Parameter optimization of orthonormal basis functions for efficient rational approximations

International audienceIn this paper, the authors present an efficient procedure for optimal placement of poles in rational approximations by Müntz-Laguerre functions. The technique is formulated as the minimization of a quadratic criterion and the linear equations involved are efficiently expressed using the orthonormal basis functions. The presented technique has direct application in rational approximation and model order reduction of large-degree or infinite-dimensional systems. An efficient choice of parameters in orthogonal Müntz-Laguerre approximation Model order reduction of large-degree or infinite-dimensional systems The choice of Müntz-Laguerre parameters is based on a least squares optimization Abstract: In this paper, the authors present an efficient procedure for optimal placement of poles in rational approximations by Müntz-Laguerre functions. The technique is formulated as the minimization of a quadratic criterion and the linear equations involved are efficiently expressed using the orthonormal basis functions. The presented technique has direct application in rational approximation and model order reduction of large-degree or infinite-dimensional systems

Characterization of time-varying human operator dynamics

Human operator performance in tracking study of pilots determined by deterministic characterization theory and mathematical mode

Control Relevant System Identification Using Orthonormal Basis Filter Models

Models are extensively used in advanced process control system design and implementations. Nearly all optimal control design techniques including the widely used model predictive control techniques rely on the use of model of the system to be controlled. There are several linear model structures that are commonly used in control relevant problems in process industries. Some of these model structures are: Auto Regressive with Exogenous Input (ARX), Auto Regressive Moving Average with Exogenous Input (ARMAX), Finite Impulse Response (FIR), Output Error (OE) and Box Jenkins (BJ) models. The selection of the appropriate model structure, among other factors, depend on the consistency of the model parameters, the number of parameters required to describe a system with acceptable accuracy and the computational load in estimating the model parameters. ARX and ARMAX models suffer from inconsistency problem in most open-loop identification problems. Finite Impulse Response (FIR) models require large number of parameters to describe linear systems with acceptable accuracy. BJ, OE and ARMAX models involve nonlinear optimization in estimating their parameters. In addition, all of the above conventional linear models, except FIR, require the time delay of the system to be separately estimated and included in the estimation of the parameters. Orthonormal Basis Filter (OBF) models have several advantages over the other conventional linear models. They are consistent in parameters for most open-loop identification problems. They are parsimonious in parameters if the dominant pole(s) of the system are used in their development. The model parameters are easily estimated using the linear least square method. Moreover, the time delay estimation can be easily integrated in the model development. However, there are several problems that are not yet addressed. Some of the outstanding problems are: (i) Developing parsimonious OBF models when the dominant poles of the system are not known (ii) Obtaining a better estimate of time delay for second or higher order systems (iii) Including an explicit noise model in the framework of OBF model structures and determine the parameters and multi-step ahead predictions (iv) Closed-loop identification problems in this new OBF plus noise model frame work This study presents novel schemes that address the above problems. The first problem is addressed by formulating an iterative scheme where one or two of the dominant pole(s) of the system are estimated and used to develop parsimonious OBF models. A unified scheme is formulated where an OBF-deterministic model and an explicit AR or ARMA stochastic (noise) models are developed to address the second problem. The closed-loop identification problem is addressed by developing schemes based on the direct and indirect approaches using OBF based structures. For all the proposed OBF prediction model structures, the method for estimating the model parameters and multi-step ahead prediction are developed. All the proposed schemes are demonstrated with the help of simulation and real plant case studies. The accuracy of the developed OBF-based models is verified using appropriate validation procedures and residual analysis

Virtual reference feedback tuning of controllers parameterized using orthonormal basis functions

Supervisor : Dr. Gustavo Henrique da Costa OliveiraCo-supervisor : Dr. Prof. Gideon Villar LeandroDissertação (mestrado) - Universidade Federal do Paraná, Setor de Tecnologia, Programa de Pós-Graduação em Engenharia Elétrica. Defesa: Curitiba, 08/06/2015Inclui referênciasÁrea de concentração: Sistemas eletrônicosResumo: Projetar e determinar com exatidão controladores para sistemas dinâmicos sempre foi um desafio para a engenharia e no intuito de ampliar a aplicação de plantas controladas em sistema reais, muitas técnicas foram desenvolvidas para generalizar o método de projetar controladores e tornar essa tarefa mais fácil e assertiva. Dessa maneira, desde os primeiros estudos a respeito da teoria e prática de projeto de controladores PID, muitas outras ferramentas surgiram, dentre elas a área de controle baseado em dados, que tem por objetivo conseguir um controlador cujo sistema se comporte próximo a uma referência. Para tanto, utiliza-se um único dado de experimento com entradas e saídas coletados da planta a fim de determinar a dinâmica do sistema em malha fechada. A técnica de controle baseada em dados possui duas principais vertentes. A primeira é um processo iterativo bem representado pela técnica do Iterative Feedback Tuning (IFT). A segunda, conhecida como VRFT, ou Virtual Reference Feedback Tuning, é uma técnica não iterativa que tem por objetivo relacionar uma referência virtual a um sistema realimentado cujo controlador deseja-se determinar. Tal técnica tem a principal vantagem e característica de transformar o problema de determinação do controlador em um problema de identificação de sistemas com dados de entrada e saída virtuais calculados utilizando dados de uma planta de referência. Para tanto, é comum encontrar na literatura trabalhos que utilizar uma estrutura fixa e pré-determinada do controlador, normalmente estruturas PID. Porém, a aproximação de tal controlador apresenta falhas de identificação e de desempenho do sistema realimentado, pois nem sempre a estrutura escolhida contém a estrutura ideal, aquela cuja identificação aproxima o erro a zero ou muito próximo disso. Dentre diversos métodos de identificação de sistemas, as séries de base de função ortonormal (OBF) possuem a grande vantagem de poder generalizar tal estrutura de controlador e depender unicamente da quantidade de funções escolhidas para representar o sistema e de um polo ou um par de polos conjugado. Por fim, este trabalho apresenta a aplicação do método de base de funções ortonormais na identificação do controlador cujos dados são obtidos através da técnica de referência virtual (VRFT). A teoria foi aplicada em sistemas dinâmicos lineares e não lineares incluindo um reator químico do tipo CSTR em presença (ou não) de ruído de medição. A técnica foi testada em ambos os sistemas e sobre diversos níveis de ruído, apresentou resultados notáveis na etapa de identificação de sistemas e consequentemente produziu uma solução para o problema de determinar com precisão e facilidade o controlador para um sistema em malha fechada. A escolha da classe de controladores é então generalizada, o que permite ao sistema e à técnica do VRFT, grande aplicabilidade na solução de problemas complexos de sistemas dinâmicos reais. Palavras-chave: Bases de Funções Ortonormais. Identificação em malha fechada. Referencia Virtual. Controle Baseado em dados.Abstract: To design and determine with accuracy controllers for dynamical systems has always been a challenge for engineering. In order to extend the application of controlled plants in real system many techniques have been developed, most of them with the objective of generalizing methods and permit controller design in an easier and assertive way. Therefore, since the first studies about the theory and practice on designing of PID controllers, a new control area based on data aims to get a controller whose system behaves as close as possible to a pre-defined reference. To this end, a single set of input and output data is collected from the plant in order to finally identify the dynamics of such closed-loop system. Data-based control techniques have two main strands. The first, an iterative technique known as Iterative Feedback Tuning (IFT) and the second one, a noniterative model called Virtual Reference Feedback Tuning (VRFT) which aims to relate a virtual reference to a feedback system whose controller would be determined. The VRFT technique has the main advantage and characteristic of turning the task of the controller determination into a problem of system identification with a set of input and output data plus a virtual reference. To this end, it is common to find in literature studies that assume a fixed and pre-determined controller structures on VRFT, mainly related with the PID control structure. Still, the solution may fail to present a good performance because not always the chosen structure contains the ideal one whose identification brings the error with regards to the desired performance close to zero. Beyond several model structures used by systems identification methods, the orthonormal basis functions (OBF) models have been receiving much attention in the literature since the past decade. In the VRFT context, it has the great advantage of being able to generalize the controller structure and improve accuracy and applicability of the method. This is the main contribution of this work, which applies and analyses OBF-models to design controllers using the VRFT technique. The VRFT approach is better explained and its methodology, advantages and limitations are compared between similar procedures. In addition, it presents a potential alternative to enhance the VRFT technique and its results by using a generalized class of modeling structures described using orthonormal basis functions The theory is applied on linear and nonlinear dynamical systems including a CSTR reactor in presence (or not) of noise measurements. After all, the presented modeling technique delivered notable results on both identification and closed loop evaluations. Consequently, the problem of determining a feasible VRFT controller for expected closed-loop system behavior is solved, making wider the applicability of solving complex problems of real dynamical systems by the VRFT technique. Key-words: Orthonormal Basis Functions. Closed-loop Identification. Virtual Reference Feedback Tuning. Data-Base Controller Tuning

A Bayesian approach to robust identification: application to fault detection

In the Control Engineering field, the so-called Robust Identification techniques deal with the problem of obtaining not only a nominal model of the plant, but also an estimate of the uncertainty associated to the nominal model. Such model of uncertainty is typically characterized as a region in the parameter space or as an uncertainty band around the frequency response of the nominal model. Uncertainty models have been widely used in the design of robust controllers and, recently, their use in model-based fault detection procedures is increasing. In this later case, consistency between new measurements and the uncertainty region is checked. When an inconsistency is found, the existence of a fault is decided. There exist two main approaches to the modeling of model uncertainty: the deterministic/worst case methods and the stochastic/probabilistic methods. At present, there are a number of different methods, e.g., model error modeling, set-membership identification and non-stationary stochastic embedding. In this dissertation we summarize the main procedures and illustrate their results by means of several examples of the literature. As contribution we propose a Bayesian methodology to solve the robust identification problem. The approach is highly unifying since many robust identification techniques can be interpreted as particular cases of the Bayesian framework. Also, the methodology can deal with non-linear structures such as the ones derived from the use of observers. The obtained Bayesian uncertainty models are used to detect faults in a quadruple-tank process and in a three-bladed wind turbine