Modal contribution and state space order selection in operational modal analysis

The estimation of modal parameters of a structure from ambient measurements has attracted the attention of many researchers in the last years. The procedure is now well established and the use of state space models, stochastic system identification methods and stabilization diagrams allows to identify the modes of the structure. In this paper the contribution of each identified mode to the measured vibration is discussed. This modal contribution is computed using the Kalman filter and it is an indicator of the importance of the modes. Also the variation of the modal contribution with the order of the model is studied. This analysis suggests selecting the order for the state space model as the order that includes the modes with higher contribution. The order obtained using this method is compared to those obtained using other well known methods, like Akaike criteria for time series or the singular values of the weighted projection matrix in the Stochastic Subspace Identification method. Finally, both simulated and measured vibration data are used to show the practicability of the derived technique. Finally, it is important to remark that the method can be used with any identification method working in the state space model.

Maximum likelihood estimation of new state space models for operational modal analysis

The modal analysis of a structural system consists on computing its vibrational modes. The experimental way to estimate these modes requires to excite the system with a measured or known input and then to measure the system output at different points using sensors. Finally, system inputs and outputs are used to compute the modes of vibration. When the system refers to large structures like buildings or bridges, the tests have to be performed in situ, so it is not possible to measure system inputs such as wind, traffic, . . .Even if a known input is applied, the procedure is usually difficult and expensive, and there are still uncontrolled disturbances acting at the time of the test. These facts led to the idea of computing the modes of vibration using only the measured vibrations and regardless of the inputs that originated them, whether they are ambient vibrations (wind, earthquakes, . . . ) or operational loads (traffic, human loading, . . . ). This procedure is usually called Operational Modal Analysis (OMA), and in general consists on to fit a mathematical model to the measured data assuming the unobserved excitations are realizations of a stationary stochastic process (usually white noise processes). Then, the modes of vibration are computed from the estimated model. The first issue investigated in this thesis is the performance of the Expectation- Maximization (EM) algorithm for the maximum likelihood estimation of the state space model in the field of OMA. The algorithm is described in detail and it is analysed how to apply it to vibration data. After that, it is compared to another well known method, the Stochastic Subspace Identification algorithm. The maximum likelihood estimate enjoys some optimal properties from a statistical point of view what makes it very attractive in practice, but the most remarkable property of the EM algorithm is that it can be used to address a wide range of situations in OMA. In this work, three additional state space models are proposed and estimated using the EM algorithm: • The first model is proposed to estimate the modes of vibration when several tests are performed in the same structural system. Instead of analyse record by record and then compute averages, the EM algorithm is extended for the joint estimation of the proposed state space model using all the available data. • The second state space model is used to estimate the modes of vibration when the number of available sensors is lower than the number of points to be tested. In these cases it is usual to perform several tests changing the position of the sensors from one test to the following (multiple setups of sensors). Here, the proposed state space model and the EM algorithm are used to estimate the modal parameters taking into account the data of all setups. • And last, a state space model is proposed to estimate the modes of vibration in the presence of unmeasured inputs that cannot be modelled as white noise processes. In these cases, the frequency components of the inputs cannot be separated from the eigenfrequencies of the system, and spurious modes are obtained in the identification process. The idea is to measure the response of the structure corresponding to different inputs; then, it is assumed that the parameters common to all the data correspond to the structure (modes of vibration), and the parameters found in a specific test correspond to the input in that test. The problem is solved using the proposed state space model and the EM algorithm. Resumen El análisis modal de un sistema estructural consiste en calcular sus modos de vibración. Para estimar estos modos experimentalmente es preciso excitar el sistema con entradas conocidas y registrar las salidas del sistema en diferentes puntos por medio de sensores. Finalmente, los modos de vibración se calculan utilizando las entradas y salidas registradas. Cuando el sistema es una gran estructura como un puente o un edificio, los experimentos tienen que realizarse in situ, por lo que no es posible registrar entradas al sistema tales como viento, tráfico, . . . Incluso si se aplica una entrada conocida, el procedimiento suele ser complicado y caro, y todavía están presentes perturbaciones no controladas que excitan el sistema durante el test. Estos hechos han llevado a la idea de calcular los modos de vibración utilizando sólo las vibraciones registradas en la estructura y sin tener en cuenta las cargas que las originan, ya sean cargas ambientales (viento, terremotos, . . . ) o cargas de explotación (tráfico, cargas humanas, . . . ). Este procedimiento se conoce en la literatura especializada como Análisis Modal Operacional, y en general consiste en ajustar un modelo matemático a los datos registrados adoptando la hipótesis de que las excitaciones no conocidas son realizaciones de un proceso estocástico estacionario (generalmente ruido blanco). Posteriormente, los modos de vibración se calculan a partir del modelo estimado. El primer problema que se ha investigado en esta tesis es la utilización de máxima verosimilitud y el algoritmo EM (Expectation-Maximization) para la estimación del modelo espacio de los estados en el ámbito del Análisis Modal Operacional. El algoritmo se describe en detalle y también se analiza como aplicarlo cuando se dispone de datos de vibraciones de una estructura. A continuación se compara con otro método muy conocido, el método de los Subespacios. Los estimadores máximo verosímiles presentan una serie de propiedades que los hacen óptimos desde un punto de vista estadístico, pero la propiedad más destacable del algoritmo EM es que puede utilizarse para resolver un amplio abanico de situaciones que se presentan en el Análisis Modal Operacional. En este trabajo se proponen y estiman tres modelos en el espacio de los estados: • El primer modelo se utiliza para estimar los modos de vibración cuando se dispone de datos correspondientes a varios experimentos realizados en la misma estructura. En lugar de analizar registro a registro y calcular promedios, se utiliza algoritmo EM para la estimación conjunta del modelo propuesto utilizando todos los datos disponibles. • El segundo modelo en el espacio de los estados propuesto se utiliza para estimar los modos de vibración cuando el número de sensores disponibles es menor que vi Resumen el número de puntos que se quieren analizar en la estructura. En estos casos es usual realizar varios ensayos cambiando la posición de los sensores de un ensayo a otro (múltiples configuraciones de sensores). En este trabajo se utiliza el algoritmo EM para estimar los parámetros modales teniendo en cuenta los datos de todas las configuraciones. • Por último, se propone otro modelo en el espacio de los estados para estimar los modos de vibración en la presencia de entradas al sistema que no pueden modelarse como procesos estocásticos de ruido blanco. En estos casos, las frecuencias de las entradas no se pueden separar de las frecuencias del sistema y se obtienen modos espurios en la fase de identificación. La idea es registrar la respuesta de la estructura correspondiente a diferentes entradas; entonces se adopta la hipótesis de que los parámetros comunes a todos los registros corresponden a la estructura (modos de vibración), y los parámetros encontrados en un registro específico corresponden a la entrada en dicho ensayo. El problema se resuelve utilizando el modelo propuesto y el algoritmo EM.

Estimating the modal parameters from multiple measurement setups using a joint state space model

Computing the modal parameters of structural systems often requires processing data from multiple non-simultaneously recorded setups of sensors. These setups share some sensors in common, the so-called reference sensors, which are fixed for all measurements, while the other sensors change their position from one setup to the next. One possibility is to process the setups separately resulting in different modal parameter estimates for each setup. Then, the reference sensors are used to merge or glue the different parts of the mode shapes to obtain global mode shapes, while the natural frequencies and damping ratios are usually averaged. In this paper we present a new state space model that processes all setups at once. The result is that the global mode shapes are obtained automatically, and only a value for the natural frequency and damping ratio of each mode is estimated. We also investigate the estimation of this model using maximum likelihood and the Expectation Maximization algorithm, and apply this technique to simulated and measured data corresponding to different structures.

Using the EM algorithm to estimate the state space model for OMAX

This paper presents a time-domain stochastic system identification method based on Maximum Likelihood Estimation and the Expectation Maximization algorithm that is applied to the estimation of modal parameters from system input and output data. The effectiveness of this structural identification method is evaluated through numerical simulation. Modal parameters (eigenfrequencies, damping ratios and mode shapes) of the simulated structure are estimated applying the proposed identification method to a set of 100 simulated cases. The numerical results show that the proposed method estimates the modal parameters with precision in the presence of 20% measurement noise even. Finally, advantages and disadvantages of the method have been discussed.

State space models for multi-setup operational modal analysis

Operational Modal Analysis consists on estimate the modal parameters of a structure (natural frequencies, damping ratios and modal vectors) from output-only vibration measurements. The modal vectors can be only estimated where a sensor is placed, so when the number of available sensors is lower than the number of tested points, it is usual to perform several tests changing the position of the sensors from one test to the following (multiple setups of sensors): some sensors stay at the same position from setup to setup, and the other sensors change the position until all the tested points are covered. The permanent sensors are then used to merge the mode shape estimated at each setup (or partial modal vectors) into global modal vectors. Traditionally, the partial modal vectors are estimated independently setup by setup, and the global modal vectors are obtained in a postprocess phase. In this work we present two state space models that can be used to process all the recorded setups at the same time, and we also present how these models can be estimated using the maximum likelihood method. The result is that the global mode shape of each mode is obtained automatically, and subsequently, a single value for the natural frequency and damping ratio of the mode is computed. Finally, both models are compared using real measured data.

Fixed wages and bonuses in agency contracts: the case of a continuous state space

In this paper, we extend the state-contingent production approach to principal–agent problems to the case where the state space is an atomless continuum. The approach is modelled on the treatment of optimal tax problems. The central observation is that, under reasonable conditions, the optimal contract may involve a fixed wage with a bonus for above-normal performance. This is analogous to the phenomenon of "bunching" at the bottom in the optimal tax literature.

Constructing internationally comparable real income aggregates by combining sparse benchmark data with annual national accounts data: A state-space approach

The importance of availability of comparable real income aggregates and their components to applied economic research is highlighted by the popularity of the Penn World Tables. Any methodology designed to achieve such a task requires the combination of data from several sources. The first is purchasing power parities (PPP) data available from the International Comparisons Project roughly every five years since the 1970s. The second is national level data on a range of variables that explain the behaviour of the ratio of PPP to market exchange rates. The final source of data is the national accounts publications of different countries which include estimates of gross domestic product and various price deflators. In this paper we present a method to construct a consistent panel of comparable real incomes by specifying the problem in state-space form. We present our completed work as well as briefly indicate our work in progress.

Modelling financial time series with switching state space models

The deficiencies of stationary models applied to financial time series are well documented. A special form of non-stationarity, where the underlying generator switches between (approximately) stationary regimes, seems particularly appropriate for financial markets. We use a dynamic switching (modelled by a hidden Markov model) combined with a linear dynamical system in a hybrid switching state space model (SSSM) and discuss the practical details of training such models with a variational EM algorithm due to [Ghahramani and Hilton,1998]. The performance of the SSSM is evaluated on several financial data sets and it is shown to improve on a number of existing benchmark methods.

Urban governance under neoliberalism:New Labour and the restructuring of state-space

In the UK there has been a proliferation of agencies at differing regulatory scales as part of the rescaling and restructuring of the state by New Labour, following the neoliberal policies of previous Conservative governments. This raises questions concerning the extent to which New Labour's urban state restructuring is embedded within neoliberalism, and the local tensions and contradictions arising from emergent New Labour urban state restructuring. This paper examines these questions through the analysis of key policy features of New Labour, and the in-depth exploration of two programmes that are reshaping urban governance arrangements, namely Local Strategic Partnerships (LSPs) and New Deal for Communities (NDC) programmes. We conclude that New Labour's restructuring is best understood in terms of the extended reproduction (roll-out) of neoliberalism. While these “new institutional fixes” are only weakly established and exhibit internal contradictions and tensions, these have not led to a broader contestation of neoliberalism.

A Bayesian state space modelling approach to probabilistic quantitative precipitation forecasting

The generation of very short range forecasts of precipitation in the 0-6 h time window is traditionally referred to as nowcasting. Most existing nowcasting systems essentially extrapolate radar observations in some manner, however, very few systems account for the uncertainties involved. Thus deterministic forecast are produced, which have a limited use when decisions must be made, since they have no measure of confidence or spread of the forecast. This paper develops a Bayesian state space modelling framework for quantitative precipitation nowcasting which is probabilistic from conception. The model treats the observations (radar) as noisy realisations of the underlying true precipitation process, recognising that this process can never be completely known, and thus must be represented probabilistically. In the model presented here the dynamics of the precipitation are dominated by advection, so this is a probabilistic extrapolation forecast. The model is designed in such a way as to minimise the computational burden, while maintaining a full, joint representation of the probability density function of the precipitation process. The update and evolution equations avoid the need to sample, thus only one model needs be run as opposed to the more traditional ensemble route. It is shown that the model works well on both simulated and real data, but that further work is required before the model can be used operationally. © 2004 Elsevier B.V. All rights reserved.

Limit Theorems for Non-Critical Branching Processes with Continuous State Space

2000 Mathematics Subject Classification: Primary 60J80, Secondary 60G99.

State-space modelling of geolocation data reveals sex differences in the use of management areas by breeding northern fulmars

We thank Orkney Islands Council for access to Eynhallow and Talisman Energy (UK) Ltd and Marine Scotland for fieldwork and equipment support. Handling and tagging of fulmars was conducted under licences from the British Trust for Ornithology and the UK Home Office. EE was funded by a Marine Alliance for Science and Technology for Scotland/University of Aberdeen College of Life Sciences and Medicine studentship and LQ was supported by a NERC Studentship. Thanks also to the many colleagues who assisted with fieldwork during the project, and to Helen Bailey and Arliss Winship for advice on implementing the state-space model.

State-space modelling of geolocation data reveals sex differences in the use of management areas by breeding northern fulmars

We thank Orkney Islands Council for access to Eynhallow and Talisman Energy (UK) Ltd and Marine Scotland for fieldwork and equipment support. Handling and tagging of fulmars was conducted under licences from the British Trust for Ornithology and the UK Home Office. EE was funded by a Marine Alliance for Science and Technology for Scotland/University of Aberdeen College of Life Sciences and Medicine studentship and LQ was supported by a NERC Studentship. Thanks also to the many colleagues who assisted with fieldwork during the project, and to Helen Bailey and Arliss Winship for advice on implementing the state-space model.