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.

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.

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.

Space-state robust control of a Buck converter with amorphous core coil and variable load

Pulse-width modulation is widely used to control electronic converters. One of the most frequently used topologies for high DC voltage/low DC voltage conversion is the Buck converter. These converters are described by a second order system with an LC filter between the switching subsystem and the load. The use of a coil with an amorphous magnetic material core rather than an air core permits the design of smaller converters. If high switching frequencies are used to obtain high quality voltage output, then the value of the auto inductance L is reduced over time. Robust controllers are thus needed if the accuracy of the converter response must be preserved under auto inductance and payload variations. This paper presents a robust controller for a Buck converter based on a state space feedback control system combined with an additional virtual space variable which minimizes the effects of the inductance and load variations when a switching frequency that is not too high is applied. The system exhibits a null steady-state average error response for the entire range of parameter variations. Simulation results and a comparison with a standard PID controller are also presented.

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.

Teaching and evaluating mechanical vibrations with Matlab®

The study of the response of mechanical systems to external excitations, even in the simplest cases, involves solving second-order ordinary differential equations or systems thereof. Finding the natural frequencies of a system and understanding the effect of variations of the excitation frequencies on the response of the system are essential when designing mechanisms [1] and structures [2]. However, faced with the mathematical complexity of the problem, students tend to focus on the mathematical resolution rather than on the interpretation of the results. To overcome this difficulty, once the general theoretical problem and its solution through the state space [3] have been presented, Matlab®[4] and Simulink®[5] are used to simulate specific situations. Without them, the discussion of the effect of slight variations in input variables on the outcome of the model becomes burdensome due to the excessive calculation time required. Conversely, with the help of those simulation tools, students can easily reach practical conclusions and their evaluation can be based on their interpretation of results and not on their mathematical skills

Asymptotic optimality of RESTART estimators in highly dependable systems

Abstract We consider a wide class of models that includes the highly reliable Markovian systems (HRMS) often used to represent the evolution of multi-component systems in reliability settings. Repair times and component lifetimes are random variables that follow a general distribution, and the repair service adopts a priority repair rule based on system failure risk. Since crude simulation has proved to be inefficient for highly-dependable systems, the RESTART method is used for the estimation of steady-state unavailability and other reliability measures. In this method, a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of a rare event (e.g., a system failure) is higher. The main difficulty involved in applying this method is finding a suitable function, called the importance function, to define the regions. In this paper we introduce an importance function which, for unbalanced systems, represents a great improvement over the importance function used in previous papers. We also demonstrate the asymptotic optimality of RESTART estimators in these models. Several examples are presented to show the effectiveness of the new approach, and probabilities up to the order of 10-42 are accurately estimated with little computational effort.

Forecasting Electricity Prices and their volatilities using Unobserved Components.

The liberalization of electricity markets more than ten years ago in the vast majority of developed countries has introduced the need of modelling and forecasting electricity prices and volatilities, both in the short and long term. Thus, there is a need of providing methodology that is able to deal with the most important features of electricity price series, which are well known for presenting not only structure in conditional mean but also time-varying conditional variances. In this work we propose a new model, which allows to extract conditionally heteroskedastic common factors from the vector of electricity prices. These common factors are jointly estimated as well as their relationship with the original vector of series, and the dynamics affecting both their conditional mean and variance. The estimation of the model is carried out under the state-space formulation. The new model proposed is applied to extract seasonal common dynamic factors as well as common volatility factors for electricity prices and the estimation results are used to forecast electricity prices and their volatilities in the Spanish zone of the Iberian Market. Several simplified/alternative models are also considered as benchmarks to illustrate that the proposed approach is superior to all of them in terms of explanatory and predictive power.

PWM Control of a Buck Converter with an Amorphous Core Coil

Pulse-width modulation is widely used to control electronic converters. One of the most topologies used for high DC voltage/low DC voltage conversion is the Buck converter. It is obtained as a second order system with a LC filter between the switching subsystem and the load. The use of a coil with an amorphous magnetic material core instead of air core lets design converters with smaller size. If high switching frequencies are used for obtaining high quality voltage output, the value of the auto inductance L is reduced throughout the time. Then, robust controllers are needed if the accuracy of the converter response must not be affected by auto inductance and load variations. This paper presents a robust controller for a Buck converter based on a state space feedback control system combined with an additional virtual space variable which minimizes the effects of the inductance and load variations when a not-toohigh switching frequency is applied. The system exhibits a null steady-state average error response for the entire range of parameter variations. Simulation results are presented.

Consciousness, Action Selection, Meaning and Phenomenic Anticipation

Phenomenal states are generally considered the ultimate sources of intrinsic motivation for autonomous biological agents. In this article, we will address the issue of the necessity of exploiting these states for the design and implementation of robust goal-directed artificial systems. We will provide an analysis of consciousness in terms of a precise definition of how an agent "understands" the informational flows entering the agent and its very own action possibilities. This abstract model of consciousness and understanding will be based in the analysis and evaluation of phenomenal states along potential future trajectories in the state space of the agents. This implies that a potential strategy to follow in order to build autonomous but still customer-useful systems is to embed them with the particular, ad hoc phenomenality that captures the system-external requirements that define the system usefulness from a customer-based, requirements-strict engineering viewpoint.

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.

Operational modal analisys using joint statistical analysis of multiple records

In Operational Modal Analysis (OMA) of a structure, the data acquisition process may be repeated many times. In these cases, the analyst has several similar records for the modal analysis of the structure that have been obtained at di�erent time instants (multiple records). The solution obtained varies from one record to another, sometimes considerably. The differences are due to several reasons: statistical errors of estimation, changes in the external forces (unmeasured forces) that modify the output spectra, appearance of spurious modes, etc. Combining the results of the di�erent individual analysis is not straightforward. To solve the problem, we propose to make the joint estimation of the parameters using all the records. This can be done in a very simple way using state space models and computing the estimates by maximum-likelihood. The method provides a single result for the modal parameters that combines optimally all the records.

A unified framework for linear function approximation of value functions in stochastic control

This paper contributes with a unified formulation that merges previ- ous analysis on the prediction of the performance ( value function ) of certain sequence of actions ( policy ) when an agent operates a Markov decision process with large state-space. When the states are represented by features and the value function is linearly approxi- mated, our analysis reveals a new relationship between two common cost functions used to obtain the optimal approximation. In addition, this analysis allows us to propose an efficient adaptive algorithm that provides an unbiased linear estimate. The performance of the pro- posed algorithm is illustrated by simulation, showing competitive results when compared with the state-of-the-art solutions.