Realization Theory in Hilbert Space

Salamon, Dietmar

doi:10.21236/ada158172

Cited by 53 publications

(92 citation statements)

References 15 publications

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“…In this paper we study a certain absolute stability problem for the class of well-posed infinite-dimensional systems which are documented in Salamon [24,25], Staffans [26,27] and Weiss [29][30][31][32]. We remark that the class of well-posed, linear, infinite-dimensional systems is rather general: it includes most distributed parameter Keywords and phrases: Absolute stability, actuator nonlinearities, circle criterion, integral control, positive real, robust tracking, well-posed infinite-dimensional systems.…”

Section: Introductionmentioning

confidence: 99%

Absolute stability results for well-posed infinite-dimensional systems with applications to low-gain integral control

Logemann¹,

Curtain²

2000

ESAIM: COCV

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Abstract. We derive absolute stability results for well-posed infinite-dimensional systems which, in a sense, extend the well-known circle criterion to the case that the underlying linear system is the series interconnection of an exponentially stable well-posed infinite-dimensional system and an integrator and the nonlinearity φ satisfies a sector condition of the form φ(u), φ(u) − au ≤ 0 for some constant a > 0. These results are used to prove convergence and stability properties of low-gain integral feedback control applied to exponentially stable, linear, well-posed systems subject to actuator nonlinearities. The class of actuator nonlinearities under consideration contains standard nonlinearities which are important in control engineering such as saturation and deadzone.AMS Subject Classification. 93C10, 93C20, 93C25, 93D05, 93D09, 93D10, 93D21.

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Section: Introductionmentioning

confidence: 99%

Absolute stability results for well-posed infinite-dimensional systems with applications to low-gain integral control

Logemann¹,

Curtain²

2000

ESAIM: COCV

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“…By a result due to Salamon [35] (and apparently discovered independently by Weiss [43,44]), every well-posed linear system Ψ has a well-defined (unbounded) control operator B and a well-defined (unbounded) observation operator C, and formulas (3), (4), (8), (9), (11), (12), (13), and (15) hold in a weak sense (see Remark 30 below). In order to present this result we need some additional definitions.…”

Section: The Generators Of a Well-posed Linear Systemmentioning

confidence: 97%

“…This theory has been developed in [33], [34], [35], [8], [11], and [43], [44], [45], [46] (and many other papers), and we refer the reader to these sources for additional reading. (Salamon calls these systems "well-posed semigroup control systems" and Weiss calls them "abstract linear systems".)…”

Section: Well-posed Linear Systems and Time-invariant Operatorsmentioning

confidence: 99%

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Quadratic optimal control of stable well-posed linear systems

Staffans

1997

Trans. Amer. Math. Soc.

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Abstract. We consider the infinite horizon quadratic cost minimization problem for a stable time-invariant well-posed linear system in the sense of Salamon and Weiss, and show that it can be reduced to a spectral factorization problem in the control space. More precisely, we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution. If both the system and the spectral factor are regular, then the feedback operator can be expressed in terms of the Riccati operator, and the Riccati operator is a positive self-adjoint solution of an algebraic Riccati equation. This Riccati equation is similar to the usual algebraic Riccati equation, but one of its coefficients varies depending on the subspace in which the equation is posed. Similar results are true for unstable systems, as we have proved elsewhere.

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“…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. Recent accounts of these connections between well-posed linear systems and Lax-Phillips scattering theory are given in [23,24,25].…”

Section: Introductionmentioning

confidence: 99%

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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Realization Theory in Hilbert Space

Cited by 53 publications

References 15 publications

Absolute stability results for well-posed infinite-dimensional systems with applications to low-gain integral control

Absolute stability results for well-posed infinite-dimensional systems with applications to low-gain integral control

Quadratic optimal control of stable well-posed linear systems

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

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