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 linear time-invariant 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 hyper-planes separating the aforementioned regions and characterises 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 set-up shows the effectiveness of P-δ controllers.

KEYWORDS:

• Time-delay systems
• proportional-delayed controllers
• stability
• fragility

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Although G is a complex function, here we consider even functions in the real domain, i.e. $G\left(x\right)=G(-x),\forall x\in R$.