Relay-Based Autotuning With Algebraic Control Design

In this paper, a combination of relay feedback identification and an algebraic control design method for stable systems is studied. Models with up to three parameters are estimated by means of a single asymmetrical relay experiment. Then a stable low order transfer function is identified. Then the controller is analytically derived from general solutions of Diophantine equations in the ring of proper and stable rational functions. The controller parameters are tuned through a pole-placement problem as a desired multiple root of the characteristic closed loop equation. A first order identification yields a PI-like controllers while a second order identification generates PID ones. The approach enables a scalar tuning parameter m>0 which can be adjusted by various principles. INTRODUCTION Since industrial processes are usually complex and nonlinear the task to control their loops properly becomes difficult and challenging. Moreover, the majority of controllers used in industrial applications have been still of the PID type. The practical advantages of PID controllers can be seen in a simple structure, in an understandable principle and in control capabilities. It is widely known that PID controllers are quite resistant to changes in the controlled process without meaningful deterioration of the loop behavior. A solution for qualified choice of controller parameters can be seen in automatic tuning of PID controllers. The development of various autotuning principles was started by a simple symmetrical relay feedback experiment proposed by Åström and Hägglund, 1984. The ultimate gain and ultimate frequency are then used for adjusting of parameters by common known ZieglerNichols rules. During the period of more than two decades, many studies have been reported to extend and improve autotuners principles; see e.g. (Åströn and Hägglung, 1995; Ingimundarson and Hägglund, 2000; Majhi and Atherton, 1998; Morilla and Gonzáles, 2000). The extension in an experimental phase was performed in (Yu, 1999; Pecharromán and Pagola, 2000; Kaya and Atherton, 2001) by an asymmetry and hysteresis of a relay, see (Thyagarajan and Yu, 2002; Kaya and Atherton, 2001), and experiments with asymmetrical and dead-zone relay feedback are reported in (Vítečková and Víteček, 2004; Vyhlídal, 2000). Also, various control design principles and rules can be investigated in mentioned references. Nowadays, almost all commercial industrial PID controllers provide the feature of autotuning. This paper is focused on a novel combination for autotunig method of PI and PID controllers. The method combines an asymmetrical relay identification experiment and a control design method which is based on a pole-placement principle. The pole placement problem is formulated through a Diophantine equation and it is tuned by an equalization setting proposed in (Gorez and Klán, 2000). RELAY FEEDBACK ESTIMATION The estimation of the process or ultimate parameters is a crucial point in all autotuning principles. The relay feedback test can utilize various types of relay for the parameter estimation procedure. The classical relay feedback test (Åström and Hägglund, 1984) proposed for stable processes a symmetrical relay without hysteresis. Then the critical (ultimate) values can be estimated and a control design can follow. Asymmetrical relays with or without hysteresis bring further progress (Yu, 1999; Kaya and Atherton, 2001). After the relay feedback test, the estimation of process parameters can be performed. A typical data response of such relay experiment is depicted in Figure1. The relay asymmetry is required for the process gain estimation (2) while a symmetrical relay would cause the zero division in the appropriate formula. In this paper, an asymmetrical relay with hysteresis was used. This relay enables to estimate transfer function parameters as well as a time delay term. For the purpose of this contribution the time delay was not utilized. The model for first order (stable) systems plus dead time (FOPDT) is supposed in the form: Proceedings 23rd European Conference on Modelling and Simulation ©ECMS Javier Otamendi, Andrzej Bargiela, José Luis Montes, Luis Miguel Doncel Pedrera (Editors) ISBN: 978-0-9553018-8-9 / ISBN: 978-0-9553018-9-6 (CD) s e Ts K s G Θ − ⋅ + = 1 ) ( (1) and the process gain can be computed by the relation (Vyhlídal, 2000): ,... 3 , 2 , 1 ; ) ( ) ( 0 0 = = ∫ ∫ i dt t u dt t y K y y