Potential and kinetic shaping for control of underactuated mechanical systems

This paper combines techniques of potential shaping with those of kinetic shaping to produce some new methods for stabilization of mechanical control systems. As with each of the techniques themselves, our method employs energy methods and the LaSalle invariance principle. We give explicit criteria for asymptotic stabilization of equilibria of mechanical systems which, in the absence of controls, have a kinetic energy function that is invariant under an Abelian group

Adaptive Second-Order Synchronization of Two Heterogeneous Nonlinear Coupled Networks

This paper investigates the second-order synchronization of two heterogeneous nonlinear coupled networks by introducing controller and adaptive laws. Based on Lyapunov stability properties and LaSalle invariance principle, it is proved that the position and the velocity of two heterogeneous nonlinear coupled networks are asymptotically stable. Finally, some numerical simulations are presented to verify the analytical results

On the Hybrid Minimum Principle On Lie Groups and the Exponential Gradient HMP Algorithm

This paper provides a geometrical derivation of the Hybrid Minimum Principle (HMP) for autonomous hybrid systems whose state manifolds constitute Lie groups $(G,\star)$ which are left invariant under the controlled dynamics of the system, and whose switching manifolds are defined as smooth embedded time invariant submanifolds of $G$. The analysis is expressed in terms of extremal (i.e. optimal) trajectories on the cotangent bundle of the state manifold $G$. The Hybrid Maximum Principle (HMP) algorithm introduced in \cite{Shaikh} is extended to the so-called Exponential Gradient algorithm. The convergence analysis for the algorithm is based upon the LaSalle Invariance Principle and simulation results illustrate their efficacy

Stabilization of an arbitrary profile for an ensemble of half-spin systems

We consider the feedback stabilization of a variable profile for an ensemble of non interacting half spins described by the Bloch equations. We propose an explicit feedback law that stabilizes asymptotically the system around a given arbitrary target profile. The convergence proof is done when the target profile is entirely in the south hemisphere or in the north hemisphere of the Bloch sphere. The convergence holds for initial conditions in a H^1 neighborhood of this target profile. This convergence is shown for the weak H^1 topology. The proof relies on an adaptation of the LaSalle invariance principle to infinite dimensional systems. Numerical simulations illustrate the efficiency of these feedback laws, even for initial conditions far from the target profile.Comment: 6 pages, 2 figure

Analysis of Lyapunov Method for Control of Quantum Systems

We present a detailed analysis of the convergence properties of Lyapunov control for finite-dimensional quantum systems based on the application of the LaSalle invariance principle and stability analysis from dynamical systems and control theory. For a certain class of ideal Hamiltonians, convergence results are derived both pure-state and mixed-state control, and the effectiveness of the method for more realistic Hamiltonians is discussed.Comment: 20 pages, 1 figure, draft versio