Ziegler-nichols Type Tuning Rules for Fractional Pid Controllers

This paper presents two sets of tuning rules for fractional PIDs that rely solely on the same plant time-response data used by the first Ziegler-Nichols tuning rule for (usual, integer) PIDs. Thus no model for the plant to control is needed; only an Sshaped step response is. These rules are quadratic and their results compare well with those obtained with rule-tuned integer PIDs. INTRODUCTION Controllers whose output is a linear combination of the input, the derivative of the input and the integral of the input, known as PID (proportional—derivative—integrative) controllers, are widely used and enjoy significant popularity, because they are simple, effective and robust. One of the reasons why this is so is the existence of tuning rules for finding suitable parameters for PIDs, rules that do not require any model of the plant to control. All that is needed to apply such rules is to have a certain time response of the plant. Examples of such sets of rules are those due to Ziegler and Nichols, those due to Cohen and Coon, and the Kappa-Tau rules [1]. It is true that PIDs tuned with such rules often perform in a nonoptimal way. But even though further fine-tuning be possible and sometimes necessary, rules provide a good starting point. Their usefulness is obvious when no model of the plant is available, ∗Address all correspondence to this author. and thus no analytic means of tuning a controller exists, but rules may also be used when a model is known. Fractional PIDs are generalisations of PIDs: their output is a linear combination of the input, a fractional derivative of the input and a fractional integral of the input [2]. Fractional PIDs are also known as PIλDμ controllers, where λ and μ are the integration and differentiation orders; if both values are 1, the result is a usual PID (henceforth called “integer” PID as opposed to a fractional PID). They have been increasingly used over the last years, but methods proposed to tune them always require a model of the plant to control [3, 4]. (An exception is [5], but the proposed method is far from the simplicity of tuning rules for integer PIDs.) This paper addresses this issue proposing sets of tuning rules for fractional PIDs. Proposed rules bear similarities to the first rule proposed by Ziegler and Nichols for integer PIDs, making use of the same plant time response data. The paper is organised as follows. Next section sums up the fundamentals of fractional calculus needed to understand fractional PIDs. Then two analytical methods for tuning fractional PIDs when a plant model is available are addressed; these are used as basis for deriving the tuning rules. The last two sections give some examples of application and draw some conclusions. FRACTIONAL ORDER SYSTEMS Definitions Fractional calculus is a generalisation of ordinary calculus. The main idea is to develop a functional operator D, associated 1 Copyright c © 2005 by ASME to an order ν not restricted to integer numbers, that generalises the usual notions of derivatives (for a positive ν) and integrals (for a negative ν). The most usual definition of D is due to Riemann and Liouville (although there are others) and generalises the equalities cD x f (x) = ∫ x c (x− t)n−1 (n−1)! f (t)dt, n ∈ N (1) DnDm f (x) = Dn+m f (x) , m ∈ Z0 ∨n,m ∈ N0 (2) which are easily proved for integer orders. The full definition of D becomes