Aalborg Universitet Identification of Civil Engineering Structures using Vector ARMA Models Andersen, Palle; Brincker, Rune; Kirkegaard, Poul Henning Published in: Dynamics of Structures 1993-1997 Publication date: 1998 Document Version Publisher's PDF, also known as Version of record Link to publication from Aalborg University Citation for published version (APA): Andersen, P., Brincker, R., & Kirkegaard, P. H. (1998). Identification of Civil Engineering Structures using Vector ARMA Models. In Dynamics of Structures 1993-1997: description of the projects in the research programme (pp. 19-38). Dept. of Building Technology and Structural Engineering, Aalborg University. 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Downloaded from vbn.aau.dk on: January 28, 2023 Aalborg Universitet Identification of Civil Engineering Structures using Vector ARMA Models Andersen, Palle; Brincker, Rune; Kirkegaard, Poul Henning Published in: Dynamics of Structures 1993-1997 Publication date: 1998 Link to publication from Aalborg University Citation for published version (APA): Andersen, P., Brincker, R., & Kirkegaard, P. H. (1998). Identification of Civil Engineering Structures using Vector ARMA Models. In Dynamics of Structures 1993-1997: description of the projects in the research programme. (pp. 19-38). Chapter Project B.1.Aalborg: Dept. of Building Technology and Structural Engineering, Aalborg University. (R / Institut for Bygningsteknik; No. R9808). General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. ? Users may download and print one copy of any publication from the public portal for the purpose of private study or research. ? You may not further distribute the material or use it for any profit-making activity or commercial gain ? You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us at [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from vbn.aau.dk on: december 17, 2014 DYN AM ICS OF STRUCTURES 1993 -1997 DESCRIPTION OF PROJECTS B.l Identification of Civil Engineering Structures using Vector ARMAMorleis Palle Andersen, Rune Brincker & Poul Henning Kirkegaard Aalborg University Department of Building Technology and Structural Engineering Sohngaardsholmsvej 57, DK-9000 Aalborg, Denmark SYNOPSIS: This paper describes the work which have been carried out in the project B.l: Damage Det eetion in Structures under Random Loading. The project is a part o f the research programme Dynamics of Structures founded by the Danish Technical Research Council. The planned contents of and the requirements to the project prior to its start is deseribed together with the results obtained during the project. The project was mainly carried out as a Ph.D. project by the first author from September 1993 to May 1997 under supervision of Professor Rune B rineker and Associate Professor Poul Henning Kirkegaard both from department of Building Technology and Structural Engineering, Aalborg University. l. INTRODUCTION In the mid eighties, researchers at the Department of Building Technology and Structural Engineering at Aalborg University, Denmark, started using time domain models for system identification of civil engineering structures. A common feature of the work has been the use of the so-called auto-regressive moving average (ARMA) models for time series modelling. The reason is the ability of these models to provide an accurate estimate of the modal parameters o f a structural system on the basis o f discretely sampled response. This ability makes them suitable as tool for vibration based inspection, i.e. damage detection. In the past decade the results of this work have been reported in several papers and in three Ph.D. theses. However, the link between the ARMA model and the mathematical description of civil engineering structures has not been addressed to the same extent as mathematical description of dynarnic systems in fields such as electrical engineering and econometrics. Therefore, the model has been applied toas a greybox model in theabove references. The relation between the auto-regressive part of the model and the modal parameters has been well understood, whereas the understanding of the moving average part haS been limited. · In order to obtain a deeper understanding of the ARMA models, and how they are related to the modeiling of civil engineering struerures and damage detection, the project B.1: Damage Detection in Structures under Random Loading, with this as its primary objective, was granted as a part of the Danish Research Council frame programme "Dynamics of Structures". Another objective of the Ph.D. project was to implement the obtained knowledge as, especially designed, time domain software for system identification of civil engineering structures. The results have been reported in several papers and the Ph.D. thesis: System ldentification of Civil Engineering Structures using Vector ARMA Models. In section 9, alistof references to papers andreports prepared during the project is presented. This list together with the possibility to download some of these and the thesis can be seen at http://www.civil.auc.dk/-i6pa. A short description of the Ph.D. thesis can be seen at http://www.civil.auc.dkl-i6pa/thesis.htm. -19- DYNAMICS OF STRUCTURES 1993-1997 DESCRIPTION OF PROJECTS 2. PLAN FOR THE PROJECT The foliowing topics relating to system identification of ambient excited civil engineering structures were planned to be investigated. l. Relation between system identification using ARMA models and vibration based inspection. 2. ARMA modelling of ambient excited civil engineering structures. 3. Estimation of ARMA models from measmed system response. 4. Extraction of modal parameters and estimation of their uncertainties. 5. Software development for system identification using ARMA models. In section 3, the relation between mathematical modeiling and system identification is described. Also deseribed are the relations between non-parametric and parametric system identification, and why it is desirable to apply the ARMA models in relation to vibration based inspection of ambient excited civil engineering structures. Finally, the scope of the work of the Ph.D. project is stated. Section 4 explains how the ambient excited civil engineering structme can be modelled in discrete-time by the ARMA model. In section 5, it is explained how the ARMA model can be estimated from measmed system response. The extraction o f the modal parameters and the estimation o f their uncertainties are the topi es of section 6. Section 7 briefly describes the software package developed during the project. Finally , in section 8 condusions are made. 3. SYSTEM IDENTIFICATION OF CIVIL ENGINEBRING STRUCTURES USING ARMAV MODELS In this section the basic concepts of system identification are introduced. Also introduced, are the applications of system identification in civil engineering that are of interest to project B. l. 3.1 System ldentification A convenient way of describing·a dynamic system is by use of mathematical models. These models can either be represented ln continuous time as differential equation systems or in discrete-time as difference equation systems. There are in general two ways to construct mathematical models: w Physical model/ing. w System identification. In physical modeiling the construction of a dynamic model is based on physical knowledge and fundamentallaws, such as the Newton 2. law of motion. On the other hand, if the physical knowledge about a dynamic system is Jimited a model o f the input l output behaviom o f the system can be o b tained through system identification based on calibration of a model using experimental data. lf the structme of the calibrated model is chosen without regard to physical knowledge the calibrated model is called a black box model. lf some parts of the model are based on physical knowledge the calibrated model is called a grey box model. On the other hand, if the calibrated model is based completely on the physicallaws, i.e. if it ariginates from a physical modelling, then the calibrated model is called a w hi te box model. Thus, system identification should not be thought of as a substitute of physical modelling, since identification can be based on model structures that have physical origin. Basically there are two categories of model structures : . -20- DYNAMICS OF STRUCTURES 1993-1997 DESCRIPTION OF PROJECTS Non-parametric model structures. Q> Parametric model structures. IGf' In any case, physical modeiling will always be linked with parametric model structures. Common to both categories o f model structures is that they depend on the applied excitation, which may be one o f the foliowing Instantaneous excitation. Q> Periodic excitation. IGf' Pseudo-random periodic excitation. Q> Stochastic excitation. Q> In the case of instantaneous excitation, the system is either given an impulse or step excitation and the system is the left vibrating on its own. The excitation may or maynot be measured. It is also possible to excite the system with a known periodic excitation, such as sinusoidal, or several periodic signals mixed to obtain a pseudo-random periodic excitation. Finally, as an alternative to the delerministic excitation, the excitation might also be a stationary stochastic process with either known or unknown statistical properties. In any case, the dynamic behaviour can be conceptually deseribed as in figure l. v(t) j Disturbane e Dynamic system u( t) y(t) Input Output Pigure l: A dynamic system with input u( t), output y( t) and disturbance v(t). The system is driven by input u( t) and affected by disturbance v( t). In some cases the user can control the input u(t ) but not the disturbance v(t). It might also bethat the actual input is unknown and therefore uncontrollable in some applications. The output y( t) describes how the system reacts or responds to the applied input and disturbance. Therefore, the output will be a mixture of dynamic response of the system and characteristics of the input and disturbance as well. In general, the input at previous time instances will also affect the current output. In other words, the dynamic system has memory. 3.1.1 Non-Parametric Model Structures The non-parametric models are deseribed by curves, functional relationsbips or tables. These analysis methods are : Transient analysis. Q> Frequency analysis. Q> Correlation analysis. Q> Spectral analysis. Q> Transient analysis is applied when the system response is transient, i.e. generated on the basis of impulse or step excitation. The dynamic behaviour of the system is then identified on the basis of the -21- DYN AM ICS OF STRUCTURES 1993 - 1997 DESCRIPTION OF PROJECTS impulse or step response. Frequency analysis is applied when the excitation is deterministic and either periodic, or pseudo-random and periodic. The measured excitation and the corresponding system response are transformed to frequency domain, and the frequency response function is obtained as the ratio o f the transformed response and ex citation. Cerrelation and spectral analysis are methods that are applied to a stationary stochastically excited system. In these cases, the excitation and the system response can be characterized either by the cerrelation functions in time domain or the spectral densities in frequency domain. Having estimated the cerrelation functions of the excitation and the response the impulse response function of the system can be obtained. On the other hand, if the spectral densities of the excitation and response are estimated instead, it is possible to obtain the frequency response function. The traditional non-parametric system identification techniques are primarily basedon the Fast Fourier Transform (FFT) techniques. However, the FFT has some !imitations. The most obvious !imitation is that the FFT assumes periodic data, which is certainly not the case for sampled response from stochastically excited structures. In principle, the Fourier transform assumes that the amount of data is infinite. In this case the frequency response functions or the spectral densities will have an infinite frequency resolution. However, in practice the available data records have a finite length, resulting in a finite frequency resolution. Since sampled stochastic signals in general exhibit non-periodicity, leakage errors will certainly be introduced. Further, in the case of closely spaced modes, it might be impossible to separate these if one o f the resonance frequencies has a small amplitude compared to the other. In this case the resonance frequency with the smallest energy content may be masked completely by the resonance frequency with highest energy content. The leakage errors are compensated by windowing the data before the FFT is applied, to secure periodicity of the data by damping the discontinuities at the ends of the data record. The problem of windowing is, that i t introduces an extra damping into the system, and thus creates its own leakage problem. 3.1.2 Pararnefrie Model Structures Pararnettie models are characterized by the assumption of a mathematical model constructed from a set of parameters. These parameters are then estimated during the system identification. The mathematical model of a linear and time-invariant continuous-time system is usually in the form of a differential equation system. The equivalent discrete-time parametric model is a difference equation system. In figure l an input l output system affected by noise was shown. The appearance of the discrete-time parametric model that describes such a system depends on whether the input is measured or not. lf the input is measured, then the associated parametric model will have a deterministic term as well as a stochastic term that describes the unknown disturbance. lf the actual input is unknown, it is treated stochastically. In this case the description of input and disturbance will be deseribed by a single stochastic term. Model Structures using Delerministic Input The general input l output model structure used for modeiling of linear and time-invariant dynamic systems excited by deterministic input, is Auto-Regressive Moving Average with eXternal input (ARMAX) y(t) = G(q)u(t) + H(q)e(t) (l) where G(q) and H(q) are the transfer functions of the deterministic part and the stochastic part. The -22- - DYN!\ M ICS OF STRUCTURES 1993 - 1997 DESCRIPTION OF PROJECTS stochastic input e(t) are the innovations, which are an equivalent process of the noise and prediction errors. If H(q) ==I (l) is called an output error (OE) model. In anycase the dynarnic properties of the system are modelled by G(q). A parametric model structure is called multivariable when it ineludes several variables. If there are several outputs, it is characterized as a multivariate model structure. If i t only has one output, it is termed a univariate model structure. Model Structures using Stochastic Input If the input is an unmeasurable stationary stochastic process, the ARMAX model is no longer the correct model structure to use. In this case an Auto-Regressive Moving Average (ARMA) model should be applied y(t) = H(q)e(t) (2) The dynarnical properties as well as the noise are now modelled by the same transfer function H( q). In the multivariate case the model structure is called an Auto-Regressive Moving Average Vector (ARMAV ) modeL As observed the choice of model structure depends on whether the input is deterrninistic or stochastic, i.e. whether the excitation of the structural system is known and measured or unknown. It also depends on whether the system is stationarily excited, or excited by an impulse or step excitation. 3.2 Applications of System ldentification of Civil Enginee ring Structures In the field of civil engineering, system identification rnight be applied forseveral reasons. However, the foliowing two areas have attracted much attention in the recent years s Modal analysis. s Vibration based inspection. Modal analysis covers a variety of applications allbasedon the analysis of modal parameters. These parameters describe specific dynarnic characteristics of the structure. One o f the applications that us es the modal parameters as basis is vibration based inspection. 3.2.1 Modal Analysis Modal analysis is based on the determination of modal parameters of a structural system. These parameters represent an optimal model, or basis, which can be used to describe the dynarnics of a structural system. The modal parameters can be divided into the foliowing four categories: s Modalfrequencies. s Modal damping. s Modal vectors. s Modal scaling. The modal frequencies aremore explicitly eigenvalues, or angular or natural eigenfrequencies. Modal damping is characterized by the damping ratios, and modal vectors by the eigenvectors or mode shapes. Finally, modal masses and residues are typical parameters used to characterize modal scaling. Since the modal parameters are directly related to the impulse and frequency response functions, as well as the -23- DYNAMIC'S OF STRUCTURES 1993- 1997 DESC'RIPTION OF PROJEC'TS cerrelation functions and spectral densities, they can be extracted from the non-parametric system identification methods by applying different curve fitting procedures. In case of parametric system identification methods there are direct mathematical relationsbips between the modal parameters and the estimated model parameters. 3.2.2 Vibration Based Inspection The accumulation of damages in a civil engineering structure will cause a change in the dynamic characteristics of the structure. The basic idea in Vibration Based Inspection (VBI) is to measure these dynamic characteristics during the lifetime of the structure and use them as a basis for identification of structural damages. Typically, a VBI programme uses the modal parameters to describe the dynamic characteristics of a structure. A synonym for the dynamic characteristics used as basis for the VBI programme is damage indicators. In other words: ..." A damage indicator is a dynamic quantity, which can be used to identify the existence of damage in a structure. Often VBI has at random been referred to as damage detection. VBI can be divided into the foliowing four levels: s Level l -Detection. « Level2- Localization. s Level 3 -Assessment. s Level 4 - Consequence. Methods of the first level give a qualitative indication that a structure might be damaged. Level two methods give information about the probable location of the damage as well. Methods of the third level provide information about the size of the damage, and finall y the level four methods give information about the actual safety of the structure given a certain damage state. The use of a damage indicator primarily gives a qualitative indication of the existence of damage, and should therefore be characterized as a level l method. However, some of the damage indicators will in same cases give rough estimates for the locations of damage, which is equivalent to a primitive level 2 method. Changes in natural eigenfrequencies are no doubt the most used damage indicators. One of the reasons for this is that the natural eigenfrequencies arerather easy to determine withahigh level of accuracy. Another reason is that they are sensitive to both global and local damages. So comparison o festimates of natural eigenfrequencies is usually an effective level l method. A local damage will cause changes in the derivatives of the mode shapes at the position of the damage. This means that a mode shape having many coordinates or measurement points will be a fast way to locate the approximate position of a damage. They can therefore be characterized as a simple level 2 method. The introduetion of damage in a s trueture will usually cause changes in the damping capacity o f the structure. It has been show n that the damping ratios are extremely sensitive to the introduetion of even smal! cracks in a cantilever beam. However, dealing with real structures, the estimation of the damping ratios of the individual modes is higbly sensitive to time-varying and non-physical sources. Thus, a satisfactory accuracy of the estimates of the damping ratios will in general be impossible to obtain. Therefore, the damping is applicable as a damage indicator, but it eannot and should not be used as the only damage indicator. As explained, all modal parameters are in principle applicable as damage indicators. This means that they can be used at least for detection of damage, and as such be characterized as level l methods. However, the key to a successful VBI is the use of unbiased and low-variance modal parameter -24- DYNAMICS OF STRUCTURES 1993-1997 DESCRIPTION OF PROJECTS estimates as damage indicators. If the estimates are biased they might cause a false alarm, i. e. indicate a damage that does not exist. If the estimation inaccuracies are too dominant, it might be impossible to detect any significant changes. Thus, the existence of a damage might be hidden. So in condusion : ~& Successful VB! basedon modal parameters requires accurate and unbiased modal parameter estimates. In this context the computational effort spent in obtaining reliable estimates is not so important. Further, the !imitations and systematic errors o f the traditional FFT- based non-parametric system identification techniques motivates the use of other techniques. This motivation can be stated as: ~& The needfor a more accurate estimation ofthe modal parametersfrom sampled data, compared to what traditional FFT-based non-parametric techniques can provide. This need is basically the reason for using the parametric models in the system identification, since the physical knowledge about a dynamic system in this way is incorporated into the system identification proces s. 3.2.3 Excitation of Civil Engineering Structures In the case of civil engineering structures there will most likely be a natura! excitation of the structure such as wind or waves. These natura! forms of excitation are comrnonly called ambient excitation and the vibrations of the structure caused by them are called ambient vibrations. System identification of structural dynamics on the basis of ambient excitation is also referred to as ambient testing. From an experimental point o f view, the simplest approach to measure the dynarnic parameters of a st rueture is to de teet the response due to ambient excitation. In the case of very large structures this approach is the only practical way of performing dynarnic tests, it is simply impossible to excite such structures artificial! y. The ambient excitation is stochastic in nature. Therefore, it eannot be deseribed by an explicit time dependent function, but must be characterized by certain statistical parameters, such as its meanand covariance function. Since the structural system can be seen as a linear transformation of the applied input, this means that the response will also be stochastic, and may as such also be represented by its statistical characteristics. Several researches have shown that ambient excitation provides a quick, inexpensive and reliable way for testing of large civil engineering structures, such as buildings and offshore structures. It has been conduded that parameter estimates obtained by ambient excitation are as good as parameter estimates obtained by extemal excitation. This condusion is based on a study of several published results of ambient versus forced vibration tests of high-rise structures in the USA. Because of the nature of dynamic testing under ambient excitation conditions this method has advantages over others, such as the impulse and periodic excitation. Ambient excitation has a broad frequency range, and thus theoretically excites all relevant modes of a structure. Also, the use of ambient excitation in dynarnic testing does not disturb the normal functioning of a structure and no excitation equipment is required for ambient testing. However, the disadvantage of ambient excitation is that its characteristics eannot be controlled and measured directly. -25- DYNAMICS Of' STRUCTURES 1993 -1997 DESCRIPTION OF PROJECTS Since ambient excitation eannot be measured directly, it can be constructed fromother measurements such as the surface elevation if system identification of an offshore structure is considered. From these measurements, sea state characteristics such as significant wave height and average zero-upcrossing period, can be estimated. These characteristics can then be used as input to models, which have been developed to describe the waves either as time series or spectral densities. The connection between the theoretical description of the waves and the forceson the structure is established using a load model, whieh eould e.g. be the Morrisen equation. In the case where the ambient exeitation is generated by fluetuating wind pressure forces, numerous measurement projects have shown that the fluetuations may bedeseribed by a stationary ergodie Gaussian stoehastie proeess with regard toshort-term eonditions. 3.3 Scope of the Ph. D. Thesis From the above the foliowing ean be stated eoneerning system identifieation of civil engineering struetures : The dynamic behaviour of a civil engineering structure·is usually modelled by a linear and J& time-invariant model. ~& The excitation of civil engineering struerures is typieally unknown ambient excitation. ~& 1fthis unknown ambient excitation is e.g. the wind, it is aften modelled as a stationary Gaussian stochastie proeess. For applieations suehas VB! a high degree of estimation aceuracy ofthe modal parameters is required. Adequate pararnefrie modelstruetures are not limited by frequeney resolution, and ean as such be more aecurate than FFT-based non-pararnefrie model struetures. These statements imply that the pararnetrie models are applieable to system identification of civil engineering st ruetures wh en a high degree o f aecuracy is needed. Therefore, the main aim o f this thesis has been to investigate how to represent arnbient excited civil engineering structural systems by stochastic time-domain models, and how to estimate these on the basis of sampled struetural response data. Since measurements of the true arnbient excitation are not available, system identification using standard multivariate input l output ARMAX model eannot be applied. Therefore, the focus has been put on the use of stochastic models of the Auto-Regressive Moving Average Vector (ARMAV) type. Particular emphasis has been be put on computationally accurate system identification methods, since the intention is to provide a more accurate alternative to the traditional non-parametric system identification methods typically applied in the field of civil engineering. A secondary purpose of the thesis has been to make the theory of system identification using s tachastic time domain models more accessible to civil engineers. Since system identification is a relatively new field for civil engineers, it has been natural to search for applicable theory and methods within disciplines that are at the research front, such as automatic control engineering, mathematical system theory, econometrics, and aerospace engineering. It is shown how an ARMA V model equivalent to the continuous-time mathematical model of a stochastically excited structural system arises. In this context, an explanation of the purpose o f the moving average is emphasized. It has also been shown how to account for the presence of disturbance, and how ARMA V models are directly related to the so-ealled stochastic state space systems. Since modal analysis is one of the main reasons for system identification of civil engineering structures, a thorough treatment of the modal deearnposition and extraction of modal pararneters has been given together with guidelines for estimation of the uncertainties of the estimated modal pararneters will be given. During the Ph. D. project emphasis has been put on the practical implementation of user-friendly system identification software. -26-