Scattering Systems with Several Evolutions and Multidimensional Input/State/Output Systems

Ball, Joseph A.; Sadosky, Cora; Vinnikov, Victor

doi:10.1007/s00020-005-1351-y

Cited by 78 publications

(113 citation statements)

References 19 publications

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“…Moreover, given any Schur-class function S ∈ S(U, Y), there is an associated functional-model scattering system S S based on the scattering function S so that the scattering function of S S coincides with the preassigned Schur-class function S. This last statement can be verified by writing down a model Lax-Phillips scattering system associated with a given Schur-class function S; we refer to [5] for further details.…”

Section: Introductionmentioning

confidence: 76%

“…The added flexibility of this coordinate-free version of S(U) is often more convenient for analysis in more general situations (see [5] and [8]). For this purpose, suppose that we are given a unitary colligation…”

Section: 4mentioning

confidence: 99%

“…For the discrete-time case such connections were first noticed in [1,11,12]. Surveys of the discrete-time with a slant toward possible generalizations to various multivariable/multidimensional settings appear in [3,4]; details for some of these more general settings are now appearing (see [5,8]). The continuous-time version of the Lax-Phillips scattering theory influenced the search for a distributed-parameter system theory, especially in the presence of energy conservation or dissipation (see [13,19,20,2]); these ideas have now matured into the theory of well-posed linear systems (conservative or not) due essentially to Staffans and Weiss-see [26] for a comprehensive treatment.…”

Section: Introductionmentioning

confidence: 96%

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Lax-Phillips scattering theory and well-posed linear systems: a coordinate-free approach

Ball

Carroll

Uetake

2008

Math. Control Signals Syst.

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Abstract. We give a further elaboration of the fundamental connections between Lax-Phillips scattering, conservative input/state/output linear systems and Sz.-Nagy-Foias model theory for both the discrete-and continuous-time settings. In particular, for the continuous-time setting, we show how to locate a scattering-conservative L 2 -well-posed linear system (in the sense of Staffans and Weiss) embedded in a Lax-Phillips scattering system presented in axiomatic form; conversely, given a scattering-conservative linear system, we show how one can view the space of finite-energy input-state-output trajectories of the system as the ambient space for an associated Lax-Phillips scattering system. We use these connections to give a simple, conceptual proof of the identity of the scattering function of the scattering system with the transfer function of the input-state-output linear system. As an application we show how system-theoretic ideas can be used to arrive at the spectral analysis of the scattering function.

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Section: Introductionmentioning

confidence: 76%

Section: 4mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

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Lax-Phillips scattering theory and well-posed linear systems: a coordinate-free approach

Ball

Carroll

Uetake

2008

Math. Control Signals Syst.

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“…satisfying the left-and right interpolation conditions G(λ Remark 7 (Operator-theoretic approaches to bivariate interpolation). Agler, McCarthy and others worked on discrete nD metric interpolation problems (see [1,4,6]) using operator-theoretic techniques, see also [5]. Interesting similarities exist between their formulas and ours, compare e.g.…”

Section: Remark 5 (Data Dualization Via Mirroring)mentioning

confidence: 99%

“…We generate from such primal system vector-exponential data at the frequencies (4, 3), (5,4) where g ij ∈ R is a free parameter, i, j = 1, . .…”

Section: Example and Commentsmentioning

confidence: 99%

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

Rapisarda¹,

Antoulas²

2016

SIAM J. Control Optim.

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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...

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Robust Control, Multidimensional Systems and Multivariable Nevanlinna-Pick Interpolation

Ball¹,

Horst²

2010

Topics in Operator Theory

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Abstract. The connection between the standard H ∞ -problem in control theory and Nevanlinna-Pick interpolation in operator theory was established in the 1980s, and has led to a fruitful cross-pollination between the two fields since. In the meantime, research in H ∞ -control theory has moved on to the study of robust control for systems with structured uncertainties and to various types of multidimensional systems, while Nevanlinna-Pick interpolation theory has moved on independently to a variety of multivariable settings. Here we review these developments and indicate the precise connections which survive in the more general multidimensional/multivariable incarnations of the two theories.

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Scattering Systems with Several Evolutions and Multidimensional Input/State/Output Systems

Cited by 78 publications

References 19 publications

Lax-Phillips scattering theory and well-posed linear systems: a coordinate-free approach

Lax-Phillips scattering theory and well-posed linear systems: a coordinate-free approach

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

Robust Control, Multidimensional Systems and Multivariable Nevanlinna-Pick Interpolation

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