# Online SOPDT Model Identification Method Using a Relay
## 1 Introduction
A modified identification method based on step characteristics using a real-coded genetic algorithm is presented in [7]. However, these methods are not suitable for online use, in contrast to the relay feedback approach, which was shown to be very useful for system identification. Several articles were also published in recent years that describe methods for determining the parameters of plant models with an assumed model class, including second-order time-delayed models [8; 9; 10]. [PERSON] and [PERSON] [11; 12] proposed state-space-based exact analytical expressions for the identification of second-order plus dead time models via asymmetrical relay tests. The use of the relay method to determine the model parameters (2) is presented in the work of [PERSON] and [PERSON] [13]. The test described in this paper was performed with an asymmetric relay, and the output and input signal transforms were calculated using integration. [PERSON] and co-authors [14; 15] developed a method to use the process input and output transients that result from relay feedback for transfer function modeling with the help of a fast Fourier transform. To estimate the parameters of the SOPDT model, [PERSON] and [PERSON] [16] used a complex A-locus function. [PERSON] and coworkers [17] carried out two identification tests and additionally used the least-squares method to determine the unknown parameters. [PERSON] et al. [18] used online measurements of the control signal and the output process to determine the harmonics needed to solve linear equations. [PERSON] and coworkers [19] presented an identification method for stable and unstable SOPDT processes using a state space approach. A relatively simple method for determining different models using an equivalent gain of the relay was presented by [PERSON] et al. [20]. [PERSON] and [PERSON] used a combined identification technique to recover a model that matches the time response while capturing the true dynamics of the system around the frequencies of interest [21]. The abovementioned identification techniques require the use of more advanced mathematical tools. The motivation for this work was the need to develop a simple online identification method that can be used without taking the control system out of operation for a time-varying plant, which is known to be best achieved by the SOPDT model. Using this method,a second-order inertial model with a time delay was determined in order to autotune a PID controller according to the rules based on the parameters of this model. In addition to carrying out the necessary calculations without the use of complex and time-consuming computational methods, such as numerical methods, the objectives set for this work included several further expectations. These included achieving a model accuracy such that, with appropriately set model parameters, the behavior of the process would be fully reproduced. This applies, of course, only if there is no model uncertainty. In order for the proposed method to be used effectively in real control systems, where this condition will not be met, it should be robust so that measurement errors of the input variable, which are unavoidable, do not lead to an instability of the closed-loop system as long as they do not exceed acceptable levels, but at most lead to a certain decrease of the control performance.