An approach to operational modal analysis using the expectation maximization algorithm

This paper presents the Expectation Maximization algorithm (EM) applied to operational modal analysis of structures. The EM algorithm is a general-purpose method for maximum likelihood estimation (MLE) that in this work is used to estimate state space models. As it is well known, the MLE enjoys some optimal properties from a statistical point of view, which make it very attractive in practice. However, the EM algorithm has two main drawbacks: its slow convergence and the dependence of the solution on the initial values used. This paper proposes two different strategies to choose initial values for the EM algorithm when used for operational modal analysis: to begin with the parameters estimated by Stochastic Subspace Identification method (SSI) and to start using random points. The effectiveness of the proposed identification method has been evaluated through numerical simulation and measured vibration data in the context of a benchmark problem. Modal parameters (natural frequencies, damping ratios and mode shapes) of the benchmark structure have been estimated using SSI and the EM algorithm. On the whole, the results show that the application of the EM algorithm starting from the solution given by SSI is very useful to identify the vibration modes of a structure, discarding the spurious modes that appear in high order models and discovering other hidden modes. Similar results are obtained using random starting values, although this strategy allows us to analyze the solution of several starting points what overcome the dependence on the initial values used.

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 Modal Parameters in Structures Using the Expectation Maximization Algorithm

This paper presents a time-domain stochastic system identification method based on maximum likelihood estimation (MLE) with the expectation maximization (EM) algorithm. The effectiveness of this structural identification method is evaluated through numerical simulation in the context of the ASCE benchmark problem on structural health monitoring. The benchmark structure is a four-story, two-bay by two-bay steel-frame scale model structure built in the Earthquake Engineering Research Laboratory at the University of British Columbia, Canada. This paper focuses on Phase I of the analytical benchmark studies. A MATLAB-based finite element analysis code obtained from the IASC-ASCE SHM Task Group web site is used to calculate the dynamic response of the prototype structure. A number of 100 simulations have been made using this MATLAB-based finite element analysis code in order to evaluate the proposed identification method. There are several techniques to realize system identification. In this work, stochastic subspace identification (SSI)method has been used for comparison. SSI identification method is a well known method and computes accurate estimates of the modal parameters. The principles of the SSI identification method has been introduced in the paper and next the proposed MLE with EM algorithm has been explained in detail. The advantages of the proposed structural identification method can be summarized as follows: (i) the method is based on maximum likelihood, that implies minimum variance estimates; (ii) EM is a computational simpler estimation procedure than other optimization algorithms; (iii) estimate more parameters than SSI, and these estimates are accurate. On the contrary, the main disadvantages of the method are: (i) EM algorithm is an iterative procedure and it consumes time until convergence is reached; and (ii) this method needs starting values for the parameters. Modal parameters (eigenfrequencies, damping ratios and mode shapes) of the benchmark structure have been estimated using both the SSI method and the proposed MLE + EM method. The numerical results show that the proposed method identifies eigenfrequencies, damping ratios and mode shapes reasonably well even in the presence of 10% measurement noises. These modal parameters are more accurate than the SSI estimated modal parameters.

Computational Methods for Identification of Vibrating Structures

System identification deals with the problem of building mathematical models of dynamical systems based on observed data from the system" [1]. In the context of civil engineering, the system refers to a large scale structure such as a building, bridge, or an offshore structure, and identification mostly involves the determination of modal parameters (the natural frequencies, damping ratios, and mode shapes). This paper presents some modal identification results obtained using a state-of-the-art time domain system identification method (data-driven stochastic subspace algorithms [2]) applied to the output-only data measured in a steel arch bridge. First, a three dimensional finite element model was developed for the numerical analysis of the structure using ANSYS. Modal analysis was carried out and modal parameters were extracted in the frequency range of interest, 0-10 Hz. The results obtained from the finite element modal analysis were used to determine the location of the sensors. After that, ambient vibration tests were conducted during April 23-24, 2009. The response of the structure was measured using eight accelerometers. Two stations of three sensors were formed (triaxial stations). These sensors were held stationary for reference during the test. The two remaining sensors were placed at the different measurement points along the bridge deck, in which only vertical and transversal measurements were conducted (biaxial stations). Point estimate and interval estimate have been carried out in the state space model using these ambient vibration measurements. In the case of parametric models (like state space), the dynamic behaviour of a system is described using mathematical models. Then, mathematical relationships can be established between modal parameters and estimated point parameters (thus, it is common to use experimental modal analysis as a synonym for system identification). Stable modal parameters are found using a stabilization diagram. Furthermore, this paper proposes a method for assessing the precision of estimates of the parameters of state-space models (confidence interval). This approach employs the nonparametric bootstrap procedure [3] and is applied to subspace parameter estimation algorithm. Using bootstrap results, a plot similar to a stabilization diagram is developed. These graphics differentiate system modes from spurious noise modes for a given order system. Additionally, using the modal assurance criterion, the experimental modes obtained have been compared with those evaluated from a finite element analysis. A quite good agreement between numerical and experimental results is observed.

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.

Estimation of modal parameters in structures using multiple time history records

In Operational Modal Analysis of structures we often have multiple time history records of vibrations measured at different time instants. This work presents a procedure for estimating the modal parameters of the structure processing all the records, that is, using all available information to obtain a single estimate of the modal parameters. The method uses Maximum Likelihood Estimation and the Expectation Maximization algorithm. Finally, it has been applied to various problems for both simulated and real structures and the results show the advantage of the joint analysis proposed.

Operational Modal Analysis using Expectation Maximization Algorithm

This paper presents a time-domain stochastic system identification method based on Maximum Likelihood Estimation and the Expectation Maximization algorithm. The effectiveness of this structural identification method is evaluated through numerical simulation in the context of the ASCE benchmark problem on structural health monitoring. Modal parameters (eigenfrequencies, damping ratios and mode shapes) of the benchmark structure have been estimated applying the proposed identification method to a set of 100 simulated cases. The numerical results show that the proposed method estimates all the modal parameters reasonably well in the presence of 30% measurement noise even. Finally, advantages and disadvantages of the method have been discussed.

Joint statistical analysis of multiple measurement setups for the estimation of vibrational modes

Computing the modal parameters of large structures in Operational Modal Analysis often requires to process data from multiple non simultaneously recorded setups of sensors. These setups share some sensors in common, the so-called reference sensors that are fixed for all the measurements, while the other sensors are moved from one setup to the next. One possibility is to process the setups separately what result 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 modes, while the natural frequencies and damping ratios are usually averaged. In this paper we present a state space model that can be used to process all setups at once so the global mode shapes are obtained automatically and subsequently only a value for the natural frequency and damping ratio of each mode is computed. We also present how this model can be estimated using maximum likelihood and the Expectation Maximization algorithm. We apply this technique to real data measured at a footbridge.

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.

De la ciudad a la estancia: casas con patio en la vivienda social madrileña (1956-1961). Saénz de Oíza y Vázquez de Castro

La tesis aborda el estudio de la casa patio y su relación con todas las escalas del proyecto, dentro del ámbito de la vivienda colectiva desarrollada en Madrid desde el comienzo de la década de los cincuenta hasta principio de los sesenta. Se investiga su vínculo con la ciudad a partir de la configuración de la trama y los espacios públicos, la organización de la casa en torno a un vacío y las relaciones espaciales entre el patio exterior y las estancias interiores. Mediante el análisis de los diferentes ejemplos existentes de casas con patio en el contexto señalado, se realiza la selección de los casos de estudio de esta investigación. Los proyectos analizados se organizan a partir del trabajo desarrollado por los arquitectos Francisco Javier Sáenz de Oíza y Antonio Vázquez de Castro, liderando equipos más amplios y como representantes de la construcción definitiva de un verdadero lenguaje moderno en la arquitectura española. Cada uno de los proyectos, estructurados en dos apartados para cada uno de los autores, muestran de forma particular, la evolución de las herramientas arquitectónicas empleadas, a partir de la casa patio, en las distintas escalas de trabajo. En los proyectos de Oíza, se analiza en primer lugar la definición de un modelo mínimo y racional de vivienda con patio en el poblado de absorción de Fuencarral “A” (1956) y con el prototipo del concurso de vivienda experimental (1956). A continuación se estudia, en el Poblado Dirigido de Entrevías (1956), la construcción de la ciudad horizontal a partir del patio como elemento estructurante de las tramas urbanas. Las obras de Vázquez de Castro muestran inicialmente la incorporación del patio como una estancia privada más de la casa con la definición de un modelo tipológico en “L” en el Poblado Dirigido de Ca��o Roto (1956). Finalmente, se analiza el desarrollo de nuevas y complejas estructuras urbanas en forma de tapiz en el proyecto de la Unidad Vecinal de Costa Rica (1961). Los ejemplos, antes descritos, protagonizan un proceso delimitado desde la importación y adaptación de los modelos de vivienda europeos producidos en el periodo de entreguerras, con un significativo retraso; hasta la apuesta por sistemas complejos como el “mat-building”, en diálogo directo, con los proyectos de las nuevas corrientes arquitectónicas internacionales de principios de los sesenta. El estudio y análisis de los ejemplos se realiza mediante la disección pormenorizada de cada uno de ellos, en los sistemas arquitectónicos que establecen la relación del patio con las tres escalas objeto de estudio: la ciudad, la casa y la estancia. Para ello se realiza una reconstrucción de los proyectos mediante la elaboración de una nueva documentación gráfica a partir de los planos y fotografías originales. Finalmente, a partir de los temas arquitectónicos más significativos extraídos del análisis de los casos de estudio y organizados en función de las distintas escalas de proyecto, se realiza una comparación transversal, a modo de conclusiones, respecto a otros ejemplos de similares características del panorama internacional de casas con patio. De esta forma se produce una síntesis sobre los distintos mecanismos que sirvieron para construir, a partir de la tradición del patio en la casa, la vivienda moderna. ABSTRACT The thesis deals the study of the house with courtyard and its relation with all the scales of the project inside the ambience of the collective housing developed in Madrid from the beginning of the fifties up to beginning of the sixties. This document investigates its link with the city from the configuration of the weave and the public spaces, the organization of the house around a void and the spatial relations between the exterior courtyard and the interior spaces. The selection of the case studies of this research is made Through the analysis of the different existing examples of courtyard houses in this defined context. The analyzed projects are organized from the work developed by the architects Francisco Javier Sáenz de Oíza and Antonio Vázquez de Castro, leading wider teams and as representatives of the definitive construction of a real modern language in the Spanish architecture. Each of the projects, structured in two sections to each one of the authors, show in a particular way, the evolution of the architectural tools employed, from the house with courtyard, at different scales of work. In the projects of Oiza, it´s analyced the definition of a minimum and rational model of a house with courtyard in the absorption village of Fuencarral “A” (1956) and with the experimental housing contest prototype (1956). Next, in “el Poblado Dirigido de Entrevías” (1956), the construction of the horinzontal city with the courtyard as a structuring element of the urban weave will be studied. The works of Vázquez de Castro initially show the incorporation of the courtyard as another private area of the house with the typological model definition as an “L” in El Poblado Dirigido de Ca��o Roto (1956). Finally, the thesis analyzes the development of new and complex urban structures with a mat-building shape in the project of Unidad Vecinal de Costa Rica (1961). The examples described above, starring a process delimited from the import and the adaptation of european housing models produced in the interwar period, with a significant delay; until the opt for complex systems such as the “mat-building”, in direct dialog, with the projects of the new architectural international flows of the beginning of the sixties. The study and analysis of the examples is made by the dissection of them, in the architectural systems that establish the relation of the courtyard with the three scales of the study: the city, the house and the stay. To reach this objetive this document reconstructs the projects making a new graphic documentation from the original maps and photos. Finally, from the most important architectural issues drawn from the analysis of the case studies and organised by the different scales of project, as conclussions, a cross comparison of similar examples of houses with courtyard around the world is made. This produces a synthesis about the different mechanisms that served to build, from the tradition of the courtyard in the house, the modern housing.

La vivienda plurifamiliar de Juan Manuel Ruiz de la Prada

El objetivo de este trabajo es conocer la arquitectura de Juan Manuel Ruiz de la Prada, un arquitecto cuya obra ha dejado una impronta en la ciudad de Madrid, pero sobre el que no hay apenas información objetiva que ponga en valor su trabajo. Lo más relevante dentro de su trayectoria como arquitecto, es una serie de edificios en los barrios de Salamanca y Chamberí, con un programa similar y ejecutados en un corto periodo de tiempo. Este trabajo analiza las circunstancias del Madrid de los 60 junto con la arquitectura de la vivienda plurifamiliar destinada a las clases acomodadas de la época. Se han estudiado con mayor profundidad, dos de los edificios de Juan Manuel Ruiz de la Prada, en los que tuvo un doble papel de promotor y de arquitecto. El primero situado en José Abascal nº50 con Zurbano nº73 y el segundo situado en Martínez Campos con Zurbano nº51 que, a su vez, es el primero de una serie de cuatro.

ITER fast plant system controller prototype based on PXIe platform

The ITER CODAC design identifies slow and fast plant system controllers (PSC). The gast OSCs are based on embedded technologies, permit sampling rates greater than 1 KHz, meet stringent real-time requirements, and will be devoted to data acquisition tasks and control purposes. CIEMAT and UPM have implemented a prototype of a fast PSC based on commercial off-the-shelf (COTS) technologies with PXI hardware and software based on EPICS

NetCDF based data archiving system applied to ITER Fast Plant System Control prototype

EURATOM/CIEMAT and Technical University of Madrid (UPM) have been involved in the development of a FPSC [1] (Fast Plant System Control) prototype for ITER, based on PXIe (PCI eXtensions for Instrumentation). One of the main focuses of this project has been data acquisition and all the related issues, including scientific data archiving. Additionally, a new data archiving solution has been developed to demonstrate the obtainable performances and possible bottlenecks of scientific data archiving in Fast Plant System Control. The presented system implements a fault tolerant architecture over a GEthernet network where FPSC data are reliably archived on remote, while remaining accessible to be redistributed, within the duration of a pulse. The storing service is supported by a clustering solution to guaranty scalability, so that FPSC management and configuration may be simplified, and a unique view of all archived data provided. All the involved components have been integrated under EPICS [2] (Experimental Physics and Industrial Control System), implementing in each case the necessary extensions, state machines and configuration process variables. The prototyped solution is based on the NetCDF-4 [3] and [4] (Network Common Data Format) file format in order to incorporate important features, such as scientific data models support, huge size files management, platform independent codification, or single-writer/multiple-readers concurrency. In this contribution, a complete description of the above mentioned solution is presented, together with the most relevant results of the tests performed, while focusing in the benefits and limitations of the applied technologies.

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.