### Optimization and Systems Theory Seminar May 8, 1998, 11.00-12.00

Jorge Marí
Division of Optimization and Systems Theory
Department of Mathematics
KTH

#### Rational modeling of time series and applications of geometric control

This thesis consists of four independent articles in systems theory, plus an introductory chapter which summarizes the main results and explains the reasons that led us to consider the different topics. The first paper deals with identification of time series, i.e., the estimation of the parameters of a dynamical stochastic model, the output of which approximates the given time series. The last three papers apply geometric control theory to concrete engineering problems. All papers, except the third, employ semidefinite programming, a branch of optimization very common in systems theory.

In the first paper we start by pointing out why some recent stochastic subspace identification methods may fail for theoretical reasons related to positive realness. In an appendix to the paper we outline three different methods which might be used to correct the lack of positive realness. The bulk of the paper is devoted to an alternative identification paradigm which does not suffer from these problems. This paradigm consists in first performing a positive extension of an estimated partial covariance sequence to a valid high-order model, and then reducing the model by a carefully selected reduction method. We exhaustively study a particular implementation of the paradigm which only requires linear algebra operations and is non-iterative in nature. We prove that if gathered data comes from a time series with rational coercive spectral density, the transfer function of the estimated system tends in $\mathcal{H}_{\infty}$ to the minimum-phase model of the time series as the data length tends to infinity. The special structure of high order autoregressive models and their stochastically balanced models are also extensively studied.

In the second paper we show how under ideal conditions state-derivatives can be used to completely reject from a given output the effect of disturbances affecting a system. In particular we study in full details the application of this concept to a car and show how an accelerometer mounted at the rear end of the vehicle, together with some other sensors, can be used to create considerably safer riding conditions for the car. The stability analysis of the system under non-ideal conditions is also performed.

In the third paper we consider linear systems with system matrices depending on completely unknown parameters. We show how to select an output for the system, and a robust feedback, in such a way that a mode of the resulting dynamics becomes decoupled from the rest of the dynamics for all possible values of the unknown parameters. This is very useful for analytical studies of low order, physically motivated models.

In the fourth paper we show how to use the theory of asymptotic output tracking for linear systems to design optimized tracking controllers for certain pursuit-evasion actions. In particular we motivate how the results could be used as a basis for a better understanding of the neuromuscular activity when a human head follows an object with the sight.

Calendar of seminars
Last update: April 30, 1998 by Anders Forsgren, andersf@math.kth.se.