Analytic Expressions of Ultimate Gains and Ultimate Periods with Phase‐Optimal Approximations of Time Delays

Abstract

R&D NOTET here are many tuning rules currently available for PI/PID controllers (Astrom and Hagglund, 1988;Seborg et al., 1989;O'Dwyer, 2003) and new tuning rules (e.g. Zhang et al., 2002) are still being reported. Among them, tuning rules based on ultimate gains and ultimate periods (Ziegler and Nichols, 1942;Zhuang and Atherton, 1993) have several advantages over tuning rules based on specific process models. For example, tuning rules based on the first order plus time delay model cannot be applied directly to the process, G(s)=1/(s+1) 6 . The process should be reduced to a first order plus time delay model before applying those tuning rules. On the other hand, tuning rules based on the ultimate gain and ultimate period can be applied without such a model reduction step. The classical Ziegler-Nichols tuning rule based on the ultimate gain and the ultimate period is often criticized for its oscillatory response. The tuning rule by Zhuang and Atherton (1993) relieves this and is comparable with tuning rules based on specific process models. Recent relay feedback auto tuning method (Astrom and Hagglund, 1984) makes tuning rules based on ultimate parameters be practical. Here, analytic expressions for ultimate periods and ultimate gains have been proposed. For this, the Routh stability method (Seborg et al., 1989) with phase-optimal approximations of the time delay term is utilized. Our analytic equations for ultimate frequencies and ultimate gains show relative errors below 1% for processes, for which analytic equations by Ho et al. (1995) have relative errors over 10%. Analytic Expressions of Ultimate Gains andUltimate gains and ultimate periods have long been used to tune PI/PID controllers since the Ziegler-Nichols tuning. Recently, very simple analytic equations for ultimate gains and ultimate periods by approximating the arctangent function have been presented, which can be used to obtain simple tuning rules. Here, by applying the Routh stability method and phase-optimal approximations of the time delay term, we present analytic expressions of ultimate parameters that are as simple as the existing ones, showing better accuracy.Il est classique d'utiliser des gains et des périodes ultimes pour régler les contrôleurs PI/PID depuis les travaux de Ziegler-Nichols. Récemment, des équations analytiques simples pour des gains et des périodes ultimes par approximation de la fonction arctangente, qui peuvent être utilisées pour obtenir des règles de réglage simple, ont été présentées. En appliquant la méthode de stabilité de Routh et les approximations de phases optimales du terme de délai, on présente ici des expressions analytiques des paramètres ultimes qui sont aussi simples que les paramètres existants, avec une meilleure précision.

Half order plus time delay (HOPTD) models to tune PI controllers

“…The angular frequency of Eq. becomes the ultimate frequency of process and its approximate solution can be obtained as …”