Jean-Baptiste Pomet

Adaptive nonlinear regulation: equation error from the Lyapunov equation

An adaptive controller for nonlinear, linearly parameterized systems is presented. The new features introduced in the design are: (1) parameter estimation is performed on the scalar Lyapunov equation instead of the n-dimensional equation of the system itself. This allows non-Lipschitz uncertainties to be tolerated, especially when the stabilizing laws are not feedback linearization plus linear control. (2) Double estimation is used. One estimate is used for the stabilizing control, and the other is used to cancel the perturbation terms introduced by the adaptation, if possible. This is proposed to solve the problem of the implicit definition of the controller which arises when one tries to do this cancellation.<<ETX>>

A REMARK ON THE DESIGN OF TIME-VARYING STABILIZING FEEDBACK LAWS FOR CONTROLLABLE SYSTEMS WITHOUT DRIFT

Abstract This paper gives an approach to design time-varying feedback laws for controllable systems without drift. This approach is based on a time-varying Lyapunov function.

Indirect adaptive nonlinear control

The authors consider how to make a nonlinear control law adaptive. Assuming that the uncertainties interfere linearly, they answer this question for a certain class of systems. In particular, they propose an estimation-based adaptive controller with corrective terms for more general families than robot arms. Assuming some structural properties, they obtain global convergence under far less restrictive growth assumptions than previously.<<ETX>>

Adaptive Stabilization for Nonlinear Systems in the Plane

Abstract We propose an adaptive controller to globally stabilize a compact set for a planar system depending linearly on unknown parameters. Global rec tifiabili ty of the cont rol vector field is assumed. Our approach is based on the Control Lyapunov Function and Lyapunov design technique.

Design of control Lyapunov functions for “Jurdjevic-Quinn” systems

This paper presents briefly a method to design explicit control Lyapunov functions for control systems that satisfy the so-called “Jurdjevic-Quinn conditions”, i.e. posses an “energy-like” function that is naturally non-increasing for the un-forced system. The results with proof will appear in a future paper. The present note rather focuses on the method, and on its application to the model of a mechanical system, the translational oscillator with rotation actuator (TORA) (also known as RTAC).

Strict Control Lyapunov Functions for Homogeneous Jurdjevic-Quinn Type Systems

Abstract The decomposition of nonlinear output feedback control into an observer and a state feedback control This paper presents a method to design an explicit control Lyapunov function for affine and homogeneous systems that satisfy the so-called “Jurdjevic-Quinn conditions”. For these systems a positive definite function is known that can only be made non increasing by feedback; a control Lyapunov function is obtained via a deformation of this function. As an example of its applications, this method allows one to construct an explicit control Lyapunov function for the stabilization of the angular velocity of an underactuated rigid body.

On the differentiability of feedback linearization and the Hartman-Grobman theorem for control systems

We consider a natural extension of the Hartman-Grobmann theorem to control systems, namely we investigate when is it possible to topologically conjugate the trajectories of a smooth single-input nonlinear control system, near an equilibrium, to those of its linear approximation when the latter is reachable. This turns out to be equivalent to smooth static-feedback linearizability which is highly non-generic. In the positive direction, we state a weak form of the Hartman-Grobman theorem for control systems whose input has a prescribed dynamics.

On Periodic Solutions of Adaptive Systems in the Presence of Periodic Forcing Terms

We consider a discrete-time system consisting of a linear plant and a periodically forced feedback controller whose parameters are slowly adapted. Using degree theory, we give sufficient conditions for the existence of periodic solutions. Using linearization methods, we give sufficient conditions for their (in)stability provided the adaptation is slow enough. We then study when the degree theoretic conditions for the existence are satisfied byd-steps-ahead adaptive controllers in the presence of unmodeled dynamics and a persistently exciting periodic reference output.