In [3] can be read that a limit cycle can be caused by the design of, in this case, the pipet. If friction is involved, the limit cycle is not caused by the level of friction, but by the difference between the friction in the stick and the friction in the slip phase. In transition from stick

39


- O 8 8 -i 20 22 24 26 28 30 32 34 36 38 40

Figure 4.13: Limit cycle behaviour for low (-) and high (- -) I-action

to slip, the friction decreases as is shown in Figure 2.3. This means that at that point, the momentum, build up to overcome the static friction, is larger than the momentum needed to move the inertia in slip phase. Because of this surplus of momentum, the inertia overshoots its reference position resulting in a new position error. However, if the difference between the two friction level diminishes, so will the amplitude of the resulting limit cycle. This can be done by reducing the friction in the stick phase by applying a different kind of lubricant. In most cases this lubricant will also effect the friction in the slip phase, and therefore the difference between the two friction levels will not get any smaller. In [3] a lubricant is mentioned which enlarges the friction level in the slip phase. The moment the system starts moving, the lubricants properties change in such a way that the friction in slip phase is larger than in the case where another lubricant was used. Because this lubricant does not influence the friction level in the stick phase, the difference between the two friction levels becomes smaller. Therefore, the momentum to overcome the static friction will not be that much larger than the momentum needed in the slip phase to move the inertia. This means, the inertia will not overshoot the reference position as much as in the earlier case. In the Figure 4.14, two situations are plotted.

Figure 4.14: Limit cycle behaviour for small (-) and big (- -) difference between friction levels

40


Firstly, the difference between Tstick and T,l+ equals 0.005 [Nm]. The result is a limit cycle with an amplitude of about 6 . [rad]. In the dashed plot the difference between the two friction levels is increased to 0.01 [Nm]. The amplitude of the resulting limit cycle is larger than the one previous situation, about 1 .3 . [rad].

Bandwidth

In FigUïe 4.15 the infiUence of the bar,&width 3:: t h e limit cycle behw.ri9ur is checked hv U J

monitoring the long-term response of the system for an initial position 20 = 37r * [rad].

Figure 4.15: Long-term response of the system for ZO = 37r. lop4 [rad]

The figure above shows a rapid decline in limit cycle amplitude and at a bandwidth of around 18 [Hz], the limit cycle no longer appears. This is a very interesting development from which the question arises what the difference is between the controller used here and the one used in [i]. With the controller used in [i], a PID controller, with a bandwidth of around 10 [Hz] a limit cycle appears. To see if with this Controller there also is a bandwidth range in which the limit cycle does not appear, the shape of this controller is frozen at a bandwidth of 10 [Hz]. The derivation of the controller parameters can be seen in Appendix B.2. With these controllers parameters the equation for the gain K is derived from Equation 3.2 in the same way as is done in Chapter 3.

frequency [Hr]

Figure 4.16: Openloop FRFs of controllers A (-) and B (- -)

41


In Figure 4.16, the openloop FRFs of the two controllers are plotted to show the difference in shape. The controller used throughout this report will now be referred to as Controller A and the controller derived in Appendix B.2 as Controller B. Just as before, the influence of the controllers bandwidth on the limit cycle behaviour is looked at by monitoring the long- term response of the system for the same initial position zo = 3n . [rad], but now using Controller B. The results are shown in Figure 4.17.

Figure 4.17: Long-term response of the system for zo = 3n. [rad]

With Controller B, the amplitude of limit cycle decreases as the bandwidth of the controller increases. Not until the bandwidth has reached a value near 48 [Hz], does the limit cycle disappear. This is quite a difference with Controller A, where a bandwidth of 18 [Hz] was sufficient to make the limit cycle disappear. By tuning the controller parameters , the shape of the controller is determined. From the results of the simulations it is clear that the shape of the controller influences the bandwidth values for which the system no longer limit cycles. One always tries to control a system with the smallest possible control effort. In Chapter 3, where the controller is derived, can be read that a higher bandwidth value leads to more P-action. Thus, the lowest bandwidth value for which the system fulfils all its technical specifications is preferred. From Figures 4.15 and 4.17 can be seen that in the situation where Controller A is used, the limit cycle no longer appears for lower bandwidth values than iri the situation with Controller B. Under the restriction that the other technical specifications are met for these low bandwidth values, Controller A is preferred over Controller B.

By changing the shape of the controller, the designer has more possibilities to find a controller with which the system is stabilised, a limit cycle does not appear and other technical specifications such as settling time, gain and phase margin might be met.

42

Chapter 5

Conclusions and Recommendations

In the final chapter of this report, the conclusions drawn from the research described in this report are summed up. Furthermore, some recommendations are given for further research. These recommendations are based partially on the progress made in the field of the bifurcation theory and on the results of the simulations done using a controller with a chosen predescribed shape.

In this report, the main topic is how the limit cycle behaviour of the mechanical servo-system, i.e., the pipet of the ACM, can be influenced using control techniques. The limit cycling of the process is caused by a combination of controller and nonlinearities. Before the system can be forced into a periodic solution with better amplitude and period time values, co-existing periodic solutions have to be found. Using a 'weak' controller in combination with an encoder as nonlinearity, different periodic solutions co-exist under the same controller properties depending on the initial position of the inertia. Unfortunately, this effort is unsuccessful. None of the co-existing periodic solutions found have better limit cycle properties than the one currently found at the pipet.

To find these different co-existing periodic solutions, two software tools are tried. It is not possible to find any periodic solution of the system at hand with the DIANA package, because DIANA demands that the system is smooth and can be translated to a system consisting of only mass-spring-damper systems. The other tool, the Shooting Method aigorithm, does find periodic solutions of the system. A drawback of this algorithm, as used in this report, is that no unstable periodic solutions of a system with an encoder as nonlinearity are found with it. How the periodic solutions behave between two branches of stable solutions has to be derived from the behaviour of the Floquet multipliers in the vicinity of the bifurcation.

The system configuration with a PD-controller and friction as nonlinearity does not result in a Limit cycle but in a constant offset from the reference position. To move the inertia, the static friction has to be overwon, which is impossible with only a PD-action.

To enable the controller to generate enough effort to overcome the static friction in such a case, an I-action is added to it. With a PID-controller, the system does limit cycle. This is due to a lower friction level in the slip-mode (Coulomb friction) than in the stick-mode (static friction).

Another point of interest is the influence which the shape of the controller has on the limit

43

5. Conclusions and Recommendations

cycle behaviour. To prevent the system becoming unstable if the controller parameters are changed, the shape of the controller is frozen by making the controller parameters depend on the controllers bandwidth. From the result of the simulations it is clear that the chosen shape of the controller has a distinct influence on the limit cycle behaviour of the system. Many different controller shapes can eliminate the friction induced limit cycle, but the controller bandwidth at which this can occur differs from one to the other. By carefully choosing the shape of the controller, a bandwidth area can be found in which not only the limit cycle no longer appears, Silt the other technica! specifications of the system migM Se E e t 2s we!!.

5.2 Recommendations

At the end of my final project, a plausible explanation and a possible solution came to my attention for the fact that the Shooting Method algorithm was unable to find the unstable periodic solutions of the system with encoder as nonlinearity. The Shooting Method algorithm finds stable and 'weak' unstable periodic solutions of the system. If an encoder is included in the system, this leads to an infinite instability. Every unstable periodic solution has a convergence area. The more unstable a solution gets, the smaller this convergence area becomes. At a transition from stable to unstable periodic solution, one of the Floquet Multipliers will shoot to infinity. Because of this, the Shooting Method algorithm is unable to find the unstable periodic solutions of the system with encoder as nonlinearity. By integrating backwards in time, the Floquet Multiplier will change from co to -& and the corresponding periodic solutions can be calculated with the Shooting Method algorithm. This way the unstable periodic solutions of the system with encoder as nonlinearity can be calculated and the bifurcationdiagram can be completed.

It is very interesting to see what influence the shape of the controller has on the bandwidth area in which a limit cycle does no longer appear. By adjusting the shape of the controller, this bandwidth area can be expanded to the area in which the system fulfils all the technical specifications required for a proper functioning of the process. By including the determination of the controllers shape into the design of the controller, a bandwidth area can be found in which the limit cycle no longer appears and the other technical specifications, such as settling time and stability demands, of the system might be met.

The difference between the friction in the stick and slip phase is part of the reason for the limit cycle occurring. An interesting point is to see to what extend this difference infiuences the limit cycle behaviour of the system. Does a minor difference for instance immediately cause the system to limit cycle.

In this report several systems with different controller and nonlinearity configurations are used. In this view, the next step would be to use a system in which both the encoder and the friction are included.

In the case of an infinite instability, this convergence area equals zero.

44

Bibliography

[i] M. van der Laan and P.M.R. Wortelboer. Limit cycles in mechanical servo-systems. Master’s thesis, Philips Nat.Lab. Technical Note nr. 269/95, October 1996.

[2] M.F. Heertjes and M.J.G. van de Molengraft J.J. Kok D.H. van Campen. Vibration reduction of a harmonically excited beam with one-sided spring using sliding computed torque control. Dynamics and Control, 7:361-375, 1997.

[3] B. Armstrong-Helouvry and P. Dupont C. Canudas de Wit. A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica, 30(7):1083-1138, 1994.

[4] J-J.E. Sloiine and W. Li. Applied Nonlinear Control. Pretence-Hall, 1991.

[5] R.H.B. Fey. Steady-state behaviour of reduced dynamic systems with local non-linearities. PhD thesis, Eindhoven University of Technology, 1992. Ph.D.thesis.

[6] E.L.B. van de Vorst. Long term dynamics and stabilization of nonlinear mechanical systems. PhD thesis, Eindhoven University of Technology, 1996. Ph.D.thesis.

[7] R.I. Leine and D.H. van Campen A. de Kraker L. van de Steen. Stick-slip vibrations induced by alternate friction models. Nonlinear Dynamics, 16:41-54, 1998.

[8] U.M. Ascher and R.M.M. Mattheij R.D. Russell. Numerical Solution of Boundary Value Problems for Ordinary Diflerential Equations. SIAM Classics in Applied Mathematics, 1995.

[9] The Mathworks, Inc. Matlab, version 4 . 2 ~ . i, December 1994.

45

Appendix A

Monodromy matrix

In order to use the iterative scheme presented in Equations 2.6 and 2.7, the monodromy matrix @T must be calculated. The monodromy matrix is the fundamental solution matrix @t at t = T , T being the period time of the periodic solution. Consider an nth-order, autonomous, nonlinear system represented by the state equation

x = f ( z ) (AA)

where ri: = ib z is a column with the n state variables of the system, t is time and f is a column of nonlinear functions of the components of z. The system is called autonomous because f does not depend on t . In an initial value problem, the initial condition is usually given at t = t o . Because f is independent of t , solutions based at time t o # O, can be translated to to, therefore the initial condition reads

dt ’

z(t = O) = zo ( A 4

The solution of the state Equation (A.l) with initial condition (A.2) is often written as $t(zO), to explicitly show the dependence on the initial condition. If an initial state zo is fed to the system, a state dt(z0) is reached after t seconds, as is visualised in the figure below.

time t

Figure A.l: System response

The elements of <pt show how sensitive the elements of &(ZO) are to an infinitely small

46

A. Monodromy matrix

perturbation of the elements of xo. Next, an example is given of how the fundamental solution matrix is calculated.

6$& = 6x0 ; 6x0 << 1

Firstly, the first element of xo is changed with an infinitely small perturbation AZO and the corresponding x, is calculated by integration. The integration is done by using one of the ode-functions of Matlab.

Next, the same is done using the second element of 20.

With (A.3), (A.4) and (A.5) the fundamentai soiution matrix can be filled,

X e i - zei X e , - Xe1

X e 2 - Xe, “2, - X e 2

Element (2 , l ) of at shows how the second element of x, changes if the first element of 20

is pertubated. Here, the fundamental solution matrix is a 2 x 2 matrix. In general the fundamental solution matrix is an n x n matrix, where n represents the number of states of the system.

47

Appendix B

Derivation of controller parameters

B.l Equation for the gain K

In Chapter 3 the standard shape of the controller, as it is used at Philips, is discussed. In Figure 3.5, a schematic picture of the standard shape of the controller is shown. The control parameters T;, T d and wz all depend on the bandwidth of the controller f b . By choosing a value for this bandwidth, the gain K can then be calculated by solving equation

By doing this, the shape of the controller will remain the same, which benefits the controller behaviour. In this appendix, the equation for the gain K is derived with the given factors for the different control parameters.

,û = 0.5

With these expressions for the controller parameters, Equation 3.2 can be rewritten to an expression for the gain K .

l ~ C ( S > l l ~ P ( ~ > l = 1

48

B. Derivation of controller Darameters

I I1 I11 c

IV

With s = j 2 s r f b this leads to,

With these results, Equation 3.2 can be written as:

= l * mJ10 36 KaKm K - 6 JG J144p2 + 352 4 ~ ~ f t J m

_ - 6 1 / m J144p2 + 352 47r2 f t J , K = ” ’ -

36 J370 Ka Km

This can be simplified to

The parameters p, A, Ka, Km and Jm are all known. The gain value K can now be calculated by implementing the desired bandwidth f b [$] into Equation B.1.

49

B. Derivation of controller Darameters

B.2 Controller parameters of Controller B

The controller used for simulation in [l], is frozen at a bandwidth of 10 [Hz] in order to derive factors with which the controller parameters can be written in terms of fb .

1 s ' 7i = =[;;;;;i1

1 314 [3-] Wl = xul2Tfb rad } + X W l = 5 [-]

f b = 10 [Hz]

In the table below the resulting controller parameters of Controller B are summed up.

I 1 1

Table B.l: Controller B parameter values

The equation for the gain value K is derived from Equation 3.2 in the same way as is done in Appendix B. l and reads,

The parameters p, A, Ka, K, and J, are all known. The gain value K can now be calculated by implementing the desired bandwidth f b [?$I into Equation B.2.

50

eindhoven university of technology master limit cycle ...limit cycles occur, is the pipet of...

Documents