This paper focuses on the fragility analysis of PIcontrollers for single-input-single-output (SISO) systems subject to input (or output) delays. Using a geometric approach, we present a simple and user-friendly approach not only to analyze the fragility of PI controllers, but also to provide practical guidelines for the design of non-fragile PI controllers. The proposed methodology is illustrated by analyzing several examples encountered in the control literature. Index Terms-PI-controller,

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... ility, Fragility, Delay. I. INTRODUCTION As reported in the literature [25] , [27] , more than 98% of the control-loops in the paper industries are controlled by SISO PI controllers. The "popularity" of PI and PID controllers can be attributed to their particular distinct features: simplicity and easy implementation. A long list of PI and PID tuning methods for controlling processes can be found in [25], [2] . As mentioned by [1], such controllers have to be designed by considering: (a) performance criteria; (b) robustness issues and, finally, (c) fragility. Roughly speaking, a controller for which the closed-loop system is destabilized by small perturbations in the controller parameters is called "fragile". In other words, the fragility describes the deterioration of closed-loop stability due to small variations of the controller parameters. This paper focuses on the fragility of PI controllers for SISO systems in the presence of I/O delays. The problem received a lot of attention in delay free systems, see, e.g., [16] (robustness techniques design leading to fragile controllers), [10] (non-fragile PID control design procedure), [1] (appropriate index to measure the fragility of PID controllers). In this context of delay free systems, some remarks concerning the controller robustness via coprime factorization and robustness optimization tools can be found in [17] , [15] . However, there exists only a few results in the delay case: [28] , where only (stable) first-order systems were considered, and more recently, [18] , where the authors proposed a robust non-fragile control design for a TCP/AQM models and, to the best of the authors' knowledge, there does C. Méndez-Barrios is with L2S (