Generalized port-Hamiltonian DAE systems

Schaft, A.J. van der; Maschke, Bernhard

doi:10.1016/j.sysconle.2018.09.008

Cited by 62 publications

(88 citation statements)

References 17 publications

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“…Note that, if we want to retrieve a pHDAE system from a Dirac structure, the additional conditions (7) and the definition of H(x) are needed. These can also be lifted to a geometric interpretation, by the means of a Lagrangian submanifold and of a dissipative structure [12].…”

Section: Dirac Structurementioning

confidence: 99%

Structure-preserving discretization for port-Hamiltonian descriptor systems

Mehrmann¹,

Morandin²

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

111

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We extend the modeling framework of port-Hamiltonian descriptor systems to include under-and over-determined systems and arbitrary differentiable Hamiltonian functions. This structure is associated with a Dirac structure that encloses its energy balance properties. In particular, port-Hamiltonian systems are naturally passive and Lyapunov stable, because the Hamiltonian defines a Lyapunov function. The explicit representation of input and dissipation in the structure make these systems particularly suitable for output feedback control. It is shown that this structure is invariant under a wide class of nonlinear transformations, and that it can be naturally modularized, making it adequate for automated modeling. We investigate then the application of time-discretization schemes to these systems and we show that, under certain assumptions on the Hamiltonian, structure preservation is achieved for some methods. Relevant examples are provided.

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Section: Dirac Structurementioning

confidence: 99%

Structure-preserving discretization for port-Hamiltonian descriptor systems

Mehrmann¹,

Morandin²

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

111

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“…Dynamic systems, which result from variational principles, can usually be modeled by a port-Hamiltonian system. A system theoretical and geometric treatment of port-Hamiltonian ordinary differential systems goes back to van der Schaft, and there is by now a well-established theory (see van der Schaft 2 and Jeltsema & van der Schaft 3 for an overview), which has been applied to electrical circuits, Gernandt et al 4 Only recently, the concept has been generalized to port-Hamiltonian differentialalgebraic systems, that is, ordinary differential equations with algebraic constrains (see van der Schaft 5 and Maschke and van der Schaft 6,7 ). In Beattie et al, 8 linear time-varying port-Hamiltonian DAEs have been studied, and the notion has been generalized to quasilinear systems in Mehrmann and Morandin.…”

Section: Introductionmentioning

confidence: 99%

Dynamic iteration schemes and port‐Hamiltonian formulation in coupled differential‐algebraic equation circuit simulation

Günther

Bartel

Jacob

et al. 2020

Circuit Theory & Apps

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Electric circuits are usually described by charge/flux-oriented modified nodal analysis. Here, we derive models as port-Hamiltonian systems on several levels: overall systems, multiply coupled systems, and systems within dynamic iteration procedures. To this end, we introduce new classes of port-Hamiltonian differential-algebraic equations. Thereby, we additionally allow for nonlinear dissipation on a subspace of the state space. Both, each subsystem and the overall system possess a port-Hamiltonian structure. A structural analysis is performed for the new setups. Dynamic iteration schemes are investigated, and we show that the Jacobi approach as well as an adapted Gauss-Seidel approach lead to port-Hamiltonian differential-algebraic equations.

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“…Very recently, see especially [3], another type of algebraic constraint equations in linear physical system models was studied. In [3,18], these were identified as arising from generalized port-based modeling with a state space that has higher dimension than the minimal number of energy variables, corresponding to implicit energy storage relations which can be formulated as Lagrangian subspaces. For linear time-varying systems this formulation has led to various interesting results on their index, regularization and numerical properties [10]; see also [4] for related developments.…”

Section: Introductionmentioning

confidence: 99%

“…The precise relation between the algebraic constraints as arising in linear standard port-Hamiltonian systems (called Dirac algebraic constraints) and those in linear generalized port-Hamiltonian systems with implicit storage (called Lagrange algebraic constraints) was studied in [3,18]. In particular, in [18] it was shown how in this linear case Dirac algebraic constraints can be converted into Lagrange algebraic constraints, and conversely. In both cases this is done by extending the state space (by Lagrange multipliers).…”

Section: Introductionmentioning

confidence: 99%

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Dirac and Lagrange Algebraic Constraints in Nonlinear Port-Hamiltonian Systems

Schaft

Maschke

2020

Vietnam J. Math.

Self Cite

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After recalling the definitions of standard port-Hamiltonian systems and their algebraic constraints, called here Dirac algebraic constraints, an extended class of port-Hamiltonian systems is introduced. This is based on replacing the Hamiltonian function by a general Lagrangian submanifold of the cotangent bundle of the state space manifold, motivated by developments in (Barbero-Linan et al., J. Geom. Mech. 11, 487–510, 2019) and extending the linear theory as developed in (van der Schaft and Maschke, Syst. Control Lett. 121, 31–37, 2018) and (Beattie et al., Math. Control Signals Syst. 30, 17, 2018). The resulting new type of algebraic constraints equations are called Lagrange algebraic constraints. It is shown how Dirac algebraic constraints can be converted into Lagrange algebraic constraints by the introduction of extra state variables, and, conversely, how Lagrange algebraic constraints can be converted into Dirac algebraic constraints by the use of Morse families.

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Generalized port-Hamiltonian DAE systems

Cited by 62 publications

References 17 publications

Structure-preserving discretization for port-Hamiltonian descriptor systems

Structure-preserving discretization for port-Hamiltonian descriptor systems

Dynamic iteration schemes and port‐Hamiltonian formulation in coupled differential‐algebraic equation circuit simulation

Dirac and Lagrange Algebraic Constraints in Nonlinear Port-Hamiltonian Systems

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