Delayed controllers for time-delay systems

Abstract

In this paper, an analysis to design, tune and apply delayed controllers on a general class of first-order up to fourth-order LTI systems with or without dead-time is presented. This analysis is based on a study of the corresponding characteristic equation for obtaining σ-stabilization regions and to place dominant roots in the left-half complex plane. Obtaining σ-stabilization regions allows to characterize a parametric plane, which leads to obtain stabilizing conditions for LTI systems with two non-commensurable delays. The placement of dominant roots due to parametric variations (including delays) allows the tuning of delayed controllers. For this, analytic expressions to place three dominant real roots are given guaranteeing an exponential decay rate in the system response. The scope of the methodology is extended to nonlinear delayed controllers using recursive techniques for a class of nonlinear systems with dead-time in tracking tasks. To illustrate the obtained theoretical results, the following cases are considered: first, the tuning and application of a novel delayed controller with two delays are presented to stabilize a delay-free LTI model of fourth-order of the underactuated mechanical system called cart-pendulum using only pendulum position and cart position. Second, the design and tuning of a tracking delayed controller for a class of nonlinear systems with dead-time are proposed. Here, a delay is inherent to the system and another delay is artificially used to stabilize.