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.

This article focuses on the root behavior of the characteristic function of some LTI SISO systems controlled by a PD-control when a delay-difference operator is used to approximate the derivative action. We will focus on the cases when the corresponding stability problem is improperly posed for "small" delay values. In this context, we will express the solutions of the corresponding characteristic function as power series and exploit their structure in deriving the asymptotic behavior of the characteristic roots. This analysis will allow detecting and explicitly characterizing the cases when delay-difference approximations lead to improperly posed stability problems. Furthermore, in the case when the derivative approximation guarantees the properly posedness of the closed-loop system, the robustness of the scheme with respect to the delay margin of the delay-difference approximation and control gains parameters are explicitly addressed. More precisely, upper bounds on the delays as well as the robustness of the corresponding controller are computed. Some illustrative examples complete the presentation.

This article presents a simple procedure that allows a practical design of fractional –[Formula: see text] controllers for single-input single-output linear time-invariant fractional-order systems subject to a constant time delay. The methodology is based on a geometric approach, which provides practical guidelines to design stabilizing and non-fragile PDμ controllers. The simplicity of the proposed approach is illustrated by considering several numerical examples encountered in the control literature. Moreover, with the aim of showing the performance of the PDμ over a classical PD controller, both controllers were implemented at the end of the article in an experimental test-bench consisting a teleoperated robotic system.

The first author is enrolled in a joint PhD program between l'Université Paris-Saclay and the UASLP.

This paper addresses the closed-loop stability analysis problem of a bilateral local teleoperation system in the presence of delays, with the purpose of maintaining a kinematic correspondence between a master and a slave device. The stability analysis is proposed as a general analysis in the controller's parameter-space under the assumption of two fixed delays, considering one due to the signal and communication processing and one defined as a design delay for the controller. Furthermore, a method for measuring the fragility of the controllers is also proposed. Finally, experimental results obtained from an experimental platform consisting of two Phantom Omni haptic devices and the Matlab-Simulink toolkit Phansim illustrate the performance of the proposed approach and the video of the experiments can be downloaded from the authors' dedicated website 1 .