# Low-Cost Implementation of an Adaptive Neural Network Controller for a Drive with an Elastic Shaft
## 3 Simulation Results  
The numerical tests were conducted to determine the behavior of the proposed control structure. The calculations were carried out with a calculation step of 0.1 ms. The initial weight values were randomly assigned in the range of (0, 0.1). The adaptation of widths started with the value of \(\sigma_{i}=0.5\). The initial center matrix--\(\mathbf{\mu}\)--was established empirically (the arrangement is symmetrical). There are five radial neurons in the hidden layer of the RBF neural network. The input vector of the network consists of two signals: the value of the speed error in the current and the previous iteration. The behavior of the system (the rotational speed of the load and the electromagnetic and torsional torques) that works under nominal conditions is presented in Figures 3 and 4. The speed of the load machine follows the reference value \(\omega_{refm}\) with high precision. The response to the load disturbance is dynamic and does not generate significant oscillations. The control system operates in a stable manner. Figure 3: Operation of the drive under nominal conditions—rotational speed of the load machine. To additionally test the adaptation efficiency of the proposed control scheme, a change in drive parameters was introduced. In applications such as paper winding machines, the inertia of the load machine increases over time. Therefore, the operation of the drive with an increased time constant \(T_{2}\) was verified. The simulation results obtained for this case are presented in Figure 5. The adaptation of all parameters can be seen. With each reversion, the overshoot is reduced, resulting in a smoother transition between the steady states. The changes in the adapted parameters: weights, centers, and widths are presented in Figure 5(a), b and c, respectively. The most significant changes can be observed in the transients of weights and widths of the radial functions. The presented centers are evenly distributed, fulfilling the role of global mapping of the whole data space. Changes in all parameters of the network were also individually inspected. The impact of the initial values of weights, centers, widths, and also values of learning rates (\(\eta\) and \(\gamma\)) is shown. The load machine speed transients for different sets of initial weights are presented in Figure 7. After the start of the drive, for most of the sets tested, significant oscillations can be observed. Later, they are dampened because of the adaptation process. Nevertheless, the oscillations of the load speed must be minimized. Therefore, optimization of these parameters is highly recommended. Figure 4: Operation of the drive under nominal conditions—electromagnetic (\(m_{e}\)) and torsional (\(m_{s}\)) torques. Figure 5: Operation of the drive with an increase in the inertia of the load machine (\(T_{2}=2T_{2n}\)) (**a**), with a zoom (**b**). The effect of the change in the initial widths of the radial functions is presented in Figure 8. The widths were selected at random in the range of (0.25, 0.75). No major changes in the load speed transients can be observed. After a few reversions, the runs are almost identical. Figure 6: Change in the adaptive parameters of the RBF neural network: weights (**a**), centers (**b**), widths (**c**). Figure 7: Impact of initial weights randomization: complete simulation (**a**), initial part of the simulations (**b**). The load speeds with different values of center of the radial functions can be seen in Figure 9. These initial parameters also do not introduce any changes that affect the operation of the drive for more than a few first reversions. The learning algorithm mitigates the inadequacy in center selection in the later stages of the simulation. The impact of learning rates associated with weights (\(\eta\)), center and widths (\(\gamma\)) is presented in Figures 10 and 11, respectively. The lower value of \(\eta\) results in slower adaptation, which results in a delayed response to load switching.