An intrinsic Hamiltonian formulation of the dynamics of LC-circuits

Maschke, Bernhard; Schaft, A.J. van der; Breedveld, Pieter C.

doi:10.1109/81.372847

Cited by 107 publications

(89 citation statements)

References 15 publications

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“…The ideas underlying interconnected systems, which appear in energy-conserving systems such as L-C circuits and nonholonomic systems, were put into the context of Poisson structures by van der Schaft and Maschke [1994]; Maschke, van der Schaft and Breedveld [1995], and later in the general context of Dirac structures by van der Schaft and Maschke [1995]; Bloch and Crouch [1997]. The idea of interconnections was investigated in the context of implicit Hamiltonian systems by Dalsmo and van der Schaft [1998]; van der Schaft [1998] and Blankenstein [2000].…”

Section: Introductionmentioning

confidence: 99%

Dirac structures in Lagrangian mechanics Part II: Variational structures

Yoshimura

Marsden

2006

Journal of Geometry and Physics

115

117

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This paper develops the notion of implicit Lagrangian systems and presents some of their basic properties in the context of Dirac structures. This setting includes degenerate Lagrangian systems and systems with both holonomic and nonholonomic constraints, as well as networks of Lagrangian mechanical systems. The definition of implicit Lagrangian systems with a configuration space Q makes use of Dirac structures on T * Q that are induced from a constraint distribution on Q as well as natural symplectomorphisms between the spaces T * T Q, T T * Q, and T * T * Q. Two illustrative examples are presented; the first is a nonholonomic system, namely a vertical disk rolling on a plane and the second is an L-C circuit, a degenerate Lagrangian system with holonomic constraints.

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Section: Introductionmentioning

confidence: 99%

Dirac structures in Lagrangian mechanics Part II: Variational structures

Yoshimura

Marsden

2006

Journal of Geometry and Physics

115

117

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“…A direct constructive method for obtaining Hamiltonian models for LC circuits has been proposed in [1]. Following the suggestion of [18], in this section we extend this method by adding ports to account for voltage and current sources and resistive elements (the inclusion of resistive ports was independently proposed by B. Maschke and published in [17] without a proof). This is shown in Fig.…”

Section: Port-hamiltonian Formulation Of Nonlinear Rlc Networkmentioning

confidence: 99%

“…A critical assumption in this procedure is that the characteristics of the network components are bijective. To avoid this limitation we prefer, in the spirit of network modeling, to proceed from the port-Hamiltonian lossless models of [18], see also [1], and add the required sources and dissipation terminations.…”

mentioning

confidence: 99%

Proportional Plus Integral Control for Set-Point Regulation of a Class of Nonlinear RLC Circuits

Castaños

Jayawardhana

García–Canseco

2009

Circuits Syst Signal Process

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In this paper we identify graph-theoretic conditions which allow us to write a nonlinear RLC circuit as port-Hamiltonian with constant input matrices. We show that under additional monotonicity conditions on the network's components, the circuit enjoys the property of relative passivity, an extended notion of classical passivity. The property of relative passivity is then used to build simple, yet robust and globally stable, proportional plus integral controllers.

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“…Accordingly the dynamics are unstable (i.e., not asymptotically stable) in the directions orthogonal to the level sets of the θ k (z 1 ). The system is still completely controllable, though, and it is easy to see that we can transform the equations of motion to assume the following form [16]:…”

Section: All Eigenvalues Of P Lie In the Open Left Half-planementioning

confidence: 99%

“…Preserving the Hamiltonian structure plus stability and passivity, however, has value in its own right, as Hamiltonian models are prevalently used in, e.g., multibody dynamics [11] or circuit design [16]. Especially for the circuit systems, passivitypreserving reduction strategies based on (split) congruence transformations have been proposed in [17,18] and, more recently, in [8].…”

mentioning

confidence: 99%

Balanced Truncation of Linear Second-Order Systems: A Hamiltonian Approach

Hartmann¹,

Vulcanov²,

Schütte³

2010

Multiscale Model. Simul.

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We present a formal procedure for structure-preserving model reduction of linear second-order and Hamiltonian control problems that appear in a variety of physical contexts, e.g., vibromechanical systems or electrical circuit design. Typical balanced truncation methods that project onto the subspace of the largest Hankel singular values fail to preserve the problem's physical structure and may suffer from lack of stability. In this paper, we adopt the framework of generalized Hamiltonian systems that covers the class of relevant problems and that allows for a generalization of balanced truncation to second-order problems. It turns out that the Hamiltonian structure, stability, and passivity are preserved if the truncation is done by imposing a holonomic constraint on the system rather than standard Galerkin projection. Abstract. We present a formal procedure for structure-preserving model reduction of linear second-order and Hamiltonian control problems that appear in a variety of physical contexts, e.g., vibromechanical systems or electrical circuit design. Typical balanced truncation methods that project onto the subspace of the largest Hankel singular values fail to preserve the problem's physical structure and may suffer from lack of stability. In this paper, we adopt the framework of generalized Hamiltonian systems that covers the class of relevant problems and that allows for a generalization of balanced truncation to second-order problems. It turns out that the Hamiltonian structure, stability, and passivity are preserved if the truncation is done by imposing a holonomic constraint on the system rather than standard Galerkin projection.Key words. structure-preserving model reduction, generalized Hamiltonian systems, balanced truncation, strong confinement, invariant manifolds, Hankel norm approximation AMS subject classifications. 70Q05, 70J50, 93B11, 93C05, 93C70. DOI. 10.1137/0807327171. Introduction. Model reduction is a major issue for control, optimization, and simulation of large-scale systems. In particular, for linear time-invariant systems, balanced truncation is a well-established tool for deriving reduced (i.e., low-dimensional) models that have an input-output behavior similar to the original model [1, 2]; see also [3] and the references therein. The general idea of balanced truncation is to restrict the system onto the subspace of easily controllable and observable states which can be determined by the computing Hankel singular values associated with the system. Moreover the method is known to preserve certain properties of the original system such as stability or passivity and gives an error bound that is easily computable. One drawback of balanced truncation is that there is no straightforward generalization to second-order systems; see, e.g., [4] or the recent articles [5,6,7,8] for a discussion of various possible strategies. Second-order equations occur in modeling and control of many physical systems, e.g., electrical circuits, structural mechanics, or vibroacoustic models (see...

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An intrinsic Hamiltonian formulation of the dynamics of LC-circuits

Cited by 107 publications

References 15 publications

Dirac structures in Lagrangian mechanics Part II: Variational structures

Dirac structures in Lagrangian mechanics Part II: Variational structures

Proportional Plus Integral Control for Set-Point Regulation of a Class of Nonlinear RLC Circuits

Balanced Truncation of Linear Second-Order Systems: A Hamiltonian Approach

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