26 research outputs found
Hamiltonian and self-adjoint control systems
This paper outlines results recently obtained in the problem of determining when an input-output map has a Hamiltonian realization. The results are obtained in terms of variations of the system trajectories, as in the solution of the Inverse Problem in Classical Mechanics. The variational and adjoint systems are introduced for any given nonlinear system, and self-adjointness defined. Under appropriate conditions self-adjointness characterizes Hamiltonian systems. A further characterization is given directly in terms of variations in the input and output trajectories, proving an earlier conjecture by the first author
Geometry of Thermodynamic Processes
Since the 1970s contact geometry has been recognized as an appropriate
framework for the geometric formulation of the state properties of
thermodynamic systems, without, however, addressing the formulation of
non-equilibrium thermodynamic processes. In Balian & Valentin (2001) it was
shown how the symplectization of contact manifolds provides a new vantage
point; enabling, among others, to switch between the energy and entropy
representations of a thermodynamic system. In the present paper this is
continued towards the global geometric definition of a degenerate Riemannian
metric on the homogeneous Lagrangian submanifold describing the state
properties, which is overarching the locally defined metrics of Weinhold and
Ruppeiner. Next, a geometric formulation is given of non-equilibrium
thermodynamic processes, in terms of Hamiltonian dynamics defined by
Hamiltonian functions that are homogeneous of degree one in the co-extensive
variables and zero on the homogeneous Lagrangian submanifold. The
correspondence between objects in contact geometry and their homogeneous
counterparts in symplectic geometry, as already largely present in the
literature, appears to be elegant and effective. This culminates in the
definition of port-thermodynamic systems, and the formulation of
interconnection ports. The resulting geometric framework is illustrated on a
number of simple examples, already indicating its potential for analysis and
control.Comment: 23 page
Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem
We develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled Lagrangians. The procedure involves making structured modifications to the Lagrangian for the uncontrolled system, thereby constructing the controlled Lagrangian. The Euler-Lagrange equations derived from the controlled Lagrangian describe the closed-loop system, where new terms in these equations are identified with control forces. Since the controlled system is Lagrangian by construction, energy methods can be used to find control gains that yield closed-loop stability. We use kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group. The procedure is demonstrated for several underactuated balance problems, including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane
Controlled invariance for hamiltonian systems
A notion of controlled invariance is developed which is suited to Hamiltonian control systems. This is done by replacing the controlled invariantdistribution, as used for general nonlinear control systems, by the controlled invariantfunction group. It is shown how Lagrangian or coisotropic controlled invariant function groups can be made invariant by static, respectively dynamic, Hamiltonian feedback. This constitutes a first step in the development of a geometric control theory for Hamiltonian systems that explicitly uses the given structure
Linear Hamiltonian behaviors and bilinear differential forms
We study linear Hamiltonian systems using bilinear and quadratic differential forms. Such a representation-free approach allows us to use the same concepts and techniques to deal with systems isolated from their environment and with systems subject to external influences and allows us to study systems described by higher-order differential equations, thus dispensing with the usual point of view in classical mechanics of considering first- and second-order differential equations only