Adaptive control scheme for uncertain time-delay systems

The paper exposes some of the potential of a recently introduced backstepping transformation for linear uncertain time-delay systems to address the classic problems of equilibrium regulation under partial measurements, disturbance rejection, parameter or delay adaptation. For each of these problems, an implementable control strategy is proposed. It is analyzed through a convergence analysis of infinite dimensional dynamics, based on a transport partial differential equation representation of the estimated input delay and the mentioned backstepping transformation. The considered controllers contain this representation of the system, under the form of a distributed parameter system, and use it to determine relevant adaptation strategies. An illustration example is proposed. 1. Introduction This paper addresses the general problem of equilibrium regulation of (potentially) unstable linear systems with an uncertain input time-delay. For such systems, predictor approaches (see e.g., the lack of robustness of this technique with respect to the uncertainty on the delay is still a major concern in automatic control theory, especially in view of implementation , in which it often appears as a performance bottleneck a new class of predictor-based techniques has been proposed to address this uncertainty for single input time-delay systems. This methodology is based on a modeling of the actuator delay as a transport partial differential equation (PDE), assuming that the full actuator state (i.e. the past values of the input) is known over an interval of length equal to the delay, as in Bresch-Pietri and Krstic (2009) for example. In this paper, we follow this methodology and develop an implementable form of the resulting controller in the spirit of ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Gang Tao under the direction of Editor Miroslav Krstic. Bresch-Pietri and Krstic (2010). In details, we use a backstepping boundary control method on a transport PDE introduced to model the delay. This transformation allows to use systematic Lyapunov design tools for robust stabilization and adaptation. Several classic cases are treated. While the resulting controllers could seem intuitive in their formulation, the results stated here and the corresponding proofs are, up to our knowledge, totally new. A series of classic issues in linear automatic control is considered: model uncertainties, disturbance rejection and partial state measurement. We address these difficulties separately and propose a dedicated implementable solution for each of them, along with analysis of convergence that stress …