Hamiltonian Control Research Articles

Dynamical suppression of decoherence in two-state quantum systems

The dynamics of a decohering two-level system driven by a suitable control Hamiltonian is studied. The control procedure is implemented as a sequence of radiofrequency pulses that repetitively flip the state of the system, a technique that can be termed quantum "bang-bang" control after its classical analog. Decoherence introduced by the system's interaction with a quantum environment is shown to be washed out completely in the limit of continuous flipping and greatly suppressed provided the interval between the pulses is made comparable to the correlation time of the environment. The model suggests a strategy to fight against decoherence that complements existing quantum error-correction techniques.

Nonlinear overtaking optimal control: sufficiency, stability, and approximation

The authors consider infinite-horizon nonlinear optimal control problems with unbounded performance indexes using notions of overtaking optimality. In the framework of optimal control, new sufficient conditions of Caratheodory-Hamilton-Jacobi type for both weak overtaking and overtaking optimality are presented and compared with known results from the calculus of variations setting. Using local structural properties of a related control Hamiltonian, conclusions are drawn on aspects of existence of overtaking and weak overtaking optimal controls, stability of optimal trajectories, and approximation of weak overtaking optimal controls. Both similarities and differences among alternative approaches to overtaking, weak overtaking, and strong optimal control problems are discussed. The formulation and results are illustrated in the classical linear-quadratic tracking problem and in an optimization approach to gain scheduling.

Adaptation of centers of approximation for nonlinear tracking control

Sufficient conditions for the asymptotic stability and exponential convergence of tracking control for nonlinear systems have proliferated in recent research. Various theories for stability have been related to, and rely upon, the functional form of the governing nonlinear equations, the linearizability of the governing equations by canonical (Lie) transformations, the linear appearance of adaptive parameters, and the richness of the input excitation. Previously, sufficient conditions have been derived for a class of Hamiltonian adaptive tracking control problems using radial basis function approximants. We demonstrate that a crucial element determining the performance of this class of tracking control methodologies is the location of the centers of approximation in the radial basis approximant/network. A learning vector quantization algorithm is employed to simultaneously adapt both the centers of approximation and the feedback gains. The simultaneous adaptation of both approximant centers in configuration space and feedback gains can demonstrate remarkable improvements in convergence rate, as compared to adaptation of feedback gains alone.

Adjoint and Hamiltonian input-output differential equations

Based on developments in the theory of variational and Hamiltonian control systems by Crouch and van der Schaft (1987), this paper answers two questions: given an input-output differential equation description of a nonlinear system, what is the adjoint variational system in input-output differential form and what are the conditions for the system to be Hamiltonian, i.e., such that the variational and the adjoint variational systems coincide? This resulting set of conditions is then used to generalize classical conditions such as the well-known Helmholtz conditions for the inverse problem in classical mechanics. >

Structure-preserving stabilization of Hamiltonian control systems

Sufficient conditions for nonasymptotical stabilization of equilibrium points in Hamiltonian control systems using the Lyapunov technique and feedback transformation which preserve the Hamiltonian structure are presented.

Equivalence and Normal Form for Nonlinear Hamiltonian Control Systems

Equivalence and Normal Form for Nonlinear Hamiltonian Control Systems

ハミルトン制御系の不変性と無散逸化制御

ハミルトン制御系の不変性と無散逸化制御

Comments on "Hamiltonian adaptive control of spacecraft" by J.J.E. Slotine and M.D. Di Benedetto

In the adaptive scheme presented in the original paper (ibid., vol.35, no.7, p.848-52, 1990) for attitude tracking control of rigid spacecraft, the spacecraft is parametrized in terms of the inertial frame. The commenter shows how a parameterization in body coordinates considerably simplifies the representation of the adaptation scheme. The new symbolic expression for the regressor matrix is easy to find even for six-degree-of-freedom (DOF) Hamiltonian systems with a large number of unknown parameters. If the symbolic expression for the regressor matrix is known in advance, the computational complexity is approximately equal for both representations. An example illustrates the advantage of the new representation when modeling a simple three-DOF model of the lateral motion of a space shuttle.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stabilizability of nonholonomic control systems

We consider here the problem of stabilization of nonlinear Hamiltonian control systems with nonholonomic constraints. Such systems are of interest in various applications including the control of robotic systems. We discuss the formulation of Hamiltonian systems with nonholonomic constraints, in particular considering the differences between the Dirac theory of constraints, the classical theory of nonholonomic constraints, and Weber's theory of generalized nonholonomic constraints. Stabilization of systems with these constraints is discussed using the theory of Hamiltonian control systems, and a reduced stabilization criterion is given.

On Structure Preserving Transformations in Hamiltonian Control Systems

Hamiltonian control systems with natural output in the sense of [12] are considered. We investigate state-dependent co-ordinate transformations in the space of controls u and observables y which preserve the Hamiltonian structure.These transformations can be characterized and constructed by canonical transformations in the ( y, u )-pace.The results generalize known statements on Hamiltonian systems affine in the control and hold for general gradient-like control systems as well.

Characterization of allowable perturbations for robust decoupling of affine non-linear systems

For affine non-linear systems, we study the effects of uncertainties in systems vector fields on decoupled linear input-output behaviour. A characterization of an allowable class of perturbations which preserve the decoupling properties of the nominal system is given. An immediate consequence of this characterization indicates that decoupling position control of rigid robotic manipulators is not robust under any perturbations in robot dynamics. The result is applied to simple hamiltonian control systems to characterize the allowable perturbation of total energy function.

Hamiltonian adaptive control of spacecraft

A new approach to the accurate attitude tracking control of rigid spacecraft handling large loads of unknown mass properties is proposed. The method, based on the construction of a physically motivated Lyapunov-like function, is inspired by the adaptive robot control algorithm of J.J.E. Slotine and W. Li (Int. J. Robot. Res., vol.6, no.3, 1987), and presents similar advantages over techniques based on inverse dynamics in terms of simplicity, easier approach to robustness issues, and adaptive capabilities. The approach is illustrated in simulations. >

Hamiltonian description and dynamic control of flexible robots

AbstractThe article is concerned with the problem of modeling and control of flexible robots. This problem is faced by using the Hamiltonian approach and the assumed modes method to obtain an approximate finite dimensional model of the robot dynamics. The pseudo‐linearization method is then applied to design a nonlinear control law that is effective in the neighborhood of the operating points. The article ends with an application to a single‐link robot arm.

Stabilizability of Nonholonomic Control Systems

We consider here the problem of stabilization of nonlinear Hamiltonian control systems with nonholonomic constraints. Such systems are of interest in various applications including the control of robotic systems. We discuss the formulation of Hamiltonian systems with nonholonomic constraints, in particular considering the differences between the Dirac theory of constraints, the classical theory of nonholonomic constraints, and Weber's theory of generalized nonholonomic constraints. Stabilization of systems with these constraints is discussed using the theory of Hamiltonian control systems, and a reduced stabilization criterion is given. The effects of symmetries in the presence of constraints are also discussed.

Feedback stabilization of nonlinear Hamiltonian control systems

Abstract Conditions for feedback stabilization of smooth affine nonlinear Hamiltonian systems such as robotic manipulators are given. Some results existing in the literature are unified and extended. Application to nonconservative bilinear systems in elasticity is illustrated.

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.

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.

Manifolds, Tensor Analysis, and Applications

The purpose of this book is to provide core material in nonlinear analysis for mathematicians, physicists, engineers, and mathematical biologists. The main goal is to provide a working knowledge of manifolds, dynamical systems, tensors, and differential forms. Some applications to Hamiltonian mechanics, fluid mechanics, electromagnetism, plasma dynamics and control theory are given using both invariant and index notation. The prerequisites required are solid undergraduate courses in linear algebra and advanced calculus.

Convex Control Problems and Hamiltonian Systems of Infinite Intervals

This paper presents results on optimal control problems over $[ {0, + \infty } [$ associated with a linear process in a Hilbert space and with convex cost criterion. Results on synthesis of optimal control and necessary conditions for optimality in Hamiltonian form are obtained.Corollaries to the main result include the behavior of Hamiltonian systems of the form: $x' - Ax \in \partial _p H(x,p)$, $p' + A^ * p \in - \partial _x H(x,p)$, in a neighborhood of a saddle point.

On Suboptimal Performance Sensitivity

A known expression for the performance sensitivity of optimal systems as related to the gradient of the control Hamiltonian has been shown to hold good for a class of suboptimal feedback control systems. The actual percentage performance changes are computed with reference to a third order linear regulator problem, when a plant parameter is varied, for both suboptimal and truly optimal control implementations and the results are compared.