TITLE OF THE ARTICLE - AdBoostersiomac.eu/iomac/2013/IOMAC_Guimaraes/files/papers/Load Estimation...The advantage of such an approach is the immediate possibility of its application

IOMAC'13

5th International Operational Modal Analysis Conference

2013 May 13-15 Guimarães - Portugal

APPLICATION OF ADVANCED LINEAR AND

NONLINEAR PARAMETRIC MODELS TO LOAD

RECONSTRUCTION

Piotr Czop1, Krzysztof Mendrok

2 and Tadeusz Uhl

3

ABSTRACT

The paper presents applications of the loading identification method, based on the inversion of linear

and nonlinear regressive parametric models, namely the reconstruction of (i) rail irregularities, (ii)

wheel movement in the vehicle suspension, and (iii) acoustic pressure inside the power plant steam

generator. The fidelity and adequacy of various model structures were evaluated in time or frequency

domain with the use of correlation coefficient between the measured and reconstructed load. The

experimental test results showed that, in the case of measurement data corrupted by noise, advanced

model structures (e.g. BJ) provided a slightly better accuracy of inversion than simple ones (e.g.

ARX). Moreover, in the case of objects with a nonlinear relationship between the input and output,

nonlinear model structures (e.g. NARX) had advantage compared to linear ones, as it was expected.

Keywords: linear model, nonlinear model, data-driven model, inverse model analysis

1. INTRODUCTION

Reconstruction of the input of the system by inverting the system’s model is important in multiple

applications. Input reconstruction is a technique frequently used in the Internal Model Control (IMC)

strategies [1] to invert data-driven parametric models and compensate the dynamics of the tracking

process [2], or for metrological purposes [3]. The literature, however, rarely addresses the problem of

dynamic inversion based on data-driven parametric models of mechanical structures and systems.

Nonetheless, the technique (model inversion) is applicable to the problems of load reconstruction in

mechanical systems in order to modify the dynamics of a structure or a system and to achieve better

performance, e.g. to lower the level of loading forces [4-5]. Load prediction in systems for which the

force signal cannot be directly measured due to constructional constraints, as in the case of forces

being exchanged by a wheel and the road or rail, is considered to be one of the most practical

applications of the inverse approach [6-7].

1 AGH University of Science and Technology, Krakow, [email protected]



P. Czop, K. Mendrok, T. Uhl

2

A model and its inverse can have either a parametric or a non-parametric representation in time or in

the frequency domain. A non-parametric representation uses a frequency response function (FRF),

called the spectral transfer function, or an impulse response function, while parametric representation

uses a transfer function or state-space equations. Examples of applications of the frequency response

function method are discussed in [8], while an impulse response function method is presented in [9,4].

The accuracy of inverse non-parametric models depends on the time or frequency resolution, i.e. the

number of samples available in a given signal realization. Non-parametric methods are not capable of

handling systems with closed-loop feedback [10], or those which are unstable or generate drift caused

by the presence of physical or geometrical nonlinearities, like nonlinear stiffness characteristics.

Moreover, the windowing technique used in minimizing the frequency leakage causes inevitable

distortions of the signals. An example illustrating the application of this approach to railway tracks,

described in [4], reveals some of its shortcomings, i.e. questionable validity at high frequencies and ill-

conditioning in multiple input-output inverse models. Parametric methods overcome these

shortcomings by providing better robustness in the case of short signal realizations and lower variance

of estimated parameters for systems with closed-loop configurations [10]. Moreover, the parametric

approach provides a significant advantage in reconstructing an input in time-varying systems

represented by a linear model of variable parameters, which are recursively adjusted based on real-

time data [11]. For example, the inverse problem of load reconstruction of vehicles moving on a

bridge requires time-varying models to properly reconstruct the load [12-13]. This approach is

supported by recursive techniques well-known in system identification theory, e.g. the recursive least

square (RLS) or the Kalman-filter approach.

This paper considers a transfer function or state-space approach which are adequate for tracking load

changes in the time or frequency domain. The advantage of such an approach is the immediate

possibility of its application as a hardware pole-zero filter. The drawback of the parametric approach,

compared to the non-parametric one, is the requirement to define a model structure. The model

structure has to be parameterized to reflect the dynamics of the system under consideration. This is a

challenge in advanced model structures where a priori knowledge concerning disturbances affecting

the systems is additionally required. If any a priori knowledge is not available, a blind search

procedure for the best structure can be applied using key measures of the model quality, e.g. best fit,

AIC. To conclude, the parametric approach is recommended when first-principle knowledge of the

system under investigation exists and the system is stationary, in the sense that its mass, damping and

stiffness properties do not significantly vary. If the stationarity conditions can not be fulfilled, an

adaptive system identification approach has to be applied to obtain an inverse adaptive model as

presented in [13].

In a parametric approach, a model can have a linear or nonlinear form. The linear parametric models

are discussed in [14] regarding their simple (ARX) and advanced model structures (ARMAX, BJ, ..,

PEM). The nonlinear parametric models are used for systems which show strong non-negligible

nonlinear distortion of the input-to-output signal. Such systems produce an output with a frequency

content which has many significant side harmonics in the case of a sinusoidal excitation. The most

popular representation of nonlinear models is a neural network [15-16]. The nonlinear inverse model

applications is demonstrated in [17].

2. MODEL INVERSION

Linear and nonlinear models have the form of a transfer function or a system of state-space equations.

The inverse model is obtained differently for linear and nonlinear cases [17]. A block diagram of the

procedure of inverting a linear model is presented in Fig 2.

5th International Operational Modal Analysis Conference, Guimarães 13-15 May 2013

3

Figure 1 Block diagram of the inversion procedure of a linear model [14]

The process of selecting an adequate model structure and an algorithm for estimating model

parameters is the initial step. The model structure is selected using the quality indicators of the direct

and inverse models’ fit to the data in the time and frequency domain, respectively. The selection

process is supported by models quality measures and visual inspection as proposed in Table 1.

Table 1 Criteria of the model structure optimality

Criterion Measure Values

Fit in the frequency

domain

Fit measure Reconstructed input (output of the

inverse model)

Fit in the time domain Fit measure Output of the direct model

Statistical properties of

model residuals

FPE and AIC Residuals of the direct model

Representation of modal

properties

Subjective

evaluation

Bode diagram and spectrum of

residuals; both obtained from the

direct model

The next step [14], estimation of the parameters of the selected model, is performed using the

available input-output data, and a one-step-forward prediction of a direct model output is computed.

The adequacy of model structures is then evaluated by means of two measures, referred to in the

literature as the Final Prediction Error (FPE) and the Akaike Information Criterion (AIC). The more

accurate the model, the smaller the values of the FPE and the AIC measures. Additionally, in order to

detect the presence of abnormalities in the frequency domain, a visual inspection of the Bode plot of

the input-to-output transfer path and the spectra of model residuals (the disturbance-to-output transfer

path) was performed for each identified model structure. Analysis of the extensive quantity of such

visual indicators (not presented here due to a lack of space) indicates the presence of no abnormalities.

Visual evaluation of model quality might also be supported by pole-stability diagrams, plotted as a

function of the orders of selected polynomials. The major criterion for model order selection is,

however, the comparison of the fit quality of the reconstructed inputs. The purpose of selection is to

obtain a suitable inverse filter capable of providing the best possible reconstruction of the input signals

with respect to the optimality criteria listed in Table 1. The strategy implemented for optimizing


4

selection of the model structure is the systematic search for a set of model structures that would satisfy

the criteria listed in Table 1.

A block diagram of the procedure of inverting a nonlinear model is presented in Fig. 2. The procedure

is similar to the procedure presented in Fig. 1.

Figure 2 Block diagram of the inversion procedure of a nonlinear model [17]

The nonlinear model does not need to be inverted in an analytical way, since it is directly identified to

map the output into the input. The model structure selection is performed by searching the most

adequate structure of the model (e.g. NARX, NARMAX, and nonlinear state-space), the orders, and

the configuration of the nonlinear regressor (i.e. number of neurons, types of their activation functions,

and the number of layers of neurons), see [16]. The Bode diagrams can only be used when the

nonlinear model is linearized around the operating point. However, such linearization should be

performed for a few operating points depending on the kind of nonlinearity, thus the approach would

be time-consuming. Therefore, only visual inspection of a time response is the most effective way to

evaluate the accuracy of the direct and inverse nonlinear model.

The inverse linear model is unstable if at least one of the zeros of the direct transfer function is located

outside the unit circle or inside the unit circle, for z-1

or z operators respectively. These zeros create a

non-minimum phase transfer function, and hereafter are referred to as non-minimum phase zeros. The

inverse transfer function can be stabilized, however, by factorization of the numerator B(z), as

discussed in [18]. The advantage of such a stabilization method is the lack of phase error and delay,

while the only disadvantage is a small gain error that is, moreover, negligible if the output signal

consists of low frequency components. This stabilization technique was applied to load reconstruction

by [6-7]. The stability of the inverse nonlinear model can not be proved since there is no general

stability theory for discrete nonlinear models. However, the model stability can be quantified for some

local operating regimes and when the input and output signals, as well as the activation functions of

the artificial neurons, are saturated. Under the assumption that the time-variation is sufficiently slow,

the stability of the neural network can be established by evaluating the eigenvalues of the linearized

system [16].

The procedure of inverting a model does not correspond to any physical phenomenon and, therefore,

inverse models always have a tendency to be unphysical. As all physical systems have a time delay as

well as limited bandwidth, an exact inverse advances the signal as a result of the delay and amplifies

high frequency noise without any bound, if the bandwidth is not restricted to the upper frequency band

of the inversing load. This phenomenon is the so-called ill-posedness of the inversion and requires

regularization techniques to be used to correctly estimate the load up to a certain bandwidth, typically

given by the frequency at which amplitudes of the signal and the noise are equal. A standard solution

of the regularization problem is to use low-pass ‘noise’ filters, which leads to a model and its realized


5

approximation to the prototype of inversion that are both strictly proper. This fact suggests that the

sampling rate of input and output signals has to be adapted to the maximal frequency of the

reconstructed signal, although a very low sampling rate can result in severe aliasing. On the other

hand, the sampling rate should allow the most important dynamics represented by vibration modes of

the structure to be captured correctly. The significance of identified modes can be classified according

to their energy levels while the structure is operating; which allows the most powerful ones to be

selected and the sampling rate to be defined.

3. THE MODELS

A linear time-invariant (LTI) system, mapping a single input onto a single output (SISO) and in

discrete time-steps, is represented by difference equations [10]. These equations take the form of (1),

where G(z-1

) and H(z-1

) are discrete-time transfer functions containing adjustable coefficients and

represent the input-to-output dynamics and the disturbance-to-output dynamics, respectively. The

transfer functions G(z-1

) and H(z-1

) are rational functions of the operator z-1

that take the form shown

on the right-hand-side of the equation [10]

)()()(

)()(

)()(

)()()()()()(

11

1

11

111 ie

zDzA

zCiu

zFzA

zBiezHiuzGiy

(1)

The polynomials A(z-1

), B(z-1

), C(z-1

), D(z-1

) and F(z-1

) are used for model parametrization. Special

cases of the LTI SISO general model structure (2) are listed below as predefined model structures

using the function notation to state their characteristic structural numbers [10]:

),,(ARX

),,(OE

),,,(ARARX

),,,(ARMAX

),,,,(BJ

),,,,,(PEM

knBnA

knFnB

kdDdBdA

knCnBnA

knDnCnFnB

knFnDnCnBnA

(2)

where nA, nB, nC, nD and nF are polynomial orders and k is the input-to-output delay [10]. A

nonlinear time-invariant system, mapping a single input onto a single output (SISO) and in discrete

time-steps, is governed by the following direct equation [10,16]

)(,),(),1(,),1()1( nBiuiunAiydiygiy (3)

The system can be identified as the inverse model [10,16]

)(,),1(),(,),(,),1(),()( 10 nBiyiunAdiyiydiydiygiu (4)

Special cases of the nonlinear SISO general model structure (3) are listed below as predefined model

structures using the function notation to state their characteristic structural numbers [10,16]:

),,(NARX

),,(NOE

),,,(NARMAX

knBnA

knFnB

knCnBnA

(5)

4. FEASIBILITY STUDY – STEAM BOILER

Cracked pipes are one of the most frequently occurring malfunctions of internal installations of steam

boiler applied in power plants. Analytical description of the process of steam flow through the crack is

a complex model involving partial differential equations [20]. It is assumed that the model can be

simplified to take the form of a black box model that is a mapping of the input onto the output. In such

a case, only specific oscillations related to the occurrence of the crack are considered however no

physical interpretation concerning the mechanism can be drawn. A typical power plant steam boiler


6

consists of a complex system of pipelines (Fig. 3 left). The microphones are indicated as follows: 1, 2

and 3.

Figure 3 Steam boiler system (left) and the experimental setup (right)

The malfunction occurring most frequently is pipe leakage caused by local cracks due to internal or

external factors, the basic ones are corrosion, material fatigue and vibrations during transient operation

[19]. Pipelines and associated vessels and valves constitute an important part of a complete installation

of a power station and cost approximately 10-15 % of the total costs of a power station [19]. The

system of pipelines is very complex and so its durability can only be estimated. Precautionary

inspections are frequently insufficient to find cracks. Running repairs of pipework are conducted

during overhauls. It is important to know the scope, i.e. the location and severity of the leakage and the

damage to the installation before reparation in order to make a proper order of replacement parts. The

ratio of product faults to operational errors is assessed to be approximately like 5-to-1 [19]. More than

10% of emergency shut-downs of power units are caused by leakage in pipes and other installations.

Currently, pressure and flow rate gauges have to be used frequently by the operational staff to monitor

pipeline systems. First generation of analog Acoustic Monitoring Systems (AMS) were developed in

the 1980s as an improvement of manual detection methods. Since, steam leakage noises propagate

through boiler gases and metal parts, the AMS system uses sensitive microphones or other sensors,

e.g. vibration sensors mounted to the metal parts, to record such noise. Filtration within variable

frequency bands is used by AMS systems to detect when acoustic energy generated by the leakage

exceeds a preset threshold value; an alarm is then activated. Such simple monitoring methods however

are only capable of indicating occurrence of the failure but do not provide early warning and therefore

is incapable of preventing emergency boiler shutdowns. A high temperature and dustiness, are the

main problem during internal measurements. In this case expensive microphones, designed for high

temperatures, are required. It is possible to avoid the internal measurements if the acoustic transfer

functions through the steam boiler shell is identified using a parametric model.

This feasibility study builds a theoretical foundation for creating a microprocessor-based detector

dedicated for locating pipeline system cracks. Active experiment was conducted based on explosion of

small charges inside the steam boiler at the height of 15 [m] during the shut-down of the power unit

(Fig. 3 right). The internal acoustic pressure (sound) level close to the location of the explosion was

measured and assumed as an input (Table 2). The external acoustic pressure measurements are the

model output (Table 2). The input-output acoustic pressure transfer functions were formulated and

identified as summarized in Table 2. The operational measurements were conducted outside the boiler

shell and during power unit normal operation. The operational measurements were used to reconstruct


7

the internal acoustic pressure during the power unit operation as showed in Table 2. The complete

experiment was described in [20].

Table 2 Results of the load identification in the frequency domain

Input Output Model structure Acoustic pressure correlation

measured vs. reconstructed

h=13.5 [m] h=40 [m] PEM(7,8) 0.44

h=13.5 [m] h=36 [m] PEM(7,8) 0.60

h=13.5 [m] h=40 [m] NARX(1,19,10) 0.51

h=13.5 [m] h=36 [m] NARX(1,19,10) 0.74

The feasibility study indicates that nonlinear models do not show significant advantage against the

linear ones. The explanation is that the acoustic path through the boiler shell has the similar

transmission properties under cold and hot conditions. The transmission is characterized as a random

process of normal distribution, therefore it is better captured by advanced model structures, e.g.

ARMAX, BJ, PEM.

5. FEASIBILITY STUDY – HYDRAULIC DAMPER

This feasibility study considers a method for reconstructing wheel movement according to the road-

load profile by means of an inverse, parametric model [21]. The signal of the vertical wheel movement

is required in semi-active suspension systems [22], which control the movement of the wheels via an

onboard system, rather than the movement determined entirely by the surface on which the passenger

vehicle is driving. The system, therefore, virtually eliminates body roll and pitch variation in many

driving situations, including cornering, accelerating and braking [22]. The road profile generates a

kinematic displacement excitation to the tire, which is transferred through the wheel assembly (Fig. 4)

to the lower arm, and further to the bottom mount (bracket) of the hydraulic damper. A hydraulic

damper consists of a piston moving inside a liquid-filled cylinder to which the fixing bracket is

attached. The top-mount is a rubber bearing element attached to the piston and the body of the vehicle.

The excitation to the vehicle body which is transferred through the entire path consists of equivalent

stiffness and damping sub-systems, corresponding to the tire, the wheel hub fixture, the hydraulic

damper, and the top-mount (Fig. 4).

Figure 4 A schematic view of a McPherson front strut suspension system

A typical semi-active system uses displacement sensors which measure the relative distance between

the rod and the hydraulic damper housing using electro-mechanical (LVDT) sensors. The velocity


8

signal is obtained by differentiation of the displacement signal. The goal is to indirectly reconstruct

wheel vertical movement, based on the force measurement above the top-mount by means of dynamic

model inversion. In turn, the displacement sensor is replaced by a load cell sensor, in which the force

is converted into an electrical signal through deformation of a strain gauge. The strain changes the

effective electrical resistance of the wire (e.g. four strain gauges in a Wheatstone bridge

configuration). The electrical signal output is typically in the order of a few millivolts and requires

calibration and amplification. This kind of sensor is a low-cost alternative to displacement sensors

which are highly sensitive to dirt. The proposed approach was validated in the laboratory environment

using road signals and a servo-hydraulic excitation system.

Experimental tests [21] were performed on a servo-hydraulic test-rig Hydropuls® MSP25 equipped

with the electronic controller IST8000. The test rig was used to load a hydraulic damper strut module

(wheel hub fixture, hydraulic damper, spring, and top-mount) and capture its dynamical characteristic,

i.e. displacement vs. force. Data acquisition was performed with the 8-channel amplifier. The road

load data were used to simulate the ride conditions.

Inversion of the model was conducted according to the scenario shown in Fig. 1 and 2, respectively for

linear and nonlinear models. The measured random displacement is required at the input of a direct

model and measured force is required at its output [21]. The selected model structures are listed in

Table 3 with results presented therein.

Table 3 Results of the load identification in the time domain

Model Displacement correlation


BJ(13,7,7,15,1) 0.88

ARX(17,15,1) 0.76

ARMAX(10,10,6,1) 0.78

PEM(8,8,8,8,8,1) 0.79

NARX(3,8,1) 0.92

The nonlinear model showed slightly better performance compared to a linear ones. However, the

advanced linear models shows also significantly better performance compared to the simple linear

structures, i.e. ARX. The shock absorber inherits significant nonlinearities between the force and

displacement, or especially the velocity signal which are better captured by the nonlinear models.

6. FEASIBILITY STUDY - RAIL IRREGULARITIES

Rail irregularities (roughness) are generated by a number of factors arising from both operation and

ordinary rail degradation [24-25]. According to railway regulations listed in [25], both the amplitude

and frequency content of the irregularities are taken into account during assessment of the rail's

technical condition. Therefore, frequency domain analysis is required to evaluate both the magnitude

of irregularity and the frequency content which corresponds to the curvature of the rails.

This feasibility study, presented already in [17,25 ] allows to validate the method to reconstruct the rail

- wheel contact forces in time and frequency domain. Because kinematic excitation in the form of rail

irregularities is directly interconnected with rail - wheel contact forces [23-25]. The measurements

were taken on a self-dumping cargo vehicle of the Fals series, type 665 4 011-4 [25]. During tests the

cargo vehicle was empty. In Fig 5, the placement of accelerometers on the axle-boxes is shown.


9

Figure 5 Measurements of the vibrations accelerations on the axle box of the vehicle

During the test rides, the time histories of two forces were recorded: vertical and horizontal, both

acting in the rail – wheel contact point in the first wheel set on the right-hand side. Together with the

forces, 6 vibration accelerations measured on the axle boxes and the frame of the vehicle were stored.

Additionally, information regarding vehicle velocity, and its gyroscopic moments, was collected for

the ride profile recognition. Forces were measured in an indirect form during the tests. Displacements

of the vehicle axles were the measured quantity. They were measured using the strain gauges set and

further recalculated to obtain the bending moments of the axles. These moments were next

transformed into the rail – wheel contact forces. The values obtained in the presented method were

compared with the identified ones. Eight data sets, which corresponded to the eight rides of the vehicle

with different velocities, were collected on the rail with varying quality. Each test ride took less than

20 s. Sampling frequency was set to 150 Hz.

Inversion of the model was conducted according to the scenario shown in Fig. 1 or Fig 2, for the linear

and nonlinear models respectively. The measured random force is required at the input of a direct

model and measured acceleration is required at its output. The orders of particular polynomials of a

linear model and orders of a nonlinear model were selected using the quality indicators from Table 1.

To compare the results of direct measurements and those obtained with the procedure of model

inversion, the correlation coefficient were computed for both the lateral and vertical forces. The results

of this comparison of measured and reconstructed rail-wheel contact force are presented in Tables 4.

Table 2 Result of the load identification in the frequency domain at vehicle velocity v=60km/h

Input Output Model structure Force correlation


Lateral force right side of the vehicle ARX(33,29,1) 0.65

Lateral force left side of the vehicle ARX(33,29,1) 0.63

Vertical force right side of the vehicle ARX(33,29,1) 0.71

Vertical force left side of the vehicle ARX(33,29,1) 0.72

Lateral force right side of the vehicle NARX(17,17,1) 0.71

Lateral force left side of the vehicle NARX(17,17,1) 0.72

Vertical force right side of the vehicle NARX(17,17,1) 0.79

Vertical force left side of the vehicle NARX(17,17,1) 0.82


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The nonlinear inverse model provides a better accuracy of 10% compared to the linear one. Model

orders are lower in the case of nonlinear model structures (i.e. NARX) compared to linear ones.

7. CONCLUSIONS

The paper addresses questions concerning the feasibility of reconstructing the load of three technical

systems, which are difficult for direct measurement. This work advocates the model inversion based

on a parametric linear and nonlinear model, as an alternative method for approaching this class of

problems to a non-parametric model.

The feasibility study on reconstruction of wheel movement in the vehicle suspension showed that, in

the case of measurement data corrupted by noise, advanced model structures (e.g. BJ) provided better

accuracy of inversion than simple ones (e.g. ARX). The shock absorber is the object with a nonlinear

relationship between the input and output, therefore nonlinear model structures (e.g. NARX) had

advantage compared to linear ones. The feasibility study on reconstruction of wheel-rail contact forces

confirmed that the nonlinear relations in the cargo car suspension system are nonlinear due to spring

and damping ratio characteristics. The feasibility study on reconstruction of an acoustic pressure level

inside the steam boiler showed again better characterization with the use of the nonlinear model

compared to the linear one. The presented studies are only the selected experimental cases and

required further investigations based on the simulation model that allow to involve theoretical

nonlinearities to simulate different operating points and noise disturbances.

Results provided by linear parametric model structures are sufficient to constitute foundations for

implementing them as inverse models in the form of fixed-point linear filters on a DSP platform. The

model implemented in such a form is capable of filtering the responses of an observed object into a

reconstructed input, which is further converted into the frequency domain by a standard DFFT

algorithm. The nonlinear model can be implemented as a nonlinear real-time filter as well, however

without the proof of the filter stability.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the financial support of the research project

2011/01/B/ST8/07210 funded by the Polish Ministry of Science (MNiI).

NOMENCLATURE

i - discrete time

A,B,C,D,E,F - polynomials used for the representation of the transfer function

nA,nB,nC,nE,nF - order of polynomials used for the representation of the transfer function

z - operator of the Z transformation

e - disturbance variables in the model

u - input variables in the model

u0 - inverse input

y - output variables in the model

G(z), G(z-1

) - transfer function

ABBREVIATIONS

AIC – Akaike Information Criterion

ARMAX – AutoRegressive Moving Average with eXogeneous input

NARMAX – Nonlinear AutoRegressive Moving Average with eXogeneous input

ARX – AutoRegressive with eXogeneous input

NARX – Nonlinear AutoRegressive with eXogeneous input

BIC – Bayesian Information Criterion

BJ – Box-Jenkins


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DSP – Digital Signal Processing

FPE – Final Prediction Error

FRF – Frequency Response Function

LTI – Linear and Time-Invariant model/system

OE – Output Error

NOE – Nonlinear Output Error

PEM – Prediction Error Method

SISO – Single-Input Single-Output

REFERENCES

[1] Piazzi, A., Visioli, A., 2001, Robust set-point constrained regulation via dynamic inversion, Int. J.

Robust Nonlinear Control, 11 (1), 1–22.

[2] Moscinski, J., Ogonowski, Z., 1995, Advanced control with MATLAB & SIMULINK, Ellis

Horwood Ltd, UK.

[3] Elster, C., Link, A., Bruns, T, 2007, Analysis of dynamic measurements and determination of

measurement uncertainty using a second-order model, Meas. Sci. Technol. 18 3682-7.

[4] Uhl, T., 2007, The inverse identification problem and its technical application, Archive of

Applied Mechanics, 77 (5), 325–337.

[5] Mendrok K., Uhl T. (2004), Overview of modal model based damage detection methods,

Proceedings of the 29th International Conference on Noise and Vibration Engineering (ISMA),

Leuven, Belgium, 561–576.

[6] Bracciali, G. Cascini, G., 1998, High-frequency mobile input reconstruction algorithm (HF-

MIRA) applied to forces acting on a damped linear mechanical system, Mechanical Systems and

Signal Processing, 21 (2), 255–268.

[7] Raath, D., Van Waveren, C.C., 1998, A time domain approach to load reconstruction for

durability testing, Engineering Failure Analysis, 4 (1), 113–119.

[8] Hundhausen R.J., Adams D.E., Derriso M., Kukuche P., Alloway R. (2005) Transient Loads

Identification for a Standoff Metallic Thermal Protection System Panel, in: 23rd International,

Modal Analysis Conference (IMAC XXIII), Orlando, Florida.

[9] Hollandsworth, P.E., Busby, H.R., 1989, Impact force identification using the general inverse

technique, International Journal of Impact Engineering, 8, 315–322.

[10] Ljung, L., 1999, System Identification – Theory for the User, Prentice-Hall.

[11] Allena, M.S., Carne, T.G., 2008, Multi-step inverse structural filter for robust force identification,

Mechanical Systems and Signal Processing, 22 (5),1036–1054.

[12] Chan, T.H.T., Law, S.S., Zeng, Q.H., 1997, Moving force identification: a time domain method,

Journal of Sound and Vibration, 201, 1–22.

[13] Chen, T.C., Lee, M.H., 2008, Research on Moving Force Estimation of the Bridge Structure

using the Adaptive Input Estimation Method, Journal of Structural Engineering, 8.

[14] Czop P., 2011, Application of Advanced Data-Driven Parametric Models to Load Reconstruction

in Mechanical Structures and Systems, Journal of Vibration and Control, 17 (9) 2011, 1424-1438.

[15] Billings, S.A., Chen, S., 1989, Representation of nonlinear systems: the NARMAX model, Int. J.

of Control, 49 (3), 1013–1032.

[16] Norgaard, M., Ravn, O., Poulsen, N.K., Hansen, L.K., 2000, Neural network for modelling and

control of dynamic systems, Springer-Verlag, London.


12

[17] Czop P., Mendrok K., 2010 Application of Inverse Linear and Nonlinear Models in the

Identification of Rail Track Irregularities, Machine Dynamics Problems, 34 (3), 46-61.

[18] Torfs D., Swevers J., De Schutter J. (1991) Quasi-Perfect Tracking Control Of Non-Minimal

Phase Systems, Proceedings of the 30th Conference on Decision and Control Brighton, England,

241-244.

[19] Allianz, Handbook of Loss Prevention. Allianz Versicherung AG, Berlin and Munich, 1997

[20] Czop P., Miękina L. (2005) Inverse Modeling in Application to Diagnostics of Steam Boiler

Pipework System. The 11th IEEE International Conference Methods and Models in Automation

and Robotics, Międzyzdroje , Poland, 881-884.

[21] Czop P., 2012, Application of Inverse Data-Driven Parametric Models in the Reconstruction of

Passenger Vehicle Wheel Vertical Movement under Ride Conditions, Journal of Vibration and

Control, 18 (8) 1133-1140.

[22] Czop P., Sławik D., 2011, A High-Frequency Model of a Shock Absorber and Servo-Hydraulic

Tester, Mechanical Systems and Signal Processing, 25 (6) 1937-1955.

[23] Gullersa, P., Anderssona, L., Lundénb, R., 2008., High-frequency vertical wheel - rail contact

forces - Field measurements and influence of track irregularitie, Wear, Elsevier, 265 (9-10), 1488-

1496.

[24] Nielsen, J.C.O., 2008, Rail Roughness Level Assessment Based on High-Frequency Wheel-Rail

Contact Force Measurements, Springer Berlin/Heidelberg, 99.

[25] Czop P., Mendrok K. Uhl. T., 2011, Application of Inverse Linear Parametric Models in the

Identification of Rail Track Irregularities, Archive of Applied Mechanics, 81 (11), 1541-1554.

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TITLE OF THE ARTICLE - AdBoostersiomac.eu/iomac/2013/IOMAC_Guimaraes/files/papers/Load Estimation...The advantage of such an approach is the immediate possibility of its application