# Design and Optimization of a Neuro-Fuzzy System for the Control of an Electromechanical Plant
## 2 Electromechanical Plant
### Approximate Model  
In some applications, a complete dynamic model of the plant may be impractical due to the difficulty in establishing the values of its parameter (in this case, the moment of inertia and the friction of the mechanical part). An alternative consists of a simplified model identifying its parameters considering the dynamic behavior of the plant [31].  
According to [31], when considering \(L_{a}\approx 0\) (very small) there is a simplified model of the plant; in this way, the diagram in Figure 3 can be represented as shown in Figure 4.  
Organizing the diagram of Figure 4, it is possible to obtain the representation of Figure 5 where \(T_{L}(s)\) can be seen as a system disturbance.  
Equivalently, the diagram shown in Figure 6 is obtained where in the time domain the input \(u(t)=\mathcal{L}^{-1}\{U(s)\}\) corresponds to the voltage; the output is the angular position \(\theta(t)=\mathcal{L}^{-1}\{\Theta(s)\}\) and \(p(t)=\mathcal{L}^{-1}\{P(s)\}\) is the disturbance given by the product  
Figure 4: Block diagram of the electromechanical system considering \(L_{a}\approx 0\).  
Figure 5: Simplified block diagram of the electromechanical system.  
Figure 3: Electromechanical system block diagram.  
\((NR_{a}/K_{t})T_{L}(t)\). In this way, the variable to sense and control corresponds to the angular position \(\theta(t)\).  
On the other hand, considering the location of the poles, the criterion to establish the simplified model is shown in Figure 7 where the pole located in \(s=-a\) corresponding to the mechanical part is the dominant pole (it governs the system dynamics) since \(b\gg a\); therefore, to establish the approximate model, the pole located at \(s=-b\) associated with the electromagnetic part can be neglected [31].  
Equation (13) describes the simplified plant transfer function corresponding to a first-order system and an integrator, where the variable \(s\) is associated with the Laplace transform.  
According to [31], the model of Equation (13) considers the parameters shown in Figure 5, namely \(J_{e}\), \(B_{e}\), \(K_{e}\), \(K_{t}\), \(N\), and \(R_{a}\). In this way, through the data measured from the open-loop plant, the effect of the parameters on the plant operation is being directly considered in the identification process.  
\[G(s)=\frac{K}{s(\tau s+1)} \tag{13}\]  
For this plant, the identification of parameters is carried out considering the average values of the speed and the settling time as shown in [27; 28].  
For the transfer function, in Equation (13) parameter \(K\) corresponds to the steady-state input-output relationship, which for this case is \(K=\Delta\omega/\Delta u\) where \(\omega\) is the angular velocity. Taking a voltage of 12 V, the average time taken by the system to travel an angle of \(\pi/2\) is 0.06 rad/s; therefore, \(K\) parameter value corresponds to 0.005 rad/(V s) [28].  
Parameter \(\tau\) can be determined as one third of the system settling time (considering a band of 5%), which is 1.13 s, hence, \(\tau\) corresponds to 0.38 s [28]. In this way, the plant model is given by Equation (14). It should be noted that the proposed control system is designed to regulate the angular position of the electromechanical system \(\theta(t)\).  
\[G(s)=\frac{\Theta(s)}{U(s)}=\frac{0.005}{s(0.38s+1)} \tag{14}\]