APPLIED MECHANICS, PUBLICATION U72, ISSN 0349-8123

Linear System Theory in Vibration Engineering

THOMAS ABRAHAMSSON

i

PREFACE

Working with practical vibrational problems I have come to realize theimportance of linear system theory in vibrational engineering. In mechanicalvibration engineering education material, much of the linear system theory isleft unnoticed, since the focus is usually on modeling and analysis of linearor non-linear structural elements and built-up structures. The recent book1 byLeonard Meirovitch is an exception from the rule. On the other hand, muchof practical vibrational engineering is related to dynamic testing. Here theexperimental modal analysis and system identification play important roles.To understand the underlain techniques and principles on which these relies,a dose of vibrational theory is crucial.

In this treatise I have not even tried to be complete, i.e. to include and com-pare all available methods and techniques, neither in a historical sense or incurrent practice. I am restricting the presentation to those tools that I havefound to be efficient and robust for real life vibrational problems in my ownwork. Those interested in comparative studies must seek such in othersources or do experiments by themselves.

The book is aimed for those already familiar with vibrational theory. Hence,those who are not may find it hard to follow. I have tried to keep a good bal-ance, and neither over-explaining nor under-explaining matters. My strivehas been to produce a material which I would have appreciated as a post-graduate student eager to learn more on vibrations and related matters. Thefamiliarity with matrices is, of course, crucial.

1. Meirovitch L., Principles and Techniques of Vibrations, Prentice-Hall, 1997

PREFACE

ii

Theories by themselves may be interesting, but when implemented in practi-cal useful tools they may become valuable. The practical use of linear systemtheories has been strongly related to available computational resources. Theimplementation of those has, in the last ten years, been very much simplifiedby the introduction of high level computer languages such as MATLAB�.The reader of this book is strongly advised to test the methods presented herein a MATLAB or MATLAB-like environment to increase his insight.

Göteborg in May 1998Thomas Abrahamsson

Extensions to and a revision of the first edition has been made.

Göteborg in May 1999Thomas Abrahamsson

My students did not fully appreciate the matrix notation used in previous edi-tions. Therefore, I have made matrices and vectors bold-faced to distinguishthem from scalars. Hopefully this makes the reading easier.

Göteborg in March 2000Thomas Abrahamsson

PREFACE I

1 INTRODUCTION 1Linearity of systems 1

2 STATE-SPACE FORMULATION AND REALIZATIONS 3The state-space formulation 3Realizations 6Canonical and minimal realizations 8

3 TIME DOMAIN SOLUTION METHODS 11The convolution integral solution 11The state transition matrix 12Discrete time models 14Markov parameters and the Hankel matrix 16Stability of zero-order-hold time integration 17Computation of the discrete-time transition matrix 18

4 MODAL DECOMPOSITION 23Modal decomposition and the Jordan normal form 23The Gerschgorin disks for eigenvalue enclosure 26Givens’ method with Sturm sequence checking 28The Wittrick-Williams eigenvalue counting technique 32*The Lanczos’ method 34*The Arnoldi method 34

5 OBSERVABILITY AND CONTROLLABILITY 35State observability 35

State controllability 37Testing observability and controllability 39

6 GRAMIANS AND BALANCED REALIZATION 43Controllability and observability Gramians 43Transient solution bounds 45The worst-case forcing function 47A balanced realization 50

7 MODEL REDUCTION 53*Modal truncation 53State reduction by use of Gramians 53*Component mode synthesis 55

8 *STATE OBSERVABILITY 57*State observers 57*The Kalman filter 57*The modal filter 57*The calibrated modal filter 57

9 EXPERIMENTAL MODAL ANALYSIS 59Introduction 59Pretest planning 61Simple circle curve fitting 63*Complex exponential 66Mode indicator functions 66*Stabilization diagram 68Correlation indicators 68Experimental mode expansion methods 71

10 SYSTEM IDENTIFICATION 75Introduction 75

*The eigensystem realization algorithm 77A state-space sub-space method for deterministic systems 77*Recursive system identification 81

.

1

1 INTRODUCTION

1.1 Linearity of systemsTo my knowledge all physical systems are non-linear. Stressed by a sufficiently

strong stimuli, all systems react in a way that the superposition property of the linearsystems is violated. For weak stimuli however, the system’s behavior may be suchthat a linear mathematical model captures its essential characteristics. Such, so-called linear, systems will be treated in the following. The onset to nonlinear behav-ior is often sudden and drastic and should not be ignored. However, a good under-standing of linear system characteristics is a good foundation for further studies ofnon-linear phenomena. Non-linear behavior, such as chaotic motion, mode satura-tion and sub- or superharmonic resonance are thus not treated in this text.

A further sub-classification of linear systems, into time-invariant and time-vary-ing linear such, is usually made. This classification is justified, since all the solutionmethods applicable to time-invariant systems do not apply for time-varying sys-tems. Physical systems are often slowly time-varying such that a short time-scalegoverns their vibration characteristics and a different and longer time-scale governstheir time-varying properties. For a given short time range such systems can beapproximated as time-invariant. The experience from experimental vibrational stud-ies is that results can rarely be reproduced from one day to another, except undervery controlled environmental conditions. The reason is usually that temperatureand humidity, which varies slowly with time, affects the mechanical properties ofthe object under test which thus becomes time-varying with respect to vibrations.The focus in this text is on time-invariant systems.

Mathematically, a system is said to be linear if it satisfies the homogeneity andadditivity properties such that

- The response to αu(t) is αy(t), where α is a constant.

- The response to u1(t) + u2(t) is y1(t) + y2(t), where y1 is the response to u1 andy2 is the response to u2.

INTRODUCTION

2

It is these properties, also known as superposition properties, that may be exploitedso efficiently for linear systems. Also, there is a vast amount of results from numer-ical linear algebra that supports the development of methods for those. All thesehave contributed to the development in fields such as vibrational engineering, con-trol engineering, signal processing and system identification.

3

2 STATE-SPACE FORMULATION AND REALIZATIONS

2.1 The state-space formulation

In the state-space formulation, the governing linear differential equations are writtenin first order form. For integer order differential equations, this can always be made.State-space models are often fitted to experimental data. To capture the noisy char-acteristics of such data, noise models are generally included in the state-space mod-els. A common state-space model is

(2.1a,b)

Here is the state-space vector, is the excitation and is theoutput. The system order is thus n, the number of stimuli elements in the excitationvector is s, and the number of response elements in the output vector is r. The matri-ces A, B, C and D are state-space coefficient matrices, constant for time-invariantsystems. The column vectors w and v represent process and output noise respec-tively. The frequency domain steady-state counterpart to the time domain model fol-lows from the harmonic assumption and is

(2.2a,b)

Here the caret (^) notation is used for the Fourier transform quantities. A block dia-gram representation of equation (2.2a,b) is shown in figure 2.1.1.

x· t( ) Ax t( ) Bu t( ) w t( )+ +=

y t( ) Cx t( ) Du t( ) v t( )+ +=

x ℜn∈ u ℜs∈ y ℜr∈

x t( ) xest=

x s( ) sI A–[ ] 1– Bu s( ) w s( )+[ ]=

y s( ) Cx s( ) Du s( ) v s( )+ +=

STATE-SPACE FORMULATION AND REALIZATIONS

4

It is an interesting exercise to relate the coefficient matrices of the state-space modelto the coefficient matrices of a discretized linear mechanical system. The latter isoften characterized by the mass matrix M, the stiffness matrix K, the viscous damp-ing matrix V (note the un-orthodox notation, which is made here to avoid confusionwith the state-space matrix C), the gyroscopic matrix G and the circulatory matrixH. The discretized system’s governing second-order equation is

(2.3)

The quantities of interest in mechanical analyses, are often selected sets of displace-ments, velocities, accelerations or quantities that can be derived from such, e.g.stresses and strains. Let the quantities of interest be called the outputs of the system.The second-order equation (2.3) may be recast in first-order form as

(2.4)

Let the state vector be composed by the displacement q and velocities such that , then equation (2.4) can be rearranged into

(2.5)

Mq·· t( ) V G+[ ]q· t( ) K H+[ ]q t( )+ + Q t( )=

V G+[ ] MI 0

q·

q··K H+[ ] 0

0 I–qq·

+ Q0

=

q·

xT qT q· T[ ]=

+++B

D

CsI A–[ ] 1–

Figure 2.1.1. Block diagram representation of equation (2.2). Process noise will be colored, i.e. its spectrum will be distorted before it enters the out-

put, by the system’s dynamics while output noise is directly transmitted tothe output. The process noise may represent unknown environmental excita-tion of the system. The output noise may be introduced by the experimentalsystem measuring the output .

w·

v·

y·

u·

y·

v·

w·

x· V G+[ ] MI 0

K H+[ ] 00 I–

x– V G+[ ] MI 0

Q0

+=

0 I–

M 1– K H+[ ] M 1– V G+[ ]x–

0

M 1–Q+=

5

THE STATE-SPACE FORMULATION

The similarity with equation (2.1a,b) should now be obvious. Let the non-zero ele-ments of the load vector Q be collected in the excitation vector u. The rectangularmatrix Pu relates the vectors, such that . One can thus identify the state-space matrices A and B to be

(2.6a,b)

One may note that for singular mass matrices M, corresponding to massless general-ized degrees-of-freedom, the state-space matrices A and B cannot be calculatedusing equations (2.6a,b). For such degrees-of-freedom however, the first order formis already present and no transformation is needed (see problem 2.2). No furtherelaboration on such systems is made here.

Let us dwell with the model outputs for a moment. In finite element vibrationalanalysis, the outputs of the model are often given after post-processing of analysisresults. The analysis results in this case are all nodal displacements and velocities ofthe model. The analysist then specifies what quantities are of his interest and lets thepost-processor calculate these, using analysis results together with complementarymodel data. Such are for example stress and strain quantities. Often the quantities ofinterest are linearly related to the displacements and velocities given by the analysis.Such quantities are natural to include in a linear state-space model.

Displacement and velocity output elements may easily be extracted. This can bemade by letting and selection matrix operate on the state vector, which containsnodal displacement and velocities. Relative displacement or velocities, or displace-ments and velocities projected upon a different coordinate system, may by com-puted by linear transformations. Let the combined selection/transformation matricesbe called Pd and Pv. For displacement and velocity outputs yd and yv we then have

(2.7a,b)

For acceleration output elements ya, we note that the time-derivative of the state-vector holds such elements. A rectangular selection/transformation matrix Pa helpsin the extraction of acceleration output elements. One thus has

(2.8)

Q Puu=

A0 I–

M 1– K H+[ ] M 1– V G+[ ]–= B

0

M 1–Pu=

yd Pd 0[ ]x= yv 0 Pv[ ]x=

ya 0 Pa[ ]x· 0 Pa[ ] Ax Bu+[ ] 0 Pa[ ]Ax 0 Pa[ ]Bu+= = =

STATE-SPACE FORMULATION AND REALIZATIONS

6

For a combined output vector, consisting of displacement, velocity and accelerationelements we then have

(2.9)

Here it may be noted that only acceleration output has static contribution, i.e. contri-bution from the excitation without the dynamics of the system via the dynamicequation . For other output quantities which relate linearly to theabove quantities, e.g. stresses and strains, the corresponding modifications of thestate-space matrices C and D should be obvious.

2.2 Realizations

A mathematical state-space model, on the form given by equation (2.1), attemptingto mimic the behavior of a real system may be called a realization of a system. Themodel relates the input u to the output y. For a multi-input multi-output system ofsay s inputs, r outputs and n states, the coefficient matrices A, B, C and D in generalhold n2+n(r+s)+rs elements in total. However, for a given input/output relationthere are an infinite number of realizations possible. A state-space description isthus not unique. However, unique state-space descriptions can be defined if oneimpose other constraints in addition to the given input-output relation. Such state-space descriptions, of which the realization known as balanced realization probablyis the most useful, will be considered in a following chapter.

To elaborate on the non-uniqueness a little further, we give examples of how differ-ent manipulations may affect individual elements of the state-space description butdo not influence the model’s input-output relation. A first example that show thatinfinitely many realizations of a given input/output relation are possible, is given bythe change of variables x = Tz with a non-singular coordinate transformation matrixT. Setting out from equation (2.1a,b) we then have

(2.10a,b)

y t( )

yd

yv

ya

Pd 0[ ]

0 Pv[ ]

0 Pa[ ]A

x00

0 Pa[ ]Bu+ Cx t( ) Du t( )+≡= =

x· Ax Bu+=

Tz· t( ) ATz t( ) Bu t( )+=

y t( ) CTz t( ) Du t( )+=

7

REALIZATIONS

as a realization free from noise. Multiply equation (2.10a) with the inverse of T andwe receive

(2.11a,b)

which gives the corresponding frequency domain relation

(2.12)

We use the matrix inversion formula + to see that the transfer function H(s) can be written

(2.13)

Here has been introduced to simplify the expression. We note that thetransfer function is independent of the transformatin matrix. Both the original andtransformed realizations obviously yield the same input/output relation. A similaritytransformation, such as x = Tz, which preserves the eigenvalues of the system thusalso preserves its input/output relation.

As a second example we may also note that if we replace B with and C with we have a different realization but again the same input/output relation.

A third example can be set out from the useful realization called the diagonal real-ization, in which the matrix A is put on diagonal form by a suitable similarity trans-formation. This realization will be considered in more detail in the next chapter. Toshow the non-uniqueness of diagonal realizations, we multiply the matrix B with anon-singular diagonal matrix and the matrix C by the correspondinginverse . We thus study the transfer function of the realization

(2.14)

z· t( ) T 1– ATz t( ) T 1– Bu t( )+=

y t( ) CTz t( ) Du t( )+=

y CT sI T 1– AT–[ ]1–T 1– Bu H s( )u≡=

a bcd+[ ] 1– a 1– a 1– b[da 1– b–=c 1– ] 1– da 1–

H s( ) CT sI T 1– AT–[ ]1–T 1– B=

CT σI σT 1– σTT 1– A 1––[ ]1–Tσ–[ ]T 1– B=

C σI σ2 σI A 1––[ ]1–

–[ ]B=

σ 1 s⁄=

γB1 γ⁄( )C

diag γi( )diag 1 γi⁄( )

x· t( ) Ax t( ) diag γi( )Bu t( )y t( )

+Cdiag 1 γi⁄( )x t( ) Du t( )+

==

STATE-SPACE FORMULATION AND REALIZATIONS

8

in which A is diagonal, i.e. . The realization’s transfer function is

(2.15)

Again, we see that we have yet other possible realizations with the same input/out-put characteristics.

2.3 Canonical and minimal realizations

Realizations are often put on special forms for different purposes. We have seen thatthere are infinitely many realizations that give identical input-output relations. If weput additional restrictions on the realization we may make one of the possiblechoices unique, than we have created a canonical realization. Symmetric matrices Awe can always diagonalize by a similarity transformation. Thus, if we put the extrarestrictions on the realization that it should be diagonal with increasing elementsalong the diagonal, we have defined one canonical realization. Other canonical real-izations are the controllability and observability canonical forms, see Kailath[2.1] formore information.

A minimal realization is such that it has the smallest number of state-variables (thesmallest order) among all realizations having the same transfer function. Let usillustrate by a simple example. We study the single-input single-output diagonalrealization. Its transfer function H(s) is

(2.16)

We note that if two diagonal elements are equal, i.e. , the correspondingtwo terms may be represented by

(2.17)

We thus note that the series expansion, equation (2.16), may be reduced to an equiv-alent lower order system with state variables. This is the so-called minimalrealization.

A diag ai( )=

H s( ) C diag 1 γi⁄( ) sI diag ai( )–[ ] 1–diag γi( )B

C sI diag an( )–[ ] 1– B

=

=

H s( ) c sI diag σi( )–[ ] 1– bcibi

s σi–-------------

i 1=

n

�= =

σi σj=

cibis σi–-------------

cjbjs σj–-------------+

cibi cjbj+

s σi–------------------------

ci˜ bi

˜

s σi–-------------≡=

n 1–

9

CANONICAL AND MINIMAL REALIZATIONS

PROBLEMS

2.1 Write down the state-space matrices A, B, C and D of the system in the fig-ure. The output vector y is here composed of the acceleration of thesecond carriage and the support reaction R(t) i.e.

2.2 The three-degree-of-freedom viscously damped system in the figure is seento have one massless degree-of-freedom and therefore a singular mass matrix.Write down a state-space model of the system. Input and outputs are as inproblem 2.1.

BIBLIOGRAPHY

2.1 Kailath T., Linear Systems, Prentice-Hall, 1980

q··2 t( )

Problem figures

k1 k2 k3

m1 m3m2

Problem 2.1

q1 t( ) Q1 0≡, q3 t( ) Q3 0≠,q2 t( ) Q2 0≡,

R(t)

yT q··2 R[ ]=

Problem 2.2k1 k2 k3

m1 m3m2 = 0

q1 t( ) Q1 0≡, q3 t( ) Q3 0≠,q2 t( ) Q2 0≡,

R(t)

c1 c3c2

STATE-SPACE FORMULATION AND REALIZATIONS

10

11

3 TIME DOMAIN SOLUTION METHODS

3.1 The convolution integral solutionThe response to an arbitrary excitation may be obtained by utilization of the super-position principle. One can imagine the excitation as a sequence of impulse excita-tions, each giving contribution to the system response. By integrating thecontributions, the total response is obtained. Chopping the excitation into impulsesis trivial, but obtaining the system’s response to these impulses is more involved. Bymodal decomposition methods, the generally coupled system equations of highorder, may be reduced to first order de-coupled equations. The unit impulseresponse g(t) of a linear first-order system is well-known fromcalculus and is

, (3.1a)

, (3.1b)

Let the excitation impulse over an infinitesimal time increment dτ be u(τ)dτ, thenthe response contribution to this impulse is

(3.2)

Assuming that the initial conditions are homogeneous, the integrated response is

(3.3)

x· t( ) ax t( )+ u t( )=

g t( ) e at–= t 0>

g t( ) 0= t 0<

dx t τ,( ) g t τ–( )u τ( )dτ=

x t( ) g t τ–( )u τ( )dτ 0

∞

�=

TIME DOMAIN SOLUTION METHODS

12

But since is equal to zero for t - τ < 0, or equally for τ > t, the upper boundof the integral may be replaced by t. The superposition integral, or convolutionintegral, formulation for the response thus is

(3.4)

3.2 The state transition matrixSimilarly to the scalar case treated above, we now seek a matrix integral solution tothe state-space initial-value problem

, (3.5)

which can be shown to be

(3.6)

Here Φ(t) is the state-transition matrix, which is . To proof this, we firstconsider the general linear scalar initial-value problem

, (3.7)

Its homogeneous solution is well known from calculus[3.1] and is

, (3.8)

Above, we have given the time domain solution, equation (3.6), to the time-invariant multivariate problem. A convolution integral solution cannot be found forthe linear time-varying multivariate problem. Since this is possible for the scalarcase it may come as a surprise and it may be interesting to know why.

As in the scalar case, it is natural to try a (matrix) integral solution on a similar formfor the time-variant homogeneous matrix initial-value problem ,

as

(3.9)

where the exponential is defined by the series

g t τ–( )

x t( ) g t τ–( )u τ( )dτ0

t

�=

x· Ax Bu+= x 0( ) x0=

x t( ) Φ t( )x0 Φ t τ–( )Bu τ( ) τd0

t

�+=

Φ t( ) eAt=

x· t( ) a t( )x t( ) b t( )u t( )+= x 0( ) x0=

x t( ) x0ea τ( )dτ

0 t�

= t 0≥

x· t( ) A t( )x t( )=x 0( ) x0=

x t( ) x0 eA τ( )dτ

0 t�

=

13

THE STATE TRANSITION MATRIX

(3.10)

However, one notes that since

(3.11)

is not generally equal to

(3.12)

unless the matrices A(t) and commute, i.e. that =. In special cases, such as for the time-invariant or diagonal

matrices A, the commutative property is seen to hold. We have thus found a distinctproperty which distinguish the time-variant matrix differential equation from thetime-invariant. This distinction rules out closed form solutions along the suggestedroute for the time-invariant case.

We continue with the time-invariant case and utilize the commutative propertywhen appropriate. We use the transition matrix as defined by

(3.13)

which we require to obey the matrix differential equation

, (3.14)

The solution to the matrix differential equation is

(3.15)

We have already observed that the matrices A and commute, such that and therefore equation (3.14) may also be written as

(3.16)

eA τ( )dτ

0 t�

I A τ( ) τd0

t

�12--- A σ( ) σd

0

t

� A τ( ) τd0

t

� …+ + +=

ddt----- e

A τ( )dτ0 t � A t( ) 1

2---A t( ) A τ( ) τd

0

t

�12---[ Aσ σd

0

t

� ]A t( )+ …+ +=

A t( )[I 12--- Aτ τd

0

t

� …+ + ]

A τ( )dτ0 t� A t( ) A τ( )dτ

0 t�

A τ( )dτ0 t�[ ] A t( )

Φ t( )

Φ t( ) eA τ( )dτ

0 t�

eAt I tA t2

2!-----A2 …+ + += = =

Φ· 1–

t( ) AΦ 1– t( )–= Φ 1– 0( ) I=

Φ 1– t( ) e At– Φ 1– 0( ) e At– I tA– t2

2!-----A2 …–+≡= =

Φ 1–

AΦ 1– Φ 1– A=

Φ 1–.

Φ– 1– t( ) A=

TIME DOMAIN SOLUTION METHODS

14

Now let us pre-multiply the state-space equation with to obtain

(3.17)

or by using that we have

(3.18)

But in view of equation (3.16) this reduces to

(3.19)

which can be integrated readily. Taking into consideration that wehave

(3.20)

Pre-multiply this equation with the state-space transition matrix , we obtain thedesired response

(3.21)

where the state transition matrix property has been utilized. We have thusshowed that the solution to the problem is indeed the solution given by equation(3.6).

3.3 Discrete time modelsThe time domain transient solutions of the scalar and matrix differential equationshave been given in convolution integral form. For practical purpose, the use of theseis limited, unless implemented in a computational scheme. The process of deriving a

Φ 1–

Φ 1– x· Φ 1– Ax Φ 1– Bu+=

ddt----- Φ 1– x[ ] Φ

· 1–x Φ 1– x·+=

ddt----- Φ 1– x[ ] Φ

· 1–x– Φ 1– Ax Φ 1– Bu+=

ddt----- Φ 1– x[ ] Φ 1– Bu=

Φ 1– 0( ) I=

Φ 1– t( )x t( ) Φ 1– 0( )x 0( ) Φ 1– τ( )Bu τ( ) τd0

t

�+=

x0 Φ 1– τ( )Bu τ( ) τd0

t

�+=

Φ

x t( ) Φ t( )x0 Φ t( ) Φ 1– τ( )Bu τ( ) τd0

t

�+=

Φ t( )x0 Φ t τ–( )Bu τ( ) τd0

t

�+=

Φ eAt=

15

DISCRETE TIME MODELS

computational scheme is the process of time-discretization, necessitated by presentday’s computer architectures. The convolution solution conveniently lends itself todiscretization. Consider the solutions at two consequent discrete times kT and kT+T,where 1/T is often referred to as the sampling frequency

(3.22a)

(3.22b)

It is interesting to note that, for a sample interval in which no excitation take place,the state xk+1 can be computed without approximation, using the discrete timetransition matrix and the previous state xk independently of the samplingfrequency.

Using so-called zero-order-hold assumption, i.e. the excitation is assumed to beconstant over the entire sampling period, we have for the excitation term

(3.23)

Moreover, by the change of variables , we have

(3.24)

xk eAkTx0 eA kT τ–( )Bu τ( ) τd0

kT

�+=

xk 1+ eA kT T+( )x0 eA kT T τ–+( )Bu τ( ) τd0

kT T+

�+=

eAT eAkTx0 eA kT τ–( )Bu τ( ) τd0

kT

�+ eA kT T τ–+( )Bu τ( ) τdkT

kT T+

�+=

eATxk eA kT T τ–+( )Bu τ( ) τdkT

kT T+

�+=

A eAT=

eA kT T τ–+( )Bu τ( ) τdkT

kT T+

� eA kT T τ–+( ) τdkT

kT T+

� Buk≈

kT T τ–+ τ'=

eA kT T τ–+( ) τdkT

kT T+

� eAτ'τ'dT

0

�– eAτ'τ'd0

T

�= =

I τ'A τ'2

2!------A2 …+ + +� �

� � τ'd0

T

� TI T2

2!------A T3

3!------A2 …+ + += =

A 1– TA T2

2!------A2 T3

3!------A3 …+ + +� �

� � A 1– eAT I–( )= =

TIME DOMAIN SOLUTION METHODS

16

We thus have the explicit time stepping algorithm

(3.25a,b)

with

, (3.26)

We note that we are required to provide the matrix which has the seriesexpansion of equation (3.13). In a following chapter (chapter 3.6) approximatemethods for computing this are presented.

3.4 Markov parameters and the Hankel matrixAfter studying the discrete-time realization it is natural to introduce the so-calledMarkov parameters. They will appear later in the context of system identification,where they play an important role. The Markov parameters are defined as theimpulse response matrices of the system. Let us see how they relate to the state-space realization triple {A,B,C}. To this end we study the initial-value impulseresponse problem

(3.27)

Here is a vector of ones for and vectors of zeros for all , i.e. a loadsequence of initial unitary impulses. For we have

,

or generally . Thus the impulse response, or Markov parameter,matrices are for . An important matrix connected tothe Markov parameters is

xk 1+ Axk Buk+= yk Cxk Duk+=

A eAT= B A 1– eAT I–( )B=

eAT

xk 1+ Axk Buk+= x0 0= uk δk=

yk Cxk=

δk k 0= k 0>k 1 2 3, ,=

k 1 := x1 Bδ0= y1 Cx1 CBδ0 CA0Bδ0= = =

k 2 := x2 Ax1 ABδ0= = y2 Cx2 CA1ˆBδ0= =

k 3 := x3 Ax2 A2Bδ0= = y3 Cx3 CA

2Bδ0= =

yk CAk 1– Bδ0=hi hi CAk 1– B= i 1 2 …, ,=

17

STABILITY OF ZERO-ORDER-HOLD TIME INTEGRATION

(3.28)

Matrices constant along the anti-diagonals, such as this, are often called Hankelmatrices. For the single-input single-output case, in which are scalars, H is astandard Hankel matrix. In the case of multi-input multi-output systems, in whichthe Markov parameter matrices form blocks constant along the anti-diagonal, thematrix H is a so-called block Hankel matrix.

3.5 Stability of zero-order-hold time integrationBefore addressing the stability aspects of the time-stepping algorithm, we considerthe diagonalization of the continuous-time realization

(3.29a,b)

By a similarity transformation, provided that the matrix A is not deficient[3.2,3.3], thesystem may be fully decoupled by a suitable transformation x = Pz, and thus

(3.30a,b)

or equivalently

(3.31a,b)

(3.31c,d)

The discrete-time counterpart is

(3.32a,b)

where

(3.33)

H

hi hi 1+ … hi j+

hi 1+ hi 2+ … hi j 1+ +

. . … .hi j+ hi j 1+ + … hi 2j+

=

hi

x· Ax Bu , y+ Cx Du+= =

z· P 1– APz P 1– Bu , y+ CPz Du+= =

z· diag σc( )z Bu , y+ Cz Du+= =

B P 1– B , C CP= =

zk 1+ Azk Buk , yk 1++ Czk 1+ Duk 1++= =

A ediag σcT( ) I diag σcT( ) 12!-----diag2 σcT( ) …+ + += =

diag (1 σcT 12!----- σcT( )2 …)+ + + diag (e

σcT)= =

TIME DOMAIN SOLUTION METHODS

18

The eigenvalues of the discrete-time state-transition matrix are thus . For astable continuous-time system the real part of the eigenvalues are negative orzero. Therefore, the stable system’s discrete time eigenvalues are bounded by theunit circle in the complex eigenvalue plane (see figure 3.1). To see this we note that

The argument of the complex eigenvalue may be any angle, since may beany real number, large or small, positive or negative.

We note that for an undamped system, i.e. , the eigenvalues of thediscrete-time system are all located on the unit circle since then .

One should note that although the transformation of eigenvalues from continuous-time to discrete-time is unique, the reverse transformation is not so. One may alsonote that, for any integer number n, the continuous-time eigenvalues

all correspond to the same discrete-time eigenvalue .Thus no unique continuous-time eigenvalue correspond to the discrete-timeeigenvalue . However, for small time steps T such that thetransformation is unique.

3.6 Computation of the discrete-time transition matrixOne notes that the algorithm (3.32a,b) is unconditionally stable, since the magnitudeof the discrete-time eigenvalues is less than or equal to one. For a homogenous

A eσcT

σc

Figure 3.5.1 Projection of the stable continuous-time system’s eigenvalues,all on the left half-plane. The unit disc embrace the eigenvalues of thediscrete-time eigenvalues.

ℜ σc( )

ℑ σc( )

ℜ σd( )

ℑ σd( )

1

eσcT

eℜ σcT( ) iℑ σcT( )+

eℜ σcT( )

eiℑ σcT( )

= =

eℜ σcT( )

eiℑ σcT( )

≤ eℜ σcT( )

1 , ℜ σcT( ) 0≤≤=

ℑ σcT( )

ℜ σcT( ) 0=eℜ σcT( ) 1=

σc σc0 2nπi/T+= eσc0T

eσc0T ℜ σcT( ) π<

19

COMPUTATION OF THE DISCRETE-TIME TRANSITION MATRIX

state-space stable system with non-homogenous initial condition, i.e. but, the solution is bounded. The discrete-time homogenous solution is also

exact, provided that the state-transition matrix is computed without truncation. Thealgorithm is thus free from algorithmic damping. For calculation, an obviousapproach is to follow the route given in the previous chapter, e.g. first diagonalizethe system by a similarity transformation and then compute diagonal elements of thestate-transition matrix by complex exponential functions. This can be made withhigh accuracy, but will be costly. Another method is to use the series expansion ofthe state-transition matrix and truncate the series at an appropriate number of terms.The series expansion may then be calculated by a number of matrix multiplication.Using Horner’s rule we have for the series truncated at m terms

(3.34)

A question then naturally arises. How many terms are sufficient for an accuratecomputation of the transition matrix? We will not answer that question here but justaddress the stability aspect of the time-stepping algorithm with a truncatedtransition matrix. Conditions for algorithm stability for an undamped system will beestablished.

For a first-order truncated series we have

(3.35)

We now seek the stability bound for the time-stepping algorithm based upon thistruncation, i.e. we seek the eigenvalues of . We assume, for simplicity but withloss of generality1, that we have a diagonal continuous-time realization matrix

(3.36)

The discrete-time eigenvalues are thus . We know that the stabilityboundary is at the unit circle, i.e. when . Separating the real and imaginarypart of the discrete-time eigenvalues we then have for stable eigenvalues

(3.37)

1. Deficient matrices cannot be put on this form by similarity transformation

u 0≡x0 0≠

A

A I TA I TA2

------- I TA3

------- I … TAm 1–------------- I TA

m-------+� �

� �…+ +� �� �+� �

� �+� �� �+=

A I TA+=

A

A diag σc( )=

A I TA+ diag 1 σcT+( )= =

σd 1 σcT+=σd 1=

1 ℜ σcT( ) iℑ σcT( )+ + 1≤

TIME DOMAIN SOLUTION METHODS

20

or equivalently

(3.38)

This defines a stability disc in the complex eigenvalue plane, the disc having unitaryradius and center at . This is illustrated in figure3.6.1. It is interesting to see that for an undamped system, i.e. , thealgorithm is unconditionally unstable. This is usually not acceptable, particularlynot for long simulation times.

We continue the stability investigation, now for an algorithm based upon a secondorder series truncation. We then have

(3.39)

with discrete-time eigenvalues . Again, their magnitude should be less than orequal to unity. It is straightforward to express this condition into a conditioninvolving the real and imaginary part of the continuous-time eigenvalues as

(3.40)

The stability region associated with (3.40) is shown in figure 3.6.1. Again it can beseen that the truncated series based algorithm is unstable for undamped systems.

At this moment we may wonder how much further the series has to be expandedbefore we receive an algorithm which is also stable for undamped systems. Ofcourse, we wish to use the shortest possible expansion that give sufficient stabilityand accuracy. We know that an algorithm based on the full series expansion isunconditionally stable and exact, such that no algorithmic damping is present. Thisknowledge encourage us to proceed the expansion further, this time using a thirdorder expansion. We then have

(3.41)

The stability analysis becomes increasingly more involved as the expansionproceeds, but the basic principle remains. We do not carry out the analysis here but

1 ℜ σcT( )+( )2 ℑ2 σcT( ) 1≤+

ℜ σcT( ) ℑ σcT( ),[ ] 1– 0,[ ]=ℜ σcT( ) 0=

A I TA 12---T2A2+ + diag (1 σcT 1

2---σc

2T2+ + ) diag σd( )≡= =

σd

8ℜ σcT( ) 8ℜ2 σcT( ) 4ℜ3 σcT( ) 4ℜ σcT( )ℑ2 σcT( )+ + +

ℜ+ 4 σcT( ) 2ℜ2 σcT( )ℑ2 σcT( ) ℑ4 σcT( )+ + 0≤

A diag (1 σcT 12---σc

2T2 16---σc

3T3+ + + ) diag σd( )≡=

21

COMPUTATION OF THE DISCRETE-TIME TRANSITION MATRIX

show the resulting stability bounds in figure 3.6.1. It is seen from the figure that thestability region now contains part of the imaginary axis. Thus the algorithm is alsostable for undamped systems. The stability condition is that . Asimilar analysis for a fourth order series expansion algorithm give the stabilitycondition . Interestingly, for a further expansion to fifth and sixthorder we again looses the stability in the undamped case (the interested reader withenough time can prove this as outlined above). A fourth order expansion seems tobe a good compromise between speed, accuracy and stability. In that case we have

(3.42)

The corresponding discrete-time zero-order-hold matrix is

(3.43)

−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 30.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0−1

1

(a)

(b)

2-2

-1

n = 1

n = 2

n = 3

n = 4

0

σd

ℑ σcT( )

1ℜ σcT( )

ℑ σcT( )

n = 1

n = 2

n = 3

n = 4

ℑ σcT( ) 3≤

ℑ σcT( ) 8≤

A I TA I TA2

------- I TA3

------- I TA4

-------+� �� �+� �

� �+� �� �+≈

B

B A 1– eAT I–( )B A 1– A I–( )B≈ T I TA2

------- I TA3

------- I TA4

-------+� �� �+� �

� �+� �� �B= =

Figure 3.6.1. (a) Stability boundaries for increasing order (n) of the expansion ofthe state transition matrix. (b) Magnitude of discrete time eigenvalue versusimaginary part of normalized continuous time eigenvalue.

TIME DOMAIN SOLUTION METHODS

22

Since the stability border is not the entire left half-plane in the complex domain,we should take care not to use too long time steps. Generally, for a third orderexpansion, it is suggested that T should be selected as

(3.44)

where is the continuous-time eigenvalue of largest imaginary part and is theCouchy number which is recommended to be in the range of [0.95,0.98]. However,if the load varies significantly over the sample period, the zero-order holdassumption cease to be valid and shorter time steps are required. It may also happenthat one require a higher time resolution for other purposes.

PROBLEMS

3.1 Make a contour plot of the magnitude of a discrete-time eigenvalue,corresponding to a continuous-time eigenvalue , of a third-order expansion.In particular draw the contour for the unit magnitude and mark the discrete-time integration algorithm’s stable and unstable regions. Use and

as the plot axes.

3.2 Use numerical simulation to calculate the output vector yk due to problem 2.1.Use a half-sine pulse of unit magnitude as the input u. Let the duration of thepulse be half of the period of the system’s highest natural frequency. Let m1 =m2 = m3 = 1 kg, k1 = 1000 N/m and k2 = k3 = 10 N/m. Simulate for asufficiently long time with increasing number of terms in the expansion of

. Let T be one quarter of the duration of the pulse. Note the algorithmicdamping properties. Is it positive or negative for different number of terms?

BIBLIOGRAPHY

3.1. Hildebrand F.B., Advanced Calculus for Applications, Prentice-Hall, 2ndedition, 1976

3.2. Wilkinson J.H., The Algebraic Eigenvalue Problem, Clarendon Press, 1965

3.3. Golub G.H. and van Loan C.F., Matrix Computations, John HopkinsUniversity Press, 3rd edition, 1996

σc

T 3κ ℑ σc( )⁄=

σc κ

σ

ℜ σT( )ℑ σT( )

eAT

23

4 MODAL DECOMPOSITION

4.1 Modal decomposition and the Jordan normal formLet us assume that the solution to the homogeneous state-space equation isin the form

(4.1)

where and are a constant and a vector, respectively. Introduce the solution(4.1) in and divide through by , we obtain

(4.2)

Equation (4.2) represents a set of homogeneous algebraic equations and is known asthe algebraic eigenvalue problem. The problem can be stated as follows: determinethe values of the parameter for which equation (4.2) has a non-trivial solution.Recalling that A is an matrix, the eigenvalue problem can be satisfied in n dif-ferent ways, namely

, (4.3)

where and ( ) are the eigenvalues and eigenvectors of A,respectively, both complex in general. These eigenvalues are the roots of the charac-teristic polynomial . Another eigenvalue problem of interest isassociated with the matrix and is known as the adjoint eigenvalue problemdefined by

, (4.4)

x· Ax=

x t( ) eσtρ=

σ ρx· Ax= eσt

Aρ σρ=

σn n×

Aρi σiρi= i 1 2 … n, , ,=

σi ρi i 1 2 … n, , ,=

det A σI–( ) 0=AT

ATλj σjλj= j 1 2 … n, , ,=

MODAL DECOMPOSITION

24

Because det(A) = det(AT), the eigenvalues of AT and A are the same. On the otherhand, the eigenvectors ( ) are known as the adjoint of the set ofeigenvectors ( ). Equation (4.4) can also be written in the form

(4.5)

Because of their position relative to A, are called right eigenvectors of A and are known as the left eigenvectors of A.

If we pre-multiply equation (4.3) with and post-multiply equation (4.5) with we have

(4.6a,b)

or by subtracting the two

(4.7)

If the eigenvalues are distinct, i.e. , we note that the right and left eigenvec-tors corresponding to different eigenvalues are bi-orthogonal, i.e. . Wealso note that they are bi-orthogonal with respect to A since for .

It is normal practice to normalize the eigenvectors so as to satisfy , inwhich case the eigenvectors are bi-orthonormal, as expressed by

, , (4.8)

where is the Kronecker delta. Next let us introduce the modal matrices ofright and left eigenvectors1

, (4.9)

as well as the matrix of eigenvalues

, (4.10)

1. Note that and are the Greek capital letter counterparts to and

λj j 1 2 … n, , ,=ρi i 1 2 … n, , ,=

λjTA σjλj

T= j 1 2 … n, , ,=

ρi λj

λjT ρi

λjTAρi σiλj

Tρi= λjTAρi σjλj

Tρi=

σi σj–( )λjTρi 0=

σi σj≠λj

Tρi 0=λj

TAρi σiλjTρi 0= =

i j≠

λiTρi 1=

λjTρi δij= λj

TAρi σiδij= i 1 2 … n, , ,=

δij n n×

P Λ ρ λ

P ρ1 ρ2 … ρn= Λ λ1 λ2 … λn=

n n×

Σ diag ρi( )= i 1 2 … n, , ,=

25

MODAL DECOMPOSITION AND THE JORDAN NORMAL FORM

Then equation (4.8) can be written in compact matrix form

(4.11a,b)

Equation (4.11b) is seen to constitute a similarity transformation, i.e. A and are similar matrices, meaning thay share the same eigenspec-

trum.

In a general case, in which the eigenvalues of A are not distinct but appear in clus-ters of multiple eigenvalues, the diagonal form may not be obtainable. For such, so-called deficient matrices, a similar matrix of minimum number of non-zero ele-ments is in block-diagonal form. Such matrices are known as the Jordan normalform. This edition of the book does not treat those general forms. Let us instead con-sider the case of non-deficient matrices A, for which diagonalization is possible.

We note that the right and left modal matrices effectively decouples the state equa-tions. With the similarity transformation (and ) and in accor-dance with equations (2.11a,b), we have

(4.12a,b)

or

(4.13a,b)

We thus have an n-dimensional set of uncoupled first order, generally complex-val-ued, differential equations

(4.14a,b)

Since A is a real-valued matrix, two possibilities exist for the solution to the eigen-value problem with the characteristic equation det( ) = 0. Either are the equa-tion’s roots real, or do they show up as complex conjugate pairs. We introduce thecomplex conjugate operator notation ( ) in the following. We note, using rulesfrom complex algebra, that if one eigensolution { } has been found for

(4.15a,b)

ΛTP I= ΛTAP Σ=

ΛTAP P 1– AP=

x Pz= P 1– ΛT=

ΛTPz· ΛTAPz ΛTBu+= y CPz Du+=

z· Σz ΛTBu+= y CPz Du+=

z· i σizi λiTBu σizi biu+≡+= i 1 2 … n, , ,=

y t( ) ρizi t( )i 1=

n

� Du t( )+=

A σiI–

σi λi ρi, ,

Aρi σiρi= λj

TA σjλjT=

MODAL DECOMPOSITION

26

then also the complex conjugate solution { } = { } is an eigensolu-tion. For real roots also the right and left eigenvectors are real, and the corre-sponding differential equation (4.14) is thus real-valued. If the roots are complex,the corresponding pair of conjugate differential equations are

(4.16a,b)

and thus give complex conjugate solution pairs, i.e. . The contribution to theoutput of those complex conjugate states are

(4.17)

since

(4.18)

It thus suffice to calculate either of the two complex conjugate solutions for obtain-ing the contribution to the output, with potential of halving the computational effort.

4.2 The Gerschgorin disks for eigenvalue enclosureUnder certain circumstances, a theorem of Gerschgorin, provides easily obtainableestimations on the eigenvalues of the general matrix A. For heavily banded,and ultimately the tridiagonal, diagonally dominant matrices A it often gives closebounds at a very modest computational cost as will be obviously seen.

Let us consider the eigenvalue problem in index notation

, (4.19)

Then, assuming that is the component of the vector with the largest modulus, (j = 1,2,..., n), we let i = k in equation (4.19) and write

(4.20)

σj λj ρj, , σi λi ρi, ,σj

z·i σizi biu+=

z· j σjzj bju+ σizi biu+= =

zj zi=

y t( ) cizi t( ) cjzj t( )+ cizi t( ) cizi t( )+ 2ℜ cizi t( ){ }= = =

cj Cρj Cρi ci= = =

n n×

Aρ σρ=

aijρjj 1=

n

� σρi= i 1 2 … n, , ,=

ρk ρρk max ρj=

σ akk–( )ρk akjρjj 1=j k≠

n

�=

27

THE GERSCHGORIN DISKS FOR EIGENVALUE ENCLOSURE

But since

(4.21)

after dividing through by ,we may obtain

(4.22)

First, we observe that represents the distance from the point in thecomplex plane to the eigenvalue , so that the inequality (4.22) may be interpretedgeometrically as a circle in the complex plane with center at and radius (seefigure 4.2.1). Then, as equation (4.19) admits n solutions, we let k = 1,2,...,n andexpress the inequality in the form of Gerschgorin’s theorem: Every eigenvalue of thematrix A lies in at least one of the circular disks with centers at akk and radii rk.These disks are often referred to as the Gerschgorin disks.

σ akk– ρk akj ρjj 1=j k≠

n

� ρk akjj 1=j k≠

n

�≤ ≤

ρk

σ akk– akjj 1=j k≠

n

� rk≡≤

σ akk– akkσ

akk rk

rk

Figure 4.2.1. A Gerschgorin disk in the complex -plane. A possible location of aneigenvalue is indicated by a filled circle.

σσ

ℑ σ( )

ℑ akk( )

ℜ akk( )

ℜ σ( )

MODAL DECOMPOSITION

28

It may be noted that for real symmetric matrices A, the disks in the complex -plane collapse to segments along the real -axis, see figure 4.3.2. Also, we note forthe tridiagonal matrix A = T, that the sum in the inequality (4.22) involves only twoterms and that in this case

(4.23)

Therefore, for diagonally dominant, i.e. , we may get avery close bounding of the eigenvalue.

4.3 Givens’ method with Sturm sequence checkingGivens’ method is a very efficient technique for computing the eigenvalues of a realsymmetric tridiagonal positive definite matrix T. It do so by finding the roots of thecharacteristic polynomial without actually requiring the polyno-mial explicitely. As modern methods for computing eigenvalues in a narrow spec-trum of large eigenvalue problems are often based on tridiagonalization such as theLanczos’ method, such methods can be used in conjunction with Givens’ method.The Givens’ method is versatile in that respect it admits the computation of individ-ual eigenvalues, as long as their general location is known. This location may, onthe other hand, be efficiently calculated by using the Gerschgorin method with par-ticularly good approximations for diagonally dominant tridiagonal matrices whichare often obtained as the result of tridiagonalization of real symmetric matrices.

σσ

0

σ

a11 a22

a11-r1 a11+r1 a22+r2a22-r2

σ2σ1

Figure 4.3.2. The Gerschgorin disks collapsed into segments along the -axis asso-ciated to a real symmetric matrix A.

σ

σ tkk– rk≤ tk k 1–, tk k 1+,+=

tkk tk k 1–, tk k 1+,+»

det T λI–( ) 0=

29

GIVENS’ METHOD WITH STURM SEQUENCE CHECKING

The characteristic determinant associated to the tridiagonal symmetric matrix T hasthe form

(4.24)

We note that if any ( ) the problem of obtaining the determinant can bebroken down to smaller sub-problems, each having their own characteristic polyno-mials with roots common with the original problem. Without loss of generality wemay therefor assume that all . Denoting by the principle minor deter-minant of i:th order of the matrix , and with , it can be shown byinduction that

(4.25a)

, (4.25b)

Some important properties of the polynomials , out of which of course is the characteristic polynomial, may be established without actual knowledge oftheir explicit expressions.

Let us consider an interval on the real -axis where a and b are not rootsof any polynomial . The first step of Givens’ method is the determination ofthe number of roots of the characteristic polynomial lying in the interval [a,b]. Toget this we exploit the following properties of the principal determinantpolynomials:

I. If for some , then and are non-zeroand of opposite signs.

II. As passes through a zero of , the quotient / changessign from positive to negative.

det T σI–( )

t11 σ– t21 0 … 0 0

t21 t22 σ– t32 … 0 0

0 t32 t33 σ– … 0 0

… … … … … …0 0 0 … tn 1,n– 1– σ– tn,n 1–

0 0 0 … tn,n 1– tnn σ–

=

tij 0= i j≠

tij 0≠ pi σ( )T σI– p0 1≡

p1 σ( ) t11 σ–=

pi tii σ–( )pi 1– σ( ) ti,i 1–2 pi 2– σ( )–= i 2 3 … n, , ,=

pi σ( ) pn σ( )

a σ b< < σpi σ( )

pi 1– µ( ) 0= σ µ= pi µ( ) pi 2– µ( )

λ pn σ( ) pn σ( ) pn 1– σ( )

MODAL DECOMPOSITION

30

To demonstrate statement I, we let in equation (4.25b), assume that and obtain

(4.26)

If we further assume that is equal to zero, then according to equation (4.25b) must also be zero, so that three consequtive polynomials in the sequence

are zero. Then we must conclude from equation (4.25b) that = =... = = 0, which is a contradiction since = 1 by definition. Hence, if

, then and , so that we may conclude fromequation (4.26) that and must have opposite signs.

Further, to prove statement II, we use Rayleigh’s theorem that states that the eigen-values of T never decreases when subjected to a constraint and that their upperbounds are set by the sequence of eigenvalues of the unconstrained matrix. First, wedenote the roots of the characteristic polynomial by andassume that they are ordered so as to satisfy . Moreover, thepolynomial represents the determinant of the matrix obtained by con-straining by striking out its last row and column. We denote the roots of

by the ordered sequence . Then, according to Rayleigh’stheorem, the two sets of eigenvalues satisfies the inequalities

(4.27)

A typical plot of and is shown in figure 4.3.1, in which the verticallines through separate regions in which the ratios are of opposite sign. Note that, since the matrix T is positive definite, both

σ µ=pi 1– µ( ) 0=

pi µ( ) ti,i 1–2 pi 2– µ( )–=

pi µ( )pi 2– µ( )

pi 3– µ( ) pi 4– µ( )p0 µ( ) p0 µ( )

pi 1– µ( ) 0= pi µ( ) 0≠ pi 2– µ( ) 0≠pi µ( ) pi 2– µ( )

pn σ( ) σ1 σ2 … σn, , ,σ1 σ2 … σn< < <

pn 1– σ( )T σI–

pn 1– σ( ) σ'1 σ'2 … σ'n, , ,

σ1 σ'1 σ2 σ'2 σ3 … σn 1–< < < σ'n 1–< < < σn< <

pn σ( ) pn 1– σ( )

Figure 4.3.1. Regions with signs of the ratio . It is seen that thesign changes from positive to negative when passes an eigenvalue

pn σ( ) pn 1– σ( )⁄σ σi

0

0

σ

σ

+ - + + + +- - - -

pn σ( )

pn 1– σ( )

σ1 σ2 σ3 σ4 σ5

σ'1 σ'2 σ'3 σ'4

σ1 σ'1 σ2 …, , , pn σ( ) pn 1– σ( )⁄pn 0( )

31

GIVENS’ METHOD WITH STURM SEQUENCE CHECKING

and are positive. It is clear from the figure, as passes through the roots, the sign of changes from positive to negative. It

follows that the sequence of polynomials fulfilles the bothstatements. A sequence of polynomials possessing these characteristics is known asa Sturm sequence.

Now we are in the position to consider Sturm’s theorem which state that: If the poly-nomials represent a Sturm sequence on the interval [a,b]and if denotes the number of sign changes in the consequtive sequence ofnumbers , then the number of roots of the polynomial

in the interval [a,b] is equal to .

Sturm’s theorem can be proved by induction. To do so, let us assume that the num-ber of sign shanges in the sequence of numbers isequal to the number of roots of corresponding to . As an example, weshow the sequence of five polynomials in figure 4.3.2. Forthe particular value of shown, there are two sign changes in the sequence of num-bers and there are exactly two roots, and , of the

pn 1– 0( ) λσ1 σ2 … σn, , , pn σ( ) pn 1– σ( )⁄

p1 σ( ) p2 σ( ) … pn σ( ), , ,

p0 σ( ) p1 σ( ) … pn σ( ), , ,s µ( )

p0 µ( ) p1 µ( ) … pn µ( ), , ,pn σ( ) s b( ) s a( )–

s µ( ) p0 µ( ) p1 µ( ) … pn µ( ), , ,pn σ( ) σ µ<

p0 σ( ) p1 σ( ) … p4 σ( ), , ,

Figure 4.3.2. Plots of the polynomials showing the signchanges for .

p0 µ( ) p1 µ( ) … p4 µ( ), , ,σ µ=

σ

µ

First sign change

Second sign change

0

+

+

-

+

+

Two roots < µ

p0 σ( )

p1 σ( )

p2 σ( )

p3 σ( )

p4 σ( )σ1 σ2

σ3

µp0 µ( ) p1 µ( ) … p4 µ( ), , , σ1 σ2

MODAL DECOMPOSITION

32

characteristic polynomial for . As increases, the number remains the same until crosses the root , at which point increases by one.This can be explained by the fact that, according to the second property of the Sturmsequence, the number of sign changes remains the same as crosses the root of

, i = 1,2, ..., n. At the same time, according to statement II, there is an addi-tional sign change as crosses a root of . Hence, the number increasesby one every time crosses a root of , which proves Sturm’s theorem.

At this time it should be stressed that the characteristic polynomials of the principalminors are not needed in explicit form. Only their values (or actually: their signs) at

and are needed which can be calculated recursively using equation(4.25). When searching for a specific eigenvalue, the interval [a,b] containing it isusually being narrowed down by the use of the bisection method.

4.4 The Wittrick-Williams eigenvalue counting techniqueThe Sturm sequence check for discretized systems enables us to find out exactlyhow many eigenfrequencies there are in a specified frequency domain. Using theSturm sequence checking after any eigensolution extraction method thus leave uswith the possibility to encertain that no mode has been missed by the solution algo-rithm.

Under certain conditions, of which the exact representation of distributed parameter(continuous) systems are the most notable, however, the Sturm sequence checkingbreaks down. This has to do with the fact that not all of the generalized degrees-of-freedom are present in the vector of unknowns q of the eigenvalue problem

(4.28)

where D is the dynamic stiffness matrix. Since not all the system’s degrees-of-free-dom are included in q, also such vibratory solutions exist such that and

. As an example, let us consider the exact dynamic reduction of the sys-tem

(4.29)

p4 σ( ) σ µ< µ s µ( )µ σ3 s µ( )

µpi 1– σ( )

µ pn σ( ) s µ( )µ pn σ( )

µ a= µ b=

Dq 0=

q 0≡det D( ) 0≠

Kaa Kab

Kba Kbb

qa

qb� �� Maa Mab

Mba Mbb

q··aq··b� ��

+Qa

Qb 0=� ��

=

33

THE WITTRICK-WILLIAMS EIGENVALUE COUNTING TECHNIQUE

When the system is vibrating in stationary harmonic motion with amplitude atangular frequency we have

(4.30)

We may eliminate the degrees-of-freedom using that toreceive

(4.31)

In free harmonic vibration, i.e. , situations may occur in which and, see figure 4.4.1.

Similar to the Sturm sequence check, Wittrick and Williams[4.1], deviced a methodfor the exact computation of the number of natural frequencies in specified fre-quency ranges of such systems. Wittrick and Williams noted the similarity of com-puting the number of sign changes of the determinants of the principle minors ofmatrix D and the number of negative diagonal elements of the upper triangular Ugiven by LU-factorization of D, i.e. D = LU. The triangular U may be given byGauss-elimination (without column pivoting). Define as the sign count of thedynamic stiffness matrix , the number of negative diagonal elements of established at the trial frequency . Also, define as the number of eigenfre-quencies of the system with the degrees-of-freedom fixed to zero. Then the Wit-trick-Williams theorem states that: The number of eigenfrequencies of asystem below a certain trial frequency is equal to the sign count of the

q·

ω

Kaa ω2Maa– Kab ω2Mab–

Kba ω2Mba– Kbb ω2Mbb–

qa

qb� �� Daa Dab

Dba Dbb

qa

qb� ��

≡Qa

0� ��

=

q·b Dbaqa Dbbqb+ 0=

Daa DabDbb1– Dba+[ ] qa D'aaqa≡ Qa=

Qa 0= qa 0=det D'aa( ) 0≠

Figure 4.4.1. A three-degree-of-freedom system. When the degrees-of-freedom and are condensed, the system posses an eigenvalue with the displacement vec-tor (here the single element ) equal to zero. The corresponding mode is indicatedby dashed masses.

q1q3

q2

q1 q2 q3

k km mM

s µ( )D µ( ) U µ( )

µ J0 µ( )q·a

J µ( )ω µ= s µ( )

MODAL DECOMPOSITION

34

dynamic stiffness matrix plus the eigenfrequencies of the system with all ele-ments of the displacement vector fixed.

4.5 *The Lanczos’ method

4.6 *The Arnoldi method

BIBLIOGRAPHY

4.1 Wittrick W.H. and Williams F.W., A General Algorithm for Computing Natu-ral Frequencies of Elastic Structures, Quarterly Journal of Mechanics andApplied Mathematics, XXIV(3), 1971

J0 µ( )

35

5 OBSERVABILITY AND CONTROLLABILITY

State observability and state controllability play an important role in experimentalvibration engineering. Loosely speaking, the state observability condition tells uswhether the states of the system can be determined by registration of its output y. Onthe other hand, the state controllability tells us whether the model’s states may beexcited by the stimuli vector u. For example in experimental pretest planning, theobservability and controllability properties are important to consider. In the physicalsensor placement, the observability criterion should therefore be fulfilled such thatno critical states are non-observable. Likewise, state controllability of critical statesshould be maintained by proper actuator placement such that these states are excitedand therefore contribute to the measured response.

Also in analytical model reduction, the state observability and controllability playimportant roles. By indexing the states, such that high index means high observabil-ity and controllability, the states of low importance may be identified and elimi-nated. This reduces the model size without severely affecting the model quality.This is the concept of model reduction after balancing a realization, a concept thatwill be the topic of a following chapter.

5.1 State observability

The concept of state observability is linked to the outputs and states of the system.Given a state-space model and its outputs y and inputs u, the question of observabil-ity is whether the model states x are deducible from it. More specifically, the observ-ability theorem states that: A linear system is said to be observable at time t0 if thestate x(t0) can be uniquely determined from the output y(t) where . If the sys-t t0≥

OBSERVABILITY AND CONTROLLABILITY

36

tem is observable for all times t0, then the system is said to be completely observ-able. We may here note that a time invariant system which is observable is alsocompletely observable.

Thus we seek a condition for the realization to be stateobservable, provided the model and its inputs and outputs are known. When the out-put is given at times , also the time derivative of the output may be deter-mined. We note that , and therefore

(5.1)

(5.2)

and so on for higher derivative orders of y. We may put these equations in matrixform to obtain

(5.3)

with (5.4)

(5.5)

Here is the observability matrix. Note that the derivative expansion has been ter-minated at the (n-1):th time derivative. The Cayley-Hamilton theorem advise us tostop since no more independent equations can be obtained. The theorem states thatany square matrix A of order n is such that if raised to a power k it can be expressedin lower order terms, i.e. that

(5.6)

x· Ax Bu+= y, Cx Du+=

t t0≥y Cx Du+=

y· Cx· Du·+ CAx CBu Du·+ += =

y·· CAx· CBu· Du··+ + CA2x CABu CBu· Du··+ + += =

Y Ox TU+=

YT [yT y·T y··T … y n 1–( )T]=

UT [uT u· T u··T … u n 1–( )T]=

T

D 0 0 … 0CB D 0 … 0

CAB CB D … 0… … … … 0

CAn 1– B CAn 2– B CAn 3– B … D

=

T [ CT CA( )T CA2( )T

… CAn 1–( )T

]=

Ak αjkj 0=

n 1–

� Aj=

37

STATE CONTROLLABILITY

The series is finite and with no more than terms. Thus, we cannot expect tofind more independent rows of the observability matrix by adding higher orderderivatives.

Now if the matrix has a rank of less then n, it means that some linear com-bination of the n columns may add to zero, and therefore there are states that willnot appear in Y. It is thus necessary for observability that the observability matrix isof full rank.

Is the full rank condition also sufficient for observability? To examine this, we startwith multiplying equation (3.3) with the transpose of to obtain

(5.7)

If is of full rank then T is non-singular and thus the state vector x can be deter-mined as the unique solution to (3.6), which is

(5.8)

Is this also the solution to equation (3.3)? If it is not, for the two different solutionsx1 and x2 we would have

(5.9)

which means that some linear combination of the columns of is zero, which con-tradicts the assumption that is of full column rank.

In conclusion we may thus state that: A realization is uniquely observable if andonly if the observability matrix has full rank n.

5.2 State controllability

The concept of controllability relate to the input and the states of a system. Toexamine the concept, we return to the state-space first order differential equation.Then we can define controllability by the following theorem: The system

is said to be state controllable at time if there exists a piece-wise continuous input u(t) that will drive the initial state x(t0) to any final state x(tf)within a finite time interval tf - t0. If this is true for all initial times and all initialstates, the system is said to be completely state controllable. If a time-invariant sys-tem is state observable it is thus also completely state observable. For such systems

n 1–

nr n×

TY T x TTU+=

x T( )1– T Y TU–[ ]=

x1 x2–[ ] 0=

x· Ax Bu+= t t0=

OBSERVABILITY AND CONTROLLABILITY

38

we can derive a quantitative test of controllability. To this end we let, without loss ofgenerality, the initial time , and rewrite equation (3.6) in the form

(5.10)

which, for non-singular , can be reduced to

(5.11)

But, recalling the Cayley-Hamilton theorem described in chapter 5.1, i.e. usingequation (5.6) together with equation (3.15), we get

(5.12)

Here has been introduced as

(5.13)

Hence, inserting equation (5.12) into (5.11), we obtain

(5.14)

Here the column vector cj has been introduced as

Equation (5.14) can be written in matrix form as

t0 0=

x tf( ) eAtfx 0( ) e

A tf τ–( )Bu τ( ) τd

0

tf

�+=

eAtf

eAtf–

x tf( ) x 0( )– ∆x tf( )≡ e Aτ– Bu τ( ) τd0

tf

�=

e Aτ– 1–( )kτkAk k!⁄k 0=

∞

� 1–( )kτk k!⁄k 0=

∞

� αjkAj

j 0=

n 1–

�= =

Aj

j 0=

n 1–

� 1–( )kαjkτk k!⁄k 0=

∞

� αj τ( )Aj

j 0=

n 1–

�≡=

αj τ( )

αj τ( ) 1–( )kαjkτk k!⁄k 0=

∞

�=

∆x tf( ) AjB αj τ( )u τ( ) τd0

tf�

j 0=

n 1–

�= AjBcj tf( )j 0=

n 1–

�≡

cj tf( ) αj τ( )u τ( ) τd0

tf�=

39

TESTING OBSERVABILITY AND CONTROLLABILITY

(5.15)

Equation (5.15) represents a set of n equations and ns unknowns. The equationshave a solution for any provided that the matrix

(5.16)

has n independent columns. The matrix is known as the controllability matrix,sometimes the reachability matrix, and thus the system is completely state controlla-ble if has rank n.

This is a sufficient condition for controllability, but is it a necessary condition? Theanswer is both yes and no. For the so-called controllability-from-the-origin problem(which is the reachability problem), i.e. , the condition is necessary[5.1].However, for the controllability-to-the-origin problem, i.e. , it is not neces-sary. For singular transition matrices we can have controllability-to-the-origin,without the controllability matrix being of full rank. However, for all practical sys-tems in vibration engineering the transition matrix is non-singular, see problem 5.2.

5.3 Testing observability and controllability

The rank of the controllability and observability matrices may be determined byrank computation, more often than not based on QR factorization. However, forhigh-order systems these rectangular matrices may be of very high dimensionalityand therefore require much computation and available memory. Other, sometimescheaper, tests have thus been developed. Such are the tests related to the theoremsby Popov and Belovitch with applications suggested by Hautus. These tests are hereand elsewhere[5.1] called the PBH tests and are especially useful for theoretical anal-ysis and also in numerical problems whenever determination of matrix eigenvaluesand eigenvectors is computationally feasible.

∆x tf( ) [ B AB A2B … An 1– B ]

c0

c1

.

.cn 1–

=

∆x tf( ) n ns×

[ B AB A2B … An 1– B ]=

x tf( ) 0≠x tf( ) 0≡

Φ

OBSERVABILITY AND CONTROLLABILITY

40

The PBH eigenvector test:

1. A state-space realization pair {A,B} will be controllable if and only if there,to the adjoint eigenvalue problem , exists no left eigenvector λof A that is orthogonal to all the columns of B.

2. A state-space realization pair {A,C} will be observable if and only if there, tothe eigenvalue problem , exists no right eigenvector ρ of A that isorthogonal to all rows of C.

For a vibrational engineer the controllability theorem should come as no big sur-prise. It simply means that if the load distribution is orthogonal to any mode of thesystem, that mode will not be driven by the loading and is therefore not controllable.

As an alternative we have the PBH rank test:

1. A state-space realization pair {A,B} will be controllable if and only if thematrix has rank n for all generally complex driving frequen-cies s.

2. A state-space realization pair {C,A} will be observable if and only if thematrix has rank n for all generally complex driving fre-quencies s.

These conditions will clearly be met for all s that are not eigenvalues of A, becausedet for such s. The point of the theorem is that the rank must be n evenwhen s is an eigenvalue of A.

The case of multiple eigenvalues (of multiplicity m) deserves a further treatment. Itmay be found during the PBH testing that

,

and thus the test for controllability may, although here possibly falsely, be consid-ered to be fulfilled. However, for multiple eigenvalues we know that also an arbi-trary linear combination of the eigenvectors is also anleft eigenvector. We thus require that . We know that a solution to

λTA σλT=

Aρ σρ=

sI A–( ) B[ ]

CT sI A–( )T[ ]T

sI A–( ) 0≠

λiTB 0≠ i 1 … m, ,=

λ λ1 λ2 … λm[ ]α Λα≡=BTλ BTΛα= 0≠

BTΛ( )s m× αm 1× 0=

41

TESTING OBSERVABILITY AND CONTROLLABILITY

can always be found if . In the case a non-trivial solution may also befound provided should be singular. However, since B is not rank deficient, ifdesigned properly, and is never rank deficient, the quadratic cannot be sin-gular. The conclusion is thus that controllability is lost if the number of inputs s arefewer than the highest multiplicity of any of the eigenvalues of the system. A simi-lar analysis of the observability reveals that observability is lost if the highest multi-plicity of any eigenvalue is higher than the number of outputs r.

PROBLEMS

5.1 A carriage of mass M has two inverted pendulums on it of lengths l2 and l3,both with end-tip bobs of mass m (see figure). The external force u act andcause the linear displacement q1 and angular displacements q2 and q3 aboutthe vertical positions. The equation of motion, for small angular motion, is

a) Check if controllability exist for the system for all length ratios l2 / l3.b) Is the system observable with output y = q2?

5.2 Show that the discrete-time state transition matrix is non-singular (has onlynon-zero eigenvalues) for most practical systems, i.e. those systems of finitedamping. Hint: relate on the one hand the eigenvalues of the continuous-timerealization to the eigenvalues of the discrete-time realization and on the otherhand damping to the real part of the continuous-time realization’s eigenval-ues.

BIBLIOGRAPHY

5.1 Kailath T., Linear Systems, Prentice-Hall, 1980

m s> m s=BTΛ

Λ BTΛ

uM

mm

l2 l3

Problem 5.1

g

Mq··1 mgq2– mgq3– u+=

m q··1 liq··

i+( ) mgqi , i 2 3,= =

Problem figure

OBSERVABILITY AND CONTROLLABILITY

42

43

6 GRAMIANS AND BALANCED REALIZATION

6.1 Controllability and observability Gramians

Suppose we have a realization , and wish to obtain thefinite energy input u that will bring the state to zero at a desired fixed time tf. Weknow that for a given u we have the solution

(6.1)

For a non-singular we thus have

(6.2)

This is an integral equation for the sought input u. Its solution can be shown to be

(6.3)

where

(6.4)

x· Ax Bu+= x t0( ) x0=

x tf( ) eA tf t0–( )

x0 eA tf τ–( )

Bu τ( ) τdt0

tf

�+=

eA tf t0–( )

x0– eA tf t0–( )

eA tf τ–( )

Bu τ( ) τdt0

tf

� eA t0 τ–( )

Bu τ( ) τdt0

tf

�= =

u τ( ) BTeAT t0 τ–( )

Gc1– t0 tf,( )x0–=

Gc t0 tf,( ) eA t0 τ–( )

BBTeAT t0 τ–( )

τdt0

tf

�=

GRAMIANS AND BALANCED REALIZATION

44

is the controllability Gramian. To see that (6.3) really gives the solution we let itenter into the integral equation (6.2). We then have

(6.5)

since is constant. We require that the controllability Gramian be non-sin-gular and therefore invertible, which is the condition for controllability.

Similarly, for the realization , , the corresponding observabil-ity Gramian

(6.6)

must be non-singular[6.1] for the realization to be observable from the output duringover times .

To mathematicians, the Gramian singularity test is known to be a test for lineardependence of functions li and lj with the Gramian . In our case, thefunctions to be checked for linear dependence are the system’s impulse responses

.

If we let the initial time at which the control input is applied vary, i.e. we let ,we observe the following property of the controllability matrix

eA t0 τ–( )

BBTeAT t0 τ–( )

Gc1– t0 tf,( )x0 τd

t0

tf

�– =

[ eA t0 τ–( )

BBTeAT t0 τ–( )

τdt0

tf

� ] Gc1– t0 tf,( )x0– =

G– c t0 tf,( ) Gc1– t0 tf,( )x0 x0–=

Gc t0 tf,( )

x· Ax Bu+= y Cx=

Go t0 tf,( ) eAT τ t0–( )

CTCeA τ t0–( )

τdt0

tf

�=

t0 t tf≤ ≤

Gij lilj tdt0

t1�=

eAtB

t0 t=

ddt-----G

ct tf,( ) d

dt----- eA t τ–( )BBTeAT t τ–( ) τd

t

tf

�= =

ddt----- [ e

A t τ–( )BBTeAT t τ–( ) ] τd

t

tf

� e0BBTe0– =

45

TRANSIENT SOLUTION BOUNDS

(6.7)

For large time ranges and damped (asymptotically stable) systems the stateimpulses eventually die out and do not contribute more to the Gramian. Thus, forlarge control times the Gramian derivative with respect to initial time variation iszero. The infinetely long time Gramian is thus governed by

(6.8)

and the corresponding long time observability Gramian is governed by

(6.9)

which are Lyapunov equations for and . Numerical methods for solvingLyapunov equations exist. Their solution generally involve many computationaloperations. For large order systems (say ) the solution may take too long toobtain even with a fast computer (of today).

We should recall that the Gramians obtained by the Lyapunov equation are for infi-nitely long control and observation times. The general controllability Gramian, asdefined by equation (6.4), is seen to be control time dependent and is thus notunique. Also, we should observe that the Gramians are not invariant to similaritytransformations. Therefore, an equivalent realization

(6.10)

normally gives other Gramians and .

6.2 Transient solution boundsConsider the response of a linear time-invariant system excited from the start oftime. Using the matrix of impulse response h(t) we have for the causal system’sresponse

(6.11)

We want to bound the output vector norm under certain restrictions put onthe input vector norm . Before proceeding, we define what is meant by the

AGc t tf,( ) Gc t tf,( )AT BBT–+

tf t–

Gc∞

AGc∞ Gc

∞AT+ BBT=

ATGo∞ Go

∞A+ CTC=

Gc∞ Go

∞

n 100>

z· T 1– ATz T 1– Bu+ Az Bu+= =

y CTz Du+ Cz Du+= =

Gc Go

y t( ) h t τ–( )u τ( ) τd∞–

t

� CΦ t τ–( )Bu τ( ) τd∞–

t

�= =

y t( )u t( )

GRAMIANS AND BALANCED REALIZATION

46

norm of a vector-valued function of time. Starting with the r-dimensional vectorfunction y(t), we define its time-varying q-norm to be

(6.12)

Normally, only the (Euclidean) 2-norm or the (maximum component) -norm areof interest.

We then extend the definition of a vector norm to consider also the norm across timeas well. We define the (p,q)-norm of the vector-valued time-dependent function y(t)to be

(6.13)

Notice that this norm is independent of time. When p is , the (p,q)-norm is thepeak value of the q-norm of the vector function y(t) over all times. Let us considerthe two, probably most interesting, cases when q is either 2 or . First, when (p,q)is ( ,2), the norm becomes the maximum value of all times of the Euclidean normof the vector y(t)

(6.14)

Second, when (p,q) is ( , ) the norm is the maximum value over all times of themaximum value of any component of the vector y(t)

(6.15)

From now on, we restrict our attention to all inputs with a given (2,2)-norm equal to, i.e. . This is the time-domain root-mean-square value of the input.

The following theorem then give output norm bounds.

Output bound theorem: Let the matrix S be such that . The( ,2)-norm of the vector of responses to any input vector u(t) from theclass of vectors having a (2,2)-norm equal to will be bounded to

y t( ) q [ yi t( ) q

i 1=

r

� ] 1 q⁄=

∞

y t( ) p q, [ y t( ) q( )p t ]1 p⁄d∞–

∞

�=

∞

∞∞

y t( ) ∞ 2, sup ∞ t ∞≤ ≤–

y t( ) 2=

∞ ∞

y t( ) ∞ ∞, sup ∞ t ∞≤ ≤–

y t( ) ∞ sup ∞ t ∞ ≤ ≤–

max 1 t r≤ ≤

yi t( )= =

ν u t( ) 2 2, ν=

S h t( )hT t( ) td∞– ∞�=

∞ y t( ) ∞ 2,ν

47

THE WORST-CASE FORCING FUNCTION

(6.16)

and the ( , )-norm of the vector of responses to the same class of inputs isbounded to

(6.17)

A proof was given by Wilson[6.3]. This theorem let us bound the transient responseof a structure without either knowing the actual loading history or solving the initialvalue problem for a given load history. Whatever the actual peak response to anygiven load, it will be less than or equal to the bounds given. In the following sectionwe will give an expression for a load history of norm that exactly matches theworst-case bound for all loads within its class.It should be mentioned here, andalways considered in practice, that the scalings of the output vector elements is crit-ical. Thus for mixed output vectors, e.g. vectors holding both stresses and strains,the elements should be properly normalized before the bounds are actually com-puted. A normalization will affect the state-space matrix C.

We note that the matrix S is

(6.18)

with Gc, the controllability Gramian, given by equation (6.4).

6.3 The worst-case forcing function

As an analytical method, the preceding bound on the system response provides ameans for analyzing its worst-case transient response. It only requires that the classof loadings can be characterized in terms of their given (2,2)-norm. A systemresponse bound analysis would reveal the most critical stress components, the mostcritical displacement, etc., using only model data without simulation.

It is often necessary, however, to qualify a system by experimentally applying aloading that is a worst-case simulation of the true operating environment. In thiscase, it might be necessary to know at least one loading history that excites the sys-

y t( ) ∞ 2, ν max eig S[ ]≤

∞ ∞

y t( ) ∞ ∞, ν max diag S[ ]≤

ν

S h t( )hT t( ) td∞–

∞

� CΦ t( )B[ ] BTΦT t( )CT[ ] td∞–

∞

�= =

C [ Φ t( )B[ ] BTΦT t( )[ ] td∞–

∞

� ] CT CGcCT≡=

GRAMIANS AND BALANCED REALIZATION

48

tem so that its response is at its limit. This load history could then be applied know-ing that all other load histories with the same (2,2)-norm would excite the systemless. One such transient loading, considering the bound of the response’s ( ,2)bound is presented here. The response norm equation (6.16) and the response con-volution expression (6.11) indicate that we look for an loading on the form

(6.19)

in which a is a vector of yet to be determined constants. The response at an arbitrarytime is then

(6.20)

and, specifically, the response at is

(6.21)

We are looking for a loading that produces the largest ( ,2)-norm, i.e. the largestEuclidean norm of the response at a certain time of all times. Let that time be

. We thus want to maximize . On the other hand, the (2,2)-norm ofthe input is

(6.22)

The matrix S is positive semidefinite and symmetric, and so it has an eigen-decom-position given by

(6.23)

∞

u t( ) h t–( )Ta=

y t( ) h t τ–( )h τ–( )Ta τd∞–

∞

�=

t 0=

y 0( ) h τ–( )h τ–( )Ta τd∞–

∞

�=

[ h τ–( )h τ–( )T τ ] ad∞–

∞

� Sa= =

∞

t 0= y 0( ) 2

u t( ) 2 2, [ u t( )Tu t( ) td∞–

∞

� ]1 2⁄=

[ aTh t–( )h t–( )Ta td∞–

∞

� ]1 2⁄ [aTSa]1 2⁄

= = ν≡

S EΣET=

49

THE WORST-CASE FORCING FUNCTION

which is orthonormal, i.e. . We denote the ordered sequence of eigenval-ues in with : and the corresponding eigenvectors in Ewith . The response is maximized, under the condition that

, if we align the vector a with the eigenvector of S that correspondsto its largest eigenvalue ,

(6.24)

where the scalar constant must be in order to satisfy the equation(6.22). To show that is really maximized under these conditions we notethat the response to load quotient

(6.25)

should be maximized. Since E is a square matrix of full rank we may expressany vector a as a linear combination of its columns . We then have

with being arbitrary constants. The matrix product is then

(6.26)

since E is orthonormal. The quotient in equation (6.25) is then

(6.27)

Since all eigenvalues are positive, this quotient is bounded by

ETE I=Σ σi 0 σ< 1 σ2 … σr<≤ ≤ei y 0( ) 2

u t( ) 2 2, ν=σr

a αrer=

αr αr ν σr⁄=y 0( ) 2

y 0( ) 22

u t( ) 2 2,2

---------------------- aTSTSa

aTSa-------------------- aTEΣETEΣETa

aTEΣETa----------------------------------------- aTEΣ2ETa

aTEΣETa----------------------------= = =

r r×r 1× ei

a αieii 1=

r

�=

αi ETa

ETa

e1T

e2T

.

.

erT

α1e1 α2e2 … αrer

α1

α2

.

.αr

α≡= =

y 0( ) 22

u t( ) 2 2,2

---------------------- αTΣ2α

αTΣα-----------------

σ12α1

2 σ22α2

2 … σr2αr

2+ + +

σ1α12 σ2α2

2 … σrαr2+ + +

----------------------------------------------------------------= =

GRAMIANS AND BALANCED REALIZATION

50

, (6.28)

with equality if and for . At equality, the quotient is which ismaximized for the maximum eigenvalue . Therefore, we should align a with thecorresponding eigenvector The worst-case loading is then given by

(6.29)

6.4 A balanced realization

Moore[6.4] has showed that it is possible to find a similarity transformation such that both and become diagonal and equal, i.e.

. The corresponding state-space model

(6.30)

is called a balanced realization, with Gramians balanced over the control and obser-vation range . It should be noted that, although the controllability and observ-ability Gramians are both diagonalized, the realization (6.30) is generally not put ondiagonal form by the transformation.

PROBLEMS

6.1 For the system according to the figure, let the single output be the supportreaction R(t). Let the (2,2)-norm of the input be .Calculate the norm of the output. Create the worst-case loading func-tion and simulate the system s response. Does the maximum response of thesimulation match the calculation?

Also, make simulations of the responses to half-sine pulses of various dura-tion (with (2,2)-norm ). Plot the dynamic amplification factor (as

y 0( ) 22

u t( ) 2 2,2

----------------------σ1

2α12 σ2

2α22 … σr

2αr2+ + +

σiαi2

----------------------------------------------------------------≤ 1 i r≤ ≤

αi 0≠ αj 0≡ j i≠ σiσr

u t( ) νσr

---------h t–( )Terνσr

---------BTΦ t–( )TCTer= =

x Tbz= Gc∞ Gc

∞

Gc∞ Go

∞ diag gi( )≡=

z· Tb1– ATbz Tb

1– Bu+ Az Bu+= =

y CTbz Du+ Cz Du+= =

0 ∞,[ ]

u t( ) 2 2, ν 1N s1 2⁄⁄= =∞ ∞,( )

1N s1 2⁄⁄

51

A BALANCED REALIZATION

peak over reference load 1N) versus normalized duration time, i.e. normal-ized with respect to the period of the most high-frequency oscillating mode.

BIBLIOGRAPHY

6.1 Kailath T., Linear Systems, Wiley, 1980

6.2 Peterson L.D., Bounding the Transient Response of Structures to UncertainDisturbances, AIAA Journal 34(6), 1996

6.3 Wilson D.A., Convolution and Hankel Operator Norms for Linear Systems,IEEE Transactions on Automatic Control 34(2), 1989

6.4 Moore B.C., ‘Principal Component Analysis in Linear Systems: Controllabil-ity, Observability, and Model Reduction’, IEEE Transactions on AutomaticControl 26(1), 1981

k1 k2 k3

m1 m3m2

Problem 6.1

q1 t( ) Q1 0≡, q3 t( ) Q3, u t( )=q2 t( ) Q2 0≡,

R(t)c1 c2 c3

m1= m2= m3= 1kgk1= 1000N/m, k2= k3=10N/m c1= 1Ns/m, c2=0.2Ns/m, c3= 0.1Ns/m

Problem figure

GRAMIANS AND BALANCED REALIZATION

52

53

7 MODEL REDUCTION

7.1 *Modal truncation

7.2 State reduction by use of Gramians

Consider the continuous time asymptotically stable realization

, (7.1)

which is assumed to be both controllable and observable. This means that its con-trollability Gramian

(7.2)

and its observability Gramian

(7.3)

are both non-singular for any control and observation duration . As statedearlier, the Gramians are not invariant under similarity transformations of the real-ization. Moore[7.1] showed that there exists a similar system for which the Gramiansare equal and diagonal. Such realization is balanced over the interval [ ].

x· Ax Bu+= y Cx Du+=

Gc t0 tf,( ) eA t0 τ–( )

BBTeAT t0 τ–( )

τdt0

tf

�=

Go t0 tf,( ) eAT τ t0–( )

CTCeA τ t0–( )

τdt0

tf

�=

tf t0– 0>

t0 tf,

MODEL REDUCTION

54

Let us assume that the realization has been brought to balanced form and moreoverthe state variables have been permuted such that the diagonal elements of the diago-nal Gramian matrices diag( ) are in decreasing order, i.e. ,

. Let us sub-divide the state vector into two partitions and ,with output contributions and respectively, and write the partitioned balancedrealization as

(7.4a,b)

where is . Let and be the minimum norm functions that drive thestate from the origin to and respectively, in the time interval[ ]. It was shown by Moore[7.1] that

(7.5)

If and and have the same norms it follows that

(7.6)

In other words, the part of the state is much less affected by the input than thepart .

Analogously, let and be the homogeneous system’s responses from the initialstates and respectively. Then

(7.7)

if

(7.8)

and

gi gi 1+ gi≤i 1 … n 1–, ,= x1 x2

y1 y2

x· 1

x· 2

A11 A12

A21 A22

x1

x2

B1

B2u1 u2++=

y1 y2+ C1 C2x1

x2

=

A11 k k× u1 u2x1

T 0[ ]T 0 x2T[ ]T

t0 tf,

u22 τd

t0

tf�

u12 τd

t0

tf�---------------------------

gkgk 1+-------------

x2 tf( ) 2

x1 tf( ) 2---------------------≥

gk gk 1+» u1 u2

x2 tf( ) 2 x1 tf( ) 2«

x2x1

y1 y2x1

T t0( ) 0[ ]T 0 x2T t0( )[ ]T

y2 τ( ) 2 τd t0

tf

� y1 τ( ) 2 τd t0

tf

�«

gk gk 1+»

55

*COMPONENT MODE SYNTHESIS

(7.9)

This means that the states affects the output much less than the states .

It seems reasonable to assume that the states does not affect the input-outputbehavior of the system very much if . This assumption suggests that therealization triple { } may be a good lower-order approximation of thesystem (7.4). Applications have also shown this to be true also in practice.

It should be noted that, since (7.4) is not on diagonal form, a reduction of the lesscontrollable/observable states does not preserve the eigenstructure of the realiza-tion. Thus eigenvalues are not only removed, but also shifted in the reduced realiza-tion. This is truly an annoying property of the reduction procedure but also aproperty shared by other popular schemes for reduction such as the Gyuan andCraig-Bampton reductions[7.2]. Although the eigenvalues are shifted, it has beenshowed[7.3] that the resulting reduced realization is still asymptotically stable.

7.3 *Component mode synthesis

PROBLEMS

BIBLIOGRAPHY

7.1 Moore B.C., Principal Component Analysis in Linear Systems: Controllabil-ity, Observability, and Model Reduction, IEEE Transactions on AutomaticControl 26(1), 1981

7.2 Geradin M. and Rixen D., Mechanical Vibrations- Theory and Applicationsto Structural Dynamics, 2nd ed., Wiley, 1997

7.3 Pernebo L. & Silverman L.M., Model Reduction via State Space Representa-tions, IEEE Transactions on Automatic Control 27(2), 1982

x1 t0( ) x2 t0( )=

x2 x1

x2gk gk 1+»

A11 B1 C1, ,

MODEL REDUCTION

56

57

8 *STATE OBSERVABILITY

8.1 *State observers

8.2 *The Kalman filter

8.3 *The modal filter

8.4 *The calibrated modal filter

PROBLEMS

BIBLIOGRAPHY

*STATE OBSERVABILITY

58

59

9 EXPERIMENTAL MODAL ANALYSIS

9.1 IntroductionExperimental modal analysis is the process of experimentally determining theeigenstructure of the system under test. Linearity of the system, unders the loadingconditions the system will be subjected to, is then assumed and usually also verifiedby the test procedure. In most cases, the system is stationary, i.e. does not movewith gross rigid body motions, and is supposed to be undamped or viscouslydamped. The linear second order differential equations governing its vibratorymotion is then (see also equation (2.3))

(9.1)

This may be written in first order form by use of the relation as

(9.2)

The homogeneous harmonic form of (9.2), i.e. and , define aneigenvalue problem

(9.3a,b)

It is the purpose of the testing to experimentally obtain the pertinent, generallycomplex-valued, eigenvalues and associated eigenmodes of the system.

Mq·· t( ) Vq· t( ) Kq t( )+ + Q t( )=

Mq· t( ) Mq· t( )– 0=

V MM 0

q·

q··� �� K 0

0 M–

qq·� ��

+Q0�

��

=

Q t( ) 0= q t( ) qeρ t=

K 00 M–

ρ σ V MM 0

ρ–= ρqq� ��

= .

σk ρ k( )

EXPERIMENTAL MODAL ANALYSIS

60

Since the eigenmodes are determined only by their shape, and not magnitude, aproper normalization is of interest. Commonly used such are unit vector normnormalizations. These may be either infinite-norms, where the largest element of thevectors are set to one, or the 2-norms of the eigenvectors set to unity. Together withthe modal constants

(9.4a,b)

the system characteristics may now be written on first order form, with the use ofthe modal matrix and the transformation , as

(9.5)

For stationary harmonic excitation, i.e. and , we then have

(9.6)

or

(9.7)

where it has been used that . One element of the system transfermatrix H can be shown to be

(9.8)

It may be seen that the eigenfrequencies , the eigenvectors togetherwith the modal normalization constants (known as modal a:s or modal Fossdampings[9.1]) fully describes the system characteristics.

For proportionally damped systems, i.e. where , a theoretical analysismay be somewhat simplified. Motivated by this, a commonly made approximation

bk ρ k( )T K 00 M–

ρ k( )= ak ρ k( )T V MM 0

ρ k( )=

P ρ 1( ) ρ 2( ) ρ 3( ) ...[ ]= qT q· T{ }T

Pz=

PT V MM 0

Pz· t( ) PT K 00 M–

Pz t( )+ =

diag an( )z· t( ) diag bn( )z t( )+ PT Q t( )0�

��

Z t( )≡=

Z t( ) Zeiω t= z zeiω t=

diag bn iωan+( )z Z=

x Pz Pdiag 1 iak ω ωk–( )⁄( )PT Q0�

��

H Q0�

��

≡= =

bk iωkak–= Hi j

Hi jρi

k( )ρjk( )

iak ω ωk–( )----------------------------

k 1=

n

�=Ri j

k( )

ω ωk–----------------

k 1=

n

�≡

σk iωk= ρ k( )

ak

V αK βM+=

61

PRETEST PLANNING

is often to treat also non-proportionally damped systems as being proportional. Thismay be justified if the system’s damping is light and its eigenfrequencies wellseparated. Using the eigenvalues and eigenmodes of the correspondingundamped system’s eigenproblem

(9.9)

and using the orthogonality properties of the modes, we may transform equation(9.1) into decoupled second-order differential equations as

(9.10)

Here it has been used that the modal masses are and the modalstiffnesses are . In stationary harmonic vibration we have, with themodal damping and with , that

(9.11)

or, with and introducing the relative viscous damping ,we have

(9.12)

One element of the system transfer function H, here also known as the dynamicflexibility or the receptance, is

(9.13)

with N being the order of the system matrices. It may be seen that theeigenfrequencies , the eigenvectors , the modal dampings and the modalmasses fully describes the system characteristics. These are the quantities thatshould be determined by the testing.

9.2 Pretest planning In the preparation of an experimental modal analysis, important considerationsregard the layout of sensor locations. The successful outcome of the test very muchrelies on that the important vibrational characteristics of the structure is well cap-tured by the test data. The worst positions of the sensors is naturally at vibrational

ωk2 ρ k( )

Kρ ω2Mρ=

PTMPz·· t( ) PTVPz· t( ) PTKPz t( )+ + =

diag mk( )z·· t( ) diag αkk βm+ k( )z· t( ) diag kk( )z t( )+ + PTQ t( )=

mk x k( )TMx k( )=kk x k( )TKx k( )=

vk αkk βmk+= z t( ) zeiω t=

diag kk iωvk ω2mk–+( ) z Z=

kk ωk2mk= ζk vk 2 kkmk⁄=

x Pdiag 1 mk ωk2 2iζkωkω ω2–+( )⁄( )PTQ HQ≡=

Hi jρi

k( )ρjk( )

mk ωk2 2iζkωkω ω2–+( )

----------------------------------------------------------k 1=

N

�=

ωk ρ k( ) ζk

mk

EXPERIMENTAL MODAL ANALYSIS

62

nodes shared by the eigenmodes of interest and such locations, if they exist, shouldbe avoided. But what about the best locations? How should such be found in a testpreparation?

From here on, we assume that a good finite element model of the test object is athand and that the dynamic characteristics of it is suitable for the planning of sensorlayout. Under that condition the method of Effective Independence, as describedbelow, for sensor placement has been found to work well. Finite element models areusually much more detailed with respect to the number of degrees-of-freedom beingemployed as compared to the number of sensors to be used in the dynamic test. Letthe partition of the modal matrix of the finite element model that is associated withcandidate sensor locations and directions be denoted . The measureddisplacement resonse at the candidate sensors then relate to the generalizedstructural motion as (see equations (2.1b) and (2.7a))

(9.14)

where is measurement noise. Under the assumption that the measurement noiseis stationary, Gaussian and white, Kammer[9.3] advocates that the covariance matrixof the estimation error should be minimized for a good sensorplacement. The true system state is here z and the estimate, that should be as close toz as possible, is . In that case, the inverse of the covariance matrix shouldbe as large as possible. For a dynamic test in which the measurement noise is notcorrelated between sensors and for which identical statistical properties of eachsensor is possessed, the measurement noise may be characterized by the scalarvariance . Kammer shows that the inverse of the covariance matrix, the Fisherinformation matrix Q, then is

(9.15)

which defines the scaled Fisher information matrix .

Above, we have loosely discussed the maximization of the Fisher informationmatrix without defining a way of obtaining its size. Kammer uses the determinant ofthe Fisher information matrix as the scalar measure of its size. Thus, keeping thedeterminant as large as possible while removing sensors from the candidate setshould be a good strategy for sensor reduction. The quantification of thecontribution to the determinant from the sensor individuals is then a key issue. Anindex vector E was deviced by Kammer, with elements related to the i:thsensor, as

Ps

ys

z t( )

ys t( ) Ps z t( )= v t( )+

v t( )

R E z z–( ) z z–( )T[ ]≡

z Q R 1–=

σ2

Q 1σ2------Ps

TPs1

σ2------Qs≡=

Qs

E i( )

63

SIMPLE CIRCLE CURVE FITTING

(9.16)

Here is the Fisher information matrix obtained after the i:th sensor has beenremoved from the candidate set, i.e. a row has been eliminated from the modalmatrix . Since removal of a sensor does never increase the determinant, it is seenthat the indices are bounded to the interval [0,1] and they are the fractionalcontributions to the determinant. The index numbers are called the effectiveindependence of the i:th sensor.

Now, by omitting the sensor giving the least index the minimum decrease of theinformation matrix determinant is achieved. By repeating the procedure, eventuallya set of sensors remains for which a large determinant of the Fisher informationmatrix exist. However, it may be seen from the definition of the Fisher informationmatrix, equation (9.15), that the matrix becomes singular when the number of rowsof the modal matrix is lesser than the its number of columns, i.e. when thenumber of sensors are fewer than the number of target modes. Therefore, theminimum number of sensors is always equal to the number of target eigenmodessince a further sensor reduction whould render a null determinant.

Kammer also device an efficient procedure for computing the effective indepen-dence indices. Let and be the eigenmatrix and the eigenvalues of theeigenvalue problem , then the effective independence vector E is

, (9.17)

Here denote the k:th column of e and ^2 denote an element-by-elementmultiplication. The number of target modes is n. We refer to Reference 9.3 for moreinformation on the method.

9.3 Simple circle curve fittingMost simple modal parameter extraction methods set out from the observation that(for systems with little damping and well separated eigenfrequencies) close to a nat-ural frequency, the system transfer functions are totally dominated by the contribu-tion from that frequency’s associated eigenmode.

Let us consider a mobility transfer function, i.e. the function Yij relating velocityresponse at degree-of-freedom i and applied force at degre-of-freedom j, of the

E i( ) det Q( ) det Q i( )( )–det Q( )

--------------------------------------------=

Q i( )

Ps

E i( )

Ps

Φ λQsΦ Φλ=

E e.kλk

k 1=

n

�= e PsΦ( )^2=

e.k

EXPERIMENTAL MODAL ANALYSIS

64

viscoelastic system. Using the dynamic flexibility of Equation (9.13), and notingthat the velocity is in harmonic motion, we thus have

(9.18)

For frequencies close to the r:th resonance , the contributions of the other modesmay be approximated as being constant, i.e.

(9.19)

Here the frequency dependent part has the basic form of the mobility of a single-degree-of-freedom system (of mass , stiffness and relative damping )which is

(9.20a,b)

By comparing equations (9.19) and (9.20a) one notes that the eigenvector elementproduct of the multi-degree-of-freedom system is a scaling of thecorresponding single-degree-of-freedom transfer function.

It may be shown that the transfer function form a perfect circle in the Nyquistplane, i.e. the complex -plane. It can be verified by inspection that the circle isdefined by

(9.21)

I.e. the circle centre is at and its radius is , see figure9.3.1a. From geometrical considerations we can deduce from the real and imaginaryparts of equation (9.20b) that

(9.22)

q· iωq

Yij iωHi jiωρi

k( )ρjk( )

mk ωk2 2iζkωkω ω2–+( )

----------------------------------------------------------k 1=

N

�= = =

iωρir( )ρj

r( )

mr ωr2 2iζrωrω ω2–+( )

---------------------------------------------------------iωρi

k( )ρjk( )

mk ωk2 2iζkωkω ω2–+( )

----------------------------------------------------------k 1=k r≠

N

�+

ωr

Yij ω ωr≈( )iωρi

r( )ρjr( )

mr ωr2 2iζrωrω ω2–+( )

--------------------------------------------------------- Bi jr )( )+≈

Y

mr mrωr2 ζr

Y iωmrωr

2 1 ω ωr⁄( )2– 2iζrω ωr⁄+( )-------------------------------------------------------------------------------

2ζrω2 ωr⁄ iω 1 ω ωr⁄( )2–( )+

mrωr2 1 ω ωr⁄( )2– 2iζrω ωr⁄+

2-------------------------------------------------------------------------------= =

ρir( )ρj

r( )

Y

Y

ℜ2 Y 1 4mrζrωr⁄–( ) ℑ2 Y( )+ 1 4mrζrωr⁄( )2=

1 4ζrωr ⁄ 0,( ) R 1 4ζrωr⁄=

θ2---tan γtan

1 ω ωr⁄( )2–2ζrω ωr⁄

------------------------------= =

65

SIMPLE CIRCLE CURVE FITTING

It may be shown that the maximum sweep rate, i.e. when is at its peak,occurs at .

Another valuable result can be obtained by further inspection of this basic modalcircle. Suppose we have two specific points on the circle, one corresponding to afrequency below the natural frequency, and the other to one above thenatural frequency (see figure 9.3.1b). Then we can write for the correspondingangles of the circle

, (9.23a,b)

and from these two equations we can obtain an expression for the relative viscousdamping of the mode

(9.24)

Armed with the above insight into the properties of the mobility near resonance, it isa straightforward matter to devise an algorithm to extract the modal parameters of aparticular mode given experimental discrete-frequency data. The algorithm reads

I. Fit a circle to the experimentally determined mobility transfer functionaround the resonanse.

dθ dω⁄ω ωr=

ωb ωa

θb

2-----tan

1 ωb ωr⁄( )2–2ζrωb ωr⁄

--------------------------------=θa

2-----tan

ωa ωr⁄( )2 1–2ζrωa ωr⁄

--------------------------------=

Re Y

Im Y

Increasing ω

ω ωr=

ω

γ θ

Re Y

Im Y = ω ωb

θb

Figure 9.3.1. Properties of the modal mobility circle. Experimentally determineddiscrete mobilities, of equal frequency increment, are indicated by dots.

θa

= ω ωa

(a) (b)

R

ζr ωa2 ωb

2–( ) (2ωr(ωaθa

2-----tan ωb

θb

2-----))tan+⁄=

EXPERIMENTAL MODAL ANALYSIS

66

II. Locate the natural frequency by finding the maximum sweep rate by numerical differentiation.

III. Obtain damping estimates, as of equation (9.24), by considering discretemobility data on both sides of the natural frequency.

IV. Determine the eigenvector elements by fixing the modal mass (say: )and determine from the radius of the circle associated to a direct mobilityelement . The radius Rkk is .

V. Determine the remaining eigenvector elements ( ) by considering theradii of the mobilities Ykj ( ) with fixed modal mass mr and now giveneigenvector element . Here .

The first step can be performed by any curve-fitting routine which finds a circlewhich gives a least-square fit to experimental data. The second step isstraightforward for experimental data with linear frequency increments which aremost common. In the third step, there are many choices for the selection offrequencies and . Different choices should be evaluated and the scatter indamping estimates determined. If the deviation is less than 5%, reasonable gooddamping estimates has been obtained. Steps IV and V are relatively straightforwardto process.

It is advisable to synthesize the mobility functions from the extracted modalparameters and to compare it with the experimental raw mobility data. If thecorrelation between the two is not good, proper action should to be taken. Seereference [9.2] for more information.

9.4 *Complex exponential

9.5 Mode indicator functionsMany mechanical systems under test are found to be little damped. In that case thestructural eigenmodes are close to being real-valued. Such real modes, called nor-mal modes, may be identified more easily by a proper loading of the structure. It isthe purpose of pre-test methods to find such load distributions that maximizes thelikelihood to obtain normal modes and thus minimizes the real part of the displace-ment response. Consider the displacement response for which we have

(9.25)

max dθ dω⁄( )

mr 1=ρk

r( )

Ykk Rkk ρkr( )( )

24mrζrωr⁄=

ρjr( ) j k≠

Rkj j k≠ρk

r( ) Rkj ρjr( )ρk

r( ) 4mrζrωr⁄=

ωa ωb

q ω( )

q ω( ) H ω( )Q ω( )=

67

MODE INDICATOR FUNCTIONS

where the individual elements of the dynamic flexibility (receptance) matrix are (see also Equation 9.13)

(9.26)

For real-valued excitation vectors , i.e.all elements in Q is either in phase or180o out-of-phase, we may expand Equation 9.25 into real and imaginarycomponents. Henceforth, we drop the angular frequency from the notation and have

(9.27)

Now, if a normal mode could be excited at a particular frequency , i.e., a load vector Q must be found such that the real part

of the response vector is as small as possible as compared to the total response q. Wedefine the norm of the total response to be

(9.28)

By minimizing the ratio of the norm of the real part to the norm of the totalresponse1, under the condition that the norm of the loading is constant, we have

(9.29)

The minimum is thus found to be the smallest eigenvalue of the eigenvalueproblem related to the Rayleigh quotient of Equation 9.29 as

(9.30)

Plotting the smallest eigenvalues as functions of frequency (Multivariate ModeIndicator Functions, MMIF) clearly show at which frequencies normal modes exist.The load vectors that excite these modes the best are given by the associatedeigenvectors to the problem. Multiple simultaneous drops of the MMIF functions

1. Sometimes mass-weighted norms are used instead, i.e.

H ω( )

Hi jρi

k( )ρjk( )

mk ωk2 2iζkωkω ω2–+( )

----------------------------------------------------------k 1=

N

�=

Q ω( )

ℜ q( ) iℑ q( )+ ℜ H( )Q iℑ H( )Q+=

ρ k( ) ωk

Hi j ωk( ) ρikρj

k 2iζkωk⁄≈ ℜ q( )

q qHq ℜ q( )Tℜ q( ) ℑ q( )Tℑ q( )+= =

QTℜ H( )Tℜ H( )Q QTℑ H( )Tℑ H( )Q+=

QT ℜ H( )Tℜ H( ) ℑ H( )Tℑ H( )+[ ]Q=

q M qHMq=

min Q = const

ℜ q( )q

----------------- min Q = const

QTℜ H( )Tℜ H( )QQT ℜ H( )Tℜ H( ) ℑ H( )Tℑ H( )+[ ]Q--------------------------------------------------------------------------------------- λ= =

λ

ℜ H( )Tℜ H( )Q λ ℜ H( )Tℜ H( ) ℑ H( )Tℑ H( )+[ ]Q=

EXPERIMENTAL MODAL ANALYSIS

68

indicate the multiplicity of the structural resonances at the particular frequency (seeFigure 9.5.1).

9.6 *Stabilization diagram

9.7 Correlation indicatorsIdeally, when an experimental modal analysis is done and a finite element eigen-value analysis of the test object has been made, the results of the two should closelyresemble. Comparison of experimentally found and theoretically obtained modalproperties may be compared in various manners. The most obvious is probably thecomparison of resonance frequencies from the test with eigenvalues of the finite ele-ment analysis. If they match to magnitude and number, in the frequency range of thetest, it is most likely that the experiment has been performed well and that the ana-lytical model closely match the test object and its supporting conditions.

Another way of comparing is by means of eigenvector comparison. Since the test isnormally not conducted with measurement of all the degrees-of-freedom of theanalytical model, the associated eigenvector elements have to be extracted from thefinite element model’s modal matrix P. Let us call the partition of P that correspondto the test sensor locations and orientations holding the eigenvectors .

0 0.5 1 1.5 2 2.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 9.5.1. Typical graph of MMIF functions. The significant simultaneous dropof two MMIF functions indicate eigenfrequency doublett.

MM

IF (t

wo

low

est)

ωω1 ω2, ω3 ω4

PA ρ k( )A

69

CORRELATION INDICATORS

Similarly, let us call the experimentally obtained eigenvectors , which may becollected as columns in the experimental mode matrix . Since the eigenmodesmay be determined to shape but not to magnitude, any eigenvector comparisonshould be made independent of eigenvector scaling. A graphical means forcomparing eigenvectors is shown in Figure 9.7.1. When experimental eigenvectorelements are plotted against their analytical counterparts, they should ideally belined up along a straight line in the plot. Another, non-graphical, correlationmeasure is to calculate the angle between the experimental and analyticaleigenvectors. For co-linear and therefor similar eigenvectors, a such angle should bezero (or 180o). If instead we take the cosine squared of this angle, we end up withthe Modal Assurance Criterion (MAC) correlation number

(9.31)

If the modes and are co-linear, this index is unity. If the modes areorthogonal, and thus fully un-correlated, it is zero. For partially correlatedeigenvector pairs, the number is between zero and unity. The MAC(i,j) indices maybe considered as elements of a matrix.

ρ k( )X

PX

−2 −1.5 −1 −0.5 0 0.5 1

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0

0

-1

-1

EMA

FEA

Figure 9.7.1. Comparison of mode shapes from experimental modal analysis (EMA)and finite element analysis (FEA). Experimental vector elements are plotted versusanalytically obtained elements. Good correlation but different mode shape scalingcan be observed.

MAC i j,( ) ρ i( )X( )Tρ j( )A( )

2ρ i( )X ρ j( )A( )

2⁄=

ρ i( )X ρ j( )A

EXPERIMENTAL MODAL ANALYSIS

70

If no modes are missing in the experimental and analytical mode sets, and theeigenmodes are ordered according to increasing frequency order, the MAC matrixshould ideally have ones along its diagonal. Note, however, that although theeigenmodes should be mutually orthogonal (in some sense) does not mean that theideal MAC matrix has zeroes everywhere outside its diagonal. This is so becausenot all structural degrees-of-freedom are measured and thus present in the modematrices.

In practice it is often found that the diagonal elements of the MAC matrix is farfrom being ones. One reason for this is that, also for good analytical models,multiple eigenvalues or close-to-multiple eigenvalues exist. For such models, it iswell known that the corresponding eigenvectors may change drastically for smallperturbations in model parameters, see Figure 9.7.2. Experimental/analytical modepairs that give a MAC index greater than 0.95 should be considered as closelycorrelated, while those giving an index less than 0.8 should be considered as poorlycorrelated.

Figure 9.7.2. Eigenvalues and two highest MAC numbers versus a modelparameter . Correlation is made to the second eigenmode of the system with

.θ

θ 1.4=

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

0.5

1

1.5

2

2.5

3

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.0 1.5 2.0

1.0

0.5

MAC

σi

71

EXPERIMENTAL MODE EXPANSION METHODS

Normally, when computing the MAC, it is seen that a simple re-arrangement of themodes (in either the experimental or analytical mode sets) would give a MACmatrix with diagonal elements much closer to unity. The analytical model and thetest object is thus in some respect similar, but the order of their eigenvalues are notthe same. When such re-arrangement has been made, pairs of modes that are wellcorrelated has been identified. Let us denote the associated mode matrices of suchre-arranged modes (in which some of the original eigenmodes may have been leftout, but m modes remain) and , for the experimental and analytical modesrespectively. Another index, known as the CO-ordinate Modal Assurance Criterion(COMAC) number could be constructed by considering rows, corresponding toindividual degrees-of-freedom, of the mode matrices. Before such indices arecalculated, the eigenvectors have to be normalized to be of unit length. They alsoshould be rotated, by a multiplication of a complex number of unit magnitude, suchthat the norm of the eigenvector pair difference is minimized. Definingthe COMAC number as

(9.32)

a set of numbers, of value between zero and unity, are obtained. For a set of pairedmodes, which give great MAC indices, a COMAC(i) index far from being zeroindicated an experimental error for the i:th degree-of-freedom. Reasons for this isoften loose accelerometers or badly connected wires, but it may be caused by manyother sources.

9.8 Experimental mode expansion methodsIn most situations the modal test need to be limited with respect to the number ofused sensors, usually because of cost or time constraints. The aim of the testing thenhas to be to give sufficiently rich information for subsequent analysis. When thespatial resolution of the test is low, sometimes the structural dynamic characteristicsare required also for locations which were not instrumented. In that case, an analyti-cal model may be of help to interpolate the experimental data to such locations.

A popular scheme for eigenmode expansion is the System Equivalent ReductionExpansion Process (SEREP). It sets out from the truncated modal matrix of theanalytical model. If we partition the modal matrix according to measured andomitted degrees-of-freedom of the test, we have

(9.33)

PX PA

ρ i( )X ρ i( )A–

COMAC i( ) 12m------- Pi .

X Pi .A

–=

PA

PA PmA

PoA

n M×

= PmA Pm

A( )r M×=

EXPERIMENTAL MODAL ANALYSIS

72

where n is the number of finite element degrees-of-freedom, M is the number ofretained modes to be used and r is the number of measured responses. Theeigenfrequency spectrum of the retained modes should span the frequency range ofinterest of the test.

Now, if we have an experimentally determined eigenmode , we can find the bestlinear combination, in a least square sense, of the retained analytical modes thatapproximate the experimental mode as

(9.34)

The combination factors in x can be used for expansion to the full structural size as

(9.35)

and therefore, the interpolated eigenvector of the un-measured degrees-of-freedomis

(9.36)

One also obtains an approximation, smoothed through the use of the analyticaleigenvectors, as

(9.37)

Another popular scheme for eigenvector expansion is through the use of the Guyanreduction matrix. In the Guyan reduction method, the reduction matrix is given bystatic condensation. Again, let us partition matrices according to the measured andomitted degrees-of-freedom and consider the static problem

(9.38)

which gives that the quasi-static response of the unmeasured degrees-of-freedom to the loading as

(9.39)

ρmX

PmA x ρm

X≈ x PmA( )

TPm

A[ ]1– Pm

A( )Tρm

X=

ρX PmA

PoA

x≈

ρoX Po

A PmA( )

TPm

A[ ]1– Pm

A( )Tρm

X=

ρmX Pm

A PmA( )

TPm

A[ ]1– Pm

A( )Tρm

X=

Kmm Kmo

Kom Koo

qm

qo� �� Qm

0� ��

=

qo

Qm

qo Koo1– Komqm– Sqm≡=

73

EXPERIMENTAL MODE EXPANSION METHODS

The total response q thus is

(9.40)

Now, if we apply as the response , the experimental eigenmode , we have anapproximation to the unmeasured partition of the eigenvector

(9.41)

as an alternative formulation to the SEREP method.

BIBLIOGRAPHY

9.1 Abrahamsson T.J.S., Modal Analysis and Synthesis in Transient Vibrationand Structural Optimization Problems, Dissertation at Chalmers Universityof Technology, 1990

9.2 Ewins D.J., Modal Testing: Theory and Practice, Research Studies PressLtd., 1986

9.3 Kammer D.C., Sensor Placement for On-Orbit Modal Identification and Cor-relation of Large Space Structures, Journal of Guidance, Control, andDynamics, 14(2), 1991

q IS

qm=

qm ρmX

ρoX Sρm

X K– oo1– Komρm

X= =

EXPERIMENTAL MODAL ANALYSIS

74

75

10 SYSTEM IDENTIFICATION

10.1 IntroductionThe number of methods for system identification from discrete time series inputoutput data of linear systems is plentiful. Such are the AR, ARX, ARMA, ARMAX,IV and PEM methods1 (see reference 10.1). Most of them are iterative in nature andrequire that initial estimates of the model parameters are provided, and increasinglymore difficult task as the model complexity (more states, more inputs and more out-puts) grows. Recent methods, more useful in vibrational engineering, are non-itera-tive and rely heavily on numerical linear algebra. A major advantage of thesemethods is that the user are left with only a few choices, of which the estimation ofthe number of significant states, or the appropriate model order, is the most impor-tant. For this choice, however, there are methods and correlation numbers support-ing the user. Two methods will be described here, the ERA and N4SID algorithms.The ERA method is basically a method for system identification of free decaymotion. It is included here because of its widespread use in the vibrational engineer-ing society. A more recent and general method is the N4SID algorithm, which is forthe system identification from input/output data of combined deterministic and sto-chastic systems, such as given by equations (2.1a,b).

Before we go into details about the state-space identification procedures, let us for amoment assume that not only the inputs and outputs of the realization are known but

1. AR - AutoRegressive, ARX - AutoRegressive with eXtended input, ARMA - AutoRegressive with Moving Average, ARMAX - AutoRegressive with Moving Average and eXtended input, IV - Instrumental Variable method, PEM - Prediction Error Method

SYSTEM IDENTIFICATION

76

also the state sequence. Later we will see that what the state-space methods do is toprovide us with a such sequence. By formulating the discrete-time realization

(10.1a,b)

on matrix form we have

(10.2)

We note that the problem of obtaining the state-space matrices is basically a linearregression problem. The least squares solution to (10.2) is

(10.3)

Here denotes the pseudo inverse. The residual to the least squares problemdefine the noise sequence, i.e.

(10.4)

The covariance matrix for can now also be determined easily by as thesample sum of the squared residuals. That will give the covariance and crosscovari-ance matrices for w and v. These matrices will allow us to compute the Kalman filtergains for the realization (10.1a,b).

Although, theoretically, we are able to extract the state-space matrices from a singleleast squares solution, in practice this procedure has been found to produce lowquality estimates of the and matrices. The and matrices, however, areusually good. One notes that, for fixed and , the estimation problem is linear in

and . We may calculate these from input/output data by use of simulations.The procedure is as follows. Let be a column vector of the output vectors,

xk 1+ Axk Buk wk+ +=

yk Cxk Duk vk+ +=

xk 1+

yk� � � �� ��

A BC D

xk

uk� � � �� �� wk

vk� � � �� ��

+=

A BC D

xk 1+

yk� � � �� ��

xk

uk� � � �� ��

†

=

()†

wk

vk� � � �� ��

xk 1+

yk� � � �� ��

A BC D

xk

uk� � � �� ��

–=

wkT vk

T{ }T

B D A CA C

B DY yk

77

*THE EIGENSYSTEM REALIZATION ALGORITHM

, stacked on top of each other. Furthermore, let and be columnvectors in which the columns of the and matrices are stacked on top of eachother, respectively. We then have

(10.5)

in which the columns of the influence matrix in found by simulating, in turn, theresponse contribution to unitary elements of and direct calculation of theresponse contributions to unitary elements of . These calculations has to be car-ried out with all but the unitary elements set to zero and with fixed and . The and matrices may then be extracted from the least squares solution

(10.6)

This procedure has been found to produce high quality estimates of the state-spacematrices. The simulation work may be decreased by transforming the realization todiagonal form. Also, the procedure may be extended to estimate a non-homoge-neous initial state besides the and matrices.

10.2 *The eigensystem realization algorithm...

10.3 A state-space sub-space method for deterministic systemsIn the following, the N4SID state-space sub-space method[10.3] for the deterministicsystem is presented. The state-space sub-space methods utilize the shift property ofthe observability matrix . We notethat the upper partition of holds the C matrix. We also note that a block shiftdown (r rows) in the observability matrix corresponds to a post-multiplication of thematrix . Let us denote the top rows of with , and the bottom rows with and we have

(10.7)

We may thus obtain the matrix by use of the pseudo-inverse of as

(10.8)

k 1 2 …, ,= B DB D

Y ΓB

D� ��

=

ΓB

DA C B

D

B

D� �� ÆY=

x0 B D

[ CT CA( )T CA2( )T … CAn 1–( )T ]T=r n×

A rn r– rn r–

A=

A

A †=

SYSTEM IDENTIFICATION

78

We thus have the necessary tools to obtain all state-space matrices from input/outputdata, provided the observability matrix is known. The establishing of that matrix isthe fundamental step in the sub-space identification.

Before we state the solution to the identification of the observability matrix, weneed a few definitions and some concepts from linear algebra. We introduce theinput and output block Hankel matrices U and Y, respectively, as

(10.9)

(10.10)

As can be seen, the upper and lower partitions of the block Hankel matrices arecalled the “past” and “future” block Hankel matrices. We may note that the notationof past and future partitions of the Hankel matrices is somewhat dubious. The parti-tions are seen to hold many block elements in common. However, the notation maybe justified by the fact that for individual columns there is a distinct border linebetween past and future data in the sense that they are ordered consecutively fromtop to bottom. The notation has been introduced to support intuitive conceptual dis-cussions about the method.

Typically, all available samples are included in the matrices which makes samples in total.

UUp

Uf

≡

u0 u1 u2 … uj 1–

u1 u2 u3 … uj

… … … … …ui 1– ui ui 1+ … ui j 2–+

ui ui 1+ ui 2+ … ui j 1–+

ui 1+ ui 2+ ui 3+ … ui j+

… … … … …u2i 1– u2i u2i 1+ … u2i j 2–+

=

“past” inputs

i block rows

“future” inputs

i block rows

YYp

Yf

≡

y0 y1 y2 … yj 1–

y1 y2 y3 … yj

… … … … …yi 1– yi yi 1+ … yi j 2–+

yi yi 1+ yi 2+ … yi j 1–+

yi 1+ yi 2+ yi 3+ … yi j+

… … … … …y2i 1– y2i y2i 1+ … y2i j 2–+

=

“past” outputs

i block rows

“future” outputs

i block rows

2i j 1–+

79

A STATE-SPACE SUB-SPACE METHOD FOR DETERMINISTIC SYSTEMS

Similarly to the input and output Hankel matrices, we may introduce the combinedinput/output block Hankel matrix. The upper partition of it (the “past” inputs/out-puts) is

(10.11)

Also, the oblique projection of matrices need to be defined for the following presen-tation. It is defined via the orthogonal projection of a matrix. We introduce twomatrices and of the same column dimension, with being of full rank.The orthogonal projection of along the row space of is then

(10.12)

A graphical illustration is given in figure 10.3.1a.

Now let be yet another matrix of the same column dimension as and ,with being of full rank. The orthogonal projection of on the jointrowspace of and is

WpUp

Yp

=

Mα Mβ MβMα Mβ

proj Mα Mβ,( ) MαMβT MβMβ

T( )†Mβ≡

Figure 10.3.1a. Interpretation of the orthogonal projection in the 2-dimen-sional space. is formed by projecting the row space of onthe row space of .

Figure 10.3.1b. Interpretation of the oblique projection in the 2-dimensionalspace. The oblique projection is formed by projecting, on the row space of

, the row space of along the row space of .

proj Mα Mβ,( ) MαMβ

Mγ Mα Mβ

Mα

Mβproj Mα Mβ,( )

(a)(b)

MαMβ

Mγ

proj Mα Mγ Mβ, ,( )

Mγ Mα MβMβ

T MγT[ ]T Mα

Mβ Mγ

SYSTEM IDENTIFICATION

80

(10.13)

The oblique projection on the row space of of the row space of along therow space of is then defined as

(10.14)

with as defined by equation (10.13). The oblique projection is illustrated in fig-ure 10.3.1b.

By now we have the necessary tools to state the sub-space identification theorem.Let the projected future outputs be the oblique projection, on the row space ofthe combined past input/output Hankel matrix , of the future output Hankelmatrix along the row space of the future input Hankel matrix , i.e.

(10.15)

Let its singular value decomposition be

(10.16)

Then the theorem states that, for a noise-free system, the system order is equal to thenumber of non-zero singular values S1 of equation (10.16). Furthermore, it statesthat the extended observability matrix

, (10.17)

proj Mα MβT Mγ

T[ ]T,( ) Mα MβT Mγ

T[ ]Mβ

MγMβ

T MγT[ ]T

†Mβ

Mγ=

Nβ Nγ[ ]Mβ

Mγ≡ NβMβ NγMγ+=

Mγ MαMβ

proj Mα Mγ Mβ, ,( ) NγMγ≡

Nγ

YfWp

Yf Uf

Yf proj Yf Wp Uf, ,( )=

Yf U1 U2[ ]S1 0

0 S2

V1T

V2T

=

x

CCA. .

CAi 1–

= i n>

81

*RECURSIVE SYSTEM IDENTIFICATION

is equal to

(10.18)

For proof of the theorem, see reference 10.6. All ingredients for a sub-space state-space algorithm is now in place. The algorithm (N4SID) can be formulated as:

1. Establish with given input/output data the output, input and combined input/out-put block Hankel matrices and compute the projected output Hankel matrix .

2. Compute the SVD of the projected output Hankel matrix. Determine the systemorder by counting the number of non-zero singular values.

3. Compute the extended observability matrix by use of equation (10.18).

4. Extract the C matrix from the first r rows of the extended observability matrix.

5. Compute the matrix by use of equation (10.8).

6. Compute, by use of simulation, the elements of the matrices and D usingequation (10.6).

There are restrictions on the use of the above state-space sub-space algorithm. First,the number of block rows i in the past and future input and output Hankel matricesmust be greater than the system order. Here the user must make an estimation of thesystem order and provide the algorithm with large enough data matrices. Second,the input sequence must be persistently exciting of order 2i. That is that the inputcovariance matrix must be of full rank, see reference 10.3. It is a user action to cre-ate such input sequence for the test. Third, the intersection of the row space of thefuture input Hankel matrix and the row space of the past states must be empty,see reference 10.6.

10.4 *Recursive system identification

x U1S11 2⁄=

Yf

x

A

B

Xp

SYSTEM IDENTIFICATION

82

BIBLIOGRAPHY

10.1 Ljung L., System Idenfification - Theory for the User, Prentice Hall, 1987

10.2 Juang J.N. and Pappa R.S., An Eigensystem Realization Algorithm for ModalParameter Identification and Model Reduction, Journal of Guidance andControl 8, 1985

10.3 Van Overschee P. & De Moor B., ‘N4SID: Subspace Algorithms for the Iden-tification of Combined Deterministic-Stochastic Systems’, Automatica 30(1),1994

10.4 Verhaegen M, ‘Identification of the Deterministic Part of MIMO State SpaceModels Given in Innovations Form from Input-Output Data’, Automatica30(1), 1994

10.5 Larimore W.E., ‘Canonical Variate Analysis in Identification, Filtering andAdaptive Control’, Proc. 29th Conference on Decision and Control, 1990

10.6 Van Overschee P. and De Moor B., Subspace Identification for Linear Sys-tems, Theory-Implementation-Applications, Kluwer, 1996

Aadditivity 1adjoint

eigenvalue problem 23algebraic

eigenvalue problem 23algorithmic damping 19

Bbalanced

realization 50bisection method 32

Ccanonical

realization 8characteristic equation 25controllability

Gramian 44matrix 39

convolution integral 12

Ddeficient 17deficient matrix 25diagonal realization 7discrete time transition matrix 15dynamic flexibility 61

Eeigenvalue

problem 23eigenvalue problem

adjoint 23eigenvectors

bi-orthogonal 24left 24right 24

FFoss dampings 60

GGerschgorin disks 27Gerschgorin’s theorem 27Gramian

controllability 44observability 44

HHankel matrix 17

block 17homogeneity 1

JJordan form 25

MMarkov parameters 16modal a

s 60modal damping 61modal masses 61modal stiffnesses 61

NNyquist plane 64

Oobservability

complete 36Gramian 44matrix 36theorem 35

observability matrixextended 80

outputs 4

Pproportionally damped 60

RRayleigh’s theorem 30reachability

matrix 39problem 39

realization 6balanced 50canonical 8

receptance 61relative viscous damping 61

Ssampling frequency 15similar matrices 25similarity transformation 25similary transformation 7state controllable 37

completely 37state-transition matrix 12Sturm sequence 31superposition integral 12

Zzero-order-hold 15