## 3 Description of the Modified Method
### Sopdt under PID Control  
Carrying out a similar derivation as in Section 3.2, after determining the simplified open-loop transfer function (27), we derive the relation for the closed-loop system with unitary feedback:  
\[G_{c}(s)=\frac{KK_{P}\big{(}T_{i}T_{d}s^{2}+T_{i}s+1\big{)}\cdot e^{-\theta s}}{T _{i}s(T_{1}s+1)(T_{2}s+1)+KK_{P}(T_{i}T_{d}s^{2}+T_{i}s+1)\cdot e^{-\theta s}} \tag{49}\]  
If we assume that the integral time constant of the PID controller is equal to the sum of the time constants of the model  
\[T_{i}=T_{1}+T_{2} \tag{50}\]  
and derivative time constant has a value  
\[T_{d}=\frac{T_{1}T_{2}}{T_{1}+T_{2}} \tag{51}\]  
then after simplifying (49), we derive  
\[G_{c}(s)=\frac{KK_{P}\cdot e^{-\theta s}}{(T_{1}+T_{2})s+KK_{P}\cdot e^{- \theta s}} \tag{52}\]  
In order for the transfer function of the closed-loop system (52) to be equal to the one desired (30), in addition to the assumed conditions (50) and (51), the proportional gain of the controller should be equal to  
\[K_{P}=\frac{T_{1}+T_{2}}{Ke\theta} \tag{53}\]  
If \(K_{p}\), similarly considered in Section Minimization of Settling Time, will be increased to a value  
\[K_{P1}=\frac{1.25(T_{1}+T_{2})}{Ke\theta} \tag{54}\]  
then the overshoot \(\chi\) will remain smaller than the width of the error band equal to 2% of the set-point change, and the settling time will be shorter than it is in a critically damped system.  
Control of a Critical Damped Sopdt Object  
There are quite a few methods in the literature for determining the Sopdt model (26) [24; 25; 26; 27], but generally, these are very complicated procedures. It is much simpler to create a critical damped Sopdt model, which can be representative of many inertial plants with fairly high accuracy. A model of this class described by the transfer function  
\[G_{p}(s)=\frac{Ke^{-\theta s}}{(1+T_{m2}s)^{2}} \tag{55}\]  
can be obtained using the Astrom-Hagglund relay method [4].  
The control objective given by the desired closed-loop transfer function (30) remains unchanged; that is, by set-point tracking, the non-oscillatory response of the system should be ensured in the shortest possible time. If a PID controller is included in the loop, the closed-loop transfer function will then be equal to  
\[G(s)=\frac{K_{ob}\cdot K_{p}(T_{i}T_{d}s^{2}+T_{i}s+1)\cdot e^{-T_{d}s^{2}}}{ T_{i}s\cdot(1+T_{m2}s)^{2}+K_{ob}\cdot K_{p}(T_{i}T_{d}s^{2}+T_{i}s+1)\cdot e ^{-T_{d}s}} \tag{56}\]  
If the priority of the designed control system is the set-point tracking, then we can tune the system so that the poles compensate the zeros introduced by the controller intothe system. If we assume that the integral and the derivative time constants will be equal to, respectively,  
\[T_{i}=2T_{m2}\ \ \ T_{d}=0.5T_{m2} \tag{57}\]  
then the closed-loop transfer function will take the following form:  
\[G(s)=\frac{K_{P}K\cdot e^{-\theta s}}{2T_{m2}s^{+}+K_{P}K\cdot e^{-\theta s}} \tag{58}\]  
For a closed-loop system to be critically damped, the proportional gain of the PID controller should be equal to  
\[K_{P}=\frac{2\ T_{m2}}{e\cdot\theta K} \tag{59}\]  
If while maintaining the condition of overshoot of the response, the settling time was to be the shortest possible, the proportional gain can be increased to the value of  
\[K_{P2}=\frac{2.5\ T_{m2}}{e\cdot\theta K} \tag{60}\]