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Laboratory Session – Identification techniques – Dr. Ing. G. Totis 1

UNIVERSITA’ DEGLI STUDI DI UDINE - DIEGM DOTTORATO IN INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE UNIVERSITA’ DEGLI STUDI DI UDINE

DOTTORATO DI RICERCA IN INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE

MODULE 3 – Laboratories for engineering research

Laboratory Session S6: Experimental identification of dynamic systems

and manufacturing processes

Dr.Ing. G. Totis

09 Marzo 2016 Udine – Polo scientifico

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UNIVERSITA’ DEGLI STUDI DI UDINE - DIEGM DOTTORATO IN INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE Prerequisites

• Fundamentals of system and control theory (dynamic systems, stability, …)

• Fundamentals of discrete and random signals processing

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UNIVERSITA’ DEGLI STUDI DI UDINE - DIEGM DOTTORATO IN INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE


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Why identification and modeling?

• Gain increased knowledge about the process behavior

• Validation of theoretical models• Process supervision for exception/fault detection• (Possibly on-line) optimization of system or

process• Signal forecast – make predictions• Design of digital control algorithms and tuning of

basic controller parameters• Development of adaptive control algorithms• …



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Theoretical vs experimental modeling

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Model simplification



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Different kinds of models

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Observations

• Theoretical analysis can become quite complex even for simple systems;

• Mostly, model coefficients derived from the theoretical considerations are not precise enough;

• Not all actions taking place inside the system are known;

• Some systems are very complex, making the theoretical analysis too time-consuming;

• Semi-empirical models can be obtained in shorter time with less effort compared to pure theoretical modeling.

Thus, grey-box or black-box models are often preferred to pure white-box models!



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The problem of identification

Identification is the experimental determination of the input-output behavior of a process or system. One uses measuredsignals and determines the input-output behavior within a classof mathematical models. The error (respectively deviation)between the real process or system and its mathematical modelshall be as small as possible.

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Error calculation

Identification methods can be classified according to the way the error between model and real process is calculated during the identification phase



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SIMO, MISO, MIMO systems

For MIMO systems, one has to apply special identification techniques!

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Identification procedure

Identification methods must be able to determine the input-output behavior under the constraints imposed by• known (adjustable control) inputs• uncontrollable unknown disturbances• uncontrollable but measurable disturbances• process (performance) outputs• other measurable or observable outputs • limited measurement time• confined test input signal amplitude• constrained output signal amplitude• purpose of the identification• available budget.



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Then, a priori knowledge must be collected, such as• recently observed behavior of the process, rough

models from previous experiments;• recommended operating conditions for conduction

of measurements;• physical laws governing the process behavior, hints

concerning linear/non-linear behavior time-invariant/time-variant behavior proportional/integral/derivative behavior order of the process (how can you determine it a

priori or from preliminary measurements?).

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Now, the measurement can be planned. One has to define• The type of input signals (normal operating signals or

artificial test signals and their shape);• sampling frequency (Shannon theorem!) and measurement

time horizon;• measurements in closed-loop or open-loop operation of the

process, on-line or off-line identification, real-time or not;• necessary equipment (e.g. oscilloscope, PC, ...);• filtering for elimination of noise;• limitations imposed by the actuators (saturation, ...).

Measurements are then executed. Signals are pre-processed and then analyzed. Model is identified and validated by direct comparison with

experimental results and (possibly) with theoretical models. The identification process is iterated if necessary.



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Classification of identification methods

The different identification methods can be classified according to the following criteria: class of mathematical model class of employed input (test) signals way the error between process and model is

calculated adopted approaches for experimental execution

and data evaluation (online, offline,…) employed algorithms for data processing (e.g.

time or frequency domain) adequacy for SISO, SIMO or MIMO models generated model complexity (SDOF or MDOF).



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Classes of mathematical models

• Non-parametric vs parametric

• Continuous vs discrete time

• Deterministic and/or stochastic

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Non-parametric models

• Non-parametric models are models without specific structure and basedon a (theoretically) infinite number of parameters. In this case, the relationbetween a certain input and the corresponding response can berepresented by a simple table or by a sampled characteristic curve.Examples are impulse responses, step responses, or frequency responsespresented in tabular or graphical form.

𝑊 𝑗𝜔𝑘 =1

𝑁

𝑖=1

𝑁

𝑊𝑖 𝑗𝜔𝑘 ℎ 𝑘𝑇 =

1

𝑁

𝑖=1

𝑁

ℎ𝑖 𝑘𝑇

Inverse Fourier Transform

Empirical Transfer Function Estimate - ETFE

Empirical Impulse Response Estimate - EIRE



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Parametric models

• Parametric models are models with a structure and based ona finite number of parameters. For example equations whichexplicitly contain the process parameters, such as

Continuous time Discrete time

nth order differential eq. / finite difference eq.

state space representation

transfer function

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Stochastic models

Auto Regressive - AR

Auto Regressive Moving Average -ARMA

Moving Average - MA

ν(k) is white noise (the stochastic term)y(k) is (disturbed) process output u(k) is process (known control) input



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Deterministic models with stochastic disturbances

Auto Regressive with eXogenous (control) input -ARX

Auto Regressive Moving Average with eXogenous input - ARMAX

Infinite Impulse Response model with output error – IIR+OE

𝑦 𝑘 + 𝑎1𝑦 𝑘 − 1 +⋯+ 𝑎𝑛𝑎𝑦 𝑘 − 𝑛𝑎 =

𝑏1𝑢 𝑘 − 1 +⋯+ 𝑏𝑛𝑏𝑢 𝑘 − 𝑛𝑏 + 𝜐 𝑘

𝑦 𝑘 + 𝑎1𝑦 𝑘 − 1 +⋯+ 𝑎𝑛𝑎𝑦 𝑘 − 𝑛𝑎 =

𝑏1𝑢 𝑘 − 1 +⋯+ 𝑏𝑛𝑏𝑢 𝑘 − 𝑛𝑏 +

+𝜐 𝑘 + 𝑑1𝜐 𝑘 − 1 +⋯+ 𝑑𝑛𝑑𝜐 𝑘 − 𝑛𝑑

𝑤 𝑘 + 𝑎1𝑤 𝑘 − 1 +⋯+ 𝑎𝑛𝑎𝑤 𝑘 − 𝑛𝑎 =

𝑏1𝑢 𝑘 − 1 +⋯+ 𝑏𝑛𝑏𝑢 𝑘 − 𝑛𝑏𝑦 𝑘 = 𝑤 𝑘 + 𝜐 𝑘

Finite time-duration Impulse Response model with output error – FIR+OE

𝑦 𝑘 = 𝑏1𝑢 𝑘 − 1 +⋯+𝑏𝑛𝑏𝑢 𝑘 − 𝑛𝑏 + 𝜐 𝑘

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Classes of excitation signals

For identification purposes, one can supply the system either with the operational input signals or with artificially created signals (test signals) satisfying the following criteria:

• they should be simple and reproducible (with or without signal generator);

• simple mathematical description of the signal and its properties;

• realizable with the given actuators

• applicable to the process, and goodexcitation of the interesting system dynamics.

They can be

• periodic or non-periodic

• transitory or stationary

• deterministic or stochastic

Overview of some excitation signals. (a) non-periodic: step and square pulse.(b) periodic: sine wave and square

wave. (c) stochastic: discrete binary noiseOther important examples: impulse, chirp



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Different setups for model identification

Offline identification: the measured data are first stored and are later transferred to the computer utilized for data evaluation and model identification. The algorithm employed is generally based on batch processing (the previously stored measurements will be processed in one shot), and it is non-recursive. It can be further subdivided into: direct processing (model is determined in one pass); iterative processing (model is determined step-

wise, i.e. iteratively and data must be processed multiple times).

Online identification: model identification is performed simultaneously parallelly to the experiment. The computer is coupled with the process and the data points are operated on as they become available.Online identification usually implies real-time processing: data are processed immediately after they become available, which necessitates a direct coupling between the computer and the process. The model is updated and improved as soon as a new measurement becomes available (recursive processing ), until parameters estimates are sufficiently accurate.

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Overview of identification methods



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Overview of identification methods

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Overview of signal analysis techniques



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Determination of characteristic values

T

Example: thermometer

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Second order system - harmonic oscillator

u

2

2

2

2

1

2 1

1gain

natural frequency

damping

1 damped natural frequency

1 2 resonancefrequency

n n

n

d n

r n

mu cu ku F

U jW j

F j m j c j k

G

j j

Gk



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Determination of characteristic values

Approximation by a second order systemIdentification from step response (response to unitary step input)

2

max

max

1Step response: 1 sin

1

2

1

1ln 1

ntd

n d

P

u t G e t

G u

T

ue

G

u

G

Ou

tpu

t u

(t)

umax

G

NTP

u∞

Maximum relative overshoot

Integer number of oscillation periods Time t [s]

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Approximation by a second order system

*0per 0

sin per 0ntd

th t

e t t

*h t

(t1,u1)

(t4,u4)

1

4

1 1ln

2 4

/ 2

u

u

c mk

Identification from impulse response



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Approximation by a second order system

2r

G

W j

Identification from frequency response

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UNIVERSITA’ DEGLI STUDI DI UDINE - DIEGM DOTTORATO IN INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE Experimental identification

System dynamics are generally unknown. When analyzingelectro-mechanical systems, main approaches forexperimental identification of their frequency response are:

• frequency sweep, based on the application of anelectromagnetic/piezoelectric actuator and other sensors,which excite the system with sinusoidal inputs (with constantor slowly time varying frequency);

• pulse test, based on impact hammer and other sensors,which excite all system harmonics at the same time by meansof an impulsive input.

After measurements and pre-processing, an Empirical TransferFunction Estimate (ETFE) or an Empirical Impulse ResponseEstimate (EIRE) are obtained, which are the main (non-parametric) inputs for mathematical interpolation withparametric models.

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UNIVERSITA’ DEGLI STUDI DI UDINE - DIEGM DOTTORATO IN INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE Frequency sweep

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Example: internal turningM

OD

ULE

3 –

Lab

ora

tori

es f

or

engi

nee

rin

g re

sear

ch


UNIVERSITA’ DEGLI STUDI DI UDINE - DIEGM DOTTORATO IN INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE Pulse test on tooling system

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UNIVERSITA’ DEGLI STUDI DI UDINE - DIEGM DOTTORATO IN INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE Pulse test on tooling system

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UNIVERSITA’ DEGLI STUDI DI UDINE - DIEGM DOTTORATO IN INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE ETFEBoring bar made of steel (Young modulus E=210 Gpa, density ρ=7.87g/cm3), diameter D=16 mm, apsect ratio, L/D=6.5.

250 500 1000 20000.01

0.1

1

10

100

X: 1507

Y: 90

frequency [Hz]

Am

pli

tude [

m/N

]

62.5 125 250 500 1000 2000

-180

0

frequency [Hz]

Phase

[deg]

0.8

830

0.013

n r

mG

N

Hz

Modal parameters from ETFE:

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UNIVERSITA’ DEGLI STUDI DI UDINE - DIEGM DOTTORATO IN INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE A crucial question

• Are the obtained results realistic?• It is highly recommmeded to compare intermediate and final

results with the outcome of (over)simplified models possibly based on well known physical principles.

• By an engineering point of view, a relative error of ±10% is wonderful! ±50% would be acceptable in some cases! At least the right order of magnitude is mandatory!

4

3

64

3

DJ

LG

EJ

Moment of inertia

Tip static compliance of a cantilever bar

In the previous example• D = 16 mm• E=210000 Mpa• Leff = (6.5+0.5)x16 mm→G = 0.854 μm/N OK



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Advanced identification methods

Identification methods (modal analysis techniques) for generation of parametric models are classified according to the characteristics of the identification algorithm, i.e. whether• it is based on time or frequency domain;• it is direct (the coefficients of polynomials composing

the frequency response are first determined and modal parameters are derived afterwards) or indirect(modal parameters are straightforwardly determined).

• it can treat only SDOF or it can be extended to MDOFsystems;

• it can treat only SISO or it can be extended to SIMOor MIMO systems.

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Main techniques for SISO-MDOF systems

• Linear Least Squares Method in the Frequency Domain (~Zobel, Levy)

• Linear Least Squares Method in the Time Domain (~Prony, typical for IIE+OE or ARX models)

• Least Squares Complex Exponential Method (Time domain)

• Ibrahim Time Domain Method (SIMO) or Single Station Time Domain (SISO)

• Inverse Method (Ewins, Kennedy-Pancu)

• …



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Linear Least Squares Method in the Frequency Domain

Cost function to minimize

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unknown parameter vector

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Alternatively, other choices are possible. For instance, one can use |G(iω)| as weight. In this manner, resonance peaks should be correctly interpolated.

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UNIVERSITA’ DEGLI STUDI DI UDINE - DIEGM DOTTORATO IN INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE Important recommendations

• Simple low-order models should be tested first;• Model complexity can be increased until the interpolation

error becomes low in the frequency range of interest, or until undesired anomalies appear, such as

• zeros or poles with positive real part are unacceptable and they must be rejected, because zeros with positive real part would represent the

unrealistic situation of exponentially exploding inputs producing extinguishing outputs; besides, if a filter had to be derived from the model, they would cause the filter to be unstable;

poles with positive real part would cause the system to be unstable (which is false for most electro-mechanical systems which are dissipative and stable).

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MARTELLO

ACCELEROMETRICO

DINAMOMETRO

FRESA

MANDRINO

ACCELEROMETRO

Example 1: rotating dynamometer



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Example 1: model interpolation

Linear Least Squares Method in the Frequency Domain was applied;Best compromise was obtained with n=6, m=6-2=4 (sum of 3 harmonic oscillators);

This method works well when the modes are only lightly damped (high resonance peaks), they are well separated, and model order is not too high. Otherwise, undesired zeros or poles may appear!

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UNIVERSITA’ DEGLI STUDI DI UDINE - DIEGM DOTTORATO IN INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE Example 1: filter construction

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UNIVERSITA’ DEGLI STUDI DI UDINE - DIEGM DOTTORATO IN INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE Example 1: filter application

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Least squares for discrete time linear processes (Prony)

IIR+OE modelor ARX



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Example 2: plate dynamometer

|Fy,out/Fy,in|[N/N]

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Example 2: model interpolation

• After trial and error, best result (without anomalies) was obtained with 32 poles and 31 zeros. However, it is still not satisfactory. Interpolation is poor. Moreover, it resembles to a pure mathematical interpolation of transfer function shape, without grasping the underlying physical structure of the dynamic system.

• This method is not suitable for modeling complicated dynamic systems composed of several vibration modes. Low-frequency resonances are not correctly recognized, especially when high-frequency energetic resonances are present, which tend to hide the effect of the former ones.

• One reason of this failure is that the system is ill conditioned.



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Least Squares Complex Exponential Method

• It is an improvement of the Prony method• Main steps:

1. a system of discrete time equations is expressed in matrix form, whence the transfer function denominator coefficients ai (i=1,..,2N) are determined by linear regression;

2. natural pulsation and damping of each of the N modes are extracted from such coefficients;

3. a new system of discrete time equations is obtained, in order to derive mode gains Gi.

• This approach is more effective than the Prony method; poles are first identified by expoliting their good signal to noise ratio; gains are determined afterwards; eventually zeros are derived as a consequence of modes sum.

• Disadvantages: it is not possible to separate the effect of low and high-frequency modes from the effect of middle-frequency modes which are of main interest; a good compromise would be to apply the Least Squares Method in Frequency Domain only to determine mode gains.

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Single Station Time Domain Method

• Improvement of Prony method• Main steps:

– System of discrete time equations is expressed in matrix form, by shifting the exponential modes composing the impulse response of a wide time span

– Modal parameters are determined– The method is iterated and the results are evaluated through

the Modal Confidence Factor approach, in order to distinguish structural modes (good, true modes) from computational modes (ghosts arising from pure shape interpolation).

• Advantages: this method is quite effective for the detection of closely spaced modes; the verification of results robustness is implicit in the algorithm

• Disadvantages: it is rather complicated, it has the tendency to give nonconservative damping estimates with noisy data.

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UNIVERSITA’ DEGLI STUDI DI UDINE - DIEGM DOTTORATO IN INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE Inverse method (Ewins)It is based on the assumption of the following model structure

𝑊𝑚𝑜𝑑 𝑗𝜔 ≅ −𝐴𝐿𝜔2 +

𝑘=ℎ1

ℎ2𝑅𝑘

𝑗𝜔 2 + 2𝜉𝑘 𝑗𝜔 + 𝜔𝑛,𝑘2 +

𝐺𝐻

𝑗𝜔 𝜔𝑛,𝐻2+ 2𝜉𝐻 𝑗𝜔 𝜔𝑛,𝐻 + 1

𝜔𝐿 < 𝜔 < 𝜔𝐻

Frequency range is split into 3 regions (Low-frequency, Medium freqeuency –that of interest and High Frequency). Thus,

The identification procedure is based on this assumption: each resonance peak is highly influenced by the corresponding mode, whereas other modes give only a constant contribution, i.e.

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UNIVERSITA’ DEGLI STUDI DI UDINE - DIEGM DOTTORATO IN INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE Inverse method (Ewins)

1. Each mode should be preliminarly identified by hand or by using another method, such as the peak picking technique)

2. For each mode, modal parameters (Rk, ξk, ωn,k) are estimated (for algebraic details see [Ewins])

3. The other unknown coefficients AL and GH are also estimated by regression.

General remarks on the method:• It is very effective, except for highly damped modes or for very close

resonances. • The main advantage of this approach is that a realizable (causal) and stable

model transfer function is obtained, with a minimum number of poles and zeros with negative real part.

• This is a direct consequence of the assumed model structure. In this case, poles are easily identified from their resonance peaks, whereas zeros are automatically determined from the sum of the previously identified modes. This way, zeros detection is not influenced by the bad signal to noise ratio affecting the frequency ranges where zeros are located.

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UNIVERSITA’ DEGLI STUDI DI UDINE - DIEGM DOTTORATO IN INGEGNERIA INDUSTRIALE E DELL’INFORMAZIONE Example 2: plate dynamometer

OK!

OK!

(Here the Inverse Method was applied)

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Example 3: machining system dynamics in milling

Mode # Gk [μm/N] ωn,k [Hz] ξk [ ]

1 0.073 281 0.038

2 0.057 449 0.073

3 0.311 533 0.026

4 0.033 1250 0.037

5 0.065 >1500 ≈1

OK!

(Here the Inverse Method was applied)



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References

• R. Isermann, M. Münchhof, Identification of Dynamic Systems, An Introductionwith Applications, (2011), Springer-Verlag Berlin Heidelberg.

• D. J. Ewins, Modal testing: theory, practice and application, 2nd edition, (2000),Research Studies Press LTD, England Ewins

• L. Ljung, System Identification: Theory for the user, 2nd edition, (1999), Prentice-Hall PTR

• N.M.M. Maia, Extraction of valid modal properties from measured data instructural vibrations, PhD Thesis, (1988) University of London

• G. Totis, M. Sortino, S. Belfio, Wavelet-like Analysis in the Frequency-DampingDomain for Modal Parameters Identification, 26th DAAAM InternationalSymposium on Intelligent Manufacturing and Automation, DAAAM 2015, in presson Procedia Engineering (2016).

Thank you for attention!

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