Semidefinite cone-invariant systems : theory and applications

Abstract

Semidefinite cone-invariant (SCI) systems are defined as a class of linear time-invariant (LTI) systems which possess the spectrahedral cone-invariance property. Such systems have a rich structure and include a large class of LTI systems, e.g., positive systems, as special cases. However, they have not received enough attention from the control community. In this thesis, we make some preliminary attempts to study the theory and applications of SCI systems. The first part of this thesis is the study of the theory of SCI systems. We define weak notions of spectrahedral cone-stability, spectrahedral cone-observability and spectrahedral cone-detectability of SCI systems and some criteria of these properties are given in terms of the distinguished eigenvalues and linear matrix inequalities. More importantly, we investigate the SCI realization problem, which was our initial motivation to study SCI systems. For discrete-time LTI systems with nonnegative impulse responses, much research has been devoted to studying their positive realization. However, the eigenvalues of nonnegative matrices, which obey the Perron-Frobenius theorem and Karpelevich theorem, cannot cover all possible modes of a nonnegative impulse response. The limitations in the eigenvalue positions suggest that positive systems are not adequately powerful as a modeling tool. Hence we propose to employ the more powerful systems, namely, SCI systems, to do the SCI realization of nonnegative impulse responses. This is a novel idea and has not previously appeared in the literature. At present, we can find SCI realizations for a large class of systems with nonnegative impulse responses, whose positive realizations may not exist. Moreover, a minimal SCI realization with the dimension equal to the order of the transfer function is obtained for a subclass of these systems. The current results indeed demonstrate the potential power of SCI realization. We have faith that an SCI realization of any nonnegative impulse response, even with the dimension equal to the order of the transfer function, can be obtained. The second part of this thesis is the study of the linear quadratic (LQ) optimal control problem of discrete-time networked control systems (NCSs) with random input gains. It is shown that the solvability of this LQ optimal control problem depends on the existence of a mean-square stabilizing solution to a modified algebraic Riccati equation (MARE). With the help of the theory of SCI systems, we provide a necessary and sufficient condition, which is given directly in terms of the system parameters, to ensure the existence of such a mean-square stabilizing solution. Such a condition is derived for the very first time, and it indicates that the common assumption or condition of the observability or detectability of certain stochastic systems is unnecessary. The other highlight is that we put the problem under a channel/controller co-design framework, which differentiates our work from a certain pure stochastic optimal controller design problem. The controller designer has the freedom to participate in the channel design by allocating the given overall channel capacity, as desired, to the individual input channels. Under this framework, the stabilization issue involved can be analytically solved.

Date of Award	2016
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology