Dirac Structure Research Articles

Interconnections of port-Hamiltonian systems: generating new passive outputs and feedback stabilization

We consider port-Hamiltonian system models whose underlying geometric structure is the Dirac structure which consists of energy storing ports, resistive ports and external ports which are interconnected to each other by a power conserving relationship. A port-Hamiltonian system with dissipation (PHSD) is formed by the composition of a Dirac and a resistive structure. We show how the choice of a new passive output for a PHSD is reflected in a new Dirac structure. We generate a new class of passive outputs for a PHSD by changing the Hamiltonian function, interconnection and damping matrices and subsequently compute the new underlying Dirac structure. We next consider power conserving interconnections of two port-Hamiltonian systems and extend the (well known) standard feedback interconnection by considering interconnections between port-Hamiltonian systems which involve their energy storing ports, resistive ports and external ports. For port-Hamiltonian systems with dissipation, we demonstrate how these interconnections can be used for their closed-loop stabilization by the control-by-interconnection (CbI) method. Finally, we discuss the similarities between our (new) control by interconnection method and the (well-known) interconnection and damping assignment passivity-based control (IDA-PBC) method.

Hamiltonian approach to the stabilization of systems of two conservation laws

This paper aims at providing some synthesis between two alternative representations of systems of two conservation laws and interprets different conditions on stabilizing boundary control laws. The first is the representation in Riemann invariants coordinates whose representation has been applied successfully for the stabilization of linear and non-linear of such hyperbolic systems. The second representation is based on physical modelling and leads to port Hamiltonian systems which are extensions of infinite-dimensional Hamiltonian systems defined on Dirac structure. The stability conditions on the boundary feedback relations derived with respect to the Riemann invariants are interpreted in terms of the dissipation inequality of the Hamiltonian functional.

Passivity-based control of spatially discretized port-Hamiltonian system

The main contribution of this paper is a procedure for the passivity-based control of high-order port-Hamiltonian systems obtained from the spatial discretization of infinite dimensional dynamics. Beside the intrinsic difficulties related to the large number of state variables, the finite element model is generally given in terms of a Dirac structure and is completely a-causal, which implies that the plant dynamics is not given in standard input-state-output form, but as a set of DAEs. Consequently, the passivity-based methods have to be extended in order to deal with dynamical systems with constraints, usually appearing in the form of Lagrangian multipliers.

Dirac structures and their composition on Hilbert spaces

Dirac structures appear naturally in the study of certain classes of physical models described by partial differential equations and they can be regarded as the underlying power conserving structures. We study these structures and their properties from an operator-theoretic point of view. In particular, we find necessary and sufficient conditions for the composition of two Dirac structures to be a Dirac structure and we show that they can be seen as Lagrangian (hyper-maximal neutral) subspaces of Kreĭn spaces. Moreover, special emphasis is laid on Dirac structures associated with operator colligations. It turns out that this class of Dirac structures is linked to boundary triplets and that this class is closed under composition.

Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids

We study holomorphic Poisson manifolds and holomorphic Lie algebroids from the viewpoint of real Poisson geometry. We give a characterization of holomorphic Poisson structures in terms of the Poisson Nijenhuis structures of Magri-Morosi and describe a double complex which computes the holomorphic Poisson cohomology. A holomorphic Lie algebroid structure on a vector bundle $A\to X$ is shown to be equivalent to a matched pair of complex Lie algebroids $(T^{0,1}X,A^{1,0})$, in the sense of Lu. The holomorphic Lie algebroid cohomology of $A$ is isomorphic to the cohomology of the elliptic Lie algebroid $T^{0,1}X\bowtie A^{1,0}$. In the case when $(X,\pi)$ is a holomorphic Poisson manifold and $A=(T^*X)_\pi$, such an elliptic Lie algebroid coincides with the Dirac structure corresponding to the associated generalized complex structure of the holomorphic Poisson manifold.

Dirac Structures in Pseudo-Gradient Systems With an Emphasis on Electrical Networks

The relationship between implicitly defined Hamiltonian systems and pseudo-gradient systems is investigated. It is shown that under certain conditions a Hamiltonian system, implicitly defined with respect to a Dirac structure on the state space manifold, can be used to generate a pseudo-gradient system on the Lagrangian submanifold of the cotangent bundle of the state space. This elucidates the relationship between the Smale pseudo-gradient and the Bloch-Couch Hamiltonian formulations of lossless circuit dynamics. A more general situation is also considered where the implicit Hamiltonian system gives rise to a foliation of the Lagrangian submanifold in which each leaf is a pseudo-Riemannian manifold which in turn gives rise to a pseudo-gradient system.

Memristive port-Hamiltonian Systems

The port-Hamiltonian modelling framework is extended to a class of systems containing memristive elements and phenomena. First, the concept of memristance is generalised to the same generic level as the port-Hamiltonian framework. Second, the underlying Dirac structure is augmented with a memristive port. The inclusion of memristive elements in the port-Hamiltonian framework turns out to be almost as straightforward as the inclusion of resistive elements. Although a memristor is a resistive element, it is also a dynamic element since the associated Ohmian laws are rather expressed in terms of differential equations. This means that the state space manifold, as naturally defined by the storage elements, is augmented by the states associated with the memristive elements. Hence the order of complexity is, in general, defined by the number of storage elements plus the number of memristors in the system. Apart from enlarging our repertoire of modelling building blocks, the inclusion of memristive elements in the existing port-Hamiltonian formalism possibly opens up new ideas for controller synthesis and design.

Deformation of Dirac structures along isotropic subbundles

Given a Dirac subbundle and an isotropic subbundle of a Courant algebroid, we provide a canonical method to obtain a new Dirac subbundle. When the original Dirac subbundle is involutive (i.e. a Dirac structure) this construction has interesting applications, for instance to Dirac's theory of constraints and to the Marsden—Ratiu reduction in Poisson geometry.

$L_∞$-algebras and higher analogues of Dirac structures and Courant algebroids

We define a higher analogue of Dirac structures on a manifold M. Under a regularity assumption, higher Dirac structures can be described by a foliation and a (not necessarily closed, non-unique) differential form on M, and are equivalent to (and simpler to handle than) the Multi-Dirac structures recently introduced in the context of field theory by Vankerschaver, Yoshimura and Marsden. We associate an L-infinity algebra of 'observables' to every higher Dirac structure, extending work of Baez, Hoffnung and Rogers on multisymplectic forms. Further, applying a recent result of Getzler, we associate an L-infinity algebra to any manifold endowed with a closed differential form H, via a 'higher analogue of Courant algebroid twisted by H'. Finally, we study the relations between the L-infinity algebras appearing above.

Omni-Lie algebroids

A generalized Courant algebroid structure is defined on the direct sum bundle D E ⊕ J E , where D E and J E are, respectively, the gauge Lie algebroid and the jet bundle of a vector bundle E . Such a structure is called an omni-Lie algebroid since it is reduced to the omni-Lie algebra introduced by A. Weinstein if the base manifold is a point. We prove that there is a one-to-one correspondence between Dirac structures coming from bundle maps J E → D E and Lie algebroid (local Lie algebra) structures on E when rank ( E ) ≥ 2 ( E is a line bundle).

Energy Shaping of Port-Hamiltonian Systems by Using Alternate Passive Input-Output Pairs

We consider port-Hamiltonian systems with dissipation (PHSD) whose underlying geometric structure is represented as the composition of a Dirac and a resistive structure. We show how the choice of a new passive input-output pair for a PHSD is reflected in a new Dirac structure. We define a general class of new passive inputoutput pairs for a PHSD and subsequently compute (in a constructive manner) the resulting new Dirac structure and examine the achievable Casimirs for this new Dirac structure. We focus on the special case where only the passive output is changed (while retaining the original input) and subsequently define a general class of new passive outputs for the PHSD. We then identify (on the basis of the achievable Casimirs) the precise form of the so-called dissipation obstacle, and how this obstacle may be removed by changing the passive output.We also review the “swapping the damping” procedure for computing a new passive output, and show how this can be obtained as a special case within our approach. We finally consider the examples of the RLC-circuit and MEMS optical switch to investigate the role played by the new class of passive outputs in shaping the system's energy.

Discussion on: “Energy Shaping of Port-Hamiltonian Systems by Using Alternate Passive Input-Output Pairs”

The paper further elaborates on a passive output that was constructed in [3] with the purpose of providing an energybalance (EB) interpretation for basic interconnection and damping assignment (BIDA [6, 7]). Such output has its roots in power shaping [4], an alternative method to stabilize nonlinear RLC circuits subjected to the dissipation obstacle [6]. It has been recently shown in [5] that this particular output is also useful in the context of control by interconnection (CbI [8, 6]). The output constructed by swapping the damping (also called the power shaping output) is without doubt worth investigating, as it plays an important role in a somewhat convoluted interplay between: energy-balance, interconnection and damping assignment, control by interconnection and the dissipation obstacle (see [5] for details). Venkatraman and van der Schaft study the power shaping output and its connection to the set of achievable Casimir functions from the more general perspective of Dirac structures [2]. Among other things, the authors show that the process of generating new passive outputs can be understood as a ‘decomposition’ and further ‘re-composition’ of the plant’s Dirac and resistive underlying structures.

Port-Hamiltonian Dynamics on Graphs: Consensus and Coordination Control Algorithms

Directed graphs are shown to be endowed with a canonical Dirac structure, which is used for formulating standard consensus and coordination control algorithms as port-Hamiltonian systems.

Poisson vertex algebras in the theory of Hamiltonian equations

We lay down the foundations of the theory of Poisson vertex algebras aimed at its applications to integrability of Hamiltonian partial differential equations. Such an equation is called integrable if it can be included in an infinite hierarchy of compatible Hamiltonian equations, which admit an infinite sequence of linearly independent integrals of motion in involution. The construction of a hierarchy and its integrals of motion is achieved by making use of the so called Lenard scheme. We find simple conditions which guarantee that the scheme produces an infinite sequence of closed 1-forms $$\omega_j, j \in {\mathbb {Z}}_+$$ , of the variational complex Ω. If these forms are exact, i.e., ω j are variational derivatives of some local functionals ∫ h j , then the latter are integrals of motion in involution of the hierarchy formed by the corresponding Hamiltonian vector fields. We show that the complex Ω is exact, provided that the algebra of functions is $$\fancyscript {V}$$ is “normal”; in particular, for arbitrary $$\fancyscript {V}$$ , any closed form in Ω becomes exact if we add to $$\fancyscript {V}$$ a finite number of antiderivatives. We demonstrate on the examples of the KdV, HD and CNW hierarchies how the Lenard scheme works. We also discover a new integrable hierarchy, which we call the CNW hierarchy of HD type. Developing the ideas of Dorfman, we extend the Lenard scheme to arbitrary Dirac structures, and demonstrate its applicability on the examples of the NLS, pKdV and KN hierarchies.

Caractérisation et existence de structures de Dirac multiplicatives

We define the product of two Dirac manifolds and introduce the notion of a Dirac–Lie group of Poisson type. This notion is equivalent to that of multiplicative Dirac structure and any real simply-connected Lie group carries a no trivial multiplicative Dirac structure when its dimension is at least 2. To cite this article: A. Affane, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

On the Geometric Structure of Hamiltonian Systems with Ports

In this article the geometric structure of Hamiltonian systems with ports is clarified. Particularly, it is shown that the theory of Hamiltonian systems with ports fits into the paradigm that the state space of a mechanical system should be modeled by a manifold M endowed with a(n almost) Dirac structure in a Courant algebroid.

On the Algebraic Index for Riemannian Étale Groupoids

In this paper we construct an explicit quasi-isomorphism to study the cyclic cohomology of a deformation quantization over a riemannian \'etale groupoid. Such a quasi-isomorphism allows us to propose a general algebraic index problem for riemannian \'etale groupoids. We discuss solutions to that index problem when the groupoid is proper or defined by a constant Dirac structure on a 3-dim torus.

DECAY CONSTANTS OF PSEUDOSCALAR MESONS IN BETHE–SALPETER FRAMEWORK WITH GENERALIZED STRUCTURE OF HADRON-QUARK VERTEX

We employ the framework of Bethe–Salpeter equation under Covariant Instantaneous Ansatz to study the leptonic decays of pseudoscalar mesons. The Dirac structure of hadron-quark vertex function Γ is generalized to include various Dirac covariants besides γ5 from their complete set. The covariants are incorporated in accordance with a power counting rule, order-by-order in powers of the inverse of the meson mass. The decay constants are calculated with the incorporation of leading order covariants. Most of the results are dramatically improved.

A tunable topological insulator in the spin helical Dirac transport regime

Helical Dirac fermions-charge carriers that behave as massless relativistic particles with an intrinsic angular momentum (spin) locked to its translational momentum-are proposed to be the key to realizing fundamentally new phenomena in condensed matter physics. Prominent examples include the anomalous quantization of magneto-electric coupling, half-fermion states that are their own antiparticle, and charge fractionalization in a Bose-Einstein condensate, all of which are not possible with conventional Dirac fermions of the graphene variety. Helical Dirac fermions have so far remained elusive owing to the lack of necessary spin-sensitive measurements and because such fermions are forbidden to exist in conventional materials harbouring relativistic electrons, such as graphene or bismuth. It has recently been proposed that helical Dirac fermions may exist at the edges of certain types of topologically ordered insulators-materials with a bulk insulating gap of spin-orbit origin and surface states protected against scattering by time-reversal symmetry-and that their peculiar properties may be accessed provided the insulator is tuned into the so-called topological transport regime. However, helical Dirac fermions have not been observed in existing topological insulators. Here we report the realization and characterization of a tunable topological insulator in a bismuth-based class of material by combining spin-imaging and momentum-resolved spectroscopies, bulk charge compensation, Hall transport measurements and surface quantum control. Our results reveal a spin-momentum locked Dirac cone carrying a non-trivial Berry's phase that is nearly 100 per cent spin-polarized, which exhibits a tunable topological fermion density in the vicinity of the Kramers point and can be driven to the long-sought topological spin transport regime. The observed topological nodal state is shown to be protected even up to 300 K. Our demonstration of room-temperature topological order and non-trivial spin-texture in stoichiometric Bi(2)Se(3).M(x) (M(x) indicates surface doping or gating control) paves the way for future graphene-like studies of topological insulators, and applications of the observed spin-polarized edge channels in spintronic and computing technologies possibly at room temperature.

Dirac geometry, quasi-Poisson actions and D/G-valued moment maps

We study Dirac structures associated with Manin pairs (d, g) and give a Dirac geometric approach to Hamiltonian spaces with D/G-valued moment maps, originally introduced by Alekseev and Kosmann-Schwarzbach [3] in terms of quasi-Poisson structures. We explain how these two distinct frameworks are related to each other, proving that they lead to isomorphic categories of Hamiltonian spaces. We stress the connection between the viewpoint of Dirac geometry and equivariant differential forms. The paper discusses various examples, including q-Hamiltonian spaces and Poisson-Lie group actions, explaining how presymplectic groupoids are related to the notion of “double” in each context.