Lossless and Dissipative Distributed Systems

280

297

Abstract. Associated with a skew-symmetric linear operator on the spatial domain [a, b] we define a Dirac structure which includes the port variables on the boundary of this spatial domain. This Dirac structure is a subspace of a Hilbert space. Naturally, associated to this Dirac structure is infinite dimensional system. We parameterize the boundary port variables for which the C0-semigroup associated to this system is contractive or unitary. Furthermore, this parameterization is used to split the boundary port variables into inputs and outputs. Similarly, we define a linear port controlled Hamiltonian system associated with the previously defined Dirac structure and a symmetric positive operator defining the energy of the system. We illustrate this theory on the example of the Timoshenko Beam.

Section: Resultsmentioning

confidence: 99%

Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators

Gorrec¹,

Zwart²,

Maschke³

2005

280

297

“…In view of the encouraging results of the application of quadratic differential forms in the context of infinite-dimensional systems (see [13,14]), it can be hoped that some of the results presented in this paper can be generalized also to systems described by linear constant-coefficient partial differential equations; such an area of research is presently under investigation. Another direction in which the research presented in this paper is being extended is that of the connections between Hamiltonian systems and optimal control.…”

Section: Such a Quantity Is Zero If And Only Ifmentioning

confidence: 93%

Linear Hamiltonian Behaviors and Bilinear Differential Forms

Rapisarda¹,

Trentelman²

2004

Abstract. 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.Key words. linear Hamiltonian systems, two-variable polynomial matrices, bilinear and quadratic differential forms, behavioral system theory AMS subject classifications. 93A10, 93A30, 93C05, 37J99, 70H50 DOI. 10.1137/S03630129024146641. Introduction. This paper aims to give a unified treatment of linear Hamiltonian systems using the formalism of bilinear and quadratic differential forms introduced in [24]. We consider systems with and without external influences, and we deal with both cases using the same techniques and the same concepts. Moreover, we formulate concepts and study the properties of Hamiltonian systems in a representation-free way, thus dispensing with the usual point of view in mechanics and in physics (see, for example, [1]) of concentrating on first-order representations in the (generalized) coordinates and the (generalized) momenta. Instead of postulating the existence of a function (the Lagrangian, or the Hamiltonian) on the basis of physical considerations (conservation of energy, etc.) and deducing from such a function the equations of motion, we proceed by assuming that a set of linear differential equations with constant coefficients describing the system is given, and we deduce the Hamiltonian nature of the system from such equations, by proving the existence of certain bilinear functionals of the variables of the system and of their derivatives satisfying some additional property. Our approach is of a system-theoretic nature rather than derived from the study of mechanics: as happens in optimal control theory for linear systems, we consider the interplay of (quadratic and bilinear) functionals of the system variables and of the equations of motion as the central object of study when dealing with Hamiltonian systems.In this paper we also reconcile our point of view with that of classical mechanics by showing how to construct a "generalized Lagrangian" on the basis of the equations of the system, in the sense that the trajectories of the system are stationary with respect to such a quadratic functional of the variables of the system and their derivatives. In this context, the concept of internal force also arises naturally from the equations describing the system: in this paper we show that generalized internal forces can be defined which depend on higher-order derivatives of the external variables and not only first-order ones at most, as happens in classical mechanics.

“…Background material. We give only the minimum amount of information needed; see [10,17,18] for more information, and [11,13,14,21,23] for important details and for applications of 2D bilinear-and quadratic differential forms.…”

mentioning

confidence: 99%

“…In terms of the 4-variable polynomial matrices associated with the BDF's, the relationship between a VBDFs and its divergence is expressed as (see Th. 4 of [18]):…”

mentioning

confidence: 99%

State-Space Modeling of Two-Dimensional Vector-Exponential Trajectories

Rapisarda¹,

Antoulas²

2016

Abstract. We solve two problems in modelling polynomial vector-exponential trajectories dependent on two independent variables. In the first one we assume that the data-generating system has no inputs, and we compute a state representation of the Most Powerful Unfalsified Model for this data. In the second instance we assume that the data-generating system is controllable and quarterplane causal, and we compute a Roesser i/s/o model. We provide procedures for solving these identification problems, both based on the factorization of constant matrices directly constructed from the data, from which state trajectories can be computed.Key words. multidimensional systems, most powerful unfalsified model, Roesser models, bilinear differential forms AMS subject classifications. 93A30, 93B15, 93B20, 93B30, 93C201. Introduction. We consider two problems in modelling two-dimensional (2D in the following) continuous trajectories from data. In both cases the data consistis of polynomial vector-exponential trajectories, and we seek state-space models explaining it, i.e. systems of partial differential equations of first order in an auxiliary, "state" variable, and zeroth-order in the measured, "external" variable. The two situations differ in the model class we assume the data-generating system belongs to: in the first case we seek an autonomous state model , i.e. a system without inputs; in the second one we assume that an input/output partition of the external variable is given, and we compute an input-state-output (i/s/o in the following) model.Modelling 2D polynomial vector-exponential trajectories with autonomous systems has been considered in [32,33], on whose results the first part of this paper on the computation of autonomous models heavily relies. Modelling vector-exponential trajectories with transfer-function (i.e. input-output) models is closely related to twovariable rational interpolation; the latter has been investigated in the SISO case in [2]. The approach taken in the present paper differs fundamentally from those: we use data to first compute state trajectories corresponding to it, and in a second stage we compute a state representation for the data and the identified state trajectories by solving linear equations in the unknown state-, input-and output matrices.Modelling methodologies where state-trajectories are computed from data and state-equations are subsequently computed are well-known in the 1D case as subspace identification methods (see e.g. [12]). Such ideas have been pursued much less frequently in the 2D case: see [7,19] for a pioneering subspace-identification approach to the computation of i/s/o representations of denominator-separable 2D discretesystems from data. Our modelling approach differs essentially from those, in that it does not exploit the shift-invariance properties of data trajectories, but rather the fact that external properties-i.e. properties at the level of external variables, in our case duality-are reflected into internal properties-i.e. at the level of state. To make the...