Approximation Of The Spectral Abscissa For Switched Linear Systems

Switched systems are a class of hybrid systems. Due to its success in applications and importance in theory, there is enormous growth of interest in the study of switched system. In general, a switched system is a dynamical system which is composed of finite subsystems described by differential or difference equations and a switching rule that coordinates the switchings among the subsystems. Switching signals which play an important role in the performance of the switched system make system dynamics be more complicated. It not only remains subsystems dynamics, but also may induces those dynamics that each subsystem does not possess. Hence, switched systems are accurate enough to represent complex nonlinear dynamics.For continuous-time switched linear systems, the spectral abscissa is a powerful tool for characterizing the performance of the system. However, the calculation of the spectral abscissa is a difficult task. To this end, we need the equivalent algebraic characterization of the spectral abscissa. Barabanov and Sun proved that the spectral abscissa is equal to the least measure of the (subsystem) matrix set, respectively. This equality leads to a way for verifying the spectral abscissa by calculating the least common matrix set measure of the subsystems. On the other hand, Blanchini pointed out that, for switched linear systems, the least measure could be approximated at an arbitrary accuracy by the ?1 measure of the transformed matrix set. In other words, the spectral abscissa can be approximated by the the least ?1 measure of the matrices obtained by generalized coordinate transformations. It should be noticed that, however, this idea is rather theoretical because the required dimension of the transformation matrix is unknown a prior. Since the numerical search for a proper transformation matrix is not tractable, it is important and challenging to solve this problem. Along this line, this dissertation aims to estimate the spectral abscissa of continuous-time switched linear system. The main contents are summarized as follows:1. The transformation matrix is square and thus invertible. As an invertible square matrix can be expressed as a product of elementary matrices, here we examine the case that the transformation matrix is a product of elementary matrices. That is, we examine the property of the least ?1 measure obtained by transformations of types ?, ? and ?, respectively, and then find the recursive transformations to de-crease the ?1 measure. The property to the transformation of type ?, together with definition of matrix set measure, deduces that the matrix set measure obtained after each transformation of type ? is invariant. By applying a series of coordinate trans-formations of type ? in an iterative manner, we obtained a sequence of minimums of matrix set measure which is convergent to the least ?1 measure that can be seen as an upper-bound estimate of the spectral abscissa. Also, the least ?1 measure obtained by a series of transformations of type ? can be used to approximate the spectral abscissa.2. The least ?1 measure is reached via coordinate transformations of type ? if and only if the column sums of the transformed matrix set,{P1*,P2*}, satisfy thatTherefore, to reduce computational effort, transformations are performed to the minimum column sum and then the least ?1 measure is obtained to estimate the spectral abscissa. Since the continuousness of the objective functions, i.e., the ?1 measure, obtained by transformations of type ?, a new algorithm is designed to search for the minimum of the matrix set measure after each transformation. The least ?1 measure obtained by recursive rounds of transformations of type ? is used to estimate the spectral abscissa. However, the problem on the connection between transformations of types ? and ? is still unsolved in theory. Numerical examples are presented to illustrate that, for matrices with some property, the least ?1 measure obtained by transformations of types ? (or ?) can approximate the spectral abscissa better than that obtained by the other type of transformations. A preliminary discussion on the general square transformation is presented.3. In general, these square transformations are not enough to guarantee the approxi-mation with a desired accuracy. Therefore, we aims to derive a proper non-square matrix of full row rank. The key idea here is to get the minimum of matrix set mea-sure by determining free unknowns of the transformation matrix and independent parameters obtained by increasing the dimension of the subsystem matrices. By performing the transformations iteratively, we obtain a series of the minimum ?1 measures which is decreasing and convergent,and the limit can be used to estimate the spectral abscissa.