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A long term goal in the theory of systems and control is to develop a systematic methodology for the design of feedback control schemes capable of shaping the response of complex dynamical systems, in both an equilibrium and a nonequilibrium setting. In this talk, we will focus primarily on periodic steady-state behavior, a phenomenon that is pervasive in nature and in man-made systems. We will begin with an analysis of the asymptotically stable oscillation in the classical voltage controlled oscillator (VCO), followed by an analysis of Brockett’s recent design of a feedback law which creates an asymptotically stable oscillation in a three dimensional, nonholonomic model of an AC controlled rotor with a constant steady-state angular velocity. We will show how to design feedback laws for stabilizable n-dimensional systems so that the existence, periods and stability of periodic responses can be analyzed and shaped when the nonlinear feedback system is driven with an arbitrary periodic input. This design is one application of a general theory developed, jointly with R. Brockett, for the existence of oscillations in a nonlinear dynamical system. The sufficient conditions use a multi-valued analogue of Liapunov functions, in much the same way as the angular variable in polar coordinates is multi-valued. For the VCO the angular variable is the output of an integral controller, while for the AC motor it measures the rotation of the magnetic field. In the general case, the sufficient conditions can be checked point-wise, just as in Liapunov theory, and therefore do not require the knowledge of the trajectories of the system or a cross-section for the dynamics. Moreover, these results can be readily used in the theory of output regulation to shape the nonequilibrium steady-state response of dissipative nonlinear feedback systems. Finally, using the recent solution of the Poincaré Conjecture and more, we show these sufficient conditions are necessary for the - - existence of an asymptotically stable oscillation - a satisfying result in the spirit of the converse theorems of Liapunov theory.
Date of Conference: 9-11 Dec. 2008