2,949 research outputs found
Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping
For pt.I, see ibid., vol.45, p.2253-70 (2000). We extend the method of controlled Lagrangians (CL) to include potential shaping, which achieves complete state-space asymptotic stabilization of mechanical systems. The CL method deals with mechanical systems with symmetry and provides symmetry-preserving kinetic shaping and feedback-controlled dissipation for state-space stabilization in all but the symmetry variables. Potential shaping complements the kinetic shaping by breaking symmetry and stabilizing the remaining state variables. The approach also extends the method of controlled Lagrangians to include a class of mechanical systems without symmetry such as the inverted pendulum on a cart that travels along an incline
The geometric structure of nonholonomic mechanics
Many important problems in multibody dynamics, the dynamics of wheeled vehicles and motion generation, involve nonholonomic mechanics. Many of these systems have symmetry, such as the group of Euclidean motions in the plane or in space and this symmetry plays an important role in the theory. Despite considerable advances on both Hamiltonian and Lagrangian sides of the theory, there remains much to do. We report on progress on two of these fronts. The first is a Poisson description of the equations that is equivalent to those given by Lagrangian reduction, and second, a deeper understanding of holonomy for such systems. These results promise to lead to further progress on the stability issues and on locomotion generatio
Explicit Simplicial Discretization of Distributed-Parameter Port-Hamiltonian Systems
Simplicial Dirac structures as finite analogues of the canonical Stokes-Dirac
structure, capturing the topological laws of the system, are defined on
simplicial manifolds in terms of primal and dual cochains related by the
coboundary operators. These finite-dimensional Dirac structures offer a
framework for the formulation of standard input-output finite-dimensional
port-Hamiltonian systems that emulate the behavior of distributed-parameter
port-Hamiltonian systems. This paper elaborates on the matrix representations
of simplicial Dirac structures and the resulting port-Hamiltonian systems on
simplicial manifolds. Employing these representations, we consider the
existence of structural invariants and demonstrate how they pertain to the
energy shaping of port-Hamiltonian systems on simplicial manifolds
Evolutionary Laws, Initial Conditions, and Gauge Fixing in Constrained Systems
We describe in detail how to eliminate nonphysical degrees of freedom in the
Lagrangian and Hamiltonian formulations of a constrained system. Two important
and distinct steps in our method are the fixing of ambiguities in the dynamics
and the determination of inequivalent initial data. The Lagrangian discussion
is novel, and a proof is given that the final number of degrees of freedom in
the two formulations agrees. We give applications to reparameterization
invariant theories, where we prove that one of the constraints must be
explicitly time dependent. We illustrate our procedure with the examples of
trajectories in spacetime and with spatially homogeneous cosmological models.
Finally, we comment briefly on Dirac's extended Hamiltonian technique.Comment: 23 pages; plain TeX. To appear: Classical & Quantum Gravit
Hamiltonian formulation of distributed-parameter systems with boundary energy flow
A Hamiltonian formulation of classes of distributed-parameter systems is presented, which incorporates the energy flow through the boundary of the spatial domain of the system, and which allows to represent the system as a boundary control Hamiltonian system. The system is Hamiltonian with respect to an infinite-dimensional Dirac structure associated with the exterior derivative and based on Stokes' theorem. The theory is applied to the telegraph equations for an ideal transmission line, Maxwell's equations on a bounded domain with non-zero Poynting vector at its boundary, and a vibrating string with traction forces at its ends. Furthermore the framework is extended to cover Euler's equations for an ideal fluid on a domain with permeable boundary. Finally, some properties of the Stokes-Dirac structure are investigated, including the analysis of conservation laws. \u
Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraints
Many successful methods to learn dynamical systems from data have recently
been introduced. However, ensuring that the inferred dynamics preserve known
constraints, such as conservation laws or restrictions on the allowed system
states, remains challenging. We propose stabilized neural differential
equations (SNDEs), a method to enforce arbitrary manifold constraints for
neural differential equations. Our approach is based on a stabilization term
that, when added to the original dynamics, renders the constraint manifold
provably asymptotically stable. Due to its simplicity, our method is compatible
with all common neural differential equation (NDE) models and broadly
applicable. In extensive empirical evaluations, we demonstrate that SNDEs
outperform existing methods while broadening the types of constraints that can
be incorporated into NDE training.Comment: 22 pages, 8 figures. Accepted at NeurIPS 202
Port controlled Hamiltonian representation of distributed parameter systems
A port controlled Hamiltonian formulation of the dynamics of distributed parameter systems is presented, which incorporates the energy flow through the boundary of the domain of the system, and which allows to represent the system as a boundary control Hamiltonian system. This port controlled Hamiltonian system is defined with respect to a Dirac structure associated with the exterior derivative and based on Stokes' theorem. The definition is illustrated on the examples of the telegrapher's equations, Maxwell's equations and the vibrating string. \u
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