Methods for Frequency Domain Estimation of Continuous-Time Models

Linkoping Studies in Science and TechnologyThesis No. 1133

Methods for Frequency DomainEstimation of Continuous-Time

Models

Jonas Gillberg

REGLERTEKNIK

AUTOMATIC CONTROL

LINKÖPING

Division of Automatic ControlDepartment of Electrical Engineering

Linkopings universitet, SE–581 83 Linkoping, SwedenWWW: http://www.control.isy.liu.se

Email: [email protected]

Linkoping 2004

Methods for Frequency Domain Estimation of Continuous-TimeModels

c© 2004 Jonas Gillberg

Department of Electrical Engineering,Linkopings universitet,SE–581 83 Linkoping,

Sweden.

ISBN 91-85295-91-4ISSN 0280-7971

LiU-TEK-LIC-2004:62

Printed by UniTryck, Linkoping, Sweden 2004

To Anna, Lisbeth, Anders, Malin, Frida, Dinh, Lars-Erik & Christina

Abstract

Approaching parameter estimation from the discrete-time domain is the dominatingparadigm in system identification. Identification of continuous-time models on theother hand is motivated by the fact that modelling of physical systems often takeplace in continuous-time. For many practical applications there is also a genuineinterest in the parameters connected to these physical models. In the black-boxdiscrete-time modelling framework however, the identified parameters often lack aphysical interpretation.

Uniform sampling has also been a standard assumption. A single sensor deliv-ering measurements at a constant rate has been considered as the ideal situation.With the advent of networked asynchronous sensors the validity of this assumptionhas however changed. In fields such as economics and finance, uniform samplingmight not be practically possible. This indicates a need for methods coping withnon-uniform sampling

In the first part of this thesis the problem of estimation of irregularly sampledcontinuous-time ARMA models in the frequency domain is treated. In this process,the model output is assumed to be piecewise constant or piecewise linear, and anapproximation of the continuous-time spectral density is calculated. MaximumLikelihood estimation in the frequency domain is then used to obtain parameterestimates. Rules of thumb concerning the model bias and variance are derivedand used in order to select the frequencies to be used in estimation. Finally, themethods are applied to a tire pressure estimation problem.

The second part of the thesis treats frequency domain identification of continuous-time ARMA and OE models for uniformly sampled data. Here the end objective isto inspire improved interpolation schemes which excel over the piecewise-linear andpiecewise-constant approximations used in the first part. The result is a methodwhich estimates the continuous-time spectrum/Fourier transform from its discrete-time counterpart.

i

Acknowledgments

”No man is an island, entire of itself; every man is a piece of thecontinent, a part of the main.”

John Donne (1572 - 1631)

First of all I would like to thank my supervisors Lennart Ljung and Fredrik Gustafs-son for their guidance and support throughout my research. I have learned toappreciate this combination since the personality and research profiles of Lennartand Fredrik seem to complement each other very well.

Several other persons but myself and my supervisors have contributed to themaking this thesis. I would like to mention Frida Eng with whom I have had manyinteresting discussions about problems connected to non-uniform sampling. Severalother people have also helped me with the writing.First, I would like to mention thelocal LATEXguru at Automatic Control, Gustav Hendeby, who have answered evenmy most imbecilic questions on typesetting. Then I would like to thank MarcusGerdin, Jonas Jansson and Frida Eng for proofreading parts of the manuscript.

Finally, I would like to express my most sincere gratitude to my, soon to bewife, Anna who have shown an extraordinary patience with me during my work.I would also like to thank my father Anders and my mother Lisbeth who havebeen my most loyal supporters, cheering me up all the time. Finally, I would liketo thank Lars-Erik and Christina, parents of Anna, who have almost provided mewith a second home in Norrkoping and Arkosund. I would also like to thank thefamilies of Ragnar Wallin and Fredrik Tjarnstrom who have enriched my and Annassocial life during this brief period of time. Of course I should not forget to thankErik Geijer Lundin for taking me pike fishing by Norra Finno and for keeping mecompany during Annas telephone conferences.

Finally a quotation which captures the relaxed and humble culture which ispromoted at the department of Automatic Control

”It is the Free Diving World Championship. I am by the way worldchampion!”

Enzo Molinari, Le Grand Bleu, 1988

ii

Contents

1 Introduction 11.1 Problem Specification . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 A Short Review of Continuous-Time System Identification 92.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Exact Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Indirect Exact Approach . . . . . . . . . . . . . . . . . . . . . 102.3 Direct Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Modulating Functions Methods . . . . . . . . . . . . . . . . . 122.3.2 Linear Filter Methods . . . . . . . . . . . . . . . . . . . . . . 132.3.3 Integration Methods . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Frequency-Domain Methods . . . . . . . . . . . . . . . . . . . . . . . 192.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Frequency-Domain Identification of Noise Models 233.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1.1 Continuous-Time ARMA Model . . . . . . . . . . . . . . . . 263.1.2 State-Space Representation . . . . . . . . . . . . . . . . . . . 273.1.3 Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Maximum Likelihood Estimation in the Time Domain . . . . . . . . 283.2.1 Efficient Computation . . . . . . . . . . . . . . . . . . . . . . 29

iii

iv Abstract

3.2.2 Cramer-Rao Lower Bound . . . . . . . . . . . . . . . . . . . . 303.3 Approximate Frequency Domain Maximum Likelihood Estimation . 30

3.3.1 Frequency Domain Model . . . . . . . . . . . . . . . . . . . . 31

4 Estimation of Power Spectrum 394.1 Interpolation and the Fourier Transform . . . . . . . . . . . . . . . . 404.2 Spectral Estimates for Uniform Sampling . . . . . . . . . . . . . . . 414.3 Effects of Interpolation on Spectral Estimates . . . . . . . . . . . . . 45

4.3.1 Periodogram Bias . . . . . . . . . . . . . . . . . . . . . . . . 474.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Properties of Bias and Variance 515.1 Asymptotic Expressions for Bias and Variance . . . . . . . . . . . . 52

5.1.1 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.1.2 Bias Expression . . . . . . . . . . . . . . . . . . . . . . . . . . 545.1.3 Variance Expressions . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Practical Considerations for Frequency Selection . . . . . . . . . . . 565.2.1 Minimizing the Variance . . . . . . . . . . . . . . . . . . . . . 565.2.2 Minimizing the Bias . . . . . . . . . . . . . . . . . . . . . . . 585.2.3 Re-Parametrization . . . . . . . . . . . . . . . . . . . . . . . 59

5.3 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 595.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6 Application to Estimation of Tire Pressure 636.1 Tire Pressure Modelling . . . . . . . . . . . . . . . . . . . . . . . . . 646.2 Problem Specifics and Objectives . . . . . . . . . . . . . . . . . . . . 656.3 Comparing Time and Frequency Domain Approaches . . . . . . . . . 656.4 Frequency Domain Estimation . . . . . . . . . . . . . . . . . . . . . 676.5 Properties of Bias and Variance . . . . . . . . . . . . . . . . . . . . . 686.6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7 Identification of CARMA Models 737.1 CARMA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747.2 Indirect Frequency-Domain Estimation . . . . . . . . . . . . . . . . . 74

7.2.1 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . 767.3 Direct Continuous-Time Estimation . . . . . . . . . . . . . . . . . . 76

7.3.1 Estimation Method . . . . . . . . . . . . . . . . . . . . . . . . 777.3.2 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . 80

7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8 Identification of COE Models 818.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828.2 Exact Indirect Frequency Domain Estimation . . . . . . . . . . . . . 828.3 Approximating Gd for Indirect Frequency Domain Estimation . . . . 84

8.3.1 Pulse Transfer Function . . . . . . . . . . . . . . . . . . . . . 84

Contents v

8.3.2 Estimation Method . . . . . . . . . . . . . . . . . . . . . . . . 868.4 Indirect Method with Modified Objective . . . . . . . . . . . . . . . 87

8.4.1 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . 898.5 Direct Method with Modified Transform . . . . . . . . . . . . . . . . 90

8.5.1 Estimation Method . . . . . . . . . . . . . . . . . . . . . . . . 918.5.2 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . 93

8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

9 Conclusions and Further Research 95

Bibliography 97

vi Contents

Notation

Symbols, Operators and Functions

p differentiation operatorΦc(iω) continuous-time spectrumΦd(eiωTs) discrete-time spectrumˆΦT

c (iω) continuous-time periodogramˆΦd(eiωTs) discrete-time periodogramθ parameter vectorθc parameters of continuous-time modelθd parameters of discrete -time modelθ0 true parameters vectorθ estimated parameter vectoru(t) continuous-time input signale(t) continuous-time white noisey(t) continuous-time output signalUT (iω) truncated Fourier transform of input signalET (iω) truncated Fourier transform of input signalYT (iω) truncated Fourier transform of output signalYd(eiωTs) discrete-time Fourier transform of output signalY

(k)T (iω) estimate of the continuous-time Fourier transform by kth order

interpolation

vii

viii Notation

δ(t) Dirac delta distributionδl Kronecker delta function.

Abbreviations

CARMA continuous-time autoregressive moving-averageCAR continuous-time autoregressiveCOE continuous-time output errorZOH zero-order holdFOH first-order hold

1Introduction

”The sciences do not try to explain, they hardly even try to inter-pret, they mainly make models. By a model is meant a mathematicalconstruct which, with the addition of certain verbal interpretations, de-scribes observed phenomena. The justification for such a mathematicalconstruct is solely and precisely that it is expected to work”

John von Neumann (1903 - 1957)

Complex industrial products such as automobiles, aircraft etc. integrate a largenumber of components of different physical nature such as mechanics, electronics,hydraulics and fluids. Take for instance an automotive engine as the one in Fig-ure 1.1. The number of components within devices is growing and their desiredbehavior is getting more and more complex. Apart from the products themselves,the process of product development is becoming more demanding. Greater func-tionality and better quality are to be implemented in less time, with less resourcesand less environmental impact. An effective product development process is there-fore becoming more of a necessary condition for market success than the sufficientone it used to be. This is even more apparent when the market gets globalized andthe competition gets tougher every single day.

In this context it is easy to understand why mathematical modelling and simu-lation have grown from a technical novelty fifty years ago, into a crucial componentof product design. Through the increased competition and the un-parallelled de-velopment of computer power, mathematical modelling has earned an importantand respected role in modern engineering. Today many industrial companies arereluctant to create systems with a behavior that cannot be modelled and simulatedin advance.

1

2 Chapter 1 Introduction

Figure 1.1 A car engine.

The use of models has also added value to science and engineering by facilitatingnew and improved functionality. Good models can provide frameworks that canbe used to interpret and make sense of measurements from various systems. Thismaterial can then be employed in order to make accurate decisions and to takeappropriate actions in order to control the state or output of the system. If properlydesigned, models can also make it possible extract as much information as possiblefrom available data, which can be important when gathering data is exceedinglydifficult and/or expensive.

Figure 1.2 A saturated steam heat exchanger [Bittanti(1997)].

Now, if models are just that important, where are they used in practice? Thefollowing example might illuminate the issue. A typical device that can be found

3

in a private house or an apartment building is a saturated steam heat exchangersuch as the one in Figure 1.2. This piece of apparatus is connected to the localdistrict heating system at one end while the other end is connected to the buildingwater heating system. As a home owner using the exchanger to heat your houseit is important that the hot water flowing through radiators etc. has the correcttemperature. In this setup control of the temperature T is achieved by varyingthe rate of the water flow q through the exchanger. This causal relationship isillustrated in Figure 1.3. In order to control the temperature properly we need

G- -q T

Figure 1.3 Input-output diagram of the heat exchanger.

to know how changing the rate of flow affects the temperature. The objective ofthe model is to capture this relationship mathematically as accurately as possible.When a mathematical model is available, a device that automatically controls thewater temperature can be readily manufactured.

How do we then make the models we need? Producing models is more or less,as John von Neumann so eloquently put it in the quote at the beginning of thischapter, the work of scientists. Models are ultimately the product of observationsand they can basically be devised in two different ways: by the so-called systemsapproach or the analytical approach [von Bertalanffy(1968)]. In the systematicapproach the object of study, the system, is decomposed into subsystems whichare again decomposed into subsystems themselves. This process of reduction mustof course end at some level of detail where some first principle rules. This canbe an assumption or an empirically established fact. The analytical approach isthe method for determining the cause and effect relationship of a system withoutdecomposing it further. This is done from observed data with very little regard tointernal structure of the system. The analytical approach can be said to be theessence of system identification.

In system identification the user has retrieved a set of input and output data,such as the one in Figure 1.4, which consists of a sequence of measurements offlow rate and temperature readings. The classical approach, e.g. [Ljung(1999)] toconstructing a model, is to describe the input-output relationship as a differenceequation

T (k + 1) + a1T (k) = b1q(k) (1.1)

which relates the different measurements to each other. The so called model param-eters θ, a1 and b1, are then determined by minimizing the difference between the


0 500 1000 1500 2000 2500 3000 3500 400090

95

100

105

y1

Input and output signals

0 500 1000 1500 2000 2500 3000 3500 40000

0.2

0.4

0.6

0.8

1

Time

u1

Figure 1.4 Input and output data from heat exchanger [Bittanti(1997)].

temperature predicted by the model, T , and the measured temperature T providedby the measured flow rates q.

θ = arg minθ

N∑

k=1

(T (k)− T (θ, k))2.

This is a very successful approach with numerous practical applications. The modelclass in (1.1) is called time-discrete because it relates the measurements at discrete-time instances to each other. The model does however not capture what happensin between measurements.

Differential equations such as

dT (t)dt

+ a1T (t) = b1q(t) (1.2)

is on the other hand a class of models which describes the input-output relation-ship continuously in time. Often theses types of models can be devised from firstprinciples and it is possible to attach physical meaning to the parameters. Theparameters can for instance, in the case of the heat exchanger, be a heat transfercoefficient relating the rate of transfer of heat from the saturated steam to thewater. For some reason it might be important for the user to know this value.A continuous-time model would then be more appropriate than an a discrete onewhere there is no or little direct physical information. This thesis concentrates onthe identification of continuous-time models.

1.1 Problem Specification 5

1.1 Problem Specification

The research on identification of continuous-time models has mainly concentratedon the time-domain with approaches such as: Poisson moment functionals, inte-grated sampling, orthogonal functions etc. A few researchers on the other hand,have tackled the problem in the frequency domain. An early reference is by Shinbrot[Shinbrot(1957)] followed by the Fourier modulating function approach introducedby Pearson et al. [Pearson(1993)]. Frequency-domain analysis for periodic andarbitrary signals have also been the starting point for the work by Pintelon et al.[Pintelon(1997)].

Recently there has been a renewed interest in continuous-time system identi-fication in general [Rao(2002)],[Ljung(2003)],[Larsson(2003)] and noise models inparticular. See for instance [Larsson(2003)] on continuous-time AR [Larsson(2002)]and [Larsson(2003)] on ARMA parameter estimation. The work on hybrid Box-Jenkins and ARMAX modeling by Pintelon et al.[Pintelon(2000)] and Johansson[Johansson(1994)] also concerns this problem.

The – at least in theory – optimal way of estimating continuous time modelsfrom sampled data is to use the Maximum likelihood (ML) method. In the casewhere the intersample behavior of the input can be constructed from the values atthe sampling instants (like piecewise constant or piecewise linear or bandlimitedinputs) the ML criterion can be derived readily: Solve for the predicted value ofthe output at the next sampling instant using the nominal parameter values (inthe continuous time model), and then minimize the distance between measuredand predicted outputs with respect to the parameters. For uniformly sampleddata this is a standard method that is implemented, e.g., in MATLAB’s SystemIdentification toolbox [Mathworks(2004)].

For non-uniformly sampled data such an approach becomes rather complex,and it may be of interest to find approximations that behave well, at least forsufficiently fast sampled data.

1.2 Goals

One of the goals of this thesis have been to understand identification of continuous-time systems in general and frequency-domain methods in particular. A purposehave also been to identify potential methods for coping with irregularly sampleddata. Eventually, approaching and applying developed methods to a real life prob-lem has been an important objective.

1.3 Outline

The thesis is structured as follows. First, in Chapter 2 there will be a brief review ofprevious publications on continuous-time system identification. Then, in Chapter3, the frequency-domain identification framework for continuous-time noise mod-els is erected. The method presented in Chapter 3 needs the continuous-time


power spectral density (PSD) in order to operate properly. Since only discreteand non-uniformly sampled measurements are available, non-parameteric methodsfor estimating the continuous-time PSD by basic interpolation are investigated inChapter 4. Finding ways to improve these spectral estimate is then the purposeof the rest of the thesis. In Chapter 5 it is investigated how parameter estimatesare affected by the quality of the spectral estimates. Asymptotic expressions re-lating bias and variance of spectral estimates to bias and variance of parameterestimates are derived. In Chapter 6 methods from Chapter 3 and 4 are used toprove that it is possible to do continuous-time frequency-domain estimation of theresonance frequency of automobile tires from a set of refined real-life data. Even-tually, the thesis ends with conclusions and ideas on how to extend the improvedinterpolation to the non-equidistant case. The need for methods for dealing withfrequency-domain outlier is also treated. In Chapter 7 the perspective moves backfrom non-uniform sampling to uniform-sampling. The product is an approximateestimation of the continuous-time spectrum from the discrete-time spectrum. Thisline of thought is continued in Chapter 8 where a series of approximations producea way of estimating the continuous-time Fourier transform from its discrete-timecounterpart.

1.4 Contributions

Most of the material in this thesis can be found in a number of earlier publishedreports. The parts on equidistantly sampled CARMA models in Section 7.2 and7.3 of Chapter 7 are also treated in

J. Gillberg and L. Ljung. Frequency-domain identification of continuous-time ARMA models from sampled data. Technical Report LiTH-ISY-R-2642, Department of Electrical Engineering, Linkoping University,SE-581 83 Linkoping, Sweden, November 2004a

The central result is the direct continuous-time frequency domain estimation methoddescribed in this chapter. The ideas in Section 8.2, 8.3, 8.4 and 8.5 of Chapter 8pertaining to the identification of COE models can be found in

J. Gillberg and L. Ljung. Frequency-domain identification of continuous-time OE models from sampled data. Technical Report LiTH-ISY-R-2643, Department of Electrical Engineering, Linkoping University, SE-581 83 Linkoping, Sweden, November 2004c

Here the indirect and direct continuous-time frequency domain estimation methodshave not been seen elsewhere.

Material related to the tire pressure identification problem in Chapter 6 is alsopresented in

J. Gillberg and F. Gustafsson. Frequency-domain continuous-time ARmodelling using non-equidistantly sampled measurements. In Proc. of

1.4 Contributions 7

the 2005 IEEE International Conference on Acoustics, Speech and Sig-nal Processing, Philadelphia, PA, March 2005. To appear

Finally, the issues in Section 4.1, 4.2, 4.3 and 5.1 of Chapter 4 and Chapter 5regarding interpolation, bias and variance can be found in

J. Gillberg and L. Ljung. Frequency-domain identification of continuous-time ARMA models: Interpolation and non-uniform sampling. Techni-cal Report LiTH-ISY-R-2625, Department of Electrical Engineering,Linkoping University, SE-581 83 Linkoping, Sweden, September 2004b

Here the the important results are the asymptotic expressions for the parameterbias and variance.

Publications on convex optimization for robust control which, for the sake ofuniformity of subject, have been left out from the thesis are

J. Gillberg and A. Hansson. Polynomial complexity for a Nesterov-Toddpotential-reduction method with inexact search directions: Examplesrelated to the KYP lemma. In Proc. of the 42nd Conference on Decisionand Control, Maui, Hawaii, December 2003

and

Hansson A. Wallin, R. and J. Gillberg. A decomposition approach forsolving KYP-SDPs. Technical Report LiTH-ISY-R-2621, Departmentof Electrical Engineering, Linkoping University, SE-581 83 Linkoping,Sweden, August 2004


2A Short Review of

Continuous-Time SystemIdentification

”Progress, far from consisting of change, depends on retentiveness.Those who cannot remember the past are condemned to repeat it.”

George Santayana (1863-1952)

Parameter identification of continuous-time systems is of course not a new sub-ject. In the old days when computers were not around, the continuous-time perspec-tive was dominating. Research on the subject of continuous-identification has notbeen intense until lately, but has instead been slowly going on since the early fifties.There are a number of excellent surveys of the area available [Unbehauen(1990)],[Sinha(1991)] and [Mensler(2003)] and the present chapter will only contain a smalland not at all comprehensive review. The purpose is instead to make the readerfamiliar with the subject and the problems associated with it.

2.1 Outline

In this chapter a number of approaches to continuous-time identification will bedescribed. First, in Section 2.2 two related exact methods will be explained. Bothhave in common that they use sampled versions of the continuous-time system. Thefirst method parameterizes the discrete-time system in terms of the continuous-timeparameters and then estimates them. The other algorithm identifies discrete-timeparameters and then transforms them to continuous-time.

In Section 2.3 a set of methods which are called direct are explained. Herethe continuous-time system or the input and output signals are transformed in

9

10 Chapter 2 A Short Review of Continuous-Time System Identification

order to produce a set of algebraic equations. From these equations approximateparameters can then be readily estimated.

The methods in the previous two sections mentioned above operate primarilyin the time-domain. Therefore the chapter is closed with a short exposition oncontinuous-time frequency-domain parameter estimation methods in Section 2.4.

2.2 Exact Approach

An exact approach to identifying continuous-time systems from sampled data is torepresent the system in state space form

x(t) = A(θc)x(t) + B(θc)u(t) (2.1)y(t) = C(θc)x(t)

where θc is the continuous-time parameters to be identified [Ljung(1999)]. If theinput is assumed to be zero-order hold (piecewise constant), then the relationshipbetween y and u sampled equidistantly with sampling time Ts can be described by

x(kTs + Ts) = Φ(θc)x(kTs) + Γ(θc)u(kTs)y(kTs) = C(θc)x(kTs) + D(θc)u(kTs)

where

Φ(θc) = eA(θc)Ts

Γ(θc) =∫ Ts

0

eA(θc)τB(θc)dτ.

If the output y is disturbed by white noise the discrete-time predictor in input-output form will be

y(t|θc) = C(θc)(qI − eA(θc)Ts

)−1∫ Ts

0

eA(θc)τB(θc)dτu(t)

and continuous-time parameters can be estimated as

θc = arg minθc

N∑

k=0

(y(kTs)− y(kTs|θc))2.

If there is process noise present the setup will be somewhat more complicatedinvolving the stationary Kalman filter, but the approach is still feasible. See Section4.3 in [Ljung(1999)] for more details on the issue.

2.2.1 Indirect Exact Approach

An alternative to the exact approach described previously is to go via discrete-time parameters θd. If time-discrete data is available a discrete-time model can be

2.3 Direct Approach 11

estimated using standard software [Mathworks(2004)]. This model can then putinto the state-space form

x(kTs + Ts) = Φ(θd)x(kTs) + Γ(θd)u(kTs)y(kTs) = C(θd)x(kTs) + D(θs)u(kTs).

If the matrix Φ(θd) has no eigenvalues on the negative real axis there exists acorresponding continuous-time system [Astrom(1984)]

x(t) = Ax(t) + Bu(t)y(t) = Cx(t) + Du(t)

where

A =lnΦTs

B = (Φ− I)−1AΓ.

The corresponding continuous-time transfer functions is then

G(s) = C(sI −A)−1B + D

and continuous-time parameters can be acquired from this expression.

2.3 Direct Approach

Consider the linear continuous-time system

y(t) + a1y(t) + a0y(t) = b1u(t) + b0u(t) + v(t) (2.2)

where the parameters a1,a0,b1 and b0 are supposed to be estimated from a set ofdiscrete-time measurements y(t1), . . . , y(tN ) and u(t1), . . . , u(tN ) of the input andoutput. The fundamental problem with this setup lies in the estimation of the time-derivatives. Most direct methods for continuous-time system identification sharethe common feature that they in some way wish to convert the continuous-timerepresentation into a set of algebraic equations indexed by k such as

y2(k) + a1y1(k) + a0y0(k) = b1u1(k) + b0u0(k) + v(k).

Here y1(k), y2(k) , y0(k), u1(k) and u0(k) are real or complex values resulting fromsystematic transformation of the derivatives.

There are basically three different direct approaches for doing the transforma-tion [Mensler(2003)]: the method of modulating functions, linear filtering methodsand integration methods. These approaches can in general be seen to consist of twostages. First, a primary stage, where the differential equations for the continuous-time system is transformed into a systems of algebraic equations as described above.


Then, a secondary stage, where the parameters are estimated by an adequate sta-tistical estimation procedure from the system of algebraic equations. A classicalapproach in the second stage has been to use the least-squares method, whichalmost always yields biased estimates. The reason for this has been that the esti-mation of the derivatives almost always corrupted the noise sequence and made itcolored. Instrumental variable methods have therefore become popular as a meansfor reducing this bias [Young(1981)].

2.3.1 Modulating Functions Methods

An nth order modulating function φ(t) is a smooth n − 1 times differentiablefunction such that

φ(i)(0) = 0

φ(i)(T ) = 0

for i = 0, 1, . . . , n− 1 on the interval [0, T ] [Shinbrot(1957)]. Here φ(i)(t) denotesthe ith derivative with respect to time.

The purpose of choosing the functions as above is to avoid the effect of initialconditions. If the differential equation in (2.2) is multiplied by such a function andis then integrated over the interval it will yield

T∫

0

y(t)φ(t)dt + a1

T∫

0

y(t)φ(t)dt + a0

T∫

0

y(t)φ(t)dt

= b1

T∫

0

u(t)φ(t)dt + b0

T∫

0

u(t)φ(t)dt +

T∫

0

v(t)φ(t)dt.

However, because of the conditions put on the modulating function, the expressionabove can be rewritten by partial integration as

T∫

0

y(t)φ(t)dt− a1

T∫

0

y(t)φ(t)dt + a0

T∫

0

y(t)φ(t)dt (2.3)

= −b1

T∫

0

u(t)φ(t)dt + a0

T∫

0

u(t)φ(t)dt +

T∫

0

v(t)φ(t)dt (2.4)

since the modulating functions are supposed to be n times differentiable. It isthen possible to evaluate the integrals numerically using the fast Fourier transformto produce an algebraic equation [Pearson(1993)]. If the integral expressions are


defined as

γyn−i = (−1)i

∫ T

0

y(t)φ(i)n (t)dt

εn =∫ T

0

v(t)φn(t)dt

the equation in (2.3) can be written as the regression equation

γy0 = γθ + εn.

where

γ = [−γyi · · · − γy

n γun−m . . . γu

n−m]θ = [an . . . a0bm . . . b0]

Doing this over number of different modulating functions of the same class, aset of regression equations can be formed and parameters can then be estimated[Mensler(2003)]. Two such classes will be described below. One of two classes offunctions classes that are usually mentioned in connection with the modulatingfunctions method is the class of Fourier modulating functions

φµ,n(t) =1T

e−iµω0t(e−iµω0t − 1)n

The other class is the Hartley modulating functions

ψµ,n(t) =n∑

k=0

(−1)k

(k

n

)(cos(n + µ− k)ω0t + sin(n + µ− k)ω0t)

2.3.2 Linear Filter Methods

The linear filter approaches have in common that they apply linear filters to theinput and output signals in order to obtain a set of algebraic equations. Thisprocedure is illustrated as

F (p)y(t) + a1F (p)y(t) + a0F (p)y(t) = b1F (p)u(t) + b0F (p)u(t) + F (p)v(t) (2.5)

where the filter F has been applied to (2.2). Here, F (p) is a continuous-time all-polefilter where p is the differentiation operator. It is usually defined as

F (p) =1

L(p)(2.6)

where L is a polynomial with order greater than or equal to n to avoid directdifferentiation. The parameter n is the maximum number of differentiations present


in the differential equation. In this case n = 2. The polynomial L can easily becompared to an observer polynomial, but the design uses a minimum of a prioriinformation. Basically, the approximate bandwidth of the system together withthe order of the differential equation [Young(1970)]. A particular choice of filter is

F (p) =(

λ

p + λ

)n

which allows the user to tune the bandwidth of the filter through the parameterλ [Garnier(2003)]. Here n is selected equal to the order of the system. For such aclass of filters (2.5) can now be written as

p2

L(p)y(t) + a1

p

L(p)y(t) + a0

1L(p)

y(t) = b1p

L(p)u(t) + b0

1L(p)

u(t) +1

L(p)v(t)

where

pn

L(p).

are known as the state variable filters. If the outputs of these filters are denotedyn(t), un(t) and vn(t) and the effects of initial conditions are ignored then

y2(t) + a1y1(t) + a0y0(t) = b1u1(t) + b0u0(t) + v0(t). (2.7)

This equation can be put in regression form

y2(t) = φ(t)T θ + v0(t)

where

φ(t)T = [−y1(t) − y0(t) u1(t) u0(t)]

θT = [a1 a0 b1 b0].

The objective function

Vc =∫ T

0

(yn(t)− φ(t)T θ)2dt

can then be minimized in order to obtain a (usually biased) least-squares parameterestimate. Since only time-discrete measurements of the signals are available inpractice the discrete a version of the criterion

Jd =N∑

k=0

(y2(tk)− φ(tk)T θ)2.

is used for estimation.


Poisson Moment Functionals

A close relative to the state variable filter approach is the method of generalizedPoisson moment functionals. Here a signal such as y is treated as a generalizedfunction, a distribution, and can be expanded about instant t0 as

y(t′) =∞∑

k=0

Mk[y(t)]eλ(t−t0)δ(k)(t− t0)

where δ(k) are derivatives of the Dirac distribution [Saha(1983)]. The coefficientsof this representation are then computed as

Mk[y] ,∫ t0

0

y(t)mk(t0 − t)dt

where

mk(t0) , βl+1 tk0k!

e−λt.

and β and λ are tuning parameters. Here Mk is called the Poisson moment func-tional of order k. An attractive feature of the method is that the integrals involvedin the functionals can be obtained as the outputs of a cascade of filters as illustratedin Figure 2.1. For these filters there is a relationship between the derivatives of the

βs+λ

- -

?

y(t)

M0[y(t)]

βs+λ

-

?

M1[y(t)]

βs+λ

-

Figure 2.1 Poisson Filter Chain

signal y such that

Ml[y(n)(t)] =i∑

j=0

(−1)j

(j

i

)βi−jλjMl−i+j [y(t)] (2.8)

−i∑

q=1

y(q−1)(0)

i−q∑

j=0

(−1)j

(j

i− q

)βi−q−jλjpl−i+q+j(t)

. (2.9)

Moment functionals Ml of a derivative of y or u can hence be written in terms oflower order moment functionals of the original signals. Ignoring the effect if initial


conditions we would for instance have [Mensler(2003)].

Ml[y(2)(t)]Ml[y(1)(t)]Ml[y(t)]

=

β2 −2λβ λ2

0 β −λ0 0 1

Ml−2[y(t)]Ml−1[y(t)]Ml[y(t)]

If the moment functional Mn is applied to the test equation such that

Ml[y(t)] + a1Ml[y(t)] + a0Ml[y(t)] = b1Ml[u(t)] + b0Ml[u(t)] + Ml[v(t)]

it is by (2.8) possible to transform this system into a regression form

Ml[y(2)(t)] = φ(t)θ + Ml[v(t)]

which can be used to produce a least squares estimate. The procedure is as men-tioned earlier very similar to the state variable filter approach.

2.3.3 Integration Methods

In the integration approach the entire differential equation in (2.2) is integrated,say n = 2 times creating an expression such as

y(t) + a1y[1](t) + a0y

[2](t) = b1u[1](t) + b[2]

u (t) + c1t1

1!+ c0

t2

2!+ v[2](t) (2.10)

where

y[k](t) =∫· · ·

∫

k times

y(t)dt

and c0 and c1 account for the effect of initial conditions [Whitfield(1987)]. Inorder to produce algebraic equations that can be used to estimate parameters,the integrals have to be computed numerically. The different approaches below arebasically different ways to perform this evaluation from samples of the input-outputdata.

Orthogonal Functions

In the orthogonal function approach, the input u(t) and output signals y(t) arerepresented as

y(t) =∞∑

i=0

yiφi(t).

in a base of functions φi which are orthogonal with respect to a weight functionw(x) ≥ 0 [Rade and Westergren(1995)]. The coefficients yi are identified by pro-jecting the function y(t) onto the basis by evaluating the scalar products like

yi =

∫ T

0w(t)y(t)φi(t)dt

∫ T

0w(t)φi(t)φi(t)dt


numerically from samples of y(t). The series representation is always truncatedsuch that

y(t) ≈N∑

i=0

yiφi(t) = yT φ(t)

with

y = [y0 y1 . . . yN ]T

φ(t) = [φ0(t) φ1(t) . . . φN (t)]T .

This operation automatically reduces the high frequency content of the represen-tation and will therefore be an important design choice.

When the differential equation is in integrated form as below

y(t) + a1y[1](t) + a0y

[2](t) = b1u[1](t) + b0u

[2](t) + v[2](t)

the new representation allows it to be converted into an algebraic equation. Thedevice which facilitates this is the so called operational matrix [Chen(1975)] forintegration P which has the property that

∫ t

0

φ(t)dt ≈ Pφ(t).

This relation shows that integrals of the basis functions can be written as linearcombinations of the basis functions themselves. If applied to the integrated differ-ential equation above the operational matrix will produce

yT φ(t) + a1yT Pφ(t) + a0y

T P 2φ(t) = b1yT Pφ(t) + b0y

T P 2φ(t) + vT P 2φ(t).

Equating the coefficients will then give a set of algebraic equations

yT + a1yT P + a0y

T P 2 = b1yT P + b0y

T P 2 + vT P 2.

which can be written in the form of linear regression

y = Ψw

where w = [a1 a0 b1 b0] and Ψ depends on y and P . That equation can then beused to estimate the parameters by least-squares or instrumental variable meth-ods. Some of the orthogonal basis functions found in the literature are the La-guerre polynomial [Hwang(1982)], Fourier series [Chung(1987)], block pulse func-tions [Palanisamy(1981)] and Legendre polynomials [Chang(1982)].

Numerical Integration Methods

Since the continuous-time realizations of the input and output are not available,discrete-time signals are integrated numerically using piecewise constant or piece-wise linear interpolation. These two types of interpolation also come under the


name of the block pulse function(BPF) or the trapezoidal pulse function (TPF)methods. For the trapezoidal rule with equidistant sampling interval Ts the inte-gration can be realized by the following expression

y[k](ti) = y[k](ti−1) + Tsy[k−1](ti) + y[k−1](ti−1)

2

u[k](ti) = u[k](ti−1) + Tsu[k−1](ti) + u[k−1](ti−1)

2

A set of algebraic equations in a linear regression form can ultimately be derivedby evaluating the integrals over different time intervals [Whitfield(1987)]. Whenalgebraic equations are present parameters can be readily estimated by for instancea least squares or instrumental variable approach [Soderstrom and Stoica(1983)].

Linear Integral Filters

The linear integral filter approach is closely related to the numerical integrationmethods [Sagara(1990)]. Numerically integrating a function over a small interval[t, t− lTs] can be written as

y[1] =∫ t

t−lTs

y(τ)dτ =l∑

i=0

fiy(t− iTs)

where fi originate from a quadrature scheme such as the trapezoidal rule. Usingthe delay operator

q−1y(t) = y(t− Ts)

this relation can be represented as

y[1] =l∑

i=0

fiq−iy(t). (2.11)

The central result of the theory of linear integral filters is that if y(j) denotes thej:th derivative of y(t) then y(j)(t) integrated n times can be written as

Iny(j) ≈ Pjy(t) =nl∑

i=0

pji q−iy(t)

where In is the operator for integrating n times and

Pj = (1− q−1)j(l∑

i=0

fiq−i)n−j .

2.4 Frequency-Domain Methods 19

. Here pji are real valued coefficients that have little to do with the differentiation

operator p.The method above avoids the problems of initial values encountered in the nu-

merical integration approach. It can also be interpreted as linear filtering approachsince multiple integration over the small interval is equivalent to pre-filtering with

L(s) =1− e−sTsl

sn.

This view supplies the user with l as a tuning parameter for a filter. As for statevariable filters and Poisson modulating functions it should be chosen in such a waythat the bandwidth of the filter closely matches the bandwidth of the system.

By applying multiple integration to the system to be estimated a linear regres-sion form can be derived. By varying the interval of integration [t − Tsl] a set ofalgebraic equations can be produced and parameters can subsequently be estimatedwithin a least squares or instrumental variable framework.

2.4 Frequency-Domain Methods

System identification of continuous-time systems can also be performed in thefrequency-domain. Suppose that the we are provided with good approximations ofthe Fourier transforms of the input and output

Y (iωk) =∫ ∞

0

y(t)e−iωktdt (2.12)

U(iωk) =∫ ∞

0

u(t)e−iωktdt.

If the data were generated by

y(t) = G(p, θ)u(t) + H(p, θ)e(t)

the Fourier transforms would be related as

Y (iω) = G(iω, θ)U(iω) + H(iω, θ)E(iω).

If e(t) is white normal noise, its Fourier transform will have the complex Normaldistribution [Brillinger(1981)]

E(iω) ∈ Nc(0, σ2)

which means that the real and complex parts of E are independent and normallydistributed. A consequence of this is that

Y (iωk) ∈ Nc(G(iωk, θ)U(iωk), σ2|H(iωk, θ)|2).


The negative log likelihood function for estimating θ from Y (iωk), U(iωk), k =1, . . . , N provided Y (iωK) are uncorrelated, will be

VN (θ) = Nlogλ +N∑

k=1

2 log |H(iωk, θ)|+N∑

k=1

1λ|Y (iωk)−G(iωk)U(iω)|2 1

|H(iωk, θ)|2 .

(2.13)

This criterion can then be minimized by appropriate non-linear optimization meth-ods.

The method above assumes that we have the continuous-time Fourier transformsto begin with. This might not always be the case and instead the discrete-timeFourier might only be available. If the rate of sampling is high the difference mightbe small, but otherwise special measures have to be taken.

A recent approach that has been proposed consists of taking the Laplace trans-form of the differential equation (2.2) over the interval [0, T ]

Y (T )c (s) =

∫ T

0

y(t)e−stdt

U (T )c (s) =

∫ T

0

u(t)e−stdt

producing the expression

A(s)Yc(s) = B(s)Uc(s) + T1(s)− e−TsT2(s) (2.14)

where

A(s) = s2 + a1s + a0

B(s) = b1s + b0

and T1 and T2 account for initial and endpoint conditions [Pintelon(1997)]. Letthe frequencies used be

ωk =2π

Tk

where Ts is the sampling time, k = 1, . . . , N and T = NTs. Then the relationshipbetween the discrete-time Fourier transform in (4.11) Yd(eiωkTs) and continuous-time Fourier transform in (3.16) Yc(iωk) is governed by the Poisson summationformula e.g. [Gasquet(1998)]

Yd(eiωkTs) =∞∑

n=−∞Yc(iωk + i

2π

Tsn).

2.4 Frequency-Domain Methods 21

The expression in (2.14) can then be written as

A(iωk)Yd(eiωkTs) = B(iωk)Ud(eiωkTs) + T (iωk) + δ(iωk) (2.15)

where

δ(iωk) =n=∞∑

n=−∞,n 6=0

B(iωk)Uc(iωk + i2π

Tsn)−A(iωk)Yc(iωk + i

2π

Tsn)

and

T (iωk) = T1(iωk)− T2(iωk).

The terms T and δ are then approximated by a high order polynomial P . Pa-rameters are then estimated in the frequency-domain maximum likelihood fashion[Ljung(1999)]

θ = arg minθ

N/2∑

k=1

|Yd(iωk)− B(iωk)A(iωk)

Ud(iω)− P (iωk)A(iωk)

|2

when the output error modelling framework is used [Pintelon(1997)].Work on how to incorporate a noise model into the frequency-domain frame-

work described previously can also be found [Pintelon(2000)]. This approach, thenenables Box-Jenkins continuous-time modelling such as the one illustrated in Fig-ure 2.2. Here the continuous-time noise sequence e(t) is assumed to be piecewise-

G(s)- - m

H(s)

?

?-

u(t) y(t)

e(t)

Figure 2.2 Hybrid Box-Jenkins model structure[Pintelon(2000)].

constant. Thereby sampling of the continuous-time noise model Hc(s) will yieldthe discrete-time representation Hd(q). While the discrete-time Fourier transform


of the output of the system can be modelled as in (2.15) the corresponding entityfor the noise model can be described as

Vd(iω) = H(e−iωTs , θ)Ed(eiωTs) + TH(e−iω, θ)

where TH accounts for the effect of noise model initial conditions. A frequencydomain estimate of the Hybrid-Box Jenkins parameters can then be acquired as

θ = arg minθ

N−1∑

k=0

∣∣∣∣Yd(iωk)−G(iωk)Ud(iωk)− TH(e−iωk , θ)− T (iωk, θ)

H(e−iωk , θ)

∣∣∣∣2

.

2.5 Summary

In this chapter a variety of different approaches for identifying continuous-timesystems have been illustrated. In Section 2.2 two different exact methods for theidentification of continuous-time systems were presented. Either by parameteriz-ing a discrete-time model in terms of continuous-time parameters, or identifyingdiscrete-time parameters and transform them to continuous-time.

A common denominator for the time-domain methods in Section 2.3 seems tobe an effort to avoid estimating the continuous-time derivatives directly. Almostall approaches presented there include, directly or indirectly, some form of low passfiltering, mainly in order to attenuate high frequency noise that would otherwisebe much amplified by the differentiation operation.

Finally, in Section 2.4, two frequency domain approaches to parameter identi-fication were illustrated.

3Frequency-Domain Identification

of Noise Models

”Le plus court chemin entre deux enonces reels passe par le complex””The shortest path between two truths in the real domain passes

through the complex”

Jacques Hadamard (1865-1963)

System identification, as described earlier, can quite simply be described as theart of building dynamical models from input and output data. It is therefor naturalthat the perspective taken on the data will influence the modelling that takes place.In the engineering world the two different aspects that manifest themselves tothe practitioner of identification are the time-domain view and frequency-domainperspective. In the time-domain each separate measurement of the input or outputquantity x is associated with the time instant t when it is taken . This relationshipis represented as x = x(t) and is illustrated in Figure 3.1. Here time evolves alongthe horizontal axis and fluctuations in air pressure are illustrated along the verticalaxis.

When the set of measurements or signal in Figure 3.1 is replicated on an audioloudspeaker we immediately identify it as a sound composed of three different tonesC, E and G. In the beginning of the 18th century the French mathematician JeanBaptiste Joseph Fourier discovered that almost all time domain signals can bemathematically decomposed into an infinite set of tones or periodic oscillations.This principle is captured in the so called Fourier transform which relates thetime-domain signal to its tone or more accurately, frequency domain description

23

24 Chapter 3 Frequency-Domain Identification of Noise Models

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3

−2

−1

0

1

2

3

s

Figure 3.1 Air pressure as a function of time for a chord involving thetones C (261 Hz), E (330 Hz) and G (392 Hz).

X(iω).

x(t) =∫ ∞

−∞X(iω)(cos ωt + i sin ωt)dω.

In Figure 3.2 the same chord is illustrated with the tone pitch or frequency on thehorizontal axis and magnitude on the vertical axis. As expected there are threedistinct peaks found at frequencies where one would expect to find the C, E andG tones.

In continuous-system identification one wishes to identify parameters of deter-ministic or stochastic differential equation

y(t) + ay(t) = e(t)

The relationship between the input e and the output y in this equation can beexpressed explicitly as

y(t) =∫ ∞

−∞h(τ)e(t− τ)dτ

where h(τ) is called the impulse response. The impulse response is the solutionwhen the input is the Dirac delta function δ(t)

h(t) + ah(t) = δ(t).

In the frequency-domain on the other hand, the relationship between the Fouriertransforms of the input E(iω) and output Y (iω) is governed by the straightforward

25

240 260 280 300 320 340 360 380 400 420

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x 104

Hz

Figure 3.2 Power of oscillation as a function of frequency for a chord in-volving the tones C (261 Hz), E (330 Hz) and G (392 Hz).

equation

Y (iω) = H(iω)E(iω)

where H(iω) is the Fourier transform of the impulse response h(t), which for theexample above looks like

H(iω) =1

iω + a. (3.1)

The magnitude and phase of this quantity is illustrated in Figure 3.3 with theconclusion that the effect of the system is simply to amplify certain frequenciesand attenuate others which at the same time delaying them more or less. This isa more simple interpretation of the response of the system.

This chapter will focus on the identification of noise models, i.e. models wherethe input is not known but is assumed to be random. The objective is to explain andformulate an approximate maximum likelihood method in the frequency domain.

The chapter will be structured as follows. In section 3.1 CARMA models areintroduced together with basic properties for continuous-time stationary stochasticprocesses. Then, in Section 3.2, the exact method in the time-domain and its asso-ciated properties are illustrated. Then, the frequency-domain method is describedand analyzed in Section 3.3.


−40

−35

−30

−25

−20

−15

−10

−5

0

Mag

nitu

de (

dB)

10−2

10−1

100

101

102

−90

−45

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Figure 3.3 Amplitude an phase of the first-order transfer function in (3.1).

3.1 Modeling

In this section the basic model and data setup are introduced. First the continuous-time autoregressive moving average (CARMA) model is introduced in an informalmanner. Then, the model is more formally introduced in a state space form. It isalso explained how this stochastic continuous-time dynamical system is simulatedand how measurements are taken.

3.1.1 Continuous-Time ARMA Model

The continuous-time autoregressive moving average (CARMA) model can infor-mally be described as

y(t) =B(p)A(p)

e(t) (3.2)

where et is continuous time white noise such that

E[e(t)] = 0

E[e(t)e(s)] = σ2δ(t− s)

The operator p is here the differentiation operator while

A(p) = pn + a1pn−1 + a2p

n−2 + · · ·+ an

B(p) = pm + b1pm−1 + · · ·+ bm.

and the vector of parameters is θ = [a1 a2 . . . an b1 b2 . . . bm λ]T where λ = σ2.

3.1 Modeling 27

3.1.2 State-Space Representation

The model in (3.2) can be formally represented in a state-space controller canonicalform

x(t) = Ax(t)dt + Be(t)y(t) = Cx(t)

(3.3)

where

A =

−a1 −a2 . . . −an−1 −an

1 0 . . . 0 00 1 . . . 0 0...

......

...0 0 . . . 1 0

B =[1 0 0 . . . 0

]T

C =[0 . . . 0 1 b1 b2 . . . bm

]

if m < n. The exact solution to the state-space representation in (3.3) can bewritten as

y(t) = CeAtx(0) +∫ t

0

CeA(t−s)Be(t)dt. (3.4)

Since the systems in this thesis are all linear the treatment of e(t) in (3.4) will serveour purpose. However, we want to remind the reader that in order to treat theintegral in (3.4) in a manner which is formally correct, one should use the definitionby Ito [Oksendal(1998)]. If yt is to be a zero mean stationary Gaussian processthe initial values x(0) must be Gaussian and distributed such that E[x(0)] = 0 andQ0 = Cov[x(0)] satisfies the Lyapunov equation

AQ0 + Q0AT + σ2BBT = 0.

For the rest of this report it will be assumed that the process yt is stationaryGaussian with zero mean value [Davis(1998)].

3.1.3 Spectral Properties

Finally, the definition of some basic concepts related to stationary stochastic pro-cesses are refreshed [Papoulis(1965)]. First of all, the continuous-time covariancefunction

r(τ) = Ey(t + τ)y(t)

and power spectrum

Φc(iω) =∫ ∞

−∞rc(τ)e−iωτdτ (3.5)


are defined. In the case of the process modelled in (3.2) the spectrum will be

Φc(iω) = σ2 B(iω)B(−iω)A(iω)A(−iω)

Moving from the spectrum Φc(iω) to the covariance rc(τ) representations is facili-tated by the well-known formula

r(τ) =12π

∫ ∞

−∞Φc(iω)eiωτdω. (3.6)

Assuming that the process y(t) is only observed at the discrete-time instancest = kTs, the covariance function of the discrete-time process y(kTs) will be

rd(k) = r(kTs).

and the discrete-time power spectrum is

Φd(eiωTs) = Ts

∞∑

k=−∞rd(k)e−iωTsk. (3.7)

Finally the following relationship, similar to the Poisson summation formula e.g.[Gasquet(1998)], exists between the discrete-time and continuous-time spectrumse.g.[Wahlberg(1988)]

Φd(eiωTs) =∞∑

k=−∞Φc(iω +

i2π

Tsk). (3.8)

3.2 Maximum Likelihood Estimation in the TimeDomain

A set of possibly non-equidistant samples y(tk), k = 1 . . . Nt of the stationaryoutput of the CARMA process in (3.2) will be Gaussian and distributed as

Y =

y(t1)...

y(tNt)

∈ N(0, R(θ)) (3.9)

where

R(θ) =

r(|t1 − t1|, θ) . . . r(|t1 − tNt |, θ)r(|t2 − t1|, θ) . . . r(|t2 − tNt |, θ)

......

r(|tNt − t1|, θ) . . . r(|tNt − tNt |, θ)

.

3.2 Maximum Likelihood Estimation in the Time Domain 29

For the stationary case of (3.2) in the state space form (3.3) the following is wellknown [Hannan(1970)] [Doob(1953)]

r(τ, θ) = CeAτPCT , τ > 0

where

AP + PA + σ2BBT = 0. (3.10)

Such knowledge paves the way for Maximum-Likelihood estimation by the criterion

θ = arg minθ

Y T R(θ)−1Y + log det R(θ). (3.11)

The reason for this is that if the data ytkis the output of the stochastic process,

the samples will be distributed as in (3.9) with the probability density

p(y(t1), . . . , y(tNt)|θ) =

1(2π)Nt/2

√det R(θ)

e−12 Y T R(θ)−1Y .

The negative log-likelihood function of this distribution will be

− log p(Y |θ) =Nt

2log 2π +

12

log det R(θ) +12Y T R(θ)−1Y.

The Maximum Likelihood (ML) method for estimating the parameters would hencebe

θ = arg minθ

Y T R(θ)−1Y + log det R(θ) (3.12)

A big obstacle in the way of exploiting this approach fully when Nt is not small isthat R(θ) will become a very large matrix.

3.2.1 Efficient Computation

Since the matrix R(θ) is symmetric, positive definite it is possible to perform anLDLT factorization, a special form of the Cholesky factorization [Golub(1996)]

R(θ) = L(θ)D(θ)LT (θ). (3.13)

The objective function in the optimization can then be rewritten as

θ = arg minθ

Y T L−T (θ)D−1(θ)L−1(θ)Y +Nt∑

k=1

log Dkk(θ)

where the Dkk(θ) are the eigenvalues λk(R(θ)). This approach tends to work onlyif we have a medium or small number of measurements Nt since the size of R(θ) isNt ×Nt.


3.2.2 Cramer-Rao Lower Bound

A property of the Maximum Likelihood estimator is that it is asymptotically un-biased [Cramer(1946)] under certain mild conditions

limNt→∞

θ → θ0

where θ is the estimate and θ0 are the true parameters.The quality of an estimator can be measured by its covariance matrix

P = E(θ − θ0)(θ − θ0)T

and it is interesting to know how good these estimates can become. A lower limitfor unbiased estimators is the Cramer-Rao lower bound (CRLB)[Cramer(1946)]which states that

P ≥ M−1.

Here

M = −Ed2

dθ2log p(Y |θ0)

is known as the Fisher information matrix and p(Y |θ) is the likelihood functionexpressing the probability of the measurements Y provided that the parametersare θ .

In the case that the parameters are estimated as in (3.12) an explicit expressionfor the CRLB known as the Slepian-Bangs formula [Slepian(1954)]

Mij =12

Tr R(θ)−1 ∂R(θ)∂θi

R(θ)−1 ∂R(θ)∂θj

can be derived. This expression will be used later on in order to evaluate theefficiency of different parameter estimators for a fairly large number of samples.We will now move on and consider a ML method in the frequency domain.

3.3 Approximate Frequency Domain Maximum Like-lihood Estimation

We shall in this section establish that when Φc is given by (3.5) and ˆΦ is an estimateof (3.5), the parameters θ of a CARMA model can be estimated by solving thefollowing minimization problem

θ , arg minθ

Nω∑

k=1

ˆΦc(iωk)Φc(iωk, θ)

+ log Φc(iωk, θ). (3.14)

3.3 Approximate Frequency Domain Maximum Likelihood Estimation 31

The frequencies ωk, k = 1, . . . , Nω are chosen such that

ωk ∈

2π

Tl, l ∈ Z

. (3.15)

The method requires the continuous-time periodogram ˆΦc, an estimate of thecontinuous-time spectrum, and is an approximate Maximum-Likelihood procedurewhere the model has been transformed into the frequency domain. The frequencieshave been deliberately selected such that the Fourier transforms of the output atdifferent frequencies are asymptotically uncorrelated. If the objective function iscompared with the time-domain expression it is apparent that the quadratic formin the time-domain criterion have been replaced by a summation in the frequencydomain. Hence, the complexity has been greatly reduced. The criterion here isthe negative log-likelihood function for the real and imaginary parts of the peri-odogram given the parameters θ. The remaining part of this chapter is dedicatedto motivating this approach.

3.3.1 Frequency Domain Model

In this subsection it is demonstrated what the result will be if we apply the trun-cated Fourier transform

Y Tc (iω) =

1√T

∫ T

0

y(t)e−iωtdt (3.16)

to the stochastic process yt defined in (3.2).

Lemma 3.1Assume that we have stochastic process yt generated by the CARMA model in(3.2). Then the truncated Fourier transform of the process from time t = 0 tot = T will be

Y Tc (iω) =

B(iω)A(iω)

1√T

∫ T

0

e−iωte(t)dt +1√T

C(iωI −A)−1(x(0)− e−iωT x(T ))

(3.17)

were x(0) and x(T ) are stochastic variables denoting the states at the initial pointand the end point.

Proof Assume that we have the following system

y(t) = CeAtx(0) +∫ t

0

CeA(t−τ)Be(τ)dτ (3.18)

Assume that this signal is observed through a window

W (t) =1√T

I[0,T ]


where I is the indicator function. Then we have the transform

Y Tc (iω) =

∫ ∞

−∞W (t)y(t)e−iωtdt

which is a stochastic variable now. The integral of the first term in (3.18) will be

1√T

∫ T

0

Ce(A−iωI)tx(0)dt

=1√T

C(iωI −A)−1(I − e(A−iωI)T )x(0)

were x(0) is a stochastic variable. Integration by parts

∫ T

0

f(t)B(t)dt = F (T )B(T )−∫ T

0

F (t)B(t)dt

f(t) = Ce(A−iωI)t

F (t) = C(A− iωI)−1e(A−iωI)t

B(t) =∫ t

0

e−AτBe(τ)dτ

will give

1√T

∫ T

0

∫ t

0

CeA(t−τ)Bdete−iωtdt =

1√T

C(A− iωI)−1e(A−iωI)T

∫ T

0

e−AτBdeτ − 1√T

∫ T

0

C(A− iωI)−1e(A−iωI)te−AtBdet

=1√T

C(A− iωI)−1e−iωT

∫ T

0

eA(T−τ)Bdeτ − 1√T

∫ T

0

C(A− iωI)−1e(A−iωI)te−AtBdet

=1√T


∫ T

0

eA(T−τ)Bdeτ − 1√T

∫ T

0

C(A− iωI)−1Be−iωtdet

=1√T


∫ T

0

eA(T−τ)Bdeτ +1√T

B(iω)A(iω)

∫ T

0

e−iωtdet

One effect of the initial and end point conditions is that

∫ T

0

eA(T−τ)Bdeτ + eAT x0 = xT

and by this the transform will become

Y Tc (iω) =

B(iω)A(iω)

1√T

∫ T

0

e−iωtdet +1√T

C(iωI −A)−1(x0 − e−iωT xT )

2


In order to determine the distribution of the Fourier transform, which is normal, itis necessary to know the correlation between different frequency components. Thefollowing theorem illuminates this relationship

Theorem 3.1If Y T

c (iω) is defined as in (3.16) and ωk and ωl are defined as in (3.15). Furtherassume that A in (3.3) is stable with no eigenvalues on the imaginary axis. Then

EY Tc (iωk)Y T

c (iωl) = Φc(iωk)δk,−l +1T

K(iωk, iωl).

where K is bounded and Φc is defined as in (3.5)

Proof First of all, since eiωTs = 1 for the particular choice of ω

Y Tc (iω) =

C(iωI −A)−1

√T

(B

∫ T

0

e−iωtdet + x0 − xT

)

=C(iωI −A)−1

√T

[∫ T

0

(e−iωt − eA(T−t)

)Bdet +

(I − eAT

)x0

].

This means that

EY Tc (iωk)Y T

c (iωl) =C(iωkI −A)−1

√T[∫ T

0

(e−iωkt − eA(T−t)

)σ2BBT

(e−iωlt − eAT (T−t)

)dt

+(I − eAT

)Ex0x

T0

(I − eAT T

)] (iωlI −AT )−1CT

√T

since Ex0et = 0 and Edetdes = δ(t − s)dtds. The term inside the brackets willbecome

∫ T

0

(e−iωkt − eA(T−t)

)σ2BBT

(e−iωlt − eAT (T−t)

)dt

+(I − eAT

)Ex0x

T0

(I − eAT T

)

=∫ T

0

e−(ωk+ωl)dt− σ2BBT

∫ T

0

e−iωkteAT (T−t)dt

−∫ T

0

e−iωlteA(T−t)BBT σ2dt +∫ T

0

eA(T−t)BBT σ2eAT (T−t)BBT dt

+ Ex0xT0 − eAT Ex0x

T0 − Ex0x

T0 eAT T + eAT Ex0x

T0 eAT T

Fortunately, since xt is a stationary continuous-time stochastic process [Davis(1998)]

P = E[xtx

Tt

]

P =∫ T

0

eA(T−t)BBT σ2eAT (T−t)dt


where P is the solution to the Lyapunov equation

AP + PAT + σ2BBT = 0.

Hence

∫ T

0

eA(T−t)BBT σ2eAT (T−t)BBT dt

+ x0xT0 − eAT x0x

T0 − x0x

T0 eAT T + eAT x0x

T0

= −eAT P − PeAT T

and

σ2BBT

∫ T

0

e−iωkteAT (T−t)dt +∫ T

0

e−iωlteA(T−t)BBT σ2

= σ2BBT(eAT T − I

)(iωkI −AT )−1 +

(eAT − I

)(iωlI −A)−1BBT σ2.

Finally, since ωk and ωl are defined on the grid (3.15) we get

∫ T

0

e−i(ωk+ωl)tdt = Tδk,l.

This means that

EY Tc (iωk)Y T

c (iωl) =C(iωkI −A)−1(σ2BBT δ(ωk + ωl)

+ σ2BBT eAT T − I

T(iωkI + AT )−1

+eAT − I

T(iωlI + A)−1BBT σ2

−eAT P + PeAT T

T

)(iωk −AT )−1CT

=Φc(iωk)δk,−l +1T

K(iωk, iωl)

where

Φc(iωk)δk,l = C(iωkI −A)−1σ2BBT δk,l(iωk −AT )−1CT


and

K(iωk, iωl) =C(iωkI −A)−1

(σ2BBT eAT T − I

T(iωkI + AT )−1

+eAT − I

T(iωlI + A)−1BBT σ2

−eAT P + PeAT T

T

)(iωlI −AT )−1CT .

Since A is a stable matrix with no eigenvalues on the imaginary axis (iωI ± A)−1

and eAT will be bounded. Hence K will also be bounded. 2

From the previous result it is possible to derive an expression for the covariancematrix of the real and imaginary parts of the frequency components

Theorem 3.2If Y T

c (iω) is defined as in (3.16) and ωk > 0 and ωl > 0 are defined as in (3.15)then

E

(Re Y T

c (iωk)Im Y T

c (iωk)

)(Re Y T

c (iωl)Im Y T

c (iωl)

)T

=

(Φc(iωl

2 ) 00 Φc(iωl)

2

)δk,l +

1T

K(iωk, iωl)

where K is a bounded 2× 2 matrix.

Proof Since

ReY Tc (iω) =

Y Tc (iω) + Y T

c (−iω)2

ImY Tc (iω) =

Y Tc (iω)− Y T

c (−iω)2i

the elements of the covariance matrix will be

ReY Tc (iωk)ReY T

c (iωl) =[Y T

c (iωk) + Y Tc (−iωk)

2

] [Y T

c (iωl) + Y Tc (−iωl)

2

]

=Y T

c (iωk)Y Tc (iωl) + Y T

c (iωk)Y Tc (−iωl)

4

+Y T

c (−iωk)Y Tc (iωl) + Y T

c (−iωk)Y Tc (−iωl)

4.


The frequencies are positive ωk > 0 and ωl > 0 and therefore Theorem 3.1 statesthat

EY Tc (iωk)Y cT (iωl) =

1T

K(iωk, iωl)

EY Tc (−iωk)Y T

c (−iωl) =1T

K(−iωk,−iωl)

EY Tc (iωk)Y T

c (−iωl) = Φc(iωk)δk,l +1T

K(iωk,−iωl)

EY Tc (−iωk)Y T

c (iωl) = Φc(iωk)δk,l +1T

K(−iωk, iωl)

Hence

EReY Tc (iωk)ReY T

c (iωl) =Φc(iωk)

2δk,l +

1T

K(iωk, iωl)

The other elements of the covariance matrix will follow analogously. 2

According to the theorem above

E

Re Y Tc (iω1)

Im Y Tc (iω1)

Re Y Tc (iω2)

Im Y Tc (iω2)

Re Y Tc (iω1)

Im Y Tc (iω1)

Re Y Tc (iω2)

Im Y Tc (iω2)

T

=

Φc(iω1)2 + 1

T K 1T K 1

T K 1T K

1T K Φc(iω1)

2 + 1T K 1

T K 1T K

1T K 1

T K Φc(iω2)2 + 1

T K 1T K

1T K 1

T K 1T K Φc(iω2)

2 + 1T K

. (3.19)

In the remaining part of the thesis it will be assume that T is large enough thatthe effect of the K terms in expression (3.19) can be ignored. This means that thetruncated Fourier transforms of y at the grid (3.15) are assumed to be independentand Gaussian with the distribution

Y Tc (iωk) ∈ N(0, Φc(iωk)).

Therefore the approximate likelihood function for the values of the truncatedFourier transform

Y Tc (iω1), Y T

c (iω2), . . . , Y Tc (iωNω )

on the grid (3.15) is

p(Y Tc (iω1), Y T

c (iω2), . . . , Y Tc (iωNω )|θ) =

Nω∏

k=1

12πΦc(iωk)

e− |Y

Tc (iωk)|2Φc(iωk) .


if the effects of the term 1T K is ignored. The negative log likelihood function will

be

L(θ) = − log p(Y Tc (iω1), Y T

c (iω2), . . . , Y Tc (iωNω )|θ)

= Nω log 2π +Nω∑

k=1

|Y Tc (iωk)|2Φc(iωk)

+ log Φc(iωk).

Suppose now that a whole continuous time realization y(t) : t ∈ [0, T ] of theoutput of a CAR model is known. Define the continuous-time periodogram of thisoutput as

ˆΦTc (iω) =

∣∣Y Tc (iω)

∣∣2 . (3.20)

where Y Tc is given by (3.16). The approximate Maximum Likelihood procedure

(approximate since the terms proportional to 1T are ignored) for estimating the

parameters is then

θ , arg minθ

V TN (θ, ˆΦT

c ) (3.21)

where

V TN (θ, ˆΦT ) ,

Nω∑

k=1

ˆΦTc (iωk)

Φc(iωk, θ)+ log Φc(iωk, θ) (3.22)

and

Φ(iω, θ) = σ2 |B(iωk)|2|A(iωk)|2 .

Before we close this chapter we would like to recommend the books by Pintelonand Schoukens [Pintelon(2001)] and Brillinger [Brillinger(1981)] which also treatthe issue of identification of noise models.


4Estimation of Power Spectrum

Science is spectral analysis. Art is light synthesis.

Karl Kraus (1874 - 1936)

As mentioned earlier, the parameter estimation method is divided into twostages as illustrated in (3.14). First the power spectrum Φc is estimated and ina second stage the estimate is used to identify the model parameters. The exactmethod for continuous-time identification requires knowledge of the continuous-time realizations of the signals involved via (3.16) and (3.20). This is not practicallypossible since it would require an infinite amount of storage. Instead, the identifierhas to be content with a number of samples distributed uniformly or non-uniformlyin time. This sampling will cause a loss of information since there may be little or noknowledge of what happens in between samples. Therefore it is necessary to resortto interpolation as illustrated in Figure 4.1 in order to obtain an approximation ofa continuous-time realization of the output. It turns out that this approach alsointerpolates the covariance function in two dimensions. The Fourier transform ofthe interpolated realization can then be computed and an estimate of the spectrumcan be formed.

The chapter will be organized as follows. First, in Section 4.1 and 4.2, the effecton and methods for the computation of the Fourier transform from interpolateddata will be discussed. Then, in Section 4.3 the effect of the resulting bias will betreated.

39

40 Chapter 4 Estimation of Power Spectrum

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

Figure 4.1 Piecewise constant and piecewise linear interpolation.

4.1 Interpolation and the Fourier Transform

Let us restate the definition of the Fourier transformation in (3.16) of the continuoustime output yt : t ∈ [0, T ] as

YT (iω) =1√T

∫ T

0

yte−iωtdt. (4.1)

A complicating element in a continuous-time estimation procedure is that we neverhave access to the entire continuous time realization of the output. Instead we have,as pointed out earlier, a finite number of samples of the continuous output yt at timeinstances t1, t2, . . . , tN. Therefore it is in some way necessary to approximate orreconstruct the continuous time realization. In this chapter we reconstruct theoutput as

y(k)(t) =N∑

i=1

ytiφki (t− ti) (4.2)

k ∈ −1, 0, 1 is the order of interpolation and φ is the interpolation kernels. Threetypes of kernels are used in this report. First we introduce

φ(−1)i (t) = (ti+1 − ti)δ(t− ti) (4.3)

which we term “Riemann interpolation”. Next we introduce the piecewise-constantinterpolation

φ(0)i (t) =

0 t < ti

1 ti ≤ t ≤ ti+1

0 ti+1 < t

(4.4)

4.2 Spectral Estimates for Uniform Sampling 41

which goes under the name Zero-Order Hold (ZOH). Finally we have piecewiselinear interpolation

φ(1)i (t) =

0 t < ti−1t−ti−1ti−ti−1

ti−1 ≤ t < titi+1−tti+1−ti

ti ≤ t < ti+1

0 ti+1 ≤ t

(4.5)

which is usually termed First-Order Hold (FOH).If we sample the interpolated output y(t) continuously and perform the Fourier

transform as in (4.1) using (4.2)

Y(k)T (iω) =

1√T

∫ T

0

N∑

i=1

ytiφk

i (t− ti)e−iωtdt.

The ”Riemann interpolation” case will yield a transform

Y(−1)T (iω) =

1√T

N−1∑

k=1

(tk+1 − tk)y(tk)e−iωtk . (4.6)

In the piecewise constant case it will be

Y(0)T (iω) =

1√T

N−1∑

k=1

y(tk)e−iωtk−1 − e−iωtk

iω(4.7)

while in the piecewise linear case we have

Y(1)T (iω) =

1√T

1iω

(y(t1)e−iωt1 − y(tN )e−iωtN ) (4.8)

+1√T

1(iω)2

N−1∑

k=1

y(tk+1)− y(tk)tk+1 − tk

(e−iωtk − e−iωtk+1).

This might seem as an awkward and expensive way of computing the Fourier trans-form. If we however do this on-line, we can restrict us to a time window of size Tand just remove old data as new samples arrive.

4.2 Spectral Estimates for Uniform Sampling

In the case of uniform sampling tk = kTs with N = T/Ts samples the truncatedFourier transform of the interpolated data will be

Y(k)T (iω) =

1√T

∫ T

0

N∑

i=1

ykTsφ(k)i (t− kTs)e−iωtdt

= Fk(iω)1√N

Yd(eiωTs) (4.9)


where

Fk(iω) =1√Ts

∫ ∞

−∞φ(k)(t)e−iωtdt. (4.10)

and the discrete-time Fourier transform is

Yd(eiωTs) =1√N

N∑

k=1

ykTse−iωkTs . (4.11)

In connection with this expression we would like to mention that when we movefrom discrete-time to continuous-time the Nyquist frequency seize to exist. Sincewe have made an assumption about the intersample behavior, frequencies abovethe Nyquist frequency can also carry information.

If we define the discrete-time periodogram as

ˆΦNd (eiωTs) = |Yd(eiωTs)|2

the continuous-time periodogram estimate can be written as

ˆΦT,(k)c (iω) = |Fk(iω)|2 ˆΦd(eiωTs) (4.12)

for uniform sampling and interpolated data.The ”Riemann interpolation” will be proportional to the ordinary discrete-time

Fourier transform as

Y(−1)T (iω) =

√Ts√N

N∑

k=1

y(kTs)e−iωkTs =√

TsYd

(eiωTs

).

It is therefore related to the discrete-time power spectrum Φd(eiωTs) defined in(3.7) as

limT→∞

EˆΦT,(−1)

T (iω) = limT→∞

E|Y (−1)T (iω)|2 = Φd(eiωTs) =

∞∑

k=−∞Φc(iω + i

2π

Tsk).

where Φc(iω) is the continuous-time spectrum as defined in (3.5)[Wahlberg(1988)].Here the relationship

limT→∞

E|Y (−1)T (iω)|2 = Φd(eiωTs)

follows from Lemma 6.2 in [Ljung(1999)] while the last equality follows from (3.8).Zero-order hold and first order hold sampling on the other hand will provide

filtered versions of the discrete-time Fourier transform.

4.2 Spectral Estimates for Uniform Sampling 43

Lemma 4.1Let F0 and F1 be given by (4.10), (4.4) and (4.5). Then

F0(iω) = Ts sinc(

ωTs

2

)e−

ωTs2 − π

Ts< ω <

π

Ts

F1(iω) = Ts sinc2

(ω

Ts

2

)

where

sincω =sin ω

ω

Proof Since for piecewise constant

φ(0)(t) =∫ t

−∞(δ(τ)− δ(τ − Ts))dτ

the Fourier transform will be

F0(iω) =1− e−iωTs

iω

= Ts sinc(

ωTs

2

)e−

ωTs2 .

On the other hand

φ(1)(t) =∫ t

−∞

∫ t1

−∞

(δ(t2 + Ts)

Ts− 2

δ(t2)Ts

+δ(t2 − Ts)

Ts

)dt2dt1

with the transform

F1(iω) =eiωTs − 2 + e−iωTs

Ts(iω)2

=

(ei ωTs

Ts − e−i ωTs2

)2

Ts(iω)2

= Ts sinc2

(ω

Ts

2

)

2

This result automatically translates to the spectrum where the expected value ofthe continuous-time periodogram in (3.20) will approach

limT→∞

EˆΦT,(0)

c (iω) = sinc2

(ω

Ts

2

) ∞∑

k=−∞Φc(iω + i

2π

Tsk)

limT→∞

EˆΦT,(1)

c (iω) = sinc4

(ω

Ts

2

) ∞∑

k=−∞Φc(iω + i

2π

Tsk).


The objective of the interpolation is to make the estimate consistent with thecontinuous-time power spectrum. Therefore we would like to remove the effectsof folding. By folding we mean that the discrete-time spectrum/Fourier transformconsists of the continuous-time spectrum/Fourier transform plus frequency shiftedor ”folded” versions of the continuous-time spectrum/Fourier transform. In viewof what has been said earlier with respect to interpolation and filtering we realizethat the optimal interpolation kernel φOPT should have a Fourier transform withsquare amplitude

|FOPT (iω)|2 =Φc(iω)

∞∑k=−∞

Φc(iω + i 2πTs

k).(4.13)

where Φc is the continuous-time power spectrum for the true parameters. In Figure4.2 we have compared the spectrum of the optimal kernel with ZOH and FOH.From this perspective we notice that FOH as an approximation of the optimal

−300 −200 −100 0 100 200 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ω

|FOPT

|2

|FZOH

|2

|FFOH

|2

Figure 4.2 Absolute square of the Fourier transforms of the interpolationkernels for Ts = 0.1. The optimal kernel is for a second ordermodel where A(p) = p2 + 2p + 1, B(p) = 1 and σ = 1.

kernel seems to be worse than ZOH for the second order CAR model used in thefigure.

4.3 Effects of Interpolation on Spectral Estimates 45

4.3 Effects of Interpolation on Spectral Estimates

The approximation y(k)(t) of a realization of the stationary process yt will have thefollowing second order properties

m(k)(t) , E[y(k)(t)] = 0

r(k)(t, s) , E[y(k)(t)− m(k)(t)][y(k)(s)− m(k)(s)] (4.14)

=N∑

i=1

N∑

j=1

Ey(ti)y(tj)φ(k)i (t− ti)φ

(k)j (t− tj) (4.15)

=N∑

i=1

N∑

j=1

r(ti − tj)φ(k)i (t− ti)φ

(k)j (s− tj) (4.16)

Where r is the covariance function defined in (3.6).An interesting question is how well r(k) approximates r. From the representa-

tion in (4.16) we see that piecewise constant interpolation of y(t) will result in apiecewise constant interpolation of r(t − s). Piecewise linear interpolation of y(t)on the other hand, will result in piecewise bilinear interpolation of r(t− s).

Interpolation in Two Dimensions

In the following lemma we explain how we can bound the interpolation error.

Lemma 4.2Let f : R2 → R be a continuous once differentiable function with bounded first

derivatives. Assume that the function is interpolated by a constant function f :R2 → R in a rectangle Ωij = x, y : 0 ≤ x ≤ hi, 0 ≤ y ≤ hj. Then we have forsome constant C > 0

max(x,y)∈Ωij

|f(x, y)− f(x, y)| ≤ Chmax (4.17)

If we interpolate f with a bilinear polynomial and assume it is twice differentiablewith bounded second derivatives we get for some C > 0

max(x,y)∈Ωij

|f(x, y)− f(x, y)| ≤ Ch2max (4.18)

Proof Both results come from Taylor’s theorem. In the case of constant interpo-lation we have

f(x, y)− f(0, 0) = fx(0, 0)x + fy(0, 0)y +O(x2 + y3)

In the bilinear case we have the interpolant

f(x, y) =f(0, 0) +f(h1, 0)− f(0, 0)

h1x +

f(0, h2)− f(0, 0)h2

y

+f(h1, h2)− f(h1, 0)− f(0, h2) + f(0, 0)

h1h2xy


Define f(x, y) = f(x, y)− f(x, y). Then

f(x, y) =f(0, 0) + fx(0, 0)h1 + fy(0, 0)h2

+12

(xy

)T (fxx(0, 0) fxy(0, 0)fyx(0, 0) fyy(0, 0)

)(xy

)+O(

√x2 + y2

3)

Now

f(0, 0) = 0

fx(0, 0) = fx(0, 0)− f(h1, 0)− f(0, 0)h1

= O(h21)

fy(0, 0) = O(h22)

Similarly we have

fxx(0, 0) = fxx(0, 0)

fyy(0, 0) = fyy(0, 0)

fxy(0, 0) = fxy(0, 0)− f(h1, h2)− f(h1, 0)− f(0, h2) + f(0, 0)h1h2

=O(

√h2

1 + h22

3)

h1h2.

Thus we have

f(x, y) =O(h21)x +O(h2

2)y +12

(xy

)T fxx(0, 0) O(

√h21+h2

23)

h1h2

O(√

h21+h2

23)

h1h2fyy(0, 0)

(xy

)

+O(√

x2 + y23)

From this we get the inequality

|f(x, y)| ≤ 12|fxx(0, 0)|h2

1 +12|fyy(0, 0)|h2

2 +O(√

h21 + h2

2

3

)

≤ |fxx(0, 0)|+ |fyy(0, 0)|2

h2max +O(h3

max)

2

This result can be used to bound the covariance function approximation error andits contribution to the bias in the periodogram.

Covariance Function Interpolation Error

From Lemma 4.2 we can now prove that the interpolation error in the whole areaΩ = t, s : 0 ≤ t ≤ T, 0 ≤ s ≤ T is bounded. Let

hmax = max1≤i<Nt−1

(ti+1 − ti). (4.19)

Then we have the following result.


Corollary 4.1Assume that r is bounded, continuous and has bounded first derivatives. Let r(0)

be defined as in (4.14). Then

maxt,s∈[0,T ]

∣∣∣r(t− s)− r(0)(t, s)∣∣∣ ≤ Chmax (4.20)

Proof Follows directly from Lemma 4.2. 2

In the case of piecewise linear interpolation of yt there will be a bilinear interpola-tion of r(t− s) where the approximation error is given by the following lemma

Lemma 4.3Assume that ry(t, s) is bounded, continuous and has bounded second derivatives.

Let rT,1y be defined as in (4.14). Then

maxt,s∈[0,T ]

∣∣∣r(t− s)− r(1)(t, s)∣∣∣ ≤ Ch2

max (4.21)

Proof Follows directly from Lemma 4.2. 2

We will now continue to treat the bias in the periodogram.

4.3.1 Periodogram Bias

Let us define the approximate continuous-time periodogram as

ˆΦT,(k)c (iω) =

∣∣∣Y (k)T (iω)

∣∣∣2

(4.22)

when the signal y is interpolated. This entity, which is an estimate of the powerspectrum, will be biased due to interpolation and limited time of observation T .This bias, as we will see later on, translates into the bias in the parameter estimates.We are therefore interested in controlling the size of the quantity

EˆΦT,(k)

c (iω)− Φc(iω)

Interpolation Contribution to the Periodogram Bias

The following expression relates the bias in the approximate periodogram to theerror from the interpolation of the covariance function.

Lemma 4.4Let y(t) be a stationary stochastic process. Let

ˆΦTc (iω) be defined by (3.20) and

letˆΦT,(k)

c (iω) be defined by (4.22). Then, for

∆Φ1 =∣∣∣E

[ ˆΦT,(k)c (iω)− ˆΦT

c (iω)]∣∣∣ (4.23)


we have∆Φ1(iω) ≤ C1 max

t,s∈[0,T ]

∣∣r(t− s)− rT,k(t, s)∣∣ , ∀ω (4.24)

Proof From the definitions we get

∆Φ1 =∣∣∣E|Y T,(k)

c (iω)|2 − E|Y Tc (iω)|2

∣∣∣

=

∣∣∣∣∣1T

∫ T

0

∫ T

0

(r(k)(t, s)− r(t− s)

)eiω(t−s)dtds

∣∣∣∣∣

≤ 1T

∫ T

0

∫ T

0

∣∣∣r(k)(t, s)− r(t− s)∣∣∣ dtds.

Let ∆r(t, s) =∣∣r(k)(t, s)− r(t− s)

∣∣ and make a change of variables. Then we get

1T

∫ T

0

∫ T

0

∆r(t, s)dtds =1T

∫ T

0

∫ T−t

−t

∆r(t, t + τ)dτdt

=1T

∫ T

0

∫ minα,T−t

max−α,−t∆r(t, t + τ)dτdt

+1T

∫ T

α

∫ −α

−t


+1T

∫ T−α

0

∫ T−t

α


These terms can be separately bounded. For the first term we have

1T

∫ T

0

∫ minα,T−t

max−α,−t∆r(t, t + τ)dτdt ≤ 1

T

∫ T

0

∫ minα,T−t

max−α,−tChm

maxdτdt

≤ αChmmax

For some α > 0 and λ > 0 we have

∆r(τ) ≤ |r(τ)|+ |r(t, t + τ)| < e−λ|τ |.

The second term can then be bounded the following way

1T

∫ T

α

∫ −α

−t

∆r(t, t + τ)dτdt ≤ 1T

∫ T

α

∫ −α

−t

e−λ|τ |dτdt

≤∫ −α

−∞e−λ|τ |dτ.

We then choose α > 0 such that

αChmmax ≥

∫ −α

−∞e−λ|τ |dτ

and we have our result. 2


Leakage Contribution to Bias

Since we only observe our process during a finite time interval [0, T ] the expectedperiodogram and the spectrum will be slightly different. The following lemma byLjung [Ljung(1999)] quantifies this difference.

Lemma 4.5Let y(t) be a stationary stochastic process with spectrum Φc and let

ˆΦTc (iω) be

defined by (3.20), then

∆Φ2(iω) =∣∣∣E ˆΦT

c (iω)− Φc(iω)∣∣∣ ≤ C2

T, ∀ω (4.25)

Proof Lemma 6.1 in [Ljung(1999)] adopted to continuous-time.

EYT (iω)YT (−iξ) =1T

∫ T

0

∫ T

0

Ey(r)y(s)eiω(ωr−ξs)dsdr

=1T

∫ T

0

∫ T

0

r(t− s)e−iω(ωt−ξs)dsdt

=1T

∫ T

0

e−i(ω−ξ)t

∫ t

t−T

r(τ)e−iξτdτdt

Now

∫ t

t−T

r(τ)e−iξτdτ = Φy(iξ)−∫ t−T

−∞r(τ)e−iξτdτ

−∫ ∞

t

R(τ)e−iξτdτ

and

1T

∫ T

0

e−i(ω−ξ)tdt =

1, if ω = ξ

0, if (ω − ξ) = 2πT k, k = ±1,±2, . . . ,±∞

Consider∣∣∣∣∣1T

∫ T

0

e−i(ω−ξ)t

∫ t−T

−∞r(τ)e−iξτdτdt

∣∣∣∣∣ ≤1T

∫ T

0

∫ t−T

−∞|r(τ)| dτdt

≤ 1T

∫ 0

−∞|τ | |r(τ)| dτ ≤ C

T

provided

∫ ∞

−∞|τry(τ)| dτ


Similarly

∣∣∣∣∣1T

∫ T

0

e−i(ω−ξ)t

∫ t−T

−∞r(τ)e−iξτdτdt

∣∣∣∣∣ ≤1T

∫ ∞

0

|τ | |r(τ)| dτ ≤ C

T

2

Power Spectrum Bias Expression

The previous two lemmas can now be used to estimate the difference between theexpected approximate periodogram and the spectrum

Theorem 4.1Let

∆Φ(iω) =∣∣∣E ˆΦT,(k)

c (iω)− Φc(iω)∣∣∣

where k = 0, 1 and

hmax = max1≤i≤N−1

ti+1 − ti

Then

∆Φ(iω) ≤ C1hk+1max +

C2

T, ∀ω (4.26)

where C1 and C2 are positive numbers.

Proof By Lemma 4.5, Lemma 4.4 and the triangle inequality the result

∆Φ =∣∣∣E ˆΦT,k − Φ

∣∣∣≤

∣∣∣E ˆΦT,k − EˆΦT + E

ˆΦT − Φ∣∣∣

≤∣∣∣E

[ ˆΦT,k − ˆΦT]∣∣∣ +

∣∣∣E[ ˆΦT − Φ

]∣∣∣≤ C1 maxr,s∈[0,T ]

∣∣∣RT,k(r, s)−R(r − s)∣∣∣ + C2

T

will follow. 2

4.4 Conclusions

In this chapter estimation of the continuous-time spectrum from sampled datahas been treated. The main approach has been the use of piecewise-constant andpiecewise-linear interpolation in order to bridge the gap between the continuousand discrete domains. Improved estimates of the continuous-time spectrum will bethe topic of the rest of this thesis, but before we move to this are issues concerningbias and variance of parameter estimates will be treated in the following chapter.

5Properties of Bias and Variance

”Compromise, if not the spice of life, is its solidity. It is what makesnations great and marriages happy.”

Phyllis McGinley (1905-1978)

In some practical identification applications memory and computational re-sources are scarce. The amount and quality of available data can also vary fromsituation to situation. This makes tools and rules of thumb valuable if they canmake the procedure of designing identification algorithms easier. The followingchapter includes a set of such qualitative tools aiding the user in his efforts. Forinstance selecting the set of frequencies to be used in the frequency-domain identi-fication in order to minimize the impacts of errors while maximizing the precisionof the estimates.

The chapter will be organized as follows. First, theoretical asymptotic expres-sions for the bias and variance of parameter estimates are derived in Section 5.1.Then, in Section 5.2 more practical rules of thumb are explained and illustrated bya set of examples. Finally, in Section 5.3, an example which illustrates of the biasand variance tradeoff is given.

In the chapter a number of simplifications with respect to notations are made.The subscripts denoting continuous-time will be omitted in for example ˆΦc yieldingˆΦ. Superscripts indicating degree of interpolation and time of observation are alsodropped.

51

52 Chapter 5 Properties of Bias and Variance

5.1 Asymptotic Expressions for Bias and Variance

In this section, it will be shown how the bias and variance of the parameter esti-mates obtained by the method in (3.21) will be related to the bias and variance ofˆΦ. In the case of bias the result will be that

E(θ − θ0) ≈Nω∑

k=1

S(iωk)∆Φ(iωk). (5.1)

where

∆Φ(iωk) =E

ˆΦ(iωk)− Φ(iωk, θ0)Φ(iωk, θ0)

. (5.2)

denotes the the relative bias of the periodogram. In this expression we also have

S(iωk) = Ψ(θ0, Φ)−1Ψk(θ0, Φ) (5.3)

where

Ψ(θ0,Φ) =Nω∑

k=1

Ψk(θ0, Φ)Ψk(θ0,Φ)T (5.4)

and

Ψk(θ0, Φ) =Φ′θ(iωk, θ0)Φ(iωk, θ0)

. (5.5)

where the last quantity is called the relative sensitivity. In the formulas above θdenotes the estimated parameters, θ0 denotes the true parameters and Φ′θ is thegradient of the spectrum with respect to the parameters in θ.

In the case of variance we resort to asymptotic reasoning where

E(θ − θ0)(θ − θ0)T → Ψ(θ0, Φ)−1

when the time of observation goes to infinity and the sampling intervals convergeto zero In the subsections below we derive these expressions for the derivatives,bias and variance.

5.1.1 Derivatives

In both the expressions for the bias and the asymptotic variance second orderderivatives with respect to the parameters are needed.

5.1 Asymptotic Expressions for Bias and Variance 53

Lemma 5.1Let V be defined as in (3.22).Then

V ′′θθ(θ,

ˆΦ) =Nω∑

k=1

Φ′′θθ(iωk, θ)− Φ′θ(iωk, θ)Φ′θ(iωk, θ)T

Φ(iωk, θ)2

(1−

ˆΦ(iωk)Φ(iωk, θ)

)

+Nω∑

k=1

Φ′θ(iωk, θ) ˆΦ(iωk)Φ′θ(iωk, θ)T

Φ(iωk, θ)3

Proof From the definition in (3.22) we have

V ′θ (θ, ˆΦ) =

Nω∑

k=1

Φ′θ(iωk, θ)Φ(iωk, θ)

(1−


)

and therefore

V ′′θθ(θ,

ˆΦ) =Nω∑

k=1

Φ′′θθ(iωk, θ)− Φ′θ(iωk, θ)Φ′θ(iωk, θ)T

Φ(iωk, θ)2

(1−


)

+Nω∑

k=1

Φ′θ(iωk, θ) ˆΦ(iωk)Φ′θ(iωk, θ)T

Φ(iωk, θ)3

2

We also take a look at the derivatives of the criterion V with respect to the pa-rameters and the information from the estimated spectrum. This is summarizedin the following lemma

Lemma 5.2Let

ˆΦk = ˆΦ(iωk) and V be defined as in (3.22), then

V ′′θˆΦk

(θ0,ˆΦ) = −Φ′θ(iωk, θ0)

Φ(iωk, θ0)2

Proof As in the previous lemma we have

V ′θ (θ, ˆΦ) =

Nω∑

k=1

Φ′θ(iωk, θ)Φ(iωk, θ)

(1−


)

and hence

V ′′θˆΦk

(θ, ˆΦ) = −Φ′θ(iωk, θ)Φ(iωk, θ)2

2


5.1.2 Bias Expression

If we define the estimated parameters as

θ , arg minθ

V (θ, ˆΦ)

and the true

θ0 , arg minθ

V (θ, Φ) (5.6)

we get the following result for the bias.

Theorem 5.1Let V be defined as in (3.22). Then

E(θ − θ0) ≈ V ′′θθ(θ0,Φ)−1

Nω∑

k=1

V ′′θΦk

(θ0, Φ)(Φ(iωk, θ0)− E

ˆΦ(iωk))

. (5.7)

Proof From the definition in (3.14) we have

V′θ (θ, ˆΦ) = 0

V′θ (θ0, Φ) = 0

A Taylor expansion of V′θ (θ, Φ) around θ0 and Φk = Φ(iωk) yields

V′θ (θ, ˆΦ) ≈ V

′θ (θ0, Φ) + V

′′

θθ(θ0, Φ)(θ − θ0)

+Nω∑

k=1

V ′′θ,Φk

(θ0, Φ)( ˆΦ(iωk)− Φ(iωk)

).

From this we get

θ − θ0 ≈ (V ′′θθ(θ0, Φ))−1

Nω∑

k=1

V ′′θΦk

(θ0, Φ)(Φ(iωk, θ0)− ˆΦ(iωk)

).

By taking expectations on both sides we get the conclusion. 2

From Lemmas 5.1 and 5.2 it then follows that

V ′′θθ(θ0, Φ) =

Nω∑

k=1

Φ′θ(iωk, θ0)Φ(iωk, θ0)

(Φ′θ(iωk, θ0)Φ(iωk, θ0)

)T

and

V ′′θΦk

(θ0, Φ) = −Φ′θ(iωk, θ0)Φ(iωk, θ0)2

.

5.1 Asymptotic Expressions for Bias and Variance 55

Using the definitions of ∆Φ in (5.2), S in (5.3), Ψ in (5.4) and Ψk in (5.5) we canwrite

E(θ − θ0) ≈Nω∑

k=1

S(iωk)∆Φ(iωk).

5.1.3 Variance Expressions

Using the expression for bias developed in the previous subsection, it is now possibleto develop an expression for the asymptotic variance. Since

θ − θ0 ≈ (V ′′θθ(θ0, Φ))−1

Nω∑

k=1

V ′′θΦk

(θ0, Φ)( ˆΦ(iωk)− Φ(iωk, θ0)

).

we have

E(θ − θ0)(θ − θ0)T ≈Nω∑

k=1

Nω∑

l=1

Σ(iωk, θ0)E[δΦ(iωk, θ0)δΦ(iωl, θ0)]Σ(iωl, θ0)T

where

Σ(iωk, θ0) = V ′′θθ(θ0,Φ)−1V ′′

θΦk(θ0, Φ)

and

δΦ(iωk, θ0) = ˆΦ(iωk)− Φ(iωk, θ0)

In analogy with Lemma 6.2 in [Ljung(1999)]

EδΦ(iωk, θ0)δΦ(iωl, θ0) →

Φ(iωk, θ0)2 if k = l

0 if k 6= l.

when T → ∞ and the time interval between samples approach zero. This meansthat

Nω∑

k=1

Nω∑

l=1

Σ(iωk, θ0)E[δΦ(iωk, θ0)δΦ(iωl, θ0)]Σ(iωl, θ0)T

→Nω∑

k=1

Σ(iωk, θ0)Φ(iωk, θ0)2Σ(iωk, θ0)T .

Expanding Σ and using the definition of V ′′θΦk

in Lemma 5.2 we get

Nω∑

k=1

Σ(iωk, θ0)Φ(iωk, θ0)2Σ(iωk, θ0)T

=Nω∑

k=1

V ′′θθ(θ0, Φ)−1 Φ′θ(iωk, θ0)

Φ(iωk, θ0)2Φ(iωk, θ0)2

Φ′θ(iωk, θ0)T

Φ(iωk, θ0)2V ′′

θθ(θ0, Φ)−1.

= V ′′θθ(θ0,Φ)−1


where the last equality follows from the fact that

V ′′θθ(θ0, Φ) =

Nω∑

k=1


Φ′θ(iωk, θ0)T

Φ(iωk, θ0).

Hence if we use the definition of Ψ in (5.4)

E(θ − θ0)(θ − θ0)T → Ψ(θ0, Φ)−1 (5.8)

as T →∞ and the maximum time interval between samples goes to zero.

5.2 Practical Considerations for Frequency Selec-tion

So far we have said very little about which frequencies to use in the criterion (3.22)in order to make a tradeoff-between bias and variance. We have only pointed outearlier that we are restricted to the frequencies

ωk =2π

Tk, k ∈ Z

where k = 1, . . . , Nω. As will be explained below it is important to avoid high fre-quencies, because they will produce a considerable parameter bias if the estimatedspectrum is biased. We will also explain why some frequencies have a significanteffect on the bias while others have very little effect

It is intuitive that as many frequencies as possible should be used if the peri-odogram estimate is unbiased. More information reduces the uncertainty. At highfrequencies the periodogram is however often biased due to the use of discrete mea-surements. Including high frequencies will then reduce the quality of the estimatesand a bias-variance tradeoff has to be made.

5.2.1 Minimizing the Variance

According the asymptotic expression in (5.8) the variance is roughly inversely pro-portional to the quantity

Ψ(θ0,Φ) =Nω∑

k=1

Ψk(θ0, Φ)Ψk(θ0,Φ)T

where

Ψ(θ0,Φ) =Φ′θ(iω, θ0)Φ(iω, θ0)

5.2 Practical Considerations for Frequency Selection 57

is called the relative sensitivity. For a CAR model where θ = [aT σ2]T , a =[a1 . . . an]T and

Φ(iωk, θ) =σ2

|A(iωk, θ)|2

we have

Ψk(θ, Φ) =1

|A(iωk)|2

− ∂∂a1

|A(iωk)|2...

− ∂∂an

|A(iωk)|2A(iω)|2

σ2

In Figure 5.1 we show the first two elements of Ψk(θ0, Φ) for a CAR model fordifferent ωk. From the figure we see that these quantities are only significantly

−5 0 5−1

−0.5

0

ω

Φa 1/ Φ

−5 0 5−2

−1

0

1

ω

Φa 2/ Φ

Figure 5.1 Relative sensitivity for a1 (upper) and a2 (lower) for a CARmodel. Here A(p) = p2 + a1p + a2, a1 = 2 and a2=1.

different from zero on a small frequency interval. In theory we should includeas many frequencies as possible. Sometimes it can however be costly to computeperiodogram estimates. In that case a rule of thumb would be to give priority tofrequencies where the magnitude of the elements of Ψk(θ0,Φ) are large in order tominimize the variance of the parameter estimates.


5.2.2 Minimizing the Bias

From expression (5.7) we know that the contribution to the bias from each indi-vidual frequency component of the power spectrum is roughly proportional to

(Nω∑

k=0

(Φ′θ(iωk, θ0)Φ(iωk, θ0)

)(Φ′θ(iωk, θ0)Φ(iωk, θ0)

)T)−1 Nω∑

k=0


EˆΦ(iωk)− Φ(iωk, θ)

Φ(iωk, θ0).

In Figure 5.2 the term

EˆΦ(iωk)− Φ(iωk, θ)

Φ(iωk, θ0).

is plotted as a function of the frequency ω. The system is the same as in the pre-vious figure and the output is uniformly sampled with sampling interval Ts = 0.5.The solid line is the relative spectral bias for the Discrete-Time Fourier Transform.The dotted line indicates the same bias for ZOH interpolation and the dash-dottedfor FOH. The contribution from an individual frequency to the total bias is pro-

−6 −4 −2 0 2 4 6−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

ω

Figure 5.2 Relative spectral bias for an uniformly sampled CAR model.Here A(p) = p2 + 2p + 1, σ = 1 and Ts = 0.5. Here the solidline is ”Riemann” interpolation while the dotted is for ZOHhold and the dash-dotted is for FOH. All curves rise sharplyfor high frequencies even if this is not evident for ZOH andFOH in the figure.

portional to

Φ′θ0(iωk, θ0)

Φ(iωk, θ0)E

ˆΦ(iωk)− Φ(iωk, θ)Φ(iωk, θ0)

.

In Figure 5.3 we illustrate the contribution to the bias as a function of ω for thesame system and circumstances as above.

5.3 Numerical Experiments 59

−6 −4 −2 0 2 4 6−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

ω−6 −4 −2 0 2 4 6

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

ω

Figure 5.3 Bias contribution for each frequency for a1 (left) and a2 (right).The model is uniformly sampled and of CAR type. Here A(p) =p2 + a1p + a2, σ = 1, a1 = 2, a2 = 1 and Ts = 0.5. Here thesolid line is ”Riemann” interpolation while the dotted is forzero-order hold and the dash-dotted is for first order hold.

5.2.3 Re-Parametrization

Consider the second order CAR model

yt =σ

p2 + a1p + a2et. (5.9)

We can also parameterize this model as

yt =ω2

0

p2 + 2ζω0p + ω20

σet (5.10)

where ζ and ω0 are the damping ratio and the undamped natural frequency.For this special parametrization the relative sensitivity functions with respect

to respective parameters are shown in Figure 5.4. Here we have chosen a1 = 2and a2 = 2 in order to get a more pronounced resonance peak. This means thatw0 = 1/

√2 and ζ =

√2. From this figure, we conclude that the relative damping

ζ is sensitive near the natural resonance frequency ω0 of the system. The naturalfrequency ω0 on the other hand is sensitive at high frequencies.

5.3 Numerical Experiments

In Figure 5.5 we have estimated the parameters in the model in (5.9) where a1 = 2and a2 = 1 and σ = 1. The time of observation was T = 200 and the samplinginterval was Ts = 0.1. First we used the frequency span 2π 1

200 , . . . , 2π 200200 and

indicated those with dots. In the second case we used 2π 8200 , . . . , 2π 208

200, indicatedby plus signs. We used the same number of frequencies, but a slightly different set.As can be seen from the picture this doubled the variance of the a2 parameterestimate. Standard deviation for a2 is 0.0201 in the first case which is indicated


0

−5 0 5−3

−2

−1

0

ω

Φζ/Φ

−5 0 50

1

2

3

ω

Φω

0/ Φ

Figure 5.4 Relative sensitivities for ζ (upper) and ω0 (lower). Here A(p) =p2 + 2ζω0p + ω2

0 , σ = ω0 and ζ = 1/√

2 and ω0 =√

2.

by dots. In the second case which is indicated by plus signs we have 0.0491 . Thereason is as we can see from Figure 5.1 that the parameter a2 is especially sensitiveat low frequencies.

While there are negative effects of the exclusion of low frequencies the use ofhigh frequencies can be equally detrimental. This is illustrated in Figure 5.6 wherewe have used frequencies up to the one indicated by the x-axis. The reason for thelarge RMSE

RMSE =

√√√√ 1NMC

NMC∑

j=1

(θ − θ0)2 (5.11)

at high frequencies is that there is a very large relative bias at these frequencies.This suggests that there is a tradeoff to make, at least in theory. In order to reducethe variance we should include higher and higher frequencies. On the other handthere will be bias at higher frequencies due to discretization effect. Therefore, asillustrated in Figure 5.6, there will be a point where including higher frequencieswill increase the RMSE of the parameter estimates.

5.4 Summary

In this chapter the focus has been on bias and variance issues related to the parame-ter estimates. First, variance and bias were proved to rely on the relative sensitivityof the parameters and relative bias of the spectral estimate. Then, practical con-siderations concerning which frequencies to use in the estimation procedure were

5.4 Summary 61

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

a1

a 2

Figure 5.5 Parameter estimate dependence on frequency choice. Parame-ters denoted by plus signs are identified using slightly differentfrequencies than those indicated by dots. Thereby causing atwice as large standard deviation in a2 for the plus sign esti-mates. Here NMC = 200, T = 200, Ts = 0.1.

discussed. Finally, the discussions in the previous sections were illustrated by twoexamples. The first one showing that some frequencies are more valuable when itcomes to reducing the bias of certain parameters. The other example illustratedthe bias-variance tradeoff which is connected to the selection of frequencies.


0 10 20 30 40 50 600

2

4

6

8

10

12

ω

RM

SE

Figure 5.6 RMSE for parameter estimates for a second order CAR modelwith respect to maximum frequency. Here NMC = 200, T =200, Ts = 0.1.

6Application to Estimation of Tire

Pressure

If real is what you can feel, smell, taste and see, then ’real’ is simplyelectrical signals interpreted by your brain.

Morpheus, The Matrix (1999)

That an accurate tire pressure is important is probably known to almost every-one with a driver’s licence. The wrong pressure can make the vehicle one drivesvery difficult to handle and induce an unnecessary amount of tire wear. It can evencause deadly accidents if the tire breaks down due to low pressure when the vehicletravels at high speed.

A way to monitor the tire would be to attach an electro-mechanical device toit which would simultaneously measure the pressure and display the value to thedriver. This is however a costly solution since it includes additional hardware.Increased cost due to extra hardware is not an attractive option in these daysof overcapacity in the automobile industry. Therefore, innovators are left withextracting as much data as possible from existing measurements. One importantsource of information is the wheel speed measurements used in connection with theAnti-lock Braking System (ABS). As pointed out in Chapter 1 a possible way toextract more information out of existing data is by using a model. It also requiresan understanding of the physics of the tire, i.e. a physically parameterized model.

In Chapter 3 and Chapter 4 a theoretical framework for system identificationof continuous-time noise models has been constructed. The methods developedin these chapters were then analyzed in Chapter 5 with respect to the quality ofthe estimates. The focus in the following chapter will be on applications of thesemethods to the estimation of tire pressure or axel vibrations.

63

64 Chapter 6 Application to Estimation of Tire Pressure

6.1 Tire Pressure Modelling

In this chapter the tire is modelled as a spring-damper system in the torsionaldirection of the tire [Wong(1993)]. Road irregularities which can be modelled as aforce with a white normal noise characteristics excite vibrations in the tire whichaffect the wheel speed directly. Sensors measuring the velocity of the wheel canthen pick up these vibrations which are usually in the area of 0-100 Hz for a normalcar.

Certain vibrations in the range of 30 - 60 Hz can be modelled as a mass-dampersystem which motivates the following relationship between the torsional oscillationsy(t) and the effect from road irregularities e(t)

y(t) +b

my(t) +

k

my(t) = e(t)

where m is the mass of a piece of the tire rubber rim and k is a spring constantconnected to the elasticity of the material. The model can also be rewritten as

y(t) + 2γy(t) + ω20(t) = e(t)

where

γ =b

2m

ω0 =

√k

m.

and therefore the oscillations can be modelled as

y(t) = H(p)e(t) =1

p2 + 2γp + ω20

e(t). (6.1)

The tire pressure will affect the spring constant and the damping coefficient andhence also the resonance peak of the system. Tracking changes in this peak willthen make it possible to track the tire pressure.

The continuous-time spectrum of the oscillatory process will be

ΦH(iω) =σ2

|H(iω)|2 =σ2

(ω2 − ω20)2 + 4γ2ω2

(6.2)

where σ is the standard deviation of the driving white noise. This means that thelocation of the resonance peak is

ωres =√

ω20 − 2γ2. (6.3)

The original model in (6.1) is given in continuous-time, and the objective will beto estimate the continuous-time parameters γ and ω0. Then, an estimate of theresonance frequency ωres can be aquired through (6.3) [Persson(2002)].

6.2 Problem Specifics and Objectives 65

6.2 Problem Specifics and Objectives

A characteristic problem with signal processing using tachometer measurementson rotating axles is that the measurements are uniform in the angle domain butnon-uniform (speed dependent) in the time domain. This comes from the fact thatmost common sensors for such applications measure the time between certain angledisplacements, which is thus speed dependent. One can for instance illustrate thiswith the ABS sensors in a car, which give between 50 and 100 pulses per revolu-tion of each wheel. If tire vibration analysis and other similar problems are to beapproached in the time sampled domain, either one has to rely on data interpola-tion to uniform time sampling, or derive dedicated algorithms. Motivated by thedevelopments in previous chapters a frequency domain approach to the problemis presented and compared theoretically to a time domain algorithm proposed in[Persson(2002), Persson(2001)].

The main specifications on a procedure aimed at high-sensitivity tire vibrationanalysis are the following:

1. Being based on parametric physical models of the vibrations such as the onedescribed in (6.1).

2. Operate on short data batches in a pre-specified speed interval where thedata pass several quality checks.

3. Potential to efficiently reject wide band disturbances that are non-interferingwith the important vibrations.

4. Potential to reject narrow band disturbances that are interfering with thevibrations.

The method given in [Persson(2002), Persson(2001)] successfully solves the firstthree specifications, but not the last one. The method proposed here, on theother had, has the potential to be modified to eliminate outliers in the frequencynarrow band disturbance. This general disturbance problem occurs in several otherapplications as in an automotive drive-line where the vibration indicates engineknocks, or in robotics where vibrations come from the load, just to mention afew. This chapter will however focus on Point 1, 2 and 3 in the list above. Inthe subsection below, there will only be a short discussion of time and frequency-domain methods with respect to Point 4.

6.3 Comparing Time and Frequency Domain Ap-proaches

Table 6.1 summarizes the notation and signal models that are used in the timeand frequency domain. Basically, the tire vibration analysis is approached bythe continuous-time autoregressive (CAR) model A(p; θ) motivated by the spring-damper model of the axel in (6.1). Superimposed on this signal are other vibrations


and external disturbances d(t), and the speed signal itself. Measurements are takeneach time tk a pulse is recieved from the ABS signal. These pulses represent acertain fixed angle displacement, which explains the special appearance of y[k] =y(tk) in Table 6.1.

Time domain Frequency domain

y(t) =1

A(p; θ)e(t) + d(t)

y[k] = y(tk) =2πk

L+

∫ tk

tk−1

d(t)dt

e(t) white noise,A(p; θ) AR model,θ parameters in the AR model,d(t) disturbance,y[k] measured non-uniform samples ofangle,L number of cogs per revolution,Angle uniform sampling, not timeuniform sampling.

Φy(iω) = ΦH(iω)Φe(iω) + Φd(iω)

ΦH(iω) =1

|A(iω, θ)|2

Φe(iω) = σ white noise spectrum,A(iω; θ) AR model,θ parameters in the AR model,Φd(iω) disturbance spectrum,Φy(iω) ’measured’ spectrum.

Table 6.1 Signal models and assumption in time and frequency domains,respectively.

The time domain algorithm proposed in [Persson(2002)] estimates the param-eters of the autoregressive model in the following steps:

1. interpolate data to a high sampling rate to avoid aliasing,

2. band-pass filter the signal to get rid of broad band disturbances and to focuson the region 30 - 60 Hz,

3. down-convert the signal utilizing deliberate aliasing,

4. estimate a discrete-time AR model and

5. extract vibration data from this model.

It is not easy to modify this algorithm to remove the narrow-band interference d(t),so the most practical solution is to turn off the algorithm when such a disturbanceis detected.

Table 6.2 on the other hand outlines the frequency domain approach which willbe described in the section below. Since the method operates in the frequencydomain, narrow band disturbances Φd(iω), as illustrated in Table 6.1, can be seenas statistical outliers in the frequency domain. Outliers in the time-domain have

6.4 Frequency Domain Estimation 67

traditionally been dealt with by introducing more robust norms in the estimationcriteria e.g. [Huber(1981)]. It is our opinion that this approach will be transferrableto the case of frequency domain data. Here, on the other hand, the focus will beon rejection of wide band disturbances, short data batches and physical modeling.Managing these issues is a necessary condition for the overall usefulness of themethod in the context of tire vibration analysis.

1. Approximate continuous timeFourier transform with

Y (iωk) =∫ T

0

y(t)eiωktdt,

y(t) =N∑

i=1

y(ti)φi(t− ti),

wk =2π

Tk, k ∈ N .

2. Average the periodogramˆΦy(iω) =

∣∣∣Y (iω)∣∣∣2

over batcheswith similar vehicle speed.

3. Maximum likelihood estimatethe CAR-model

a = arg mina

∑

k∈N

ˆΦy(iωk)ΦH(iωk, θ)

+ log ΦH(iωk, θ)

ΦH(iωk, θ) =σ2

|(iωk)2 + 2γiωk + ω20 |2

Table 6.2 Frequency-Domain algorithm.

6.4 Frequency Domain Estimation

The approach to frequency domain parameter estimation in this section is similarto the framework described in Chapter 3 and 4. Since the entire continuous-timerealization of the output is unavailable and only a finite number of samples at timeinstances t1, t2, . . . , tN are available piecewise-constant interpolation as in (4.2)

y(t) =N∑

i=1

y(ti)φ(0)i (t− ti)

will be used to reconstruct the realization. In order to reduce the variance ofthe periodogram data, the output data set is split into Nb batches of durationTn, n = 1, . . . , Nb. Then a periodogram

ˆΦ(n)y = |Y (0)

Tn(iω)|2

is calculated for each batch. The truncated Fourier transform for ZOH is computedas in (4.7)

Y(0)Tn

(iω) =1√Tn

Nn−1∑

k=1

y(tk)e−iωtk−1 − e−iωtk

iω


where Nn is the number of samples in each batch. A periodogram with reducedvariance is then formed as

ˆΦy(iω) =1

Nb − 1

Nb∑n=1

ˆΦ(n)y (iω).

This method is analogous to the method by Welch [Welch(1967)] for the smoothingof spectral estimates.

When the periodogram estimate ˆΦy of the wheel speed y(t) is available acontinuous-time AR model can be identified by using the approximate Maximum-Likelihood procedure described in (3.21) and (3.22).

θ = arg minθ

∑

k∈N

ˆΦy(iωk)ΦH(iωk, θ)

+ log ΦH(iωk, θ). (6.4)

Here ΦH is defined as in Table 6.2. The frequencies are ωk, k = 1, . . . , Nω whereωk = 2πk/T, k ∈ N . The index set N denotes those frequencies which are destinedto be used in the estimation procedure.

6.5 Properties of Bias and Variance

In Chapter 5 statistical properties of estimators such as the one in (6.4) werediscussed. In this section we will use the results from that chapter, but in a slightlydifferent manner.

In the case of tire vibration analysis there can be wide band disturbances cor-rupting the measurements of the outut y. From (5.1) and (5.7) we know that

E(θ − θ0) ≈∑

k∈NS(iωk)∆Φy(iωk).

where θ are the estimated and θ0 are the true parameter values. The relative biasin the periodogram estimate of the power spectrum is defined as

∆Φ(iωk) =E

ˆΦy(iωk)− Φ(iωk, θ0)Φ(iωk, θ0)

.

Here we have for simplicity defined Φ = ΦH . The sensitivity of the parameterestimates to the relative bias in the periodogram is

S(iωk) = Ψ(θ0, Φ)−1Ψk(θ0, Φ)

where the relative sensitivity is

Ψk(θ0, Φ) =Φ′θ(iωk, θ0)Φ(iωk, θ0)

.

6.6 Experimental Results 69

and

Ψ(θ∗, Φ) =∑

k∈NΨk(θ0, Φ)Ψk(θ0, Φ)T

In the case of tire vibration analysis there can be wide band disturbances corruptingthe spectrum of the output y. Those disturbances are in some cases not interferingwith with the important vibrations. Therefore, the spectrum ΦH of the model isvery small in the area of the disturbancees and the relative bias in the spectrumcan be quite significant. Hence, in order to avoid bias it is necessary to ignoreinformation from frequencies where the relative bias and sensitivity are large.

In an online automotive application computational power and available mem-ory will always be an important design constraint. Calculating the periodogramfor a particular frequency can be cumbersome and therefore it is important toknow which frequencies carry the most information. In the case of variance it wasestablished in (5.8) in Chapter 5 that

E(θ − θ0)(θ − θ0)T → Ψ(θ0, Φ)−1

as T → ∞ and the largest sampling interval goes to zero. Again the relativesensitivity plays an important role. Therefore, in order to reduce the variance,information from frequencies where the relative sensitivity is large should be pri-oritized.

6.6 Experimental Results

In this section the theory presented above is applied to the estimation of the reso-nance peak of the torsional vibrations of a pneumatic tire. The samples y(tk) arepre-processed measurements from an axel-angle measurement device in the ABSsystem of a car. The frequency spectrum of y(t) is approximately divided as sum-marized in Table 6.3.

0-10 10-15 15-30 30-60 60-80 80-100 100–

Speed Mode 1 Noise Mode 2 Noise Mode 3 Noise

Narrow-band noise components

Table 6.3 Frequency spectrum with approximate limits in Hz. We willconcentrate on Mode 2 which operates on the interval 30-60 Hz.

The vibrations in the range 30-60 Hertz can be modelled as a spring-dampersystem excited by white noise e(t) as in (6.1).

For the special parametrization of the spectrum in (6.2) the relative sensitivityfunctions with respect to the parameters are shown in Figure 6.1. Here we havechosen γ = 33.88 and ω0 = 289.687. This means that wres = 285 rad/s or fres =45.47Hz. From this figure, we conclude that γ in (6.1) is sensitive to periodogram


−100 −50 0 50 100−0.06

−0.04

−0.02

0

f

Φγ/Φ

−100 −50 0 50 100−0.04

−0.02

0

0.02

0.04

f

Φω

0/ Φ

Figure 6.1 Relative sensitivities Φk in (5.5)for γ (upper) and ω0 (lower).

bias near the natural resonance frequency of the system. The frequency ω0 in (6.1)on the other hand, is particularly sensitive to periodogram bias at low frequencies.According to Table 6.3 there is noise between 15 and 30 Hertz and 60 and 80 Hertz.Therefore we restrict the frequencies used to those between 30 and 60 Hertz.

In Figure 6.2 we have estimated the resonance frequencies from the refined setof real life data from an ABS sensor. The data has been divided into four parts.These parts have then been subdivided into a set of batches with a duration of acertain number of revolutions or laps of the tire. The number of laps per sub-batchis indicated on the x-axis of the figure. Periodograms have been estimated usingZOH for each batch and subsequently averaged in order to yield four estimates ofthe spectrum. The mean value and estimated standard deviation are then plotted.The figure indicates that the method is feasible and that a sub-batch size of about10 laps is sufficient to yield a stable estimate of the resonance frequency withmoderate variance. Below 10 laps per sub-batch the mean value of parameterestimates decrease rapidly due to bias from the the small number of frequencydomain samples.

6.7 Summary

In this chapter a frequency-domain alternative to time-domain estimation of tirepressure or axel vibrations has been developed. The method is thought to beeasy to extended in order to reject narrow band disturbances interfering with thevibration. The method has also proved feasible on real-life data. It is based on aphysical model, can use fairly short batches of data and is able to reject wide banddisturbances.

6.7 Summary 71

5 10 15 2044.4

44.6

44.8

45

45.2

45.4

45.6

45.8

Laps

f res

Figure 6.2 Resonance frequency fres in Hertz versus batch size in numberof tire laps. Bars indicates one standard deviation.


7Identification of CARMA Models

Creativity is the ability to introduce order into the randomness ofnature.

Eric Hoffer (1902 - 1983)

In the previous chapters the focus has been on frequency-domain identification ofnoise models. This topic will also be continued in the present chapter but therewill be a slight shift in focus. The spotlight will move back from the generalnon-equidistant setting in Chapter 4 to the simpler but more workable setting ofequidistant sampling in search for alternative identification methods. The goal inthis chapter is to derive a method for the identification of continuous-time autore-gressive moving average (CARMA) noise models from equidistantly sampled data.Instead of using piecewise linear and piecewise constant interpolation as in Chap-ter 4 we exploit the high-frequency properties of the linear system while derivingmethods.

First, in Section 7.1 a framework for the continuous-time ARMA model will berepeated. Then, in Section 7.2, the indirect frequency domain estimator, where thecontinuous-time parameters are obtained from the discrete-time spectrum, is intro-duced. Following this, in Section 7.3 the direct frequency domain type estimationprocedure is presented. Here, an estimate of the continuous-time output spectrumis produced by modifying the discrete-time output spectrum. The continuous-timespectrum is then directly fitted to this estimate.

73

74 Chapter 7 Identification of CARMA Models

7.1 CARMA Model

In this chapter we will again estimate the parameters of the continuous-time au-toregressive moving average (CARMA) model introduced in Chapter 3. Therefore,we restate its definition here since it will be referred to several times in the text.The model can is described as

yt =B(p)A(p)

et (7.1)

where et is continuous-time white noise such that

E[et] = 0

E[etes] = σ2δ(t− s)

The operator p is here the differentiation operator while

A(p) = pn + a1pn−1 + a2p

n−2 + · · ·+ an

B(p) = pm + b1pm−1 + · · ·+ bm.

and the vector of parameters is θ = [a1 a2 . . . an b1 b2 . . . bm λ]T where λ = σ2.Here m < n. The continuous-time spectrum for this process will be

Φc(iω, θ) = σ2 B(iω)A(iω)

(7.2)

We will now move on to two frequency domain methods for estimating theparameters of this model.

7.2 Indirect Frequency-Domain Estimation

The common way of modelling a noise sequence of a time series is to use a discrete-time model and estimate it in the time domain e.g [Ljung(1999)]. Suppose thatthe continuous-time output y(t) : t ∈ [0, T ] of the CARMA model is known atthe discrete-time instances tk = kTs, k = 1, . . . , N T with T = NTs and let

ˆΦd(eiωTs) =∣∣∣Yd(eiωTs)

∣∣∣2

. (7.3)

where

Yd

(eiωkTs

)=

1√N

N∑

k=1

y(kTs)e−iωkTs (7.4)

and

ωk =2π

Tsk.

7.2 Indirect Frequency-Domain Estimation 75

A frequency domain procedure for estimating the discrete-time parameters θd ofthe spectrum is then

θd , arg minθd

Vd(θd)

where

Vd(θd) ,Nω∑

k=1

ˆΦd(eiωkTs)Φd(eiωkTs , θd)

+ log Φd(eiωkTs , θd)

as compared to the continuous-time expression in (3.14). This method is known

−100

−80

−60

−40

−20

0

Mag

nitu

de (

dB)

10−2

10−1

100

101

102

−180

−135

−90

−45

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Figure 7.1 Bode diagram comparing the Whittle likelihood estimatorwithout (dashdot) and with (dashed) folding of the continuous-time spectrum to the true system (solid). The dashed line isalmost identical to the solid.

as the Whittle likelihood estimator [Whittle(1961)] and is an approximation of thecorresponding time-domain ML-method. It has long been used to estimate param-eters of discrete-time ARMA models. Here, we are on the other hand interestedin the estimation of continuous-time ARMA models. This is made possible by therelationship between the discrete-time and continuous-time spectrum

Φd(eiωTs , θ) =∞∑

k=−∞Φc(iω + i

2π

Tsk, θ) (7.5)

presented in expression (3.8) in Chapter 3. Here θ are the continuous-time param-eters in (7.2). A practical benefit is that only a limited number of terms in (7.5)may be needed to produce a good approximation of its discrete-time counterpartin the frequency range of interest.


7.2.1 Numerical Illustration

In Figure 7.1 we have estimated the second-order continuous-time AR-model

yt =σ

p2 + a1p + a2et (7.6)

with σ = 1, a1 = 3 and a2 = 2. The duration of the data set was T = 1000s withthe sampling time Ts = 1s. The figure illustrates the frequency-domain bias whichcould occur if only the term k = 0 in 7.5 is used in the expression for Φd, that isif the folding is not taken into account. In Table 7.1 the mean parameter valuesfor NMC = 250 Monte Carlo simulations are illustrated. Here Nf = 0 and Nf = 5terms around k = 0 have been included in the sum (7.5). From the figure and thetable we see that ignoring the effects of folding can produce very biased estimates.

Table 7.1 Mean Values of Parameters EstimatesSystem/Method a1 a2 σ

True System 3 2 1Folded (Nf = 5) 3.090567 2.037784 1.023398

Unfolded (Nf = 0) 4.978773 3.141028 1.639744

7.3 Direct Continuous-Time Estimation

In the previous section we have estimated continuous-time parameter by param-eterizing the discrete-time spectrum in terms of the continuous-time parameters.Then, we used the discrete-time Whittle estimator to estimate the parameters.In this section, the method is slightly different. Instead of using a discrete-timeidentification method we use the continuous-time method developed in Chapter 3.This time, the continuous-time spectrum is estimated from discrete-time data.

The continuous-time frequency domain estimation method developed in Chap-ter 3 can be summarized as (see (3.14) and (3.20))

θ , arg minθ

V TN (θ, ˆΦT

c )

where

V TN (θ, ˆΦT

c ) ,Nω∑

k=1

ˆΦTc (iωk)

Φc(iωk, θ)+ log Φc(iωk, θ). (7.7)

The continuous-time spectrum is estimated as

ˆΦTc (iω) =

∣∣Y Tc (iω)

∣∣2 .

7.3 Direct Continuous-Time Estimation 77

where the truncated continuous-time Fourier transform is

Y Tc (iω) =

1√T

∫ T

0

y(t)e−iωtdt.

The problem is that the sampled data periodogram ˆΦd(eiωTs) can be readily foundfrom the sampled data in (7.3) and (7.4), while we need the continuous-time peri-

odogram ˆΦc(iω) in the criterion (7.7). How can the latter be estimated or computed

from ˆΦd(eiωTs) ? We will be looking for relationships like

ˆΦc(iω) = H(iω) ˆΦd(eiωTs) (7.8)

for a suitable function H. We have already in Chapter 4 studied such transforma-tions that were based on the assumptions that the signal is ZOH or FOH betweenthe samples, see (4.12). To do better than that we could argue as follows: If thetrue signal parameters θ0 were known, we would have

ˆΦd(eiωTs) ≈ Φd(eiωTs , θ0)

and

ˆΦc(iω) ≈ Φc(eiωTs , θ0).

This means that the ideal transformation filter H in (7.8) would be

H(eiωTs) =Φc(iω, θ0)

Φd(eiωT s, θ0). (7.9)

Since θ0 is unknown, we cannot construct H in this way, but the point is that whenTs → 0, H(eiωTs) in (7.9) will approach something that does not depend on thesignal parameters θ0, but only on the relative degree (pole excess) l = n−m of thesignal model in (7.1). This is what we will now show.

7.3.1 Estimation Method

Let the system in (7.1) be strictly proper, stable and l = n − m be its relativedegree (or pole excess), i.e. the difference between the number of poles and zerosof the system. Further assume that ω is below the Nyquist frequency. Define

Φf (eiωTs) ,

∣∣∣ eiωTs−1iωTs

∣∣∣2l

∣∣∣B2l−1(eiωTs )(2l−1)!

∣∣∣(7.10)

where B2l−1(z) are the so called Euler-Frobenius polynomials. The polynomialsare defined as [Astrom and Sternby(1984)]

Bl(z) = bl1z

l−1 + bl2z

l−2 + · · ·+ bll (7.11)


where

blk =

k∑m=1

(−1)k−mml

(n + 1k −m

), k = 1, . . . , l.

Below is a list of polynomials for different values of l

B1(z) = 1B2(z) = z + 1

B3(z) = z2 + 4z + 1

B4(z) = z3 + 11z2 + 11z + 1.

In Figure 7.2 we see that there is a very good correspondence between Φc(iω, θ0)/Φd(iω, θ0)and Φf (iω) for the system in (7.6). This observation is verified by the followingtheoretical result.

Theorem 7.1Assume Φf is defined as in (7.10), that ω is less than the Nyquist frequency andthat l ≥ 1. Then

Φc(iω)Φd(eiωTs)

→ Φf (eiωTs)

as Ts → 0.

Proof Since ω is less than the Nyquist frequency

Φc(iω + i 2πTs

k)σ2

|iω+i 2πTs

k|2l

→ 1

as Ts → 0 if k 6= 0. This has the consequence that

Φc(iω)Φd(eiωTs)

→ Φc(iω)Φc(iω) +

∑k 6=0

σ2

|iω+i 2πTs

k|2l

as Ts → 0.From Lemma 3.2 in [Wahlberg(1988)]

Φf (eiωTs) =1

|iω|2l

1|iω|2l +

∑k 6=0

1|iω+i 2π

Tsk|2l

.

By putting the two previous expressions on a common denominator, we get the thefollowing relation

Φc(iω)Φd(eiωTs)

− Φf (eiωTs) → Φf (eiωTs)Φr(iω)Φs(iω)

7.3 Direct Continuous-Time Estimation 79

−10 −8 −6 −4 −2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ω

Figure 7.2 Comparison of Φc/Φd (solid) to Φf (dashdot) for the system in(7.6). In the inner plot Ts = 2 and in the outer plot Ts = 0.5.

where

Φr(iω) =1− Φc(iω) |iω|

2l

σ2

Φc(iω) +∑

k 6=0σ2

|iω+i 2πTs

k|2l

and

Φs(iω) =∑

k 6=0

σ2

|iω + i 2πTs

k|2l.

Since Φf and Φr are bounded and

1|iω + i 2π

Tsk|2l

≤ C

(Ts

k

)2l

if k 6= 0, the result∣∣∣∣

Φc(iω)Φd(eiωTs)

− Φf (eiωTs)∣∣∣∣ ≤ CT 2l

s

will follow. 2

Therefore a reasonable estimate of the continuous-time spectrum would be

ˆΦc(iω) =

∣∣∣ eiωTs−1iωTs

∣∣∣2l

∣∣∣B2l−1(eiωTs )(2l−1)!

∣∣∣ˆΦd(iω).



In Table 7.3.2 we have compared the performance of this estimator, Method 2,compared to the one described in Section 7.2, Method 1, for different samplingintervals. The mean for each different sampling interval have been estimated byNMC = 250 Monte Carlo simulations. The system is the one in (7.6) and thecorrespondence between the mean parameter estimates in the table seems to begood.

Ts Method a1 a2 σTrue 3 2 1

0.6 1 3.0355 2.0189 1.00710.6 2 2.9935 2.0166 1.00070.5 1 3.0186 2.0206 1.00540.5 2 3.0476 2.0376 1.01500.4 1 3.0266 2.0150 1.00550.4 2 3.0449 2.0240 1.01100.3 1 3.0276 2.0143 1.00710.3 2 3.0327 2.0168 1.00860.2 1 3.0009 2.0118 1.00090.2 2 3.0017 2.0122 1.00110.1 1 3.0245 2.0210 1.00710.1 2 3.0246 2.0211 1.0072

Table 7.2 Comparison of mean values of parameter estimates from the in-direct (Method 1) and direct(Method 2) estimators versus thesample time Ts. The system and circumstances are the sameas in (7.6). The statistics for each sampling time has been esti-mated by NMC = 250 Monte-Carlo Simulations.

7.4 Conclusions

Two parametric frequency-domain identification algorithms for continuous-timeARMA noise models have been presented. For low sampling rates, the Poissonsummation formula is used to establish an expression for the discrete-time Fouriertransform of the output of the CARMA model. In the case of rapid sampling thecontinuous-time spectrum is estimated from the discrete-time spectrum by meansof a certain compensator. Further, numerical examples illustrate the propertiesof the different approaches. A possible generalization of the methods would beestimation of continuous-time output error models and the case of non-uniformlysampled data.

8Identification of COE Models

”If I could offer you only one tip for the future, sunscreen would beit. The long term-benefits of sunscreen have been proved by scientists,whereas the rest of my advise has no basis more reliable than my ownmeandering experience”

Baz Lurman, Everybody’s Free to Wear Sunscreen

In the previous chapters up to now the focus has mainly been on the identifi-cation of continuous-time noise models. An assumption has been that we are notable to measure the input which is modelled as a Gaussian white noise sequence.There can however be a measurable input which dictates the output of the system.Precisely this situation will be the object of study in the present chapter. Theoutput will be disturbed by a white Gaussian noise sequence.

The underlying ideas presented in this chapter are basically the same as thoseintroduced in the previous chapter. After introducing the output error setup inSection 8.1 a procedure for parameterizing the discrete-time transfer function interms of the continuous-time transfer function is introduced in Sections 8.2 and 8.3.By assuming that the rate of sampling is relatively high (well above the systembandwidth) a series of approximations is then introduced in Sections 8.4 and 8.5.The last approximation can be interpreted as a modification of the discrete-timeFourier transform in order to estimate the continuous-time Fourier transform. Sucha transform estimate will then allow approximate but direct estimation of thecontinuous-time transfer function.

81

82 Chapter 8 Identification of COE Models

8.1 Preliminaries

Assume that the continuous-time output error (COE) model is represented as

y(t) = Gc(p)u(t) + e(t) (8.1)

where e(t) is a white noise process such that Ee(t)e(s) = σ2δ(t− s). The operatorp is the differentiation operator and the transfer operator is defined as

Gc(p) =b0p

m + b1pm−1 + · · ·+ bm

pn + a1pn−1 + a2pn−2 + · · ·+ an. (8.2)

where m < n. The variables y(t) and u(t) are measured inputs and outputs. Alsodenote the vector of parameters as

θ = [a1 a2 . . . an b0 b1 . . . bm]T .

The transfer function Gc parameterized in terms of these parameters will be de-noted Gc(p, θ). In the following sections three frequency-domain methods for theidentification of the model structure are demonstrated. First the exact approachis presented.

8.2 Exact Indirect Frequency Domain Estimation

Assume that the continuous-time system Gc is written in the state space form

x(t) = A(θ)x(t) + B(θ)u(t)y(t) = C(θ)x(t)

and the input u(t) is assumed to be zero-order hold. Suppose that the input-outputdata have been sampled with the constant sampling interval Ts. The relationshipbetween the discrete-time input and output can then be characterized by the pulsetransfer function Gd

Gd(eiωTs , θ) = C(θ)(eiωITs − eA(θ)Ts)Γ(θ) (8.3)

Γ(θ) =∫ Ts

0

eA(θ)tB(θ)dt

as described in Section 2.2 of Chapter 2. Define the discrete-time Fourier transformof the output yt as

Yd(eiωTs) =1√N

N∑

k=1

y(kTs)e−iωkTs . (8.4)

and the discrete Fourier transform at the frequencies ω1, . . . , ωNω as

YDFT = Yd(eiω1Ts), . . . , Yd(eiωNω Ts)

8.2 Exact Indirect Frequency Domain Estimation 83

where ωk = 2πTs

k. Let, analogously Ud denote the discrete-time Fourier transform ofthe input. Then, we may treat YDFT and UDFT as observations, neglect transient(non-periodic) effects, and assume that the Fourier transforms are independentat different frequencies. All of this may be more or less good approximations –see p 230 in [Ljung(1999)]. Under these assumptions, the negative log-likelihoodfunction for values of the discrete-time Fourier transform at the frequencies is

L(θ) = − log p(YDFT |θ)

=Nω

2log 2π +

Nω∑

k=1

12

∣∣Yd(eiωkTs)−Gd(eiωkTs , θ)Ud(eiωkTs)∣∣2 .

where Gd is the pulse-transfer function (8.3) for the system Gc in (8.1). The ML-procedure for estimating the parameters is then

θ , arg minθ

Vd(θ)

where

Vd(θ) ,Nω∑

k=1

∣∣Yd(eiωkTs)−Gd(eiωkTs , θ)Ud(eiωkTs)∣∣2 . (8.5)

This is a method that has long been used to estimate parameters of OE models

−80

−60

−40

−20

0

Mag

nitu

de (

dB)

10−1

100

101

102

103

−90

0

90

180

270

360

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Figure 8.1 Bode diagram comparing the estimator without (dashdot) andwith (dashed) folding of the continuous-time spectrum to thetrue system (solid). The dashed line is almost identical to thesolid.

[Mathworks(2004)]. The transfer function Gd can also be parameterized in terms


of the discrete-time parameters. These can then be estimated and transformedto continuous-time using the matrix exponential as illustrated in Section 2.2.1 ofChapter 2.

8.3 Approximating Gd for Indirect Frequency Do-main Estimation

Instead of going via the state space formulas, an expression in input-output formfor the sampling of transfer functions (with piecewise constant inputs) can be used.

8.3.1 Pulse Transfer Function

Consider a discrete-time system consisting of a zero-order hold circuit at theinput, a continuous-time system Gc as in (8.2) and a sampler at the output.Then the discrete-time pulse transfer function Gd in (8.3) can also be writtenas [Astrom(1984)]

Gd(eiωTs , θ) =(

1− e−iωTs

Ts

) ∞∑

k=−∞

Gc(iω + i 2πTs

k, θ)

iω + i2πTs

k. (8.6)

This result can be found in [Astrom(1984)] and since it is central for the chaptera brief summary of the thoughts leading to it will follow below. First define theDirac ”comb” function as

m(t) ,∞∑

k=−∞δ(t− Tsk)

where δ is the Dirac delta function. Denote

u∗(t) = u(t)m(t)y∗(t) = y(t)m(t)

as the sampled versions of the input and output. If the input is assumed to bezero-order hold its sampled version will pass through a hold circuit with transferfunction

H(s) =1− e−sTs

s

before entering the system G. Let the combination of the hold circuit and thesystem be defined as

F (s) = G(s)H(s).

8.3 Approximating Gd for Indirect Frequency Domain Estimation 85

Then, the relationship between the sampled versions of the output is

y∗(t) = m(t)F (p)u∗(t)

where p is the differentiation operator. If f is the impulse response of the systemF , the sampled versions of the input and output will be related as

y∗(t) = (f(t) ∗ u∗(t))∗ = m(t)∫ ∞

−∞f(t− τ)m(τ)u(τ)dτ.

At the same time

(f∗(t) ∗ u∗(t)) =∫ ∞

−∞m(t− τ)f(t− τ)m(τ)u(τ)dτ

=∫ ∞

−∞m(t)f(t− τ)m(τ)u(τ)dτ

since m(t− kTs) = m(t) and m(τ) 6= 0 only for τ = kTs [Astrom(1984)]. Hence

y∗(t) = (f(t) ∗ u∗(t))∗ = (f∗(t) ∗ u∗(t))

and an analogous expression holds for the Laplace transform where

Y ∗(s) = [F (s) ∗ U∗(s)]∗ = F ∗(s)U∗(s).

Since

U∗(s) =∫ ∞

0

u(t)m(t)e−stdt =∞∑

k=0

u(kTs)e−skTs

Y ∗(s) =∫ ∞

0

y(t)m(t)e−stdt =∞∑

k=0

y(kTs)e−skTs

the only term of real interest is F ∗ for which

F ∗(s) =∫ ∞

0

f(t)m(t)e−stdt =1Ts

∞∑

k=−∞F (s +

2π

Tsk). (8.7)

since

F ∗(s) =1

i2π

∫ γ+i∞

γ−i∞F (v)M(s− v)dv =

1i2π

∫ γ+i∞

γ−i∞F (v)

11− e−(s−v)Ts

dv.

Placing the path of integration between the poles of F and M and completing itby a large semi-circle enclosing the poles of M , residue calculus can be used under


some mild conditions. The poles of M will be located at the zeros of eTs(s−v) = 1which are

v = s + i2π

Tsk, k = . . . ,−1, 0, 1, . . . .

The residues can be proved to be

1Ts

F (s +2π

Ts).

Since F (s) = Gc(s)H(s) and

H(s + i2π

Tsk) =

1− e−(s+i 2πTs

k)Ts

s + i 2πTs

k

1− e−sTsei2πk

s + i2πTs

k

1− e−sTs

s + i 2πTs

k.

Hence the expression

Gd(eiωTs , θ) =(

1− e−iωTs

Ts

) ∞∑

k=−∞

Gc(iω + i 2πTs

k, θ)

iω + i 2πTs

k

follows from (8.7). For a more detailed but similar discussion we refer the readerto Theorems 4.1 and 4.2 in the book by Astrom and Wittenmark [Astrom(1984)].


A drawback connected with using the formula in (8.6) is of course the infinite sum.Good approximations can however be achieved with a few terms at fast samplingwhen l ≥ 1.

In Figure 8.1 the second-order continuous-time OE-model

y(t) =b0p + b1

p2 + a1p + a2u(t) + e(t) (8.8)

with true parameters a1 = 3, a2 = 2, b0 = 1 and b1 = 0.5 was estimated. Theduration of the data set was T = 1000s, the sampling time was Ts = 1s and arandom binary signal was used. The figure illustrates the frequency-domain biaswhich could occur if the folding is not taken into account, that is, if only the centralterm k = 0 is included in expression (8.9). In Table 8.1 the mean parameter valuesfor NMC = 100 Monte Carlo simulations with SNR = 40 dB are illustrated. Henceaccording to Table 8.1 we will get very biased estimates if the folding is ignored. InFigure 8.2 the model in (8.8) have been estimated with different number of terms

8.4 Indirect Method with Modified Objective 87

Table 8.1 Mean Values of Parameters EstimatesMethod a1 a2 b0 b1

True System 3 2 1 0.5Folded 3.0206 2.0293 1.0104 0.5113

Unfolded 3.1396 15.5759 -1.1265 4.8789

−2 0 2 4 6 8 10 122

3

4

a1

−2 0 2 4 6 8 10 120

10

20

a2

−2 0 2 4 6 8 10 12−2

0

2

b1

−2 0 2 4 6 8 10 120

5

10

Nfolds

b2

Figure 8.2 Parameter estimates for the model (8.8) with different degreesof folding.

included in the expression in (8.6). Besides the central term, Nf = 20 terms havebeen used in the way described in the expression below

Gd(eiωTs , θ) =(

1− e−iωTs

Ts

) Nf∑

k=−Nf

Gc(iω + i 2πTs

k, θ)

iω + i2πTs

k. (8.9)

The same parameter values are also illustrated in Table 8.2. In the next section asimplified approximate procedure for parameter identification when the samplingrate is high will be illustrated.

8.4 Indirect Method with Modified Objective

In the previous section we used the summation formula (8.9) in order to estimateparameters. In this section we try to find a good approximation of the sum whichis easier to calculate. Define

F(l)dc (iω) = T l+1

s

Bl(eiωTs)

l! (eiωTs − 1)l− 1

(iω)lH(iω) (8.10)


Table 8.2 Mean Values of Parameters EstimatesNfold a1 a2 b0 b1

True System 3 2 1 0.50 3.1209 15.4465 -1.1158 4.83691 2.4350 5.1814 0.3892 1.54972 2.6098 2.5186 0.8580 0.69983 2.7516 2.2177 0.9280 0.59254 2.8373 2.1252 0.9585 0.55855 2.8519 2.1213 0.9607 0.55546 2.8964 2.1291 0.9744 0.55387 2.9349 2.0750 0.9871 0.53738 2.9629 2.0575 0.9970 0.53069 3.0124 1.9930 1.0169 0.5074

where Bl(z) are the Euler-Frobenius polynomials described in (7.11) of Chapter 7and

H(iω) =1− e−iωTs

iω(8.11)

is the continuous-time transfer function of a hold circuit. Assume that the systemin (8.1) has relative degree l = n−m, i.e. the number of poles minus the numberof zeros of the system. Then, we know from the transfer operator in (8.2) that theparameter

b0 = limω→∞

Gc(iω)(iω)l. (8.12)

The following theorem will show that for moderately high sampling rates the ob-jective function Vd in (8.5) will be approximately equal to

Vdc(θ) ,Nω∑

k=1

∣∣∣∣∣Yd(eiωkTs)− (Gc(iωk, θ)H(iω)

Ts+ b0

F(l)dc (iω)Ts

Ud(eiωkTs)

∣∣∣∣∣

2

. (8.13)

Theorem 8.1Assume that l ≥ 1 and that ω is below the Nyquist frequency, then as Ts → 0

Gd(eiωTs , θ) → Gc(iω, θ)H(iω)

Ts+

b0

TsF

(l)dc (iω)

where F(l)dc (iω) is defined as in (8.10) and b0 as in (8.12). Also, H and F

(l)dc are

completely independent of the parameters in θ.

Proof Since ω is below the Nyquist frequency

b0

(iω+i 2πTs

k)l

Gc(iω + i 2πTs

k)→ 1

8.4 Indirect Method with Modified Objective 89

as Ts → 0 for k 6= 0. An intuitive explanation is that only the high frequency partof the transfer function will be visible in this frequency range. This means that thediscrete-time pulse transfer function in (8.6) will be

Gd(eiωTs , θ) → 1Ts

Gc(iω, θ)H(iω)

+b0

Ts

∑

k 6=0

1− e−iωTs

(iω + i2πTs

k)l+1

as Ts → 0. Here H is defined as in (8.11). Then, from Lemma 3.2 in [Wahlberg(1988)]

eiωTsT l+1s

Bl(eiωTs)

l! (eiωTs − 1)l+1=

∞∑

k=−∞

1(iω + i2π

Tsk)l+1

and by multiplying by 1− e−iωTs and subtract the term where k = 0 we get

F(l)dc (iω) = T l

s

Bl(eiωTs)

l! (eiωTs − 1)l− 1

(iω)l

H(iω)Ts

.

This completes the proof. 2

The terms on the right-hand side of expression (8.10) allow a few interesting re-flections. Let

F(l)dc (iω) = F1(eiωTs)− F2(iω).

The first term

F1(eiωTs) = T ls

Bl(eiωTs)

l! (eiωTs − 1)l

can be interpreted as the frequency response of a system made up of a chain of lintegrators where the input is subject to zero-order hold and the input and outputare sampled. For more details the reader is referred to Lemma 1 in the paperon sampling zeros by Astrom and Wittenmark [Astrom and Sternby(1984)]. Thesecond term

F2(iω) =1

(iω)lH(iω)

is the continuous-time transfer function of a hold circuit followed by the same chainof integrators.


In Table 8.3 the effect of the method described in (8.13) is illustrated. For moderatesampling rates the performance is illustrated, compared to using no compensationat all. Method 1 in this table is the indirect method while method 2 is the indirectmethod with modified objective. The following section will explain how the addi-tive compensation described above can be changed into a multiplicative one if thesampling is fast enough.


Table 8.3 Mean Values of Parameters Estimates.Ts Method a1 a2 b0 b1

True 3 2 1 0.50.1 1 3.0102 2.0005 1.0032 0.5023

2 3.0040 2.0122 1.0001 0.50523 3.1611 1.8367 1.0807 0.4702

0.2 1 3.0193 2.0241 1.0046 0.50922 2.9994 2.0662 0.9950 0.52023 3.2184 1.9481 1.1265 0.5201

0.3 1 3.0054 1.9926 1.0024 0.49692 2.9569 2.0811 0.9804 0.52033 3.1585 2.2402 1.1238 0.6256

0.4 1 3.0153 2.0014 1.0055 0.50132 2.9160 2.1552 0.9620 0.54263 3.1002 2.9902 1.0777 0.8951

Table 8.4 Comparison of the performance of the indirect method (Method1), the indirect method with modified objective (Method 2) andthe direct method without modification (Method 3). The statis-tics for each sampling time has been estimated by NMC = 100Monte-Carlo simulations with SNR=40 dB. The input u is arandom binary signal with energy up to the Nyquist frequency.

8.5 Direct Method with Modified Transform

Assume that there is no output noise in the continuous-time model (8.1). Thefrequency domain relationship between the sampled input and output will then be

Yd(eiωTs) = Gd(eiωTs , θ)Ud(eiωTs).

where Gd is defined by (8.3) if the input is assumed to be ZOH. In continuous-timethe relationship would be characterized by

Yc(iω) = Gc(iω, θ)Uc(iω) = Gc(iω, θ)H(iω)Ud(eiωTs)

since the relationship between the continuous- and discrete-time Fourier transformsof the input is

Uc(iω) = H(iω)Ud(eiωTs)

because of the hold circuit in (8.11). This would open for estimating the continuous-time Fourier transform of the output of the systems as

Yc(iω) =Gc(iω, θ0)H(iω)

Gd(eiωTs , θ0)Yd(iω). (8.14)

See also Section 7.3 in Chapter 7 for a similar discussion.

8.5 Direct Method with Modified Transform 91


Let the system in (8.1) be strictly proper and stable, l = n − m be its relativedegree. Further assume that the sampling time Ts is such that the rate of samplingis above the system bandwidth ωb and that ω is below the Nyquist frequency.Define

F (l)c (iω) =

(eiωTs−1

iωTs

)l+1

eiωTs Bl(eiωTs )(l+1)!

(8.15)

where Bl(z) are the Euler-Frobenius polynomials defined in expression (7.11) ofChapter 7. In Figure 8.3 we see that there is a good correspondence between

10−3

10−2

10−1

100

101

102

100

w

0 5 10 15 20 25 30 35−1

0

1

2

3

4

5

w

Deg

rees

Figure 8.3 Comparison of GcH/Gd (8.14) (solid) and F (8.15)(dash-dot).The sampling rate is Ts = 0.1.

GcH/Gd in (8.14) and F (8.15). This is also verified by the following theoreticalresult.

Theorem 8.2Assume F

(l)c is defined as in (8.15) with l ≥ 1 and that ω is less than the Nyquist

frequency. Then

Gc(iω, θ0)H(iω)Gd(eiωTs), θ0

→ F (l)c (iω)

as Ts → 0.


Proof Since ω is below the Nyquist frequency

Gc

(iω + i 2π

Tsk)

b0

(iω+i 2πTs

k)l

→ 1 (8.16)

as Ts → 0 if k 6= 0 and b0 is defined as in (8.12). This has the consequence that

Gc(iω)H(iω)Gd(eiωTs)

→Gc(iω)

iωGc(iω)

iω +∑

k 6=0b0

(iω+i 2πTs

k)l+1

.

as Ts → 0 if we insert (8.16) in (8.6). From Lemma 3.2 in [Wahlberg(1988)]

Fc(iω) =1

(iω)l+1

1(iω)l+1 +

∑k 6=0

1(iω+i 2π

Tsk)l+1

.

By putting the two previous expressions on a common denominator, we get the thefollowing relation

Gc(iω)H(iω)Gd(eiωTs)

− Fc(iω) → Fc(iω)R(iω)S(iω)

where

R(iω) =1−Gc(iω) (iω)l

b0Gc(iω)

iω +∑

k 6=0b0

(iω+i 2πTs

k)l+1

and

S(iω) =∑

k 6=0

b0

(iω + i 2πTs

k)l+1.

Since F and R are bounded in ω, ω is below the Nyquist frequency and the termsof S are bounded as

∣∣∣∣∣∣∣1(

iω + i2πTs

k)l+1

∣∣∣∣∣∣∣≤

∣∣∣∣T l+1

s

(iωTs + i2πk)l+1

∣∣∣∣ ≤ C

(Ts

k

)l+1

if k 6= 0, the result

∣∣∣∣Gc(iω)H(iω)

Gd(eiωTs)− F (iω)

∣∣∣∣ ≤ |F | |R| |S| ≤ CT l+1s

follows. 2

8.6 Conclusions 93

This opens for the estimation of the continuous-time Fourier transform of the out-put as

Yc(iω) = F (l)c (iω)Yd(eiωTs)

which can be interpreted as y being integrated l times between the sampling in-stants. The parameters can then be estimated from Yc using the objective function

Vc(θ) ,Nω∑

k=1

∣∣∣Yc(iωk)−Gc(iωk, θ)Uc(iωk)∣∣∣2

.

This two-stage process of first estimating the continuous-time spectrum and thenestimate the parameters is an advantage of the direct method.


In Table 8.5.2 the parameters of the system in (8.8) have been estimated with re-spect to the sampling time Ts using the methods described in this chapter. Method1 in this table is the indirect method while method 2 is the indirect method withmodified objective. Method 3 is the direct method with modified transform whilemethod 4 is the direct method without modification of the transform. As canbe seen, the approximate method performs very well in comparison to the exactmethod at high sampling rates.

8.6 Conclusions

We have studied different approaches to direct estimation of continuous-time trans-fer functions based on sampled data observations by transforming the data to thefrequency domain. If the input intersample behavior is known this can be donewithout approximation. In particular, for piece-wise constant and piece-wise linearinputs, there are well known, but somewhat complicated formulas for this. Wehave investigated various frequency domain approximations of the exact transfor-mation that are simpler to use and give good approximations, at least for suffi-ciently fast sampling. Essentially, the approximations are based on replacing thetrue parameter-dependent system with a number of integrators that equal the poleexcess of the system. An important benefit of this study is that it is probablypossible to use these experiences in the non-uniformly sampled case.


Ts Method a1 a2 b0 b1

True 3 2 1 0.50.1 1 3.0102 2.0005 1.0032 0.5023

2 3.0040 2.0122 1.0001 0.50523 2.9808 2.0512 0.9923 0.51844 3.1611 1.8367 1.0807 0.4702

0.2 1 3.0193 2.0241 1.0046 0.50922 2.9994 2.0662 0.9950 0.52023 2.9220 2.1871 0.9670 0.56244 3.2184 1.9481 1.1265 0.5201

0.3 1 3.0054 1.9926 1.0024 0.49692 2.9569 2.0811 0.9804 0.52033 2.8099 2.2975 0.9224 0.59784 3.1585 2.2402 1.1238 0.6256

0.4 1 3.0153 2.0014 1.0055 0.50132 2.9160 2.1552 0.9620 0.54263 2.7023 2.4803 0.8682 0.66114 3.1002 2.9902 1.0777 0.8951

Table 8.5 Comparison of the performance of the indirect method (Method1), the indirect method with modified objective (Method 2),the direct method with modified Fourier transform (Method 3)and the direct method without modification (Method 4). Thestatistics for each sampling time has been estimated by NMC =100 Monte-Carlo simulations with SNR = 40 dB.

9Conclusions and Further Research

”Be careful whose advice you buy, but be patient with those whosupply it. Advice is a form of nostalgia. Dispensing it is a way of fishingthe past from the disposal, wiping it off, painting over the ugly partsand recycling it for more than it was worth”

Baz Lurman, Everybody’s Free to Wear Sunscreen

This thesis has been propelled by the will to find alternative ways of identifyingcontinuous-time noise models in the frequency-domain. In Chapter 3 an approx-imate maximum-likelihood method for doing this was identified. Unfortunatelythe method required an accurate estimate of the continuous-time spectrum of theoutput while there are only discrete-time measurements of the output available.

Producing an estimate of the continuous-time spectrum for uniformly and non-uniformly sampled data then became the objective of Chapter 5. Using piecewiseconstant and piecewise linear interpolation estimates of the continuous-time spec-trum with good asymptotic properties were produced. In Chapter 5 issues relatedto bias and variance within this framework was analyzed. The method was alsoput to test in Chapter 6. Here parameters were estimated from real-life data froma tire pressure monitoring application.

Numerical experiments on the other hand showed seemingly disappointing re-sults for models of different orders at medium sampling rates. Spurred by thesefindings further investigations into how to improve estimates of the spectrum werecarried out in Chapter 7. In order to isolate central issues the focus was moved backto equidistant sampling. Here it was found that the relative degree of the modelplays a pivotal role and a connection to the sampling zeros was made. This led to

95

96 Chapter 9 Conclusions and Further Research

a way to estimate the continuous-time spectrum from the discrete-time spectrum.In Chapter 8 the estimation approach from the previous chapter was extendedto continuous-time output error models. Here a similar way of estimating thecontinuous-time Fourier transform of the output was discovered.

Another issue that arose in connection with estimation of continuous-time noisemodels from vibration data is that of frequency-domain outliers. In the time-domain outliers have traditionally handled by robust norms of various kind. Thequestion is if it is possible to take a similar approach in order to avoid the effect offrequency-domain outliers.

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Methods for Frequency Domain Estimation of Continuous-Time Models