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Motion Control of Systems with Backlash

Master Team Project

R.M.R. Bruns 0534792

J.F.P.B. Diepstraten 0529909

X.G.P. Schuurbiers 0529176

J.A.G. Wouters 0529393

DCT number: 2006.075

August 29, 2006

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Contents

Summary v

Introduction vii

1 Previous Work Done 1

1.1 Backlash Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Physical Model for Backlash . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Deadzone Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Modeling with Describing Functions . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Linear Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Describing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.3 Adaptive and Non-Linear Control from the Motor Side . . . . . . . . . . . . 51.2.4 Adaptive Control and Non-Linear Control State Feedback . . . . . . . . . . 6

2 Analysis 7

2.1 Analysis Torque Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Describing Function Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Stability of Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Describing Function of Backlash . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Simulations 13

3.1 Different Backlash models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 PI-Controller with Torque Observer . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Soft Switching Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Experiments 21

4.1 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Design of a Shaft with Backlash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Implementation of the Two Controllers . . . . . . . . . . . . . . . . . . . . . . . . . 244.4 Designed Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

A M-files 29

B Simulink Models 35

C Shaft Design 37

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Summary

Backlash in systems is a non-linear behavior that can affect the set-up in two ways:

1. stability, backlash may cause (high frequent) limit cycles;

2. performance, the load is for some time autonomous, which affects the absolute value of theerror.

The major part of systems with backlash has to do with gears and transmissions. Speed control,which is important, in such systems is investigated in this report. After some research in theexisting literature, two major control strategies for this kind of systems, are recognized. On onehand, there is a strategy, which is designed to keep the time that the load is in the backlash,as small as possible. This methodology is called "strong action in the backlash gap". On theother hand we have "weak action in the backlash gap", which contains two controllers: onefor stability, when the system is in the backlash gap and one for performance, when the back-lash gap is closed. Switching between these two modes is based upon the value of the motor torque.

For each of these two general control strategies, one example is analyzed further. For the"strong action in the backlash gap" we have the PI-controller with a torque observer, whereas forthe "weak action in the backlash gap", a soft switching method is examined. Both controllers are

tested with respect to the prediction of limit cycles, based on the describing function theory.

These theoretical results are also examined using simulations in Simulink. We aim at aconstant velocity on the motor, when a step-disturbance on the load side of the system, is added.Some configurations of the system indeed give rise to limit cycles, which are a serious problem ingears.

In experiments, speed control appeared to be impossible, due to the absence of speed sen-sors on the experimental set-up. Numerical differentiation of the position was not goodenough to determine a well-measured FRF. So position control was done, using a ramp as areference trajectory. Implementation however of the two example controllers gave some draw-backs, because were designed on speed control and due to the non-linearity of the controllersthey can not be simply integrated. So only a manually tuned PD-controller was well tested on

the experimental set-up. A special shaft with backlash was designed to perform these experiments.

To do actual experiments using both example controllers, some remarks are made to im-prove the experimental set-up. This boils down to be able to do speed control on the experiments.

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Introduction

Backlash is a common phenomenon in many systems. Especially in gear systems this is a problemwhich very often occurs, and is a drawback when constant angular velocity is desired. In the pastmany kinds of mechanical solutions are thought of, but it seems impossible to completely reducebacklash in a mechanically way.Another solution to deal with the backlash is by controlling the movement in a clever way. With

the right control strategy it would be very nice, if the existing backlash partly or even completelycould be deleted from the system.For a long time this problem was strictly avoided by control-engineers. The common approachwas to control the system with backlash, like a system without any backlash. When the backlashis very small, no problems will occur, but when high performance is desired, this approach is notgood enough. In the control world several, approaches exists to tackle the backlash problem.

In Chapter 1 of this report a research of backlash and several approaches are examined.Different control strategies in backlash problems are studied; whereas two control strategies areanalyzed further. The stability of those controllers are mentioned in Chapter 2, where also aprediction of limit cycles is given.The two control strategies are compared by simulation in a Simulink environment in Chapter 3.

In Chapter 4 the control strategies are implemented in an experimental set-up which containsbacklash. Followed by some recommendations which would lead to satisfying results.

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Chapter 1

Previous Work Done

When backlash occurs in a system, the stability (in the form of limit cycles) and the performancecan change. There are different models and controllers mentioned in the literature to counteractthese phenomena. Here is a summary given of these techniques. Starting with the models, thereare three different models of backlash considered, three possibilities of the position of the backlash.It can be situated at the input of the plant, i.e. at the motor side, at the output of the plant, i.e.at the load side or between the motor and the load which is called "sandwich backlash", [1].Concerning the controllers, there are different methods, with various results in performance andstability. These are given in a short form. There is a distinction made between three controlmethods; linear control, control using describing functions and adaptive and non-linear controlfrom the motor side, from load side and from both sides. But first three methods of modelingbacklash are discussed.

A motor tracking experiment with the PATO and a tuned controller, see section 4.4, shows theperformance limitations when backlash enters the system. Here the major oscillations are due

to the backlash and the performance is decreased with a factor of 1.8 when a simple sinusoid isapplied to the system, see Fig. 1.1. Between 0.17 and 0.27 seconds the system is in the backlashgap and the performance is decreased. Because it is a repetitive input signal, this phenomenonwill also be repetitive.

Figure 1.1: Position error of a system with backlash

1

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1.1. BACKLASH MODELS CHAPTER 1. PREVIOUS WORK DONE

1.1 Backlash Models

Backlash models or gear play is a common non-linear behavior in mechanical systems. Accordingto the place where the backlash occurs, it will behave differently in the system. Depending on themechanical surroundings and the machines operating conditions, different mathematical modelsmust be used.

1.1.1 Physical Model for Backlash

A physical system, that consists of an inertia-free shaft with a backlash gap 2 , a spring withelasticity ks and viscous damping cs, is modeled, see Fig. 1.2, [2].

Figure 1.2: Two mass rotating system

The exact expression for the torque T is given by:

T = kss + cs

s = ks(d b) + cs(d

b) (1.1)

s = d b (1.2)

Where s represents the shaft twist, d is the difference angle of the motor and the load side andb represents the backlash angle, b | |.

There are three different cases, contact at a backlash angle of , no contact (T = 0) andcontact at a backlash angle of. When there is no contact, Eq. 1.1 gives:

d b =k(d b)

cs(1.3)

which solution is:

d b = (d b(t0))expks(tt0)

cs (1.4)

Since the shaft twist is given, or measured, Eq. 1.4 constructs the backlash angle. When the deriv-ative of the backlash angle is computed, the torque can be computed using Eq. 1.1. This results ina non-linear dynamical system that gives the torque with given shaft twist angle and its derivative.

The expression for the derivative backlash angle is:

b =

max(0, d +kscs

(d b)), if b = ;

d +kscs

(d b), if |b| < ;

min(0, d +kscs

(d b)), if b = .

(1.5)

Master team project 2 Motion Control of Systems with Backlash

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CHAPTER 1. PREVIOUS WORK DONE 1.1. BACKLASH MODELS

1.1.2 Deadzone Model

This model is a simplification of the exact physical model. It neglects the inner shaft dampingwhich leads to a simpeler model, but it is only a valid model if there is no or small damping in

the shaft. The dead zone model is the most common used model in practice, [2]. In this model,the shaft torque Ts , is proportional to the shaft twist s:

Ts = kss = ksD(d) (1.6)

The deadzone function D is then defined as:

D =

d , if d > ;0, if |d| ;d + , if d < .

(1.7)

In this case, the shaft is modeled as a pure spring having no internal damping and no inertia.

When there is no backlash contact, the shaft is assumed to be in steady state, see Fig. 1.3.

Figure 1.3: Deadzone model

If the inner shaft damping is neglectable, this model can be used for adaptive backlash inverse.The parameters of the deadzone model (ks1, ks2 and b) can be estimated using an adaptation law.This approximation for the deadzone model can be used as a compensation for the real backlashand the system can be controlled in a linear way.

1.1.3 Modeling with Describing FunctionsIn case of modeling a non-linear system, the system can be divided into two parts; a linear partand a non-linear part [18]. This non-linearity, like backlash, can be described with a describingfunction.To derive describing functions the following steps are taken: first at the input of a non-linearelement a sine wave is given with constant offset B:

d = B + A sin(t + ) (1.8)

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1.2. CONTROLLERS CHAPTER 1. PREVIOUS WORK DONE

harmonic NAA:

s = NBB + NAA sin(t + ) (1.9)

NA(A,B,) = NP(A,B,) +jNq(A , B, ) (1.10)

NB = NB(A , B, ) (1.11)

These two parameters are called the dual input describing functions, DIDFs. The DIDF can alsobe reproduced as:

Ts(d, d) = NBB + NPA sin(t) + NqA sin(t) (1.12)

The operation condition can be described as:

T0 = BNB(A , B, ) (1.13)

With the unique solution:

B(A, T0, ) (1.14)

When T0 = 0, the describing functions reduces to a sinusoidal input describing function, SIDF.In most cases the backlash is described with a SIDF (see section 1.2.2), so the only describingfunction is the following:

N(X, ) =Y1X

expj1 (1.15)

With:X = Amplitude of input sineY1 = Amplitude of the first harmonic component

F1 = Phase shift of the first harmonic componentThe describing function can be dependent of the frequency but this is not necessary. Forthe nonlinear control system, we have a limit cycle if the sinusoid at the nonlinearity inputregenerates itself in the loop, that is:

G = 1

N(X, )(1.16)

1.2 Controllers

The suggested controllers differ a lot in usability and have different results. There are not manysolutions for the backlash control problem that will work perfectly. But still there is a lot of effortdone, trying to solve this problem with various control strategies.

1.2.1 Linear Controllers

Backlash introduces nonlinearity in the system. If a linear controller must control this system, itis necessary that the controller must be robust enough to handle the non-linearity, [3].If there is no backlash in the system, a simple PID-controller can be used to control the system.To increase the performance a higher bandwidth is needed. This is realized by adding two notches,see section 2.4. Without backlash this gives no problem, but when the backlash is introduced thehigh bandwidth controller causes large limit cycles for an operating point with no load torque.The PID-controller does not have limit cycles, however the performance is worse, [4].A solution for this problem is to switch between these two controllers, a better performance isrealized and the limit cycles do not exist. The controller, "weak action", is now non-linear and has


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CHAPTER 1. PREVIOUS WORK DONE 1.2. CONTROLLERS

a tremendous performance, this controller, [4], will be analyzed and discussed further in chapter2.

Another way of designing a controller is proposed in [5], especially in gears with backlash.

They started with the assumption that only the position and the velocity of the driving gearare available for feedback. The controller used was a simple PD-controller. With the Lyapunovstability theorem they simply could prove stability of the position and angular velocity of themotor and the angular velocity of the shaft, but they needed Lasalles theorem to prove stabilityfor the position of the driven gear. However the results were not confirm the theory; the positionof the driving gear did go to zero, but the position of the driven gear did not. It converges to asmall, but non-zero value.

Another linear control method is composed of a PID-controller and a feedback compensa-tion of the load torque, "strong action". This load torque is estimated by a torque observer.When there is no load torque, the backlash gap is opened and the compensation influencesthe PID-controller. Without this influence limit cycles will occur. The controller with feedback

compensation can suppress the vibrations of the system well, [6], however the performance of thesystem is worse. To improve this, a feed-forward compensation is added. Now the performancecan be improved well, [7].

In [3] is shown that linear controllers increase the torque when the backlash is entered,which results in an impact, and is obviously not wanted. Therefore linear control is not a greatsolution to the backlash control problem.

1.2.2 Describing Functions

A logical way to integrate a non-linear element into a linear control design problem is to describethis non-linear part by a describing function, [8]. Such a non-linear element can be for examplebacklash or coulomb friction. It expresses how it depends on certain values like position or velocity.

A describing function is always an approximation of the real situation. Often, the non-linearitymust be described by a set of describing functions and can contain dynamics, which makes thedescribing function frequency dependent, [9].With an inverse of the describing function the backlash can be neutralized. The control strategyis the following: the linear part of the open loop must not intersect with the negative inverse ofany of the describing functions, then no limit cycles are predicted, see section 2.3. Furthermore adescribing function approach can be used for two purposes: limit cycles can be analyzed and theinput/output behavior of a non-linear plant can be characterized, [10].

Describing functions can be divided in two parts, the memoryless element describing func-tion and the integrated plant/backlash describing function. The memoryless element has nointernal state, only real numbers and is never frequency dependent. The integrated plant/backlashis the opposite of the memoryless element. It gives an approximation of influence of the whole

plant state on the backlash. And can therefore be imaginary and frequency dependent, [11].

1.2.3 Adaptive and Non-Linear Control from the Motor Side

Many industrial drive systems can be reasonably well modeled as a two mass system withbacklash. In this report this model will also be used. The first mass corresponds to that partof the machine that contains the engines, whereas the second mass corresponds to the loadside. To model the actuator dynamics we include non-minimum phase dynamics. The controlproblem in this case, is the speed regulation of the motor with torque disturbances on thedriven load. In [13] a model reference adaptive controller (MRAC) is proposed to do this job:


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1.2. CONTROLLERS CHAPTER 1. PREVIOUS WORK DONE

a PI controller that is only active when a disturbance occurs and it compensates for steady-state speed errors. This PI-controller suffices to use a linear speed controller of P-type in steadof PI-type, which makes that the order of the system is reduced and thus creating a faster response.

In [4] it is stated that it was possible to get almost the same performance for a systemwithout backlash when something smart was done. Backlash has two major drawbacks:

1. Performance the load is for some time autonomous.

2. Stability backlash introduces (high frequent) limit cycles.

The idea was to make two separate linear controllers that counteract these two drawbacks. So onecontroller was tuned optimally for the system without backlash, and one for robust performance.By smart switching it was possible to get almost the same performance as for the systemwithout backlash. This switching was done by looking at the controller-output, the motor torquereference, in stead of the current backlash angle, [14], [15]. The reason for this was very sim-ple; it is far easier to measure the control-output, than the exact value of the current backlash gap.

Something almost the same, was done by [16], in stead of switching between two linearcontrollers, two linear observers, one for the system with backlash and the other one for thesystem without backlash were used. Switching in this case is done by looking at the state of thebacklash element. Two non-linear schemes were also proposed, which differ with the amount ofsensors available:

1. When only a motor sensor is available, a compensation of the steady state error is proposedby using special elements in the controller;

2. If there is also a sensor at the load side and the system is in the backlash, a trajectory ispredicted which ensures that the gap is closed smoothly.

1.2.4 Adaptive Control and Non-Linear Control State Feedback

Another way of computing a controller for systems with backlash is the state-dependent algebraicRiccati equation for optimal control, SDARE method, proposed by [15]. This method has severaladvantages; it is easy to implement, secondly it does not require that the backlash (and otherparasitic effects) are smooth functions and the underlying theory is intuitive. To use the SDAREmethod, it is important that the model have a non-zero slope at the origin. If the slope in the modelis zero, the system is not locally controllable and many design methods, including the SDAREfail.The SDARE method leads to a two gain state feedback control law. Similar to the section 1.2.3,switching is needed to change the controller but here switching is applied if the difference anglebetween the motor and the load is larger then the backlash angle. In [15] an observer to constructthe backlash angle, if it is not possible to measure it, was proposed. Simulation show that theswitched gain control law achieves a rapid response without overshoot and eliminates limit cycles.

The performance of this controller, if an observer is used to determine the backlash gap does notdiffer much, only a high frequency oscillation is added to the response.


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Chapter 2

Analysis

In the previous chapter two main strategies for controlling backlash are proposed. The first strategyis the strong action, in the backlash gap, based on [3], which is meant to close the backlash gapas fast as possible by applying an increase in torque (strong action) while the system is in thebacklash gap. When the system is in the backlash gap the load behaves autonomously, the strongaction strategy is meant to minimize this autonomous time. The second strategy is the weak action,in the backlash gap, based on [4], which is a method that deals with the two major drawbacks ofbacklash:

stability;

performance.

The weak action has a high performance controller when the backlash gap is closed, but switchesto a controller that provides stability, when the backlash gap is opened. In this chapter oneexample of both methods is analyzed: the PI-controller with load torque observer for the strong

action strategy and the soft switching controller for the weak action strategy. When the linearsystem is stable, the stability problem of the system with backlash is restricted to limit cycles,[20] and [19]. Describing function techniques are used to investigated the limit cycle behaviorof the non-linear system. The describing function analysis is an approximate method that isbest suited for the discontinuous non-linearities in control systems. The method gives a reliableprediction concerning limit cycle behavior, [18].

2.1 Analysis Torque Observer

To handle the backlash in a system a torque disturbance, estimated by the observer, is added.This torque disturbance enters the system on the location of the backlash. An observer is used to

estimate the state of the system.

In the figure, 2.1, the layout of the system is showed. The input of this is motor torqueTm. It also contains motor velocity m, shaft torque Ts, disturbance torque Td, load torque Tland load velocity l.The torque observer has two inputs; the motor velocity and motor torque. The equations in thetorque observer are:

T =Tm Jmms

tfilts + 1(2.1)

Ucomp = K01

tfilts + 1T (2.2)

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2.1. ANALYSIS TORQUE OBSERVER CHAPTER 2. ANALYSIS

Figure 2.1: Layout of the system

In the first equation, the so called shaft torque estimate is calculated. The ms term representsthe angular acceleration. When the system drives at constant speed this term does not contributeto this equation. When the system enters the backlash, the velocity changes and therefore alsohas an acceleration. So the term ms does contribute to the equation. This term, multiplied withJm, the motor inertia, leads to a virtual torque, and is compared with the motor torque in Eq.2.1. So when the system enters the backlash gap, an acceleration is present and and extra torqueis given to the system. This leads to a shorter time in the backlash gap. With the parameter tfiltthe engine dynamics is taken into account.

Figure 2.2: Frf of Ucomp over T

In Eq. 2.2 the control signal is calculated. The compensator gain K0 has to be manually tuned.The frf of this equation can be seen in Fig. 2.2. The system can now be analyzed as a linear systemand therefore can be linear controlled with a PI controller. By this strategy a strong control isdesired, this means that the backlash must be traveled as quickly as possible. So that the motorside and load side are disconnected for the shortest possible time.


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CHAPTER 2. ANALYSIS 2.2. DESCRIBING FUNCTION METHODOLOGY

2.2 Describing Function Methodology

The presented system is a single-input-single-output nonlinear system represented by the feedbackconnection of Fig. 2.3.

Figure 2.3: feedback connection

The linear part, g(s), is the system without backlash and the non-linear part, f( ) is the backlashwhich is modeled as a deadzone backlash, see section 1.1.2. For simplicity the input r(t) is set to 0.The existence of periodic solutions, limit cycles, needs to be proved. A periodic solution satisfiesy(t + 2

) = y(t) for all t, where is the frequency of oscillation. The condition for harmonic

balance can be derived through Fourier analysis with the trial solution of Eq. 2.3.

e(t) = A sin(t) (2.3)

This results in the condition for harmonic balance, Eq. 2.4. With the describing function, Eq.2.5, the condition for harmonic balance is written into Eq. 2.6. This condition has an interestinggraphical interpretation. Let the Nyquist diagram g(j) of the linear system be plotted in thecomplex plane, with in the same diagram the describing function 1

N(A) .

0 = g(j)(U1 +jV1) + A (2.4)

N(A) =U1 +jV1

A(2.5)

g(j) =1

N(A)(2.6)

A limit cycle is predicted if there is an intersection of the two functions.

2.3 Stability of Limit Cycles

The stability of the limit cycles are not jet predicted, therefore the similarity between the above

Eq. 2.6 and the Nyquist criterion for limit cycles is used. In Fig. 2.4 a fictitious system (G) withdescribing function criterium 1

Gdis plotted, where Gd is the non-linear backlash part of the system.

Here are two limit cycles with characteristics (MA, A) and (MB, B) with MA < MB, where Mrepresents the amplitude A. Consider the intersection A of G(j) and 1

Gdand assume a small

decrease in amplitude MA. The representative point on the describing function criterium willmove to a new point, D. This point is not encircled by the system, the system will move furtherand further away from the intersection and the oscillations will eventually stop. Therefore, pointA has divergent characteristics and is an unstable limit cycle. A similar analysis can be made withrespect to point B, with the difference that B has convergent characteristics and is a stable limitcycle.In general, the limit cycle is predicted to be stable or unstable depending on the direction ofcrossing with respect to the linear system function in the Nyquist diagram.


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2.4. DESCRIBING FUNCTION OF BACKLASH CHAPTER 2. ANALYSIS

Figure 2.4: Stability of limit cycles

2.4 Describing Function of Backlash

To predict the limit cycles and their stability, the non-linear part of the system is modeledas a describing function. The different backlash models (section 1.1) have different describingfunctions. Here a classical (deadzone) model, section 1.1.2, is chosen to predict the existence oflimit cycles with the describing function analysis, [21]. The result of this backlash describingfunction with the controlled systems, total system with performance and with stability controller,is presented in Fig. 2.6.

N(A) = 0, if |x| ;

N(A) = k

1 Ns

A

, if x .

(2.7)

where:

Ns(z) =2

arcsin

1

z

+

1

zcos

arcsin

1

z

(2.8)

To illustrated the prediction of limit cycles, the later on used controllers Ghigh and Glow, seesection 3.3, of the switching controller are used. These controllers have a similar form, see Eq. 2.9,but different parameters, [4] and appendix A.

G(s) = kp

sTi + 1

sTi

s2/21 + 2/1 + 1

s2/22 + 2/2 + 1

s2/23 + 2/3 + 1

s2/24 + 2/4 + 1

(2.9)

In this equation three elements can be analyzed. First a PI controller is used to increase theperformance of this system. Due to the fact that it is a speed controlled system and thus lowfrequent a -1 slope in amplitude and -90 degrees in phase and a anti-resonance and resonancewhich lead to a phase lead of 135 degrees, a PI controller can be used. The first notch-filteris used to decrease the phase between the anti-resonance and resonance, in order to decreasethe amplitude after the resonance. This prevents the system to become unstable, because the


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CHAPTER 2. ANALYSIS 2.4. DESCRIBING FUNCTION OF BACKLASH

cross-over frequency after the resonance occurs at a lower frequency, where there is still somephase advance, see Fig. 2.5. The second notch-filter is needed to gain more phase in that frequencyregion. This is a notch with a phase lead after a phase lag.

(a) With and without notch 1 (b) With and without notch 2

Figure 2.5: System with and without one of the notches

In Fig. 2.6 the high performance controlled system, Ghigh, there is an intersection with thedescribing function. The controller with high stability properties doesnt has this intersectionand therefore doesnt predict a limit cycle. The describing function goes to zero with increase ofamplitude of the sinusoidal input. Thus the predicted limit cycle is stable. The describing functiondepends on the shaft stiffness, another value for the stiffness result in a change and it might beinevitable to get limit cycles with this type of controller. The system has still an intersection withthe backlash describing function, but this is at a very low amplitude and such a high frequencythat it is neglectable.

Figure 2.6: Prediction of limit cycles


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Chapter 3

Simulations

The two main backlash models, the exact and the deadzone model, are simulated with the softswitching controller to verify the output of both models w.r.t each other. The deadzone model

behavior should be close to the exact model behavior when there is no or small damping in theshaft (section 1.1.2). In simulation this damping can be adjusted easily by changing a constantvalue.Furthermore, the two controllers analyzed in Chapter 2 are implemented in Simulink and testedwith a constant angular velocity and disturbance.

3.1 Different Backlash models

The difference between a deadzone and exact backlash model is evaluated with a simulation. Theused controller is a PI controller with relative low bandwidth. In theory the deadzone modelrepresents the exact model if the shaft damping is low. The results show indeed huge differenceswhen the shaft damping is high.

(a) Angular velocity (b) Motor torque

Figure 3.1: Simulated results with low shaft damping

In Fig. 3.1 it can be seen that the deadzone model behaves like the exact model. The main erroroccurs when there is no load disturbance torque (after 2 seconds) and the system jumps throughits backlash. Fig. 3.2 shows, like predicted, that with large shaft damping the differences betweenthe two models are huge.

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3.2. PI-CONTROLLER WITH TORQUE OBSERVER CHAPTER 3. SIMULATIONS

(a) Angular velocity (b) Motor torque

Figure 3.2: Simulated results with high shaft damping

3.2 PI-Controller with Torque Observer

To simulate, the system is built in Simulink and tested. The model contains a two mass-systemwith backlash. Further more, it contains a block with actuator dynamics, the controller-blockand torque observer-block. The motor velocity is controlled, and the load torque is added asdisturbance. The used parameters are given in PI-torqueParameters.mand the simulink model inAppendix B.The simulation is done by controlling the motor velocity at a constant 10 rad/s. The input torqueat the motor side is 10 Nm. At 5 sec a load torque, disturbance, of 400 Nm is applied on thesystem. In Fig. 3.3 a comparison between a system with and without torque observer. The systemwith observer has a slightly better performance.

Figure 3.3: Motor velocity for system with and without observer

In Fig. 3.4 the motor velocity and load velocity are plotted. Because of the disturbance the loadvelocity decreased and also the motor velocity decreased. This disturbance is not big enough toget the load through the backlash, so the gap remains closed. The load velocity decreases, becauseof this disturbance and through the stiffness of the shaft and the motor velocity decreases also,but with a time delay. The controller tries to get the velocity back up to 10 rad/s, so it sends out


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CHAPTER 3. SIMULATIONS 3.2. PI-CONTROLLER WITH TORQUE OBSERVER

Figure 3.4: Motor velocity and load velocity

a torque to speed-up the motor velocity to the correct value. The backlash stays closed and themotor and the load velocity are increased at almost the same time.At t = 7s the load torque is set to zero. The load and the motor accelerate, because of theextra available torque. The magnitude of the step is not big enough to open the backlash. Thecontroller decreases the motor velocity whereas the load maintains its velocity. The backlash gapnow opens. At t = 7.3 s the backlash gap is closed again on the other side. Here the load has ahigher velocity, so the motor velocity is increased again. Then the controller sets both motor andload speed to the steady state value of 10 rad/s. Between t = 7.7s and 8s the situation is preciselythe other way around, but the system does not jump completely through the backlash.

Figure 3.5: Backlash gap

In Fig. 3.5 the backlash gap is shown. The shifted position in the backlash can be seen. From 7.3still 7.7s the backlash is closed on the other side. Also the difference between a system with- andwithout observer is shown. The desired approach of a strong controller is to be in the backlashgap as short as possible. A strong controller (with observer) runs faster through the backlash gap


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3.3. SOFT SWITCHING CONTROLLER CHAPTER 3. SIMULATIONS

then a normal controller (without observer).

Figure 3.6: Shaft torque and motor torque

In Fig. 3.6 the shaft torque and the motor torque are shown. From 5s to 7s nothing specialhappens, the backlash gap remains closed. When the backlash gap opens a small increase ordecrease in torque appears. This is caused in the same way explained earlier. When the loadtorque is set to zero, with a step, the backlash gap opens two times. At the moment the positionof the backlash is shifted, the torque changes.

3.3 Soft Switching Controller

The soft switching controller has three parts: two controllers and a switching function. Thesetwo controllers have different properties. The first controller Ghigh is a controller with highperformance, but this controller causes limit cycles around the backlash, see section 2.4. ThisGhigh is the controller with the highest performance. The second controller, Glow is globallystable, but has reduced performance. These two linear controllers are combined into one non-linearcontroller using switching such that the system has the performance of Ghigh when the sys-tem is not in the backlash and it has the stability ofGlow, when the system is around the backlash.

Like stated before, the motor angular velocity, subject to a load disturbance, needs to becontrolled. When the plant with the backlash gap is closed, due to a large load torque Td, theplant behaves like a linear plant without backlash. For this linear system, the transfer functionbetween the input Tm and output m, the transfer function in which we are interested, can becomputed. The bode diagram of this system with actuator dynamics (modeled as a delay of 10ms) and without backlash is presented in Fig. 3.7.The controllers Ghigh and Glow are suggested as a PI controller with two notches, this results ina bandwidth of 18 rad/s for the high performance controller and a bandwidth of 12 rad/s of thelow performance controller Glow. The closed loop specifications are:

1. Global stability (for the linear system the controller Ghigh is globally stable, but with back-lash the controlled system will show limit cycles when the backlash gap is open).

2. Minimal speed error of the system subjected to a load torque step.

3. Sensitivity modulus < 6 dB.


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CHAPTER 3. SIMULATIONS 3.3. SOFT SWITCHING CONTROLLER

Figure 3.7: Bode diagram of system with and without controllers

Both systems do not exceed the 6 dB criterion, the performance of the high performance controlleris indeed better, then the stability controller, see Fig. 3.8. And both controlled systems areglobally stable for the linear case.

Figure 3.8: Sensitivity of controlled system with both controllers

Some simulations are made to verify and predict results of the Soft Switching controller.The used parameters are given in SystemParameters.m and SoftSwitchingController.m. Speed-control is desired at an angular velocity of 10 rad/s. There is a switching between the twocontrollers that is based on the motor torque. When the torque is high, the backlash gap is closedand the high performance controller is used. When the torque is zero, the backlash gap is openand the stability controller is used. In between the value for which the high performance controller


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is used, dependant of upper bound Tmax, and a motor torque of zero (or a lower bound, Tmin)there is a soft switching. This is done by the following equation:

v(T) =

0, if |T| Tmin;

|T|TminTmaxTmin , if Tmin < |T| < Tmax;1, if Tmax |T|.

(3.1)

For this simulation the values for Tmin and Tmax are respectively 0 and 165. These can bespecified experimentally. A load disturbance of 400 Nm is initially put on the system and aftertwo seconds this value is set to 0. And thus switching is needed after two seconds. Because thesystem does not start up in steady state, there is some switching in the first two seconds.First the two controllers separately are considered, it can be seen that there occurs limit cycleswhen there is no load disturbance applied to the system for the GHigh system, see Fig. 3.9 and3.10, like predicted. The results of the GLow system, Fig. 3.11, show, like predicted, no limitcycles, Fig. 3.12. In initialization fase, the first 2 seconds, the performance of this controller isless than GHigh. It takes more time and with a larger error to achieve the convergence. When the

motor torque of the two systems are compared, it can be seen that the controller with the bestperformance demands more torque. After the two seconds of initialization with a load torque of400 Nm, the limit cycles occur and the GLow is desirable. These simulation results are expected,there are limit cycles when only GHigh is applied.

(a) Angular velo city and applied load torque (b) Motor torque

Figure 3.9: GHigh controlled system with applied load disturbance

When the soft-switching controller is applied, the following results are simulated. In Fig.3.13 the controlled m with the same disturbance as in the previous results is shown. Here the

error in initialization fase is roughly the same as in Fig. 3.9, but when the disturbance is takenof the load there are no limit cycles like predicted, Fig. 3.14, and the performance tends to thestability controller.

The main difference in this area are some oscillations in the time interval from 2 to 2.5 seconds.This can be explained with the switching parameter v, Fig. 3.14. Here between 2 and 2.5 secondsthere is a huge switch from 0 to 1 and back. This because the motor slows to much down and isshifting from one side of the backlash gap to the other.The constant value of v at the of the simulation is a result of the low value for Tmin. Because thevalue of the motor torque with only the second mass as load is higher than this parameter. Thefinal controller in rest is thus a combination of the two controllers.


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CHAPTER 3. SIMULATIONS 3.3. SOFT SWITCHING CONTROLLER

Figure 3.10: Phase diagram of GHigh controlled system


Figure 3.11: GLow controlled system with applied load disturbance

Figure 3.12: Phase diagram of GLow controlled system


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Figure 3.13: Soft Switching controlled system with applied load disturbance

(a) Value of switching parameter v (b) Phase diagram of switching controller

Figure 3.14: Soft switching controller v and phase diagram


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Chapter 4

Experiments

To test the controllers and to verify the simulation results, experiments are done. The PATOset-up is used to track a reference. Due to problems with differentiating, position control is the

strategy. To introduce backlash into the system a special shaft is designed. Where after the twodiscussed controllers are implemented. Because of the occurring problems the results will differfrom the expectations.To encounter the problems that occur when a system has backlash and to compare differentmethods of control using one backlash model a manually tuned controller is analyzed within aexperimental set-up.

4.1 Experimental Set-up

To test the two controllers that were selected in chapter 1 and simulated in chapter 3, a simpleexperimental set-up is used. This set-up is used many times in literature and it is also chosen here.It contains two masses of different weight, connected with a shaft. In this shaft a special design

is mounted to create the necessary backlash. This design is discussed in section 4.2 of this report.The set-up contains two position encoders, the first one to measure the position on the motor side,the second one to measure on the load side. The input of the test system is torque, created througha current. The output of the system is the position in radians. Because this report is about speedcontrol, the output of the system needs to be differentiated using a 8th order butterworth-filter tosuppress the influence of the obtained high frequencies due to the numerical differentiated.

(a) PSD of the output signal (b) PSD of the differentiated output signal

Figure 4.1: Power spectral density of output and differentiaited output

With the knowledge of the power spectral densities of the output and the differentiated output,

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4.1. EXPERIMENTAL SET-UP CHAPTER 4. EXPERIMENTS

Fig. 4.1, a butterworth filter with cut-off frequency of 300 Hz is constructed, see Fig. 4.2. The highmagnitude at the low frequencies present in the output signal y, is still present in the differentiatedsignal, therefore it is not necessary to filter this low frequency of the measured signal. The noise-peak at 350 Hz, present because of numerical differentiation of the output y, is the lowest frequency

that should be filtered out of the signal. Therefore fc is chosen at 300 Hz.

Figure 4.2: Butterworth filter

The results of the frf-measurement of the linear system without backlash are plotted in Fig. 4.3.The coherence of the process-sensitivity is by far not sufficient to achieve a reliable frf of the plantwith this measurement. The measurement of this transfer-function of the angular velocity over thetorque is therefore not useful with this experimental set-up. To analyse the influence of backlashwith this experimental set-up, position control is used. The reference trajectory however will bea ramp, so although position control is done, also some conclusions can be drawn about speed

control for systems with backlash.

(a) Sensitivity (b) Process-sensitivity

Figure 4.3: Measurement of angular velocity over torque

Another frf-measurement is made to determine the transfer-function of the position over the torque.Here, it is not necessary to differentiate the output and thus is the coherency of the sensitivityand process-sensitivity, Fig. 4.4, much better. The achieved frf, Fig. 4.5, is therefore reliable (inthe frequency domain of the high coherence).


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CHAPTER 4. EXPERIMENTS 4.2. DESIGN OF A SHAFT WITH BACKLASH

(a) Sensitivity (b) Process-sensitivity

Figure 4.4: Measurement of position over torque

Figure 4.5: Frf of the system; position over torque

4.2 Design of a Shaft with Backlash

To test the soft-switching and PI-controller with torque observer an experiment will be done. Forthis experiment the already existing PATO-machine will be used. In this PATO a flexible shaftconnects the motor and load with each other. This flexible shaft must be replaced by a new shaftwith backlash.This new shaft has to fit in the PATO-machine and contains a backlash gap of 2 degrees. It is infavorite that the shaft has also a little flexibility in it. The design is attempted to be as easy aspossible, so that the production of it takes as little time as possible.

The design consists of two parts that can be slide into each other. Each part has a solidrod in the middle. This rod is the same as in the old shaft, so the flexibility of the shaft is partlymaintained. At the end of this rod a tube is glued around it. This tube has a diameter of 2.5 mmand a wall thickness of 0.5 mm. The end of this tube is cut in longitudinal direction and one halfof the tube is cut off. Because part one has a cut off of 180 degrees and part two (see Fig. B.2)has a cut off of 178 degrees, a backlash gap of two degrees has been created.To make sure that the two parts stay in line with each other, a second tube is fitted around it.This tube has a diameter of 3 mm and a wall-thickness of 0.25 mm and is glued to part one.

If the two parts slide into each other, the shaft contains a backlash gap of 2 degrees. If


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4.3. IMPLEMENTATION OF THE TWO CONTROLLERS CHAPTER 4. EXPERIMENTS

(a) Part 1 of the shaft (b) Part 2 of the shaft

(c) Assemblage of the two parts

Figure 4.6: Shaft design for experimental setup

part two is replaced by another part that has a cut off of 180 degrees instead of 178 degrees, thebacklash gap is set to zero. In this way it is possible to do a comparison between the shaft withand without backlash. The total drawing of the shaft is visible in appendix C.

Although the design of the shaft is made as simple as possible it took to much time toproduce the shaft. Therefore another, less favorite, solution has been used. The already existingshaft in the PATO is loosen a little bit. This is done by putting a small tube around the shaftwhich is fixed in the mass. This tube has a larger diameter than the shaft so that the shaft is notfixed to the mass; this tube also makes sure that the shaft does not come out of the mass.The main disadvantage of this solution is that the backlash gap is not constant, because of thefact that the fixation of the tube is not robust. Due to the vibrations, which occur during theexperiments the tube can come loose a little bit, which will result in a larger backlash gap.Another disadvantage of this solution is the position of the shaft fixation, because the tube has a

larger diameter than the shaft, the center line of the shaft is not aligned with the center line ofthe motor anymore.

4.3 Implementation of the Two Controllers

The implementation of both suggested controllers in the experimental set-up lead to someproblems. In order to apply the torque observer controller the angular acceleration is needed,which is a problem, because the encoder only measures angular position and as stated in section4.1 it is impossible to differentiate the position to velocity numerically, and therefore also fromposition to acceleration. So the contribution of the angular acceleration can not taken intoaccount.


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CHAPTER 4. EXPERIMENTS 4.4. DESIGNED CONTROLLER

The second problem of the torque observer implementation is the input from motor torque. Thedesired input is not available, so the assumption is made that the torque is proportional with thevoltage, whereas actually the torque is more or less proportionally with the current.It is uncertain if the remaining torque observer is still able to detect the state of the system, and

therefore still can be used in the experimental set-up with backlash.

In the experiment a ramp in position, a constant velocity, is set as input. On a certaintime (10 sec) a negative step in position is put on the ramp. This causes the opening of thebacklash gap.The parameters of the implemented torque observer need to be (manually) tuned. In Fig. 4.7 theerror of the system with and without observer can be seen. These results show that the systemwith observer has almost the same error values than the system without observer.

Figure 4.7: Experimental results of Torque Observer controller

It seams clear that the observer cannot be used in this system. The system with observer shouldbe more damped than the system without backlash, this is not the case. So maybe too muchinformation is lost in the remaining of the original torque observer. The acceleration term isomitted, and also the motor torque is not the exactly right value. This means that it is notpossible to implement a torque observer in this system, with this layout.

With the usage of a velocity encoder the angular velocity of the motor can be measured.Still the angular acceleration is needed to observe the torque, which makes another numericaldifferentiation needed. Another problem is the motor torque estimation, this could be done bymeasuring the ingoing current to the motor. The current is proportional with the torque becausea DC-motor is used in the experimental set-up.

The soft switching controller can not be implemented because the control is based uponposition and not upon speed. Therefore an extra 90 degrees phase lag is "added", which causesthe system to become unstable with the type of used control. To gain 90 degrees phase the PIis substituted with a PD controller. The result is a controller which does not behave at all likeour expectations. In order to investigate the influence of backlash in our experimental set-up, amanually tuned controller is implemented, section 4.4.

4.4 Designed Controller

To design the manually tuned controller, the frf-measurement is used. In order to actually seethe difference between the error of a system with and without backlash, a controller needsto be tuned, which leaves a small error. In Fig. 4.5, the frf-measurement of the experimental


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4.4. DESIGNED CONTROLLER CHAPTER 4. EXPERIMENTS

set-up, without backlash, is plotted, which is the basics of loop-shaping of the controller. Thebandwith of the controller needs to be as high as possible, to suppress as many disturbancesas possible. Theoretically the bandwith of a second order system can be placed anywhere, butin this experiment it is not the case. Due to the noise, present from 1000 Hz and the phase lag

it contains, the maximum bandwith is 30 Hz. When the bandwith is increased from that point,the slope -2 after the antiresonance and the resonance and the phase lag in that region, gets thesystem unstable. Implementing this controller leads to a error of 0.1 rad.

This amount of error is not enough to see the backlash in the error signal. So a accelera-tion feedforward is added to increase the performance. The "mass" is calculated using the dataof the frf measurement, see Fig. 4.4. Where the coherence is high the mass can be determined ina point of the -2 slope. Using this mass feedforward the error decreases to 0.04 rad.

In Fig. 4.8, the error is plotted for the system with and without backlash. This controllerand the mass feedforward is now also implemented in the system with backlash. In this figure itis wel seen that the system has backlash:

Figure 4.8: Position error of a system with and without backlash

In Fig. 4.8, the error signal of a sinus-wave scaled reference signal is shown. The backlash occurseach time the motor is decelerating. This is at the moment the reference signal is going throughzero. At this point the load is getting a higher angular velocity than the motor, due to the inertiaof the load. Consequently the backlash gap is entered. At t = 0.17 the load reaches the other

side of the backlash gap, which results in an impact. This impact causes the sudden increase ofthe error signal, of a factor 1.8. Only 0.02 seconds later another impact is visible, here the loadreaches the side of the backlash again. This process of impact repeats itself one more time, butwhit smaller amplitude, because of the friction of the system. At t = 0.27 no more impacts appearand the load remains at the other side of the backlash gap.


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Conclusion and Recommendations

In this report it has been shown that backlash indeed gives a problem when only a PID controllerhas been used. In general the backlash will result in a performance limiting effect. To prevent thiseffect two controllers are investigated, one example from each of the two major control strategies.The PI-controller with a torque observer for the "strong action in the backlash gap" and the softswitching controller for the "weak action in the backlash gap". Both controllers were analyzed

theoretically whether limit cycles did occur. Also both controllers were simulated in Simulinkusing a constant velocity and a load disturbance. The soft-switching controller had the bestperformance, because it has the ability to switch between a stability and a performance mode.Also the impact of closing the backlash gap is much smaller in the soft switching controller, whichleads to a better performance and less wear. The implementation of both controllers however didnot succeed, due to the differences between speed and position control. Our PATO set-up didonly have position encoders, so no speed control could be done. Also a manually controller wasdesigned, which uncovered the effect of backlash in the total error behavior.

A few recommendations exist to avoid the problem in this set-up. The encoder should be-come an acceleration encoder. The velocity can be calculated by integration, this would notlead to any problem. Another way of dealing with the problem is to think of a clever way todifferentiate the position signal to velocity and acceleration signals.

Also the motor torque should be well estimated. This could be done by measuring the ingoingcurrent to the motor. The current is more or less proportional to the torque. So from the currentthe motor torque could be calculated and the torque observer would work better.When there will be more experiments the new shaft design should be used. In this case, thebacklash gap would be a constant so better results would be expected.

In the future, research should be done in getting a reliable and easy way in implementingthe two proposed controllers. Secondly there should be a more accurate test set-up where such ahigh bandwith can be achieved, that limit cycles experimentally occur.

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Appendix A

M-files

SystemParameters.m

c l e a r al l ; c l o s e al l ; cl c

% D e f i ni n g p a r a me t e r s f o r S o f t S wi t c h i ng s i m u la t i o n .

J m = 0 .4 ; % M o to r m o me n t o f i n er t ia [ k g m2 ]

J l = 5 .6 ; % L oa d m o me n t o f i n er t ia [ k gm 2 ]

c m = 0 .1 ; % V i s c ou s m o t or f r i c ti o n [ N m / ( r a d / s ) ]

c l = 1 ; % V i sc o us l oa d f r ic t io n [ N m /( r a d /s ) ]

k s = 3 30 0; % S h a ft e l a s ti c i t y [ N m / r a d ]

c s = 1 ; % I n ne r d a mp i ng c o ef f ic i en t o f t he s h af t [ N m /( r a d /s ) ]

t d = 0 . 00 4 ; % T im e d el ay , f ro m r e fe r en c e t o m o to r t o rq u e [ s ]

t e = 0 . 00 6 ; % T im e c o ef f ic i en t a c tu t or [ s ]

a lf a = 0 . 05 ; % B a ck l as h a ng l e [ r a d ] 0 .0 5

I n i t i al a n g l e = 0 . 0 5; % I n it i al b a ck l as h a ng l e [ r ad ]

I n i t i al s p e e d = 1 0 ; % I n it i al m ot o r s p ee d [ r ad / s ]

29

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APPENDIX A. M-FILES

SoftSwitchingController.m

% S o f t s wi t c h i ng p a r a m et e r s

T mi n = 0 ;

T ma x = k s * al f a ;

% C o n t r ol l e r s

% F o r P I - c o n t r o l l er

K p = 2 6;

T i = 0 .2 ;

% P a ra m et e rs o f G _ hi g h

K ph = 1 60 ;

T ih = 0 .2 ;

z 1h = 0 .2 5 ;

z 2h = 0 .5 5 ;

z 3h = 0 .2 ;

z 4h = 0 .2 5 ;

w 1h = 8 0;

w 2h = 8 0;

w 3h = 8 5;

w 4h = 4 5;

% P a ra m et e rs o f G _ lo w

K pl = 8 0;

T il = 0 .2 ;

w 1l = 1 00 ;

w 2l = 1 00 ;

w 3l = 4 5;

w 4l = 3 5;

z 1l = 0 .1 5 ;

z 2l = 0 .3 5 ;

z 3l = 0 .4 5 ;

z 4l = 0 .5 ;


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APPENDIX A. M-FILES

Analytic.m

c l e a r al l ; cl c

% D e f i ni n g p a r a me t e r s u s i n g S y s t e mP a r a m e te r s . m a n d

% S o f t S w it c h i n g Co n t r o l le r . m

S y s t e m P a r a m e t e r s

S o f t S w i t c h i n g C o n t r o l l e r

% D e te r mi n g t he t r an s fe r f u nc t io n b e tw e en T m a nd w _m

d e n = [ J m * J l ( J l * ( c m + cs ) + J m * ( c m + cs ) ) ( ( J l + Jm ) * k s + c m * c l + cm * c s + c l * c s ) .. .

( c m + c l ) * k s ] ;

n um = [ Jl ( cl + cs ) k s ];

G_tm = tf ( num ,den );

Act = tf ([1] ,[ te 1] );

SYS = Act * G_ tm ;

n u m = [ c e l l 2 m at ( S Y S . n u m ) ] ;

d e n = [ c e l l 2 m at ( S Y S . d e n ) ] ;

[ A ,B , C , D] = t f 2s s ( nu m , d en ) ; % S t at e - s p a c e r e p r e se n t a t io n o f l i n ea r % s y st e m

sys = ss ( A , B , C , D );

s y s . i n pu t d = 0 . 0 08 ; % I n pu t de l ay - > A c tu a to r d y na m ic s

f i g u r e

b o d e ( G _ t m , s y s , g ,{1,1000})

grid

s e t ( g c f , c o l o r , [1 1 1 ]) ;

% G _ h ig h c o n t r ol l e r :

i _a ct _h ig h = t f( [ Ti h 1] , [ Ti h 0] );

n o t c h _1 _ h i g h = t f ( [ 1 / w 1h ^ 2 2 * z 1 h / w 1h 1 ] , [ 1/ w 2 h ^ 2 2 * z 2 h / w 2h 1 ] ) ;

n o t c h _2 _ h i g h = t f ( [ 1 / w 3h ^ 2 2 * z 3 h / w 3h 1 ] , [ 1/ w 4 h ^ 2 2 * z 4 h / w 4h 1 ] ) ;

G _h ig h = K ph * i _a ct _h ig h * no tc h_ 1_ hi gh * n ot ch _2 _h ig h ;

% G _ l ow c o n t ro l l e r :i _a ct _l ow = tf ( [ Ti l 1] , [ Ti l 0] );

n o t c h _1 _ l o w = t f ( [ 1/ w 1 l ^ 2 2 * z 1 l / w 1l 1 ] , [ 1/ w 2 l ^ 2 2 * z 2 l / w 2l 1 ] );

n o t c h _2 _ l o w = t f ( [ 1/ w 3 l ^ 2 2 * z 3 l / w 3l 1 ] , [ 1/ w 4 l ^ 2 2 * z 4 l / w 4l 1 ] );

G _l ow = K pl * i _a ct _l ow * n ot ch _1 _l ow * n ot ch _2 _l ow ;

% P I - c o n tr o l le r , s t r o ng a c t io n :

P I = 0 . 75 * K p * tf ( [ Ti 1 ] ,[ T i 0 ] );

% M ak in g a s ys te m o f g ai n 1 w it h t he s am e d el ay a s o ur s ys te m .

o ne = s s (0 , 0 , 0 ,1 );

o n e . i n pu t d = 0 . 0 08 ;

% P l ot t in g t he r e su l ts

f i g u r e

b o d e ( G _ h i g h , G _ l o w , P I )

l e g e n d ( G _ { h i g h } , G _ { l o w } , P I , L o c a t i o n , N o r t h W e s t )grid

s e t ( g c f , c o l o r , [1 1 1 ]) ;

f i g u r e

b o d e ( G _ h i g h * s y s , G _ l o w * s y s , P I * s y s )

l e g e n d ( G _ { t ot h i g h } , G _ { t ot l o w } , G _ { P I } , L o c a t i o n , B e s t )

grid

s e t ( g c f , c o l o r , [1 1 1 ]) ;

f i g u r e

b o d e m a g ( o n e / ( o n e + G _ h i g h * s y s ) , o n e / ( o n e + G _ l o w * s y s ) , o n e / ( o n e + P I * s y s ) )

l e g e n d ( G _ { t ot h i g h } , G _ { t ot l o w } , G _ { P I } , L o c a t i o n , B e s t )

grid

s e t ( g c f , c o l o r , [1 1 1 ]) ;

f i g u r e

n y q u i s t ( G _ h i g h * s y s , G _ l o w * s y s , P I * s y s , { 2 , 1 0 ^ 1 5 } )


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APPENDIX A. M-FILES

l e g e n d ( G _ { s ys h i g h } , G _ { s ys l o w } , G _ { P I } , L o c a t i o n , B e s t )

grid

s e t ( g c f , c o l o r , [1 1 1 ]) ;


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APPENDIX A. M-FILES

PI-torqueParameters.m

c l e a r al l ; cl c ;

% P a ra m et e rs f or B a ck l as h m o de l .

J m = 0 .4 ; % M o to r m o me n t o f i n er t ia [ k g m2 ]

J l = 5 .6 ; % L oa d m o me n t o f i n er t ia [ k gm 2 ]

c m = 0 .1 ; % V i s c ou s m o t or f r i c ti o n [ N m / ( r a d / s ) ]

c l = 1 ; % V i sc o us l oa d f r ic t io n [ N m /( r a d /s ) ]

k s = 3 30 0; % S h a ft e l a s ti c i t y [ N m / r a d ]

c s = 1 ; % I n ne r d a mp i ng c o ef f ic i en t o f t he s h af t [ N m /( r a d /s ) ]

t d = 0 . 00 0 6; % T im e d el ay , f ro m r e fe r en c e t o m o to r t o rq u e [ s ]

t e = 0 . 00 0 8; % T i me c o e f f ic i e n t a c t u a t o r

a lf a = 0 . 05 ; % B a ck l as h a ng l e [ r a d ]

% % P a r a me t e r s P I c o n t ro l e r

K p = 4 0; % P - a c t i on

K v = 4 00 ; % I - a c t i on

% % P a ra m et e rs f or t o rq u e o b se r ev e r

Ko = 0. 1; % C o m p e ns a t i o n g a i n

T fi lt = t e; % T i me d e l ay c o m p e ns a t i o n


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Appendix B

Simulink Models

Soft Switching Model

(a) Upper model (b) Controller part

(c) Backlash model (d) Model

(e) Actuator dynamics

Figure B.1: Soft switching model

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APPENDIX B. SIMULINK MODELS

Torque Observer Model

(a) Upper model (b) Controller part

(c) Backlash model

Figure B.2: Torque observer model


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Appendix C

Shaft Design

Figure C.1: Assembly of the shaft, part 1

Figure C.2: Assembly of the shaft, part 2

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List of Figures

1.1 Position error of a system with backlash . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Two mass rotating system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Deadzone model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Layout of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Frf ofUcomp over T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 feedback connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Stability of limit cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 System with and without one of the notches . . . . . . . . . . . . . . . . . . . . . . 112.6 Prediction of limit cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1 Simulated results with low shaft damping . . . . . . . . . . . . . . . . . . . . . . . 133.2 Simulated results with high shaft damping . . . . . . . . . . . . . . . . . . . . . . . 143.3 Motor velocity for system with and without observer . . . . . . . . . . . . . . . . . 143.4 Motor velocity and load velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5 Backlash gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6 Shaft torque and motor torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.7 Bode diagram of system with and without controllers . . . . . . . . . . . . . . . . . 17

3.8 Sensitivity of controlled system with both controllers . . . . . . . . . . . . . . . . . 173.9 GHigh controlled system with applied load disturbance . . . . . . . . . . . . . . . . 183.10 Phase diagram ofGHigh controlled system . . . . . . . . . . . . . . . . . . . . . . . 193.11 GLow controlled system with applied load disturbance . . . . . . . . . . . . . . . . 193.12 Phase diagram ofGLow controlled system . . . . . . . . . . . . . . . . . . . . . . . 193.13 Soft Switching controlled system with applied load disturbance . . . . . . . . . . . 203.14 Soft switching controller v and phase diagram . . . . . . . . . . . . . . . . . . . . . 20

4.1 Power spectral density of output and differentiaited output . . . . . . . . . . . . . 214.2 Butterworth filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 Measurement of angular velocity over torque . . . . . . . . . . . . . . . . . . . . . 224.4 Measurement of position over torque . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 Frf of the system; position over torque . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.6 Shaft design for experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.7 Experimental results of Torque Observer controller . . . . . . . . . . . . . . . . . . 254.8 Position error of a system with and without backlash . . . . . . . . . . . . . . . . . 26

B.1 Soft switching model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35B.2 Torque observer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

C.1 Assembly of the shaft, part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37C.2 Assembly of the shaft, part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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