Identiﬁcation and Analysis of Nonlinear Systems829311/FULLTEXT01.pdfthe frequency based methods Reverse Path and a Frequency Domain Structure Selection Method (FDSSA). The time domain

Master Thesis MEE03:32

Identification and Analysis of Nonlinear Systems

5 10 15 2010

−7

10−6

10−5

10−4

10−3

Henrik ÅkessonBenny Sällberg

Degree of Master of Science in Electrical EngineeringExaminer: Prof. Ingvar ClaessonSupervisor: Prof. Kjell AhlinDepartment of Telecommunication and Signal ProcessingBlekinge Institute of TechnologyDecember, 2003

Abstract

In classical mechanical engineering the predominant group of system analysis and identification tools relieson Linear Systems, where research have been carried out for over half a century. Usage of Linear Systems ismost widely spread, often due to its simple mathematics and formulation for many engineering problems. Al-though linearizing is a means for simplifying a problem, it will introduce more or less severe modelling errors.In some cases the errors due to linearizing are too large to be practically acceptable, and therefore nonlinearstructures and models are sometimes introduced.

This thesis aims in implementing and evaluating some popular methods and algorithms for nonlinear struc-ture analysis and identification, with emphasis on systems having nonlinear terms. Preferably the algorithmsshould be optimized in their computational load.

The result are several algorithms for nonlinear analysis and identification. The ones giving best results werethe frequency based methods Reverse Path and a Frequency Domain Structure Selection Method (FDSSA). Thetime domain based method, Nonlinear Autoregressive Moving Average with Exogenous Input (NARMAX), inwhich a lot of hope had been put, did perform very well in giving good system descriptions, but due to itsnonphysical representation it was not suitable for usage in this thesis.

The algorithms and methods were finally applied for two cases, a four system black-box case and an ex-perimental test-rig case. The methods did perform well in three out of four systems in the first case, but themethods did not perform well for the second case, due to problems in applying correct levels of excitation forceat the test-rig’s resonance frequencies.

Contents

1 Introduction 1

2 Theoretic Background 22.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Degree of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.2 Basic mechanical system SDOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.3 Larger mechanical systems MDOF . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.1 Brief Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Theoretical Representation of Nonlinearities . . . . . . . . . . . . . . . . . . . . . . 92.2.3 Presentation of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 System Identification and Analysis 103.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Linear System Identification and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.1 Non-Parametrical Spectrum Identification . . . . . . . . . . . . . . . . . . . . . . . . 103.2.2 SISO Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.3 Time Series Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.4 Special Types of Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.5 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Nonlinear System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.1 NARMAX Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.2 Reverse Path Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.3 Frequency Domain Structure Selection Algorithm . . . . . . . . . . . . . . . . . . . . 203.3.4 Finding the Nonlinear Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Nonlinear System Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4.1 Harmonic Balance Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4.2 Hilbert Transformation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4.3 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 System Synthesis 304.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Linear System Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.1 Time Response Synthesis using Laplace Transformation . . . . . . . . . . . . . . . . 304.2.2 Synthesis by Digital Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Nonlinear Systems Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3.1 Analytical Time Responses Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3.2 Synthesis by Ordinary Differential Equation Solvers . . . . . . . . . . . . . . . . . . 364.3.3 Synthesis by Extended Digital Filter Structures . . . . . . . . . . . . . . . . . . . . . 38

4.4 Synthesis Quality Assessments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

ii

5 Experimental Evaluation of Black-box Systems 405.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2 The Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2.1 The First System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2.2 The Second System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.2.3 The Third System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2.4 The Fourth System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2.5 The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6 Experimental Evaluation of Test-Rig 576.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.3.1 Linear Part of System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.3.2 Nonlinear Part of System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.3.3 Nonlinear Property Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.4.1 Measurement Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.4.2 Equipment List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.4.3 Work Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.5 Choice of Performance and Excitation Signals . . . . . . . . . . . . . . . . . . . . . . . . . . 616.6 Measurement Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.6.1 Measurement Settings, Collection from Signal Calc when using build-in functions . . 616.6.2 Measurement Settings, Collection from Signal Calc when using excitation signal pro-

duced in matalab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.6.3 Channels Settings for measurement 1-3 . . . . . . . . . . . . . . . . . . . . . . . . . 616.6.4 Channels Settings for measurement 4-11 . . . . . . . . . . . . . . . . . . . . . . . . 62

6.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7 Summary and Conclusions 817.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.2 Conclusions and Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

A Derivations of Impulse-, Step-, and Ramp Invariance 82A.1 Derivation of Impulse Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82A.2 Derivation of Step Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83A.3 Derivation of Ramp Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

B Modified Bootstrap Structure Detection 86

iii

Nomenclature

Matrix Notation

.. Column Vector..T Row Vector[..] Matrix[..]T Transpose of a matrix[..]−1 Inverse of a matrix[..]H Complex conjugate transpose, or Hermitian transpose, of a matrix

Operator Notation

x∗ Complex conjugatex First derivative with respect to timex Second derivative with respect to timex Estimated value of xx Mean value of xF Fourier transformF −1 Inverse Fourier transformL Laplace transformL−1 Inverse Laplace transformZ Z transformZ−1 Inverse Z transformH Hilbert transformH −1 Inverse Hilbert transformR Real part of complex numberI Imaginary part of complex number∂∂t Partial derivative with respect to independent variablet

Mechanical Notation

a Acceleration m/s2

v Velocity m/sx Displacement mm Mass kgc Viscous damping kg/sk Stiffness N/mf Force Nλ Eigenvalue, pole of transfer function rad/sζ Relative damping DLR Residue -r mode number (subscript) DLp response point (subscript) DLq reference point (subscript) DL

iv

Signal Analysis Notation

t Time sT Time period sF Frequency Hzf Normalized frequency DLFs Sampling frequency HzΩ Angle frequency rad/sω Normalized angle frequency DLΩ Instantaneous frequency rad/s2

θ Instantaneous Phase rad/sτ Time delay, Independent variable sA Instantaneous amplitude -s Laplace domain variable -z Z domain variable -x,x(t) Input signal -y,y(t) Output signal -x(n) Time discrete signal -y(n) Time discrete signal -X( f ) Spectrum ofx(t) -Y( f ) Spectrum ofy(t) -Pxx Auto spectrum ofx(t) -Pyy Auto spectrum ofy(t) -Pyx Cross Spectrum ofx(t) with y(t) -Rxx Autocorrelation ofx(t) -H( f ) Frequency response function -h(t) Impulse response function -w(n) Window function -δ(t) Impulse function -δ(n) Discrete impulse function -n Sample number DLN Integer number, order DLM Integer number, order DL

Statistical Notation

Mi Central Moment -M2 Second Moment -sx Skewness -kx Kurtosis -σx Standard deviation of x -σ2

x Variance of x -n(t) White Gaussian noise -

v

General Notation

[I ] Identity matrix DLJ The Jacobian -f.. Nonlinear function -G[..] Nonlinear system -kn Nonlinear coefficient constant -Ω Harmonic excitation -e(n) Model error -e(t) Error -L Integer number, order DLj

√−1 -4 Independent variable -

Notes:- The unit varies with variables.DL Dimensionless variable.

Abbreviations

DOF Degree of FreedomSDOF Single Degree of FreedomMDOF Multi Degree of FreedomFEM Finite Element MethodFRF Frequency Response FunctionGFRF General Frequency Response FunctionSISO Single Input Single OutputSIMO Single Input Multiple OutputMISO Multiple Input Singe OutputCSD Cross Spectrum DensityPSD Power Spectrum DensityAR Auto RegressiveMA Moving AverageARMA Auto Regressive Moving AverageARX Auto Regressive with External InputARMAX Auto Regressive Moving Average with External InputNARMA Nonlinear Auto Regressive Moving AverageNARMAX Nonlinear Auto Regressive Moving Average with External InputHBM Harmonic Balance MethodODE Ordinary Differential EquationCEA Complex Exponential AlgorithmLSCE Least Squares Complex ExponentialPTD Polyreference Time DomainITD Ibrahim Time DomainMRITD Multiple Reference ITDERA Eigensystem Realization AlgorithmPFD Polyreference Frequency DomainSFD Simultaneous Frequency DomainMRFD Multi-Reference Frequency DomainRFP Rational Fraction PolynomialOP Orthogonal PolynomialCMIF Complex Mode Indicator FunctionMBSP Modified Bootstrap Structure DetectionDBG Data Block GeneratorLR Linear RegressorSD Structure Discriminator

vi

Chapter 1

Introduction

This master thesis is divided into eight chapters. The initial chapters act as brief introductory to the huge fieldof nonlinear system analysis, identification and synthesis. The later chapters presents operations and results onfour black-box systems and on a physical test-rig. The chapters are as follows:

Chapter 2 - Theoretic Background Gives a theoretical background to the field of mechanics and modal anal-ysis. Single or multi degree of freedom dynamical systems, pure linear systems or systems having non-linear elements are explained and presented.

Chapter 3 - System Identification and AnalysisPresents briefly linear system identification and analysis meth-ods, and more thoroughly for nonlinear system identification and analysis methods.

Chapter 4 - System SynthesisIn this chapter tools for system synthesis are presented, both for linear systemsand for nonlinear systems.

Chapter 5 - Experimental Evaluation of Black-box systemsThe analysis and identification tools are usedfor experimental evaluation of four black-box systems.

Chapter 6 - Experimental Evaluation of Test-Rig The analysis and identification tools are used for experi-mental evaluation of a nonlinear mechanical system.

Chapter 7 - Summary and ConclusionsSummarizes and concludes the work and results of this thesis.

1

Chapter 2

Theoretic Background

2.1 Linear Systems

2.1.1 Degree of freedom

The understanding of the degree of freedom is a condition for understanding the concept of modal analysis.The number of degree of freedom is the number of independent coordinates and independent motions in eachcoordinate. In one single point there exists six degrees of freedom; the motions in each direction (x,y,z),and the rotational motion (xθ,yθ,zθ) of each axis. A mechanical system has an infinite number of degrees offreedom, because the system is continuous and must be described by an infinite number of coordinates. Theobserved degrees of freedom is in reality of course a finite number, limited by different physical causes. Thefollowing parameters reduces the number of degree of freedom; the frequency range of interest, the dynamicsin the measurement system and the in reality possible degrees of freedom able to reach and measure. All theseparameters makes it possible to connect theoretical analysis with real measurements.

Further definitions

A coordinate (x,y,z) of the mechanical system is hereafter referred to as a node, together with the motions, thenode forms a mode in each of these direction. With the fourth dimension time, mode shapes will appear and allmode shapes together describes the motion behavior of the whole system.

Assumptions

To be able to calculate and analyze the system, following assumptions will be made about the systems in thischapter.

• The system is linear:

a1H x1(t)+a2H x2(t)= H a1x1(t)+a2x2(t) (2.1)

H)(1 tx

)(2 tx

)( ty1a

2a

)(1 tx

)(2 tx

1aHH

2a)( tyÛ

Figure 2.1: Condition for system to be linear.

2

CHAPTER 2. THEORETIC BACKGROUND

• The system is causal:If the output from the system only depends on the current value or the previous values, the system isdenoted causal. That means the systems lacks of termszk, wherek > 0 which is same as the condition inequation (2.2), whereh(n) is the discrete impulse response.

h(n) = 0, n < 0 (2.2)

• The system is time invariant:That means the system acts independent over time.

y(t) = H x(t)⇔ y(t−k) = H x(t−k) (2.3)

H )( ty Û)( tx kZ - H kZ -)( ty)( tx

Figure 2.2: Condition for system to be time-invariant.

• Reciprocity:If the system fulfils the reciprocity condition then the transfer function is the same, even if the input isapplied into nodeq and the output measured in nodep as the opposite.

Hpq(s) = Hqp(s) (2.4)

)( sH p q)( sH q pq

p

)( tx

)( tfFigure 2.3: Condition for system to fulfil the reciprocity.f (t) is the force into nodeq of the system,x(t) is thedisplacement response from the system in nodep.

2.1.2 Basic mechanical system SDOF

Simple systems can be modelled as a mass-damper-spring system in a single point and direction which aredenoted as single degree of freedom system (SDOF). This is described by Newton’s equation, equation (2.5),where m is the mass, c the damping coefficient and k is the stiffness coefficient.

mx(t)+cx(t)+kx(t) = f (t) (2.5)

3

2.1. LINEAR SYSTEMS

k c

)( tx )( tf

m

Figure 2.4: An example single-degree-of-freedom system, described using massm, springk and damperc. Thesystem excitation forcef (t) and responsex(t) are displayed as well.

This equation has two solutions: A transient solution and a steady state solution. The transient solution is ahomogenous solution and is the one of interest when investigating the properties of the system.

To formulate the problem, a good start would be to set up an equation, based on Newtons second law,equation (2.5). This equation of motion describes the behavior of the system when a force is present. By usingthe Laplace transform the problem becomes more convenient to solve.

L mx(t)+cx(t)+kx(t)= L f (t) (2.6)

m(s2X(s)−sx(0)− x(0))+c(sX(s)−x(0))+kX(s) = F(s) (2.7)

Assuming initials conditionx(0) = 0, x(0) = 0, the equation system is rewritten into

ms2X(s)+csX(s)+kX(s) = F(s) =(ms2 +cs+k)X(s) = F(s) (2.8)

⇒1

(ms2 +cs+k)=

X(s)F(s)

= H(s) (2.9)

The Homogenous solution is solved withf (t) = 0.

ms2X(s)+csX(s)+kX(s) = 0 (2.10)

Solving the non trivial solutionL x(t) 6= 0, will give the characteristic system. This is a simple seconddegree equation solved by the general equation (2.11)

x1,2 =− p2±

√p2

2−q (2.11)

for the expressionx2 + px+ q = 0. The parameters from above used in equation (2.11) gives the naturalfrequencies.

λ1,2 =− c2m

±√( c

2m

)2− k

m(2.12)

4


Another common representation of the system is the sum of partial fraction of the system equation (2.9).

H(s) =R1

s−λ1+

R2

s−λ2(2.13)

whereR1 andR2 are the residues belonging to each rootλ1 andλ2. For underdamped(( c

2m)2 < km

)systems

the roots from equation (2.13) will be complex conjugated (λ1 = λ∗2). By definition the residues will also becomplex conjugate, equation (2.13) is rewritten into equation (2.14).

H(s) =R

s−λ+

R∗

s−λ∗(2.14)

2.1.3 Larger mechanical systems MDOF

For higher degree of freedom than the order one, the system becomes a multi degree of freedom (MDOF)system.

It is more convenient to first rewrite the equation system onto matrix form and then solve the homogenouspart.

[M ]x(t)+[C]x(t)+[K ]x(t)= f (t) (2.15)

[M ] =

m11 m12 · · · m1m

m21 m22 · · · m2m...

. . ....

mm1 · · · mmm

(2.16)

[C] =

c11 c12 · · · c1m

c21 c22 · · · c2m...

.. ....

cm1 · · · cmm

(2.17)

[K ] =

k11 k12 · · · k1m

k21 k22 · · · k2m...

.. ....

km1 · · · kmm

(2.18)

x(t)=

x1(t)x2(t)

...xm(t)

. . . x(t)=

x1(t)x2(t)

...xm(t)

f (t)=

f1(t)f2(t)

...fm(t)

(2.19)

The matrix notation in equation (2.17-2.18) is the general expression, but more common is the expression fromthe FEM (Finite Element Method) where the mass matrix is a diagonal matrix and the damping and stiffnessmatrices are symmetric. The equation (2.20) is an example of a lumped system calculated from a FEM modelwith the special case where the damping matrix equals zero. This special case is called a non-damped system.

[M ] =

m1 0 0 00 m2 0 0

0 0... 0

0 0 0 mm

[K ] =

k1 +k2 −k2 · · ·−k2 k2 +k3

..... .

(2.20)

5

2.1. LINEAR SYSTEMS

When the equation system is rewritten, the problem could be seen as an eigenvalue problem. There arethree different systems that will be mentioned, non-damped system, proportionally damped systems and nonproportionally damped system. The first system with no damping,[C] = [0] formulates the easiest problem tosolve.

([M ]s2 +[K ]

)X= 0 (2.21)∣∣[M ]α+[K ]∣∣ = 0 (2.22)∣∣[M−1][K ]+α[I ]∣∣ = 0 (2.23)

The systems poles are the roots of the eigenvalues calculated from equation (2.23),√

λ = α. Putting eacheigenvalue into the same equation and calculating the corresponding eigenvector will result in a mode shape,which belongs to the natural frequency with the relationshipλr = σ± jω.

Proportionally damped systemIt is shown that proportionally damped systems can be diagonalized in the same way as for non damped systems.The proportionally damped systems can thereby be represented as a set of uncoupled single-degree-of-freedomsystems.

The definition of a proportionally damped system[[M−1][C]

]s[[M−1][K ]]r =

[[M−1][K ]

]r[[M−1][C]]s

(2.24)

s, r ∈ ZA more simple definition which works in many cases is the definition in equation (2.25).

[C] = α[M ]+β[K ] (2.25)

Proportionally non damped systemWhen it comes to the non proportional damped system, which does not fulfil the above conditions, one gets amore complicated formulation to solve. But it is a common problem which has been solved by reformulatingthe problem using state space formulation. This is done by adding equation (2.27) and then reformulating thematrix definition as in equation (2.29-2.29).

[M ]x(t)+[C]x(t)+[K ]x(t)= 0 (2.26)

[M ]x(t)− [M ]x(t)= 0 (2.27)

[A] =[

0 MM C

](2.28)

[B] =[ −M 0

0 K

](2.29)

y= xx

y=

xx

(2.30)

Equation (2.31) gives the final expression and can be solved as the eigenvalue problem previously shown forthe non-damped system and proportionally damped system.

[A]y+[B]y= 0 (2.31)

The solution differs in some aspect from the previous solution, as a result from the state space formulation.The eigenvalues and eigenvectors will be twice as many as before. And natural frequencies are not the squareroot of the eigenvalues, but directly given from the eigenvalues. This could be noticed from the order of thedifferential equation (2.31).

6


2.2 Nonlinear Systems

2.2.1 Brief Introduction

In many areas of science, nonlinear systems are often linearized. Linearization is applied for making calcu-lations more manageable and less tedious. But in some cases, when neglecting the nonlinearities, errors toolarge to be acceptable are introduced. An example of errors due to linearization are presented in figure (2.5).Therefore, nonlinear models are developed for representing the reality a bit more accurate, and thus minimizing

0 20 40 60 80 100 120 140 160 180 200−130

−120

−110

−100

−90

−80

−70

−60

−50

F − Frequency [Hz]

Frequency Response Functions

Dis

plac

emen

t [N

/m] (

dB r

el. 1

)

H1(F)

H2(F)

H3(F)

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1


Coherence Functions

γ12(F)

γ22(F)

γ32(F)

a) b)

Figure 2.5: Example of system errors that could occur due to linearization. In plot a) the estimated frequencyresponse functions of three systems, the systems are:H1(F) - The underlying linear SDOF-system,H2(F) - Anonlinear system having a cubic term with the constant1 ·109 N

m3 , H3(F) - A nonlinear system having a cubic

term with the constant1 · 1010 Nm3 . In plot b) the coherence functions of the systems. The underlying linear

SDOF system is represented with the modal parametersM = 1 kg, C = 10 kg/s, K = 1 · 104 N/m using thesampling frequencyFs = 400Hz.

such modelling errors. Even though a lot of models are developed, this thesis will only deal with some fewselected models or methods. Special concern will be taken for the frequency spectrum based method ReversePath and a time series method, the Nonlinear Autoregressive Moving Average model with eXogenous inputs(NARMAX).

Following in this section are some important properties, concepts and ideas concerning nonlinear systemssuch as: nonlinear structures, stability considerations, frequency modulation, bifurcations and briefly aboutnonlinear resonance.

Nonlinear Structures

Nonlinear structures are often divided into three main types: zero memory, finite memory and infinite memorysystems. The zero memory types of systems are perhaps the most simple of the three types as it only appliesthe nonlinear operator at system input, whereas the infinite memory types of systems applies also at the sys-tem response, coupled back to system input. This thesis will emphasize on nonlinear systems having infinitememory, i.e. the nonlinearity of such a system is recursively coupled back to system input. A typical infinitememory type of system is a MDOF system as in equation (2.32), this system is often referred to as the DuffingOscillator System in literature.

[M ]x(t)+[C]x(t)+[K ]x(t)+[Kn]x3(t) = f (t) (2.32)

7

2.2. NONLINEAR SYSTEMS

Stability Considerations

Stable stationary solutions to dynamical systems are called equilibrium points. Stable but dynamic (time-variant) solutions are called trajectories or orbits. A nonlinear system can have more than one equilibriumsolution as opposed to linear systems which always have only one such equilibrium point. Later in this thesis themethod Harmonic Balancing will be described, which is a method for experimentally finding such equilibriumsolutions.

Frequency Modulation

The concept of frequency modulation is that in nonlinear systems, poles are no longer fix entities. The polesof nonlinear systems can be modified by for an example different excitation force amplitudes due to internalforce feedback of nonlinear responses, and thus changing the modal mass, stiffness or damping properties ofthe nonlinear system. One of the modifications a pole could get is change in its resonance frequency. Thisproperty is called frequency modulation in nonlinear systems. See section (3.4.2) on Hilbert transformationtechniques for a more thorough discussion on nonlinear frequency modulation.

Bifurcations

Linear systems only have a single equilibrium point, but nonlinear systems exhibits one or more equilibriumpoints. The number of equilibrium points can change, as well as the stability of the points as the system or ex-citation signal properties changes. These sorts of events are called bifurcations. There are a manyfold of typesof bifurcations, where some adds and some removes equilibrium points. A common feature is that they alterthe stability of the equilibrium points. Of special interest in the duffing oscillator configuration is the pitchforkbifurcation, where equilibrium points are created or destroyed symmetrically, with a change of stability of thesepoints. To display the nature of bifurcations, bifurcation diagrams are used.

Another special type of bifurcation is the Hopf Bifurcation. The Hopf bifurcation occurs when a stableequilibrium point is intersected by a stable trajectory, i.e. when a stable solution is intersected with a dynamicalstable solution. The Hopf Bifurcation makes the system to translate from a stable solution into a dynamicallystable solution. In literature, the bifurcation diagram of the Hopf Bifurcation is identical to the PitchforkBifurcation.

Nonlinear Resonance

The term nonlinear resonance means that a nonlinear structure will be excited by frequencies lower than theunderlying linear system’s resonance frequencies. This happen as the response is fed back to the system inputthrough a nonlinearity. Following is an intuitive example of the nonlinear resonance phenomenon, howevernot containing any proofs whatsoever. Assuming a simple SDOF duffing oscillator system, having a resonanceangular frequency equal toΩr of the underlying linear system, we let the excitation force be a sinusoidal as inequation (2.33). Initially when the feedback response is dorment, the response becomes of sinusoidal order asin equation (2.34) andh(t) is the underlying linear system impulse response function. At timet +µ immediatelyafter the nonlinear feedback has been initiated the excitation force could be approximated as in equation (2.35)and thus giving the response as in equation (2.36) having a response frequency three times higher than theexcitation force frequency. Now, let the excitation force frequency beΩ = 1

3Ωr . This means that the responsehas a periodic component atΩr and thus, the duffing oscillator system will be excited at resonance frequency.By this simple yet intuitive example the phenomenon of nonlinear resonance is displayed, but not proved byany means.

f (t) = Asin(Ωt) (2.33)

x(t) = h(t)∗ f (t) = Bsin(Ωt +θ) (2.34)

f (t +µ) ≈ f (t +µ)−knx3(t) (2.35)

x(t +µ) ≈ h(t +µ)∗ f (t +µ) = h(t +µ)∗ (f (t +µ)−knx3(t)

)=

= h(t +µ)∗(

f (t +µ)−knB3(−1

4sin(3Ωt +θ1)+

34

sin(Ωt +θ2)))

(2.36)

8


2.2.2 Theoretical Representation of Nonlinearities

Volterra Series

Purely linear systems can be fully represented by a single convolution integral, for calculation of the system’stime response. But in the case of systems having nonlinearities, the time response is a combination of overtonesthus rendering a higher order convolution integral. The higher order expansion of the convolution integral iscalled a Volterra series expansion, as in equation (2.37). The functionshk(τ1, . . . ,τk) are called Volterra serieskernels.

x(t) =∫ +∞

−∞h1(τ1) f (t− τ1)dτ1 +

∫ +∞

−∞

∫ +∞

−∞h2(τ1,τ2) f (t− τ1) f (t− τ2)dτ1dτ2 + . . . (2.37)

The description of a linear system could be seen as a special case of the Volterra series expansion, having onlyh1(τ). This in turn shows that a nonlinear system always has an underlying linear system. See [11] for furtherintroduction on theoretical backgrounds to nonlinear systems.

2.2.3 Presentation of Nonlinear Systems

In the case of linear dynamical systems, FRF:s are often enough to show the characteristics of such systems.But, when dealing with nonlinear systems one has to use different presentation techniques to fully show thecharacteristics of the nonlinear dynamical system.

Generalized Frequency Response Function

The generalised FRF (GFRF) is defined as the multidimensional Fourier transformation of the Volterra serieskernel of ordern as in equation (2.38).

Hn(ω1,ω2, . . . ,ωn) =∫ ∞

−∞. . .

∫ ∞

−∞hn(τ1,τ2, . . . ,τn)e− j(ω1τ1+...+ωnτn)dτa . . .dτn (2.38)

The GFRF for linear systems are all zero forn> 1 and equal to the linear system description, or FRF, forn= 1.

Backbone of Nonlinear Systems

The backbone of a nonlinear system is a plot of the resonance frequencies as functions of the excitation forceamplitude. Since nonlinear terms affects the structure itself, making the structure more or less stiff or dampedthis property changes in most cases as the excitation force amplitude is altered.

9

Chapter 3

System Identification and Analysis

3.1 Introduction

System identification and analysis is a vast topic in the area of signal processing; this thesis will only brieflytouch selected areas of interest and present some suitable methods and algorithms for nonlinear system iden-tification and analysis. The emphasis is on so called Black-box system identification methods, i.e. methodswith few and often non-strict system assumptions. Black-box methods operates directly on measured data andshould preferably be robust to measurement noise.

When identifying or analyzing a nonlinear system some basic questions need to be answered

Is Nonlinear analysis necessaryCould the system be approximated by a linear system description, or does alinearization render too large errors?

Where is a nonlinearity situated Between which of the nodes in an approximated system is the nonlinearfunctional mounted, ground node included?

Partly a linear system Estimate the underlying linear system.

The nonlinear functional What is the type of nonlinearity? Approximate the nonlinear functional.

3.2 Linear System Identification and Analysis

3.2.1 Non-Parametrical Spectrum Identification

Cross Spectral Density

The Cross (Power) Spectral Density (CSD) shows the power transfer between two signals as a function offrequency. Following are some popular intuitive estimators of the CSD.X

(e− jω)

andY(e− jω)

are the discretefourier transforms ofx(n) andy(n) respectively, wheren∈ [0,N−1] andω ∈ [−π,π]. FurthermoreFS is thesampling frequency in[Hz] andw(n) is the window function.

Periodogram The periodogram calculates a non-weighted CSD estimate for the signalsx(n) andy(n). Anestimate of the periodogram could be expressed as:

PYX(ω) =1

FSNY

(e− jω)

X∗(e− jω)

(3.1)

Modified Periodogram The Modified Periodogram is basically the Periodogram where a window is appliedto the time-signals in advance. Thus minimizing the leakage that would appear due to abrupt endings in

10

CHAPTER 3. SYSTEM IDENTIFICATION AND ANALYSIS

the data vectors. One could see the Periodogram as a special case of the Modified Periodogram, wherethe rectangular window has been applied.

PYX(ω) =1

FSNU

(Y

(e− jω)∗W

(e− jω))(

X∗(e− jω)∗W

(e− jω))

(3.2)

U =1N

N−1

∑n=0

|w(n)|2 (3.3)

Bartlett Barlett’s method for estimating the CSD is an extension of the Periodogram. The signals are dividedinto blocks, and for each block the periodogram is estimated. All sub-periodograms are averaged to formthe final averaged CSD estimate.

PYX(ω) =1

FSKN

K−1

∑k=0

Yk(e− jω)

X∗k(e− jω)

(3.4)

Welch Welch’s method uses averaging like Bartlett’s method except that it also uses an overlap technique. Byoverlapping, the number of effective blocks are increased a lot and thereby lowering the variance of theestimate even more. Depending on which window is used different values of the overlap exists for givingoptimal performance.

PYX(ω) =1

FSKNU

K−1

∑k=0

(Yk

(e− jω)∗W

(e− jω))(

X∗k(e− jω ∗W

(e− jω)))

(3.5)

U =1N

N−1

∑n=0

|w(n)|2 (3.6)

Auto Power Spectral Density

(Auto) Power Spectral Density (PSD) could be seen as a special case of the CSD, as the PSD is calculated foronly one signal. For an example, the Periodogram for thex(n) signal would become as in equation (3.7).

PXX(ω) =1

FSN

∣∣X (e− jω)∣∣2 (3.7)

The unit ofPXX(ω) depends on the signal you are measuring, ifx(n) is a force signal with the unit[N] then thePSD of the signal will have the unit[N2/Hz].

Integration for all frequencies of a PSDω ∈ [−π,π] gives the total signal power. This means that the PSDis very useful for estimating the power for signals having wideband components.

3.2.2 SISO Systems

There are different ways to calculate systems when the input and output of the system is known. But in realmeasurements these signals are always more or less contaminated with noise and can thereby not be calculatedbut instead estimated. There are different methods that models different situations with respect to the noise.The first estimatorH1 equation (3.8) assumes only noise at the output, the second estimatorH2 equation (3.9)assumes only noise at the input. The third estimatorH3 equation (3.10) is the mean of the two previously esti-mators thou theH1 estimator underestimates the trueH( f ), while theH2 estimator overestimates the trueH( f ).

H1( f ) =Pyx( f )Pxx( f )

(3.8)

H2( f ) =Pyy( f )Pxy( f )

(3.9)

H3( f ) =H1( f )+ H2( f )

2(3.10)

11

3.2. LINEAR SYSTEM IDENTIFICATION AND ANALYSIS

WherePxx( f ) andPyy( f ) are the autospectrums of the input and output signals respectively.Pxy( f ) andPyx( f )are the crosspectrum of the input and output signals respectively. These spectrums could be calculated with oneof the following methods;Periodogram, Modified Periodogram, Bartlett or theWelchmethod. Se the previouschapter for the definition and description of the different methods.

The last estimatorHc is developed for shaker excitations where the force and e.q. the acceleration ismeasured. It is reasonable to assume noise at both the input signal (force) and the output signal (acceleration).The estimator is based on the possibility to measure the signalv(t) to shaker and the assumption that this signalis free from contamination. If that’s the case, then the systemH f v( f ) could be estimated using theH1 estimator.Also the total systemHav( f ) with the inputv(t) and the contaminated outputa(t) could be estimated with theH1 estimator.

- H f v( f ) - Ha f ( f ) - ?

na(t)

-

?- -inf (t) f (t)

a(t)v(t) f ′(t) i

Figure 3.1: A model for theHc estimator. The first block represents the shaker and the second block representsthe system which is examined.

By using equation (3.11) the system of interestHa f is derived from the estimated systemsHav andH f v.

Ha f ( f ) =Hav( f )H f v( f )

(3.11)

3.2.3 Time Series Models

3.2.4 Special Types of Random Processes

A brief description of different model assumptions, where the input signal to the system havingp poles andqzeros is white gaussian noise for identification purpose.

ARMA - Auto Regressive Moving Average

The ARMA model is as the general model with both poles and zeros. The coefficienta0 could be set one in theequation that follows.

Time domain

a0y(n) = b0x(n)+b1x(n−1)+ . . .+bqx(n−q)−a1x(n−1)− . . .−apx(n− p) =

=q

∑k=0

bkx(n−k)−p

∑l=1

al y(n− l) (3.12)

Z domain

H(z) =B(z)A(z)

=

q

∑k=0

bkz−1

a0 +p

∑l=1

al z−k

(3.13)

12


AR - Auto Regressive

The AR model is special case of the ARMA model where the order of the b-polynomial equals zeros.

Time domain

a0y(n) = b0x(n)−a1y(n−1)− . . .−apy(n− p) =

= x(n)−p

∑k=1

aky(n−k) (3.14)

Z domain

H(z) =b0

A(z)=

b0p

∑k=0

akz−k

(3.15)

wherep is the order of the AR-model.

MA - Moving Average

The MA model is also a special case of the ARMA model but where the order of the a-polynomial equals zerosinstead. Under this sectiona0 is not written into the equations.

Time domain

y(n) = b0x(n)+b1x(n−1)+ . . .+bqx(n−q) =

=q

∑k=0

bkx(n−k) (3.16)

Z domain

H(z) = B(z) =q

∑k=0

bkz−1 (3.17)

whereq is the order of the MA-model.

ARMAX - Auto Regressive Moving Average with Exogenous input

The ARMAX is an extended version of the ARMA model as previously mentioned, but with an extra, measur-able signalx(t). n(t) is white gaussian noise, which has been filtered through an ARMA model. The movingaverage part of the model uses unknown sampled noise sequence to reduce the biased error and the exogenousinput is the past samples of the input time record. A general presentation of the ARMAX model is given infigure 3.2.

- B( f )A( f )

- C( f )A( f )

6

- -y(t)x(t)

n(t)

i

Figure 3.2: General description of the Auto Regressive Moving Average model with exogenous input.x(t) isseen as the exogenous input,n(t) is the modelled noise.

13


a0y(n) = b0x(n)+b1x(n−1)+ . . .+bqx(n−q)+−a1y(n−1)−a2y(n−2)− . . .−apy(n− p)++c0n(n−1)+c1n(n−1)+ . . .+crn(n− r) =

=q

∑k=0

bkx(n−k)−p

∑l=1

al y(n− l)+r

∑m=0

cl n(n−m) (3.18)

wherer is the grade of the c-polynomial.

ARX - Auto Regressive with Exogenous input

The ARX model is a special case of the ARMAX where the moving average is not modelled. The truncatedmodel could result in a biased error.

- B( f )A( f )

- 1A( f )

6

- -y(t)x(t)

n(t)

i

Figure 3.3: General description of the Auto Regressive model with exogenous input.x(t) is seen as the exoge-nous input,n(t) is the modelled noise.

a0y(n) = b0x(n)+b1x(n−1)+ . . .+bqx(n−q)+−a1y(n−1)− . . .−apy(n− p)+n(n) =

=q

∑k=0

bkx(n−k)−p

∑l=1

al y(n− l)+n(n) (3.19)

3.2.5 Modal Analysis

Modal analysis is defined as the process of characterizing the dynamics of a structure in terms of its modesof vibration [10]. As described in chapter 2.1.1 the eigenvalues and eigenvectors of the mathematical modelare also the parameters which defines the resonant frequency and the mode shape of the modes of vibration.Modal analysis could be performed by the analytical finite element modelling as previously shown. Modalanalysis could also be performed by modal testing, also referred to as experimental modal analysis. Theexperimental modal analysis is performed to either confirm an analytical finite element model or to characterizean unknown structure e.g. for troubleshooting. The process of characterizing a structure or system is calledmodal parameter estimating, also referred to as curve fitting. There are many methods for modal analysis withdifferent advantages and disadvantages, see table (3.2.5) for the most common algorithms [12].

The two most popular curve fitting methods [10], either curve fit to measured Frequency Response Functiondata using a parametric model of the FRF, this model can be written as in equation (3.20),

H( jω) =N

∑r=1

Rr

jω−λr+

R∗rjω−λ∗r

(3.20)

or they curve fit to measured Impulse Response Function data using a parametrical model of the IRF, whichcan be written as in equation (3.21).

h(t) =N

∑r=1

Rreλr t +R∗r eλ∗r t (3.21)

Most algorithms are variants of these methods, but also other methods exists, i.e. curve fitting based on state-space models.

14


Algorithm Full name Domain Polynomial Order CoefficientsCEA Complex Exponential Algorithm Time High ScalarLSCE Least Squares Complex Exponential Time High ScalarPTD Polyreference Time Domain Time High MatrixITD Ibrahim Time Domain Time Low Matrix

MRITD Multiple Reference ITD Time Low MatrixERA Eigensystem Realization Algorithm Time Low MatrixPFD Polyreference Frequency Domain Frequency Low MatrixSFD Simultaneous Frequency Domain Frequency Low Matrix

MRFD Multi-Reference Frequency Domain Frequency Low MatrixRFP Rational Fraction Polynomial Frequency High BothOP Orthogonal Polynomial Frequency High Both

CMIF Complex Mode Indicator Function Frequency Zero Matrix

Table 3.1: A collection of the most commonly used algorithms.

Proposed Procedure for Curve Fitting

The first step is to estimate the poles from the measured data, the pole describes the resonance frequencies anddamping values e.g. equation (3.22).

λr =−ζrωr ± jωr

√1− ς2

r (3.22)

The estimation of the poles can be done by the Least Square Complex Exponential Algorithm which isan extension of the Complex Exponential Algorithm. The CEA, also referred to as the Prony algorithm, wasdiscovered in 1795 [10] by R. Prony and is a Pole-Zero model that minimizes the errore′(n) = x(n)−h(n),whereh(n) is the unit sample response of the system. Prony’s method is a very simple method which onlyrequires solving a set of linear equations, the extension LSCEA is known as Shank’s method which differsin the approach of finding the zeros of the system function. The approach is to formulate a moving averageparameter estimation problem as a pair of auto regressive parameter estimations using Durbin’s method [7].While Prony’s method bases the moving average coefficients on a model error equal to zero forn= 0,1,2. . . ,q,see equation (3.23) whereq is the grade of the numerator andp is the grade of the denominator. In other words,Shank suggest a least square minimization of the model errore′(n) = x(n)− x(n) according to figure 3.5.

e(n) = ap(n)∗x(n)−bq(n) (3.23)

-+ +I

)( nb q

)( nb q

)( ne)( zA p

)( nx

Figure 3.4: Signal model for Prony’s method.x(n) = h(n) is the unit sample response of the system,bq(n) isthe systems numerator coefficients andAp(z) the denominator.e(n) is the error which is to be minimized.

A least square minimization of Prony’s error is given in equation (3.28), the roots of the estimated denomi-

15


)( nd )( ng)( zB q

)( nx

)( nx

)(' ne- +)(

1zA p

+I

Figure 3.5: Signal model for Shanks method.x(n) = h(n) is the unit sample response of the system,δ(n) is theimpulse function,Bq(z) is the systems numerator andAp(z) the denominator.e′(n) is the error which is to beminimized.

nator are the poles.

Xq =

x(q) x(q−1) · · · x(q− p+1)x(q+1) x(q) · · · x(q− p+2)x(q+2) x(q+1) · · · x(q− p+3)

......

...

(3.24)

xq+1 = [x(q+1),x(q+2),x(q+3), · · · ]T (3.25)

rx = XHq xq+1 (3.26)

Rxx = XHq Xq (3.27)

ap = −R−1xx rx (3.28)

During this procedure, the number of poles has to be determined in order to get the best estimation possible.The number of poles chosen is often more than the exact number of poles in the examined frequency span.Those extra poles are called computational poles which is a result from measurement errors of the FRF. As atool for choosing the number of poles a stability diagram is used, such a diagram is shown in figure 3.6. Thenext step is to calculate the residues. This is often done in frequency domain using a least square method, but itcould also be calculated in time domain from the estimated nominator. The estimation of the nominator usingShank’s method is presented in equation (3.31).

rxg(k) =∞

∑l=0

x(n)g∗(n−k) (3.29)

Rgg(k,l) =∞

∑n=0

g(n− l)g∗(n−k) (3.30)

bq = −R−1gg rxg (3.31)

16


0 5 10 15 20 25 30 35 40−150

−140

−130

−120

−110

−100

−90

−80

−70

4*

5*

6*

7*

8*

9*

10*

11*

12*

13*

14*

15*

16*

Log

Mag

nitu

d

Frequency [Hz]

Stability diagram

Figure 3.6: Stability diagram of a MDOF-system where the number of poles has been estimated from 4 to16 and plotted as red crosses in increasing order where the grade of each estimation is presented on the lefttogether with an asterisk.

17

3.3. NONLINEAR SYSTEM IDENTIFICATION

3.3 Nonlinear System Identification

3.3.1 NARMAX Modelling

NARMAX is an abbreviation for Nonlinear Auto Regressive Moving Average Model with eXogenous inputs.Exogenous means that the model is excited not only by noise as in the ARMA, AR and MA cases but also bya measurable input signal. The nonlinear elements of the NARMAX model consists of polynomial terms ofoutputs, inputs and noise, cross terms as well. The general formulation for a NARMAX model is as in equation(3.32).

a0y(n)+ . . .+aNyy(n−Ny) = f (y(n−1), . . . ,y(n−Ny);x(n), . . . ,x(n−Nx);e(n), . . . ,e(n−Ne))++b0x(n)+ . . .+bNxx(n−Nx)+c0e(n)+ . . .+cNee(n−Ne) (3.32)

f. . . is a nonlinear function of the input, output and error signals respectively.

On Structure Selection

The NARMAX model, as well as other time-based models have one very important drawback, that is that thestructure of the model must be either known in advance or estimated for a given set of data. The goal for anydiscrete time model is to use as small model as possible yet having the capability to describe the system withminimal modelling errors. In this thesis two methods were used to truncate models for getting optimally smallmodels with minimized modelling errors. The first method was to guess the orders of the model by investi-gating the frequency response functions for an example, the second method was to use the Bootstrap structuredetection algorithm (see appendix B). The most reliable of the two methods were the first method, to guessmodel orders. The reason of that seems to lie in that the choice of correct initial parameters is crucial for theBootstrap method to give good performance.

Summarize of the NARMAX model

The NARMAX structure itself has a limitation in describing many physical quantities and scenarios. This ori-gins from the fact that many systems has nonlinearities that operates directly on system output, and then feedback to system input. The NARMAX model has no such correspondence in its structure. This is a problem inthe context of this thesis, which aims to get physically interpretable parameters, but does not have to constitutea problem in other applications.

To empicture this problem, a single degree of freedom system were used, as in equation 3.33, the system isoften denoted as the duffing oscillator system in literature.

mx(t)+cx(t)+kx(t)+knx3(t) = f (t) (3.33)

The Modified Bootstrap Structure Detection Algorithm proposed a NARMAX structure which did not onlyconsist of the cubic term, but also of lagged input and output terms. By definition, a NARMAX differentialequation cannot contain the expressiony(n) more than once, and thus the infinite memory system cannot bemodelled exactly using the NARMAX model without modelling errors.

One should not neglect the NARMAX model out of the results of this thesis, there are certainly manyother applications where the NARMAX model is more suitable. If the NARMAX model is to be used in anapplication. It is also important to understand that for the NARMAX method to perform well, a suitable choiceof parameters is needed. The classical question always exists; Do we fit our model to the current available data,or do we fit our model to the real, physical model? The later is of course the desired case.

3.3.2 Reverse Path Method

The nonlinear systems discussed in this thesis has mostly been concentrated on systems with feedback. And aspreviously mentioned this causes a system with infinite memory as shown in figure 3.7. The method proposed

18


- - H( f )6

G[y(t)] ¾

-y(t)x(t) i

Figure 3.7: General description of a system with nonlinear feedback, i.e. having a nonlinearity with infinitememory.

by Bendat is based on system with zero or finite memory. To be able to use Bendats method so called DirectMISO technique (Multiple Input Single Output) on systems with feedback, a reverse operation of the signalhas to be applied as in figure 3.8. This operation makes it possible to regard the system as a zero memory orfinite memory system depending on the feedback.

- H−1( f )

- G[y(t)]

6

- -x(t)y(t) i

Figure 3.8: By reversing a system with infinite memory nonlinearity, a system with zero memory or finitememory nonlinearity is given.

Direct MISO

Systems described as in figure 3.9, two parallel SISO systems could be solved with theH1 estimator. Note thatthe estimated systemH( f ) is the true estimate ofH( f ) if the outputs fromH( f ) andA( f ) are uncorrelated.When the systemA( f ) equals a constant the nonlinear system is said to be a zero memory system, otherwise itis a finite memory system.

- H( f )?

n(t)

- G[x(t)] A( f )-

6

- -y(t)x(t)

x2(t)

i

Figure 3.9: General description of a single-input-single-output system with arbitrary finite or zero memorysystem given byA( f ).

To be able to use the estimator for both systemH( f ) andA( f ), the system model is simplified into the systemmodel in figure 3.10. To do this,x2(t) has to be calculated which means thatG[x(t)] has to be known. To beable to draw any conclusions when the systemsA1( f ) andA2( f ) are estimated, some measure of how goodthese results are is needed. This measure is the ordinary coherence function from both systems compared to themultiple coherence function. Which show how much linear dependency each systems has to the output signal.

19

3.3. NONLINEAR SYSTEM IDENTIFICATION

- A1( f )

- A2( f )

6

- -y(t)x1(t)

x2(t)

?

n(t)

i

Figure 3.10: Multiple-input-single-output system with correlated inputsx1(t) andx2(t), the outputs ofA1( f )andA2( f ) are uncorrelated.

Conclusion

The reverse path method together with the direct MISO is a good method to estimate both the linear system andthe nonlinear system when the nonlinearity is known. Or perhaps assumed to be known. When using the directMISO from the reverse path method the linear system is of course the inverse of the estimated system. Usingdirect MISO method for identifying the nonlinearity could only give a measure of how good different guesseswas. By testing different common nonlinearities there is a chance to draw good conclusions about the systemmodel. It is perfect method to identify where nonlinearities are located by testing all available permutations ofthe inputs and outputs and compare the results to each other.

3.3.3 Frequency Domain Structure Selection Algorithm

The Frequency Domain Structure Selection Algorithm (FDSSA) is based on the modified SIMO infinite mem-ory nonlinear system as in equation (3.34).

xi(t) =∫ +∞

τ=−∞hi, j(τ− t)

f j(t)−g(x1(t), . . . ,xI (t))

dτ (3.34)

Wherei is a response index,i ∈ 1, . . . , I . j is the index of the excitation force node. Finally,g(·) is the nonlinearfunction of one response or a combination of responses. Note thatg(·) is a fix quantity for a specific system,independent of the indicesi and j.Let for an example the nonlinear functiong(·) be a polynomial of orderP+1 as in equation (3.35).

g(x1(t), . . . ,xI (t)) =P

∑p=1

λpgp(t) (3.35)

gp(t) = (cp,1x1(t)+ . . .+cp,I xI (t))p+1 (3.36)

Wherecp,i are fixed on-off weighting terms for each response signali and polynomial orderp, cp,i ∈ −1,0,1.λp are unknown weighting terms for the nonlinear polynomial combinations, and are of particular interest andto be found in the FDSSA method.

The fourier transformation is calculated for the terms in the equation and a corresponding frequency domainexpression is evaluated for frequenciesω ∈ [−π,π], as in equation (3.37).

Xi(ω) = Hi, j(ω)

(Fj(ω)−

P

∑p=1

λpGp(ω)

)(3.37)

Gp(ω) = F gp(t)ω (3.38)

By multiplication ofX∗i (ω) to each side and expressing for the expected value equation (3.39) is found.

EXi(ω)X∗i (ω) = Hi, j(ω)

(E

Fj(ω)X∗i (ω)

−P

∑p=1

λpEGp(ω)X∗i (ω))

(3.39)

The left side expectation term in equation (3.40) is the definition of the Power Spectral Density (PSD) of thesignalxi(n), the right side expectation terms are the definitions of the cross spectral densities (CSD) between

20


the signalsf j(n), xi(n) andgp(n) respectively. Robust estimates of the PSD and CSD functions are applied andthe final expression is evaluated as in equation (3.40).

PXiXi (ωk) = Hi, j(ωk)

(PFj Xi (ωk)−

P

∑p=1

λpPGpXi (ωk)

)(3.40)

The final expression in equation (3.40) could now easily be arranged with respect to the polynomial termsλp

for suiting an optimizing algorithm.

3.3.4 Finding the Nonlinear Nodes

This section will present a brute force methodology for finding the node or the nodes where the nonlinearfunctional is mounted.

The Brute Force Methodology

Assume we have a system ofN nodes. The brutal force methodology uses data from each of the system responsenodes to find the node that is housing the nonlinearity. The following list shows the brutal force methodology.

1. Make an assumption of where the nonlinearity is mounted. Assume that the nonlinearity operates on thedifferencex1(t)−x2(t), such asg(x1(t),x2(t)).

2. Estimate the nonlinearity of the system when it is assumed thatg(x1(t),x2(t)) is a valid assumption.Calculate a goodness-of-fit value, for an example the ordinary coherence function.

3. Repeat 1. and 2. for all possible nodes of combinations ofxi(t).

4. For all combinations, find the combination that has the best goodness-of-fit value. This combinationcould then be seen as the node where the nonlinearity is mounted.

The withdraw with the Brutal Force methodology is that it does not render well when the true combinationof input signals is not included. Also, questions like; does another set of input combinations render bettergoodness-of-fit values than the true combination; thus giving rise to need of other, more robust methods forfinding the nonlinear nodes.

21

3.4. NONLINEAR SYSTEM ANALYSIS

0 20 40 60 80 100 120 140 160−110

−100

−90

−80

−70

−60

−50


Harmonic Balance Simulation

Underlying Linear SystemFirst set of HBM SolutionsSecond set of HBM SolutionsThird set of HBM Solutions

Figure 3.11: A harmonic balance plot for an example SDOF duffing oscillator system having the modal param-etersm= 1kg, c = 10kg/s, k = 1 ·104N/m and the cubic constant1 ·1010N/m3. Note the harmonic balancemethod’s ability to find the unstable solutions (green dots). The sampling frequency was set to 400 Hz.

3.4 Nonlinear System Analysis

The analysis methods suggested in this thesis does not always reveal the true type of nonlinearity, but themethods will give the reader a sense of how the nonlinearities operates in the linear system environment.

3.4.1 Harmonic Balance Method

Introduction

The Harmonic Balance Method, further denoted as HBM, is a method that is used for experimentally findingthe steady state equilibrium solutions to a dynamical system. The solutions are calculated per frequency andpresented as a FRF. See figure 3.11 for an example of how the harmonic balance solutions could be presented.

As mentioned earlier, a nonlinear system could have one or more equilibrium points per frequency, thiscould easily be seen with the HBM. Notable is that chaotic behavior, such as bifurcations are not considered inthe HBM, but they could render errors in the FRFs.

The method originates as many other from the definition of linear systems, but could easily be expandedfor nonlinear systems as could be seen in upcoming sections. The implementation in this thesis works for anyinfinite memory nonlinearity, as long as it could be formulated as equation (3.41).

Derivation of the Method

To derive the Harmonic Balance Method, one should expand the newtonian dynamical system description as in3.41. In the linear casegx(t) should be set to0.

mx(t)+cx(t)+kx(t)+gx(t) = f (t) (3.41)

22


Given that the responses and excitation forces all have bounded energy, the fourier transformation could beapplied for the expression in (3.41) giving equation (3.42).

m[( jΩ)2X( jΩ)− jΩx(0)− x(0)

]+c[( jΩ)X( jΩ)−x(0)]+kX( jΩ)+G( jΩ) = F( jΩ) (3.42)

WhereG( jΩ) =∫ +∞−∞ gx(t)ejΩtdt, i.e. the fourier transformation of the nonlinear functionalgx(t) and

F( jΩ) is the fourier transformation of the excitation forcef (t). x(0) and x(0) are initial conditions of theresponse signal. Simplifying and rearranging the expression gives equation (3.43).

[−mΩ2 + jΩc+k]X( jΩ) = F( jΩ)−G( jΩ)+( jΩm+c)x(0)+mx(0) (3.43)

Let 1H( jΩ) =−mΩ2 + jΩc+k and simplifying gives the final expression in equation (3.44).

X( jΩ) = H( jΩ)F( jΩ)−H( jΩ)G( jΩ)+H( jΩ) [( jΩm+c)x(0)+mx(0)] (3.44)

As mentioned in the introduction; the HBM calculates the steady state equilibrium solutions at a given fre-quency for a dynamical system. But for nonlinear systems, due to for an example nonlinear resonance orfrequency modulation, there could be several over- or under tones to a singular frequency response. Thereforedifferent versions of the HBM exists; versions which only investigates the ground tone and versions whichinvestigates the ground- as well as the over-/under tones. The implementation described in this thesis onlyinvestigates the ground tone of the responses for simplicity, but could be extended for over-/under tone analysisas well. Therefore a simplification has to be made; the time response of the excitation force and correspond-ing responses for a given frequency is assumed to be as in equations (3.45) and (3.46) respectively, havingfourier transformations as in (3.47) and (3.48). Note that for multi tone analysis, equations (3.46) and (3.48)are expanded to include energy contributions for the over- and/or under tones as well.

f (t) = Fampsin(Ωkt) (3.45)

xl (t) = Xamp,l sin(Ωkt) (3.46)

F( jΩ) =πFamp

jδ(Ω−Ωk)−δ(Ω+Ωk) (3.47)

Xl ( jΩ) =πXamp,l

jδ(Ω−Ωk)−δ(Ω+Ωk) (3.48)

Hl ( jΩ) =Xamp,l

Famp, l ∈ [1,L] (3.49)

WhereFamp∈C andXl ∈C. Note thatXl is one of (l ∈ [1,L]) L roots such that equation (3.44) is in equilibrium,i.e in Harmonic Balance. The Harmonic Balance basis function should preferably be of sinusoidal order suchas sin(Ωkt). The solutions to the FRF of the dynamical system for which the HBM was applied could beexpressed, per frequency, as equation (3.49).L is of order1 in the linear case and of orderL ≥ 1 in thenonlinear case.

3.4.2 Hilbert Transformation Techniques

The Hilbert Transform and its properties is commonly used among different application and has been pro-posed also to be used in the non-linear vibration analysis [4],[5]. Common areas are envelope calculations andmodulation analyzing. The Hilbert Transform properties of analyzing modulation is one of the features usedwhen diagnose and characterize frequency modulation in nonlinear vibrating systems. The goal of the Hilberttransform time domain technique is to detect and classify the change in the natural frequency. Classifying withrespect to the type oscillation from the nonlinear system which could be noticed as a modulation. The Hilberttransform of a signal is defined according to equation (3.50) whereH is the Hilbert transform and∗ denotesconvolution.

x(t) = H [x(t)] =∫ ∞

−∞

x(u)π(t−u)

du= x(t)∗ 1πt

(3.50)

23


As could be seen in the definition of the Hilbert transform, the transformation is in the domain, that is the timedomain. In other words the transform is a phased shifted version of the signal which is easy to see when lookingat the Fourier transformF of 1

πt in equation (3.50).

F[

1πt

]=− j sgnf =

− j , f > 0

0 , f = 0

j , f < 0

(3.51)

The Hilbert transform weights the signal aroundt according to equation (3.52) which makes the transformsuitable for analyzing local behavior in opposite to the Fourier transform which base is an exponential functione− jωt and has to be used on the whole time record.

x(t) = x(t)∗ 1πt

(3.52)

The main properties in the algorithm is the instantaneous characteristics of amplitudeA(t), phaseθ(t) and fre-quencyf (t). Those are all derived from the functionz(t) which is defined by equation (3.53).

z(t) = x(t)+ j x(t) (3.53)

x(t)

x(t) z(t)

θ(t) Real

Imag

Figure 3.12: The analytic functionz(t) used in the Hilbert transform time domain algorithm.

Instantaneous Phase:

θ(t) = anglex(t), x(t) (3.54)

Instantaneous Frequency:

Ω(t) =dθ(t)

dt(3.55)

Instantaneous Amplitude:

A(t) =√

x2(t)+ x2(t) (3.56)

These tools which instantaneous analyzes the signal, could now be used in the Hilbert transform time domainalgorithm. To see the differences between instantaneous and non-instantaneous analyzing tools the instanta-neous frequency will be compared with the Fourier transform. The signal should be of a non-stationary typethat shows the advantage of the Hilbert transform and the weakness of the Fourier transform. Such a signal is

24


the sinus sweep which is shown in figure 3.13 and defined by equation (3.59).

x(t) = sin(2π f (t)t) (3.57)

f (t) = 1.25t (3.58)

(3.59)

0 1 2 3 4 5 6 7 8 9 10−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time t [s]

x(t)

[m]

Sinus sweep 0−25 Hz

Figure 3.13: A sinus sweepx(n) defined in equation (3.59) , starting at 0 Hz, sweeping up to 25 Hz.

As previously mentioned the goal is to find oscillation caused by the nonlinear structure near the resonancefrequency. It is thereby important to only examine the frequencies in a narrow band where the resonancefrequency of interest is. The following listing are the steps of the algorithm:

• FilteringExtracting a narrow frequency band where no other modes are interfering with the mode of interest.

• Hilbert transformCalculate thez(t).

• Instantaneous QuantitiesCalculate the different instantaneous signals.

• AnalyzingCompare the results with known discrimination diagrams for classifying the nonlinearity.

25


0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

time t [s]

Ω(t

)/(2

π) [H

z]

Instantaneous Frequency

Simulated Ω(t)/(2π)Ideal Ω(t)/(2π)

Figure 3.14: The blue curve is instantaneous frequencyΩ(t) of the sinus-sweep signalx(t) defined in equation(3.59). The red curve is the frequency vector used when manufacturing the signalx(t).

−40 −30 −20 −10 0 10 20 30 405

10

15

20

25

30

35

40

45

50

Frequency f [Hz]

Mag

nitu

d

Fourier Transform of x(t)

Figure 3.15: The Fourier transform of the sinus-sweep signalx(t) defined in equation (3.59), based on the wholetime record. It is apparent that the Fourier transform is not an appropriate analyzing tool for non-stationarysignals.

26


3.4.3 Statistical Analysis

Introduction

Statistical analysis is mainly used for indicating if systems are having nonlinearities. The statistical analysisorigins from Random theory [8] and [1]. Random theory stems that one of many important properties of linearsystems is the linearity property. The linearity property states that a signal into a linear system retains its statis-tical properties, though with an eventual change in amplitude, phase or mean. But, the symmetrical propertiesof a signal must be kept unchanged. Thus by estimating and comparing certain parameters that measure thesymmetry of the probability density functions, before and after system application, one could get a hint whethera system is linear or not.

It is important to see that an infinite memory nonlinear system often has less noticeable nonlinear effectscompared to a zero memory nonlinear system for example. This comes from the fact that the infinite memorysystem’s nonlinearity is filtered through the system itself. Where the system eventually are damping the feed-back nonlinearity signal, the zero memory nonlinear system has no such filtering of the nonlinear function andthe nonlinear signal could therefore be transferred directly to the system’s response.

It is recommended that the histograms of excitation signals and responses are investigated at an early stagewhen starting a new measurement session of a system, to get an overview of the system’s level of linearity.Many other abnormalities concerning signal quality could be discovered by the histogram investigation as wellsuch as signal clipping, DC-offset problems, undesired signal glitches due to improper inducer mounting foran example, exterior sinusoidal disturbances et cetera.

Central Moments

The central momentMi of a signal is defined as equation (3.60).

Mi = E[(x− x)i

](3.60)

The second central moment is also referred to as the sample variance, asM2 = σ2x.

Skewness

The skewness could be seen as a measure of in which extent a signal is symmetric around its mean, wheresymmetric refers to the signal’s probability density function. If a signal is symmetric the skewness should beas close as possible to zero. The gaussian distribution is an example of signals having skewness equal to zero.The skewness is defined as the normalized third order central moment, as in equation (3.61).

Sx =M3

σ3x

(3.61)

The skewness is a dimensionless parameter.

Kurtosis

Kurtosis on the other hand could be seen as a comparative measure of the probability density function’s tailswith respect to the gaussian distribution’s tails. A fully gaussian distribution has a kurtosis value equal to three.The kurtosis is defined as the normalized fourth order central moment, as equation (3.62).

Kx =M4

σ4x

(3.62)

The kurtosis is a dimensionless parameter.

27


Example Statistical Analysis

Assume a infinite memory nonlinear system defined by the modal massm= 1kg, the dampingc = 10kg/s, thestiffnessk = 1 · 104N/m and the nonlinear polynomial1 · 109x3(t) + 1 · 106x2(t), let this be the first system.Now, assume a zero memory nonlinear system, having the same modal parameters and nonlinear polynomialas the first system, let this be the second system. Let the excitation force be gaussian distributed noise havingzero mean and the standard deviation10N. Let the sampling frequency be400Hz.

The histogram of the excitation force, the first system’s nonlinear response, the second system’s nonlin-ear response and their linear responses are presented in figure 3.16. The statistical parameters skewness andkurtosis for the signals are presented in table 3.2.

−15 −10 −5 0 5 10 150

1000

2000

3000

4000

5000

6000

Num

ber

of s

ampl

es

Force [N]

Histogram of excitation force

−1.5 −1 −0.5 0 0.5 1 1.5

x 10−3

0

1000

2000

3000

4000

5000

6000

Num

ber

of s

ampl

es

Response [m]

Histogram of linear sub−system response

a) b)

−1.5 −1 −0.5 0 0.5 1 1.5

x 10−3

0

1000

2000

3000

4000

5000

6000

Num

ber

of s

ampl

es

Response [m]

Histogram of first nonlinear system response

−1.5 −1 −0.5 0 0.5 1 1.5

x 10−3

0

1000

2000

3000

4000

5000

6000

Num

ber

of s

ampl

es

Response [m]

Histogram of second nonlinear system response

c) d)

Figure 3.16: Histograms for a sample system given by parametersm= 1kg, c = 10kg/s, k = 1 ·104N/m andthe nonlinear polynomial1 ·109x3(t)+ 1 ·106x2(t). Plot a) is the histogram of the excitation force, b) is thehistogram of the linear subsystem response, c) is the histogram of the first system’s nonlinear response, and d)is the histogram of the second system’s nonlinear response.

28


Statistical Parameter Excitation Force Nonlinear System 1 Nonlinear System 2 Linear SubsystemMean 0.0285N −4.7·10−6m 1.7·10−6m 2.9·10−6mSkewness −0.014 −0.038 −0.039 −0.016Kurtosis 3.03 2.67 3.83 3.04

Table 3.2: Statistical analysis of an excitation force, the response from a nonlinear system, and the linearresponse of the system’s linear subsystem. The first nonlinear system and its linear subsystem is given by theparametersm= 1kg, c = 10kg/s, k = 1 ·104N/m and the nonlinear polynomial1 ·109x3(t)+1 ·106x2(t). Thesecond nonlinear system has the same linear subsystem as the first, but instead having the nonlinear polynomial4 · 10−2F3(t) + 4 · 10−6F2(t). Note the difference between the two nonlinear response’s parameters and thelinear response’s parameters.

29

Chapter 4

System Synthesis

4.1 Introduction

One of the goals during this master thesis was to develop powerful tools for analysis and identification of me-chanical systems, as well as getting tools for dynamical system synthesis. Synthesis means reproducing orpredicting the system response given a specific system excitation force, of course under the assumption thatthe system is correctly identified and analyzed. Synthesis is thereby very useful for simulation of algorithmsfor system control. It saves a lot of money and time, but the synthesis itself is only useful if it introducesmanageable and limited errors to the predicted output. Having too large errors in the synthesis gives data that isuseless to the end user. Therefore the synthesis tools presented in the following section aims to minimize sucherrors. Each of the synthesis tools are explained such that the area of usage is clearly stated.

This chapter includes the following sections:

Linear System SynthesisSome popular methods for linear system synthesis, such as; analytical time responsesynthesis using laplace transformations, synthesis by digital filters.

Nonlinear System SynthesisSome methods for nonlinear system synthesis, such as; analytical time responsesynthesis, synthesis by ordinary differential equation solvers, synthesis by extended digital filter struc-tures.

Synthesis Quality AssessmentsHow to assess the errors of the synthesis methods, which references one coulduse and which quality assessments are available.

4.2 Linear System Synthesis

4.2.1 Time Response Synthesis using Laplace Transformation

The most common method for time response synthesis when having analytical excitation forces is by using theLaplace transformation. This method is not suitable when having experimental data. The standard definitionof the Laplace transform is as in equation 4.1. It is notable that the Laplace transform is only defined fort ≥ 0.

X(s) = Lx(t)=∫ +∞

0x(t)e−stdt (4.1)

Note that there are double sided laplace transforms as well, not regarded in this thesis. Some important, basicproperties of the Laplace transformation when determining the analytical time response to linear dynamicalsystems are presented in equation (4.2) to (4.4).

L

x(n)(t)

= snX(s)−n

∑k=1

sn−kx(k−1)(0) (4.2)

Lx(t) = sX(s)−x(0) (4.3)

Lx(t) = s2X(s)−sx(0)− x(0) (4.4)

30

CHAPTER 4. SYSTEM SYNTHESIS

Wherex(i)(0), i ∈ (0,1, . . .) are initial conditions of the response signal.

An intuitive example

Assume a SDOF linear dynamical system, described as the newtonian equation (4.5).

mx(t)+cx(t)+kx(t) = f (t) (4.5)

Laplace transformation of the SDOF system description gives equation (4.6).

m(s2X(s)−sx(0)− x(0)

)+c(sX(s)−x(0))+kX(s) = F(s) (4.6)

Solving forX(s) in equation (4.6) gives equation (4.7).

X(s) =1

(ms2 +cs+k)(F(s)+mx(0)s+mx(0)+cx(0)) (4.7)

Partial fraction expansion gives equation (4.8).

X(s) =(

As−λ

+A∗

s−λ∗

)(F(s)+mx(0)s+mx(0)+cx(0)) (4.8)

Inverse laplace transformation gives equation (4.9).

x(t) =(

Aeλt +A∗eλ∗t)∗ ( f (t)+mx(0)δ(t)+mx(0)+cx(0)) (4.9)

4.2.2 Synthesis by Digital Filters

Introduction

The main topic of synthesis by using digital filters is how to perform the transformation from the continuoustime- into discrete time representation of a dynamical system. With frequency notation; transformation fromcontinuous time system description as in equation (4.10) to corresponding sampled time system description asin equation (4.11).

H(s) =N

∑r=1

(Rr

s−λr+

R∗rs−λ∗r

)(4.10)

H(z) =N

∑r=1

br,0 +br,1z−1 +br,2z−2

1+ar,1z−1 +ar,2z−2 (4.11)

WhereH(s) is a general Laplace domain dynamical system description represented as the sum of pairs ofmodal poles and residues.H(z) is the corresponding Z-domain representation as the sum of second order filtersections, wherebk,i andak, j are filter coefficients for each section,k.

There are a lot of different methods for performing such a transformation, where the bilinear transform andthe impulse invariance methods are popular examples of such transformations, often reoccurring in literature.The advantage of those methods are that they are often quite easy to comprehend and to use. The main dis-advantage for the impulse invariance methods is that its gain at low frequencies is not constant for differentsystem resonance frequencies, due to aliasing effects for an example. The main disadvantage of the bilineartransformation method is its nonlinear frequency translation, thus making eventual errors in the transformationphase. As the Nyquist sampling theorem states; that the maximal signal bandwidth has to be less than half thesampling frequency; problems occur with the example methods.

The impulse invariance method uses the unit impulse function for transforming from continuous time todiscrete time, i.e for sampling, thus rendering synthesis errors. This thesis will present two extensions tothe impulse invariance method; the step invariance and the ramp invariance method. Both having very good

31

4.2. LINEAR SYSTEM SYNTHESIS

0 5 10 15 20 25 30 35 40 45 50−60

−50

−40

−30

−20

−10

0

10

20

30

FR

/Fs − Resonance frequency rel. to sampling frequency [%]

|H(F

)|2 (

dB r

el. 1

)

FRF of SDOFs having different resonance frequencies, Q = 10

Figure 4.1: Example SDOF systems having different resonance frequencies, ranging from2% up to 38% ofthe sampling frequency. The SDOF systems all have a Q-value equal to10. The sampling frequency was set to400 Hz.

DC gain characteristics, but not as good resonance frequency gain. But, since system information often is infrequencies lower than half the sampling frequency, the resonance frequency gain will not cause such a problemas the DC gain problem would. Example SDOF systems having different resonance frequencies is presentedin figure 4.1. The DC gain problem with the impulse invariance and the improvements with the step- andramp invariance methods is presented in figure 4.2. The resonance frequency gain for the different methods ispresented in figure 4.3.

32


5 10 15 20 25 30 35

0

5

10

15

20

25

30

35

40

45

FR


|H*(

0) −

H(0

)|/|H

(0)|

[%]

SDOF DC gain error as function of resonance frequency

H*(F) ~ Impulse invarianceH*(F) ~ Step invarianceH*(F) ~ Ramp invariance

Figure 4.2: DC gain error characteristics for the methods impulse invariance, step invariance and ramp invari-ance for the example SDOF systems having different resonance frequencies. Ideal DC gain for the sampleSDOF systems is1. The sampling frequency was set to 400 Hz.

5 10 15 20 25 30 35

0

5

10

15

20

25

30

35

40

45

FR


|H*(

FR

) −

H(F

R)|

/|H(F

R)|

[%]

SDOF resonance frequency gain as function of resonance frequency

H*(F) ~ Impulse invarianceH*(F) ~ Step invarianceH*(F) ~ Ramp invariance

Figure 4.3: Resonance frequency gain error characteristics for the methods impulse invariance, step invarianceand ramp invariance for the example SDOF systems having different resonance frequencies. Ideal resonancefrequency gain for the sample SDOF systems is10. The sampling frequency was set to 400 Hz.

33

4.2. LINEAR SYSTEM SYNTHESIS

Suggested Methodology for Deriving Transformation Methods

The suggested methodology for deriving a transformation method, is by using a catalyzer function in the trans-formation steps. The transformation derivation methodology can be generalized into five steps, assuming ageneral analogous system descriptionH(s) as in equation (4.10).

1. Multiply a Laplace domain catalyzer functionC(s) to the analogue system description.

HC(s) = C(s)H(s) (4.12)

2. Calculate the continuous time response function, by inverse Laplace-transforming the catalyzed Laplacedomain system.

hC(t) = L−1HC(s) (4.13)

3. Apply equidistant sampling to the continuous time response function, assuming the sample intervalT =1/FS.

hC(nT) = hC(t)|t=nT (4.14)

4. Z-transformation of the sampled time response function.

HC(z) = Z

hC(n)

(4.15)

5. Compensating for the catalyzer functionC(s) by dividing its corresponding z-domain representation.

H(z) =HC(z)C(z)

(4.16)

The methods of impulse invariance-, step invariance- and ramp invariance transformation are derived in ap-pendix A.

34


4.3 Nonlinear Systems Synthesis

4.3.1 Analytical Time Responses Synthesis

The time responsey(t) from a linear system is simply expressed as the convolution between the inputx(t) andthe systemh(t) as in equation (4.17).

y(t) =∞∫

τ=−∞

h(τ)x(t− τ)dτ = h(t)∗x(t) (4.17)

For nonlinear systems a first order convolution is not enough in order to describe the output signal. But asshown in [2] also nonlinear systems can be modelled, but via higher order convolution. This by a sort ofsuperposition stems from the Volterra series representing the nonlinear system according to equation (4.18).

y(t) = y1(t)+y2(t)+ . . .+yn(t). (4.18)

Wherey(t) is the response from the system. Depending on the kind of system, zero memory, finite memory orinfinite memory and in the two first cases the polynomial order of the nonlinear system the one needs differentamount of from the Volterra series in order to represent the system. For the general case with infinity terms theresponse would look like equation (4.19).

y(t) =∞∫

τ1=−∞

h1(τ1)x1(t− τ1)dτ1 +

+∞∫

τ1=−∞

∞∫

τ2=−∞

h(τ1,τ2)x1(t− τ1)x2(t− τ2)dτ1dτ2 + . . .

+∞∫

τ1=−∞

∞∫

τ2=−∞

∞∫

τ3=−∞

h(τ1,τ2,τ3)x1(t− τ1)x2(t− τ2)x3(t− τ3)dτ1dτ2dτ3 + . . . (4.19)

The most basic nonlinear Volterra series are the bilinear system equation (4.20) and the trilinear system equation(4.21), see [2]. These systems is of second order and third order respectively, example of such systems is thesquarer and the cuber.

∞∫

τ1=−∞

∞∫

τ2=−∞

h(τ1,τ2)x1(t− τ1)x2(t− τ2)dτ1dτ2 (4.20)

∞∫

τ1=−∞

∞∫

τ2=−∞

∞∫

τ3=−∞

h(τ1,τ2,τ3)x1(t− τ1)x2(t− τ2)x3(t− τ3)dτ1dτ2dτ3 (4.21)

Whether system is of kind zero memory or finite memory is determined by the transfer functionh(t).

35

4.3. NONLINEAR SYSTEMS SYNTHESIS

4.3.2 Synthesis by Ordinary Differential Equation Solvers

Ordinary differential equation (ODE) solvers are a group of system solvers that are using recursive algorithmssuch as Runge-Kutta for solving differential equations, linear or nonlinear. This section will emphasize onODE versions as available in the standard MATLAB1 5 software package. For further reading on MATLABODE solvers, please refer to [9].

ODE solvers in general are not intended for real time applications due to their tedious and computationallyexpensive nature. ODE solvers are thereby not really interesting regarding the goals of this thesis, but are in-cluded for general knowledge in system synthesis.

The MATLAB 5 Software package does include ODE solvers with applications as presented in table 4.1.

ODE solver Application Method Orderode23 For nonstiff systems Runge-Kutta 2 and 3ode45 For nonstiff systems Runge-Kutta of Dormand-Prince 4 and 5ode113 For nonstiff systems Adams-Bashforth-Moulton predictor-corrector 1 to 13ode23s For stiff systems Runge-kutta 2 and 3ode15s For stiff systems Numerical differentiation formulas 1 to 5

Table 4.1: Matlab 5 ordinary differential equation solvers with applications, solving method and specific orders.

ODE Solver Methods

Following is a brief overview of the solver methods used by MATLAB implementations of ODE solvers.

Runge-Kutta Runge-Kutta methods are referred to as one-step multistage methods. The update algorithm ofa fourth order Runge-Kutta method is presented in equation (4.22).

xn+1 = xn +h6

(k1 +2k2 +2k3 +k4) (4.22)

k1 = f (tn,yn) (4.23)

k2 = f (tn +12

h,yn +12

hk1) (4.24)

k3 = f (tn +12

h,yn +12

hk2) (4.25)

k4 = f (tn +h,yn +hk3) (4.26)

The Runge-Kutta of Dormand-Prince is an extended version of the Runge-Kutta method, using numericalinterpolation for increased performance, and minimal error.

Adams-Bashforth-Moulton The Adams-Bashforth-Moulton method is a linear multistep method defined instandard form as in equation (4.27).

k

∑j=0

α jyn+ j−k+1 = hk

∑j=0

β j fn+ j−k+1 (4.27)

Numerical differentiation formulas The numerical differential formulas are extensions to the backward dif-ferential formulas and are expressed as in equation (4.28).

k

∑m=1

1m

∇myn+1 = h fn+1 +κγk

(yn+1−

k

∑m=0

1m

∇myn

)(4.28)

1MATLAB is a trademark of The MathWorks, Inc.

36


ODE Solver Applications

The definition of whether a system is said to be stiff or nonstiff is rather advanced, but two of the requirementsfor stiff systems is that the real parts of the eigenvalues of the jacobian to the system must be negative, andthat the stiffness ratio is large. Let us apply the two requirements to a SDOF duffing oscillator system as anexample system for testing for stiffness or nonstiffness. The SDOF is stated as in equation (4.29) but rewritteninto the first order differential equation system functionalu(x,y, t) as in equation (4.30).

f (t) = mx(t)+cx(t)+kx(t)+αx3(t) (4.29)

u(x,y, t) =

u1(x,y, t)u2(x,y, t)

=

y(t)x(t)

=

1m

(f (t)−cy(t)−kx(t)−αx3(t)

)y(t)

(4.30)

The jacobian matrix of the first order ODE system is defined as in equation (4.31).

[J] =

[ ∂u1(x,y,t)∂x(t)

∂u1(x,y,t)∂y(t)

∂u2(x,y,t)∂x(t)

∂u2(x,y,t)∂y(t)

]=

[−k−3αx2(t)

m−cm

0 1

](4.31)

The eigenvalues of the jacobian to the example system are in equation (4.33).

det|J−λI | = 0 (4.32)

λ1

λ2=

1

− k+3αx2(t)m

(4.33)

Under the assumption that any response signalx(t) is real-valued and that the stiffnessk, cubic stiffnessα andmassmare positive constants, the second eigenvalue is always negative, but the first eigenvalue is positive. Theexample system is therefore said to be non-stiff.

37

4.3. NONLINEAR SYSTEMS SYNTHESIS

- Hi,1(ω)x1(n)

- Hi,2(ω) -x2(n)

...

- Hi,L(ω) -xL(n)

-Fi(n)

Figure 4.4: Standard SIMO linear system description.

- i+ - Hi,1(ω)x1(n)

- Hi,2(ω) -x2(n)

...

- Hi,L(ω) -xL(n)

g(x j(n))

-

¾

?+−Fi(n) zi, j(n)

x j(n)

Figure 4.5: Expansion of the standard SIMO linear system description into SIMO nonlinear system description,having infinite memory.

4.3.3 Synthesis by Extended Digital Filter Structures

When synthesizing the output of a Single Input Multiple Output (SIMO) nonlinear dynamical system using dig-ital filters, this thesis proposes a method which is an expansion of the standard linear SIMO system description.The standard linear SIMO system description as in figure 4.4 could be formulated as in equation (4.34).

x j(n) = −N

∑k=1

akx j(n−k)+M

∑m=0

bmFi(n−m) (4.34)

H(e− jω)

=B

(e− jω)

A(e− jω)=

b0 +b1e− jω + . . .+bMe− jωM

1+a1e− jω + . . .+aNe− jωN (4.35)

For an infinite memory nonlinear system having a nonlinear function of responsex j(n) as presented in figure4.5, this description is reformulated as in equation (4.36).

x j(n) = −N

∑k=1

akx j(n−k)+M

∑m=0

bm(Fi(n−m)−g(x j(n−m))) (4.36)

b0g(x j(n))+x j(n) = −N

∑k=1

akx j(n−k)+b0Fi(n)+M

∑m=1

bm(Fi(n−m)−g(x j(n−m))) (4.37)

Solving forx j(n) for each point in time,n, gives the expected output for the nonlinear dynamical system. Thecurrent implementation uses the Newton-Raphson iteration for solvingx j(n). Other schemes for solving couldbe used as well.

38


4.4 Synthesis Quality Assessments

This thesis have sofar only presented different synthesis algorithms. No effort in quality assessments have beenmade. Following is an introductory on how to assess the quality of system synthesis.

The Method

Following is a suggested four-step procedure for calculating synthesis goodness estimators. The suggestedmethodology assumes that the system is a infinite memory nonlinear system, as presented earlier in figure 4.5,for variable names and declarations please refer to the previously mentioned figure 4.5.

1. Apply the synthesis to the system, i.e calculate the responsex j(n) given an excitation forceFi(n) using asynthesis algorithm.

2. Calculate the fictive excitation force as it is seen from the linear subsystem as in equation (4.38).

3. Recalculate a new estimate of the responsex j(n) by filtering the fictive excitation force through the linearsubsystem as in equation (4.39).

4. Finally express the quality function as the standard deviation of difference between the synthesized re-sponse signal and the estimated response signal, divided by the standard deviation of the synthesizedresponse signal as presented in equation (4.40) to (4.42).

zi, j(n) = Fi(n)−x j(n) (4.38)

x j(n) = Hi, j

zi, j(n)

(4.39)

Qi, j =D

x j(n)− x j(n)

D

x j(n) (4.40)

Dx(n) =

√1N

N−1

∑n=0

(x(n)−x)2 (4.41)

x =1N

N−1

∑k=0

x(k) (4.42)

39

Chapter 5

Experimental Evaluation of Black-boxSystems

5.1 Background

Four systems of arbitrary high degrees of freedom, containing unknown nonlinearities of unknown mountingnodes where challenged to the master thesis students. The aim was for the students to use the, in thesis de-veloped black-box system identification methods and thus identifying the unknown systems. The only priorsystem knowledge was as follows:

• There is only one nonlinearity present in each system.

• The nonlinearity of a system (if any) must be mounted from one of the system nodes to ground.

• A nonlinearity should not have internal memory.

The four systems are presented in table 5.1. Their respective true underlying linear systems are presented infigure 5.1.

It should be emphasized that the third system diverged from the experiment prerequisites in that its nonlin-earity did have internal memory. The third system housed a nonlinearity called Stick-Slip. Stick-Slip is closelyrelated to the hysteresis nonlinearity, i.e a nonlinearity with internal memory.

System DOF’s Nonlinearity Description Mounted node1 8 0.8·109x3

3(t) Cubic stiffness node three to ground2 8 0 Linear -3 8 unknown Stick-slip node five to ground4 8 5arctan(4000x4(t)) Arcus tangens node four to ground

Table 5.1: The contents of the four challenged Black-box systems.

40

CHAPTER 5. EXPERIMENTAL EVALUATION OF BLACK-BOX SYSTEMS

0 10 20 30 40 50 60−105

−100

−95

−90

−85

−80

−75

−70

−65

−60


|H1(F

)| −

FR

F (

dB r

el. 1

) [m

/N]

Linear subsystem of System 1 (Force node=1, Response node=1)

0 10 20 30 40 50 60−105

−100

−95

−90

−85

−80

−75

−70

−65

−60


|H2(F

)| −

FR

F (

dB r

el. 1

) [m

/N]


a) b)

0 10 20 30 40 50 60−105

−100

−95

−90

−85

−80

−75

−70

−65

−60


|H3(F

)| −

FR

F (

dB r

el. 1

) [m

/N]


0 10 20 30 40 50 60−105

−100

−95

−90

−85

−80

−75

−70

−65

−60

−55

−50


|H4(F

)| −

FR

F (

dB r

el. 1

) [m

/N]


c) d)

Figure 5.1: True frequency response functions of the underlying linear system of the four challenged black-boxsystems. a) FRF of system 1, b) FRF of system 2, c) FRF of system 3 and d) FRF of system 4.

41

5.2. THE EXPERIMENT

5.2 The Experiment

The four black-box systems are to be identified and thoroughly analyzed during this experimental evaluation.The following questions will be answered for each of the four systems:

1. Is the system linear or nonlinear? For answering this question one could for an example excite thesystem with different excitation force amplitudes within the operating range. If the frequency responsefunctions of the different excitation levels does not change, besides eventual measurement noise, thesystem could be seen as linear. Specially, when the system is nonlinear, the anti-resonances of thesystem tends to change.

2. Where is the nonlinearity mounted? Preferably one could use the Brute Force method for systemnonlinear node identification.

3. Estimate the nonlinearity and the linear subsystem.Estimation using the suggested methods; Linearsystem identification, Reverse Path Method and the Frequency Domain Structure Selection Algorithm.

5.2.1 The First System

Is the system linear or nonlinear?

For testing whether the system is linear or nonlinear five different force excitation levels where used. Theresulting frequency response functions are presented in figure 5.2 and clearly indicates that the system is notlinear.

0 5 10 15 20 25 30 35 40−105

−100

−95

−90

−85

−80

−75

−70

−65

−60


|H* 1(F

)|2 −

FR

F (

dB r

el. 1

) [m

/N]

Estimated FRF of System 1 for five excitation levels F1,RMS

to F5,RMS

Excitation force F1,RMS

= 0.1 NExcitation force F

2,RMS = 0.5 N


= 1 NExcitation force F

4,RMS = 5 N


= 10 N

Figure 5.2: Estimated frequency response functions between force node one and response node one for System1, having five different excitation force levels. The system’s anti-resonances as well as its resonances arechanged as the excitation force level changes, thus the system is assumed to be nonlinear.

Where is the nonlinearity mounted?

The brute force method applied to the first system results in figure 5.3. It indicates that the nonlinearity ismounted in node three.

42


12

34

56

78

910

1

2

3

4

5

6

0

0.2

0.4

0.6

0.8

1

Non−linear function xk

Nonlinear−Linear Coherence System1

Nodes

Figure 5.3: Estimated nodes of where the nonlinearity could be mounted in the first system. Polynomial ordersfrom 1 up to order 10 were individually tested. As obvious in the figure, node three holds the nonlinearity.

43

5.2. THE EXPERIMENT

Estimate the nonlinearity and the linear subsystem

The FDSSA method were used for evaluating which linear subsystem and nonlinearity the first system contains.The resulting nonlinear function is plotted in figure 5.4 in comparison to the true nonlinearity. The resultinglinear subsystem estimate is plotted in comparison to the true linear subsystem in figure 5.5.

−4 −3 −2 −1 0 1 2 3 4

x 10−3

−60

−40

−20

0

20

40

60Polynomial functions of the nonlinearity in the first system

x

p(x)

− p

olyn

omia

l fun

ctio

n

True polynomialFDSSA estimate, Q

err=3.2136%

Figure 5.4: Polynomial functions of the first system’s nonlinearity. (Blue) the true nonlinear polynomial, and(Red) the estimated nonlinear polynomial. The estimation errorQerr is defined as the standard deviation of thepolynomial fit error divided by the standard deviation of the true polynomial.

44


0 5 10 15 20 25 30 35 40−110

−100

−90

−80

−70

−60

−50

−40


Dis

plac

emen

t (dB

rel

. 1)

[N/m

]

Frequency Response Function of the underlying linear system for system one

FRF of true linear subsystemEstimated FRF of linear subsystem

Figure 5.5: The linear subsystem estimate of the first system. (Blue) the true linear subsystem, and (Red) theestimated linear subsystem.

45

5.2. THE EXPERIMENT

5.2.2 The Second System


For testing whether the system is linear or nonlinear five different force excitation levels where used. The result-ing frequency response functions are presented in figure 5.6 and clearly indicates that eventual nonlinearitiesare not detectable in the system, for the applied excitation force levels.

0 5 10 15 20 25 30 35 40−105

−100

−95

−90

−85

−80

−75

−70

−65

−60


|H* 2(F

)|2 −

FR

F (

dB r

el. 1

) [m

/N]


to F5,RMS



2,RMS = 0.5 N



4,RMS = 5 N


= 10 N

Figure 5.6: Estimated frequency response functions between force node one and response node one for System2, having five different excitation force levels. All FRFs are overlaying each other, and the system is thereforesaid to be linear, for this specific range of operation.


The first test reveals that there is no nonlinearity in the second system, the brute force test seems thereforeunnecessary. But, the brute force method is although evaluated for the second system for clarity. The results ofthe brute force method is presented in figure 5.7.

46


12

34

56

78

910

1

2

3

4

5

6

0

0.2

0.4

0.6

0.8

1


Nonlinear−Linear Coherence System 2

Nodes

Figure 5.7: Estimated nodes of where the nonlinearity could be mounted in the second system. Polynomialorders from 1 up to order 10 were individually tested. As obvious in the figure, no node holds any of the testednonlinearities.

47

5.2. THE EXPERIMENT

5.2.3 The Third System



0 5 10 15 20 25 30 35 40−105

−100

−95

−90

−85

−80

−75

−70

−65

−60


|H* 3(F

)|2 −

FR

F (

dB r

el. 1

) [m

/N]


to F5,RMS



2,RMS = 0.5 N



4,RMS = 5 N


= 10 N



The brute force method applied to the third system results in figure 5.9. It indicates that the nonlinearity ismounted in node five.

48


12

34

56

78

910

1

2

3

4

5

6

0

0.2

0.4

0.6

0.8

1

Non−linear function v k

Nonlinear−Linear Coherence System3

Nodes

Figure 5.9: Estimated nodes of where the nonlinearity could be mounted in the third system. Polynomial ordersfrom 1 up to order 10 were individually tested. As obvious in the figure, node five holds the nonlinearity.

49

5.2. THE EXPERIMENT


The FDSSA method were used for evaluating which linear subsystem and nonlinearity the third system con-tains. The resulting nonlinear function is plotted in figure 5.10 in comparison to the true nonlinearity. Theresulting linear subsystem estimate is plotted in comparison to the true linear subsystem in figure 5.11.

−5 −4 −3 −2 −1 0 1 2 3 4 5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Polynomial functions of the nonlinearity in the third system

x´(t)

p(x´

(t))

− p

olyn

omia

l fun

ctio

n

FDSSA estimate

Figure 5.10: Polynomial functions of the third system’s nonlinearity. (Blue) the true nonlinear polynomial, and(Red) the estimated nonlinear polynomial. The estimation errorQerr is defined as the standard deviation of thepolynomial fit error divided by the standard deviation of the true polynomial.

50


0 5 10 15 20 25 30 35 40−110

−100

−90

−80

−70

−60

−50

−40


Dis

plac

emen

t (dB

rel

. 1)

[N/m

]

Frequency Response Function of the underlying linear system for system three


Figure 5.11: The linear subsystem estimate of the third system. (Blue) the true linear subsystem, and (Red) theestimated linear subsystem.

51

5.2. THE EXPERIMENT

5.2.4 The Fourth System



0 5 10 15 20 25 30 35 40−105

−100

−95

−90

−85

−80

−75

−70

−65

−60


|H* 4(F

)|2 −

FR

F (

dB r

el. 1

) [m

/N]


to F5,RMS



2,RMS = 0.5 N



4,RMS = 5 N


= 10 N



The brute force method applied to the fourth system results in figure 5.13. It indicates that the nonlinearity ismounted in node four.

52


12

34

56

78

910

1

2

3

4

5

6

0

0.2

0.4

0.6

0.8

1


Nonlinear−Linear Coherence System 4

Nodes

Figure 5.13: Estimated nodes of where the nonlinearity could be mounted in the fourth system. Polynomialorders from 1 up to order 10 were individually tested. As obvious in the figure, node four holds the nonlinearity.

53

5.2. THE EXPERIMENT


The FDSSA method were used for evaluating which linear subsystem and nonlinearity the fourth system con-tains. The resulting nonlinear function is plotted in figure 5.14 in comparison to the true nonlinearity. Theresulting linear subsystem estimate is plotted in comparison to the true linear subsystem in figure 5.15.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 10−3

−8

−6

−4

−2

0

2

4

6

8Polynomial functions of the nonlinearity in the fourth system

x

p(x)

− p

olyn

omia

l fun

ctio

n

True polynomialFDSSA estimate, Q

err=6.5009%

Figure 5.14: Polynomial functions of the fourth system’s nonlinearity. (Blue) the true nonlinear polynomial,and (Red) the estimated nonlinear polynomial. The estimation errorQerr is defined as the standard deviation ofthe polynomial fit error divided by the standard deviation of the true polynomial.

54


0 5 10 15 20 25 30 35 40−110

−100

−90

−80

−70

−60

−50

−40


Dis

plac

emen

t (dB

rel

. 1)

[N/m

]

Frequency Response Function of the underlying linear system for system four


Figure 5.15: The linear subsystem estimate of the fourth system. (Blue) the true linear subsystem, and (Red)the estimated linear subsystem.

55

5.2. THE EXPERIMENT

5.2.5 The Results

The results for the Experimental Evaluation of Black-box systems give good system descriptions for three outof four cases. The first, second and fourth system were identified with acceptable error levels, but the third sys-tem could not be estimated correctly at all, due to the fact that the third system diverged from the experimentprerequisites.

It is very important to emphasize that the experimental evaluation was conducted without any present noise.

56

Chapter 6

Experimental Evaluation of Test-Rig

6.1 Background

After developing different methods to identify the nonlinearities and analytical evaluation of the those meth-ods, some practical tests are in place. Different known simulated system with and with out nonlinearities andalso with and with out noise has been applied on the methods with different results. There has only been onenonlinearity a time, the nonlinearity has been coupled either between different nodes towards ground or nodeto node. Also unknown systems have been identified with variable outcomes for the different methods, but totalimpression gives a quit good view over the systems behavior.

6.2 Introduction

To be able to test the different methods on measured data a known system with a nonlinearity property isneeded. A common system with a simple nonlinearity such as cubic stiffness could be built according to acertain mechanical model. Besides this, another property of the system is wanted. Which is to have a firstresonance frequency (further referred to the first mode) easily identifiable meaning sufficient separated fromthe next mode.Most important is the nonlinear effect on the first mode.

This kind of system could be found in a report [6] about identification of nonlinear systems, which alsoinclude results from both the analytical model and measurements done on the system. With this report as afoundation for the measurements done on an almost identical model, the aim was to get similar results.

6.3 System Model

6.3.1 Linear Part of System Model

The linear system with an appropriate first mode around 50 Hz was constructed by a firmed clamped beam,only clamped on the one side of the beam with the dimensions according to figure 6.1.

6.3.2 Nonlinear Part of System Model

To cause a cubic stiffness coupled to first mode, two thinner beams are applied under and over the first beamrepresenting the linear system, this at the free end of the bream and in right angles to it. The two beams arefirmly clamped in both sides of the beams and having dimensions which results in resonance frequencies muchhigher than the first mode, according to figure 6.2. The cubic stiffness is now applied due to the increasedtension under large amplitudes and should effect the first mode. Figure 6.3 is a closeup on the cubic stiffnessmodel. The previous linear systems properties has been changed somewhat, but the total system now has the

57

6.3. SYSTEM MODEL

Figure 6.1: The underlying linear system of the experimental test-rig.

Figure 6.2: By adding two parallel thin beams, perpendicular to the linear system beam, a nonlinear functionis added to the structure.

wanted properties.

8 mm

1 2 m m

Ø 5 m m

8 mm

1 2 m m

Ø 5 m m

Figure 6.3: A close-up of the mechanical system that modulates the nonlinearity property of the system usedin this experiment setup.

6.3.3 Nonlinear Property Verification

Verification of the systems nonlinearity property was made with the assumption of cubic stiffness according tothe parametersαx+ βx3, which was done by a static load test. The test is done by applying different amountforce i.e. applying different weights and measure the displacement. The result of this measurements could beeseen in figure 6.4 as blue dots. The test will also give a hint of what level of the force is needed to get enoughinfluence from the nonlinearity. The result from the static load test indicates a property of nonlinearity, which

58

CHAPTER 6. EXPERIMENTAL EVALUATION OF TEST-RIG

−3 −2 −1 0 1 2 3

x 10−3

−100

−80

−60

−40

−20

0

20

40

60

80

100Static Load Test

For

ce F

[N]

Displacement x [m]

Figure 6.4: A curvefit from static load test confirms the presence of at least a cubic stiffness. The equationαx+βx3 is only fitted the an interval where the system acts according to the model. The estimated parametersareα = 12.242·103N/m andβ = 24.962·108N/m3, The blue dots are the measured results and the red curveis the fitted result.

could be a cubic stiffness. If the cubic stiffness is a correct assumption the estimated parametersα andβ willbe12.242·103N/m and24.962·108N/m3 respectively. The resulting function could bee seen as the red curvein figure 6.4. According to the estimated parameters a force of50−70N should be enough to get measureddata effected by the nonlinearity, which could be used later on, to identify the nonlinearity.

6.4 Experimental Setup

According to the report [6] previous mentioned in the introduction, is an appropriate node to measure in is theintersection of the first beam and the two thinner beams. This node will be a drivingpoint of the measurementsdone on the system.

6.4.1 Measurement Equipment

The equipment used to perform this experiment is an analyzer, an amplifier, a shaker, an accelerometer, a forcesensor and finally a computer connected to the analyzer with the task of managing the analyzer.

6.4.2 Equipment List

• Analyzer, "SignalCalc Mobilyzer" a portable multi-channel Dynamic Signal Analyzer

• Amplifier

• Shaker

• Force Transducer,ICP Sensor, Model PCB 208C01, SN 17157, 116.51mV/N

• Acceleromter, ICP Sensor, Model PCB 353B03, SN 45862, 9.51mV/ ms2

6.4.3 Work Materials

• An enormous block of solid steel, which the rig is attached to, denoted "ground".

• Three smaller underlying blocks of solid steel, the frame of the rig attached to the ground.

59

6.4. EXPERIMENTAL SETUP

• Three smaller overlying blocks of solid steel, which together with the underlying blocks gives a firmclamped properties to each end of the beams.

• One beam of type A, this beam constitute the linear model with the mode of interest.

• Two thinner beams of type B, contributing with the nonlinearity property to the system.

• Two smaller blocks, making the two thinner beams firmly clamped.

• 6 screws of type M12x100, to attach the blocks towards ground.

• 6 screws of type M10x70, to attach the blocks towards each other.

• one special threaded screw of type UNF 10-32 with the length of 18mm, to attach the sensors with.

• Two holders for the sting wire, one attached to the shaker and the other attached to the force transducer.

• Two power cables with laboration connectors, for the amplifier to drive the shaker.

• A coaxial cable BNC to BNC, from the analyzer to the amplifier.

• Two coaxial cables 10-32 UNF-2A to BNC, from the sensors to the analyzer.

• One crossed TP cable, for network communication between the analyzer and computer.

• One big solid steel module, for holding the shaker.

C o m p u t e r M o b y l i z e r A m p l i f i e r

S h a k e r

F o r c e T r a n s d u c e r

A c c e l e r o m t e r

Figure 6.5: The measurement setup of the test-rig system.

The force transducer is mounted on top of the top beam and the accelerometer is mounted under the bottombeam. The sensors are clamping the three beams together with a special threaded screw. The force transduceris physically connected to the shaker trough a wire of steel. The shaker is connected to the amplifier withpower cables able to drive the shaker with the excitation signal, and with the correct amount of force. Thisexcitation signal comes from the analyzer output channel trough a coaxial cable. The sensors are connected tothe two first input channels on the analyzer. The connection between the analyzer and the computer is a networkconnection with capacity of 100 Mbit/s. The cable has to be a crossed TP network cable. For managing theanalyzer a program called "Signal Calc Mobilyzer" is installed on the host. This program supplies with manytools of controlling excitation signals and and analyzing tools for the measured data. For an overview over theexperiment setup se figure 6.5.

60


6.5 Choice of Performance and Excitation Signals

The first thing to do when the experimental rig setup is complete, is to check that signals seems to be correct.Using a random noise low excitation level and adjusting the amplifier to a suitable level.

When all setting are checked and adjusted to the correct value the real measurement can commence. Tobegin with some noise signal with different excitation levels and different frequency spans [625, 320, 160] Hz,is a simple way to get an overview over the behavior of systems. One could after those measurements decidewhich span that is of interest. Information of this was already available from the report [6], but an unwrittenrule is to decide this from experimental measurements. The goal of this noise measurements is to show someinfluence from the nonlinearity property when overlaying the FRF’s in the same plot.

Further measurements of interest is the sinus sweep, up and down with different excitation levels, to pointout any presence of the cubic stiffness.

After a quick evaluation of the results from the measurement which lacks indication of the cubic stiffness,the conclusion is to increase the frequency resolution. This resulted in an excitation signal produced inMAT-LAB , since the program Signal Calc did not supply a resolution high enough.

New random noise signals where with different excitation levels and after that even longer signals wheresynthesized, now with a resolution which should be more than enough.

6.6 Measurement Settings

6.6.1 Measurement Settings, Collection from Signal Calc when using build-in func-tions

Measurement 1-5 have been done with 7 different excitation levels, but because of non-uniformed force distri-bution over time, a specification of the force level is not presented here.

Meas. Output Avgs. Avg type: Overlap Trigger Freq. Span [Hz] ∆F [Hz] Fs [Hz]1 noise 50 Stable 50 % Free Run 625 0.635 26042 noise 50 Stable 50 % Free Run 320 0.317 13023 noise 50 Stable 50 % Free Run 160 0.158 6514 sweep up 10 Stable 0 Free Run 160 0.158 6515 sweep down 10 Stable 0 Free Run 160 0.158 651

6.6.2 Measurement Settings, Collection from Signal Calc when using excitation signalproduced in matalab

Measurement 6 is done with 3 different excitation levels and measurement 9 has been done with 4 differentexcitation levels. The same reason as for the measurement 1-5 also applies here.

Measurement Excitation Freq. Span [Hz] Fs [Hz] Time span [Sec]6 noise 160 651 1547 sweep up 50-70 651 1548 sweep down 70-50 651 1549 noise 160 651 30010 sweep up 50-70 651 90011 sweep down 70-50 651 900

6.6.3 Channels Settings for measurement 1-3

Channel Window Coupling EU Other1 Hanning ICP 4mA m/s2

2 Hanning ICP 4mA N reference

61

6.7. RESULTS

6.6.4 Channels Settings for measurement 4-11

Channel Window Coupling EU Other1 Rectangular ICP 4mA m/s2

2 Rectangular ICP 4mA N reference

6.7 Results

0 100 200 300 400 500 600−70

−60

−50

−40

−30

−20

−10

0

10

20

30

Frequency F [Hz]

FR

F M

agni

tude

[dB

(m

/s2 )/

N]

FRFs, from measurement 1 with 5 different excitation levels

RMS=0.0561 NRMS=0.147 NRMS=0.298 NRMS=0.459 NRMS=0.657 N

Figure 6.6: FRFs from the first measurement with the frequency range 0-625 Hz to get an overview of thesystem. Five different excitation levels were measured in order to see any abnormalities. The different levelsare presented as the RMS of the force signal.

62


0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency F [Hz]

Coh

eren

ce γ

af 2

Coherence, from measurement 1 with 5 different excitation levels


Figure 6.7: Coherence function of the first measurement shows the need of higher excitation level in order toget a valid FRF

63

6.7. RESULTS

0 50 100 150 200 250 300−80

−60

−40

−20

0

20

40

Frequency F [Hz]

FR

F M

agni

tude

[dB

(m

/s2 )/

N]



Figure 6.8: FRFs from measurement 2, a smaller frequency spawn than previous measurement.

64


0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency F [Hz]

Coh

eren

ce γ

af 2



Figure 6.9: Coherence function from measurement 2, almost the same result as measurement 1 but with ahigher resolution.

65

6.7. RESULTS

0 50 100 150−60

−50

−40

−30

−20

−10

0

10

20

30

40

Frequency F [Hz]

FR

F M

agni

tude

[dB

(m

/s2 )/

N]



Figure 6.10: FRFs from measurement 3, the highest frequency resolution with the program Signal Calc.

66


0 50 100 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency F [Hz]

Coh

eren

ce γ

af 2



Figure 6.11: Coherence function from measurement 3, some abnormalities could be seen at approximate 54 Hzwhich could bee an indication of some nonlinearity or the shakers inability to drive the structure.

67

6.7. RESULTS

0 50 100 150−60

−50

−40

−30

−20

−10

0

10

20

30

40

Frequency F [Hz]

FR

F M

agni

tude

[dB

(m

/s2 )/

N]

FRFs, from measurement 4 and 5

RMS=0.702 NRMS=1.18 NRMS=1.66 N

Figure 6.12: FRFs from frequency sweep up and down, measurement 4 and 5. Has almost the same result asmeasurement 3. There are three sweep up and three sweep using three different excitation levels.

68


0 50 100 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency F [Hz]

Coh

eren

ce γ

af 2

Coherence, from measurement 4 and 5

RMS=0.702 NRMS=1.18 NRMS=1.66 N

Figure 6.13: Coherence function from measurement 4 and 5.

69

6.7. RESULTS

0 20 40 60 80 100 120 140 160−100

−80

−60

−40

−20

0

20

40FRF from measurement 6

Frequency F [Hz]

FR

F M

agni

tude

[dB

(m

/s2 )/

N]

RMS=0.418 NRMS=0.862 NRMS=1.37 N

Figure 6.14: Three FRFs with white gaussian noise synthesized in matlab as excitation signals.

70


0 20 40 60 80 100 120 140 1600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Coherence from measurement 6

Frequency F [Hz]

Coh

eren

ce γ

af 2

RMS=0.418 NRMS=0.862 NRMS=1.37 N

Figure 6.15: Coherence function from measure 6.

71

6.7. RESULTS

50 52 54 56 58 60 62 64 66 68 70−5

0

5

10

15

20

25

30

35FRF from measurement 7 and 8

Frequency F [Hz]

FR

F M

agni

tude

[dB

(m

/s2 )/

N]

Sweep upSweep down

Figure 6.16: Two FRFs from measurement 7 and 8 where the blue plot is the sweep up and the magenta plotis the sweep down. The excitation level has been changed due to properties of the system combined with theshaker.

72


0 20 40 60 80 100 120 140 160−40

−30

−20

−10

0

10

20

30FRF from measurement 9

Frequency F [Hz]

FR

F M

agni

tude

[dB

(m

/s2 )/

N]

Figure 6.17: FRF with the highest excitation level with gaussian noise, which was performed in measurement9.

73

6.7. RESULTS

0 20 40 60 80 100 120 140 1600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Coherence from measurement 9

Frequency F [Hz]

Coh

eren

ce γ

af 2

Figure 6.18: Coherence function from measurement 9.

74


50 52 54 56 58 60 62 64 66 68 70−10

−5

0

5

10

15

20

25

30

35

40FRF from measurement 10 and 11

Frequency F [Hz]

FR

F M

agni

tude

[dB

(m

/s2 )/

N]

Sweep upSweep down

Figure 6.19: The highest excitation level with frequency sweep was performed in measurement 10 and 11. Thelevel of the force over time could be seen in the waterfall diagram figure6.21.

75

6.7. RESULTS

Figure 6.20: Acceleration signal from measurement 10, the acceleration signal show an amplitude, constant inthe first harmonic component during the sweep from 50 to 70 Hz.

76


Figure 6.21: Force signal from measurement 10, the force has an excitation which effects more than the in-tended frequency and is sweeping from 50 to 70 Hz. The force was supposed to have constant amplitude, butthe shaker was unable to fulfill the desired input.

77

6.7. RESULTS

Figure 6.22: Acceleration signal from measurement 11 shows a constant amplitude in the first harmonic com-ponent during the sweep from 70 to 50 Hz.

78


Figure 6.23: Force signal from measurement 10, the force has an excitation which effects more than the in-tended frequency, sweeping from 70 to 50 Hz. During the frequency sweep up the force is decreasing inamplitude over the first mode.

79

6.8. CONCLUSION

6.8 Conclusion

The results from the measurements which have been presented in the previous section indicates an undesiredforce signal rather than it indicates the desired nonlinear property. The first problem in the measurement wasto keep constant excitation force level during the frequency sweep, shown in figure 6.21 and 6.23. The secondproblem was the excitation of more than one frequency during the sweeps, shown in figure 6.20 - 6.23. Theseproblems are a result from the weak structure combined with the shaker’s inability to drive the desired forcesignal into the structure. The shaker should instead of a voltage amplifier been driven by a current amplifier,controlled by a feedback system which task is to ensure that the applied excited force is identical to the desiredforce. It is extremely important that the applied signal is the same as the desired for the system to excite itsnonlinearities.

80

Chapter 7

Summary and Conclusions

7.1 Summary

In this thesis, an overview of the theoretics of linear systems as well as nonlinear systems have been presented.The theoretics have given brief information to both analysis and synthesis of linear and nonlinear systems.The theoretical background has constituted a knowledge-base for understanding, implementing and evaluatingseveral parametric and nonparametric methods and algorithms.

In the later part of the thesis, the methods and algorithms have been applied for two experimental cases.Case one constituted of four unknown black-box systems and the second case constituted of a test-rig. Thetask for the given cases was to identify if the systems hosted nonlinearities, and where the nonlinearities weremounted, and finally estimation of the nonlinearities as well as the linear subsystems, thus giving a total descrip-tion of the systems. The idea of using the test-rig comes from [6], where the test-rig is said to contain nonlinearelements. But due to problems in applying the correct level of excitation force at resonance frequencies thesystem did not show its nonlinear behavior as expected.

7.2 Conclusions and Further Research

The results from the first experimental case indicate that the frequency-domain methods performs better thantime-domain methods such as the NARMAX method, for the given thesis prerequisites. The prerequisites statesthat the methods and algorithms in the thesis should be optimized for time-efficiency and should be suitable forphysically applicable scenarios. The successful methods do both perform well in system identification as wellas being computationally efficient.

For getting success in the second case, the setup should be armed with a force control system. Such thatthe desired applied force really is the applied force, even at system resonance frequencies.

A summary of the conclusions that could be drawn from this thesis is that it clearly shows that nonlinearsystem identification and analysis is a huge, yet important area of research within the field of signal processing.There are many pitfalls and hinders within the subject of nonlinear systems, and a researcher in the subjectmust be extremely critical to the material and information that is presented.

81

Appendix A

Derivations of Impulse-, Step-, and RampInvariance

A.1 Derivation of Impulse Invariance

The method of impulse invariance uses the dirac delta function (A.1) as catalyzer function.

c(t) =

+∞ , t = 00 , t 6= 0

,

∫ +∞

0c(t)dt = 1 (A.1)

C(s) = 1 (A.2)

c(nT) =

T ,nT = 00 ,nT 6= 0

(A.3)

C(z) = T (A.4)

By using the suggested methodology the method of impulse invariance could be formulated as:

HC(s) = C(s)H(s) = 1H(s) =N

∑r=1

(Rr

s−λr+

R∗rs−λ∗r

)(A.5)

hC(t) = L−1HC(s)=N

∑r=1

(Rre

λr t +R∗r eλ∗r t)

(A.6)

hC(nT) = hC(t)|t=nT = TN

∑r=1

(Rre

λr nT +R∗r eλ∗r nT)

(A.7)

HC(z) = ZhC(nT)=N

∑r=1

(Rrz

z−eλr T+

R∗r z

z−eλ∗r T

)=

N

∑r=1

(Rr

1−eλr Tz−1+

R∗r1−eλ∗r Tz−1

)(A.8)

H(z) =HC(z)C(z)

=HC(z)

T=

N

∑r=1

(Rr

1−eλr Tz−1+

R∗r1−eλ∗r Tz−1

)=

=N

∑r=1

br,0 +br,1z−1 +br,2z−2

1+ar,1z−1 +ar,2z−2 (A.9)

82

APPENDIX A. DERIVATIONS OF IMPULSE-, STEP-, AND RAMP INVARIANCE

The filter coefficientsbr,i andar, j per second order section have the following expressions

br,0 = Rr +R∗r (A.10)

br,1 = −(

Rreλ∗r T +R∗r eλr T

)(A.11)

br,2 = 0 (A.12)

ar,1 = −(

eλr T +eλ∗r T)

(A.13)

ar,2 = e(λr+λ∗r )T (A.14)

A.2 Derivation of Step Invariance

The method of step invariance uses the unit step function as catalyzer.

c(t) =

1 , t ≥ 00 , t < 0

(A.15)

C(s) =1s

(A.16)

c(nT) =

T ,nT ≥ 00 ,nT < 0

(A.17)

C(z) = Tz

z−1(A.18)

By using the suggested methodology the method of step invariance could be formulated as

HC(s) = C(s)H(s) =H(s)

s=

N

∑r=1

(Rr

s(s−λr)+

R∗rs(s−λ∗r )

)(A.19)

A general inverse Laplace-transform of a functionH(s)s is stated by equation (A.20).

L−1

H(s)s

=

∫ t

0−h(τ)dτ (A.20)

h(t) = L−1H(s) (A.21)

Applying the inverse Laplace-transform to equation (A.19).

hc(t) =N

∑r=1

(∫ t

0−Rre

λr τ +R∗r eλ∗r τdτ)

=

=N

∑r=1

[Rreλr τ

λr+

R∗r eλ∗r τ

λ∗r

]τ=t

0−=

=N

∑r=1

(Rr

λr

(eλr t −1

)+

R∗rλ∗r

(eλ∗r t −1

))(A.22)

Sampling of the continuous time response function.

hc(nT) = TN

∑r=1

(Rr

λr

(eλr nT−1

)+

R∗rλ∗r

(eλ∗r nT−1

))(A.23)

Z-transformation of the sampled time response function.

HC(z) = ζhc(nT)=

= TN

∑r=1

(Rr

λr

(z

z−eλr T− z

z−1

)+

R∗rλ∗r

(z

z−eλ∗r T− z

z−1

))=

= TN

∑r=1

(Rr

λr

(1

1−eλr Tz−1− 1

1−z−1

)+

R∗rλ∗r

(1

1−eλ∗r Tz−1− 1

1−z−1

))(A.24)

83

A.3. DERIVATION OF RAMP INVARIANCE

Compensation for the catalyzer function and simplifying the frequency response function.

H(z) =HC(z)C(z)

=

=1−z−1

THC(z) =

=N

∑r=1

(Rr

λr

(1−z−1

1−eλr Tz−1−1

)+

R∗rλ∗r

(1−z−1

1−eλ∗r Tz−1−1

))=

=br,0 +br,1z−1 +br,2z−2

1+ar,1z−1 +ar,2z−2 (A.25)

The filter coefficient for the second order sections could be expressed as in following equations.

br,0 = 0 (A.26)

br,1 =Rr

λr

(eλr T −1

)+

R∗rλ∗r

(eλ∗r T −1

)(A.27)

br,2 = −(

Rr

λr

(eλr T −1

)eλ∗r T +

R∗rλ∗r

(eλ∗r T −1

)eλr T

)(A.28)

ar,1 = −(

eλr T +eλ∗r T)

(A.29)

ar,2 = e(λr+λ∗r )T (A.30)

A.3 Derivation of Ramp Invariance

The method of ramp invariance uses the ramp function as catalyzer, as defined by equation (A.31).

c(t) =

t , t ≥ 00 , t < 0

(A.31)

C(s) =1s2 (A.32)

c(nT) =

Tn ,nT ≥ 00 ,nT < 0

(A.33)

C(z) = Tz

(z−1)2 (A.34)

By using the suggested methodology the method of ramp invariance could be formulated as:

HC(s) = C(s)H(s) =H(s)

s2 =N

∑r=1

(Rr

s2(s−λr)+

R∗rs2(s−λ∗r )

)(A.35)

A general inverse Laplace-transform of a functionH(s)s2 is stated by equation (A.36), i.e convolving the time

response functionh(t) with a ramp functiont.

L−1

H(s)s2

=

∫ t

0−(t− τ)h(τ)dτ (A.36)

h(t) = L−1H(s) (A.37)

84

APPENDIX A. DERIVATIONS OF IMPULSE-, STEP-, AND RAMP INVARIANCE

Applying the inverse Laplace-transform to equation (A.38).

hc(t) =N

∑r=1

(∫ t

0−(t− τ)Rre

λr τ +(t− τ)R∗r eλ∗r τdτ)

=

=N

∑r=1

[Rrtλr

eλr τ− Rr

λrτeλr τ +

Rr

λ2reλr τ +

R∗r tλ∗r

eλ∗r τ− R∗rλ∗r

τeλ∗r τ +R∗rλ∗r

2 eλ∗r τ

]τ=t

0−=

=N

∑r=1

(Rr

λ2r

(eλr t −λrt−1

)+

R∗rλ∗r

2

(eλ∗r t −λ∗r t−1

))(A.38)

Sampling of the analogue time response function.

hc(nT) = TN

∑r=1

(Rr

λ2r

(eλr nT−λrnT−1

)+

R∗rλ∗r

2

(eλ∗r nT−λ∗r nT−1

))(A.39)

Z-transformation of the discrete time response function.

HC(z) = Zhc(n)=

=N

∑r=1

(Rr

λ2r

(z

z−eλr T− λrTz

(z−1)2 −z

z−1

)+

R∗rλ∗r

2

(z

z−eλ∗r T− λ∗r Tz

(z−1)2 −z

z−1

))=

=N

∑r=1

(Rr

λ2r

(1

1−eλr Tz−1− λrTz−1

(1−z−1)2 −1

1−z−1

)+

+R∗rλ∗r

2

(1

1−eλ∗r Tz−1− λ∗r Tz−1

(1−z−1)2 −1

1−z−1

))(A.40)

Compensation for the catalyzer function and simplifying the frequency response function.

H(z) =HC(z)C(z)

=

=(1−z−1)2

z−1 HC(z) =

=(1−z−1)2

z−1

N

∑r=1

(Rr

λ2r

(1

1−eλr Tz−1− λrTz−1

(1−z−1)2 −1

1−z−1

)+

+R∗rλ∗r

2

(1

1−eλ∗r Tz−1− λ∗r Tz−1

(1−z−1)2 −1

1−z−1

))(A.41)

Which could be simplified into a sum of second order sections

H(z) =N

∑r=1

br,0 +br,1z−1 +br,2z−2

1+ar,1z−1 +ar,2z−2 (A.42)

Where the coefficientsbr,0, br,1, br,2, ar,1 andar,2 for each second order section can be expressed as

br,0 =Rr

λ2r

(eλr T −1−λrT

)+

R∗rλ∗2

r

(eλ∗r T −1−λ∗rT

)(A.43)

br,1 =Rr

λ2r

(1+(λrT−1)eλr T −e(λr+λ∗r ) +(1+λrT)eλ∗r T

)+

+R∗rλ∗2

r

(1+(λ∗rT−1)eλ∗r T −e(λr+λ∗r ) +(1+λ∗rT)eλr T

)(A.44)

br,2 = −Rr

λ2r

(1+(λrT−1)eλr T

)eλ∗r T − R∗r

λ∗2r

(1+(λ∗rT−1)eλ∗r T

)eλr T (A.45)

ar,1 = −eλ∗r T −eλr T (A.46)

ar,2 = e(λr+λ∗r )T (A.47)

85

Appendix B

Modified Bootstrap Structure Detection

The MBSD is a recursive algorithm that tries to identify the structure and parameters of a time series model,such as defined by the NARMAX, ARMAX, ARMA structures for an example. The core element of the MBSDis a linear in the parameters regressor, see [3]. The bootstrap method is used in parallel to the linear regressorfor identifying false and true parameters in the regression model. False parameters are removed from the linearregressor and the algorithm repeats until convergence, and the true parameters are found.

An overview schematic of the MBSD is presented in figure B.1. The individual blocks of the overviewschematic are more thoroughly specified in the following listing.

1.) DBG - Data Block Generator The data block generator takes or reads input signals and divide them intoblocks ofN data samples.

2.) LR - Linear in the parameters regressor The linear regressor uses the least squares method for fittingoptimal regressor coefficients. Derivation of the least squares linear regression model;

x = [Ψ]copt (B.1)

e = x−x (B.2)

∇c(eTe) = 2[Ψ]T [Ψ]c−2[Ψ]Tx= 0 (B.3)

copt =([Ψ]T [Ψ]

)−1 [Ψ]Tx (B.4)

TheΨ matrix is modified in the current implementation with respect to the original implementation ofthe bootstrap method, as proposed in [3], and does only contain the following terms for an assumed

1.) DBG

N6

2.) LR

I (r)6

F -x - 3.) BS

B6

--x -

4.) SD[C(r)] -

I (r)6

-I (r+1)

Figure B.1: Overview Schematics of the Modified Bootstrap Structure Detection method. The abbreviationsof the blocks are; 1.) DBG - Data block generator, 2.)LR - Linear Regressor, 3.) BS - Bootstrapper, 4.) SD- Structure Discriminator. The signal wires are;F - The excitation force,x - The response,N - Datablock length in samples,I (r) - Valid regressor term indicator vector for recursionr, x - Linear in theparameters regressed response,B - Number of bootstrap replicas,[C(r)] - Bootstrap replicas of the regressorterms for recursionr, r - The recursion index,r ∈ (0,R). The recursion stops as the method converges;r = R,I (R−1)= I (R).

86

APPENDIX B. MODIFIED BOOTSTRAP STRUCTURE DETECTION

polynomial orderP;

[Ψ] = [[Ψx], [ΨF ]] (B.5)

[Ψx] =[∆x, . . . ,∆Nxx, · · · ,∆xP, . . . ,∆NxxP] (B.6)

[ΨF ] =[F,∆F, . . . ,∆NFF, · · · ,FP,∆FP, . . . ,∆NF FP] (B.7)

In the matrices above; when a vector is raised to the power ofp means element wise operation, such asif xT = 1,2,3 ⇒ x3T = 1,8,27 for an example. The∆ is the delay operator, such as∆3x(n) =x(n−3) for an example. It is also assumed thatx(n) = 0, x(n) = 0 andF(n) = 0 for n < 0.

3.) BS - Bootstrapper The bootstrapper uses assumptions of the linear regressor error residuals, specificallythat they are independent and identically distributed and has zero mean. This works for the least squaresmethod. The error residuals of an initial linear regression are resampled with replacement for each boot-strap replication. The resampled error residual is added to the initally regressed data and a new set ofregressor parameters are calculated. This is repeatedB times, and thus renderingB bootstrap replicas ofthe regressor parameters.

e(b) = RESAMPLE(e) (B.8)

x(b) = x+e(b) (B.9)[Ψ(b)

]=

[[Ψx(b) ], [ΨF ]

](B.10)

c(b) =([Ψ(b)]T [Ψ(b)]

)−1[Ψ(b)]Tx(b) (B.11)

[C]b,· = c(b)T (B.12)

4.) SD - Structure Discriminator The structure discriminator takes a matrix ofB bootstrap replicas of theregressor parameters. It estimates the Probability density function for each of the regressor parametersand establishes an interval of confidence for each regressor parameter. If the bounds of the interval ofconfidence for a regressor parameter surrounds zero, that parameters is regarded as false and is therebyremoved from the regression model, by update of the valid regressor term indicator vector. If zero is notwithin the bounds of the confidence interval it is regarded as a true parameter and is still in the regressionmodel.

Convergence Rate

The convergence rate of the MBSD algorithm is in high degree bound to: the number of bootstrap replicas, thedata block size, the input-/output signal memory and the degree of the polynomial of the nonlinearity.

87

Bibliography

[1] Saven EduTech AB.Noise and Vibration Analysis III. Täby Sweden, 2002.

[2] Julius S. Bendat.Nonlinear system analysis and identification from random data. John Wiley and Sons,Inc., 1990.

[3] Hans Peter Geering Esfandiar Shafai, Mikael Bianchi. detectnarmax: A graphical user interface forstructure detection of narmax models using the bootstrap method.SYSID 2003, 13th IFAC Symposiumon System Identification(13), August 2003.

[4] Michael Feldman. Non-linear system vibration analysis using hilbert transform - ii. forced vibrationanalysis method ’forevib’.Mechanical System and Signal Processing, 8(3):309–318, 3 May 1994.

[5] Michael Feldman. Non-linear free vibration identification via the hilbert transform.Journal of Sound andVibration, 208(3):475–489, 11 July 1997.

[6] Janito Vaquerio Ferreira.Dynamic Response Analysis of Structures with Nonlinear Components. PhDthesis, University of London, May.

[7] Monson H. Hayes.Statistical digital signal processing and modelling. Wiley, 1996.

[8] Peyton Z. Peebles Jr.Probability, Random Variables, and Random Signal Principles. McGraw-HillHigher Education, 4th edition, 2001.

[9] R Vaillancourt R Ashino, M Nagase. Behind and beyond the matlab ode suite.Computers and Mathe-matics with Applications, 40(4-5):491–512, 2000.

[10] Mark Richardson and Brian Schwarz. Modal parameter estimation from operating data. Sound andVibration, January 2003.

[11] Structural Dynamics Research Lab University of Cincinatti. 20-263-781: Non-linear vibrations. CourseLiterature, 27 Marsch 2000.

[12] Structural Dynamics Research Laboratory University of Cincinatti. Modal parameter estimation. CourseLiterature, 5 February 2002.

88

Documents

Identiﬁcation and Analysis of Nonlinear Systems829311/FULLTEXT01.pdfthe frequency based methods Reverse Path and a Frequency Domain Structure Selection Method (FDSSA). The time domain