# Online SOPDT Model Identification Method Using a Relay
## 3 Simulation Results and Discussion
### Identification of the Second-Order Time-Delayed Plant
Figure 4: Step characteristics of the plant \(G_{\mathrm{ob1}}\) and its models. \(G_{m22}\)—overdamped SOPDT model; \(G_{m}\)—SOPDT model with parameters determined during a single relay test; \(G_{2}\)—critically damped SOPDT model. In the considered example, the model obtained only on the basis of the values measured using only the standard identification tests also better reflected the properties of the plant than the critically damped SOPDT. It should be emphasized that the value of \(\theta_{\rm c}\) is a limiting value for the time delay, i.e., when the measured value of the delay was greater than \(\theta_{\rm c}\), we assumed that \(\theta=\theta_{\rm c}\) for the model, as determined using the described method. Example 2  
The transfer function of the plant subjected to the identification test was  
\[G_{2}(s)=\frac{e^{-2s}}{(10s+1)(s+1)} \tag{30}\]  
The steady-state gain of the model was measured in the pretest as 0.996 and the delay time as equal to 2.01 s. When the ultimate gain \(K_{u}=7.07275\) was reached, the ultimate frequency \(\omega_{u}=\omega_{1}=0.5985\), and, after the introduction of the ideal integral term, the frequency of the sustained oscillation changed the value to \(\omega_{2}=0.1742\). From the measurement of the limit cycle using a saturation relay with a gain of \(k=1.16\cdot K_{\rm{ue}}\), the critical frequencies \(\omega_{1}=0.599\) and \(\omega_{2}=0.176\) were obtained, which made it possible to determine the time constants \(T_{1}=0.992\) s and \(T_{2}=9.775\) s.  
Example 3: SOPDT model with uncertainty  
The examples presented above concern the creation of second-order models of objects of the same order, where the quality of the model is only related to the accuracy with which its parameters are determined. The next examples deal with higher-order plants, where the problem of modeling the influence of poles further from the imaginary axis arises. Now let us consider a high-order inertial plant, where a second-order model is not able to fully represent its properties. Its transfer function is given as  
\[G_{3}(s)=\frac{10\;e^{-4s}}{\left(2s+1\right)\left(4s+1\right)\;\left(s+3 \right)^{3}} \tag{31}\]  
From the measurement of the limit cycle using a saturation relay with a gain of \(k=1.16\cdot K_{\rm{ue}}\), the critical frequencies were \(\omega_{1}=0.327\) rad/s and \(\omega_{2}=0.148\) rad/s. \[G_{m3}(s)=\frac{0.37e^{-4.48s}}{11.4s^{2}+6.42s+1} \tag{32}\]  
The following measurement results were obtained by simulating the identification process of this plant: the time delay was \(\theta=4.48\) s, the increase in the controlled variable at \(B=0.5\) after 40 s was \(\Delta y=0.185\), oscillation period was \(T_{\infty 1}=19.2\) s, and the oscillation period after adding the integrator was \(T_{\infty 2}=42.3\) s. On the basis of the measured values, the steady-state gain \(K=0.37\) and the product \(a\) and sum \(b\) of the time constants were calculated, which were equal to \(a=11.4\) and \(b=6.42\). Figure 5 shows the step response of the plant and the model created via the proposed method. \[G_{m22}(s)=\frac{1.995\:e^{-3.01s}}{(10.954s+1)(1.907s+1)} \tag{25}\]0 Step responses of the plant from example 3 and its model. Example 4: Plant with a low \(\theta/T_{dom}\) ratio  
In the previous example, \(\theta/T_{dom}\), which captures the ratio of the dead time to the dominant time constant, was large at 1. Let us now consider the case when it is small.