Relay Feedback Autotuning – A Polynomial Design Approach

Abstract

A method of autotuning using an asymmetric relay with hysteresis feedback test is proposed and developed. Then, three parameters for aperiodic first or second order transfer functions can be obtained. After the identification relay experiment, controller parameters are computed through linear diophantine equation in the ring of proper and stable rational functions. This algebraic approach for a traditional 1-DOF feedback structure generates a class of PI or PID controllers. The pole placement principle in the methodology brings a scalar positive “tuning knob” for additional controller tuning. A Matlab-Simulink program implementation was developed for simulation and verification of the studied approach. Two illustrative examples support simplicity and efficiency of the proposed methodology. INTRODUCTION Despite of expressive evolution of control hardware, PID controllers remain the main tool in industrial feedback loops and they survived many changes in technology. 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. However, real industrial processes suffer from many unpleasant drawbacks as non-linearity, complexity and time variance. These features induce difficulties to control their loops properly. Adequate and sufficient tuning of controllers needs to know relevant process parameters. One way how to overcome the mentioned problems consists in automatic tuning of controllers. The development of various autotuning principles was started by a simple symmetrical relay feedback experiment (Astrom and Hagglund 1984) for a PID structure. Ultimate gain and ultimate frequency are then used for adjusting of parameters by common known Ziegler-Nichols rules. During the period of more than two decades, many studies have been reported to extend and improve autotuners principles; see e.g. (Astrom and Hagglund 1995; Ingimundarson and Hagglund 2000; Majhi and Atherton 1998; Morilla at al. 2000). The extension in an experimental phase was performed in (Yu 1999; Pecharroman and Pagola 2000; Kaya and Atherton 2001) by an asymmetry and hysteresis of a relay, see (Thyagarajan and Yu 2002), and experiments with asymmetrical and dead-zone relay feedback are reported in (Viteckova and Vitecek 2004; Vyhlidal 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 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 Klan 2000). PROCESS PARAMETERS IDENTIFICATION System identification of the process parameters is a crucial point for many auto-tuning principles. The identification rules with relay in feedback loops can utilize various types of relays. The relay feedback proposed by Astrom in 1984 used a symmetrical relay without hysteresis. The identification procedure with a relay in the feedback loop can utilize various types of relays. The relay feedback proposed by Astrom in 1984 utilizes symmetrical relay without hysteresis. This procedure gives the ultimate parameters of the process and control design may follow. Unfortunately, the process gain must be known in advance because the symmetrical relay test cannot identify it. On the other hand, the process gain can be obtained during the relay feedback test when an asymmetrical relay is utilized. A typical data response from the relay experiment can be seen in Figure 1. In this paper, an asymmetric relay with hysteresis is used. It enables to identify the parameters of the transfer function, such as proportional gain, time constant, as well as time delay term. Time delay is approximated by Pade approximation before the algebraic controller design. Proceedings 24th European Conference on Modelling and Simulation ©ECMS Andrzej Bargiela, Sayed Azam Ali David Crowley, Eugene J.H. Kerckhoffs (Editors) ISBN: 978-0-9564944-0-5 / ISBN: 978-0-9564944-1-2 (CD) 0 5 10 15 20 25 30 35 40 45 50 -0.2 -0.1 0 0.1 0.2 0.3 Figure 1: Relay feedback test of stable process First order system The most popular and simplest approximation of aperiodic industrial processes can be characterized by the first order transfer function with time delay (FOPDT) in the form: Ls e Ts K s G − ⋅ + = 1 ) ( (1) When the asymmetric relay is used for the relay feedback test, it is shown in Figure 1, the output y converges to the stationary oscillation in one period. These oscillations are characterized by equations (Hang et al. 2001):