disturbance characterization for mimo controlmate.tue.nl/mate/pdfs/9382.pdf · characterized....

Disturbance Characterization forMIMO Control

T. ten Dam

DCT 2008.090

Master’s thesis

Coach(es): M. Boerlage

Supervisor: A.G. de JagerM. Steinbuch

Committee: M. SteinbuchM.J.G. van de MolengraftD. KosticM. Boerlage

Technische Universiteit EindhovenDepartment Mechanical EngineeringDynamics and Control Technology Group

Eindhoven, July, 2008

Summary

In today’s demanding industry, servo systems are pushed to their limits. Techniques like dis-turbance accommodated control and the internal model principle have shown that knowledge ofthe disturbances is essential for the control design in order to increase performance. In orderto apply these techniques to multivariate systems, knowledge of multivariate disturbances is re-quired. Therefore, a control relevant disturbance identification procedure for MIMO systems isdeveloped. The developed procedure consists of a couple of separate steps and is tested on theAVIS experimental setup.

First, a frequency response model of the system is obtained, using spectral analysis tech-niques. The obtained model is validated on basis of the coherence, which can be seen as thesignal to noise ratio. The model can be improved by choosing a sum of sinusoids as input signal.

Next, a controller is designed on basis of the frequency response model, that stabilizes theclosed loop system. The stability of the closed loop system is verified on basis of a couple ofstability criteria that are derived from the Nyquist theorem. Alternatives for the Nyquist theoremare used, since this theorem is difficult to interpret for MIMO systems and the effect of a changein the controller on the stability of the closed loop system is not straightforward. Stability crite-ria that are derived are the characteristic loci, the spectral radius, the structured singular valuestability criterium and the Gershgorin bounds.

Subsequently, a parametric model of the sensitivity, the transfer function from output distur-bances to the output, is derived using the n4sid subspace technique. The power of the subspacetechniques is that it is possible to derive a MIMO model on basis of a single experiment. Theobtained model is validated on basis of cross correlation analysis of the residu and the input.

The output disturbances are reconstructed on basis of a time measurement of the outputs,using a Toeplitz matrix, which is a by-product of the subspace identification technique.

Finally, the disturbances are characterized and modeled. The characterization is done on ba-sis of the multivariate power spectral density and the bicoherence, which is a third order statistic.Analysis of the multivariate power spectral density has lead to an improved control design. Theoutput disturbances are modeled, using a stochastic realization subspace technique and by theselection of some basis functions for a waveform model. It is shown that both models are able toreproduce the second order statistics of the disturbances reasonable good.

Using the presented procedure, it is possible to identify and characterize the multivariatedisturbances acting on a MIMO system, which can be used in an improved control design.

iii

iv SUMMARY

Contents

Summary iii

1 General Introduction 11.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

I Theory 5

2 Signal Descriptions 72.1 Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Higher Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 First order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Second order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 Third order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.4 Fourth order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Waveform Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 System Identification 273.1 Frequency Response Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.1 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1.2 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Subspace Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.1 N4SID subspace identification . . . . . . . . . . . . . . . . . . . . . . . . 303.2.2 Time Domain Prediction and Reconstruction . . . . . . . . . . . . . . . . 343.2.3 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Stochastic Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.1 Canonical Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

v

vi CONTENTS

4 MIMO Control and Stability 434.1 Decentralized Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 Stability Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

II AVIS application 51

5 AVIS System Identification 535.1 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.1.1 Input Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2 N4SID Subspace Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2.1 Input Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2.2 AVIS Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2.3 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6 AVIS Control Design 616.1 Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2.1 Independent Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.2.2 Sequential Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.2.3 Combined Sequential and Independent Design . . . . . . . . . . . . . . . 696.2.4 Stability verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7 AVIS Disturbance Characterization 777.1 Disturbance Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.2 Bicoherence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807.3 External Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.4 Multivariate Power Spectral Density Analysis . . . . . . . . . . . . . . . . . . . . 817.5 Parametric Disturbance Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.5.1 Stochastic Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.5.2 Waveform Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867.5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7.6 Improved Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

8 Conclusion en Recommendations 918.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

A Power spectral Density Estimator 99

B Identification and Estimator information 103

CONTENTS vii

C Calibration of the AVIS experimental setup 105

D AVIS model Figures 107

E AVIS PSD Figures 123

F MATLAB scripts 135

viii CONTENTS

Chapter 1

General Introduction

In today’s demanding industry, servo systems are pushed to their limits. Techniques like distur-bance accommodated control [10] and the internal model principle [6] have shown that knowledgeof the disturbance is essential for the control design in order to increase performance. These con-trol strategies however assume that knowledge of the disturbances is available. Other research hasshown that a disturbance signal can be reconstructed by backwards filtering of a measurementsignal [1] given an accurate model of the corresponding transfer functions. These models can beobtained using system identification techniques like spectral analysis [8] or subspace methods[12].

The disturbance signal that can be obtained in this way needs further interpretation, char-acterization and modeling before it can be used in a control design. General signal descriptiontechniques can be used for that purpose on basis of various model assumptions. The definitionof another signal class, the waveform structure [11, 24], can provide additional insight in signalcharacteristics. Further characterization can also be reached by examining the (higher order) stat-ics of the signal [29, 16, 18, 15].

The problem is that most of these techniques hold for SISO (single input, single output) sys-tems and scalar signals, but most applications are MIMO (multi input multi output). Therefore,this work will focus on MIMO systems and multivariate signals. Since these MIMO systemsbecome more and more complex with increasing input output relations and accompanying in-teractions the aforementioned techniques need to be studied on their applicability on MIMOsystems and multivariate signals.

The interaction in a MIMO system increases the complexity, but the interaction in a multi-variate disturbance signal might help in the control design. When at this moment for example amultivariate disturbance at a closed loop MIMO system is examined, all disturbances are consid-ered independently, without considering their interactions. In the case that there is one source,for example a pump working at a certain frequency, that causes distortions in all channels of thesystem, then the interaction is not considered. If this problem is treated in a MIMO framework,the control design may benefit from the knowledge of interaction due to a single source and theperformance may increase with a decreased control effort.

1.1 Objective

The main objective of this report can be formulated as follows:

1

2 CHAPTER 1. GENERAL INTRODUCTION

Develop a control relevant disturbance identification procedure for MIMO systems.Illustrate this procedure on the AVIS experimental setup.

In order to fulfill this objective, a strategy is embraced consisting of the following sub objectives.

• Present a characterization and identification of multivariate disturbances.

• Present a system identification technique that is capable of handling MIMO systems.

• Present a control design and stability analysis which guarantees that the closed loop MIMOsystem is stable considering interactions.

The Dynamics and Control lab hosts a MIMO system, the AVIS [21], that can be used for testingand validating the multivariable theories.

The intended disturbance identification procedure must be applicable to closed loop systems,since in practise most systems are in closed loop. Therefore, a stabilizing control design is nec-essary to let the AVIS operate in closed loop. When the system is in closed loop, the transferfunctions from disturbances to outputs can be identified with the multivariable system identifi-cation technique. The disturbances can then be reconstructed on the basis of the inverse of theidentified transfer functions. As a result of this reconstruction, the disturbances can be furthercharacterized. Important in this analysis is that no simplifications to scalar expressions is made.In that way, the intended procedure is able to handle MIMO systems as such and may benefitfrom multivariable properties.

1.2 Outline

The report is divided into two parts. In Part I the theory necessary to answer the sub objectives,in order to fulfill the main objective is treated, whereafter this theory is applied to the AVIS ex-perimental setup in Part II. The theory is clarified with examples where necessary.

Part I starts with a signal description in Chapter 2 which can be used for the characteriza-tion of disturbance signals. A distinction is made between deterministic and stochastic signalsand the waveform description is introduced, which is a combination of these two classes. Fur-thermore a detailed statistical analysis of stochastic signals is presented. Next in Chapter 3 thesystem identification is treated. First, a spectral analysis technique is presented in order to ob-tain frequency response models for control design. Next, a subspace identification technique ispresented in order to obtain a parametric time-domain model for output prediction. Finally, astochastic realization technique is shown for the identification of disturbance models. Finally,in Chapter 4 the control design is described. The decentralized feedback control approach is ex-plained and stability measures for multi variable systems are presented.

Part II starts with the identification of the AVIS in Chapter 5. A non parametric model isderived using spectral analysis, after which a state space model is derived using a subspace tech-nique. Next the control design for the AVIS is described in Chapter 6 after a rigid body decouplingof the system. Moreover, stability of the closed loop system is proven given the stability measures.Finally in Chapter 7, the disturbance identification is treated. First the relation between measur-able signals and disturbances is derived, after which these disturbance signals are reconstructed.

1.2. OUTLINE 3

These signals are analyzed using the signal description tools presented in Chapter 2 and 3. Todemonstrate this approach a synthetic external disturbance is applied to the plant during thesemeasurements.

In Chapter 8 conclusions are drawn and recommendations for future research are given.

Figure 1.1: Overview of the AVIS experimental setup and a close-up of the isolators

4 CHAPTER 1. GENERAL INTRODUCTION

Part I

Theory

5

Chapter 2

Signal Descriptions

When an experimental signal is available for further analysis, e.g. modeling of the dynamics,control design, fault detection, etc., it is convenient to know more about its properties. In order todeal with these experimental signals, it is important to make some distinction between differenttypes of signals. Commonly it is assumed that there are two classes of signals: stochastic anddeterministic signals. This difference is shown on basis of the "spectrum of uncertainty" inFigure 2.1, where the stochastic signals (white and colored noise) are on the uncertain side of the"spectrum" and the deterministic signals on the certain side of the spectrum.

Stochastic signals are random in nature, i.e. the exact value at a certain moment in the futurecannot be predicted. Future values of a stochastic signal can only be predicted with a certain prob-ability. Therefore the statistics of these signals should be determined in both time and frequencydomain. Gaussian processes are completely described by their first and second order statistics,but if the process deviates from Gaussianity, higher order statistics [9, 17] can be studied to pro-vide additional information.

Deterministic behavior of signals on the other hand can be well predicted at a certain momentin the future, since their structure is known beforehand.

Disturbance signals are in general combinations of a deterministic and a stochastic parts.These signals are thus situated more to the center of the "spectrum of uncertainty". The questionarises if it is possible to model these disturbance signals properly with a waveform structure[11, 24], since a stochastic description is most of the time too general, whereas a deterministicdescription is too strict.

In this chapter, first the different types of processes will be discussed that are considered inthis chapter. Next the definitions of the first four orders of statistics will be given, whereafterfor each statistic will be explained what its interpretation is and an example will show how thesequantities can be used in practise. Finally the waveform structure is presented as an alternativemodel for the stochastic and deterministic signal descriptions and some special choices for theselection of the basis functions are discussed.

7

8 CHAPTER 2. SIGNAL DESCRIPTIONS

?

?

?

? whitenoise

colorednoise

waveformstructure

deterministic

Spectrum of Uncertainty

Figure 2.1: Relative position of different signal types in the spectrum of uncertainty

2.1 Processes

In this work, a couple of different processes are considered. A schematic representation of theseprocesses is given in Figure 2.2. It is assumed that all system are either linear time invariant (LTI)or non linear time invariant (NLTI). The input signal or initial state is not known and the outputof the process, x(k), is measured in discrete time.

The first process is a LTI system with an initial condition, x0. The second process is a NLTIsystem with an initial condition. The third process is a LTI system with a Gaussian input signal,wG. The fourth process is a LTI system with a non Gaussian input signal, wNG. The last processis a NLTI system with a Gaussian input signal.

Process one and three are Gaussian processes which can completely be described by first andsecond order statistics. However, on basis of output information it is not possible to distinguishbetween these two systems. Process two, four and five deviate from Gaussianity due to a nonGaussian input or non linearities in the system. In the next example will be shown that a nonGaussian signal can be considered as the output of a NLTI system with a Gaussian input (pro-cess five). If this holds and the input is unknown, process four and five are undistinguishable.Therefore all further processes discussed in this chapter are considered to have a Gaussian input.

An example of a non Gaussian distribution is the χ2 distribution, defined as:

χ2 =ν∑

k=1

X2k , (2.1)

with ν the degree of freedom and Xk a single realization of ν different independent normaldistributions. Equation (2.1) can be seen as a realization of process five, with: wG = Xk andNLTI = f(x2). This is a single example of a non Gaussian signal that is the output of a realizationof process five, but it is assumed that this holds for all non Gaussian signals.

2.1. PROCESSES 9

LTI

LTI

LTI

NLTI

wG

wNG

wG

x0

assumed measured

x(k)

x(k)

x(k)

x(k)

(1)

(3)

(4)

(5)

NLTI

x0

x(k)

(2)

Figure 2.2: Schematic representation of the considered processes


2.2 Higher Order Statistics

In this section the definitions of the first four orders of statistics will be given. Then for eachstatistic it will be explained what its interpretation is. At the end an example will show how thesequantities can be used in practise.

2.2.1 First order

The most evident characteristics of a signal are the mean and maximum values of the signal.These quantities can for example be used to detect offsets and amplifier saturations.

The mean value, µ, is defined as the first moment, m1, or first cumulant, c1, of a randomsignal:

µx = m1x = c1x = E{x}, (2.2)

with x a realization of a certain process.

The estimation of expectations is not considered explicitly in this chapter [29]. The readerhowever has to be aware of the difficulties considering the estimation. The estimation can forexample become unreliable due to non stationarity or outliers in the signal and the variance ofthe expectation reduces for larger data sequences. The digital motion system we consider in thiswork have high sample rates, so large data sequences can easily be gathered. All this has beenconsidered when an estimation is given in this chapter.

2.2.2 Second order

When the first order quantities do not provide sufficient information one can estimate and studythe second order statistics. These are the variance, correlation and the power spectrum. Thesequantities are useful for example to detect channel interactions, signal fluctuations and the en-ergy of the signal at certain frequencies.

The joint second moment, m2, for a scalar process is defined as in Equation (2.3) and isknown as the correlation, r. In practise the variance or the squared standard deviation, σ2, ismostly used. This is defined as the joint second central moment, which equals the joint secondcumulant, c2, and is described in Equation (2.4). For zero mean signals the second order mo-ment and cumulant become the same as can be concluded from Equation (2.5). Because this avery useful advantage that saves computation time, most signals will be made zero mean beforefurther analysis. This can be done without losing generality.

rx = m2x = E{x2} (2.3)

σ2x = c2x = E{(x− µx)2} (2.4)

c2x = m2x −m21x. (2.5)

The second moment sequence, m2(τ1), known as the correlation sequence, r(τ1), is definedin Equation (2.6). The second cumulant sequence, c2(τ1), known as the covariance sequence isdefined in Equation (2.7) and is related to the moment sequences as in Equation (2.8). For zero

2.2. HIGHER ORDER STATISTICS 11

mean signals, the covariance sequence equals the correlation sequence. These sequences areconsidered in order to obtain more information from a signal on basis of second order statistics.

rx(τ1) = m2x(τ1) = E{x(k)x(k + τ1)} (2.6)

σ2x(τ1) = c2x(τ1) = E{(x(k)− µx)(x(k + τ1)− µx)} (2.7)

c2x(τ1) = m2x(τ1)−m21x, (2.8)

with τ1 the time lag. For zero lag, these Equations reduce to Equation (2.3) to (2.5). Thesesequences provide additional information about the dependency on past values, which can beuseful for detecting the non stationarity of a process.

To complete the second order analysis the discrete step fourier transform of the second ordercumulant sequence, C2x(ω1), as defined in Equation (2.9), can be estimated. This quantity isknown as the power spectral density (PSD), Φx(ω1),

Φx(ω1) = C2x(ω1) =∞∑

τ1=−∞c2x(τ1)e−jω1τ1 . (2.9)

Since the cumulant sequence and the fourier transform can not be calculated exactly severaladvanced estimators [29] are described in detail in Appendix A. The PSD describes the frequencycontent of a process, showing which frequencies are present and how much power they possess.This can be very useful in i.e. fault detection or control design.

Multivariate

So far we have only looked at single random processes and their auto correlations. This can easilybe extended to multi variable processes and their cross correlation functions. The correlationchanges then to the cross correlation matrix, Rxy, as defined in Equation (2.10) and the variancechanges to the cross covariance matrix, Σ2

xy, Equation (2.11). If x = y Equation (2.10) and (2.11)describe the auto correlation matrix and auto covariance matrix respectively. The same extensioncan be made for the correlation sequence (2.12) and covariance sequence (2.13),

Rxy = M2xy = E{xyT } (2.10)

Σ2xy = C2xy = E{(x− µx)(y − µy)T } (2.11)

Rxy(τ1) = M2xy(τ1) = E{x(k)y(k + τ1)T } (2.12)Σ2

xy(τ1) = C2xy(τ1) = E{(x(k)− µx)(y(k + τ1)− µy)T }. (2.13)The boldface in this equation indicates that we are dealing with a vector for lower case or a matrixfor capitals.

In order to calculate the multi variate PSD, Φxy(ω1), the discrete fourier transform has to betaken for each element in the covariance sequence matrix Σ2

xy(τ1), evaluated at all lags τ (2.14),

Φxy(ω1) = C2xy(ω1) =∞∑

τ1=−∞C2xy(τ1)e−jω1τ1 . (2.14)

Using the concepts shown in this section, processes can be described, but only linear gaussianprocesses are completely described by their second order statistics. In order to describe non linearsystems or processes with a non Gaussian input, higher order statistics can be studied. These willbe explained in the next sections.


Normal

Positive skewness Negative skewness

Positive kurtosis Negative kurtosis

Figure 2.3: Illustration of the skewness and kurtosis, compared to a normal distribution

2.2.3 Third order

When a realization is studied that is not generated by a Gaussian process, third or higher orderstatistics are needed. In this section the third order statistics will be described, which are theskewness, bispectrum and bicoherence. These quantities give information about the skewnessof the underlying distribution and about phase coupling. This is useful for characterizing thegenerating process.

The joint third moment, m3, is defined in Equation (2.15). The joint third cumulant, c3,which equals the joint third central moment, is defined in Equation (2.16) and is known as theskewness, γ. The joint third cumulant is related to the joint third moment in Equation (2.17), so,for zero mean signals, the third order moment equals the third order cumulant,

m3x = E{x3} (2.15)

γx = c3x = E{(x− µx)3} (2.16)

c3x = m3x − 3m2xm1x + 2m21x. (2.17)

The skewness is a measure for the skewness of the underlying distribution. This is illustratedin Figure 2.3. For symmetric distributions the skewness is zero, which can be concluded fromEquation (2.16). If γ < 0 the distribution is right or negative skewed, which means that the massof the distribution is on the right from its mean value. For γ > 0 the inverse applies.

The definitions in Equation (2.15) to (2.17) can be extended to Equation (2.18) to (2.20)in order to define the third moment, m3(τ1, τ2), and cumulant sequence, c3(τ1, τ2), and their


mutual relation,

m3x(τ1, τ2) = E{x(k)x(k + τ1)x(k + τ2)} (2.18)

c3x(τ1, τ2) = E{(x(k)− µx)(x(k + τ1)− µx)(x(k + τ2)− µx)} (2.19)

c3x(τ1, τ2) = m3x(τ1, τ2)−m1x[m2x(τ1) + m2x(τ2) + m2x(τ1 + τ2)] + 2m31x. (2.20)

For zero lag these Equations reduce again to Equation (2.15) to (2.17). The meaning of thesequantities is not straightforward, but the third order cumulant sequence is required to definethe bispectrum, C3x(ω1, ω2), as in Equation (2.21). The spectrum of the cumulant instead of themoment is estimated, since this has some additional benefits [16, 18],

C3x(ω1, ω2) =∞∑

τ1=−∞

∞∑τ2=−∞

c3x(τ1, τ2)e−j(ω1τ1+ω2τ2). (2.21)

The estimation of the bispectrum, as discussed in for example [17, 16], shows that the bispectralestimates have a variance that depend on the power spectral properties of the signal. Thereforethe bispectrum is often normalized to remove this effect [5]. There are two most commonly usednormalizations, namely the skewness function, P s

3x(ω1, ω2), and the bicoherence, P b3x(ω1, ω2), as

defined in Equation (2.22) and Equation (2.23) respectively.

P s3x(ω1, ω2) =

√|C3x(ω1, ω2)|2

C2x(ω1)C2x(ω2)C2x(ω1 + ω2)(2.22)

P b3x(ω1, ω2) =

√|C3x(ω1, ω2)|2

E{|X(ω1)X(ω2)|2}E{|X(ω1 + ω2)|2} , (2.23)

with X(ω1) the fourier transform of x(t). The most important difference between the skewnessfunction and the bicoherence is that the latter is bounded between zero and one, while the skew-ness function is unbounded.

The motivation to study the bicoherence is twofold. First, the bicoherence can be used toextract information due to deviations from Gaussianity and suppress additive (colored) Gaussiannoise. Second, the bicoherence can be used to detect and characterize asymmetric non linearitiesin signals via quadratic phase coupling or identify systems with quadratic non linearities.

The extension to multivariate processes that was made for second order statistics will not bemade for the third order statistics, since the computational effort is too large. The extension tocross cumulants on the other hand can be made. The third order cross moment, m3xyz , and cu-mulant, c3xyz , and the third order moment, m3xyz(τ1, τ2), and cumulant sequence, c3xyz(τ1, τ2),are defined in Equation (2.24) to Equation (2.27),

m3xyz = E{xyz} (2.24)

c3xyz = E{(x− µx)(y − µy)(z − µz)} (2.25)

m3xyz(τ1, τ2) = E{x(k)y(k + τ1)z(k + τ2)} (2.26)

c3xyz(τ1, τ2) = E{(x(k)− µx)(y(k + τ1)− µy)(z(k + τ2)− µz)}. (2.27)


The cross bispectrum is calculated as in Equation (2.21) only using the cross cumulant sequence,c3xyz(τ1, τ2), instead of the auto cumulant sequence, c3x(τ1, τ2). The cross bispectrum can forexample be useful in characterizing non linear systems, when both the input and output are avail-able.

It is shown in this section that third order statistics can be useful for characterizing nonGaussian signals and asymmetric quadratic non linearities in particular. Since the class of signalswith quadratic non linear behavior is very limited, the extension to fourth order statistics will bemade in the next section.

2.2.4 Fourth order

When a realization is studied that is not generated by a Gaussian process and has no asymmet-ric quadratic non linearities, one could have a look at the fourth order statistics. In this sectionthe definitions of these statistics are given, which are the kurtosis, trispectrum and tricoherence.These quantities give information about the shape of the underlying distribution and about cubicphase coupling.

The joint fourth moment, m4, is defined in Equation (2.28). The joint fourth cumulant, c4,known as the kurtosis, κ, does not equal the joint fourth central moment as was the case for lowerorder statistics. Therefore c4 is defined as function of moments in Equation (2.29),

m4x = E{x4} (2.28)

κx = c4x = m4x − 3m22x − 4m3xm1x + 12m2xm2

1x − 6m41x. (2.29)

Kurtosis is often normalized with σ4, which implies that for zero mean signals: κ = m4σ4 − 3, as

can be derived from Equation (2.29). A Gaussian distribution has zero kurtosis (mesokurtic). Ifκ < 0 (platykurtic), the distribution function is flatter than Gaussian. If κ > 0 (leptokurtic) thedistribution has a sharper peak and longer tails than a Gaussian distribution. This is illustratedin Figure 2.3.

The definitions in Equation (2.28) and (2.29) can be extended to Equation (2.30) and (2.31)in order to define the fourth moment, m4(τ1, τ2, τ3), and cumulant sequence, c4(τ1, τ2, τ3),

m4x(τ1, τ2, τ3) = E{x(k)x(k + τ1)x(k + τ2)x(k + τ3)} (2.30)

c4x(τ1, τ2, τ3) = m4x(τ1, τ2, τ3)−m2x(τ1)m2x(τ3 − τ2)−m2x(τ2)m2x(τ3 − τ1)−m2x(τ3)m2x(τ2 − τ1)−m1x[m3x(τ2 − τ1, τ3 − τ1) + m3x(τ2, τ3)+m3x(τ1, τ3) + m3x(τ1, τ2)] + m2

1x[m2x(τ1) + m2x(τ2) + m2x(τ3)+m2x(τ3 − τ1) + m2x(τ3 − τ2) + m2x(τ2 − τ1)]− 6m4

1x. (2.31)

This fourth cumulant sequence becomes very complicated. For zero mean signals, however, itsexpression reduces to Equation (2.32):

c4x(τ1, τ2, τ3) = m4x(τ1, τ2, τ3)−m2x(τ1)m2x(τ3 − τ2)−m2x(τ2)m2x(τ3 − τ1)−m2x(τ3)m2x(τ2 − τ1). (2.32)


For zero lags, Equation (2.30) to (2.32) reduce again to Equation (2.28) and (2.29). The extensionto cross cumulants will not be made for the fourth order statistics, since these expressions becometoo complex to present here. The interpretation of these sequences is not straightforward, but thecumulant sequence will be used to define the trispectrum, C4x(ω1, ω2, ω3), as in Equation (2.33):

C4x(ω1, ω2, ω3) =∞∑

τ1=−∞

∞∑τ2=−∞

∞∑τ3=−∞

c4x(τ1, τ2, τ3)e−j(ω1τ1+ω2τ2+ω3τ3). (2.33)

The trispectrum has the same advantages as the bispectrum. It can be used to extract informa-tion due to deviations from Gaussianity even in presence of additive (colored) Gaussian noise.Moreover, the trispectrum can be used to identify non minimum phase systems. The biggestadvantage of the trispectrum over the bispectrum, however, is that the trispectrum can be used todetect and characterize symmetric non linearities in signals via cubic phase coupling, or identifysystems with cubic non linearities. This is a big advantage, since in many real world problemsthe systems under consideration have (nearly) symmetric non linearities and also the input iscommonly symmetrically distributed.

The estimation of the trispectrum is described in i.e. [17] and also faces problems due to vari-ance of the estimation, like the bispectrum does. Therefore, the trispectrum is often normalizedto the kurtosis function, P k

4x(ω1, ω2, ω3), or the tricoherence, P t4x(ω1, ω2, ω3), as defined in Equation

(2.34) and (2.35):

P k4x(ω1, ω2, ω3) =

√|C4x(ω1, ω2, ω3)|2

C2x(ω1)C2x(ω2)C2x(ω3)C2x(ω1 + ω2 + ω3)(2.34)

P t4x(ω1, ω2, ω3) =

√|C4x(ω1, ω2, ω3)|2

E{|X(ω1)X(ω2)X(ω3)|2}E{|X(ω1 + ω2 + ω3)|2} . (2.35)

The most important difference between the kurtosis function and the tricoherence is that thelatter is bounded between zero and one, while the kurtosis function is unbounded.

It can be concluded from this section that next to third order statistics the fourth order statis-tics could be useful for characterizing non Gaussian signals. The extra benefit of the tricoherenceis that it can be used to detect symmetric non linearities, since it can detect cubic phase coupling.The tricoherence however will not be included in the analysis of the examples in this section,since a good estimation of this quantity costs too much computational power. If the tricoherencefor example has to be estimated on basis of 11 samples, 5 lags and 10 frequency points, alreadymore than a million calculations are needed, which takes about 3 seconds. Whereas, for a goodestimation much more samples are needed. Moreover is it hard to visualize the trispectrum,since it is a four dimensional quantity. This problem is tackled in literature by problem specificvisualization [5], a sliced spectral approach [34] or dimension reducing measures like the kurtosis.

2.2.5 Examples

In order to show how the quantities presented in this section can be applied in practise, a theo-retical example will be shown. In this example, it will be illustrated how one can characterize asystem based on the output signal. The example will start with a description of the signal generat-ing system. Next, the quantities presented in this chapter will be estimated and analyzed. Finally,


Table 2.1: Parameters for the Equation of motion of the mechanical system

M [kg] d[Ns/m] k[N/m] ν1[−] ν2[−] δ1[m] δ2[m]linear 1 2.75 5e3 9 9 0 0

asymmetric non linear 1 2.75 5e3 9 5.4 0 0symmetric non linear 1 2.75 5e3 9 9 0.1e-3 0.1e-3

will be concluded what kind of phenomena in the generating system can be characterized usingthese quantities.

Mechanical system

This example is taken from [23]. The mechanical system, as shown in Figure 2.4, is a forcedoscillator with a pair of elastic stops with clearances δ1 and δ2. The output of the system is theacceleration of the mass. The governing Equation of motion of the system is:

Mx + cx + kx + g(x) = f(t), (2.36)

where M denotes the mass, d the damping coefficient, k the linear stiffness, f(t) a randomGaussian excitation force and g(x) a non linear function of the displacement, s, defined as:

g(x) =

0, −δ2 ≤ s ≤ δ1

ν1k(s− δ1), s > δ1

ν2k(s + δ2), s < −δ2

(2.37)

The constants ν1 and ν2 are the ratio of the stiffness of an elastic stop to the linear stiffness k.Using these Equations we are able to simulate a linear system (ν1 = ν2 and δ1 = δ2 = 0), anasymmetric non linear system (ν1 6= ν2 and δ1 = δ2) or a symmetric non linear system (ν1 = ν2

and δ1 = δ2 6= 0). The parameters for these systems are collected in Table 2.1. The labelingsymmetric and asymmetric non linear is based on the relation between the reaction force andthe position as shown in Figure 2.5. A mechanical example of this asymmetric non linearity is acracked beam as shown in [22] and the symmetric non linearity represents for example a systemwith symmetric backlash.

These three systems will be used in this example to study differences between the non linear-ities. First the statistical quantities are analyzed. Whereafter the PSD and the bicoherence will bestudied to further characterize the different processes.

The statistical quantities for the linear and non linear processes are estimated and given inTable 2.2. Both the mean value and the variance are almost the same for all three processes. Onlythe mean value for the asymmetric non linear process is a little larger due to this non linearity.The effect of the asymmetric non linearity is noticeable in the skewness, while the symmetric nonlinearity is noticeable in the kurtosis. This can also be concluded when looking at the histogram ofthe output signals, as shown together with a section of the time trace in Figure 2.6. The histogramfor the output of the linear process looks like Gaussian. The histogram for the asymmetric nonlinear process shows a negative skewness and the histogram for the symmetric non linear processshows a leptokurtic behavior.


s

M

k

k,ν2 k,ν1

d

δ2 δ1

f(t)

Figure 2.4: Forced oscillator model

δ2 δ1

Linear

Asymmetricnon linear

Symmetricnon linear

F

s

Figure 2.5: Reaction force, F , as function of position of the forced oscillator, s


Table 2.2: Statistical analysis of the mechanical system

asymmetric symmetriclinear non linear non linear

mean -0.0001 0.0206 -0.0002variance 3.7585 3.8611 3.9155skewness -0.0008 -0.3040 0.0014kurtosis -0.0077 0.1223 2.3014

On basis of this analysis, it can be concluded that it is possible to detect an asymmetric nonlinearity using the skewness and a symmetric non linearity using the kurtosis. However, in thistheoretical example, it is possible to compare the values of the skewness and kurtosis of boththese non linear cases with the values in the linear case as a reference. For experimental data thiswill not be possible. Consequently, it will be harder to draw conclusions on basis of the skewnessand kurtosis in a practical example. No further characterization of the system can be made on ba-sis of this information. Therefore, the power spectral density and the bicoherence will be studied.Study of the tricoherence would make this analysis complete, but due to visualization difficultiesand large computational times this analysis will be discarded.

The power spectra and bicoherence plot for the linear, symmetric non linear and asymmetricnon linear processes are shown in Figure 2.7. More information about the estimation of thesespectra can be found in Appendix B. From the shape of the three PSD’s, it can be concluded thatwe are dealing with three different processes. The peak in the PSD of the asymmetric processis shifted to a lower frequency due to a lower total stiffness, which influences the resonant fre-quency. The peak in the PSD of the symmetric process is also shifted to a lower frequency. Thisis due to a lower average stiffness. Moreover, in this case is the resonant frequency dependent onthe displacement, so the resonant peak in the PSD widens due to frequency modulation of theestimation.

The PSD’s of the non linear processes also show a small peak. For the asymmetric case thisis at twice the resonant frequency and for the symmetric case at approximately three times theresonant frequency. This could indicate higher order components if it is a matter of quadratic orcubic phase coupling respectively at the resonant frequency. However, if there is no phase cou-pling, these extra peaks could also be a linear phenomenon. Phase coupling can not be concludedfrom the PSD’s since the phase information is lost. To prove quadratic or cubic phase couplingone should study the bicoherence or the tricoherence respectively. From the bicoherence plots inFigure 2.7 can be concluded that there is quadratic phase coupling in the asymmetric non linearprocess at the resonant frequency and there is no quadratic phase coupling in the symmetric nonlinear process. The tricoherence is not available. On basis of the kurtosis and the observationof a peak at three times the resonance frequency in the PSD, can nevertheless be concluded thatthere is cubic phase coupling in the symmetric non linear process at the resonant frequency.

From this example, it can be concluded that on basis of the power spectrum it is possible todistinguish between the different processes in this example. Based on that analysis it is not obvi-ous where these differences come from. The estimation of the skewness and kurtosis indicated


0 1 2 3 4 5−10

−5

0

5

10

Mag

nitu

de [−

]

Time [s]

−10 −5 0 5 100

50

100

150

200

Num

ber

[−]

Magnitude [−]

(a) Linear

0 1 2 3 4 5−10

−5

0

5

10

Mag

nitu

de [−

]

Time [s]

−10 −5 0 5 100

50

100

150

200

Num

ber

[−]

Magnitude [−]

(b) Asymmetric Non Linear

0 1 2 3 4 5−10

−5

0

5

10

Mag

nitu

de [−

]

Time [s]

−10 −5 0 5 100

50

100

150

200

250

300

Num

ber

[−]

Magnitude [−]

(c) Symmetric Non Linear

Figure 2.6: Time trace and histogram of the mechanical system in the linear and both nonlinear cases


0 20 40 60 80 100

10−4

10−2

100

Power spectrum

frequency [Hz]

PS

D [−

]

020

4060

80100

0

50

1000

0.2

0.4

0.6

0.8

frequency [Hz]

Bicoherence

frequency [Hz]

Mag

nitu

de [−

]

(a) Linear

0 20 40 60 80 100

10−4

10−2

100

Power spectrum

frequency [Hz]

PS

D [−

]

020

4060

80100

0

50

1000

0.2

0.4

0.6

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frequency [Hz]

Bicoherence

frequency [Hz]

Mag

nitu

de [−

]

(b) Asymmetric Non Linear

0 20 40 60 80 100

10−4

10−2

100

Power spectrum

frequency [Hz]

PS

D [−

]

020

4060

80100

0

50

1000

0.2

0.4

0.6

0.8

frequency [Hz]

Bicoherence

frequency [Hz]

Mag

nitu

de [−

]

(c) Symmetric Non Linear

Figure 2.7: Power spectrum and Bicoherence of the mechanical system in the linear and bothnon linear cases


the presence of a quadratic and cubic non linearity in the asymmetric and symmetric non linearsystem respectively. However, the location or frequency of these non linearities is not known.The bicoherence has shown that there is quadratic phase coupling at the resonant frequency inthe asymmetric non linear system and no quadratic phase coupling in the symmetric non linearsystem. The tricoherence could have indicated the cubic phase coupling in the symmetric nonlinear proces, but this quantity is not available. On basis of the kurtosis and the observation of apeak at three times the resonance frequency in the PSD, can nevertheless be concluded that thereis cubic phase coupling in the symmetric non linear process.

2.2.6 Conclusion

The first and second order statistics presented in this section can be used to describe linear gaus-sian process completely. In order to describe non linear systems or non gaussian signals the thirdand fourth order statistics, also presented in this section, can be used. The third order statisticsare useful for characterizing asymmetric quadratic non linearities in particular. However, theclass of signals with quadratic non linear behavior is very limited. The fourth order statisticsare useful for characterizing symmetric non linearities on basis of cubic phase coupling. Thetrispectrum however is not included in the analysis of the examples in this section, since a goodexpectation of this quantity costs too much computational power. Moreover, it is hard to visualizethe trispectrum.

From the example can be concluded that on basis of power spectra two process may lookequal, but differ on basis of higher order statistics. It is shown that the bispectrum and trispec-trum may be used to indicate asymmetric non linearities and symmetric non linearities respec-tively.

In Chapter 7 the analysis tools presented in this section will be used to characterize the dis-turbance signals measured on the AVIS. It is tried to use the bicoherence to indicate quadraticnon linearities, if these are present in the AVIS. Moreover, the multivariate PSD will be usedto characterize the disturbance signals. This information may be useful for improved controldesign.


2.3 Waveform Description

In this section the waveform description will be presented. Further will be discussed what howthe basis functions can be selected. Whether this signal class is applicable in the characterizationof disturbance signals will be discussed in Chapter 7.

The main idea behind the waveform mode description is that there are one or more distin-guishable deterministic patterns recognizable in the signal, at least over short time intervals.Moreover, these deterministic patterns can change in magnitude in a random but piecewise con-stant manner. The waveform structured disturbance, dw(t) can be modeled by the analyticalexpression:

dw(t) = W (f1(t), . . . , fM (t); c1(t), . . . , cM (t)), (2.38)

where fi(t), i = 1, . . . ,M are known deterministic functions. The terms ci(t), i = 1, . . . , M areunknown parameters that may jump in value in a random but piecewise constant manner. Thelength of these intervals ∆ck

for these jumps are non equidistant in time. The non linear expres-sion (2.38) can be simplified to a linear expression. This assumption is often justified in practise.Another assumption is that these deterministic functions are any direct Laplace transformablefunction (ramps, steps, sinusoids, etc.), which are most deterministic signals.

dw(t) =M∑

i=1

ci(t)fi(t). (2.39)

Following the waveform description in (2.39), the signal is viewed as a linear combination ofdeterministic signals, each weighted by a piecewise constant random value over a certain amountof time.

The next step is to find a set of differential equations that could have produced the waveform.This could be e.g. a state space model. If the basis functions are Laplace transformable, this stepis rather straightforward. The parameters ci(t) are treated as constants for the moment, but willbe dealt with as part of a time-variant output matrix. The Laplace transform of (2.39) is thengiven by:

Dw(s) =M∑

i=1

ciFi(s) =M∑

i=1

ciΨi(s)Φi(s)

, (2.40)

which can be written as a single ratio of polynomials:

Dw(s) =Ψ(s)Φ(s)

. (2.41)

Given these expressions, there is no unique state space model that generates a time realization(2.39). In order to keep up with the idea of the linear model, it is convenient to first transfer eachLaplace transformed basis function to a separate state space model, whereafter these state spacemodels can be augmented to form one single model as shown in (2.42) and (2.43):

[xdi

(t)dwi(t)

]=

[Adi

Cdi

]xdi

(t), xdi(t0) = xdi0

, i = 1, . . . , M

[xd(t)dw(t)

]=

[Ad

Cd(t)

]xd(t), xd(t0) = xd0

(2.42)

2.3. WAVEFORM DESCRIPTION 23

with:

Ad =

Ad1 ∅. . .

∅ AdM

, xd0 =

xd10

...xdM0

Cd(t) =[c1(t)Cd1 . . . cM (t)CdM

](2.43)

The piecewise constant weighting functions ci(t) are made part of the output matrix Cd(t) inorder to weigh the deterministic signals generated by the system. For multivariate disturbancesignals with similar waveform structure, the output matrix can simply be extended by appendingas many rows as signals, with other piecewise constant weighting functions ci(t).

The next step is to select a set of basis functions that can produce the same waveform as inan observed experimental recording. This has to be done, with insight of the engineer, since thechoice for the set of basis functions is not straightforward.

One possible approach is to look at the PSD of the disturbance signal in order to detect thefrequency of harmonic components present in the waveform. The state space realization of aharmonic basis function is given by (2.44).

Ah =[

0 1−(2πf)2 0

]xh0 =

[01

]

Ch(t) =[c(t)2πf 0

](2.44)

If this approach is not applicable due to a lack of harmonics or if the frequency of the harmonicschange over time, another approach should be attempted. Therefore, the waveform could alsobe approximated with a power series expansion (2.45) [25, 26, 20]. This is a particular usefulapproach if the engineer does not have a good insight in the deterministic functions which arepresent in the signal. The state space realization of the power series expansion is given by (2.46).The power series expansion approximation is only applicable if the waveform varies relativelyslowly with respect to ∆ci . This is important, because if the disturbance signal varies too much,the weighting functions cannot adapt fast enough to the changing signal.

dp(t) =M∑

i=1

ci(t)ti−1 (2.45)

Ap =

0 0 . . . 0 01 0 . . . 0 0...

......

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

xp0 =

10...00

Cp(t) =[c1(t) c2(t) . . . cM−1(t) cM (t)

]

(2.46)

It is of course also possible to combine these two approaches. This is useful if the signal is acombination of some known harmonic components and an unknown part.


Given the description of the waveform structure as presented in this section, it can be con-cluded that this model can be used to represent a wide variety of signals. The assumptions aboutlinearity and the Laplace transformability of the basis functions are often justified in practise.Moreover, it should be noticed that there is no restriction for stationarity, which means that thewaveform structure is well suited to model non stationary signals, for example signals that turnup randomly. This is a benefit of the piecewise constant weighting functions and is not possiblewith the standard deterministic or stochastic approaches. A drawback of the waveform descrip-tion for signals is the difficulty concerning the selection of the basis functions. In certain casesinformation can be found in PSD plots, to select harmonic basis functions. If this is not possible,a power series expansion can be used as basis function.

2.3.1 Conclusion

It is shown in this section that the waveform description of a signal as given in (2.39) is able todescribe not fully deterministic signals. Consequently this signal model is placed in between thestochastic and deterministic signal classes in the "spectrum of uncertainty". Important in thewaveform model is the definition of the random but piecewise constant parameters ci(t), whichmakes it possible for the signal to be non stationary. This is for example useful to model distur-bances that turn up randomly. The choice for the basis functions in the waveform model is notstraightforward. One method is to examine the PSD for harmonic components in the disturbancesignal, which will result in sinusoidal basis functions. Another method is to use a power seriesexpansion. It is also possible to combine the two methods.

The benefits of the waveform structure for control design will lie in the field of disturbanceaccommodation control [10, 14], where is assumed that the disturbances are of known waveformtype. In Chapter 7 will be investigated whether it is possible to model the disturbances workingon the AVIS using this waveform structure.

2.4 Conclusion

In this chapter, first, a distinction is made between the different processes on basis of output in-formation. It can be concluded that it is not possible to distinguish between Gaussian processeswith or without input signal, or distinguish between non gaussian processes with a NLTI systemor non gaussian input signal. Consequently all processes are assumed to have Gaussian inputsignals and LTI or NLTI systems.

Secondly, this chapter a distinction is made between different signal models, with stochasticsignals on the uncertain side of the spectrum and deterministic signals on the certain side of thespectrum. Whereas most experimental recorded signals are better described with a combinationof both signal models, a new signal model, the waveform description, is presented that can beplaced in the middle of the "spectrum of uncertainty".

This waveform model describes a signal on basis of a set of basis functions multiplied bypiecewise constant weighting functions. Consequently this waveform model can be used to rep-resent a wide variety of signals, with no restriction to stationarity, which means that the wave-form structure is well suited to model signals that turn up randomly. This may be a benefit ofthe waveform model over the stochastic or deterministic signal descriptions. The difficulty of the

2.4. CONCLUSION 25

waveform description is the selection of the basis functions. In order to select harmonic basicfunctions can in certain cases information be found in PSD plots, if this is not possible, a powerseries expansion can be used as basis function.

The first four orders of statistics are thoroughly described in this chapter. For second orderstatistics the extension to multivariate signals is made. This may be very useful for describingmultivariate disturbances. First and second order statistics, which are the mean, correlation andPSD, are used to describe LTI systems with a Gaussian input. If a NLTI system has to be de-scribed, third and fourth order statistics are needed. Third order statistics, which are the skewnessand bicoherence, are used to detect asymmetric non linearities. Fourth order statistics, which arethe kurtosis and tricoherence, are used to detect symmetric non linearities.

Chapter 3

System Identification

The control relevant identification of disturbances will finally be tested on the practical AVIS ex-perimental setup. The AVIS has to operate in closed loop, so an accurate plant model is requiredfor control design. Next, the disturbances have to be reconstructed on basis of output measure-ments, so a parametric time domain input-output model is required. Finally, the control relevantcharacteristics of the disturbances have to be modeled, so a parametric time domain output modelis required. Therefore is chosen to describe three different identification techniques that are ca-pable of handling MIMO systems and provide models that are suited for the final purposes.

3.1 Frequency Response Identification

For control design and stability analysis, a representation of the system is required that describesthe control relevant dynamics as good as possible. This representation must be able to describeinput-output relations of MIMO systems. The model should be either in the time or frequencydomain. Here is chosen for a frequency domain model, since the control design will also bein the frequency domain. Moreover, the model size is not restricted, since the control designand stability analysis are both carried out off line. For this purpose, a frequency domain, nonparametric, MIMO model is suited. These kinds of models can be obtained via fourier analysis orspectral analysis of time measurements [13, 28, 7]. The difference between these two approachesis very small and in this work is chosen to follow the spectral analysis. Models obtained viaspectral analysis describe the system as a complex value for each point in the frequency grid. Themore frequency points considered, the more detailed the obtained model becomes. An additionaladvantage of the spectral analysis is that it also provides a measure for the reliability of the model,namely the coherence, which will be used for model validation.

In this section, first the spectral analysis technique is described, whereafter the coherence isdiscussed.

3.1.1 Spectral Analysis

The problem addressed in this section is stated as follows: Estimate a frequency response transferfunction, that describes the control relevant dynamics of the real system as good as possible, onbasis of finite multivariate input-output data {u(t), y(t), t = 0, 1, . . . , N}. Consider therefore adata generating linear system described by:

y(t) = G0(q)u(t) + v(t), (3.1)

27

28 CHAPTER 3. SYSTEM IDENTIFICATION

with q the forward shift operator. The frequency response identification on basis of spectral anal-ysis is based on the assumption that the input, u(t) ∈ Rm, and noise, v(t) ∈ Rp, are uncorrelated.In that case, Equation (3.1) can be written as:

Φyu(ω) = G0(eiω)Φu(ω), (3.2)

and so:

G0(eiω) = Φyu(ω)Φu(ω)−1. (3.3)

Where Φu(ω) and Φyu(ω) are the (cross) power spectral densities of the input and output, whichcan be estimated as described in Appendix A. On basis of these estimations a direct frequencyresponse of the system can be estimated. The quality of this estimation is closely related to theestimation of the spectral densities, which depends on a number of factors like the signal length,number of averaging and choice of window.

On basis of Equation (3.3) and estimations of the (cross) power spectral densities, an estima-tion of the frequency response of the system is determined.

3.1.2 Coherence

By estimating a linear model of the real system, the system should be described by (3.1). Thisis not the case since the system can exhibit non linear behavior and the measurements will becontaminated by measurement noise and disturbances. In order to present a measure for thereliability of the system identification, in spite of these imperfections, the coherence is discussed.

A drawback of the coherence is that it is a measure for SISO systems and no extension toMIMO system can be made as will be shown next. Consequently, the coherence will be discussedfor SISO systems, since a measure for the reliability of the identification is needed.

Non linearities in the system can be considered as disturbances following:

y(t) = G0(q)u(t) + G0,nl(q)u(t) + v′(t)v(t) = G0,nl(q)u(t) + v′(t). (3.4)

Where G0,nl(q) is the non linear part of the system. The disturbance signal, v, is not directlymeasurable, thus the estimation of Φv(ω) has to be based on input and output measurements.Therefore Equation (3.1) is rewritten as follows:

Φy(ω) = |G0(eiω)|2Φu(ω) + Φv(ω). (3.5)

Substituting Equation (3.3) in Equation (3.5) yields:

Φv(ω) = Φy(ω)− |Φyu(ω)|2Φu(ω)

. (3.6)

On basis of Equation (3.6), the Coherence spectrum between output and input, Cyu(ω), can bedefined as:

Cyu(ω) :=

√|Φyu(ω)|2

Φy(ω)Φu(ω). (3.7)

3.1. FREQUENCY RESPONSE IDENTIFICATION 29

This Coherence spectrum takes positive real values between 0 and 1 and acts like a frequencydependent correlation coefficient between the input and output. It is closely related to the signalto noise ratio of the system as can be derived from Equation (3.6):

Φv(ω) = Φy(ω)[1− Cyu(ω)2]. (3.8)

For Cyu(ω) = 1, the noise spectrum is zero at this frequency and thus the output of the system iscompletely determined by the noise free, linear part, G0(eiω)Φu(ω)G∗

0(eiω). For Cyu(ω) = 0, the

output spectrum equals the noise spectrum, Φy(ω) = Φv(ω), and there is no contribution of theinput signal in the output.

For MIMO systems an output is affected by more than one input. Thus if all inputs are simul-taneously excited, the coherence is no longer related to the signal to noise ratio. It is therefore notpossible to use the coherence as a reliability measure for identification of MIMO systems in oneexperiment. Instead, multiple SIMO (single input, multi output) experiments should be carriedout, since then each output is only affected by one output at a time and the coherence can againbe interpreted as a the signal to noise ratio.

The coherence as defined in this section is a good measure for the reliability of the frequencyresponse estimation. The coherence equals 1 if the output of the system is completely determinedby the noise free, linear part and the coherence equals zero if the output of the system is com-pletely determined by noise or non linear part of the system. A drawback is that the coherence isa measure for SISO systems and no extension can be made to MIMO systems. Consequently, inorder to be able to measure the reliability of the obtained models, no MIMO experiments shouldbe carried out, but multiple SIMO experiments.

3.1.3 Conclusion

In this section, spectral analysis is used in order to estimate the frequency response of the sys-tem. This resulted in a frequency domain, non parametric, MIMO model. The quality of theestimation is closely related to the estimation of the (cross) spectral densities of input and out-put. Moreover the coherence is defined, which is a measure for the reliability of the frequencyresponse estimation. The coherence equals one if the output is completely determined by thenoise free, linear part of the system and the coherence equals zero if the output is completelydetermined by noise or the non linear part of the system. Unfortunately, the coherence can notbe extended to analyze MIMO systems. Therefore, SIMO experiments should be carried out, sothat it is nevertheless possible to measure the reliability.


3.2 Subspace Identification

For the prediction of multivariate disturbances on basis of measured outputs a time domain,parametric, MIMO model is required. These kinds of models can be obtained for example viaprediction error methods [28, 13, 7] or subspace identification techniques [12, 19]. In this work ischosen for a subspace technique, since these are well suited for MIMO identification in contraryto the prediction error methods which become pretty complex for MIMO systems. A couple ofdifferent subspace techniques are developed in the past, like the MOESP and N4SID technique[30, 31], in this work is chosen for the N4SID technique. A drawback of the subspace identificationtechniques is that there is no cost function minimization. As a result it is not straightforwardto weigh modeling errors, in order to end with an optimal model. Therefore a model validationtechnique is required to ensure whether the obtained models are a sufficiently well representationof the real system.

In order to characterize the multivariate disturbances on basis of the derived state spacemodel, time domain prediction is investigated. Furthermore, it is investigated if this predictioncan be done more efficient on basis of a Toeplitz matrix that is obtained as by-product during theidentification procedure.

In this section, first, the N4SID subspace identification technique is discussed. Second, timedomain prediction is examined and finally, a model validation technique is presented.

3.2.1 N4SID subspace identification

For subspace identification the following discrete time, LTI system is considered:

x(t + 1) = Ax(t) + Bu(t)y(t) = Cx(t) + Du(t), t = 0, 1, · · · .

(3.9)

Here is x(t) ∈ Rn the state vector, u(t) ∈ Rm the control input, y(t) ∈ Rp the output vector andA ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n, D ∈ Rp×m are constant system matrices. In the followingwe assume that the system that will be identified can be described by these equations and that(A,B) is reachable and (A,C) is observable.

Therefore, the problem addressed in this section is that given finite input-output data {u(t), y(t), t =0, 1, . . . , 2k + N − 2}, the dimension n and the system matrices (A,B, C, D), up to a similaritytransformation, of an arbitrary system need to be identified.

The matrix input-output relations can be derived by repeated use of Equation (3.9):

yk(t) = Okx(t) + Ψkuk(t), t = 0, 1, · · · , k − 1. (3.10)

Where the output vector, yk(t), input vector, uk(t), observability matrix, Ok, and block Toeplitz

3.2. SUBSPACE IDENTIFICATION 31

matrix, Ψk, are defined as:

yk(t) =

y(t)y(t + 1)

...y(t + k − 1)

∈ R

kp, uk(t) =

u(t)u(t + 1)

...u(t + k − 1)

∈ R

km

Ok =

CCA

...CAk−1

∈ R

kp×n, Ψk =

DCB D

.... . . . . .

CAk−2B · · · CB D

∈ R

kp×km

(3.11)

Equation (3.10) can also be written as function of block Hankel matrices:

Yp = OkXp + ΨkUp

Yf = OkXf + ΨkUf .(3.12)

Where the block Hankel matrices can be formed by rearranging the input-output data as follows:

Up := U0|k−1 =

u(0) u(1) · · · u(N − 1)u(1) u(2) u(N)

.... . .

...u(k − 1) u(k) · · · u(k + N − 2)

∈ R

km×N (3.13)

The indices 0 and k−1 denote the arguments of the upper left and lower left element, respectivelyand can be chosen freely for identification. The index k should be chosen larger than the modelorder n, which is not known beforehand. The number of columns of the block Hankel matrix,N , should be chosen sufficiently large.

Define Yp := Y0|k−1 ∈ Rkp×N , Xp := X0 ∈ Rn×N and Uf := Uk|2k−1 ∈ Rkm×N , Yf :=Uk|2k−1 ∈ Rkp×N , Xf := Xk ∈ Rn×N the same way, where the subscripts p and f denote thepast and future, respectively. The total number of observations necessary to form these matricesequals 2k + N − 2. Note that the past and future state matrices cannot be formed directly basedon the input-output data. Further define Wp,Wf ∈ Rk(m+p)×N as:

Wp :=[Up

Yp

], Wf :=

[Uf

Yf

]. (3.14)

Three assumptions (A1 to A3) are made about the input matrix and initial state matrix. Ifthese assumptions do not hold, no reliable identification can be achieved.

A1) rank(X0) = n

A2) rank(U0|2k−1) = 2km, where 2k > n

A3) span(X0) ∩ span(U0|2k−1) = {0}, where span(·) denotes the space spanned by the rowvectors of a matrix


Assumption A1) implies that the state vector is sufficiently excited or the system is reachable.Assumption A2) shows that the input sequence should satisfy the persistently exciting (PE) con-dition of order 2k. Assumption A3) means that the row vectors of X0 and U0|2k−1 are linearlyindependent, or there is no feedback from the states to the input.

Figure 3.1: Oblique projection of Yf onto Wp along Uf

Theorem 3.1. Suppose that assumptions A1), A2) and A3) are satisfied. Let the obliqueprojection of Yf onto Wp along Uf in Figure 3.1 be given by:

ξ = E||Uf{Yf |Wp}. (3.15)

Let the singular value decomposition (SVD) of ξ be given by:

ξ =[U1 U2

] [S1 00 0

] [V T

1

V T2

]= U1S1V

T1 , (3.16)

then, we have the following results:

n = dimS1 (3.17)ξ = OkXf ∈ Rkp×N (3.18)

Ok = U1S1/21 T ∈ Rkp×n, |T | 6= 0 (3.19)

Xf = T−1S1/21 V T

1 ∈ Rn×N . (3.20)

Proof. see [12]

Equation (3.17) only holds for noise free outputs. If the output is disturbed by noise, Yf 6= Yf ,the singular values under a certain bound should be set to zero. This bound determines the finalmodel order and should be selected with the insight of the user.

The proof of Theorem 3.1, is based on the LQ decomposition of the input and output Hankelmatrices:

Uf

Wp

Up

=

L11 0 0L21 L22 0L31 L32 L33

QT1

QT2

QT3

(3.21)


Moreover, it is proven that the LQ decomposition of known inputs and outputs can be linkedto the oblique projection, Equation (3.15), and the block Toeplitz matrix, Ψk, Equation (3.11),respectively.

ξ = OkXf = L32L†22Wp (3.22)

Ψk = (L31 − L32L†22L21)L−1

11 . (3.23)

The estimates of A and C can be derived from the observability matrix, which can be obtained viaEquations (3.22),(3.16) and (3.19). The estimates of B and D can be obtained from the Toeplitzmatrix in Equation (3.23). This approach is known as the MOESP algorithm [30, 31] and will notbe discussed any further. This relation is nevertheless shown, since it will be used in Section 3.2.2.

The N4SID method of identifying state space models uses an estimate of the state vectorgiven by Equation (3.20). The system matrices (A,B, C, D) relate to the state, input and outputas follows:

[Xk+1

Yk

]=

[A BC D

] [Xk

Uk

](3.24)

Where:

Xk+1 :=[x(k + 1) · · · x(k + N − 1)

]Xk :=

[x(k) · · · x(k + N − 2)

]Uk :=

[u(k) · · · u(k + N − 2)

]Yk :=

[y(k) · · · y(k + N − 2)

](3.25)

This is a system of linear equations for the system matrices, so these can be estimated by applyingthe least-squares method:

[A B

C D

]=

([Xk+1

Yk

] [Xk

Uk

]T)([

Xk

Uk

] [Xk

Uk

]T)−1

(3.26)

The N4SID algorithm can now be summarized by the following steps:

1. Compute ξ by using Equation (3.22) and the LQ decomposition in Equation (3.21)

2. Determine the model order, n, by examining the SVD of Equation (3.16) and compute thestate vector Xf using Equation (3.20)

3. Compute the system matrices (A,B, C, D) by solving the regression equation (3.24) usingthe least-squares method in Equation (3.26)

It is shown that when a MIMO system can be described by Equation (3.9) it is possible toidentify the system matrices (A,B, C,D) and the model order n on basis of finite input-outputdata. If the output data is not noise free, the model order can be estimated on basis of a singularvalue decomposition of the oblique projection of Yf onto Wp along Uf . This system identificationapplies if; the system is reachable, the input is persistently exciting and there is no feedbackfrom the states to the input. These assumptions are plausible if the identification experiment isproperly defined. This will be discussed in Section 5.2.


3.2.2 Time Domain Prediction and Reconstruction

In order to characterize disturbance signals, a reconstruction of these disturbances is needed.On basis of the state space model derived in Section 3.2.1, future outputs can be predicted givenarbitrary disturbances or inputs. The inverse of this model might be used to reconstruct theseinputs on basis of the outputs. Moreover will be investigated if there is an efficient method toobtain an inverse model representation.

First recall the system description in Equation (3.9) and the input-output relation in Equation(3.10),

x(t + 1) = Ax(t) + Bu(t)y(t) = Cx(t) + Du(t)

yk(t) = Okx(t) + Ψkuk(t), t = 0, 1, · · · , k − 1.

The observability matrix, Ok, and block Toeplitz matrix, Ψk, can be constructed on basis of theidentified system matrices, but these matrices can also be obtained as by-product during theidentification procedure as can be seen in Equations (3.22) and (3.23).

The state vector, x(t), is not known during a new experiment, but if the system is stable (3.27),the following equivalent equations are satisfied,

max | eig(A)| < 1 (3.27)

limi→∞

CAi−1 = limi→∞

Ok,i = 0 (3.28)

limi→∞

y(t + i) = Ψk,iu(t + i). (3.29)

Where i is the number of time steps ahead prediction and Ok,i and Ψk,i are the ith block rowsof the Observability and Toeplitz matrix respectively. Hence, when the system dynamics arereasonable fast, Equation (3.29) is satisfied for i ¿ k, the state vector can be neglected and theinput-output relation reduces to:

yk(t) = Ψkuk(t). (3.30)

On basis of this equation and the Toeplitz matrix that is either constructed on basis of an esti-mated state space model or obtained during the subspace identification, future outputs can bepredicted on basis of future inputs.

Inverse Model

In this work however, we are interested in te prediction of future disturbances on basis of futureoutputs. Consequently an inverse of the model is needed, which can be obtained in three differ-ent ways: an inverse system identification, the inverse of the state space model or the inverse ofthe input-output relations.

For inverse system identification one may exchange the inputs with the outputs. However,this is not reliable, since assumption A3) from Section 3.2.1, which states that the input may notbe correlated with the states, is not satisfied in that case. There is namely a direct relation between


the states and the output, which will than used as input.

Alternatively, the inverse of the state space model can directly be derived from Equation (3.9)by exchanging the inputs and outputs of an earlier identified model:

x(t + 1) = (A−BD−1C)x(t) + BD−1y(t)u(t) = −D−1Cx(t) + D−1y(t).

(3.31)

On basis of this relation, the inverse system matrices can be defined as: AI = A − BD−1C ∈Rn×n, BI = BD−1 ∈ Rn×p, CI = −D−1C ∈ Rn×m and DI = D−1 ∈ Rm×p. The matrix Dshould be non singular. On basis of these inverse system matrices the inverse Toeplitz matrix, ΨI

k,and inverse observability matrix, OI

k, can be constructed following Equation (3.11). The outputinput relation can be derived by repeated use of Equation (3.31):

uk(t) = OIkx(t) + ΨI

kyk(t). (3.32)

The output input relation can also be derived directly on basis of Equation (3.10) by exchang-ing the inputs and outputs after identification:

uk(t) = −Ψ−1k Okx(t) + Ψ−1

k yk(t). (3.33)

Note that OIk = −Ψ−1

k Ok and ΨIk = Ψ−1

k . Ψ−1k can be determined as the inverse of the block

Toeplitz matrix that is obtained as by-product of the identification procedure.

The state vector in Equation (3.33) is not known, but can be neglected under the assumptionthat the dynamics are reasonable fast as shown earlier in this section. The output input relationthen reduces to:

uk(t) = Ψ−1k yk(t). (3.34)

On basis of this equation, the disturbances can be reconstructed on basis of measured outputs.The benefit of using the Toeplitz matrix as by-product of the identification procedure instead

of the Toeplitz matrix constructed on basis of the estimated system matrices is that no informa-tion is lost during the estimation of the system matrices. A drawback of this approach is that thereconstruction is limited to the horizon, k, chosen during the identification procedure. However,if the system dynamics are stable, the following equivalent equations are satisfied.

max | eig(A)| < 1

limi→∞

CAi−2B = limi→∞

Ψk,i1 = 0.(3.35)

Where Ψk,i1 is the first block in the ith block row of the Toeplitz matrix. If the dynamics arereasonable fast, all the first blocks in the following block rows of the Toeplitz matrix can be ne-glected. The Toeplitz matrix can on basis of this assumption be extended to increase the horizon


to infinity as follows:

Ψ∞ =

DCB D

.... . . . . .

CAk−2B · · · CB D

CAk−2B · · · CB D. . .

... CB D

CAk−2B...

. . . . . .CAk−2B · · · CB D

(3.36)

It is shown that on basis of the input-output relations the output can be predicted given a statespace model or a Toeplitz matrix. Moreover, it is shown that it is possible to reconstruct the inputon basis of the measured output with an inverse of the state space model or the Toeplitz matrix.The use of the Toeplitz matrix is hereby preferred. There is namely information lost during theestimation of the system matrices, due to limited order of the state space model.

The initial state, which is required for prediction on basis of input-output relations, is ne-glected in this section on basis of the assumption of fast and stable dynamics, which is a reason-able assumption. Estimation of the initial state vector will increase the accuracy of the prediction.However, this requires further research, which is not discussed in this work.

3.2.3 Model Validation

The system identification technique presented in this section does not guarantee any reliabilityof the model if the system can not be described by Equation (3.9). This is always the case sincea system exhibits even in the most ideal case some non linear behavior and the output will bedisturbed by noise. Therefore the model should be validated on basis of a measure that indicatesthe reliability of the model. This can be done on basis of a comparison of the frequency responsewith the earlier identified non parametric model. Another approach is to verify whether the out-put of the model compares to the output of the real system and if the residu is not correlated tothe input. This approach is discussed in this section.

If the model is a good representation of the system, this implies that the residual, ε(t) =y(t)− ym(t), with y(t) the system output and ym(t) the model output, both based on input, u(t),asymptotically becomes uncorrelated with past input samples,

rεu(τ) = E{ε(t + τ)u(t)} = 0, τ > 0. (3.37)

It can be shown [7], that if the cross correlation, rεu(τ), drops under a certain bound, P , themodel is a good estimation of the system. The bound is determined by the auto correlation ofboth the residual and the input. The input, u(t), that is used in this analysis should be chosen insuch way that the system is excited the same way as during the intended application of the model.Hereby a model validation technique is presented that gives a measure for the reliability of themodel. The use of this method is illustrated in Section 5.2.


3.2.4 Conclusion

It is shown that when a MIMO system can be described by a discrete time LTI model, it is possibleto identify the system matrices (A,B, C,D) and the model order n on basis of finite input-outputdata. Since the output data is corrupted by noise, the model order can be estimated on basis of asingular value decomposition. The identification algorithm does not guarantee any stability of theobtained models. Moreover, no cost function is minimized, so no modeling errors are weighted.Therefore, a model validation technique is presented that gives a measure for the reliability of themodel, on basis of the cross correlation between residual and input.

The obtained model can be used to predict future outputs on basis of future inputs. Besides,an inverse model is used to reconstruct disturbances on basis of measured outputs. This is onlypossible if the D system matrix is non singular. The best way to reconstruct the disturbances, ison basis of the Toeplitz matrix, which is shown to be a by-product of the proposed identificationalgorithm.


3.3 Stochastic Realization

The multivariate disturbance signals that can be predicted using the models obtained on basisof the subspace identification presented in the previous section can be used to identify a distur-bance model. Therefore an identification technique is necessary that is able to identify a MIMOmodel on basis of output information only. Techniques that are able to do so are called stochasticrealization identification techniques [12, 19]. In this work the canonical variance algorithm (CVA)is described, since it has some benefits over other algorithms like the principal component (PC)algorithm.

3.3.1 Canonical Variance

For the stochastic realization identification, we assume that the signal can be described as theoutput of the following discrete time, LTI system, which is an interpretation of process 3 asshown in Figure 2.2:

x(t + 1) = Ax(t) + w(t)y(t) = Cx(t) + v(t), t = 0, 1, · · · .

(3.38)

With w(t) ∈ Rn and v(t) ∈ Rp zero mean signals with covariance matrix:

Σ2wv = E

{(wv

) (wT vT

)}=

(Q SST R

). (3.39)

Here is x(t) ∈ Rn the state vector, y(t) ∈ Rp the output vector, A ∈ Rn×n, C ∈ Rp×n the constantsystem matrices and Q ∈ Rn×n, S ∈ Rn×p, R ∈ Rp×p the covariance matrices. In the followingwe assume that the stochastic process is stationary, which implies that A is a stable system matrix.

Therefore, the problem addressed in this section is that given finite output data {y(t), t =0, 1, . . . , 2k + N − 2}, the dimension n, the system matrices (A,C), up to a similarity transfor-mation and the covariance matrices (Q,S, R) of an arbitrary system need to be identified so thatthe second order statistics of the output of the model and of the given output are equal.

The CVA is based on the following theorem.

Theorem 3.2 (Stochastic Identification). Under the assumptions that:

1. The process noise, w(t), and the measurement noise, v(t), are not identically zero.

2. The number of measurements goes to infinity N →∞3. The user defined weighting matrices W1 ∈ Rpk×pk and W2 ∈ RN×N are such that W1 is

of full rank and W2 obeys: rank(Yp) = rank(YpW2), where Yp is the block Hankel matrixcontaining the past outputs.

With ξk defined as the oblique projection of future outputs on the past outputs:

ξk = Yf/Yp (3.40)

and the SVD:

W1ξkW2 =[U1 U2

] [S1 00 0

] [V T

1

V T2

]= U1S1V

T1 . (3.41)

We have:

3.3. STOCHASTIC REALIZATION 39

1. The matrix ξk is equal to the product of the extended observability matrix and the forwardKalman filter state sequence:

ξk = OkXf . (3.42)

2. The order of the system (3.38) is equal to the number of singular values in Equation(3.41) different from zero.

3. The extended observability matrix Ok is equal to:

Ok = W−11 U1S

1/21 T. (3.43)

4. The part of the state sequence, Xf , that lies in the column space of W2 can be recoveredfrom:

XfW2 = T−1S1/21 V T

1 . (3.44)

5. The state sequence, Xf , is equal to:

Xf = O†kξk. (3.45)

Proof. see [19]

The matrices Yp, Yf are defined as in Equation (3.13) and Ok as in Equation (3.11).The choices for the weighting matrices W1 and W2 determines the stochastic realization al-

gorithms as described in literature. If W1 = Σ2YfYf

and W2 = I , the algorithm that follows fromTheorem 3.2 is the CVA, which can be summarized by the following steps:

1. Calculate the projections:

ξk = Yf/Yp

ξk−1 = Y −f /Y +

p .

2. Calculate the SVD of the weighted projection:

W1ξkW2 = USV T .

3. Determine the order by inspecting the singular values in S and partition the SVD accord-ingly to obtain U1 and S1

4. Determine Ok and Ok−1 as:

Ok = W−11 U1S

1/21 , Ok−1 = Ok.

5. Determine Xf and Xf+1 as:

Xf = O†kξk, Xf+1 = O†k−1ξk−1.


6. Compute the system matrices (A,C) by solving the regression equation:

[Xf+1

Yk|k

]=

[AC

]Xf +

[ρw

ρv

].

7. Determine the covariance matrices (Q,S, R) from:

E

{(ρw

ρv

)(ρT

w ρTv

)}. =

(Q SST R

)

8. Determine Σs, G and Λ0 from:

Σs = AΣsAT + Q

G = AΣsCT + S

Λ0 = CΣsCT + R.

9. Determine P and K of the forward innovation model, provided that Λ0 − CPCT is nonsingular by solving the Riccati equation:

P = APAT + (G−APCT )(Λ0 − CPCT )−1(G−APCT )T

K = (G−APCT )(Λ0 − CPCT )−1.

10. The identified forward innovation model is given by:

x(t + 1) = Ax(t) + Ke(t)y(t) = Cx(t) + e(t)Σ2

ee = R.

With Y +p = Y0|k, Y −

f = Yk+1|2k−1 and e(t) is zero mean white noise with unit variance. Forthis algorithm the estimation horizon, k, should be chosen larger than the model order and thenumber of measurements, N , should be chosen sufficiently large as is concluded next.

The benefit of the CVA over PC is that it ensures positive realness of the estimated covariancesequence. A drawback of this algorithm is that it does not lead to an asymptotically unbiased es-timate unless N →∞. In practise however, for measurements that are reasonable long, the biasis relatively small. The tradeoff between the benefits and drawbacks of the CVA are thoroughlydescribed in [19].

The reliability of these models can be measured on basis of comparison of the PSD, the PSDof the outputs of the model under influence of a white noise with zero mean and unit varianceshould namely equal the PSD of the measurements.

3.3. STOCHASTIC REALIZATION 41

3.3.2 Conclusion

In this section an identification algorithm is presented that is able to identify a LTI stochastic sys-tem on basis of output information. The algorithm, CVA, estimates the model order n, the systemmatrices A and C and the covariance matrix K. The benefit of the the CVA is that it ensures pos-itive realness of the estimated covariance sequence at the cost of a biased estimate. However, ifthe data length, N , is chosen sufficient large, this bias can be neglected. Since the CVA is able tomodel multivariate output signals it is well suited to model the disturbance signals working onthe AVIS as will be shown in Chapter 7. The reliability of these models can be measured on basisof comparison of the PSD, the PSD of the outputs of the model under influence of a white noisewith zero mean and unit variance should namely equal the PSD of the measurements.


3.4 Conclusion

In this chapter three different system identification techniques are presented for three differentpurposes. First a frequency response identification technique, the spectral analysis, is presentedthat can be used to obtain highly accurate input-output models which are useful for control de-sign. The reliability of the obtained models can be measured on basis of the coherence, whichcan be seen as the signal to noise ratio. However, the coherence can not be extended to MIMOsystems. Consequently, the system need to be identified on basis of multiple SIMO experiments,in order use the coherence as a measure for the reliability of the obtained model.

Secondly, a subspace identification technique, the N4SID method, is presented that can beused to obtain parametric time domain input-output models that can be used for the reconstruc-tion of disturbances. The reliability of the models can be measured on basis of comparison withthe frequency response models or on basis of a correlation analysis of the residu with the input.An advantage of this identification technique is that only a single experiment is needed to obtaina parametric MIMO model of the system.

Finally, another subspace identification technique, the stochastic realization, is presented.This technique can be used to obtain disturbance models on basis of output measurements,equivalent to process 3 in Figure 2.2. The reliability of these models can be measured on basis ofcomparison of the PSD, the PSD of the outputs of the model should namely equal the PSD of themeasurements. An advantage of this identification technique is that the method is well suited forthe identification of MIMO systems.

Chapter 4

MIMO Control and Stability

Many different MIMO control techniques, see [27], are developed in the past. The simplest ap-proach to multivariable controller design is to use decentralized control. Since this approachuses a diagonal controller, it works well if the plant is close to diagonal, because then the plantis essentially a collection of independent sub-plants. If the off-diagonal elements of the plantare large, and hence the interaction is relevant, this control approach may cause problems, be-cause no attempt is made to counteract these interactions. Whether this control approach is wellsuited for the AVIS is investigated in Chapter 6. In this chapter, two approaches to the design ofdecentralized controllers are discussed, namely independent design and sequential design.

After the design of a decentralized feedback controller, the closed loop system has to be stable.This can be verified on basis of a couple of stability criteria, see [27], that are applicable to MIMOsystems. Distinction between these different stability criteria is made on basis of conservatismand interpretation of relation to controller design.

K Ge yr

-

d

Figure 4.1: Control configuration

The control configuration as considered in this chapter is shown in Figure 4.1. With y themeasured output, r the reference signal, e the servo error and d the disturbances. The openloop is defined as L(s) = G(s)K(s), where G(s) is a square plant, controlled by a decentralizedcontroller, K(s).

43

44 CHAPTER 4. MIMO CONTROL AND STABILITY

4.1 Decentralized Control

The design of decentralized control systems involves two main steps: control configuration selec-tion and design of the diagonal controller elements, ki(s),

K(s) =

k1(s)k2(s)

. . .km(s)

. (4.1)

In this section we assume that the control configuration is diagonal and we will focus on thedesign of the diagonal controller elements. The independent and sequential design method arepresented here.

Independent design

In independent design, each controller element, ki(s), is designed based on the correspondingdiagonal element of G(s), such that each individual loop is stable. This control design is desirableif the individual loops can operate independently, which is a strong requirement, since the inter-actions have to be very small. However, this can be approached if the plant is defined in variablesthat have no or little mutual interactions.

Sequential design

In sequential design the controller elements are designed one at a time, with the previous de-signed controller elements implemented. This has the advantage of reducing the number ofinput output relations of the MIMO problem with each step. The sequential design approach canbe used for interaction problems if the independent approach does not work. A drawback of thisapproach is that the loops that are closed earlier are affected by the loops that are closed later.This may cause problems with robustness and performance. Moreover one has to decide on theordering to close the loops, which is not straightforward.

The sequential design control configuration is shown in Figure 4.2, where the plant is rear-ranged in such way, that the sequential closed loop equals, u2 = K2y2. The equivalent plantdescribes, y1 = G∗

1u1, and is defined as:

G∗1 = G11 + G12K2(I −G22K2)−1G21. (4.2)

The sequential control loop can be designed on basis of this equivalent plant. The interactionterms of the plant, G12 and G21, are not considered in the sequential closed loop. However, theseterms are made part of the equivalent plant and so considered in the other closed loops. So,interactions are considered, but not in an explicit manner.

4.2 Stability Criteria

The stability of the closed loop systems need to be verified. In this section stability criteria forMIMO closed loop systems are discussed. This is done on basis of a couple of theorems pre-sented in [27]. Some of these can be applied to MIMO systems in general, other are specific fordecentralized controlled MIMO systems.

4.2. STABILITY CRITERIA 45

K2

u2

u1 y1

y2

G11 G12

G21 G22

Figure 4.2: Sequential design control configuration

Theorem 4.1 (Generalized MIMO Nyquist theorem). If Pol is the number of open loop un-stable poles in G(s)K(s), then the closed loop system is stable if and only if the Nyquistplot of det(I + L(s)) makes Pol anti-clockwise encirclements of the origin and does not passthrough the origin.

Proof. see [27, p.152]

This Nyquist criterium is the most general MIMO stability criterium and is a necessary andsufficient condition for stability, but it has some drawbacks. The Nyquist stability condition giveslittle insight in the effect on the stability of the closed loop system as a result of a parameterchange in the controller. Moreover, can the Nyquist plot become very complex, which makes theinterpretation of the plot rather involved. As an alternative, the characteristic loci, λi(L(jω)), canbe examined, which can be directly derived from the Nyquist theorem.

Theorem 4.2 (Characteristic loci stability theorem). A closed loop system is stable if andonly if the Nyquist plots of the characteristic loci do not pass through (−1, 0), where thecharacteristic loci are defined as the eigenvalues of the frequency response of the open-looptransfer function, λi(L(jω)).

Proof. The proof makes use of the fact that the product of eigenvalues of a matrix is equal tothe determinant of that matrix:

det(I + L(jω)) (4.3)

=n∏

i

λi(I + L(jω)) (4.4)

=n∏

i

(1 + λi(L(jω)). (4.5)

Consequently, a system is stable if the Nyquist plots of 1+λi(L(jω)) do not pass through theorigin or equivalent, the Nyquist plots of λi(L(jω)) do not pass through (−1, 0)

This Characteristic loci stability condition is also a necessary and sufficient condition for sta-bility. These characteristic loci partly provide a generalization of the Nyquist criterium from SISOto MIMO systems. However, it is still difficult to predict the effect of a parameter change in the


controller, on the change of the characteristic loci. The change of one of the diagonal controllerelements will namely effect all the characteristic loci.

Other stability conditions, based on the Nyquist criterium, are the spectral radius stabilitycondition and the small gain theorem. The spectral radius is defined as follows:

Theorem 4.3 (Spectral radius stability condition). A system with a stable loop transferfunction L(s) is closed loop stable if:

ρ(L(jω)) < 1 (4.6)ρ(L(jω)) , max

i|λi(L(jω))|, ∀ω. (4.7)

Proof. [27, p.155] To prove condition (4.6) the reverse is proven; that is, if the system isunstable and therefore det(I + L(s)) does encircle the origin, then there is an eigenvalue,λi(L(jω)), which is larger than 1 at some frequency. If det(I + L(s)) does encircle the origin,then there must exist a gain ε ∈ (0, 1] and a frequency ω′ such that,

det(I + εL(jω′)) = 0. (4.8)

This is easily seen by geometric arguments since det(I + εL(jω′)) = 1 for ε = 0. Expression(4.8) is equivalent to:

n∏

i

λi(I + εL(jω′)) = 0 (4.9)

and

1 + ελi(L(jω′)) = 0 for some i, (4.10)

and

λi(L(jω′)) = −1ε

for some i. (4.11)

A sufficient condition for this expression is:

|λi(L(jω′))| ≥ 1 for some i, (4.12)

which is equivalent to:

ρ(L(jω′)) ≥ 1. (4.13)

The small gain theorem below follows directly from Theorem 4.3.

Theorem 4.4 (Small gain theorem). A system with a stable loop transfer function L(s) isclosed loop stable if:

||L(jω)|| < 1, ∀ω, (4.14)

where ||L|| denotes any matrix norm satisfying ||AB|| ≤ ||A|| · ||B||.

4.2. STABILITY CRITERIA 47

Proof. [27, p.156] If a matrix norm is considered, which by definition satisfies ||AB|| ≤ ||A|| ·||B||. Then, at any frequency, we have ρ(L(jω)) < ||L(jω)||.

The spectral radius stability criterium in Theorem 4.3 is only a sufficient condition. It is con-servative because phase information of L(jω) is not considered. This conservatism is introducedin the proof of the Theorem at Equation (4.12). The small gain theorem in Theorem 4.4 is asconservative or more conservative than the spectral radius criterium based on the matrix normthat is used and therefore only a sufficient condition for stability. Nevertheless, both conditionsare considered, since the effect of a change in the control design on the stability bound is easierto predict on basis of the spectral radius and the small gain stability condition than on basis ofthe Nyquist condition. Besides, these criteria are easy to interpret.

Independent design stability criteria

In order to analyse decentralized control systems, based on independent design, a factorizationof the sensitivity is presented. This will give more insight in the influence of the interactions inthe plant on the stability of the closed loop system.

Assuming that the control configuration is diagonal, the diagonal plant Gd(s) is defined asthe matrix consisting of the diagonal elements of G(s). The sensitivity, Sd, and complementarysensitivity, Td, for the individual loops are then defined as:

Sd = (I + GdK)−1 (4.15)Td = I − Sd. (4.16)

The sensitivity for the individual loops relates to the sensitivity for the overall closed loop systemas follows:

S0 = Sd(I + ETd)−1. (4.17)

Where E, schematically represented in Figure 4.3, are the interaction terms scaled by the diagonalelements of G,

E = (G−Gd)G−1d . (4.18)

Equation (4.17) factorizes the stability analysis in two parts: det(I+GK) = det(I+GdK) det(I+ETd). In order to verify stability on basis of the factorization in Equation (4.17) it is assumed that(a) the plant G is stable and (b) each individual loop is stable by itself; Sd is stable.

Theorem 4.5. With assumptions (a) and (b), the overall system, S0, is stable:(i) if and only if det(I + E(jω)Td(jω)) does not encircle the origin and does not pass

through the origin for all ω(ii) if and only if λi(E(jω)Td(jω)) does not pass through the origin for all ω(iii) if

ρ(E(jω)Td(jω)) < 1, ∀ω. (4.19)

(iv) if

σ(Td) = maxi|td,i| < 1/µTd

(E), ∀ω. (4.20)

The structured singular value µTd(E) is computed with respect to a diagonal structure of Td.


E

Gd

G

Figure 4.3: Interpretation of scaled interaction E

Proof. see [27, p.438]

Criterium (i) is derived from the Nyquist Theorem 4.1 and is therefore a necessary and suffi-cient condition for stability. Criterium (ii) is derived from the characteristic loci Theorem 4.2 andis therefore also a necessary and sufficient condition. Criterium (iii) is derived from the spectralradius Theorem 4.3 and is a sufficient condition for stability. Criterium (iv) uses the structuredsingular value to split up the spectral radius and is also a sufficient condition.

On basis of this condition it is clear how interaction cause the system to be unstable, which isvery useful for control design, in spite of the conservatism of this bound.

Another way to split up ρ(E(jω)Td(jω)) in Equation (4.19) is to use Gershgorin’s theorem[27, p.439]. A sufficient condition for stability can then be derived in terms of the rows or columnsof G respectively:

|td,i| < |gii|/∑

j 6=i

|gij |, ∀i, ∀ω (4.21)

|td,i| < |gii|/∑

j 6=i

|gji|, ∀i, ∀ω. (4.22)

These Gershgorin stability criteria are complementary to the structured singular value conditionin Equation (4.20). Since the use of the Gershgorin criteria is not always more conservative thanthe the structured singular value condition. The smallest of the upper bounds in Equation (4.21)and (4.22) is always more restrictive than the upper bound in Equation (4.20). However, Equation(4.20) imposes the same bound for each loop, whereas Equation (4.21) and (4.22) give boundsfor each individual loop, which may be less restrictive.

By examining both the structured singular value stability condition and the Gershgorin bounds,a stable controller can be designed on basis of the independent control design, taking the inter-actions into account.

Sequential design stability criterium

In order to analyse decentralized control systems, based on sequential design, another factoriza-tion of the stability criterium is required. Therefore, the equivalent open loop is defined as G∗

i ki,with G∗

i the equivalent plant (4.2), with all controllers, but ki, implemented. On basis of thesedefinitions, the following stability theorem can be applied to sequential designed control systems.

4.3. CONCLUSION 49

Theorem 4.6 (Equivalent open loop stability theorem). A closed loop system is stable if andonly if the Nyquist plot of each equivalent open loop does not pass through (-1,0).

Proof.

det(I + G(jω)K(jω)) =n∏

i

det(1 + G∗i (jω)ki(jω)). (4.23)

Consequently, a system is stable if the Nyquist plot of 1+G∗i (jω)ki(jω) does not pass through

the origin, for all i, or equivalent, the Nyquist plot of G∗i (jω)ki(jω) does not pass through

(−1, 0), for all i.

It should be noted that the equivalent plant, G∗i , in this theorem is a SISO system, so with all

controllers, but ki, implemented. Further, it should be noted that each control element influencesall equivalent open loops by means of the interactions in the system. These interactions arenot considered explicitly as in the independent design stability analysis, but in a more implicitmanner. The equivalent open loop stability theorem is a necessary and sufficient condition forstability.

4.3 Conclusion

In this chapter the stability analysis for the decentralized control approach is discussed. Firstthe independent and sequential control design are presented. Next, the stability of the resultingclosed loop system is analyzed on basis of several stability criteria.

The independent control design assumes that the individual loops can operate independently,which is a very strong requirement. However, this can be approached by a good decoupling ofthe plant. An alternative is the sequential control design. This approach analysis the equivalentplant after closing all other control loops. Consequently, the interactions are considered in theequivalent plant and the control design simplifies. However, the change of a control element willaffect all equivalent plants. This may cause problems with robustness and performance.

The complete stability analysis is based on the generalized MIMO Nyquist theorem. ThisNyquist condition and the characteristic loci condition which is derived from it, are the necessaryand sufficient stability conditions presented in this chapter. The characteristic loci conditionhas the benefit over the MIMO Nyquist condition, that it is easier to interpret. A drawback ofboth these stability conditions is that the effect of a change in the parameters of the controlleron the stability of the system is not straightforward. Therefore the spectral radius condition,the structured singular value stability condition and the Gershgorin bounds are derived, whichare easier to interpret but are unfortunately also more conservative. These conditions are onlysufficient conditions for stability.

A factorization of the closed loop system, into the individual loops, Sd, and a remainingpart including the interactions, (I + ETd)−1, can be used to analyse independently designedclosed loop systems. The individual loops are stable by choice of the controller, so the remainingpart needs to be verified stable. Analysis of this part with the spectral radius condition and thestructured singular value stability condition can indicate where the interactions cause instability.On basis of that information a stabilizing controller can be designed.


For the stability analysis of sequentially designed control systems another factorization ofthe closed loop system is proposed. This factorization is based on the equivalent open loop,G∗

i (jω)ki(jω), and stability of all these loops guarantees stability of the entire closed loop system.

Part II

AVIS application

51

Chapter 5

AVIS System Identification

This chapter illustrates two identification concepts developed in Chapter 3 on the AVIS. The AVIShas to be identified for two different purposes. One model is required for control design andanother model is required for the prediction of disturbances. In Chapter 3 was concluded that forcontrol design a frequency domain, non parametric model should be used, which can be obtainedvia spectral analysis. For prediction of disturbances a time domain, parametric model should beused, which can be obtained via a subspace identification technique. Both these techniques arealso described in Chapter 3.

In this chapter, first the system identification of the AVIS on basis of the spectral analysisis discussed, whereafter the system identification of the AVIS on basis of N4SID subspace tech-nique is discussed.

5.1 Spectral Analysis

In this section the spectral analysis identification technique as discussed in Section 3.1 is appliedto the AVIS in order to obtain a frequency response model for control design. Therefore, theconfiguration of the identification experiments, the design of the input signal and the processingof the signal measurements will be discussed. Finally, the frequency response identification ofthe AVIS will be presented and the reliability of this identification will be discussed on basis ofthe coherence.

Since the AVIS experimental setup is open-loop stable, it is possible to perform this identifi-cation procedure in open-loop. The configuration of the experiments is than straightforward asshown in Figure 5.1, following Equation (3.1), and the input is uncorrelated with the noise as wasassumed. In this figure equals G0 the AVIS after rigid body decoupling as will be discussed inSection 6.1. The input design, u, is very important for the quality of the system identification aswill be shown next.

5.1.1 Input Design

If the coherence is interpreted as the signal to noise ratio, then the goal of the input design isto obtain a coherence equal to one. Since the noise is determined by external disturbances andnon linearities in the system this design is not straightforward. For small displacements arounda point most systems behave linear, so low power input signals would be best to minimize non

53

54 CHAPTER 5. AVIS SYSTEM IDENTIFICATION

G0

u(t)

v(t)

y(t)

Figure 5.1: Configuration of the frequency response identification experiments

linear effects. However, if the external disturbances are relatively large, this will neverthelessresult in a bad signal to noise ratio. So it is important to find a good tradeoff between both effectsfor the design of the input signal.

For the AVIS is chosen to set maximum of 10V for each output, which equals approximately0.25mm/s as is derived in Appendix C. The standstill (no input signal) output signal fluctuatesaround the 0.5V, which indicates a good signal to disturbance ratio. Besides, it is assumed thatfor these velocities, the AVIS is dominated by linear behavior.

If a maximum of 10V is set for the outputs, a further point of consideration is the powerdistribution of the input signal over the frequency domain. During spectral analysis only certainfrequencies are analyzed, so it is beneficial if the power of the input signal is focussed at thesefrequencies. In that way a better signal to noise ratio can be achieved at these frequencies. Thefocus of power at certain frequencies can be achieved if a sum of sinusoids is used as input signalfor the identification. This is implemented in the idinput function in MATLAB.

The sum of sinusoids signal is constructed on basis of a number of sinusoids with frequenciesequally spread over a chosen grid. Each sinusoid is given a random phase and in a number of tri-als the phases that give the smallest signal amplitude are selected. The amplitude can afterwardsbe scaled to achieve the desired signal power.

When a sum of sinusoids is used as input signal, the frequency response should be analyzedat the frequencies of these sinusoids. As a result the power spectra in Equation (7.1) and (3.7)should also be estimated at these frequencies. This is possible on basis of the Goertzel algorithm[3, p.26].

As was concluded in Section 3.1, no MIMO validation is possible on basis of the coherence,since each output is then affected by six inputs simultaneously. As a result, six SIMO experimentsare carried out. In this way, each output is only affected by one input at a time and is it possibleto obtain a good signal to noise ratio.

A drawback of the SIMO approach is that the power of the input signal has to be optimizedconcerning all six outputs, although the magnitude of the frequency response may be muchsmaller for the off-diagonal terms then for the diagonal term. As a result, the output of the diag-onal term will quickly reach the 10V, whereas the output of the off-diagonal terms only moderatefluctuates. Consequently, the signal to noise ratio for the off-diagonal terms is worse for thediagonal terms.

This effect can be avoided if SISO experiments are carried out, but in that case six times asmuch experiments are needed. Besides, the power can not be increased unlimited, since highgains may cause non linearities. Therefore, in this work chosen is to do SIMO experiments.

5.1. SPECTRAL ANALYSIS 55

On basis of preparatory experiments, which are not shown or further discussed, it is deter-mined that the AVIS exhibit its essential dynamics approximately in the frequencies between 0.02and 200Hz. The highest frequency is chosen as such that some dominant resonance modes arein the considered domain. The lowest frequency is chosen as low as possible in order to get agood identification of the first 0dB crossing as discussed in Chapter 6. For lower frequencies it isvery likely that the AVIS shows non linear behavior due to friction. Moreover a lot of data has tobe stored for a couple of periods of these low frequencies, which can cause memory problems.

Identifying the whole frequency range in one experiment is very hard due to two problems.First of all there is a lot of difference in the magnitude of the frequency response over the entirefrequency range. Secondly there is a lot of difference in timescale between the lowest and highestfrequencies.

As a result of the difference in the magnitude of the frequency response over the entire fre-quency range, the output will quickly reach the 10V, due to the dominant suspension modes,whereas there is almost no contribution to the output from the higher and lower frequencies.Consequently will the coherence at these frequencies be very bad.

As a result of the difference in timescale between the lowest and highest frequencies will itbe impossible to store an experiment that is long enough given the available memory. In orderto identify the frequency response at 200Hz a sample frequency of at least 400Hz and preferablya lot higher, in order to prevent anti-aliasing as discussed next, is required. Conversely, in orderto identify the frequency response at 0.02Hz a measurement time of at least one period, 50s, andpreferably a lot more, in order to obtain beter estimates, is required. This results in a experimentwith a long measurement time and a high sample frequency, which causes memory problems.

A solution to both of these problems is to divide the entire frequency range in several smallerfrequency ranges. Consequently, the difference in magnitude of the frequency response and thedifference in timescale between the lowest and highest frequency will be a lot smaller. After-wards, it is possible to combine the several different frequency response measurements into onefrequency response of the AVIS.

Anti Aliasing

Shannon’s theorem states that a sampled signal with sampling frequency ωs can exactly repro-duce a continuous time signal, provided that the continuous signal has no frequency contentabove the Nyquist frequency, ω ≤ ωs/2 = ωn. If the signal has frequency content above theNyquist frequency, it is visible in the estimation of the frequency response in the form of fre-quency folding. This effect is known as aliasing. For example, if a sinusoid with a frequency of60Hz is sampled at 100Hz, than the signal will be approximated with a sinusoid of 40Hz.

In order to prevent this aliasing effect, the sample frequency during data acquisition shouldbe chosen high enough, such that the magnitude of the frequency response above the Nyquistfrequency is not significant. Next the data should be filtered with a lowpass filter with a cutofffrequency at the desired Nyquist frequency. Finally, the signal can be resampled to the desiredsample frequency.

For the frequencies chosen in this work for the various data is referred to Appendix B.


5.1.2 Results

On basis of the identification configuration and the input design as presented in this sectionis the frequency response of the AVIS identified. The magnitude of the frequency response isshown in Figure D.1, the phase in Figure D.2 and the accompanying coherence in Figure D.3.

On basis of the coherence, it can be concluded that the measurements are reliable. Especiallythe diagonal terms give good identification results, for both very low and high frequencies. Themost difficult was the identification of the sixth input for approximately 1 to 20Hz. This can bedue to the difference in magnitude of the frequency response of the diagonal term comparedto the off-diagonal terms as explained in this section. The same effect can be seen in the otherinputs, but less obvious.

The identified frequency response is characterized by a +1 slope at low frequencies. A coupleof suspension modes, depending on the axis, at frequencies between 1Hz and 5Hz. A −1 slopefor frequencies between 5Hz and 100Hz and some resonance modes above 100Hz. This is exactlywhat would be expected for a mass, spring, damper system with force as input and velocity asoutput.

If the suspension modes are closer examined, it can be concluded that there are modes withsix different frequencies. Three of these modes appear in loop 1, 2 and 6 and the other threemodes appear in the other loops. The grouping of these three loops is not unexpected, since themotion in x and y direction is closely related to the rotation around the z direction as a result ofthe position of the actuators.

5.1.3 Conclusion

Following the spectral analysis technique as presented in Section 3.1 a frequency response modelof the AVIS after decoupling is derived, that can be used for control design. This model is val-idated on basis of the coherence. For the main part is the frequency response of the AVIS de-termined on input output data with a coherence of 1, which means that the output is linearlydependent on the input and, after averaging, not disturbed by noise. This result is reached on ba-sis of an input design taking differences in magnitude and time scale of the frequency responseinto account.

In order to prevent aliasing to occur, the output data is sampled at a high frequency, filteredby a low pass filter and finally down sampled.

5.2. N4SID SUBSPACE IDENTIFICATION 57

G0

u(t)v(t)

y(t)

- K

Figure 5.2: Configuration of the subspace identification experiments

5.2 N4SID Subspace Identification

In this section the N4SID subspace identification technique as discussed in Section 3.2 is appliedto the AVIS, in order to obtain a time domain model for prediction of disturbances. Thereforethe configuration of the identification experiments and the design of the input signal will be dis-cussed. At the end will this state space model be compared to the frequency response modelderived in the previous section. Moreover is the obtained model validated on basis of anotherexperiment that correlates the residu to the input as discussed in Section 3.2

The disturbance identification will take place in closed-loop, so the subspace identificationalso has to be performed under feedback of the output. The configuration of the experimentsis shown in Figure 5.2. In this figure is G0 the AVIS after rigid body decoupling as will bediscussed in Section 6.1 and K is a stabilizing controller as derived in Section 6.2. In Chapter7 all disturbances are treated as output disturbances, v. Consequently the input, u, is injected atthe same point in the loop to be able to identify the output sensitivity, S0 in Equation (5.1), whichwill be used for disturbance reconstruction.

y(t) = (I + G0K)−1

︸︷︷︸S0

(u(t) + v(t)). (5.1)

The subspace identification theory is based on the assumption that there is no feedback from thestates to the input, which is the case in this configuration. Moreover is assumed that the statevector is sufficiently excited, thus the data should be recorded after the system is excited for awhile.

5.2.1 Input Design

Just as in the spectral analysis case is the input design very important for the final identificationresult, however some other considerations have to be made.

The only condition the input should satisfy, is that the signal should be persistently exciting(PE) of order k, where k is the identification horizon. It can be shown [12, p.337] that noise signalsare PE of order infinity and a sum of sinusoids is PE of order 2p, with p the number of sinusoids.Both signals could thus be used for identification of the AVIS. In this work is chosen for sixindependently, white noise signals as input, since the system will than equally be excited at allfrequencies. As a result is it possible to identify the complete AVIS in one experiment.


As can be concluded from Equation (5.1) is the system excited by both the input and theoutput disturbances. Consequently the input should be chosen high enough to obtain a goodsignal to noise ratio, taken into account that for high input signals the system might show nonlinear behavior. Therefore the output of the AVIS is maximized to 10V. This problem is alreadytreated in Section 5.1.1.

The subspace identification algorithm does not average the data as is the case in the frequencyresponse identification. Therefore the input signal is repeatedly injected, after which the outputsignal is averaged. One period of the input and the averaged output are then fed into the identifi-cation algorithm. As a result is the effect of the stochastic disturbances on the output minimized.

The length of the period is limited by memory. Based on the sample frequency of the requiredmodel is the period time of both input and output determined. Consequently is it difficult todo good low frequent identification experiments. If the system is dealt with as multiple SISOsystems, a longer input period can be used. In this work however, it is chosen to deal with thiscompletely as a MIMO problem, since the MIMO application is the strength of this identificationprocedure.

5.2.2 AVIS Identification

Before the N4SID subspace algorithm can be used to identify the AVIS on basis of the input-output data, three parameters should be selected, namely; the sample frequency, the identificationhorizon k and the input length N . The sample frequency should be chosen, such that all relevantdynamics are within half this sample frequency. The horizon should be chosen higher than themodel order, which is not known beforehand and the input length should be chosen bigger than:2k(m + p). Besides holds in general that the bigger k is chosen, the better the approximationbecomes. For the values of the identification parameters chosen in this work, see Appendix B.

The model order is determined during the subspace identification on basis of a singular valuedecomposition. The model order is chosen as such, that the singular values for higher orders arenegligible small. The model order for the AVIS is set at 30. During the identification, a statespace model and a Toeplitz matrix are obtained, both representing the AVIS. The magnitude andphase of the obtained model are shown in Figure D.1 and D.2 respectively. Note that this is nota plot of the identified output sensitivity, but of the plant, G0, derived on basis of the identifiedoutput sensitivity and the controller, K, in order to compare this identification result with thefrequency response identification.

G0 = (S−1O − I)K−1 (5.2)

As can be concluded from these figures is the subspace identification successful. For low fre-quencies the identification is less accurate, which is due to the limited period length. Also theoff-diagonal terms are less accurate than the diagonal terms, but this can be due to the differencein magnitude of the frequency response as explained in the previous section. Nevertheless is theidentification successful, considering that this identification is done in one time, resulting in a30th order 6× 6 MIMO model of the AVIS.

5.2.3 Model Validation

The obtained state space model of the AVIS is already compared to the frequency response modelfor validation, but to get more confidence in the obtained model, also the validation technique as

5.2. N4SID SUBSPACE IDENTIFICATION 59

−500 0 500−0.5

0

0.5

1Cross corr. function between input u1 and residuals from output y1

lag

−500 0 500−0.1

−0.05

0

0.05

0.1Cross corr. function between input u2 and residuals from output y1

lag

Figure 5.3: Correlation of the residu with the input

presented in Section 3.2.3 will be applied to the model. Therefore, an input signal should be cho-sen that actuates the system the same way as the system is actuated during its purpose. After thatthe correlation of the residu with this input signal is examined in order to determine whether themodel is reliable.

Since the AVIS is meant for vibration isolation and does not need to follow setpoints, thechoice for the input signal of the validation method is not straightforward. In this work is chosenfor a sequence of impulses, one for each input, with an interval of 5s starting at input 1, sincethis signal contains all frequencies. Consequently, only one experiment is needed for modelvalidation.

The output of the AVIS as a result of this sequence of impulses is compared with the esti-mation of the output on basis of the obtained state space model. The correlation of the residuof output channel 1 with input channel 1 and 2 is shown in Figure 5.3. The same figure can beshown for the other correlations. The black circles indicate the correlation between the residu ofthe output and the input for different lags. The gray area indicates the 99% probability bound P .So on basis of this plot can be concluded that output 1 is, for 99% certain, uncorrelated with input1 after 200 lags, or 0.5s. Moreover can be concluded that output 1 is, for 99% certain, uncorrelatedwith input 2.

5.2.4 Conclusion

In this section, it is shown that it is possible to obtain a parametric, time domain model of thecomplete 6 × 6 AVIS on basis of one experiment using the N4SID subspace identification algo-rithm. The obtained model is validated on basis of a frequency response model and a correlation


analysis of the residuals with the inputs. These indicate that the obtained models are reliable.The state space model is not as accurate as the frequency response model, but is derived on basisof a single experiment, which is very efficient. The input used for the subspace identification is awhite noise sequence, that is repeatedly injected in the system. The output is averaged and thenused for identification. As a result, the effect of the stochastic disturbances is minimized.

The obtained state space model and Toeplitz matrix will be used in the next chapter to predictthe output disturbances.

5.3 Conclusion

In this chapter, two different model identification techniques for two different purposes are suc-cessfully applied to the AVIS experimental setup.

The first identification technique, a spectral analysis, resulted in a frequency response modelthat will be used for control design. This model is very accurate for both very low and highfrequencies and is validated on basis of the coherence between the input and output.

The second identification technique, the N4SID subspace identification, resulted in a statespace model and Toeplitz matrix which will be used for the reconstruction of disturbances. Themodel is validated by comparison with the frequency response model and on basis of a corre-lation analysis between the residuals and the input. Following these validation techniques canbe concluded that this model is not as accurate as the frequency response model, but since themodel is parametric it is more widely applicable. Moreover, only one experiment is needed forthis identification.

Chapter 6

AVIS Control Design

The disturbance characterization, as discussed in Chapter 7, is based on a closed-loop system,since most systems in practise that need disturbance analysis are already working in closed loop.Therefore, the AVIS needs to be controlled by feedback of the outputs. Consequently, a controllershould be designed with as most important design requirement that it stabilizes the AVIS inclosed-loop.

In this chapter will be investigated if this is possible on basis of the decentralized controlprinciple as opposed and discussed in Chapter 4. This control principle does not take systeminteractions into account for the control design, so the system should be decoupled in order tominimize these interactions. In this work, a rigid body decoupling is discussed.

Decentralized control involves two main steps. The control configuration selection and thedesign of the decentralized control elements. The control configuration selection will be madeon basis of the rigid body decoupling of the plant and a relative gain array analysis. The design ofthe diagonal controller elements will be discussed and further analyzed in combination with thestability analysis, presented in Chapter 4.

6.1 Decoupling

The AVIS [21] consists of a table top and a chassis, interconnected by means of four isolatormodules. The top view of the AVIS is schematically represented in Figure 6.1. The table top isactuated by 8 actuators at the corners and the velocity at these corners is measured by 6 sensors. Ifno decoupling is considered, large interactions are present, which complicates the control design.Moreover, no good choice can be made on the control configuration, since there is no obviousrelation between the sensors and the actuators.

In order to decrease the interactions and relate the sensors to the actuators, a relation shouldbe found between the measurements and actuation positions and the center of mass. Which isa rigid body decoupling problem. This can be done on basis of the kinematics of the plant, [33].If this is done properly, the interactions will decrease where the system behaves like a rigid body.Also a measure, the relative gain array, is presented to analyse whether the interactions are smalland which control configuration is preferred.

The decoupling will describe the AVIS in terms of the generalized velocities, q, as shown inFigure 6.2, where Aa is the mapping matrix from generalized velocities to actuator inputs, Ai,and A−1

s is the mapping matrix from sensor outputs, So, to generalized velocities. The decoupled

61

62 CHAPTER 6. AVIS CONTROL DESIGN

x

yz

12

34

a1z

s1z

a1y

s1y

s2x

s2z

a2z

a2x

s3z s3y

a3z

a3y

a4z

a4x

Figure 6.1: Top view of the AVIS, with sensor and actuator positions

Aa As

-1Ai

So

q.

q.

G(s)~

Figure 6.2: Definition of the mapping matrices for the AVIS

plant is then defined as:

G(s) = A−1s G(s)Aa. (6.1)

The generalized velocities are defined as:

q =[x y z φ θ ψ

]T. (6.2)

Where x, y and z are the velocities of the center of mass and φ, θ and ψ are the rotational velocitiesof the center of mass around the three axis.

Kinematics

The derivation of the kinematics is described using sensor S1z as an example. The same deriva-tion holds for the other sensors and actuators. In order to describe the kinematics, some defini-tions are needed: (~) denotes a vector and (−) denotes a column.

In order to describe the kinematics of the AVIS, two axis systems should be defined. Thefirst axis system, ~e1, is fixed to the earth and the second axis system, ~e2, is fixed to the AVIS. Theposition, ~r, of sensor S1z is described relative to the center of mass as function of the two axissystems by:

~rS1z = ~rCM + ~rS1z/CM , (6.3)

6.1. DECOUPLING 63

which can be rewritten as:

r1T

CM~e1 + r2T

S1z/CM~e2 (6.4)

= r1T

CM~e1 + r2T

S1z/CMA21~e1. (6.5)

Where r2S1z/CM is defined as the position of sensor S1z , relative to the center of mass, described

in ~e2. A21 is a direction cosine matrix, that describes the rotation from ~e2 to ~e1. In this work theTait-Bryant angles are used, since these are suited for linearizations along small angles, [33]. Thelinearized cosine matrix for small angles is given by:

A21 ≈

1 ψ −θ−ψ 1 φθ −φ 1

. (6.6)

The velocity of S1z can be described as the time derivative of its position:

d

dt(~rS1z) =

d

dt

(r1T

CM

)~e1 + r1T

CM

d

dt

(~e1

)+

d

dt

(r2T

S1z/CM

)~e2 + r2T

S1z/CM

d

dt

(~e2

)(6.7)

=d

dt

(r1T

CM

)~e1 + r2T

S1z/CM

d

dt

(~e2

). (6.8)

Where ddt

(~e1

)= ~0, since ~e1 is a fixed reference axis. Moreover d

dt

(r2T

S1z/CM

)= 0, since

ddt

(r2T

S1z/CM

)is a body fixed vector. Equation (6.8) can also be written as:

~rS1z = r1T

CM~e1 + r2T

S1z/CM~e2 (6.9)

= r1T

CM~e1 + r2T

S1z/CM A21

~e1, (6.10)

with:

A21

= −A21ω1 (6.11)

ω1 =

0 ψ −θ

−ψ 0 φ

θ −φ 0

. (6.12)

It is assumed that in Equation (6.10), ~rS1z equals the sensor output in the z-direction, So1z , caused

by the velocity of the center of mass in the z-direction, r1T

CM~e1z , and a couple of rotational speed

contributions, r2T

S1z/CM A21

~e1. Equation (6.10) can thus also be written as function of the gener-alized velocities, q:

So1z~e

1z =

001

r2S1z/CM,y − r2

S1z/CM,zφ + r2S1z/CM,xψ

−r2S1z/CM,x − r2

S1z/CM,zθ + r2S1z/CM,yψ

0

T

q~e1z. (6.13)

In order to simplify this equation, the mapping from generalized velocities to sensor outputsfor zero angles, φ = θ = ψ = 0, is considered. This equation can be extended for all six


sensor outputs to obtain a mapping matrix As. An inverse of the obtained mapping matrix, A−1s ,

describes the generalized velocities as function of the sensor outputs:

q = A−1s So (6.14)

A−1s =

−0.3810 0.0158 1.0000 −0.1241 0.3810 0.10830.5000 −0.1371 0 0.1208 0.5000 0.0163

0 0.4980 0 0.0017 0 0.50020 1.3054 0 −1.1506 0 −0.15480 0.1507 0 −1.1821 0 1.0315

1.1905 0 0 0 −1.1905 0

. (6.15)

Similarly, a mapping, Aa, from generalized velocities to the eight actuator inputs, Ai, can befound:

Ai = Aaq (6.16)

Aa =

0 0.5 0 0.1570 0 0.55400 0 0.25 0.2950 −0.4760 0

0.5 0 0 0 −0.1570 0.45400 0 0.25 −0.3760 −0.4030 00 0.5 0 0.1570 0 −0.55400 0 0.25 −0.2920 0.4760 0

0.5 0 0 0 −0.1570 −0.45500 0 0.25 0.3770 0.3950 0

. (6.17)

The polarization of the sensors and actuators has to be taken into account. The effect of thedecoupling depends on how accurate the position of the sensors, actuators and center of mass isdetermined.

Relative Gain Array

On basis of the two decouple matrices as derived in this section, the pairing of outputs andinputs for the decentralized control of the AVIS is straightforward. The velocity of the center ofmass in x-direction is controlled by feedback of the measurement of the velocity in x-direction,etcetera. Whether this is a good choice, can be concluded on basis of the relative gain array (RGA),Λ(G(jω)), defined as [27]:

Λ(G(jω)) = G(jω)× (G(jω)−1)T , ∀ω. (6.18)

Where × denotes the element by element multiplication. The RGA should be computed for allfrequencies and is an indication for the interaction in the system. Large RGA indicates largeinteractions. Following the RGA, a pairing is preferred such that the rearranged system, withthe selected pairings along the diagonal, has an RGA matrix close to identity at the frequenciesaround the closed-loop bandwidth. The real value of the complex RGA is shown in Figure D.4. Itcan be concluded that for frequencies between 2Hz and 100Hz there is not much interaction inthe system and that diagonal pairing is the best choice. For lower and higher frequencies there isinteraction and some other pairings are preferable. This can be expected, since the system doesnot behave like a rigid body for low and high frequencies. This might be a problem, since thecrossover will be placed approximately around 2Hz. If these interactions limit the control design,

6.1. DECOUPLING 65

another decoupling method should be tried. This could for example be a modal decoupling tech-nique.

It is shown, that on basis of kinematic relations between sensors, actuators and the center ofmass, a rigid body decoupling can be derived. This decoupling resulted in two matrices whichdescribes the generalized velocities as function of the sensor outputs and the actuator inputs asfunction of the generalized velocities respectively. With this, the system is properly decoupled inthe frequency interval from 2Hz to 100Hz. In this domain the preferred control configuration isa diagonal pairing, as can be concluded on basis of the RGA. Outside this frequency range, thesystem is not properly decoupled. This is because the system can not be described as a rigid bodyfor high and low frequencies. If this causes problems in the control design, another decouplingmethod should be investigated, for example a modal decoupling technique.


6.2 Control Design

The diagonal controller elements, for both the independent design and sequential design, are de-signed on basis of classical loopshaping, using the FRF data of the AVIS as identified in Chapter5. The resulting closed loop system needs to be stable, which has to be shown on basis of stabilitycriteria presented in Chapter 4. On basis of the stability analysis, the conservatism of the stabilitycriteria will be discussed and a stable controller will be presented. The stability of this controllerwill be verified on basis of experimental data.

First, the stability of the AVIS is discussed on basis of the independent design. Second, thestability of the AVIS is discussed on basis of the sequential design. Finally, a combination of thetwo design strategies is applied.

6.2.1 Independent Design

Following the independent design of decentralized control, as discussed in Section 4.1, the loop-shape design will be based on the FRF data of the AVIS as shown in Figure D.1 and D.2. Eachdiagonal controller element, ki(jω), will be designed on basis of the corresponding diagonal sys-tem element, gii(jω). These resulting individual loops have to satisfy a couple of requirements.The loop has to be stable and the maximum peak in the sensitivity may not exceed 6dB:

si(jω) = (1 + ki(jω)gii(jω))−1 < 6dB. (6.19)

Satisfying these requirements, the gain should be tuned as high as possible around the sus-pension modes, in order to suppress these modes and improve the performance. No trackingrequirements are proposed, since the system does not have to follow a reference trajectory. If,on basis of these independently designed controller elements, the overal system is unstable, thecontroller elements have to be retuned.

Individual Loopshaping

The magnitude of the diagonal terms in the bode diagram of the AVIS is characterized by a +1slope at low frequencies. A couple of suspension modes, depending on the axis, at frequenciesbetween 1Hz and 5Hz. A−1 slope for frequencies between 5Hz and 100Hz and some resonancemodes above 100Hz. This means, that there will be two 0dB crossings, but these will be atapproximately −90 degrees and 90 degrees, so, considering the Nyquist stability criterium, therewill be no need for extra phase in the controller. Therefore each diagonal controller element willconsist of a low pass filter, one or more notches and an adjustable gain. The low pass filter with acut off frequency of 80Hz will decrease te effect of high frequent noise and the notches will filterout the resonance modes. Moreover, control element k4 has a notch at around 1.5Hz as will beexplained in the next section. The gain must be tuned as high as possible in order to suppressthe suspension modes but not exceeding the 6dB limit on the sensitivity.

The designed diagonal controller elements, as shown in Figure 6.3, place the 0dB crossingsof the open-loop between 0.5Hz and 5Hz and result in a sensitivity that is smaller than 6dB forall frequencies.

6.2. CONTROL DESIGN 67

−80

−60

−40

−20

Mag

nitu

de [d

B]

10−1

100

101

102

−270

−180

−90

0

90

Frequency [Hz]

Pha

se [d

eg]

k1

k2

k3

k4

k5

k6

Figure 6.3: Bode diagram of the decentralized controller elements

MIMO stability

The stability criteria, as derived in Chapter 4, can be used to analyze the stability of the indepen-dently designed controller. The Nyquist plot in Figure 6.4 and the characteristic loci plot in Figure6.5, show that the closed loop system, (I + GK)−1 is stable. As mentioned earlier, this Nyquistplot is difficult to interpret, but the characteristic loci show clearly that the closed loop systemis stable. In order to analyze the influence of the interactions, a factorization of the sensitivity,S0 = Sd(I + ETd)−1, will be investigated. On basis of the Nyquist plot of this factorization (notshown) can again be concluded that the closed loop system is stable. However, the spectral radiuscondition and the structured singular value condition, shown in Figure 6.6, are not satisfied. Onbasis of these stability criteria, it can not be concluded that the system is stable. Note that on basisof this result neither can be concluded that the system is unstable, since these stability boundsare sufficient conditions instead of sufficient and necessary conditions.

The independent design applied in this section resulted in a controller that stabilizes theclosed-loop system. This can be concluded on basis of the non conservative Nyquist criteriumand characteristic loci criterium. The sufficient spectral radius condition and structured singularvalue condition are not satisfied for the factorized problem. So on basis of these criteria noconclusion can be drawn about stability.


−500 0 500 1000

−200

0

200

400

600

800

10000 dB

Real Axis

Imag

inar

y A

xis

(a)

−10 −8 −6 −4 −2 0 2−2

−1

0

1

2

30 dB

−10 dB−6 dB

−4 dB

−2 dB

10 dB6 dB

4 dB

2 dB

Real AxisIm

agin

ary

Axi

s

(b)

Figure 6.4: Nyquist plot for det(I +G(jω)K(jω)) (a) overview and (b) zoomed in around theorigin

Figure 6.5: Nyquist plot of the characteristic loci


10−1

100

101

102

10−3

10−2

10−1

100

101

Frequency [Hz]

Spe

ctra

l Rad

ius

[−]

(a)

10−1

100

101

102

10−3

10−2

10−1

100

101

Frequency [Hz]

Mag

nitu

de [−

]

(b)

Figure 6.6: (a) Spectral Radius for E(jω)Td(jω) and (b) Structured Singular Value forE(jω)Td(jω), with σ(Td) (gray) and 1/µ(E) (black)

6.2.2 Sequential Design

In this section the sequential control design approach is followed for decentralized control as dis-cussed in Section 4.1. The difference between this approach and the independent design, is thatwith sequential design, each sequential controller is designed on basis of a the equivalent plant,instead of the original system, whereafter all equivalent plants can be analyzed for stability. Inthis work the diagonal control elements for the sequential design are the same as for the inde-pendent design.

MIMO stability

The stability of this sequential design is, following the equivalent open loop stability theorem,analyzed on basis of the equivalent open loops, G∗

i ki. The equivalent plants, G∗i are shown in

Figure 6.7. In this plot is clearly visible that each axis has one dominant suspension mode.The Nyquist plots of the equivalent open loops are shown in Figure 6.8. From this plot can beconcluded that the closed loop system is stable and there is marge to increase the gain of thefeedback controller.

6.2.3 Combined Sequential and Independent Design

In this section a combination of sequential and independent design is followed. The used strategyis as follows. First, one control loop is closed, whereafter the remaining equivalent plant is com-puted. This equivalent plant is then controlled as in the independent design case and analyzedfor stability using the factorization of the sensitivity. If the stability criteria fail, this approach isrepeated until the conservative stability criteria are satisfied.

It turned out that closing three loops was enough to satisfy the stability criteria. The decisionon the ordering of closing the loops is not straightforward. Here is chosen to close loop 1 (x) first,then loop 4 (φ) and finally loop 6 (ψ). The equivalent plant is shown in Figure 6.9 (only the (2, 2)


10−1

100

101

102

−40

−20

0

20

40

60

80

Frequency [Hz]

Mag

nitu

de [d

B]

G1*

G2*

G3*

G4*

G5*

G6*

Figure 6.7: Bode plot of the equivalent plants

Figure 6.8: Nyquist plots of the equivalent open loops


element of the system is shown to remain orderly) for all sequential steps, in order to see theinfluence of closing one loop on the remaining loops. Closing loop 1 and 3 causes for examplethe dominantly present suspension modes in these loops to damp in loop 2.

MIMO stability

Each sequential equivalent plant can be analyzed for stability on basis of the spectral radius con-dition and the structured singular value condition. The spectral radius is shown in Figure 6.10.The equivalent plant is proven stable after three loop closings. The choice for closing the first twoloops (1 and 4) is based on this spectral radius plot. It can be seen that the criterium is initiallynot satisfied, because of high peaks at respectively 1.5Hz, 3Hz and 4.25Hz. These are causedby the suspension modes in loop 1 and 4. After closing these loops the spectral radius bound isalmost satisfied.

If the structured singular value condition in Figure 6.11 is examined, it can be concluded that,after closing 2 loops, the biggest challenge in order to satisfy this bound, is the region around at0.6Hz. This is due to a zero in loop 3 and 5. Here, the diagonal terms have low gain and therefore,the non diagonal terms are relatively large and cause stability problems. The only two loops leftthat does not have this zero in the plant dynamics, are loop 2 and 6. This bound can thus besatisfied by closing both loops, or by closing one loop and reducing the gain in the other loop assuch that this bound is satisfied. Since the gain of the plant dynamics in loop 2 is smaller thanthe gain in loop 6, there is chosen to close loop 6 and reduce the gain of the controller for loop 2.After closing this third loop, both the spectral radius condition and the structured singular valuecondition are satisfied and the system is proven stable on basis of these conservative stabilitybounds.

As can be seen in Figure 6.3 has control element k4 a notch at approximately 1.5Hz. Thisnotch is necessary to meet the structured singular value condition at that frequency as can beseen in Figure 6.11. The notch is placed in control loop 4, since there is a strong interaction ofthe suspension mode at 1.5Hz in that loop. It should be noted that this notch is only necessaryto prove stability in the combined sequential and independent analysis.

The combination of the sequential design and the independent design resulted in a stableclosed-loop system that satisfies the sufficient spectral radius stability criterium and structuredsingular value stability criterium. The order of loop closing is not straightforward, but on basisof the bode plots, the spectral radius condition and structured singular value condition for eachsequential step, the order of loop closing could be determined.

The independent design and sequential design presented in this work do not differ in controlelement design, but only in stability analysis. The independent design has the stronger require-ment, that the whole system has to be stable under influence of the off-diagonal terms, while thesequential design only requires the equivalent plants to be stabilized. A drawback of the sequen-tial design is that the performance of the sequential closed loops is uncertain, since all equivalentplants are influenced by changing a control loop.

6.2.4 Stability verification

In the previous sections has theoretically been proven that, on basis of the designed controller,the AVIS is closed-loop stable. It is possible that the control design is nevertheless unstable, dueto a wrong system identification or unmodeled control implementation effects. Consequently, it


20

40

60

80M

agni

tude

[dB

]

100

−180

−90

0

90

180

Frequency [Hz]

Pha

se [d

eg]

Figure 6.9: Bode diagram of the (2, 2) element of the outer plant after closing 0 loops (black),1 loop (dark gray), 2 loops (gray), 3 loops (light gray)

10−1

100

101

102

10−3

10−2

10−1

100

Frequency [Hz]

Spe

ctra

l Rad

ius

[−]

Figure 6.10: Spectral Radius for E(jω)Td(jω) of the outer plant after closing 1 loop (black),2 loops (gray), 3 loops (light gray)


10−1

100

101

102

10−3

10−2

10−1

100

101

Frequency [Hz]

Mag

nitu

de [−

]

(a)

10−1

100

101

102

10−3

10−2

10−1

100

101

Frequency [Hz]

Mag

nitu

de [−

]

(b)

10−1

100

101

102

10−3

10−2

10−1

100

101

Frequency [Hz]

Mag

nitu

de [−

]

(c)

Figure 6.11: Structured Singular Value for E(jω)Td(jω), with σ(Td) (gray) and 1/µ(E) (black)after closing (a) 1 loop, (b) 2 loops, (c) 3 loops


need to be shown that the designed controller stabilizes the AVIS. Moreover, it need to be shownthat, if the Nyquist criterium is not satisfied, the AVIS system is indeed unstable. This will bedone on basis of experimental results of the impulse response of the closed loop system.

A time trace of the six outputs of the AVIS is shown in Figure 6.12. The input for thistime trace is an impulse in the x direction after one second. For each plot another controller isimplemented. The first plot is based on the stable controller as derived in the previous section.The second and third plot are based on the same controller, multiplied by a scaling factor α ofrespectively 7 and 10. Here is chosen, to multiply each controller element by same factor, sincethe characteristic loci are scaled by the same factor following:

λ(αG(jω)K(jω)) = αλ(G(jω)K(jω)). (6.20)

This is useful to find a scaling that will cause instability. On basis of the characteristic loci, forwhich the largest are shown in Figure 6.13, can be concluded that the closed loop system is onthe edge of stability for α = 7 and unstable for α = 10. The time plots in Figure 6.12 confirmthis analysis.

The stability analysis in this section has shown that the characteristic loci stability criteriumis a reliable analysis tool. Besides is shown that the model used for control design is a reliablerepresentation of reality and that unmodeled control implementation effects do not affect theclosed-loop stability.

6.3 Conclusion

In order to minimize the interactions in the AVIS, the system is decoupled on basis of rigid bodydecoupling. This decoupling is based on the kinematic relations of the sensor and actuator posi-tions relative to the center of mass. On basis of a RGA analysis can be concluded that the systemis properly decoupled in the frequency domain from 2Hz to 100Hz. Outside this range the sys-tem can not be described as a rigid body. Since the bandwidth of the final controller is outside thisdomain, it is expected that, the performance could be increased if another decoupling method,e.g. a modal decoupling technique, is applied. On basis of this rigid body decoupling a diagonalpairing is chosen.

The AVIS is controlled on basis of an independent loop shape design of the diagonal controllerelements. First, the stability of the closed loop system is verified on basis of the Nyquist criterium.

Second, stability analysis of the factorization of the sensitivity, which follows from the inde-pendent design approach, has shown that the sufficient structured singular value condition andspectral radius conditions were not satisfied. This indicates that the structured singular valuecondition and spectral radius conditions are no necessary conditions for stability.

Third, stability of the closed loop system is verified on basis of the equivalent open loopstability criterium, which follows from the sequential design approach. On basis of this analysiscan be concluded that there is marge to increase the gain of the feedback controller.

Finally, a combination of the sequential and independent control design is followed. Afterclosing three loops, it is possible to guarantee stability of the remaining equivalent plant on basisof the sufficient structured singular value condition and spectral radius condition.

6.3. CONCLUSION 75

0 0.5 1 1.5 2 2.5 3

−2

−1

0

1

2

x 10−4

Time [s]

Vel

ocity

[m/s

]

(a)

0 0.5 1 1.5 2 2.5 3

−2

−1

0

1

2

x 10−4

Time [s]

Vel

ocity

[m/s

]

(b)

0 0.5 1 1.5 2 2.5 3

−2

−1

0

1

2

x 10−4

Time [s]

Vel

ocity

[m/s

]

(c)

Figure 6.12: Impulse response of the closed-loop AVIS experimental setup with implementedcontroller (a) K(s), (b) 7K(s), (c) 10K(s), for all six outputs

(a) (b)

Figure 6.13: Characteristic loci plot of (a) λ1(7G(jω)K(jω)) and (b) (a) λ1(10G(jω)K(jω))


At the end of this chapter, experimental data has shown that the necessary and sufficientcharacteristic loci stability criterium is a reliable analysis tool.

Chapter 7

AVIS Disturbance Characterization

In this chapter will first be investigated whether it is possible to reconstruct the time traces of thedisturbances that act on the AVIS. These time traces can than be used for further characterizationand modeling of the disturbances. The bicoherence, described in Section 2.2, will be analyzed inorder to determine whether quadratic non linearities are present in the system. The multivari-ate PSD, described in Section 2.2, will be analyzed in order to get a good characterization of thedisturbances. On basis of this characterization, an improved, stabilizing control design will bepresented. The disturbances will be modeled using a stochastic realization technique and by theselection of basis functions for a waveform model. In order to clarify the presented methods, anexternal disturbance is applied to AVIS using a PATO setup.

7.1 Disturbance Reconstruction

In this section, it will be discussed if it is possible to reconstruct the unknown disturbances usingthe measurable signals in the control loop. The considered MIMO system is shown in Figure 7.1and consists of a feedback loop with stabilizing controller K and the AVIS experimental setup G.The measurable signals are the reference signal, r, the measured output, y, the error, e and thecontrol signal, u. The unknown signals are the input disturbances, di, the output disturbances,do and the measurement noise, n.

First, a transfer function from the unknown disturbances, w(t), to the known signals, s(t), is

K Ge u

y

di do

n

r

-

Figure 7.1: Block scheme

77

78 CHAPTER 7. AVIS DISTURBANCE CHARACTERIZATION

derived as shown in Equation (7.1). An inverse of this transfer function is necessary to identifythe unknown disturbances.

s(t) = H(z)w(t)

H(z) =

−K(z)−II

(I + G(z)K(z))−1

[G(z) I I

]

s(t) =

u(t)e(t)y(t)

w(t) =

di(t)do(t)n(t)

.

(7.1)

Since the rank of H(z) is equal to 1, as can be concluded from Equation (7.1), the matrix oftransfer functions H(z) is not invertible. The disturbances have a linear dependency equal to[G(z) I I

]. This means that it is not possible to distinguish between the three disturbance

signals on basis of the measurable signals. This is due to the fact that there is no measurementpoint between the locations where the disturbances enter the loop. To deal with this problemadditional assumptions are required. We express these assumptions as the choice of a weight-ing matrix V (z), defined in Equation (7.2). Here, ww(t) is the column of weighted unknowndisturbances. The weighting matrix can be designed on the physical understanding of the distur-bance signals, but for the weighting as used in Figure 7.1 an identity matrix should be chosen forV (z). Other choices for V (z) yield other, possible frequency depended, linear combinations ofthe disturbances.

ww(t) = V (z)w(t)

V (z) =

V1di(z) V1do(z) V1n(z)V2di(z) V2do(z) V2n(z)V3di(z) V3do(z) V3n(z)

.

(7.2)

The most simple choice for V (z) is a zero matrix, with on the second row of V (z) unity matrices.By this choice, one assumes that all the disturbance can be treated as output disturbances, d(t),as schematically represented in Figure 7.2. On basis of this assumption, Equation (7.1) can besimplified to Equation (7.3) and inverted to obtain Equation (7.4). This equation is overdeter-mined and only one relation is required to reconstruct d(t). For example, a measurement of y(t)can be used as in Equation (7.5):

s(t) =

−K(z)−II

(I + G(z)K(z))−1d(t) (7.3)

d(t) = (I + G(z)K(z))[−K(z)−1 −I I

]s(t) (7.4)

d(t) = (I + G(z)K(z))y(t). (7.5)

On basis of these relations, using the model derived in Section 5.2 and the prediction methodsderived in Section 3.2.2, it is possible to reconstruct the combined output disturbances following:

do(t) = Ψ−1N yN (t). (7.6)

7.1. DISTURBANCE RECONSTRUCTION 79

K Ge ur

-y

d

n

do

di

Figure 7.2: Block scheme with combined disturbances

0 0.5 1 1.5 2 2.5 3−3

−2

−1

0

1

2

3x 10

−5

Time [s]

Vel

ocity

[m/s

],[ra

d/s]

Figure 7.3: Time trace of the output disturbances. From black to light gray, x, y, z, φ, θ andψ

Here is N the length of the output and Ψ−1N the inverse of the extended Toeplitz matrix which

is obtained during the subspace identification. An experimental time trace of the six outputdisturbances is shown in Figure 7.3.

It is shown that on basis of the measurable signals in the control loop, it is not possibleto reconstruct all the different disturbances separately. Therefore additional assumptions arerequired. We showed that for the case that all disturbances are considered as output disturbances,it is possible to reconstruct the time trace of these disturbances. These time traces are nowavailable for further characterization.


Table 7.1: Statistical analysis of the output disturbances

x y z φ θ ψ

mean 0.0062 -0.0051 -0.0418 -0.0228 -0.0266 -0.0280variance 0.0235 0.0246 0.0374 0.0970 0.1403 0.0195skewness 0.0375 0.0225 -0.0308 -0.0420 -0.0148 -0.1574kurtosis -0.3223 0.1666 0.0391 -0.0537 -0.0767 -0.0480

050

100150

0

50

100

150

0

0.2

0.4

0.6

0.8

1

Frequency [Hz]Frequency [Hz]

Mag

nitu

de [−

]

Figure 7.4: Bicoherence of the output disturbance on the z-rotation

7.2 Bicoherence Analysis

In this section, the bicoherence will be used to analyse the output disturbances. On basis of thatanalysis will be determined whether it is possible to indicate quadratic non linearities if these arepresent.

The statistical quantities of the output disturbances are shown in Table 7.1. On basis of thesestatistical quantities it is difficult to determine if there are quadratic non linearities, since thereis no reference information. Nevertheless, the skewness of the output disturbance on ψ deviatesthe most from zero, so if there would be any quadratic non linearity, it would be on that rotation.Therefore, the bicoherence of the output disturbance on ψ is estimated and plotted in Figure 7.4.On basis of this plot can be concluded that there is some phase coupling at 50Hz and it is likely,that there is a quadratic non linearity at 150Hz. It is expected, that in spite of this non linearity,the output disturbances can be analyzed and identified using second order techniques like PSDand stochastic realization.

7.3. EXTERNAL DISTURBANCE 81

Figure 7.5: The PATO setup with unbalance on top of the AVIS

7.3 External Disturbance

In order to clarify the following characterization and identification methods, the AVIS has to bedisturbed in a controlled way. Therefore, a PATO setup is placed on top of the AVIS as shown inFigure 7.5. This PATO setup has a large unbalance and is controlled to rotate at 5Hz. This rotatingfrequency in combination with the large unbalance will result in an external disturbance. Afrequency of 5Hz is chosen in order to have a disturbance around the frequency of the suspensionmodes. The PSD of the rotational frequency of the PATO and consequently the disturbanceapplied by this setup, is shown in Figure 7.6. This plot shows clearly that not only a disturbanceis created at 5Hz, but also at the higher harmonics and at 70Hz, which is the resonance frequencyof the PATO system. This known disturbance can be used to verify the derived methods.

7.4 Multivariate Power Spectral Density Analysis

In this section, the auto multivariate PSD of the output disturbances will be estimated. This es-timate will be analyzed for characterization of the disturbances using the multivariate PSD andsome techniques that simplify the PSD analysis. On basis of this characterization a better controldesign could be realized.

In order to estimate the PSD of the disturbance signal a multivariate version of the WelchPSD estimator [29] is developed as described in Appendix A. The multivariate PSD estimate ofthe output disturbances under influence of the extra disturbance, Φdex(ω), is compared to thePSD estimate of the output disturbance during standstill, Φdss(ω), in Figure E.1. With the autospectra on the diagonal and the cross spectra in the upper and lower triangle. These PSD’s givean impression of the power of disturbance at different frequencies.

The contribution of the extra disturbance on the PSD is clearly visible. Also, it may be no-ticed that for those frequencies where the extra disturbance has very low amplitude, both PSDestimates coincide. This indicates that the background noise, which is the disturbance in thestandstill case, does not vary significantly over time.

Further, it can be noticed that the largest contributions of the output disturbances are at the


100

101

102

−160

−120

−80

−40

Frequency [Hz]

PS

D [d

B]

Figure 7.6: PSD of the rotational frequency of the Pato setup

frequencies of the AVIS suspension modes. This indicates that the assumption of only outputdisturbances acting on the closed-loop system is incorrect and that input disturbances should beconsidered for a better characterization.

These conclusions can be made on basis of close examination of the multivariate PSD plots.However, it is quite difficult to interpret the 6 × 6 plots. Therefore a couple of techniques areavailable that simplify this interpretation. In this work, the following techniques are used toanalyse the multivariate PSD: the frequency dependent sum norm (FDSN), the frequency depen-dent singular value decomposition (FDSVD) and the cumulative power spectral density (CPSD).Moreover, the principal component analysis (PCA) is shown to perform an analysis that is notfrequency based.

Frequency dependent sum norm

The FDSN of a multivariate PSD is defined in Equation (7.7):

||Φxy(ω)||sum =∑

i

∑

j

|Φxiyj (ω)|, ∀ω, (7.7)

and the FDSN of Φdex(ω) and Φdss(ω) is shown in Figure 7.7. The FDSN can be interpreted asa worst case scenario for all outputs. It gives an indication of frequencies where the disturbanceshave the most power. From the PSD plot we concluded that the sources contain much poweraround the frequencies of the suspension modes. On basis of the FDSN plot, it can be concludedthat the disturbances contain the most power at around 3.5Hz, which is the suspension mode foraxis 3, 4 and 5. Moreover, it can be seen that there is a lot of power in the disturbances at 121Hz.This can be due to the resonance frequency of the AVIS at that frequency, which again indicatesthat there are input disturbances working on the system.

7.4. MULTIVARIATE POWER SPECTRAL DENSITY ANALYSIS 83

100

101

102

−200

−150

−100

−50

0

50

Frequency [Hz]

FD

SN

[dB

]

Figure 7.7: Frequency dependent sum norm of the output disturbances for standstill data(gray) and under extra disturbance of the Pato (black)

On basis of the FDSN, it can be concluded at which frequencies the disturbances contain themost power. This simplifies the analysis of the PSD, but does not give additional information.

Frequency dependent singular value decomposition

The FDSVD of a multivariate PSD is defined in Equation (7.8) [27]:

Φxy(ω) = U(ω)S(ω)V (ω)H , ∀ω, U, V ∈ C, S ∈ R. (7.8)

Where U and V form the orthonormal bases for the input and output space of Φxy(ω). S isa diagonal matrix of singular values in descending order. The singular values of the FDSVDof Φdex(ω) and Φdss(ω) are shown in Figure 7.8. This plot gives information about the gainat different frequencies. The interpretation is more or less equal to the FDSN plot. The onlydifference is, that not only the maximal gains are shown in the FDSVD plot, but also the lowergains. As a result is for example the power contribution to the disturbance signal at 2Hz, due tothe suspension mode for axis 6, also noticeable.

The FDSVD can be used for the same purposes as the FDSN with the benefit that it alsoprovides insight in disturbance sources with lower power contributions in orthogonal directionsat that frequency.

Cumulative power spectral density

The CPSD, Ψxy(ω), is defined in Equation (7.9):

Ψxy(ω) =ω∑

ω1=ωmin

Φxy(ω1), ∀ω, (7.9)


100

101

102

−200

−150

−100

−50

0

50

Frequency [Hz]

FD

SV

D [d

B]

Figure 7.8: Frequency dependent singular value decomposition of the output disturbances forstandstill data (gray) and under extra disturbance of the Pato (black)

where ωmin is the lowest frequency. The CPSD of Φdex(ω) and Φdss(ω) are shown in FigureE.2. The CPSD can be interpreted as a total of power in the signal up to the regarding frequency.The course of the CPSD plots tells something about the contribution of the different frequenciesto the total amount of power in the signal. On basis of the CPSD can thus also be concludedthat the disturbances contain the most power at frequencies around the suspension modes andfor Φdex(ω) also at 5Hz. On basis of this information can be decided at which frequencies thedisturbances should be suppressed as much as possible. Moreover, it can be concluded on basis ofthe CPSD that disturbances on φ and θ contain the most power, which should also be consideredin control design.

On basis of the CPSD can also be concluded at which frequencies the disturbances containthe most power, but the CPSD normalizes the information over all frequencies, while the FDSNand the FDSVD normalize the information over all different axes.

Principal Component Analysis

The PCA is a linear transformation, that can be used to find the dominant directions in thedisturbances. The PSD of the principal components (PC) is related to the PSD of the disturbancesby:

Φd(ω) = WΦz(ω)W T , (7.10)

where W is a direction matrix, which can be found on basis of the SVD of the covariance matrixfollowing:

Σ2xy = USUT = WW−1 (7.11)

W = US1/2. (7.12)

7.5. PARAMETRIC DISTURBANCE MODELS 85

Note that the orthonormal basis for the input space is the transformed orthonormal basis for theoutput space, since the covariance matrix is a symmetric matrix. On basis of these equations,Φz(ω) is determined as shown in Figure E.3. On basis of this plot can be concluded that thePC’s are not Gaussian, since the off diagonal terms are not equal to zero. The directions thatcorresponds to the dominant PC’s of dss and dex are respectively:

Wdss =

−0.00810.01440.06410.27310.35720.0233

, Wdex =

−0.1727−0.01770.5606−2.0741−1.9971−0.0538

. (7.13)

On basis of these directions, it can be concluded that the disturbances are most dominant on φand θ. On basis of this information can be decided to increase the feedback gain in these loops,in order to decrease the output error caused by these disturbances.

It is shown that, on basis of a multivariate PSD, which can be estimated using the developedmultivariate version of the Welch estimator, the disturbance signals can be characterized. Anal-ysis of the PSD has shown that the background noise does not change significantly over time.Further, it can be concluded that not all disturbances are output disturbances, since the distur-bances contain lots of power at the frequencies of the suspension modes, which indicates thatthere is a significant contribution of input disturbances. Finally, it can be concluded that the dis-turbances on φ and θ are most dominant. These conclusion can be made on basis of the PSDplots, but the FDSN, the FDSVD, the CPSD and the PCA simplify this analysis.

7.5 Parametric Disturbance Models

In the previous section, a multivariate PSD of the output disturbances is estimated and analyzed.However, if the disturbance characterization will be used in parametric control design, a paramet-ric model is required. Therefore, two model techniques are applied. First, a model is estimatedusing the prediction of the output disturbances and the stochastic realization identification tech-nique as presented in Section 3.3. Second, a waveform model is chosen on basis of basis functionselection as described in Section 2.3.

7.5.1 Stochastic Realization

A stochastic realization model of the output disturbances of order 50, for both the standstill andthe extra disturbance case, is determined and shown in Figure D.5. These two models can notbe validated or compared directly, since the covariance matrix of the white input noise, R, maydiffer. Therefore the PSD of the output of the models is determined with a white noise signalwith covariance matrix R as input. These PSD’s are compared to the PSD of the reconstructeddisturbances as shown in Figure E.4 and E.5. On basis of these plots can be concluded that thestochastic realization identification method is successful.

In the standstill case, only the low frequent behavior is not modeled properly. This can be dueto memory problems. Consequently, the data length is limited and as a result, there is not enoughlow frequent behavior in the signal. In the extra disturbance case, the higher order harmonics ofthe 5Hz disturbance are not modeled properly.


7.5.2 Waveform Description

From the time plots of the output disturbances can be concluded that the signals consist of oneor more harmonic components. The exact frequency is determined on basis of the PSD. Twodominant harmonic components are distinguished of respectively 1.66 and 3.66Hz. For the extradisturbance case also 5Hz is dominantly present. Therefore, a waveform structure consistingof sinusoids with these frequencies, as defined in Equation (2.44), is chosen. The set of basisfunction is extended with a power series expansion of order 10, as defined in Equation (2.46),resulting in two waveform models of order 14 and 16 respectively. The power series expansion isadded to compensate for a possible offset and high frequent behavior.

In order to determine whether the obtained waveform model is capable of modeling the out-put disturbances, the model is used as an Luenberger observer using the following equations:

xd(t) = Adxd(t) + Le(t) (7.14)yd(t) = Cdxd(t) (7.15)e(t) = do(t)− yd(t), (7.16)

where Ad and Cd are the system matrices of the waveform model, do is the reconstruction of theoutput disturbances and L is the observer gain determined as the solution of a Riccati equation.It should be noted that the quality of the observer is dependent on the weighting functions usedin this Riccati equation. The PSD of the output of this observer, yd, is estimated and compared tothe PSD of the output disturbances as shown in Figure E.4 and E.5. As one can see is the approx-imation using this waveform model almost equal to the experimental recording. Only the highfrequent behavior of the signal is not modeled. The piecewise constant weighting function areapparently not able to change fast enough to model this high frequent behavior. The waveformmodel can be improved by adding basis functions with higher frequencies or increasing the orderof the power series expansion.

7.5.3 Conclusion

It is difficult to compare the waveform model and the stochastic realization model, since themodels are defined in a different manner. The stochastic realization model has a white noise asinput with a certain covariance. Whereas, the waveform model is used in an observer structure.However, it possible to compare the output of both models as described in the previous sectionsand shown in Figure E.4 and E.5.

It can be concluded that both models are able to reproduce the second order statistics of theestimated output disturbances reasonable good. The stochastic realization model describes thehigh frequent behavior of the disturbances better, but is not able to describe the higher orderharmonics of the extra disturbance. The waveform model describes the low frequent behavior ofthe disturbances better and is able to describe the extra disturbances, but not for high frequencies.

The choice for one of the two model structures is dependent on the control strategy that willbe used.

7.6. IMPROVED CONTROL DESIGN 87

Figure 7.9: Nyquist plot of the characteristic loci for the improved controller

7.6 Improved Control Design

In this section an improved control design for disturbance rejection will be presented, based onPSD analysis of the output disturbances.

In this work, it is assumed that the servo errors for all axis are equally important. Conse-quently, it is possible to compare disturbances on the translation of the AVIS with disturbanceson the rotation of the AVIS. PSD analysis of the output disturbances in this chapter has shownthat the disturbances on z, φ and θ are most dominant. Moreover, it has be shown that the distur-bances contain the most power at frequencies between 1 and 5Hz. The transfer function betweenthe disturbances and the output that has to be optimized, the sensitivity (I + GK)−1, is alreadyderived in Equation (7.1).

On basis of the expression for the sensitivity, it can be concluded that a large controller gainshould be chosen for the loops with the dominant disturbances. This will namely result in a dipin the sensitivity at those frequencies where the dominant disturbances have the most power. Thecontroller gain for the can not be increased unlimited, since a large loop gain can cause instability.Therefore, the controller gain in the other control loops should be chosen smaller. The improvedcontroller design is based on the controller designed in Chapter 6. Only the controller gainsare adjusted and the notch filter at 1.5Hz is omitted. The improved controller gains are 0.02,0.02, 0.04, 0.067, 0.067 and 0.02 respectively. The improved controller is also stable as can beconcluded on basis of the characteristic loci plot, shown in Figure 7.9.

The suppression of disturbances is always a tradeoff. If disturbances are rejected at a certainfrequency, they will be amplified at another frequency, the so-called waterbed effect. This is clearlyvisible, if the sensitivity of the improved designed controller is compared to the earlier designedcontroller as shown in Figure D.6. In this figure, it is also visible that the output disturbancesacting on φ and θ are more suppressed by the improved controller.

PSD analysis of the output disturbances in this chapter has also shown that there is a signif-icant contribution of input disturbances. Therefore, it is interesting to investigate the transferfunction between the input disturbances and the output, the process sensitivity (I +GK)−1G, asderived in Equation (7.1). The process sensitivity for both controllers is compared to the plant in


0 0.5 1 1.5 2 2.5 3−3

−2

−1

0

1

2

3x 10

−5

Time [s]

Vel

ocity

[rad

/s]

(a) φ

0 0.5 1 1.5 2 2.5 3−3

−2

−1

0

1

2

3x 10

−5

Time [s]

Vel

ocity

[m/s

]

(b) x

Figure 7.10: Output measurement of the AVIS with the controller derived in Chapter 6 (gray)and the improved controller design (light gray) implemented, compared to the open loop(black)

Figure D.7. In this figure, it is also visible that the input disturbances acting on φ and θ are moresuppressed by the improved controller.

A time trace of the output of the AVIS, alternately with the two controllers implemented andin open loop, is shown in Figure 7.10. On basis of the open loop time trace, it can be concludedthat the disturbances on φ are bigger than on x. In this figure it is also clear that the gain in thecontrol loop for φ is increased, while the gain in the control loop for x is decreased. These gainsshould be tuned as such, that all outputs are in the same order of magnitude and that the closedloop system is stable.

In order to compare the performance of both controllers, the FDSN is determined for thediagonal terms of the multivariate PSD of the output, as shown in Figure 7.11. Here is clearlyvisible that disturbances up to 20Hz are better suppressed and disturbances in the range of 20Hzto 60Hz are not as good suppressed.

It can be concluded that PSD analysis of the disturbances can lead to an improved controldesign with a better disturbance rejection.

7.7 Conclusion

In this chapter, the disturbances acting on the AVIS are analyzed. For better understanding ofthe procedure, an extra disturbance is applied to the AVIS, using a Pato setup with an unbalancerotating at a fixed frequency.

First, it is shown that on basis of the measurable signals in the control loop, it is not possibleto reconstruct all the different disturbances separately. However, with some additional assump-tions, it is shown that for the case that all disturbances are considered as output disturbances, itis possible to reconstruct the time trace of these disturbances. These time traces are further used

7.7. CONCLUSION 89

100

101

102

−160

−140

−120

−100

−80

−60

−40

−20

Frequency [Hz]

FD

SN

[dB

]

Figure 7.11: FDSN for the diagonal terms of the multivariate PSD of the output with thecontroller derived in Chapter 6 (black) and the improved controller design (gray) implemented

for characterization and identification of the disturbances.

Second, the reconstructed output disturbances are characterized on basis of the bicoherenceand the multivariate PSD. Analysis of the bicoherence is quite difficult, since for this experimentalsetup no linear reference information is available. Nevertheless, a non linearity was detected at150Hz in the disturbance on ψ. It is expected, that in spite of this non linearity, the outputdisturbances can be further analyzed and identified using second order techniques.

Analysis of the PSD, using the FDSN, the FDSVD, the CPSD and the PCA, has lead to moreinsight in the disturbances. It can be concluded that not all disturbances are output disturbances,since the disturbances contain lots of power at the frequencies of the suspension modes. This in-dicates that there is a significant contribution of input disturbances. Further, it can be concludedthat the background noise does not change significantly over time. Finally, it can be concludedthat the disturbances on φ and θ are most dominant.

On basis of this PSD analysis a new controller is designed, that improves the disturbancerejection on the AVIS.

Finally, two parametric models for the output disturbances are obtained. The first model is astochastic realization model, based on the reconstruction of the output disturbances. The secondmodel is a waveform model, based on the selection of 3 harmonic basis functions, extendedwith a power series expansion of order 10. Both models are able to reproduce the second orderstatistics of the reconstructed output disturbances reasonable good. The stochastic realizationmodel describes the high frequent behavior of the disturbances better, but is not able to describethe higher order harmonics of the extra disturbance. The waveform model describes the lowfrequent behavior of the disturbances better and is able to describe the extra disturbances, butnot for high frequencies.

Chapter 8

Conclusion en Recommendations

8.1 Conclusion

In this thesis, a control relevant disturbance identification procedure for MIMO systems is de-veloped. Moreover, this procedure is illustrated on the AVIS experimental setup. The proposedprocedure is as follows. If the system is already operating in closed loop, the first two steps canbe skipped.

• Obtain a frequency response model of the system, using spectral analysis.

• Design a controller that stabilizes the closed loop system.

• Obtain a parametric model of the sensitivity, the transfer function between output distur-bances and output, using the n4sid subspace identification technique.

• Reconstruct the output disturbances on basis of a time measurement of the outputs.

• Characterize the disturbances on basis of the multivariate PSD and the bicoherence.

• Obtain a parametric model of the disturbances, using a stochastic realization subspacetechnique or obtain a waveform model by selection of basis functions.

The characterization of multivariate disturbances is done on basis of second and third orderstatistics. Third order statistics, the skewness and bicoherence, are used to detect quadratic nonlinearities via quadratic phase coupling. A theoretic example has shown that it is possible to detectthese non linearities. However, in practise, it turned out that it is hard to determine whether nonlinearities are present in the signal.

In order to estimate the multivariate PSD, a multivariate welch estimator is developed. Theoutput disturbances are characterized on basis of the multivariate PSD. This characterization haslead to more insight in the disturbances. It can be concluded that not all disturbances are outputdisturbances, since the disturbances contain lots of power at the frequencies of the suspensionmodes. This indicates that there is a significant contribution of input disturbances. Further, itcan be concluded that the background noise does not change significantly over time. Finally, itcan be concluded that the disturbances on φ and θ are most dominant.

The identification of multivariate disturbances has lead to two different models. The firstmodel is a stochastic realization model, based on a time trace of the output disturbances. The

91

92 CHAPTER 8. CONCLUSION EN RECOMMENDATIONS

second model is a waveform model, based on the selection of 3 harmonic basis functions, ex-tended with a power series expansion of order 10. Both models are able to reproduce the secondorder statistics of the output disturbances reasonable good.

Two different MIMO system identification procedures are presented. First, a frequency re-sponse identification technique, the spectral analysis, is presented that can be used to obtainhighly accurate input-output models which are useful for control design. Second, a subspaceidentification technique, the N4SID method, is presented that can be used to obtain parametrictime domain input-output models that can be used for the prediction of disturbances.

The frequency response model a non parametric, frequency domain model and is very accu-rate for both very low and high frequencies. The model is validated on basis of the coherencebetween the input and output, which can be seen as the signal to noise ratio. As input signal forthe identification is chosen for a sum of sinusoids, in order to maximize the coherence at the de-sired frequencies. Unfortunately, the coherence can not be extended to analyze MIMO systems.Therefore, SIMO experiments are used, so that it is still possible to measure the reliability.

The subspace model is a parametric, time domain model. The model is validated by com-parison with the frequency response model and on basis of a correlation analysis between theresiduals and the input. As input for the subspace identification is chosen for a white noise se-quence, that is repeatedly used as input. The output is averaged and then used for identification.As a result, the effect of the stochastic disturbances is minimized. The advantage of this identifi-cation technique is that only a single experiment is needed to obtain a parametric MIMO modelof the system. The Toeplitz matrix, which is a by-product of the subspace identification, can beused for disturbance reconstruction and is more accurate than the state space description.

Two different decentralized control approaches, independent design and sequential design,are presented for MIMO control. The stability analysis is different for each approach, but thecontroller elements are equal for both approaches. The stability of the closed loop system canbe proven on basis of the MIMO Nyquist theorem or the characteristic loci, which are necessaryand sufficient stability conditions. A drawback of both stability analysis is that the effect of aparameter change in the controller on the stability of the system is not straightforward. Thereforeother stability criteria, as the spectral radius condition, the structured singular value stabilitycondition and the Gershgorin bounds, are derived from the MIMO Nyquist theorem. Thesestability conditions are easier to interpret but are only sufficient conditions for stability.

In order to analyze the closed loop system on basis of the independent control design, a fac-torization of the sensitivity, into the individual loops and a remaining part including the interac-tions, is presented. The individual loops are stable by choice of the controller elements. Analysisof the remaining part with the spectral radius condition and the structured singular value stabil-ity condition can indicate where the interactions cause instability. On basis of that information astabilizing controller can be designed.

For the stability analysis of sequentially designed control systems another factorization ofthe closed loop system is proposed. This factorization is based on the equivalent open loop andstability of all these loops guarantees stability of the entire closed loop system. Stability of theequivalent open loops can be proven on basis of SISO stability measures.

8.2. RECOMMENDATIONS 93

8.2 Recommendations

Future research may focus on the following issues, in order to improve the presented controlrelevant disturbance identification procedure for MIMO systems.

The use of higher order statistics for the characterization of disturbances is in this work lim-ited to the analysis of the bicoherence, since the tricoherence could not be estimated due to com-putational limitations. Moreover, few conclusions could be drawn on basis of the bicoherenceanalysis, since the estimator depends too much on the choice of estimator settings. Furthermore,no reference information is available. Therefore, it is recommended to improve the estimatorsfor both the bicoherence and tricoherence, in order to limit computational power and increasethe consistency of the estimates. With this, we expect that the characterization of non linearitiesin an experimental environment could be improved.

In this work, a parametric input output model is obtained using the n4sid subspace identi-fication algorithm. A drawback of this technique is that there is no cost function minimization.Consequently, no frequency weighting could be applied in order to obtain a better fit around thebandwidth crossings of the open loop system. Therefore, it is recommended to investigate thepossibilities for frequency weighting in the applied subspace identification techniques. Moreover,it should be investigated if it is possible to fit a parametric MIMO model on the frf data obtainedby the frequency response identification. Such methods exist for SISO systems and MIMO ex-tensions may contribute to improved MIMO system identification.

The rigid body decoupling of the AVIS in this work is successful in the frequency intervalfrom 2Hz to 100Hz. In order to increase the performance of the decentralized controllers, therigid body decoupling of the AVIS should be improved. Alternatively, another decoupling methodmay be investigated. If the AVIS is equipped with extra sensors, so that there are sensors at ev-ery corner of the AVIS, we expect that the rigid body decoupling can be improved. Anotherdecoupling technique that should be investigated is a modal decoupling technique, to decouplethe suspension modes of the AVIS. Again, additional sensors may improve such decoupling ap-proach.

In this work, it is assumed that all disturbances are output disturbances by choice of a weight-ing matrix. However, analysis of the PSD has shown that there are also input disturbances andit is likely, that there is some measurement noise. Therefore, it should be investigated, whetherit is possible to distinguish between the different disturbance signals by another choice of the,possible frequency dependent, weighting matrix. The directions of the multivariate disturbancesmay be used to find such a weighting matrix.

On basis of the PSD analysis in this work, an improved decentralized loopshaping controldesign is presented. However, we expect that performance improvement using decentralizedloop shaping is limited. Therefore, it is recommended to investigate other control techniquesthat have shown how knowledge of the multivariable aspects of the disturbances can be used inthe control design, in order to increase performance. Some of these techniques are for example,disturbance accommodated control [10] or techniques based on the internal model principle likedisturbance decoupling control [4] and fixed direction disturbance rejection [2].

94 CHAPTER 8. CONCLUSION EN RECOMMENDATIONS

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[2] M.L.G. Boerlage, R. Middleton, M. Steinbuch, and A.G. de Jager. Rejection of fixed directiondisturbances in multivariable electromechanical motion systems. pages 1–6. IFAC Worldcongress, 2008.

[3] C.S. Burrus and T.W. Parks. DFT/FFT and Convolution Algorithms: Theory and Implementa-tion. John Wiley & Sons, 1985.

[4] C.J. Chiang and A.G. Stefanopoulou. Control of thermal ignition in gasoline engines. pages3847–3852. ACC, 2005.

[5] W.B. Collis, P.R. White, and J.K. Hammond. Higher-order spectra: The bispectrum andtrispectrum. Mechanical Systems and Signal Processing, 12(3):375–394, May 1998.

[6] B.A. Francis and W.M. Wonham. The internal model principle of control theory. Automatica,12(5):457–465, September 1976.

[7] P. van den Hof. Lecture Notes SC4110 - System Identification. 2006.

[8] P. van den Hof. Lecture Notes wb3210 - Random Signal Analysis. fourth edition, 2006.

[9] A. Hyvärinen, J. Karhunen, and E. Oja. Independent Component Analysis. John Wiley & Sons,2001.

[10] C.D. Johnson. Accommodation of external disturbances in linear regulator and servomech-anism problems. Transactions on automatic control, AC-16(6):635–644, December 1971.

[11] C.D. Johnson. Control and Dynamic systems, advances in theory and applications, volume 12.Academic Press, 1976.

[12] T. Katayama. Subspace Methods for System Identification. Springer, 2005.

[13] L. Ljung. System Identification, Theory for the user. Information and System Sciences. PrenticeHall, second edition, 1999.

[14] T.W. Martin and W.G. Johnston. Disturbance accommodated control with fast fourier trans-form frequency detection. Proceedings of the Fortieth Midwest Symposium on Circuits and Sys-tems, 1997., 1:509–513, August 1997.

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[15] J.M. Mendel. Tutorial on higher-order statistics (spectra) in signal processing and systemtheory: Theoretical results and some applications. volume 79, pages 278–305. IEEE, 1991.

[16] C.L. Nikias and J.M. Mendel. Signal processing with higher order spectra. IEEE SignalProcessing Magazine, pages 10–37, July 1993.

[17] C.L. Nikias and A.P. Petropulu. Higher order Spectra Analysis: a nonlinear signal processingframework. Prentice Hall, 1993.

[18] C.L. Nikias and M.R. Raghuveer. Bispectrum estimation: A digital signal processing frame-work. Proceedings of the IEEE, 75(7):869–891, July 1987.

[19] P. van Overschee and B. de Moor. Subspace Identification for Linear Systems. Kluwer, 1996.

[20] J.A. Profeta, W.G. Vogt, and M.H. Mickle. Disturbance estimation and compensation inlinear systems. Transactions on aerospace and electronic systems, 26(2):225–231, March 1990.

[21] N.G.M. Rademakers. Modelling, identification and multivariable control of an active vibra-tion isolation system. Masters thesis, Eindhoven University of Technology, 2005.

[22] A. Rivola and P.R. White. Bispectral analysis of the bilinear oscillator with application to thedetection of fatigue cracks. Journal of Sound and Vibration, 1998.

[23] A. Rivola and P.R. White. Detecting system non-linearities by means of higher order statis-tics. International Conference, Acoustical and Vibratory, 1998.

[24] D. de Roover. Motion Control of a wafer stage. PhD thesis, Technische Universiteit Delft,1997.

[25] D. de Roover and O.H. Bosgra. Dualization of the internal model principle in compensatorand observer theory with application to repetitive and learning control. Proceedings of theAmerican Control Conference, pages 3902–3906, June 1997.

[26] D. de Roover and O.H. Bosgra. An internal-model-based framework for the analysis anddesign of repetitive and learning controllers. Proceedings of the Conference on Decision andControl, pages 3765–3770, December 1997.

[27] S. Skogestad and I. Postlethwaite. Multivariable Feedback Control. Wiley, second edition,2005.

[28] T. Söderström and P. Stoica. System Identification. Systems and Control Engineering. Pren-tice Hall, first edition, 1989.

[29] P. Stoica and R. Moses. Spectral Analysis of Signals. Pearson Prentice Hall, first edition,2005.

[30] M. Verhaegen and P. Dewilde. Subspace model identification, part 1: The output-error state-space model identification class of algorithms. International Journal of Control, 56(5):1187–1210, 1992.

[31] M. Verhaegen and P. Dewilde. Subspace model identification, part 2: Analysis of the el-ementary output-error state space model identification algorithm. International Journal ofControl, 56(5):1211–1241, 1992.

BIBLIOGRAPHY 97

[32] P.D. Welch. The use of fast fourier transform for the estimation of power spectra: A methodbased on time averaging over short, modified periodograms. IEEE Transactions on Audio andElectroacoustics, AU-15(2):70–76, June 1967.

[33] N. van de Wouw. Lecture Notes 4J400 - Multibody Dynamics. 2004.

[34] Y. Zhang and S.X. Wang. Slice spectra approach to nonlinear phase coupling estimation.Electronics letters, 34(4):340–341, February 1998.

98 BIBLIOGRAPHY

Appendix A

Power spectral Density Estimator

Definition

The power spectral density (PSD) of a zero mean time sequence x(k) can be defined in twoways following [29]. The first method uses a discrete time fourier transformation (DTFT) of thecovariance sequence r(τ) as in Equation (A.1) and (A.2). A second definition is given in Equation(A.3). It can be shown that this definition is equivalent to (A.1) under the mild assumption thatthe covariance sequence decays sufficiently rapid so that (A.4) holds. In this work we will followthe definition in Equation (A.1).

Φ(ω) =∞∑

τ=−∞r(τ)e−iωτ (A.1)

r(τ) = E{x(k)x(k − τ)} (A.2)

Φ(ω) = limN→∞

E

1N

∣∣∣∣∣N∑

k=1

x(k)e−iωk

∣∣∣∣∣

2 (A.3)

limN→∞

1N

N∑

τ=−N

|τ ||r(τ)| = 0 (A.4)

Approximation

The PSD can not be calculated exactly, so it should be estimated. The spectral estimator deriveddirectly from (A.1) is called the correlogram and is defined in Equation (A.5). The covariancesequence that should be inserted in the correlogram can be estimated using the standard biasedauto correlations sequence (ACS) defined in Equation (A.6).

Φc(ω) =N−1∑

τ=−(N−1)

r(τ)e−iωτ (A.5)

r(τ) =1N

N∑

k=τ+1

x(k)x(k − τ), 0 ≤ τ ≤ N − 1 (A.6)

99

100 APPENDIX A. POWER SPECTRAL DENSITY ESTIMATOR

Analysis of the statistical properties of the spectral estimators in [29] show the poor qualityof these estimators of the PSD. To quantify the quality of an estimator, the bias and variance areoften used. The motivation is that the mean squared error (MSE) of the estimate is the sum ofthe squared bias and the variance (A.7). It can be shown that for increasing N , the bias of Φp(ω)would go to zero. The main problem of the correlogram method lies in its large variance.

MSE , E{|a− a|2} = |bias{a}|2 + var{a} (A.7)

The variance/covariance of Φp(ω) equals (A.8) in the case of a general linear signal as defined in(A.9).

limN→∞

E{[

Φp(ω1)− Φ(ω1)] [

Φp(ω2)− Φ(ω2)]}

={

Φ2(ω1), ω1 = ω2

0, ω1 6= ω2(A.8)

x(k) =∞∑

τ=1

hτe(k − τ) (A.9)

It follows from these equations that, for a fairly general class of signals, the correlogram values areasymptotically (for N À 1) uncorrelated random variables whose means and standard deviationsare both equal to the corresponding true PSD values. Hence, the correlogram is an inconsistentspectral estimator, which continues to fluctuate around the true PSD, with a non zero variance,even if the length of the processed sample increases without bound.

Welch Method

To reduce the variance of the estimator and thus improve its quality, a modified correlogram-based method is used. In this case it is the Welch method [32]. The idea of this method isto reduce the variance by splitting the available sample of N observations into S subsampleswith M observations each. These subsamples may overlap each other with K observations andeach subsample may be windowed prior to computation of the covariance sequence. The Welchmethod can be described mathematical by Equation (A.14). The recommended value for K in theWelch method is K = M/2, which equals 50% overlap. The windowed correlogram is computedas in (A.10), where xj(k) is the subsample, v(k) the used window and P denotes the "power" ofthis window (A.13). The Welch estimate is found by averaging the windowed correlograms as in(A.14).

Φj(ω) =1

MP

M−1∑

τ=−(M−1)

rj(τ)e−iωτ (A.10)

xj(k) = v(k)x((j − 1)K + k),k = 1, . . . , Mj = 1, . . . , S

(A.11)

rj(τ) =1N

N∑

k=τ+1

xj(k)xj(k − τ) (A.12)

P =1M

M∑

k=1

|v(k)|2 (A.13)

ΦW (ω) =1S

S∑

j=1

Φj(ω) (A.14)

101

The selection of the window’s shape, v(t), should be based on a tradeoff between smearing andleakage effects. The choice of the window’s length should be based on a tradeoff between spectralresolution and statistical variance.

102 APPENDIX A. POWER SPECTRAL DENSITY ESTIMATOR

Appendix B

Identification and Estimatorinformation

Mechanical system, Figure 2.7

f [Hz] t [s] N [-] NS [-] NK [-] nfft [-]PSD 200 328 65536 4096 0 4096

Bicoherence 200 328 65536 128 0 128

Frequency Response Model, Figures D.1, D.2 and D.3

fg [Hz] fs [Hz] t [s] N [-] P [-] NS [-] NK [-] nfft [-]160 2 6720 13440 42 6720 0 [0.01875,0.99375]200 40 2880 115200 72 14400 0 [1.025,14.975]200 40 960 38400 24 9600 0 [2.025,14.975]1200 400 480 192000 48 23000 0 [7.1,199.9]

Subspace Identification Model, Figures D.1 and D.2

fg [Hz] fs [Hz] t [s] N [-] P [-] NS [-] k [-] n [-]1200 400 9600 3840000 240 6000 166 30

Stochastic Realization Model, Figures D.5

fs [Hz] t [s] N [-] NS [-] k [-] n [-]400 300 120000 120000 60 50

Output Disturbance, Figures E.1, E.4 and E.5

f [Hz] t [s] N [-] NS [-] NK [-] nfft [-]Measurements 400 300 120000 6000 3000 16384

Waveform model 400 300 120000 6000 3000 16384Stochastic Realization model 400 300 120000 2000 1000 4096

103

104 APPENDIX B. IDENTIFICATION AND ESTIMATOR INFORMATION

Bicoherence AVIS output disturbance, Figure 7.4

f [Hz] t [s] N [-] NS [-] NK [-] nfft [-]Bicoherence 400 300 120000 128 0 128

PSD Pato disturbance, Figure 7.6

f [Hz] t [s] N [-] NS [-] NK [-] nfft [-]PSD 1000 200 200000 8000 4000 16384

Appendix C

Calibration of the AVIS experimentalsetup

The AVIS velocity sensors will be calibrated on basis of comparison of the sensor output in thez-direction after decoupling with a calibrated acceleration sensor that is placed above the centerof gravity on the AVIS. The goal is to find a scaling factor between the output in Volts and the realvelocity in meters per second.

Two independent measurements are compared in this problem. The AVIS sensor output ofthe velocity in the z-direction and the acceleration in the same direction measured by a calibratedsensor using SIGLAB.

Both signals are synchronized by means of an impulse in the z-direction at the AVIS actuators.The response to this actuation is clearly visible in both measurements, on which they can besynchronized.

In order to remove low frequent drift and high frequent noise, both measurements are band-pass filtered with a bandpass filter between 10Hz and 20Hz. Consequently both measurementsare zero mean signals which simplifies the comparison.

The measurement of the calibrated sensor has to be integrated in order to obtain the velocity.Here is assumed that the integration constant is 0, since the AVIS is initial at rest.

For better comparison of both measurements, the AVIS is actuated in z-direction with afrequency of 15Hz. Given al this information the scaling factor, rs, is determined at rs =2.3974 · 10−4ms−1V −1.

This scaling factor is determined for the z-direction, but can also be applied for the otheroutputs. This is straightforward for the translations, but this holds also for the rotations, sincethe distance to the center of mass is considered in the decoupling matrix.

105

106 APPENDIX C. CALIBRATION OF THE AVIS EXPERIMENTAL SETUP

Appendix D

AVIS model Figures

107

108 APPENDIX D. AVIS MODEL FIGURES

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Figure D.7: Process sensitivity of the closed loop system for the controller derived in Chapter6 (gray) and the improved controller design (light gray) compared to the plant (black)

Appendix E

AVIS PSD Figures

123

124 APPENDIX E. AVIS PSD FIGURES

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Figure E.1: Multivariate PSD of the output disturbances for standstill data (gray) and underextra disturbance of the Pato (black)


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Figure E.4: Power spectral density of the output disturbances for standstill data, experimental(black), output of the stochastic realization model (gray) and the waveform model (light gray)


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Figure E.5: Power spectral density of the output disturbances under extra disturbance of thePato, experimental (black), output of the stochastic realization model (gray) and the waveformmodel (light gray)

Appendix F

MATLAB scripts

Multivariate MATLAB Implementation Welch Method

function PSD = pwelchmultivariate(x,K,M,nfft,fs)

% x = data with s channels of length N% K = percentage overlap [0-1)% M = number of observations in a subsample% nfft = number of fft points used to estimate the PSD% fs = sampling frequency

[N,s] = size(x) ; % observations in the sample, number of samplesif s>N, x = x.’ ; [N,s] = size(x) ; end

S = fix((N-M)/((1-K)*M))+1 ; % number of subsamples

if nfft < 2*M, error(’nfft too small for number of lags: non symmetric estimate’), end

h = hamming(M) ; % use Hamming windowH = zeros(length(h),s) ;for j = 1:s

H(:,j) = h ;end

Xi = 0 ;for i = 1:S

a = fix((i-1)*(1-K)*M+1) ; b=(a+M-1) ; % subsample boundsxi = x(a:b,:) ; % subsample ixi = xi.*H ; % window each sample with an Hamming window

ci = xcorr(xi) ; % auto/crosscorrelationXi = Xi+fft(ci,nfft)./(h’*h) ; % fourier transform normalized by window power

end

Xi = Xi./S ; % average over subsamples

135

136 APPENDIX F. MATLAB SCRIPTS

f = ((0:2/nfft:1)*fs/2)’ ; % define frequency vector

% select one-sided spectrumif rem(nfft,2) % Odd

select = 1:(nfft+1)/2 ;psd_un = Xi(select,:) ;psd = [psd_un(1,:) ; 2*psd_un(2:end,:)] ;

else % Evenselect = 1:nfft/2+1 ;psd_un = Xi(select,:) ;psd = [psd_un(1,:) ; 2*psd_un(2:end-1,:) ; psd_un(end,:)] ;

end

psd = psd./fs ; % normalize with the sample frequency

for i = 1:sfor j = 1:s

PSD(i,j) = frd(psd(:,(i-1)*s+j),f*2*pi) ; % frd output objectend

end

Multivariate MATLAB Implementation n4sid

function [G,Psi] = n4sidbasicMIMO(u,y,n,N,k,Ts)

% u = input - >N+2*k-1 x m% y = output - >N+2*k-1 x p% n = model order or [] for selection based on singular values% N = number of block colums% k = number of block rows% Ts = sample time

% Order data[lu,m] = size(u) ; if m > lu, u = u’ ; m = lu ; end[ly,p] = size(y) ; if p > ly, y = y’ ; p = ly ; end

% Check for data lengthif length(u) < N+2*k*m-1 || length(y) < 2*k*p-1,

error(’input/output length too short’)else

u = u(1:N+2*k*m-1,:) ; y = y(1:N+2*k*p-1,:) ;endif N < k*(2*m+p),

error(’N too small for horizon k, given the number of in- and outputs’)elseif N < 2*k*(m+p),

137

disp(’Warning: LQ decomposition not complete due to small N’),end

% Construct Hankel matrices from i/ofor i = 1:m

Up(i:m:k*m,1:N) = hankel(u(1:k,i),u(k:N+k-1,i)) ;Uf(i:m:k*m,1:N) = hankel(u(k+1:2*k,i),u(2*k:N+2*k-1,i)) ;

endfor i = 1:p

Yp(i:p:k*p,1:N) = hankel(y(1:k,i),y(k:N+k-1,i)) ;Yf(i:p:k*p,1:N) = hankel(y(k+1:2*k,i),y(2*k:N+2*k-1,i)) ;

end

% Lq and SV decomposition[L11,L21,L31,L22,L32] = lqn4sid(Up,Yp,Uf,Yf) ;[U,S,V] = svd(L32*pinv(L22)*[Up;Yp]) ; V = V’ ;

% Select model order on basis of SVDif isempty(n)

figure, semilogx(diag(S)), gridtitle(’Select model order in command window.’)disp(’Hit any key to continue’)pausen = -1 ;while n<=0

n = input(’Select model order: ’) ;n = floor(n) ;if n<=0,disp(’Order must be a postitive integer’),endpause(1)

endend

% Determine observability, block toeplitz and initial state matrixT = eye(n) ;Obs = U(:,1:n)*S(1:n,1:n)^0.5*T ;Xf = inv(T)*S(1:n,1:n)^0.5*V(1:n,:) ;Psi = (L31-L32*pinv(L22)*L21)*inv(L11) ;

% Approximate SS-model using least squares methodXk1 = Xf(:,2:end) ;Xk = Xf(:,1:end-1) ;Uk = Uf(1:m,1:end-1) ;Yk = Yf(1:p,1:end-1) ;ABCD = ([Xk1;Yk]*[Xk;Uk]’)*inv([Xk;Uk]*[Xk;Uk]’) ;G = ss(ABCD(1:n,1:n),ABCD(1:n,n+1:n+m),...

ABCD(n+1:n+p,1:n),ABCD(n+1:n+p,n+1:n+m),Ts) ;

138 APPENDIX F. MATLAB SCRIPTS

%-------------------------------------------------------------------------

function [L11,L21,L31,L22,L32] = lqn4sid(Up,Yp,Uf,Yf)

km = size(Up,1) ;kp = size(Yp,1) ;

[Q,L] = qr([Uf;Up;Yp;Yf]’,0) ;

Q = Q’ ;L = L’ ;

L11 = L(1:km,1:km) ;L21 = L(km+1:2*km+kp,1:km) ;L31 = L(2*km+kp+1:2*(km+kp),1:km) ;L22 = L(km+1:2*km+kp,km+1:2*km+kp);L32 = L(2*km+kp+1:2*(km+kp),km+1:2*km+kp);

disturbance characterization for mimo controlmate.tue.nl/mate/pdfs/9382.pdf · characterized....

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