New Stability Conditions for Time Delay Control of Nonlinear Systems

Abstract

Time Delay Control (TDC) for linear and nonlinear systems with unknown or uncertain dynamics continues to attract attention with several applications, extensions, and variations, given its relatively simple and compact nature. Time Delay Control utilizes time-delayed signals for approximating unknown dynamics at a given instant, and canceling these dynamics, and inserting desired dynamics through feedback. Despite advances involving Uncertainty and Disturbance Estimator, and Active Disturbance Rejection Control, TDC retains its appeal due to its simplicity, as well as similarity and equivalence to PI/PID controller as well as other methods. It provides the potential of a simplified and different approach to controller design for systems with uncertainties. This paper describes improved sufficient conditions for Bounded Input Bounded Output stability while being practical, for continuous Time Delay Control of generalized nonlinear, time-varying systems that are not necessarily affine in input, and for Time Delay (L) that is finite, but not necessarily infinitesimally small. The paper contrasts the methodology and conditions compared to previous results for continuous Time Delay Control for such systems, indicating the superiority of the new results while remaining practical for cases that include nonaffine nonlinear systems. It is hoped that the results of this paper, coupled with other results, lay the foundation for future advances in design methods for various versions of Time Delay Control, including discrete as well as other implementation aspects, that are relatively practical and less conservative.