The behavioral approach to systems and modeling

An introduction to behavioral system theory, and a brief review of the content of the Special Issue are given

Hamiltonian and Variational Linear Distributed Systems

We use the formalism of bilinear- and quadratic differential forms in order to study Hamiltonian and variational linear distributed systems. It was shown in [1] that a system described by ordinary linear constant-coefficient differential equations is Hamiltonian if and only if it is variational. In this paper we extend this result to systems described by linear, constant-coefficient partial differential equations. It is shown that any variational system is Hamiltonian, and that any scalar Hamiltonian system is contained (in general, properly) in a particular variational system

Pick matrix conditions for sign-definite solutions of the algebraic Riccati equation

We study the existence of positive and negative semidefinite solutions of algebraic Riccati equations (ARE) corresponding to linear quadratic problems with an indefinite cost functional. The problem to formulate reasonable necessary and sufficient conditions for the existence of such solutions is a long-standing open problem. A central role is played by certain two-variable polynomial matrices associated with the ARE. Our main result characterizes all unmixed solutions of the ARE in terms of the Pick matrices associated with these two-variable polynomial matrices. As a corollary of this result we obtain that the signatures of the extremal solutions of the ARE are determined by the signatures of particular Pick matrices

Linear Hamiltonian behaviors and bilinear differential forms

We study linear Hamiltonian systems using bilinear and quadratic differential forms. Such a representation-free approach allows us to use the same concepts and techniques to deal with systems isolated from their environment and with systems subject to external influences and allows us to study systems described by higher-order differential equations, thus dispensing with the usual point of view in classical mechanics of considering first- and second-order differential equations only

A Behavioral Approach to Passivity and Bounded Realness Preserving Balanced Truncation with Error Bounds

In this paper we revisit the problems of passivity and bounded realness preserving model reduction by balanced truncation. In the behavioral framework, these problems can be considered as special cases of balanced truncation of strictly half line dissipative system behaviors, where the number of input variables of the behavior is equal to the positive signature of the supply rate. Instead of input-state-output representations, the balancing algorithm uses normalized driving variable representations of the behavior. We show that the diagonal elements of the minimal solution of the balanced algebraic Riccati equation are the singular values of the map that assigns to each past trajectory the optimal storage extracting future continuation. Since the future behavior is only an indefinite inner product space, the term singular values should be interpreted here in a generalized sense. We establish some new error bounds for this model reduction method

Model Reduction for Controllable Systems

In the papers [1], [7] a new scheme for passivity-preserving model reduction has been proposed. We have shown in [2] that the approach can also be interpreted from a dissipativity theory point of view, and we put forward two procedures in order to compute a driving variable or output nulling representation of a reduced order model for a given behavior. In this paper we illustrate improved versions of both algorithms, which produce a controllable reduced-order model. The new algorithms are based on several original results of independent interest