# Online SOPDT Model Identification Method Using a Relay
## 4 Conclusions  
The SOPDT model identification method presented in this article, unlike the methods mentioned in the Introduction, did not require any signal transformations, solving of non-linear equations, or the use of other calculations using numerical methods. During the identification experiment, measurements of the relevant quantities were made.  
The quality of the resulting model was influenced by the correct choice of model class; the accuracy of the measurements carried out regarding the static gain, dead time, and both critical frequencies; and, indirectly, by the duration of the measurements, the presence of measurement noise, and possibly the impact of time-variant disturbances. The relatively simple method for creating the SOPDT model presented in this paper allowed for a more accurate representation of the dynamics of lower-order systems while maintaining the standard accuracy of limit cycle parameter measurements. In order to improve the quality of the model created, a two-stage procedure was employed, which used a relay with saturation and appropriately selected gain to generate sustained oscillations.  
The presented method for measuring the dead time under certain conditions may allow for a more precise determination of this model parameter. Moreover, the algorithms for the calculations needed to derive the next three of the four model parameters do not require any transformations of the process description or calculations via numerical methods, which makes this method unique with this number of unknowns. A special feature of this method is the possibility of dispensing in the second stage of the experiment with the measurement of the amplitude, which in the presence of noise may be subject to some error. The proposed SOPDT model identification method is the simplest calculation approach and produces the most accurate models within this class.  
A certain disadvantage may be the slightly longer duration of the extended identification experiment, which was ultimately intended to increase the accuracy of the created model since, during the identification test, the system lost its ability to react to load disturbances.  
In addition to the considered problem of reducing the influence of measurement noise, which to a greater or lesser extent affects all relay autotuning methods, it would be advisable in real-time systems to consider, if only by means of heuristic methods, the estimation of the time delay of higher-order models, of which part is due to dead time and the remainder to the phase lag introduced by high-frequency dynamics. The delay time of the critically damped model, which can be calculated analytically (29), can then be taken as a maximal reference value. Previous studies on model quality assessed on the basis of the step or frequency characteristics would suggest reaching for fuzzy logic when identifying the delay time, at least in the case of insufficient a priori knowledge of the controlled plant.  
This project was financed under the program Regional Initiative of Excellence of the Polish Minister of Science and Higher Education called Regionalna Iniciatywa Doskonalosci w latach 2019-2022 nr projektu 006/RID/2018/19 kwota finansowania 11 870 000 zl.  
Data Availability Statement: Data sharing not applicable.  
The author declares no conflict of interest.