# A Novel Robust Control System Design and Its Application to Servo Motor Drive
## 6 Simulation Results and Discussion  
### Simulations for Systems under the Conditions of Nominal, Load, and Uncertainty  
MATLAB/Simulink was used to create a position control system, as shown in Figure 4. The motor parameters of inertia, viscous friction, and torque constant in the nominal condition are respectively \(J_{m}=0.002\) Nm/s\({}^{2}\), \(B_{m}=0.0015\) Nm/s, and \(K_{t}=2.25\) Nm/A [34]. The uncertainties of \(J_{m}\) and \(B_{m}\) are in the range [\(-50\%\), \(+50\%\)] of their nominal values and [\(-20\%\), \(0\)] for the torque constant. The details are shown in Table 1. An external load of 2 Nm is also assumed to check the robustness of the proposed controller. Figure 4: The simulation system. The objective of the simulation system is to design the servo system with the poles at \(s_{1,2}=-10\pm 10i\) and \(s_{3}=-100\). The state feedback control gain can be determined as \(\mathbf{k}_{1}^{\mathrm{T}}=-[-17.7778\quad-1.9556\quad-0.1060]\), and the linear control law is:  
\[u_{L}=\mathbf{k}_{1}^{\mathrm{T}}\mathbf{z}=-[-17.7778\quad-1.9556\quad-0.1060] \mathbf{z} \tag{39}\]  
With the defined ranges of uncertainties, the switching control law is set as:  
\[-q\mathrm{sgn}(\sigma)=-3\frac{\sigma}{|\sigma|+0.001} \tag{40}\]  
Figure 4 shows the simulated system where the reference input is set as \([1,-2,2,-2]\) rad with a sample time of 2.8 s. External load is set as [0, 2, 0, 2, 0, 0] Nm with a sample time of 1.5 s. The signals of reference input and external load can be seen in Figures 5-7. Those settings are used to check the performance from the frustrations under the transient and steady-state conditions. Three control structures are compared: a nominal system (ideal system) without load and uncertainty included, a controlled system using state feedback and SMC, and a controlled system with only state feedback. Three kinds of control situations are set for comparison: the uncertain condition, the loaded condition, and the condition with both the uncertainty and load. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter Name** & **Value** & **Variation Range** \\ \hline Inertia \(J_{m}\) (Nm/s\({}^{2}\)) & 0.002 & [\(-\)0.001, +0.001] \\ Viscous friction \(B_{m}\) (Nm/s) & 0.0015 & [\(-\)0.00075, 0.00075] \\ Torque constant \(K_{t}\) (Nm/A) & 2.25 & [\(-\)0.45, 0] \\ Feedback gain \(\mathbf{k}_{1}\) & [\(-\)17.7778, \(-\)1.9556, \(-\)0.1060] & \\ \(q\) & 3 & \\ Position/speed loop sampling & 2 kHz & \\ frequency & 2 Nm & \\ External load & 2 Nm & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the simulated system. Figure 5: The system’s responses with parameter uncertainty: (**a**) position, (**b**) velocity, and (**c**) control law. at the transition of position command change. The system's response with SMC is closer to the ideal system than the system with only the integral state feedback. Figure 5b is the speed responses, and Figure 5c is the control law. Since the system states are controlled on the sliding surface with the control law (40), it shows no chattering on the speed and position response. In the simulated system, the default motor parameters determine the feedback gain \(\mathbf{k}_{1}\). Thus, the uncertainties of inertia, viscous friction, and torque constant lead to the system's poles being relocated to the positions [\(-10.8087\)+\(11.5237\)i, \(-10.8087\)+\(11.5237\)i, \(-42.7344\)]. They differ from the original for the dominant and non-dominant poles, they show a smaller damping ratio than the ideal condition, and the non-dominant pole is closer to the complex dominant poles. Next, the load condition is used to check the performance. A load of 2 Nm is applied to the system at 1.5 s and 4.5 s and released at 3 s and 6 s as the trace purple one shown in Figure 6a-c.