# Zero Average Dynamic Controller for Speed Control of DC Motor
## 3 Results  
In this section, the performance of the buck-motor system using the ZAD technique is discussed. The parameters of the converter, motor, and control systems are shown in Table 1. The experiment considered \(W_{m}\) with 28 bits; \(i_{a}\), \(v_{c}\), \(i_{L}\) with 12 bits; duty cycle with 10 bits. The experimental test of the buck-motor system with the ZAD control strategy is shown in Figure 6. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \(B\) & Viscous friction coefficient & 0.000138 (Nm/rad/s) \\ \(C\) & Capacitance & 46.27 \(\mu\)F \\ \(d\) & Duty cycle & 10 bits \\ \(E\) & Input voltage & 40.086 V \\ \(Fc\) & Switching frequency & 6 kHz \\ \(Fs\) & Sampling frequency & 6 kHz \\ \(i_{a}\) & Motor armature current [A] & 12 bits \\ \(i_{L}\) & Inductor current [A] & 12 bits \\ \(I_{eq}\) & Moment of inertia & 0.000115 (kg\(\cdot\)m\({}^{2}\)) \\ \(r_{s}\) & Source internal resistance & 0.84 \(\Omega\) \\ \(L\) & Inductance & 2.473 mH \\ \(r_{L}\) & Internal inductor resistance & 1.695 \(\Omega\) \\ \(R_{a}\) & Armature resistance & 2.7289 \(\Omega\) \\ \(L_{a}\) & Armature inductance & 1.17 mH \\ \(k_{l}\) & DC motor torque constant & 0.0663 (Nm/A) \\ \(k_{e}\) & DC motor voltage constant & 0.0663 (V/rad/s) \\ \(Tf_{ric}\) & Friction torque & 0.0284 (Nm) \\ \(T_{L}\) & Load torque & Variable (Nm) \\ \(W_{mref}\) & Speed reference & Variable (rad/s) \\ \(1T_{P}\) & Delay period & 166.6 us \\ \(K_{S1},K_{S2},K_{S3}\) & Control parameters & Variables \\ \(W_{m}\) & Motor speed [rad/s] & 28 bits \\ \(V_{fd}\) & Forward voltage of the diode & 1.1 V \\ \(V_{a}\) & Voltage applied to the motor & Variable (V) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the buck-motor system and the controller. The performance of the DC motor controlled with the ZAD strategy is shown in Figure 7a. Simulation is performed in discrete time with parameters \(K_{S1}=K_{S2}=2\), \(K_{S3}=40\), and neither time delay nor quantization effects. The results are discussed in what follows. To start with, the reference signal was 0 rad/s from 0 s to 0.2 s. Then, the reference signal was changed to 150 rad/s at 0.2 s, resulting in a settling time of 0.05 s with a maximum overshoot of 2.36%. Next, the reference was changed to 300 rad/s at 0.4 s, resulting in a settling time of 0.07 s with a maximum overshoot of 1.99%. The percentage of output tracking error is shown in Figure 7b. There was an overshoot with an error of 2.35% for a reference of 150 rad/s, and a maximum error of 0.66% for a reference of 300 rad/s. Therefore, the steady-state error was less than 0.48% for both cases. The evolution of the duty cycle is shown in Figure 7c. The duty cycle did not saturate and it remained in the steady-state operation at values between 0 and 1. That means that the system presented a fixed switching frequency and no saturation in the duty cycle. Moreover, the system was quite stable with neither subharmonics and unwanted noises, nor chattering. The changes in the trajectory of the input voltage \(V_{a}\) (Figure 7d) were highly related to those of the duty cycle (Figure 7c), and \(V_{a}\) exhibited no abrupt changes during steady-state operation, which is convenient for optimal motor performance. In [19], the  
Figure 6: Experimental platform for validating the ZAD control. establishment time obtained was 0.1473 s, while the one in this paper was 0.07 s, so the proposed method is two times faster while keeping a low steady-state error. The numerical and experimental results of the buck-motor system controlled with ZAD, using the parameters of Table 1, are shown in Figures 8 and 9. The simulation was performed in discrete time with \(K_{S1}=K_{S2}=2\), \(K_{S3}=40\), delay time (166.6 \(\upmu\)s), and quantization effects. The ZAD technique is capable of controlling the speed of the DC motor \(W_{m}\), both numerically (Figure 8a) and experimentally (Figure 8b). In addition, the regulation error was less than 2% (Figure 8c,d).