Kernel - based continous - time systems identification: methods and tools

2012/2013Questa tesi ha lo scopo di formalizzare un nuovo filone teorico, che deriva dall’algebra degli operatori lineari integrali di Fredholm-Volterra agenti su spazi di Hilbert, per la sintesi di stimatori dello stato e parametrici per sistemi dinamici a tempo continuo sfruttando le misure ingressi/uscite, soggetti a perturbazione tempo-varianti. In maniera da ottenere stime non-asintotiche di sistemi dinamici a tempo continuo, i metodi classici tipicamente aumentano la dimensione del vettore delle variabili di decisione con le condizioni iniziali incognite di stati non misurati. Tuttavia, questo porta ad un accrescimento della complessitá dell’algoritmo. Recentemente, diversi metodi di stima algebrici sono stati sviluppati, sfruttando un approccio algebrico piuttosto che da una prospettiva statistica o teorica. Mentre le forti fondamenta teoriche e le proprietá di convergenza non asintotiche rappresentano caratteristiche notevoli per questi metodi, il principale inconveniente é che l’implementazione pratica produce una dinamica internamente instabile. Quindi, la progettazione di metodi di stima per questi tipi di sistemi é un argomento importante ed emergente. L’obiettivo di questo lavoro é quello di presentare alcuni risultati recenti, considerando diversi aspetti e affrontando alcuni dei problemi che emergono quando si progettano algoritmi di identificazione. Lo scopo é sviluppare un’architettura di stima con proprietá di convergenza molto veloci e internamente stabile. Seguendo un ordine logico, prima di tutto verrá progettato l’algoritmo di identificazione proponendo una nuova architettura basata sui kernel, utilizzando l’algebra degli operatori lineari integrali di Fredholm-Volterra. Inoltre, la metodologia proposta sará affrontata in maniera da progettare stimatori per sistemi dinamici a tempo continuo con proprietá di convergenza molto veloci, caratterizzati da gradi relativi limitati e possibilmente affetti da perturbazioni strutturate. Piú nello specifico, il progetto di adeguati kernel di operatori lineari integrali non-anticipativi dará origine a stimatori caratterizzati da proprietá di convergenza idealmente "non- asintotiche".Le analisi delle proprietá dei kernel verrá affrontata e due classi di funzioni kernel ammissibili saranno introdotte: una per il problema di stima parametrica e uno per il problema di stima dello stato. Gli operatori che verranno indotti da tali funzioni kernel proposte, ammettono realizzazione spazio-stato implementabile (cioé a dimensione finita e internamente stabile). Allo scopo di dare maggior completezza, l’analisi del bias dello stimatore proposto verrá esaminata, derivando le proprietá asintotiche dell’algoritmo di identificazione e dimostrando che le funzioni kernel possono essere pro- gettate tenendo in debito conto i risultati ottenuti in questa analisi.This thesis is aimed at the formalization of a new theoretical framework, arising from the algebra of Fredholm-Volterra linear integral operators acting on Hilbert spaces, for the synthesis of non-asymptotic state and parameter estimators for continuous-time dynamical systems from input-output measurements subject to time-varying perturbations. In order to achieve non-asymptotic estimates of continuous-time dynamical systems, classical methods usually augment the vector of decision variables with the unknown initial conditions of the non measured states. However, this comes at the price of an increase of complexity for the algorithm. Recently, several algebraic estimation methods have been developed, arising from an algebraic setting rather than from a statistical or a systems-theoretic perspective. While the strong theoretical foundations and the non-asymptotic convergence property represent oustanding features of these methods, the major drawback is that the practical implementation ends up with an internally unstable dynamic. Therefore, the design of estimation methods for these kind of systems is an important and emergent topic. The goal of this work is to present some recent results, considering different frameworks and facing some of the issues emerging when dealing with the design of identification algorithms. The target is to develop a comprehensive estimation architecture with fast convergence properties and internally stable. Following a logical order, first of all we design the identification algorithm by proposing a novel kernel-based architecture, by means of the algebra of Fredholm-Volterra linear integral operators. Besides, the proposed methodology is addressed in order to design estimators with very fast convergence properties for continuous-time dynamic systems characterized by bounded relative degree and possibly affected by structured perturbations. More specifically, the design of suitable kernels of non-anticipative linear integral operators gives rise to estimators characterized by convergence properties ideally “non-asymptotic". The analysis of the properties of the kernels guaranteeing such a fast convergence is addressed and two classes of admissible kernel functions are introduced: one for the parameter estimation problem and one for the state estimation problem. The operators induced by the proposed kernels admit implementable (i.e., finite-dimensional and internally stable) state- space realizations. For the sake of completeness, the bias analysis of the proposed estimator is addressed, deriving the asymptotic properties of the identification algorithm and demonstrating that the kernel functions can be designed taking in account the results obtained with this analysis.XXVI Ciclo198

Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions

A truncated sequential procedure is constructed for estimating the drift coefficient at a given state point based on discrete data of ergodic diffusion process. A nonasymptotic upper bound is obtained for a pointwise absolute error risk. The optimal convergence rate and a sharp constant in the bounds are found for the asymptotic pointwise minimax risk. As a consequence, the efficiency is obtained of the proposed sequential procedure.Comment: Published at http://dx.doi.org/10.3150/14-BEJ655 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Parameter estimation of ODE's via nonparametric estimators

Ordinary differential equations (ODE's) are widespread models in physics, chemistry and biology. In particular, this mathematical formalism is used for describing the evolution of complex systems and it might consist of high-dimensional sets of coupled nonlinear differential equations. In this setting, we propose a general method for estimating the parameters indexing ODE's from times series. Our method is able to alleviate the computational difficulties encountered by the classical parametric methods. These difficulties are due to the implicit definition of the model. We propose the use of a nonparametric estimator of regression functions as a first-step in the construction of an M-estimator, and we show the consistency of the derived estimator under general conditions. In the case of spline estimators, we prove asymptotic normality, and that the rate of convergence is the usual $\sqrt{n}$-rate for parametric estimators. Some perspectives of refinements of this new family of parametric estimators are given.Comment: Published in at http://dx.doi.org/10.1214/07-EJS132 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Calculation of Generalized Polynomial-Chaos Basis Functions and Gauss Quadrature Rules in Hierarchical Uncertainty Quantification

Stochastic spectral methods are efficient techniques for uncertainty quantification. Recently they have shown excellent performance in the statistical analysis of integrated circuits. In stochastic spectral methods, one needs to determine a set of orthonormal polynomials and a proper numerical quadrature rule. The former are used as the basis functions in a generalized polynomial chaos expansion. The latter is used to compute the integrals involved in stochastic spectral methods. Obtaining such information requires knowing the density function of the random input {\it a-priori}. However, individual system components are often described by surrogate models rather than density functions. In order to apply stochastic spectral methods in hierarchical uncertainty quantification, we first propose to construct physically consistent closed-form density functions by two monotone interpolation schemes. Then, by exploiting the special forms of the obtained density functions, we determine the generalized polynomial-chaos basis functions and the Gauss quadrature rules that are required by a stochastic spectral simulator. The effectiveness of our proposed algorithm is verified by both synthetic and practical circuit examples.Comment: Published by IEEE Trans CAD in May 201