Asymptotic forecast uncertainty and the unstable subspace in the presence of additive model error

It is well understood that dynamic instability is among the primary drivers of forecast uncertainty in chaotic, physical systems. Data assimilation techniques have been designed to exploit this phenomenon, reducing the effective dimension of the data assimilation problem to the directions of rapidly growing errors. Recent mathematical work has, moreover, provided formal proofs of the central hypothesis of the assimilation in the unstable subspace methodology of Anna Trevisan and her collaborators: for filters and smoothers in perfect, linear, Gaussian models, the distribution of forecast errors asymptotically conforms to the unstable-neutral subspace. Specifically, the column span of the forecast and posterior error covariances asymptotically align with the span of backward Lyapunov vectors with nonnegative exponents. Earlier mathematical studies have focused on perfect models, and this current work now explores the relationship between dynamical instability, the precision of observations, and the evolution of forecast error in linear models with additive model error. We prove bounds for the asymptotic uncertainty, explicitly relating the rate of dynamical expansion, model precision, and observational accuracy. Formalizing this relationship, we provide a novel, necessary criterion for the boundedness of forecast errors. Furthermore, we numerically explore the relationship between observational design, dynamical instability, and filter boundedness. Additionally, we include a detailed introduction to the multiplicative ergodic theorem and to the theory and construction of Lyapunov vectors. While forecast error in the stable subspace may not generically vanish, we show that even without filtering, uncertainty remains uniformly bounded due its dynamical dissipation. However, the continuous reinjection of uncertainty from model errors may be excited by transient instabilities in the stable modes of high variance, rendering forecast uncertainty impractically large. In the context of ensemble data assimilation, this requires rectifying the rank of the ensemble-based gain to account for the growth of uncertainty beyond the unstable and neutral subspace, additionally correcting stable modes with frequent occurrences of positive local Lyapunov exponents that excite model errors

協方差型隨機子空間識別法之應用

In this research the application of output-only system identification technique known as Stochastic Subspace Identification (SSI) algorithms in civil structures is carried out. With the aim of finding accurate modal parameters of the structure in off-line analysis, a stabilization diagram is constructed by plotting the identified poles of the system with increasing the size of data matrix. A sensitivity study of the implementation of SSI through stabilization diagram is firstly carried out, different scenarios such as noise effect, nonlinearity, time-varying systems and closely-spaced frequencies are considered. Comparison between different SSI approaches was also discussed. In the following, the identification task of a real large scale structure: Canton Tower, a benchmark problem for structural health monitoring of high-rise slender structures is carried out, for which the capacity of Covariance-driven SSI algorithm (SSI-COV) will be demonstrated. The introduction of a subspace preprocessing algorithm known as Singular Spectrum Analysis (SSA) can greatly enhance the identification capacity, which in conjunction with SSI-COV is called the SSA-SSI-COV method, it also allows the determination of the best system order. The objective of the second part of this research is to develop on-line system parameter estimation and damage detection technique through Recursive Covariance-driven Stochastic Subspace identification (RSSI-COV) approach. The Extended Instrumental Variable version of Projection Approximation Subspace Tracking algorithm (EIV-PAST) is taking charge of the system-related subspace updating task. To further reduce the noise corruption in field experiments, the data pre-processing technique called recursive Singular Spectrum Analysis technique (rSSA) is developed to remove the noise contaminant measurements, so as to enhance the stability of data analysis. Through simulation study as well as the experimental research, both RSSI-COV and rSSA-SSI-COV method are applied to identify the dynamic behavior of systems with time-varying characteristics, the reliable control parameters for the model are examined. Finally, these algorithms are applied to track the evolution of modal parameters for: (1) shaking table test of a 3-story steel frame with instantaneous stiffness reduction. (2) Shaking table test of a 1-story 2-bay reinforced concrete frame, both under earthquake excitation, and at last, (3) damage detection and early warning of an experimental steel bridge under continuous scour.UCR::Vicerrectoría de Docencia::Ingeniería::Facultad de Ingeniería::Escuela de Ingeniería Civi

Stochastic Modeling and Estimation of Wireless Channels with Application to Ultra Wide Band Systems

This thesis is concerned with modeling of both space and time variations of Ultra Wide Band (UWB) indoor channels. The most common empirically determined amplitude distribution in many UWB environments is Nakagami distribution. The latter is generalized to stochastic diffusion processes which capture the dynamics of UWB channels. In contrast with the traditional models, the statistics of the proposed models are shown to be time varying, but converge in steady state to their static counterparts. System identification algorithms are used to extract various channel parameters using received signal measurement data, which are usually available at the receiver. The expectation maximization (EM) algorithm and the Kalman filter (KF) are employed in estimating channel parameters as well as the inphase and quadrature components, respectively. The proposed algorithms are recursive and therefore can be implemented in real time. Further, sufficient conditions for the convergence of the EM algorithm are provided. Comparison with recursive Least-square (LS) algorithms is carried out using experimental measurements. Distributed stochastic power control algorithms based on the fixed point theorem and stochastic approximations are used to solve for the optimal transmit power problem and numerical results are also presented. A framework which can capture the statistics of the overall received signal and a methodology to estimate parameters of the counting process based on the received signal is developed. Furthermore, second moment statistics and characteristic functions are computed explicitly and considered as an extension of Rice’s shot noise analysis. Another two important components, input design and model selection are also considered. Gel’fand n-widths and Time n-widths are used to represent the inherent error introduced by input design. Kolmogorov n-width is used to characterize the representation error introduced by model selection. In particular, it is shown that the optimal model for reducing the representation error is a finite impulse response (FIR) model and the optimal input is an impulse at the start of the observation interval

Comparative review of methods for stability monitoring in electrical power systems and vibrating structures

This study provides a review of methods used for stability monitoring in two different fields, electrical power systems and vibration analysis, with the aim of increasing awareness of and highlighting opportunities for cross-fertilisation. The nature of the problems that require stability monitoring in both fields are discussed here as well as the approaches that have been taken. The review of power systems methods is presented in two parts: methods for ambient or normal operation and methods for transient or post-fault operation. Similarly, the review of methods for vibration analysis is presented in two parts: methods for stationary or linear time-invariant data and methods for non-stationary or non-linear time-variant data. Some observations and comments are made regarding methods that have already been applied in both fields including recommendations for the use of different sets of algorithms that have not been utilised to date. Additionally, methods that have been applied to vibration analysis and have potential for power systems stability monitoring are discussed and recommended. � 2010 The Institution of Engineering and Technology

LQG Online Learning

Optimal control theory and machine learning techniques are combined to formulate and solve in closed form an optimal control formulation of online learning from supervised examples with regularization of the updates. The connections with the classical Linear Quadratic Gaussian (LQG) optimal control problem, of which the proposed learning paradigm is a non-trivial variation as it involves random matrices, are investigated. The obtained optimal solutions are compared with the Kalman-filter estimate of the parameter vector to be learned. It is shown that the proposed algorithm is less sensitive to outliers with respect to the Kalman estimate (thanks to the presence of the regularization term), thus providing smoother estimates with respect to time. The basic formulation of the proposed online-learning framework refers to a discrete-time setting with a finite learning horizon and a linear model. Various extensions are investigated, including the infinite learning horizon and, via the so-called "kernel trick", the case of nonlinear models

Coarse Brownian Dynamics for Nematic Liquid Crystals: Bifurcation Diagrams via Stochastic Simulation

We demonstrate how time-integration of stochastic differential equations (i.e. Brownian dynamics simulations) can be combined with continuum numerical bifurcation analysis techniques to analyze the dynamics of liquid crystalline polymers (LCPs). Sidestepping the necessity of obtaining explicit closures, the approach analyzes the (unavailable in closed form) coarse macroscopic equations, estimating the necessary quantities through appropriately initialized, short bursts of Brownian dynamics simulation. Through this approach, both stable and unstable branches of the equilibrium bifurcation diagram are obtained for the Doi model of LCPs and their coarse stability is estimated. Additional macroscopic computational tasks enabled through this approach, such as coarse projective integration and coarse stabilizing controller design, are also demonstrated

MODEL UPDATING AND STRUCTURAL HEALTH MONITORING OF HORIZONTAL AXIS WIND TURBINES VIA ADVANCED SPINNING FINITE ELEMENTS AND STOCHASTIC SUBSPACE IDENTIFICATION METHODS

Wind energy has been one of the most growing sectors of the nation’s renewable energy portfolio for the past decade, and the same tendency is being projected for the upcoming years given the aggressive governmental policies for the reduction of fossil fuel dependency. Great technological expectation and outstanding commercial penetration has shown the so called Horizontal Axis Wind Turbines (HAWT) technologies. Given its great acceptance, size evolution of wind turbines over time has increased exponentially. However, safety and economical concerns have emerged as a result of the newly design tendencies for massive scale wind turbine structures presenting high slenderness ratios and complex shapes, typically located in remote areas (e.g. offshore wind farms). In this regard, safety operation requires not only having first-hand information regarding actual structural dynamic conditions under aerodynamic action, but also a deep understanding of the environmental factors in which these multibody rotating structures operate. Given the cyclo-stochastic patterns of the wind loading exerting pressure on a HAWT, a probabilistic framework is appropriate to characterize the risk of failure in terms of resistance and serviceability conditions, at any given time. Furthermore, sources of uncertainty such as material imperfections, buffeting and flutter, aeroelastic damping, gyroscopic effects, turbulence, among others, have pleaded for the use of a more sophisticated mathematical framework that could properly handle all these sources of indetermination. The attainable modeling complexity that arises as a result of these characterizations demands a data-driven experimental validation methodology to calibrate and corroborate the model. For this aim, System Identification (SI) techniques offer a spectrum of well-established numerical methods appropriated for stationary, deterministic, and data-driven numerical schemes, capable of predicting actual dynamic states (eigenrealizations) of traditional time-invariant dynamic systems. As a consequence, it is proposed a modified data-driven SI metric based on the so called Subspace Realization Theory, now adapted for stochastic non-stationary and timevarying systems, as is the case of HAWT’s complex aerodynamics. Simultaneously, this investigation explores the characterization of the turbine loading and response envelopes for critical failure modes of the structural components the wind turbine is made of. In the long run, both aerodynamic framework (theoretical model) and system identification (experimental model) will be merged in a numerical engine formulated as a search algorithm for model updating, also known as Adaptive Simulated Annealing (ASA) process. This iterative engine is based on a set of function minimizations computed by a metric called Modal Assurance Criterion (MAC). In summary, the Thesis is composed of four major parts: (1) development of an analytical aerodynamic framework that predicts interacted wind-structure stochastic loads on wind turbine components; (2) development of a novel tapered-swept-corved Spinning Finite Element (SFE) that includes dampedgyroscopic effects and axial-flexural-torsional coupling; (3) a novel data-driven structural health monitoring (SHM) algorithm via stochastic subspace identification methods; and (4) a numerical search (optimization) engine based on ASA and MAC capable of updating the SFE aerodynamic model

Time-varying Autoregressive Modeling of Nonstationary Signals

Nonstationary signal modeling is a research topic of practical interest. In this thesis, we adopt a time-varying (TV) autoregressive (AR) model using the basis function (BF) parameter estimation method for nonstationary process identification and instantaneous frequency (IF) estimation. The current TVAR model in direct form (DF) with the blockwise least-squares and recursive weighted-least-squares BF methods perform equivalently well in signal modeling, but the large estimation error may cause temporary instabilities of the estimated model. To achieve convenient model stability monitoring and pole tracking, the TVAR model in cascade form (CF) was proposed through the parameterization in terms of TV poles (represented by second order section coefficients, Cartesian coordinates, Polar coordinates), where the time variation of each pole parameter is assumed to be the linear combination of BFs. The nonlinear system equations for the TVAR model in CF are solved iteratively using the Gauss-Newton algorithm. Using the CF, the model stability is easily controlled by constraining the estimated TV poles within the unit circle. The CF model shows similar performance trends to the DF model using the recursive BF method, and the TV pole representation in Cartesian coordinates outperforms all other representations. The individual frequency variation can be finely tracked using the CF model, when several frequency components are present in the signal. Simulations were carried on synthetic sinusoidal signals with different frequency variations for IF estimation. For the TVAR model in DF (blockwise), the basis dimension (BD) is an important factor on frequency estimation accuracy. For the TVAR model in DF (recursive) and CF (Cartesian), the influences of BD are negligible. The additive white noise in the observed signal degrades the estimation performance, and the the noise effects can be reduce by using higher model order. Experiments were carried on the real electromyography (EMG) data for frequency estimation in the analysis of muscle fatigue. The TVAR modeling methods show equivalent performance to the conventional Fourier transform method