# Online SOPDT Model Identification Method Using a Relay
## 1 Introduction  
Models play a very important role in automation. A model should reflect the static and dynamic properties of a plant as closely as possible. They can be directly used in many control structures, such as internal model control (IMC), model following control (MFC), Smith's predictor, or other time delay compensators. In the process of PI/PID controller tuning, a model can provide a simplified representation of the plant. In time-varying systems, model identification carried out online allows controller parameters to be adapted according to changes in process dynamics. New models of programmable logic controllers (PLCs) have an \"autotuning\" mode, which allows the controller settings to be adjusted to provide a sufficiently large margin of stability in the control system. However, a much greater opportunity in the adaptation process is provided by model-based identification of the control process. Certain tuning methods are based on model parameters of a predetermined class. Due to the needs of PI/PID controller tuning, the assumed class is often reduced to first- or second-order models [1]. Many industrial time-delayed plants are modeled in terms of the following:  
1. First-order inertia plus dead time (FOPDT);
2. An integrating model with time delay;
3. Second-order inertia plus dead time. For many years, inertial systems were estimated using a first-order model with time delay, but in many cases, these models were not adequate for the properties of the plant, which could lead to lower performance of the control system. For some PI and PID designs and other more sophisticated model-based controllers, an SOPDT model may be more appropriate [2]. SOPDT models, which can be described by the transfer function  
\[G_{m}(s)=\frac{Ke^{-\theta s}}{T^{2}s^{2}+2\xi Ts+1} \tag{1}\]  
can include underdamped, critically damped, and oerdamped dynamics. If the damping factor \(\xi\) is less than one, the model is an underdamped model; if \(\xi=1\), then it is acritically damped model; and if the coefficient \(\xi\) is greater than one, it is an overdamped SOPDT model. In this work, we limited our considerations to overdamped systems and, in particular, to those for which the transfer function can also be represented as  
\[G_{m}(s)=\frac{Ke^{-\theta s}}{(1+T_{1}s)(1+T_{2}s)} \tag{2}\]  
where \(K\) is the steady-state gain, \(T_{1}\) and \(T_{2}\) are time constants, and \(\theta\) is the time delay. Controller tuning can be carried out by staff, but the process can also be automated. To represent the properties of the control process in a more simple manner, several identification methods were developed. A very useful tool for such controller tuning based on models of the processes is the relay method formulated by [PERSON] and [PERSON] [3], which consists of exciting ultimate oscillations of limited amplitude in a closed system by means of a relay. It can be used online without having to disconnect the control system. The parameters of a predetermined model transfer function can be obtained on the basis of the limit cycle parameters using the describing function method. If the steady-state gain is known, one identification test that determines the coordinates of a critical point of the Nyquist diagram allows for further model parameters to be obtained, although [PERSON] and [PERSON] [4] presented a method in which an FOPDT model was determined using a system of three equations. [PERSON] derived complex analytical expressions from the half-cycle analysis of the relay response curve [5]. However, the determination of an overdamped SOPDT model is much more difficult, as it requires the determination of four parameters: steady-state gain \(K\), delay time, and time constants \(T_{1}\) and \(T_{2}\). The algorithms used to determine overdamped models are much more complicated for this reason. Such a model can be constructed via various methods, such as the one invented by [PERSON] and [PERSON] [6] on the basis of the step response analysis.