Stabilization of invariant sets for nonlinear systems with applications to control of oscillations

Abstract

SUMMARYThe existing results on stabilization of invariant sets for nonlinear systems based on speed}gradient method and the notion of <-detectability are overviewed and extended. Applications to control of oscillations in pendulum, cart}pendulum and spherical pendulum systems are presented.

Physics and Control: Exploring Physical Systems by Feedback

“…are discussed (Arecchi et al, 1998). Similar property of reachability of the control goal by means of arbitrarily small control was observed in a broader class of oscillatory systems (Fradkov and Pogromsky, 1998;Shiriaev and Fradkov, 2001). …”

“…Again, it looks like a standard regulation problem with an additional restriction that we seek for "small control" solutions. However, such a restriction makes the problem far from standard: even for a simple pendulum, nonlocal solutions of the stabilization problem with small control were obtained only recently, see (Shiriaev and Fradkov, 2001). The class of admissible control laws can be extended by introducing dynamic feedback described by differential or time-delayed models.…”

Natural wave control in lattices of linear oscillators

“…The proposed control design is based on speed-gradient approach [9], [10], [34], [35] with the energy-based goal function…”

Robust and Adaptive Observer-Based Partial Stabilization for a Class of Nonlinear Systems

“…The passification method [11], [27], [28] is based on a feedback design making the closed loop system passive. It allows one to solve partial stabilization problem for system (2) with respect to the zero level set of storage function.…”