Realization theory for linear hybrid systems, part I: Existence of realization

The paper is the first part of a series of papers which deal with realization theory for linear hybrid systems. Linear hybrid systems are hybrid systems in continuous-time without guards whose continuous dynamics is determined by linear control systems and whose the discrete dynamics is determined by a finite state automaton. In Part I of the current series of papers we will formulate necessary and sufficient conditions for the existence of a linear hybrid system realizing a specified set of input-output maps. We will also sketch a realization algorithm for computing a linear hybrid system from the input-output data. In Part II we will present conditions for observability and span-reachability of linear hybrid systems and we will show that minimality is equivalent to observability and span-reachability; we will also discuss algorithms for checking observability and span-reachability and for transforming a linear hybrid system to a minimal one

Realization theory for linear hybrid systems, part II: Reachability, observability and minimality

The paper is the second part of the series of papers started in [1]. The paper deals with observability, reachability and minimality of linear hybrid systems. Linear hybrid systems are continuous-time hybrid systems without guards, whose continuous dynamics is determined by time-invariant linear control systems. We will show that that if a set of input-output maps has a realization by a linear hybrid system, then it has a realization by a minimal linear hybrid system. We will present conditions for observability and span-reachability of linear hybrid systems and we will show that minimality is equivalent to observability and span-reachability. We will sketch algorithms for checking observability and span-reachability and for transforming a linear hybrid system to a minimal one

Identification of Piecewise Linear Models of Complex Dynamical Systems

The paper addresses the realization and identification problem or a subclass of piecewise-affine hybrid systems. The paper provides necessary and sufficient conditions for existence of a realization, a characterization of minimality, and an identification algorithm for this subclass of hybrid systems. The considered system class and the identification problem are motivated by applications in systems biology

Model Reduction of Linear Switched Systems by Restricting Discrete Dynamics

We present a procedure for reducing the number of continuous states of discrete-time linear switched systems, such that the reduced system has the same behavior as the original system for a subset of switching sequences. The proposed method is expected to be useful for abstraction based control synthesis methods for hybrid systems

Model Reduction by Moment Matching for Linear Switched Systems

Two moment-matching methods for model reduction of linear switched systems (LSSs) are presented. The methods are similar to the Krylov subspace methods used for moment matching for linear systems. The more general one of the two methods, is based on the so called "nice selection" of some vectors in the reachability or observability space of the LSS. The underlying theory is closely related to the (partial) realization theory of LSSs. In this paper, the connection of the methods to the realization theory of LSSs is provided, and algorithms are developed for the purpose of model reduction. Conditions for applicability of the methods for model reduction are stated and finally the results are illustrated on numerical examples.Comment: Sent for publication in IEEE TAC, on October 201