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The design and implementation of an adaptivebroadband feedback controller on a

six-degrees-of-freedom vibration isolation set-up

A.P. Duindam

Masters thesis

February 2005

University of TwenteThe Netherlands

Faculty of Engineering Technology

Department of Mechanical EngineeringMechanical Automation Laboratory


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Masters thesis in Mechanical Engineering

February 2005

University of TwenteFaculty of Engineering TechnologyDepartment of Mechanical EngineeringMechanical Automation LaboratoryP.O. Box 2177500 AE EnschedeThe Netherlands

Report no. WA-978

Committee:Chairman Prof. dr. ir. J.B. JonkerMentor Dr. ir. J. van DijkSecond mentor Ir. G. NijsseExternal member Dr. ir. A.P. Berkhoff

Typeset by the author with the LATEX 2 document preparation system.Printed in The Netherlands.

Copyright c 2005, University of Twente, Enschede, The Netherlands.All rights reserved. No part of this report may be used or reproduced in

any form or by any means, or stored in a database or retrieval system

without prior written permission of the university except in the case of

brief quotations embodied in critical articles and reviews.


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iv


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Summary

As a result of structrure born vibrations, an unwanted noise signal may be emitted. Thevibrations are usually caused by a certain source structure, which are carried over to a re-ceiving structure emitting the unwanted disturbance signal. Besides passive control, which

may sufficiently attenuate high-frequency disturbance signals, active control is applied inorder to attenuate low-frequency noise. Active control can be achieved by fixed gain aswell as adaptive control.The topic of this report is to investigate the performance obtained by controlling broad-band (0-1kHz) disturbances using adaptive control. The performance obtained by adaptivecontrol was compared to the performance obtained by the design of equivalent fixed-gaincontrol. Therefor at our laboratory an hybrid isolation vibration setup has been built.The setup consists of a source structure inducing the disturbance, connected by six hybridisolation mounts to a receiver structure. The source structure is isolated from the receiverstructure by minimizing signals from six acceleration sensor outputs and by steering sixpiezo-electric actuator inputs (which serve as hybrid isolation mounts).

First the performance obtained by implementation of the controller in feedforward arrange-ment was investigated. Adaptive control was applied using the AdjointLMS algorithmwhich is known to be a computational efficient algorithm. To speed up the convergenceof the adaptive algorithm, the postconditioning technique was applied. The controller wasregularized in order to prevent saturation of the piezo-actuator inputs. It is shown thata reduction of the disturbance signal of 9.4 dB could be achieved in realtime. After theperformance of feedforward control was investigated, an adaptive feedback controller wasdesigned using the internal model arrangement. Also the adaptive controller was designedusing the postcondtioned AdjointLMS algorithm. The controller has to be stabilized usingregularization techniques by which a disturbance rejection measured by the six accelerationsensors of 3.5 dB was achieved.

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vi 0. Summary


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Preface

During the past year I have worked on my graduation assignment at the Mechanical Au-tomation Laboratory at the Faculty of Mechanical Engineering, University of Twente. Itwas an intensive and very instructive period. By choosing this assignment the combination

of the theoretical design and simulation of active vibration control in combination with re-altime implementation on an experimental setup was the thing which attracted me most.I want to thank Ph.D. student Ir. G. Nijsse, my direct supervisor for his enthusiastic ap-proach and the fruitful discussions we had, often helping me solve problems by consideringthem in a less complicated way. Furthermore I want to thank my fellow students at thelaboratory who gave me a pleasant time working on my masters assignment during thepast year.By ending this masters thesis, I also conclude my study Mechanical Engineering. I espe-cially want to thank my parents and brothers for their continuing support during my studyand giving me the opportunity to stand where I am now.

Arjen DuindamEnschede, 17th February 2005

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viii 0. Preface


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Contents

Summary v

Preface vii

Contents x

1 Introduction 1

2 Optimal control 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 General optimal filter problem . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Optimal feedforward control . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Time domain controller . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.3 Frequency domain controller . . . . . . . . . . . . . . . . . . . . . . 112.4 Optimal feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.2 Internal model control . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.3 Time domain controller . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.4 Transform domain controller . . . . . . . . . . . . . . . . . . . . . . 152.4.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Adaptive control 19

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Adaptive feedforward control . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.2 General LMS control problem . . . . . . . . . . . . . . . . . . . . . . 203.2.3 Presenting the secondary path: FxLMS algorithm . . . . . . . . . . 233.2.4 Reducing the computational load and increasing the convergence

speed: IO factorization . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.5 Decreasing the steering signals: regularized solution . . . . . . . . . 273.2.6 Reducing the computational load: adjointLMS algorithm . . . . . . 30

3.3 Adaptive feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.2 Design of the adaptive controller . . . . . . . . . . . . . . . . . . . . 34

3.3.3 Stability and convergence properties of the adaptive feedback controller 36

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x Contents

3.3.4 Postconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Experimental results 41

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Feedforward control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3.2 Realtime results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4 Feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.4.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.4.2 Realtime results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Conclusions & recommendations 55

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A Inner-outer factorization 57

A.1 Inner-outer factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57A.2 Outer-inner factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59


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Chapter 1

Introduction

As digital signal processors have become more powerful over the past decades, the possi-

bility to active control unwanted noise and vibrations has gained an increased amount of

interest.

The impact an earthquake can have is a clear example of the effect of unwanted vibra-

tions. On a smaller scale the vibrations generated by a ship engine, propagated through

the body of the ship to the passengers inside, resulting in a disturbing noise, can be seen

as another example of unwanted vibrations. Applications where a high level of accuracy is

needed, like miniaturized manufacturing applications or high precisions sensing systems,

vibrations propagated through the surface or air can seriously influence the functioning of

the specific application. In general it can be said, that in order to reduce the effect of thevibrations induced by a certain source structure, those vibrations need to be isolated from

the receiving structure. For example, a suspension system can be implemented between

the ship body and the engine of the ship. This type of vibration control is usually called

passive control, because only passive elements are being used to reduce the vibrations.

Using passive control, high frequency disturbances can successfully be attenuated. The

isolation of lower frequency disturbances by passive control is often accompanied by prac-

tical problems like a limited minimum stiffness requirement of the suspension system. A

more suitable way of attenuating low frequency disturbances is to make use of an active

control system. The basic idea of active control is that the vibrations which need to be

attenuated are measured on an appropriate place. Using these measured signals a steeringsignal is calculated using a suitable algorithm. This steering signal is being send to the

actuators inducing a vibration which counteracts the disturbance signal. If the system is

assumed to be linear there can be made use of the superposition principle, which states

that if a wave A counteracts with wave B, the resulting wave C is just the sum of wave A

and B. The idea of active control is to isolate a disturbance signal by determining a steering

signal, which produces a secondary anti disturbance signal with the same magnitude and

opposite phase as is shown in figure 1.1. The net result is that the residual disturbing

signal will cancel out and the receiving body will be isolated from the source body. An

effective way to reduce high frequency as well as low frequency disturbances is to combine

both control methods which is commonly mentioned as hybrid control. A simple schematic

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2 1. Introduction

Primarydisturbance

signal

Secondaryanti-disturbance

signal

Residualdisturbance

signal

Figure 1.1: Principle of superposition

view of an hybrid control system of the engine of ship can be seen in figure: 1.2. The engineof the ship is placed on hybrid mounts which consists of both passive and active elements,

thereby attenuating the vibrations carried over from the engine to the body of the ship.

In active control there can be made a distinction between fixed gain and adaptive con-

Hybridmounts

Raft

GearboxCoupling Bearing

Screwpropeller

Engine

Figure 1.2: Ships engine place on hybrid mounts

trollers. Fixed gain controllers are computational efficient but are only optimized for

disturbances whose statistics are time independent. For example if the engine power of a

ship is varied during a certain amount of time, also the statistics of the vibrations produced

by the engine can change. A time independent controller would give a less optimal atten-

uation in this situation. Therefore a better approach is to use a time adaptive controller

which can produce optimal results even if the disturbance statistics change. An importantcharacteristic of an adaptive controller is the time it needs to effectively adapt to the time

varying signals.

Another question which is important in designing a suitable controller is which informa-

tion is when available. An prerequisite for every controller is the availability of a signal

which represents the signal to be attenuated. Often the residual signal is used for this.

This signal is measured and feed back to the controller to calculate an appropriate steering

signal. However the controller performance can be significantly increased if there is some

kind of time advanced signal available. For instance if the signal that drives the engine

of the ship is available, this signal can feed forward to controller, by which the controller

can make an prediction of the disturbance signal in the time to come. The former type


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1. Introduction 3

of controller arrangements is called the type of feedback controllers, the latter is called a

feedforward control arrangement.

For research and development a six-degrees-of-freedom (6DOF) vibration isolation set-up

has been built at our laboratory [13]. A photograph of the setup is depicted in figure

1.3 on the left; a schematic picture is given on the right. The set-up consists of three

(a) Six degrees of freedom hybrid isolationvibration setup.

Shaker

Spring

Receiver plate (B)

Sourceplate(A)

Sensor(D)

Actuator(C)

Mount

(b) Schematic representation of the six degrees of freedom hy-brid isolation vibration setup.

Figure 1.3: The experimental setup

mounts carrying a source plate (A). The source plate is excited with a disturbance by an

electro-dynamic shaker. The source plate is connected to the three mounts by six ceramic

piezo-electric actuators (C) (two actuators per mount), which serve as hybrid isolation

mounts. The three mounts are attached to a receiver plate (B) and every mount has two

acceleration sensors (D) on top. The objective of the set-up is to investigate if the receiver

plate can be isolated from the source plate by the six hybrid isolation mounts, such that

disturbances induced by the source plate are isolated from the receiver plate. Isolation is

established by minimizing signals from the six acceleration sensor ouputs and by steering

the six ceramic piezo-electric actuator inputs by a controller [7].

The topic of active vibration isolation control (AVIC) is being researched at the Me-

chanical Automation group in co-operation with an industrial partner (TNO) and another

research group at the faculty of Engineering Technology at the University of Twente. For-

mer research of AVIC at a one degree of freedom setup has been carried out by [11]. On

the 6 degree of freedom setup the AVIC problem with broadband disturbances has been

investigated in simulation by [4], and a realtime implementation of the AVIC problem with

tonal disturbances was carried out by [12]. The availability of a computationally fast D-

Space system makes it possible to implement a broadband controller on the experimental

6 degree of freedom setup. The objective of this thesis can be summarized as follows:

design and implement a broadband adaptive feedforward and feedback controller on the


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4 1. Introduction

six degree of freedom setup. Thereby taking into consideration the need of a stable and

computational efficient algorithm gaining a high performance in combination with a short

learning curve.

This thesis is outlined as follows: in chapter 2 the general active optimal control prob-

lem will be issued in feedforward as well as in a feedback configuration. The adaptive

counterpart will be issued in chapter 3, where also several ways are described to improve

the convergence speed as well as to decrease the computational load. Special attention will

be given to the stability of the feedback adaptive algorithm. In chapter 4 first the identifi-

cation procedure will be briefly noticed. Subsequently the obtained results on the 6-DOF

setup will be presented for the various algorithms, analyzed and compared with simulation

results. Finally in chapter 5 conclusions will be drawn and several recommendations will

be made.


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Chapter 2

Optimal control

2.1 Introduction

The main objective of this report is the design and implementation of an adaptive con-

troller. However, to gain insight in the performance of the adaptive controller, its perfor-

mance properties will be compared with the performance of the fixed gain controller, which

functions as a benchmark for the adaptive controller if fully converged. Also the fixed gain

controller can be used to determine the stability properties of the adaptive controller, es-

sentially in a feedback arrangement.

Therefor, in this chapter the design of the fixed gain controller will be described. First

in section 2.2, the general filter problem will be presented. This general problem will bedetailed out in the following section for a feedforward arrangement and subsequently in the

next section for a feedback arrangement. In both sections the controller will be described

in the time as well as in the transform domain. Special attention to the stability prop-

erties will be given if the controller is described in a feedback arrangement. A summary

concludes this chapter.

2.2 General optimal filter problem

In this section the general optimal filter problem will be described. A schematic overview

of the basic filter problem can be seen in figure 2.1. The blockdiagram consists of thetransfer paths P and S. The primary transfer path P is defined as the transfer path from

the reference signal x(n) to the disturbance signal d(n). The secondary path S is defined

as the transfer from the steering signal u(n) to the output signal y(n).

If the example of the ship is considered, the systems P and S can be physically represented

as the ships engine producing the disturbance vibration and the actuators producing the

antidisturbance signal respectively.

The systems P and S on the experimental setup are considered time invariant, linear

and stable. The time invariance property implies that the model of the systems can be

considered as constant. The property of linearity implies that the superposition principle

holds, which simplifies the model design significantly. If a discrete model of the system

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6 2. Optimal control

P

W(q1)

x(n)

d(n)

y(n) e(n)u(n)

S

+

+

Figure 2.1: Blockdiagram of general optimal filter problem

is transformed to the z-domain [2], the stability requirement is guaranteed if the poles of

the transformed discrete model will lie inside the unit circle of the z-plane. Every physicalsystem is required to be causal. The causality requirement in the time domain implies that

the output of the system is not influenced by future input values.

The controller is represented by W which drives the actuator input u(n) from the reference

signal x(n). According to the superposition principle, the residual error measured on time

instant n by the sensors is the sum of the outputsignal y(n) and the disturbance signal

d(n).

e(n) = y(n) + d(n) (2.1)

The models obtained of the physical systems as well as the controller can be described in

several ways. One way of describing it, is by representing it as a finite impulse response(FIR) filter:

y(n) =L1l=0

h(l)u(n l) (2.2)

where the inputsignal u(n) is filtered by the coefficients h(l) of the filter. The filtercoeffi-

cients of the FIR filter can be seen as the response of the system to a unit impulse input.

Equation 2.2 can be alternatively written using the unit delay operator q1 which is defined

as follows:

ql

u(n) = u(n l) (2.3)Which results in:

y(n) = H(q1)u(n) (2.4)

An advantage of a FIR filter is that it is inherently stable, because all its poles are laying in

the origin. However if a lightly damped system is described by a FIR filter, the filter may

require many coefficients to describe such a system sufficiently. Therefore another way of

describing a system can be obtained by means of a state space model (SSM):

H x(n + 1) = Ax(n) + Bu(n)y(n) = Cx(n) + Du(n) (2.5)


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2.3. Optimal feedforward control 7

By this way the state of the system on time-stamp n is described by the state vector x(n).

The state matrices A, B, C and D indicating the contribution of the states and inputsignal

to the future states and outputsignal. A state space description is a numerically robust

and computational efficient way of describing the model of a system [8].

Throughout this report in the time domain H(q1) denotes a model described as a FIRM.

A state space description is denoted as H, without the unitdelay operator. In the transform

domain, this distinction will not be made and the model description of H(z), should follow

logically from the text. Furthermore, time domain signals are dentoted italic, whereas

transform domain signals are denoted regular.

Having identified the different signals and systems of blockdiagram 2.1, the objective is

now to find the controller that will maximal attenuate the error signal. Ideally the optimal

controller which entirely cancels the error signal would be of the form:

W = S1P (2.6)

However, a physical realization of W can only be given if the controller is stable and causal.

When equation 2.6 is used, the resulting controller will likely to be unstable caused by the

non-minimum phase zeros of S 1. In that case also the controller W will be unstable

provided no zeros of P will cancel out any poles of S1 outside the unit circle. Therefore

another way of determining the controller is needed. A widely used way of doing this,

is to base the controller W on the minimisation of a predefined criterium J. A suitable

criterium appears to be the expected value of the the squared error signal. This criterium

can be seen as a measure for the energy contained by the residual error signal [2].

J = E[eT(n)e(n)] (2.7)

Minimisation of this criterium as a function of all possible causal stable controllers leads

to the optimal controller W.

Wopt = arg minW

J(W) (2.8)

In the following sections, an expression in the time as well as in the transform domain for

the optimal controller will be derived, using criterium 2.8.

2.3 Optimal feedforward control

2.3.1 Introduction

A feedforward arrangement suggests that signals upstream in the system are feedforward

to the controller to better predict the outcome of the disturbance signal. In section 2.2 a

general way of finding the optimal contoller was presented. In this section an expression

for the fixed gain controller in feedforward arrangement will be derived. First this will be

done in the time domain, subsequently the transform domain will be treated. The purpose

and effect of regularisation will be discussed in the appropriate subsections.

1

A system is said to be minimum phase if all the poles and zeros lay within the unit circle.


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2.3.2 Time domain controller

A general schematic overview of the feedforward optimal controller can be seen in figure2.2. In this figure x(n) RK,u(n) RM,d(n), y(n), e(n) RL represents the reference,steering, disturbance, antidisturbance and error signals respectively. The primary path

P consists of K inputs and L outputs, the secondary path S consists of M inputs and L

outputs and the controller W(q1) consists of K inputs and M outputs, where M Kand L M.

To derive the time domain controller according to the defined criterium 2.7, we will assume

P

W(q1) S

x(n) d(n)

u(n) y(n) e(n)

K

M L+

+

Figure 2.2: Blockdiagram of the feedforward configuration with optimal control.

that the controller can be represented by a FIR filter. As was mentioned in the previous

subsection the output of the controller u(n) will be a weighted sum of a finite number of

present and pervious input values. The transfer of the kth input signal xk to the mth

output signal um can be noted as:

um(n) = Wm,k(q1)xk(n) (2.9)

Where the k to mth FIR filter is described by I coefficients:

Wm,k(q1) = [w

(0)m,k, w

(1)m,kq

1, . . . , w(i)m,kq

i, . . . , w(I1)m,k q

I+1] (2.10)

The coefficients wim,k can be stacked as follows:

wm,k = [w(1)m,k, w

(2)m,k, . . . , w

(i)m,k, . . . , w

(I)m,k] R1I (2.11)

wm = [wm,1,wm,2, . . . ,wm,k, . . . ,wm,K]

R

1KI (2.12)

w = [w1,w2, . . . ,wm, . . . ,wM]T RM KI1 (2.13)where w

(i)m,k denotes the ith filter coefficient of the FIR filter from input xk to output um.

Subsequently a notation for the matrix of regression vectors X will be introduced:

xk(n) = [xk(n), xk(n 1), . . . , xk(n i), . . . , xk(n I)] R1I (2.14)x(n) = [x1(n),x2(n), . . . ,xk(n), . . . , xK(n)]

T RKI1 (2.15)

X(n) =

x(n) 0 00 x(n)...

. . .

0 x(n)

T

RMM KI (2.16)


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The vector of delayed inputsignals x(n) is called the regression vector. The anti-disturbance

signal y(n) can be calculated as follows:

y(n) = SW(q1)x(n) (2.17)

= S[X(n)w] (2.18)

Assuming time invariant models, the multiplication order ofS and W(q1) may be inter-

changed:

y(n) =S xT(n)w (2.19)

Where denotes the kronecker tensor product[5], and

R(n) = S xT

(n) RLM KI

(2.20)

is the matrix with past reference signals x(n) filtered by the secondary path S. It consists

of LMK sequences of filtered reference signals rlmk:

rl,mk(n) = Slmxk(n) (2.21)

Using equation 2.19 the error signal can now be expressed in terms of the filtered reference

signal:

e(n) = R(n)w+ d(n) (2.22)

Where R(n)w denotes a matrix vector product. Subsequently the criterium J can beexpressed in terms of the coefficients of the optimal controller Wopt

J = E[eT(n)e(n)] = EwTRT(n) + dT(n)

(R(n)w+ d(n))

= wTE

RT(n)R(n)

w+ 2wTE

RT(n)d(n)

+ . . .

. . . E dT(n)d(n)

(2.23)

This is a quadratic expression for each of the FIR filter coefficients of the controller w(i)m,k

and is minimized according to each of the filter coefficients by setting the derivative of the

criterium J to the corresponding coefficient to zero:

J

w(i)m,k

= 0 (2.24)

Which leads to the following expression for the optimal controller in the time domain:

Wopt =

ERT(n)R(n)

1ERT(n)d(n)

(2.25)

The matrix ERT(n)R(n)

is the autocorrelation matrix of the filtered reference signals

and may be written as Rrr. If this autocorrelation matrix is positive definite, the minimi-

sation of equation 2.23 leads to a unique global minimum. The autocorrelation matrix will

be positive definite if the filtered reference signal persistently excites the control filter W.

This means that the signal should have at least half as many spectral components as that


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there are filtercoefficients. The crosscorrelation matrix between the filtered reference sig-

nal and the disturbance signal can be rewritten as Rrd which leads, compared to equation

2.25, to the compact notation:

Wopt = Rrr1Rrd (2.26)Where Rrr and Rrd are defined as:

Rrr = ERT(n)R(n)

(2.27)

Rrd = ERT(n)d(n)

(2.28)

Regularisation

Criterium 2.7 is optimal in the sense that it tries to minimise the energy of the error signal.

It does not take into account the energy of the steering signals. In practical situations, this

steering signal can be limited for various reasons. The actuator output can be restricted to

a maximum output level. Also to comply with the linearity assumption, the displacements

caused by the actuators can be limited. So in general it is often desirable to modify the

cost function by adding an effort weighting term in addition with the error weighting term.

Such a cost function can be defined as follows:

J = E[eT(n)e(n) + uT(n)u(n)] (2.29)

However using this cost function to determine the filter coefficients in an adaptive way,

turns out to be computational rather inefficient as will be mentioned in the next chapter.

Using the fixed gain solution only for analysis purposes, the following cost function will be

introduced:

J = E[eT(n)e(n) + wTw] (2.30)

This cost function additionally weighs the squared filtercoefficients multiplied by a weight-

ing factor beta. The factor beta is used to determine how strong the coefficients should

be weighed compared to the error signal. If the reference signal is a white noise sequence,

costfunctions 2.29 and 2.30 can be considered as the same. Combining equations 2.30 and

2.23 leads to the following expression for the optimal filter coefficients:

Wopt = {

Rrr +

I}1 R

rd (2.31)The new optimal solution can simply be computed by adding a term beta to the main

diagonal of the autocorrelation matrix. Another adaventage of this regularized solution

is that by adding a small term to the main diagonal will result in an increase of the

eigenvalues, and therefore makes a potentially poorly conditioned matrix easier to convert.

The performance of the reguralized solution will be less optimal compared to the optimal

solution without regularization. It turns out however, that the introduction of a small value

beta will only have a slight impact on the performance of the controller, while a significant

reduction in effort energy can be obtained. Furthermore it turns out that regularization of

the filtercoefficients also has a positive influence on the stability and convergence properties

of the adaptive solution as will be discussed in the next chapter.


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2.3.3 Frequency domain controller

In the past subsections the fixed gain controller is derived in the time domain. In thefollowing subsections an expression for the fixed gain controller in the frequency domain

will be given. The minimization of the cost function in the time domain can be viewed as

the minimization of the H2-norm of the system in the frequency domain. To illustrate this

first the cost function is represented in the frequency domain by use of Parsevals theorem:

= E[eT(n)e(n)] = trE[e(n)eT(n)] (2.32)

=1

2tr

See(ej T)dT (2.33)

where in the second equation T denotes the sampletime and See(ejT) denotes the power

spectral density of the error signal:

See(ej T) = E[e(ejT)eT(ej T)]

= E

d(ej T) + y(ej T)

d(ejT ) + y(ejT)T

= E

P(ej T)x(ejT) + S(ej T)W(ej T)x(ejT)

. . .

. . .

P(ej T)x(ej T) + S(ej T)W(ej T)x(ejT)T

=

P(ejT) + S(ejT )W(ejT )

E

x(ejT)xT(ej T)

. . .

. . . PT(ej T) + WT(ejT)ST(ejT) (2.34)

If the reference signal is a white noise sequence with unit variance, then the expectation

of the reference signal is equal to unity:

E

x(ejT)xT(ej T)

= I (2.35)

By defining the H2-norm of the system H(z) as:

H(z) 2 =

12

tr

H(ej T)HT(ej T)dT (2.36)

The costfunction 2.7 can alternatively be written as the square of the H2-norm of the

system P(z) + S(z)W(z):

E[eT(n)e(n)] =1

2tr

P(ejT) + S(ejT)W(ejT)

. . .

. . .

PT(ejT) + WT(ej T)ST(ej T)

dT (2.37)

= P(z) + S(z)W(z) 22 (2.38)Because solving the optimization problem according to 2.7, is equivalent of minimizing the

squared H2 norm of the system, the minimization problem can now be stated as follows:

W(z) = minW P(z) + S(z)W(z) 22 (2.39)


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The derivation of the solution, which is known as the model based causal Wiener solution,

can be found in [3], and the solution is directly given:

W(z) = S1o (z)

STi (z1)P(z)

+

(2.40)

The solution is obtained by performing an inner outer factorization on the secondary path.

The inner-outerfactorization is defined in appendix as is defined in appendix A.1. Because

the outerfactor has the property that it is minimum phase, it has a stable inverse. The

anticausal 2 transposed system, also known as the adjoint system, of the causal innerfac-

tor is denoted by STi (z1). The causality operator is denoted by {}+, which is defined

as taking the causal part of the total system between the brackets. The advantage of

this solution in contrast to the solution described by correlation matrices is that by the

former a purely model based solution is obtained. The latter has the disadvantage thatan autocorrelation matrix, with potentially huge dimensions, has to be inverted. Another

disadvantage is that an error will be made by using a finite data length to calculate the

correlationmatrix. However in order to obtain the modelbased solution, the models of the

primary and secondary path should be available.

Regularisation

To regularize the solution, the cost function can be extended as in equation: 2.30:

J = E[eT(n)e(n) + wTw] (2.41)

=1

2tr

P(ej T) + S(ejT)W(ejT)

PT(ejT) + WT(ejT)ST(ejT )

+ . . .

. . .

IMW(ejT)WT(ejT)

IMdT (2.42)

This can be written, using the H2-norm:

W(z) = minW

Paug(z) + Saug(z)W(z) 22 (2.43)

With:

Paug(z) = P(z)0MK

Saug(z) = S(z)IMM

(2.44)the augmented (L + M) K primary and augmented (L + M) L secondary path. Thestructure of the optimization problem of 2.43 is now the same as the structure for the

general optimization problem as shown in equation 2.39. Therefor the regularised transform

domain fixed gain solution can now directly be written using the general structure of the

solution of the general optimization problem.

W(z) = Saug -1o (z)

Saug Ti (z

1)Paug(z)

+(2.45)

2

A system H(z) is said to be anticausal H(z1

) if its output is depending on future inputvalues


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2.4. Optimal feedback control 13

The above equation can be simplified using the fact that the the innerfactor of the aug-

mented secondary path can be split into two parts:

Saugi (z) =

S

augi,1 (z)

Saugi,2 (z)

(2.46)

With Saugi,1 (z) a L M system and Saugi,2 (z) a M M system. The product between thecausality operator can now be rewritten as:

Saug, Ti,1 (z

1) Saug, Ti,2 (z1)

P(z)

0MK

= Saug, Ti,1 (z

1)P(z) (2.47)

It is shown that the primary path has not to be augmented which leads to a simplified and

a computational more efficient way of determining the regularized controller:

W(z) = Saug -1o (z)

Saug Ti,1 (z

1)P(z)

+(2.48)

2.4 Optimal feedback control

2.4.1 Introduction

In the previous section the fixed gain feedforward controller was derived. It was assumed

that the reference signal is known a priori. However, if this knowledge is not available, the

controller can be arranged in an feedback configuration. By using the principle of internal

model control (IMC) an estimation of the disturbance signal can be obtained, which is

driving the controller like a newly obtained reference signal. The principle of IMC will

be discussed in the next section. Following the same structure as in the previous section

subsequently expressions for the fixed gain controller in the time domain and frequency

domain will be derived. Because a feedback structure does not have the property of being

inherently stable as was the case using a feedforward structure, a special section will

cover the stability properties of the feedback solution. In this section also the regularized

solutions will be treated.

2.4.2 Internal model control

A blockdiagram of a feedback configuration using IMC is shown in figure 2.3.In this configuration the disturbance signal d(n) is estimated by substracting an esti-

mation of the output signal y(n) from the error signal. This estimated output signal is

obtained by filtering the steering signal u(n) by the internal model S which is an estima-

tion of the real secondary path. By this way an estimation of the disturbance signal is

achieved:

y(n) = Su(n) (2.49)

d(n) = e(n) y(n) (2.50)This estimated disturbance signal functions as a newly obtained reference signal and is

used as the input for the controller W(q1). By defining the feedback path H as shown


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P

W(q1) S

S

H

x(n) d(n)

d(n)

u(n) y(n)

y(n)

e(n)

K

M L

+

+

+

Figure 2.3: Blockdiagram of the feedback configuration in an IMC arrangement

in figure 2.3, the feedback arrangement can be represented in the simplified form shown

in figure 2.4. Where the transfer function from the error signal to the steering signal is

defined as:

H(z) = W(z)1 + S(z)W(z)

(2.51)

Using expression 2.51, the resulting sensitivity function G(z) of this arrangement is given

SH

d(n)

u(n) y(n) e(n)

M L+

+

Figure 2.4: Simplified blockdiagram of the feedback configuration in an IMC arrangement

by:

G(z) =e(z)

d(z)=

1 + S(z)W(z)

1

S(z) S(z)

W(z)(2.52)

If perfect plant knowledge is assumed, the arrangement of the sensitivity function reduces

to a form which is linear in the filtercoefficients:

G(z) = 1 + S(z)W(z) (2.53)

and criterium 2.7 can be applied which is quadratic in the filtercoefficients. Minimization

of the criterium to each of its coefficients leads to a global minimum, giving the coefficients


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2.4. Optimal feedback control 15

of the optimal controller according to criterium 2.7. So using IMC and assuming perfect

plan knowledge will lead to an equivalent feedforward minimization problem, which is also

shown in figure 2.5: The blockdiagram of the feedback system is now given by an entirely

SWd(n) u(n) y(n) e(n)

ML +

+

Figure 2.5: Block diagram of feedback configuration with perfect plant knowledge assumed

feedfworward structure.

The strategy of using IMC and assuming perfect plant knowledge will be applied in the

next sections to calculate the optimal feedback controller.

2.4.3 Time domain controller

Using the IMC arrangement discussed in the previous subsection the optimal feedback time

domain controller is derived in the same way as was discussed in section 2.3.2. Assuming

perfect platnt knowledge the estimated disturbance signal equals the original disturbance

signal, which is playing the role of the reference signal in the feedforward case. The matrix

R(n) is now achieved by the kronecker tensor product between the secondary path S and

the disturbance signal d(n):

dl(n) = [dl(n), dl(n 1), . . . , dl(n i), . . . , dl(n I)] R1I (2.54)d(n) = [d1(n),d2(n), . . . ,dl(n), . . . ,dL(n)]

T RLI1 (2.55)R(n) = S dT(n) RLM LI (2.56)

Having specified the matrix R(n), the optimal time domain controller consisting of the

L M FIR filters can be obtained using equation 2.25.Wopt =

ERT(n)R(n)

1ERT(n)d(n)

(2.57)

In case perfect plant knowledge may not be assumed the designed controller above may

be suboptimal. A more optimal controller may then be derived by using the estimated

disturbance signal, which actually acts as the input of the controller, instead of the real

disturbance signal. However by designing the optimal controller using the estimated dis-

turbance, the dependence of the estimated disturbance with respect to the coefficients is

ignored in order to obtain a quadratic minimization problem. It is therefor not guaran-

teed to give a better performance compared to the controller obtained using perfect plant

knowledge.

2.4.4 Transform domain controller

Also a transform domain solution can be obtained using the internal model control im-

plementation and assuming perfect plant knowledge. Referering to cost function 2.33, the


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power spectrum density function of the error signal can be written as:

See(ejT

) = E[e(ejT

)eT

(ejT

)]= E

d(ej T) + y(ej T)

d(ej T) + y(ej T)

T= E

P(ejT)x(ej T) + S(ej T)W(ej T)P(ejT)x(ej T)

. . .

. . .

P(ejT)x(ej T) + S(ejT)W(ej T)P(ejT)x(ej T)T

=

P(ejT ) + S(ej T)W(ej T)P(ejT)

E

x(ej T)xT(ej T)

. . .

. . .

PT(ejT) + PT(ej T)WT(ej T)ST(ejT)

(2.58)

The problem can now be restated as minimising the following H2-norm:

W(z) = minW P(z) + S(z)W(z)P(z) 2

2 (2.59)

which has the following solution [3]:

W(z) = S1o (z)

STi (z1)P(z)PTci(z

1)

+Pco(z) (2.60)

where denotes the left inverse operation and Pco, and Pci are the co-outer and co-innerfactor respectively of the primary path as explained in appendix A.1.

Noting that:

P(z)PTci(z1) = P(z)P1ci (z) = Pco(z) (2.61)

expression 2.60 can be simplified leading to the following solution for the optimal wiener

feedback controller:

W(z) = S1o (z)

STi (z1)Pco(z)

+

Pco(z) (2.62)

2.4.5 Stability

By assuming perfect plant knowledge, the feedback loop is completely ignored. In fact

this feedback loop is still present as can be seen in blockdiagram 2.6. Actually this is

SW

S S

d(n) u(n) y(n) e(n)

ML + +

+

Figure 2.6: blockdiagram of the IMC arrangement with internal feedback loop present

just a rearranged version of the blockdiagram shown in figure 2.3. In the above figure the

looptransfer path can be defined as:

L(z) = W(z)[S(z) S(z)] (2.63)


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2.5. summary 17

In order to have a stable feedback transfer path it is required that the loopgain, which

may be expressed by the

-norm, of the loop transferpath is less then unity:

||L(z)|| < 1 for all T (2.64)

Intuitively it may be seen that the gain of the filter should be reduced at frequencies where

the gain of the filter is high, in order to stabilize an otherwise unstable feedback system.

To reduce the the peak values of the controllergain the method of regularisation can be

applied in the same way as was mentioned in the section on feedforward control. The

time domain regularized solution is then obtained using equation 2.33, using the estimated

filtered disturbance signal as defined in 2.56, instead of the filtered reference signal.

To obtain the transform domain solution, one could refer to equations: 2.45 and 2.58.

Using Parsevals theorem and combining referred equations leads to the following expressionfor the costfunction in the frequency domain:

J = E[eT(n)e(n) + wTw] (2.65)

=1

2tr

[P + SWP] [P + PWS] +

IMWW

IMdT (2.66)

Where (ejT) has been omitted for clarity reasons and denotes the complex conjugatetranspose. Because the controller W is prefactorised with the primary path, both terms

in the integral can not be combined.Therefor the transform-domain cost function can not

be obtained in the same general form as was derived in equation 2.38 and the solution cannot easily be generalized. For analysis purposes only the time-domain regularised solution

will be used.

2.5 summary

In this chapter a general method was presented to attenuate unwanted vibrations. This

method of attenuating vibrations was mainly based on the design of a time independent

controller which used the minimisation of the energy contained by the error signal as a way

to determine the optimal controller. A static time domain controller, which is commonly

denoted in literature as the class of Wiener controllers, was derived by the use of statisticalproperties of the reference and measured signals. By formulating the minimisation of a

criterium into the minimisation of the H2-norm a transform domain solution could be de-

rived. After the feedforward solution was derived a method to reduce the steering signals

was described, which is often constrained to a certain maximum level. Subsequently the

feedback problem was dealt with. By using an internal model arrangement, it was shown

that the feedback problem could be encountered as an equivalent feedforward problem,

provided perfect plant knowledge is available. The time and transform domain feedback

controllers were therefor derived using standard feedforward control laws. Finally it was

shown how imperfect plant knowledge may lead to stability problems. By using regular-

ization a method was presented to improve the stability of the feedback system.


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29/72

Chapter 3

Adaptive control

3.1 Introduction

The fixed gain Wiener controller derived in the previous chapter is optimised only for in-

putsignals which statistic properties are stationary and can become less optimal if these

properties change over time. This changing can be caused by a slowly varying primary

path. Also a change in the statistics of the reference signal may result in a less optimal

controller. Under these conditions using an adaptive controller may lead to a better reduc-

tion of the disturbance signal, because it has the possibility to adapt to a more optimal

controller when the correlation properties of the inputsignals change over time. It requires

however the calculation of a new set of control filters every sample period and thereforan important design criterium will be the efficiency of the algorithm. Another important

requirement of an adaptive controller is that it has to converge to a stable controller where

it is desirable to have a fast convergence, so that the disturbancesignal may be attenuated

over a short time period. The actuators are driven by the output of the controller, yet the

output energy of the actuators to attenuate the disturbance signal is often restricted. This

leads to the requirement that the adaptive controller should also be efficient in the sense

that maximal reduction of the disturbance energy is achieved with a minimal amount of

control effort. These demands will lead to the design of an adaptive controller, arranged

in a feedforward or a feedback structure and will be the subject of this chapter.

The organisation of this chapter is as follows, first a basic structure for an adaptive con-troller will be laid out. Principles to determine the stability and convergence speed will be

issued. Subsequently the adaptive feedforward controller will be derived, first, the FxLMS-

algorithm. It will be shown how the computational efficiency can be enhanced together

with a increase in convergence speed by using the principle of inner-outer factorization.

In the following subsection it will be shown how the robustness to modeluncertainties can

be enhanced by using regularization. In the final subsection, the adjointLMS algorithm

is introduced leading to a further decrease of the computational load. In the following

section the adaptive controller in feedback arrangement is derived. The design can mainly

be generalized from the feedforward arrangement. Yet referring to the design of the fixed

gain feedback controller, special attention will be given to the stability requirements to

19


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20 3. Adaptive control

LMS

W(q1, n)

x(n)

d(n)

y(n)

e(n)

K L

+

+

Figure 3.1: Blockdiagram of general adaptive filter problem

guarantee a stable convergence of the adaptive controller. A summary will conclude this

chapter.

3.2 Adaptive feedforward control

3.2.1 Introduction

In this section first the general feedforward adaptive problem will be discussed. The con-

straints to guarantee a stable adaptive controller will be mentioned. Subsequently several

methods will be introduced to increase the convergence speed and to reduce the computa-

tional load. Finally a way to enhance the stability will be presented, which simultaneously

reduces the maximum actuator outputs.

3.2.2 General LMS control problem

In figure 3.1 the basic blockdiagram of an adaptive control problem is shown: The refer-

ence signal x(n) is fed through the adaptive controller W(q1, n) in order to produce an

antidisturbance signal which is used to attenuate the disturbance signal d(n). The residual

error signal may then be written as:

e(n) = W(q1, n)x(n) + d(n) (3.1)

Where W(q1, n) is described using an FIR filter structure.

In the previous chapter it was mentioned that the optimal fixed gain controller de-

scribed by an FIR structure could be derived according to the minimization of a certain

costfunction J with respect to the coefficients of the control filter. The coefficients of the

optimal control filter could then be obtained by setting the derivative of the costfunction

according to the coefficients to zero:

J

w = 0 (3.2)


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3.2. Adaptive feedforward control 21

MSE

w2

w1

Figure 3.2: Graphical representation of the MSE as function of the filtercoefficients.

provided the costfunction is a quadratic function of the coefficients. In adaptive control

essentially the same procedure takes place except that the minimum of the costfunction

is not determined at once, but instead is determined iteratively by letting the controlfil-

ter coefficients change each sampleperiod in the direction of the global minimum of the

costfunction. The size and direction in which the coefficients adapt is then determined by

the negative derivative of the costfunction with respect to the coefficients. This type of

adaptive gradient descent algorithms is commonly known as steepest descent algorithms

because the coefficients are updated in the direction of the steepest descent w.r.t. the cost-

function. When the costfunction is plotted against the filtercoefficients and consideringonly two coefficients, this may be visualized in the figure: 3.2. The adaptation process

may now be imagined as a path winding around the surface from a certain initial position

to a certain space around the place defined by the set of coefficients where the costfunction

has its minimum value. When a quadratic function is used, the negative derivative points

to the global minimum of the costfunction. The set of new control filter coefficients can

then be defined by the following updatestep:

w(new) = w(old) 2

J

w(3.3)

where denotes the stepsize and is used to adjust the speed of the adaptation process.

Using the mean square error given by equation 2.7 as the costfunction the general update

equation can be written as a function of the reference and error signals:

w(new) = w(old) E[XT(n)e(n)] (3.4)

because of the expectation operation incorporated, the computational load of the calcula-

tion of the derivative in the update equation may be rather large. Therefor the use of an

instantaneous estimate of the derivative is proposed. The set of equations to update the

controlfilter coefficients can then be given as:

w(new) = w(old) XT(n)e(n) (3.5)


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The adaptation algorithm involved with this update equation appears to be simple and

numerically robust and is commonly known as the least mean squares (LMS) algorithm.

Yet compared to an actual steepest descent method, it may differ in the fact that the

instantaneous gradient may differ from the gradient according to the MSE criterium and

the path on the errorspace may differ from the one using the MSE criterium. However,

it can be shown that when the adaptation process takes place slowly over time, i.e. the

amount in which the filtercoefficients are changing is small during the time defined by

the time the filters impulseresponse needed to decay sufficiently, the coefficients of the

adaptive filter will converge in the mean to the coefficients of optimal Wiener solution [2].

stability condition

The stability property of the LMS algorithm can be conveniently analyzed considering anaveraged behaviour of the algorithm. This is expressed by taking the mean value of the

different terms of the update step of the algorithm over a number of trials:

E[w(n + 1)] = E[w(n)] + E[XT(n)d(n)] E[XT(n)X(n)w(n)] (3.6)

When considering the reference signal as statistical independent of the filtercoefficients the

last term may be split into two different factors:

E[XT(n)X(n)w(n)] = E[XT(n)X(n)]E[w(n)] (3.7)

This independence assumption is only valid for a slowly varying filter, i.e.: the coefficients

of the filter can be considered as constant w.r.t. the length of the filter. By defining the

normalized coefficients as:

(n) = E[w(n)] wopt (3.8)

and recalling the expression for the optimal filtercoefficients 2.26 where Rxx is defined

as the autocorrelationmatrix of the reference signal, equation 3.6 can be substituted in

equation 3.8 yielding:

(n + 1) = [I Rxx](n) (3.9)

which represents a set of coupled equations demonstrating the evolution of the normalizedfiltercoefficients over time. By using an eigenvalue decomposition of the autocorrelation-

matrix:

Rxx = QQT (3.10)

equation 3.9 can be written as:

v(n + 1) = [I ]v(n) (3.11)

forming a set of I independent equations:

vi(n + 1) = (1 i)vi(n) (3.12)


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Where the ith normalized averaged rotated filter coefficient, vi(n) is defined as:

vi(n) = QT

[E[w(n)] wopt] (3.13)The independent coefficients vi(n) are also referred as the different modes in which the

adaptive algorithm converges. To guarantee a stable convergence, every independent mode

has to converge to zero. This leads to the following condition for the stepsize for each

independent mode i:

|1 i| < 1 0 < < 2/i (3.14)

From this condition it becomes clear that the mode associated with the highest eigenvalue

max will be the first mode to become unstable. Therefore, the maximum stepsize is bound

by the highest eigenvalue and the condition can be more specifically written as:

0 < < 2/max (3.15)

In practical situations however the independence assumption 3.7 is often not achievable

because of a fast developing filter. In that case a smaller value of the stepsize is required

to obtain a stable adaptation process.

convergence speed

The speed of convergence of each independent mode can be described by an exponential

decay [2] with time constant:

i =1

2i(samples) (3.16)

The speed of the convergence process is determined by the mode mode with largest time

constant which corresponds to the mode with the smallest eigenvalue. Also the convergence

process is influenced by the stepsize. A larger stepsize results in a faster convergence, so

a good measure of the overall convergence process appears to be the ratio between the

largest and smallest eigenvalue maxmin

. A fast overall convergence behaviour requires a small

eigenvaluespread.

3.2.3 Presenting the secondary path: FxLMS algorithm

In the previous subsection the general LMS adaptive filter problem was discussed. It was

assumed that the output of the control filter effects the disturbance signal measured at the

errorsensors without any change in gain or delay, i.e. the influence of the secondary path

is neglected. This assumption is not valid for most practical situations where there will be

a noticeable transfer path between the output of the control filter and the place where the

effect of the output of the controlfilter is measured. Therefore the effect of the secondary

path has to be incorporated in the LMS algorithm.

In the derivation of the controller, the assumption is made that the controlfilter W(q1, n)

will only change slowly compared to the timescale of the system dynamics of the secondary


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LMS

W(q1, n)

S

Sx(n)

d(n)

y(n)

e(n)

u(n)

R(n)

K M

L

+

+

Figure 3.3: Blockdiagram of FxLMS adaptive filter problem

path. By making this assumption, the filteroperations of the secondary path and the con-

trol filter may be transpositioned and still giving an accurate output. Refereing to equation

2.22 the error signal may then be written as:

e(n) = R(n)w(n) + d(n) (3.17)

Where R(n) represents the matrix of filtered reference signals and w(n) the vector of

timedependent controlfilter coefficients as was defined in the previous chapter. Defining

again the mean square error as the used costfunction, and using the instantaneous gradient,

the update step for the controlfilter may be written as:

w(n + 1) = w(n) RT(n)e(n) (3.18)

From this update equation it can be seen that an additional operation needs to be per-

formed in order to determine the matrix of filtered reference signals defined by R(n) com-

pared with the update equation incorporated with the general LMS algorithm mentioned

in the previous subsection. When a SISO problem is considered, this operation reduces to

a simple filtering operation between the reference signal and the secondary path. There-

fore this algorithm is commonly known as the filtered-reference LMS (FxLMS) algorithm.

However when multiple inputs or outputs are concerned, the filtering of the reference signal

implies a kronecker tensor product, although this signal will still be mentioned as a filteredreference signal in the following text. An extended blockdiagram with the secondary path

added is shown in figure 3.3.

Expression 3.18 used to update the filtercoefficients assumes perfect knowledge of the

physical plant modeled by S. Usually the the actual secondary path S can not be mod-

eled exactly and an estimated version is used, denoted by S. The use of a model of the

secondary path which not exactly represents the actual plant has the implication that

the instanteous gradient will point in a slightly different direction, leading to a different

convergence path of the filtercoefficients described by the following update equation:

w(n + 1) = w(n) RT(n)e(n) (3.19)


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Provided the algorithm is stable, the implication of using the modified update equation is

that the coefficients will in the mean converge to a suboptimal solution described by:

w =

E[RT(n)R(n)]1

E[RT(n)d(n)] (3.20)

which differs from the optimal Wiener solution described by equation: 2.26.

The stability criterium for the FxLMS algorithm can be derived in a similar way as

was described for the general LMS algorithm, leading to an expression for the theoretical

maximum stepsize:

0 < < 22Re(max)

|max|2 (3.21)

Where the potentially imaginary eigenvalues are taken from the crosscorrelation matrix

consisting of the reference signal filtered by the estimated and real secondary path Rrr.

It may also be noted that if at least one of the eigenvalues has a negative real part,

the associated independent mode vi described by equation 3.12 will exponentially increase,

leading to an instable adaptation process. If perfect plant knowledge is assumed the matrix

E[RT(n)R(n)] will be guaranteed to be positive definite, having only positive eigenvalues,

provided the reference signal persistently excites the filter.

A sufficient condition to guarantee stability if perfect plant knowledge is not assumed

[2] may then be given by :

eig[SH(ejT)S(ejT ) + SH(ej T)S(ejT)] > 0 for all T (3.22)

Where H denotes the complex conjugate transpose operator. Besides using a more accurate

plant model, another solution to stabilize the controller is to make use of regularization.

This will be discussed in section 3.2.5.

3.2.4 Reducing the computational load and increasing the convergence

speed: IO factorization

In the previous subsection it was shown that the convergence properties of the FxLMS

algorithm are depending on the size of the eigenvaluespread from the crosscorrelation ma-

trix consisting of the reference signal filtered by the estimated and real secondary path

Rrr. If this eigenvaluespread approaches unity, faster convergence of the adaptive filtercoefficients to the fixed gain solution is achieved. A low eigenvaluespread therefor is desir-

able for a fast convergence. Using the FxLMS algorithm, the eigenvaluespread is limited

by the dynamical range of the power spectrum of the reference signal combined with the

dynamical range of the frequency response of the secondary path [2]. Considering a single

input single output system and assuming perfect plant knowledge this may be written as:

maxmin

|S(ej T)|2Sxx(ej T)max[|S(ejT)|2Sxx(ejT)]min

(3.23)

If the reference signal is assumed to be a white noise sequence, it has a power spectrum of

unity over the whole frequency range. The eigenvaluespread is then bounded only by the


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LMS

W(q1, n) S

Si

S1o

x(n)

d(n)

y(n)

e(n)

u(n)u(n)

R(n)

K M

L

+

+

Figure 3.4: Blockdiagram of FxLMS adaptive filter problem with postconditioning applied

ratio of maximum and minimum values of the gain of the secondary path. So as to have afast convergence it is desirable to keep this ratio as small as possible.

Now lets consider the outputsignal of the adaptive controller is to be prefiltered by the

inverse of the secondary path. Referring to the properties of the inner and outer factor

explained in appendix A.1, this is allowed because the outerfactor is a minimum phase

system and thus has a stable inverse. The errorsignal may then be expressed as:

e(n) = SS1o W(q1, n)x(n) + d(n) (3.24)

= SiW(q1, n)x(n) + d(n) (3.25)

and the blockdiagram of the FxLMS algorithm using IO-factorization, which is also refereedto as postconditioning, is shown in figure 3.4. The resulting secondary path may now be

recognized as the inner-factor of the original secondary path. Because the inner-factor of

a system is by definition an all-pass system, it has a frequency response of unity over the

whole frequency range, and therefor the power spectrum density of the reference signal

filtered with the inner-factor remains unaffected. Consequently the eigenvaluespread of

the filtered reference signal is limited by the dynamical range of the power spectrum of the

reference signal only when inner outer factorization is applied.

The prize to pay using the postconditioned FxLMS algorithm, is that the output signal

of the adaptive filter has to be filtered by the inverse of the outerfactor, which is an extra

operation. However by noticing that the order of the inner factor equals the amount ofzeros of the secondary path outside the unit circle, the order of the model used to filter the

reference signal may just be largely reduced. When the computational load of the FxLMS

algorithm is examined, it appears that kronecker tensor product between the reference

signal and the secondary path S x(n) requires the major part. Filtering the referencesignal using a reduced order secondary path may therefor lead to a far more efficient

algorithm compared to the traditional FxLMS algorithm.

A comparison of the amount of additions and multiplications referred to as floating

point operations (flops) between the two algorithms is given in table 3.1. The total amount

of flops per sample required by each version of the FxLMS consist of the amount of flops

of a number of separate operations which can be categorized as:


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1. The filtering of the reference signal by the adaptive control filter.

2. The calculation of the kronecker tensor product between the reference signal and thesecondary path.

3. The calculation of the new vector of filtercoefficients.

4. For the postconditioned FxLMS algorithm, the filtering of the inverse of the outer-

factor by the outputsignal of the adaptive control filter.

In the table, N denotes the order of the secondary path, Ni denotes the order of the

inner-factor. The order of the outerfactor is by definition equal to the order of the system

itself. K, M and L denotes respectively the amount of reference, steering and error signals.

Further the amount of tabs of the control filter is denoted by I.To have an idea of the practical reduction in the amount of flops an inner-outer factor-

Table 3.1: Comparison of the number of floating point operations required by the FxLMSalgorithm with and without postconditioning

Operation Traditional FxLMS withFxLMS algorithm postconditioning

Filtered reference signal LM K[2N2 + 3N + 1] LM K[2N2i + 3Ni + 1]

Filterupdate 2MKIL + L 2MKIL + L

Filtering controller output 0 2N[N + M + L]by outerfactor +2M L

N

L

Filtering reference signal 2IKM M 2IKM Mwith adaptive filter

Total amount of flops LM K[2N2 + 3N + 2I + 1]+ LM K[2N2i + 3Ni + 2I + 1]+2M KI M + L 2M KI M+

2N[N + M + L] + 2M L N

ization on an identified secondary path of the 6-DOF hybrid isolation vibration setup is

performed. The identified system has an order of 100, yielding an innerfacor of order 25 and

the adaptive controlfilter is assumed to be contained by I is 200 tabs. The total amount

of flops without inner-outer factorization of the FxLMS algorithm equals: 747636. Using

postconditioning, yields a total of 86902 number of flops, which is less then 12 percent of

the amount of flops required by the traditional FxLMS algorithm.

3.2.5 Decreasing the steering signals: regularized solution

The energy required by the actuators to obtain optimal reduction of the disturbance signal

can be considerably high when the adaptive controller is designed to minimize the mean

square error only. Therefore as was discussed in the previous chapter it may be desirable

to make use of a regularized solution, which also tries to minimize the output of the

controller. In this subsection the implication of regularization on the adaptive controller

will be discussed.


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Regularization of the controller output

In order to derive the regularized adaptive controller, first the costfunction described byequation 2.29 is considered. In addition to the mean squared error also a term proportional

to the mean squared actuator-input is to be minimized. First the adaptive controller

without applying postconditioning is considered. The error and actuator-input signal may

then be written as:

e(n) = R(n)w(n) + d(n) (3.26)

u(n) = X(n)w(n) (3.27)

Taking the instantaneous derivative of the mentioned costfunction according to each of the

filtercoefficient yields the following update equation for the vector of filtercoefficients:

w(n + 1) = w(n) [RT(n)e(n) + XT(n)u(n)] (3.28)

Resulting in an extra of 3M KI flops per sampletime compared to the non-regularized

solution. Next the regularized solution using IO-factorization is derived. The filtered

reference signal is obtained by taking the kronecker tensor product between the inputsignal

and the innerfactor of the secondary path as was defined in the previous subsection. The

inputsignal can now be defined as follows:

u(n) = S1o

X(n)w(n)

(3.29)

= Q(n)w(n) with Q(n) = S1

o xT(n) (3.30)

Taking again the instantaneous derivative of the costfunction to each of the filtercoefficients

the update equations is now given by:

w(n + 1) = w(n) [RT(n)e(n) + QT(n)u(n)] (3.31)

The problem arises with the calculation of the kronecker tensor product which incorpo-

rates an amount LM K[2N2 + 3N1] extra multiplications and additions which leads to an

relatively large increase of the computational load.

Regularization of the controlfiltercoefficients

Therefore another approach is taken when applying postconditioning. Instead of weighing

the controller-output signals, the weighing of the sum of the squared coefficients of the

adaptive controller in the costfunction is proposed. By doing so the coefficients of the

adaptive FIR filter of the controller will be minimized in addition with the mean square

error according to the following costfunction:

J = E[eT(n)e(n) + wT(n)w(n)] (3.32)

Which results in the following expression for the update equation of the filtercoefficients:

w(n + 1) = [1 ]w(n) [RT(n)e(n)] (3.33)


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where the factor [1 ] is called the leakage factor and so this algorithm is also knownas the leaky LMS algorithm, because the coefficients would leak away if the error signal

approaches zero. The extra flops involved with the leaky LMS algorithm are just M KI+ 2

operations.

When a similar analysis on the stability of this algorithm is performed as was discussed

in section 3.2.2 it appears that the stability and convergence properties of the algorithm

are now depending on the eigenvalues of the matrix: E[RT(n)R(n) +I], which effectively

means that to each of the eigenvalues a term is added. So besides decreasing the controller

output, another advantage of introducing a coefficient weighting factor is that eigenvalues

having otherwise a small negative real part may now become positive. In general can

be said that adding a small value to the vector of eigenvalues will increase the smallest

eigenvalue by a relative large amount and therefore reducing the eigenvaluespread. The

leaky LMS algorithm can therefor be used to decrease the steering signals, increase the

robustness as well as the convergence speed of the FxLMS algorithm. The drawback of

using coefficient weighing factor in the costfunction is that the solution will converge to

2.31 which gives a suboptimal performance compared with the optimal solution given by

equation 2.26. However experimental results have proved using only a small value of can

have a major effect on the decrease of the steering signals and resulting in only a slightly

decreased performance.

Regularization of the outerfactor

It may be mentioned that using regularization in combination with postconditioning, leads

only to the minimizing of the adaptive part of the controller. This may be seen by consid-

ering the complete controller consisting of two systems. The first system can be recognized

as the adaptive controller represented by the FIR-filter W(q1, n), the second part equals

the inverse outerfactor of the secondary path: S1o and the complete controller may thenbe represented by the following expression:

Wcombined = S1o

fixedW(q1, n)

adaptive(3.34)

When a regularization term is included in the costfunction, the sum of squared coefficients

of the adaptive filter is being minimized together with the mean square error. As a result

the gain of the adaptive part of the controller will decrease at its peak values. However

the gain of the fixed part of the controller remains unaffected, which may still lead to a

considerable large gain of the combined controller at those frequencies corresponding to

the peak values of the frequency response of the inverse outer factor.

If it is desired to decrease the gain of the total controller at the mentioned frequencies

as well, the outerfactor may also be regularized. This is done by adding a small value to

frequency response of the outerfactor, which may be expressed as [2]:

STo (z1)So(z) = STo (z1)So(z) + I (3.35)


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Where I denotes an MM identity matrix, and M denotes the number of inputsignals ofS(z). The gain of the inverse of the outerfactor is most large at those places where the gain

of the outerfactor is most small. By adding a small value to the gain of the outerfactor, it

will most largely be increased at places where its value is smallest, thereby decreasing the

gain of the inverse of the outerfactor most largely at its peak values. It was observed that

by lowering the gain at the peak values of the inverse outerfactor, the gain of the adaptive

part of the controller was increased at the same frequencies. By doing so the gain at those

frequencies may be more efficiently reduced by the adaptive controller using the standard

regularization according to equation: 3.32

Instead of filtering the errorsignal by Si(z), the errosignal now needs to be filtered by:

S(z)S1o (z). The additional computational load is determined by the increased size of the

impulse response of the latter model, compared to the impulse response of Si(z). Yet, the

advantage of this solution with respect to for example introducing a frequency dependent

weighing term in the costfunction is that the additional computational load required, may

be much smaller.

The convergence properties are determined from the eigenvalues of the autocorrela-

tionmatrix obtained by filtering the reference signal by the combined model: S(z)S1o (z).

Because this system will not anymore be an all-pass system, an increase in the eigenvalue-

spread will be the result.

3.2.6 Reducing the computational load: adjointLMS algorithm

A disadvantage of the FxLMS algorithm is that the computational load is relatively highespecially when multiple inputs and outputs are considered. As was mentioned earlier,

this is mainly caused by the kronecker tensor product involved in filtering the reference

signal by the secondary path, which requires a relatively large amount of flops. In this

subsection it will be shown that a computational much more efficient algorithm called the

AdjointLMS algorithm can be obtained by filtering the error signal instead of the reference

signal.

The algorithm will be derived considering a time averaged approach. First lets consider

the general costfunction

J = E[eTe] (3.36)

Taking the derivative of this costfunction to the filtercoefficients results in:

J

w= 2E[RT(n)e(n)] (3.37)

Written out the expectation operation, this derivative can be written apart for each coef-

ficient as follows:

J

w(i)m,k

= limN

2

N

N

n=NL

l=1 Sm,lxk(n i)el(n) (3.38)


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When approaching the state space model Sl,m by a FIR model with a sufficient number of

J coefficients s(j)l,m, the above derivative can be written as:

J

w(i)m,k

= limN

2

N

Nn=N

Ll=1

J1j=0

s(j)m,lxk(n i j)el(n) (3.39)

By introducing the dummy variable n = n j the derivative becomes:

J

w(i)m,k

= limN

2

N

Nn+j=N

Ll=1

J1j=0

s(j)m,lel(n

+ j)xk(n i) (3.40)

When considering taking the mean of the derivative from n = Nj until n = Nj as

the same as taking the mean from N until N as N goes to , n

+j may be replaced byn in the expectation operation:

J

w(i)m,k

= limN

2

N

Nn=N

Ll=1

J1j=0

s(j)m,lel(n + j)xk(n i) (3.41)

By defining the filtered error signal as:

fm(n) =

Ll=1

J1j=0

s(j)m,lel(n + j) (3.42)

The gradient may then be regarded as a multiplication of the reference and the filtered

error signal:

J

w(i)m,k

= limN

2

N

Nn=N

fm(n)xk(n i) (3.43)

The time averaged behaviour of an adaptation algorithm using the gradient based on a

filtered error signal will be the same as a steepest descent algorithm based on a filtered

reference signal [15, 2].

When taking the instantaneous version of the gradient defined by equation 3.43 and

examining the expression for the filtered error, it becomes apparent that this expression is

not causal because a time advanced error signal is required. To make the expression causala delay ofJ 1 samples will be introduced to the error and reference path and by defining

j = j + J 1 the filtering of the error signal can be written as

fm(n J + 1) =L

l=1

j=J1j=0

s(jJ1)m,l el(n j) (3.44)

Leading to the following expression for the update-step

w(n + 1) = w(n) +

2

J

w(n)(3.45)

= w(n) + XT(n J + 1)f(n J + 1) (3.46)


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LMS

W(q1, n) S

S(q1) qq

x(n)

d(n)

y(n)

e(n)

u(n)

x(n ) f(n )

K

M

M

L

+

+

Figure 3.5: Blockdiagram of adaptive filter problem using AdjointLMS alghoritm, de-

notes J 1 samples

Where X(n) is defined as in equation 2.16 and f(n) is defined as the vector ofM1 filterederror signals. It may be noticed that the filtering of the error signal actually occurs by a

delayed time reversed impulse response of the secondary path model, which z-transform

can be written as:

zJ+1Sm,l(z1) =

J1j=0

s(j)m,lz

jJ1 (3.47)

The filter ST(z1) is called the adjoint of S(z) and is defined as its anticausal transposedcounterpart. Therefore this algorithm is also known as the adjoint-LMS algorithm. In

order to implement a stable and causal approximation of the adjoint of the secondary path

the adjoint system is described by a finite impulse response. The length of the filter is

governed by the amount of samples by which the impulse response of the secondary path

is contained.

The blockdiagram of the adjointLMS algorithm is shown in figure 3.5. The stability behav-

iour of the FxLMS algorithm was described for a slowly varying filter. Because the gradient

estimate of the AdjointLMS algorithm will be similar to the gradient of the FxLMS in the

limit of a slow adaptation process, the stability conditions described for the FxLMS algo-

rithm will also apply for the AdjointLMS algorithm [2, 1]. Yet the convergence behaviourwill be somewhat slower because of the delay introduced to make the adjoint filter causal.

Because both algorithms converge to the same optimal solution the amount of reduction

obtained at the disturbance signal will ultimately be similar for both algorithms.

Having specified the algorithm, the most important reason using the adjoint LMS alogrithm

is because it is much more efficient. The reason of this, lies in the fact that the kronecker

tensor product incorporated by the filtering of the reference signal by the FxLMS algo-

rithm can be avoided. This may save a large amount of flops required per sample, especially

when multiple input and output channels are involved. The difference in the amount of

flops for the traditional FxLMS and the adjoint LMS algorithm is denoted in table 3.2

Where the symbols are defined as in section 3.2.4 and J denotes the amount of tabs by


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Table 3.2: Comparison of the number of floating point operations required by the FxLMS

algorithm and Adjoint LMS algorithm FxLMS algorithm Adjoint LMS algorithm

Filtered reference/error signal LM K[2N2 + 3N + 1] 2M LJ MFilterupdate 2MKIL + L 2M KI + M

Filtering reference signal 2IKM M 2IKM Mwith adaptive filter

Total amount of flops LM K[2N2 + 3N + 2I + 1]+ M[2JI + 4KI 1]2M KI M + L

which the impulse response of the secondary path is contained. Using the practical ex-

ample mentioned before, and noting that J can be given by 200 tabs, the total amount

flops of the FxLMS algorithm is 747636. The total amount of flops required by the Ad-

joint LMS algorithm contains 484794, which is only 65 percent compared to the traditional

FxLMS algorithm. Finally it should be noted that when the adjoint LMS algorithm, like

LMS

W(q1, n) SS1o

Si (q1) q

q

x(n)

d(n)

y(n)

e(n)

u(n)u(n)

x(n ) f(n )

K

M

M

L

+

+

Figure 3.6: Blockdiagram of adaptive filter problem using AdjointLMS alghoritm withpostconditioning applied, denotes J

1 samples

the FxLMS algorithm is arranged with postconditioning as shown if figure 3.6 it takes

advantage from the more contained size of the impulse response of the innerfactor of the

secondary path compared with the impulse response of the secondary path itself. However

the reduction in the amount of flops between the FxLMS and the AdjointLMS algorithm

is less pronounced when using postconditioning. The innerfactor of the secondary path

may now be sufficiently described by a FIR filter containing 20 tabs. The flops required by

the postconditioned FxLMS algorithm amounts 86902 compared to 75160 flops required

by the postconditioned AdjointLMS algotihm.


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3.3 Adaptive feedback control

3.3.1 Introduction

In the previous section the design of an adaptive feedforward controller was discussed. It

was mentioned in the chapter on the design of the fixed gain controller, that a feedback

arrangement can be considered as a feedforward arrangement using the principle of internal

model control and assuming perfect plant knowledge. By this way the optimal feedback

controller could be obtained using the standard feedforward design theory. This strategy

will also be applied in the design of the adaptive feedback controller, as will be shown in the

first subsection of this section. In the following subsection the influence of modeluncertainty

on the behaviour of the stability and convergence properties of the adaptive controller will

be described. Finally the adaptive controller is presented using postconditioning whichconcludes this section on adaptive feedback control.

3.3.2 Design of the adaptive controller

Following closely the discussion on the design of the fixed gain feedback controller men-

tioned in section 2.4, the adaptive feedback controller is designed according to the Ad-

jointLMS algorithm discussed in the previous section.

First perfect plant knowledge is not assumed and the according blockdiagram of the

adaptive feedback controller can be shown in figure 3.7. As a reference signal, the estimated

LMS

W(q1, n) S

S

S(q1) qq

d(n)

d(n)

y(n)

y(n)

e(n)

u(n)

x(n ) f(n )

M

M

L

+

+

+

Figure 3.7: Blockdiagram of adaptive feedback filter problem using the AdjointLMS al-ghoritm, denotes J 1 samples


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3.3. Adaptive feedback control 35

disturbance signal d(n) is used. The error signal can then be written as:

e(n) = SW(q1, n)d(n) + d(n) (3.48)

The estimated disturbance signal is reconstructed by filtering the steering signal by an

internal mode

Duidamscriptie

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