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We formulate a behavioral approach to higher-order linear port-Hamiltonian systems. We formalize constitutive laws such as power conservation, storage and (anti-)dissipative relations, and we study several properties of such systems. We also define the dual of a given port-Hamiltonian behavior.

The recently obtained approach to the construction of state maps, which is directly based on the linear differential operator describing the system, is shown to lead to an immediate and insightful relation between external and internal decompositions and symmetries of a linear system. This is applied to the decomposition of a system into its controllable and uncontrollable part (in the state space representation commonly referred to as the Kalman decomposition), and to the correspondence between external and internal symmetries.

We consider oscillatory systems consisting of identical subsystems symmetrically coupled,. We show that the time average of any quadratic functional of the variables of a subsystem and their derivatives equals the time average of the same functional on any other subsystem.

We study the H∞-partial realization problem from a behavioral point of view: we give necessary and sufficient conditions for solvability, and a characterization of all solutions. Instrumental in such analysis is the notion of time- and space- symmetrization of the data, which allows to transform the realization problem with metric- and stability constraints into an unconstrained behavioral modeling one.

A new algorithm is illustrated for J-spectral factorization of polynomial matrices. The algorithm is based on the calculus of two-variable polynomial matrices and associated quadratic differential forms, and is especially suited for spectral factorization problems arising in the polynomial and in the behavioral approaches to H∞-control.

The problem of balancing quadratic measures on the external signals of a linear, time-invariant differential system is considered from a behavioral point of view. It is shown how to compute a balanced state space representation for a system described by a set of linear, time invariant differential equations, directly from the equations. A reduced order model can be computed from the balanced state space representation by truncation.

Modelling of physical systems produces as a result a set of differentialalgebraic equations (generally of higher order). As such, state space models are not a natural starting point for modelling, while they have utmost importance in the simulation and control phase. The paper addresses the problem of computing state representations for systems of linear differential-algebraic equations from the behavioral point of view.

The possibility of a recursive in1plementation of the modeling procedures of [2]-[3] and of [7]-[8] is investigated in the light of the results of [4] on the kernel structure of Hankel matrices.