Identificação de sistemas utilizando métodos de subespaços

Abstract

Over the last two decades, subspace identification methods have attracted great attention due their potential for application in industry, especially for multivariable systems. The algorithms of subspace identification are as easy to implement as wellknown algorithms such as least squares. However, the theory behind these methods requires concepts fromlinear systems, stochastic processes, system identification, linear algebra, and others, making their understanding more difficult. As a consequence such methods are less well-known than others. This work investigates the use of subspace identification techniques applied to discrete-time, linear, time-invariant and multivariable systems. Our eorts have focused on the following objectives: (I) to geometrically interpret the methods, (II) to investigate, by means of simulations, situations in which the algorithms are best indicated, (III) to compare with other classical methods and (IV) to apply them to simulated and experimental systems. At first, basic conceptsare reviewed: data modeling in state space, block matrices and vectors, geometric and statistical tools. These concepts are critical to understanding the theory behind subspace identification. Later, in the case of deterministic subspace identification, a comprehensive study about how state matrices can be obtained from input-output datais provided. In this study, the algorithms N4SID and MOESP are presented. Subsequently, the stochastic case is treated similarly to the deterministic one. It is shown that the methods N4SID and MOESP are robust to white measurement noise. However, when the methods are exposed to colored noise, either measurement or process, these estimators are biased. One alternative is to use instrumental variable methods. Two such algorithms are presented: MOESP-PI and MOESP-PO. Demonstrations and simulated examples are presented, in order to facilitate the understanding of the characteristics of subspaces methods. Finally, the methods N4SID, MOESP, MOESP-PO, MOESP-PI are applied to three multivariable systems. The first system is a simulated model of a DC motor. The other two are experimental systems, namely water pumping system and a column flotation system. The results suggest that the subspace methods are a feasible alternative for linear systems of multiple inputs and multiple outputs.