Proportional-delayed controllers design for LTI-systems: a geometric approach

Abstract

This paper focuses on the design of P -δ controllers for single-input-single-output (SISO) linear timeinvariant (LTI) systems. The basis of this work is a geometric approach allowing to partitioning the parameter space in regions with constant number of unstable roots. This methodology defines the hyperplanes separating the aforementioned regions and characterizes the way in which the number of unstable roots changes when crossing such a hyper-plane. The main contribution of the paper is that it provides an explicit tool to find P -δ gains ensuring the stability of the closed-loop system. In addition, the proposed methodology allows to design a non-fragile controller with a desired exponential decay rate σ. Several numerical examples illustrate the results and a haptic experimental setup shows the effectiveness of P -δ controllers.

σ‐Stabilization of a Flexible Joint Robotic Arm via Delayed Controllers

“…Condition (19) follows directly from (21). Now, through direct computations, it can be verified that equations (20) and (21)…”

“…In this framework, in [17], a method for the migration of a double imaginary characteristic root to the left half-plane or the right halfplane under the variation of two parameters of a quasipolynomial is presented. e idea of deliberately introducing time delays in closed-loop systems and considering it as a control parameter is not a novel approach, but it has been intensively studied in recent years, see [18][19][20] and the references therein. e analysis of such class of controllers focuses mainly on the following topics: characterization of the stability crossing curves [21], tuning of delayed controllers to stabilize second-order systems [18,20,22] (and its noise attenuation analysis [23]), the design of proportional integral controllers for second-order linear systems [19,24], and design of maximum decay rate using elimination theory [25].…”

“…is result ultimately guarantees a desired exponential decay rate σ. On the other hand, design of nonfragile controllers with a desired exponential decay rate is proposed in [20]. Here, the authors present conditions on the parameters (k p , k δ ) such that the p − δ controller of the form k p + k δ e − τs ensure the stability of the closed-loop system.…”

“…Delay-based controllers have as their main advantages the simplicity with which control laws are designed and, as a consequence, their practical implementation facility. Common applications of such controllers focus mainly on second-order systems, e.g., regulation problems of DC servomotors [18,31], haptic virtual systems [20], underactuated mechanical system [32], and numerous academic examples. In the present proposal, a more challenging implementation is addressed: a flexible joint robotic arm (fourth order system), where trajectory tracking tasks are addressed.…”

A Novel Modified Delay-Based Control Algorithm with an Experimental Application

“…PR control application to a mechatronic system was presented in [37,38], wherein the authors present a pendulum control performance without tweaking differentiators. Yet another study elaborated the closed-loop stability analysis of voltage buck using proportional-delayed controller [17] that the is also based on the geometric approach, which allows partitioning the parameters space into regions with a constant number of unstable roots. In the work of Diez et al, a transparent bilateral control scheme for a local teleoperation system was realized by using a proportional-delayed controller [19].…”