ABSTRACT
The interesting discovery that the topmost equilibrium of a pendulum can be stabilized by oscillatory vertical movement of the suspension point has been attributed to Bogolyubov [11] and Kapitsa [26], who published papers on this subject in 1950 and 1951, respectively. In the intervening years, literature appeared analyzing the dynamics of systems with oscillatory forcing, e.g., [31]. Control designs based on oscillatory inputs have been proposed (for instance [8] and [9]) for a number of applications. Many classical results on the stability of operating points for systems with oscillatory inputs depend on the eigenvalues of the averaged system lying in the left half-plane. Recently, there has been interest in the stabilization of systems to which such classical results do not apply. Coron [20], for instance, has shown the existence of a time-varying feedback stabilizer for systems whose averaged versions have eigenvalues on the imaginary axis. This design is interesting because it provides smooth feedback stabilization for systems which Brockett [15] had previously shown were never stabilizable by smooth, time-invariant feedback. For conservative mechanical systems with oscillatory control inputs, Baillieul [7] has shown that stability of operating pointsmay be assessed in terms of an energy-like quantity known as the averaged potential. Control designs with objectives beyond stabilization have been studied in path-planning for mobile robots [40] and in other applications where the models result in “drift-free” controlled differential
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equations. Work by Sussmann and Liu [36-38], extending earlier ideas of Haynes and Hermes, [21], has shown that, for drift-free systems satisfying a certain Lie algebra rank condition (LARC discussed in Section 52.2), arbitrary smooth trajectories may be interpolated to an arbitrary accuracy by appropriate choice of oscillatory controls. Leonard and Krishnaprasad [28] have reported algorithms for generating desired trajectories when certain “depth” conditions on the brackets of the defining vector fields are satisfied.
