## 4 Case Study: Application for DC Motor Servo Systems  
In this section, to help readers understand how to apply the proposed controller, we show an application of the proposed controller for DC motor servo systems. For precision control of DC motor servo systems, studies including MPC [29], data-driven [30], fuzzy [31], neural network [32], cascade control [33], and the digital twin-based optimization [34] were introduced recently. However, except for MPC, these studies did not consider the state constraints; therefore, we compare MPC and the proposed controller for performance verification. The parameters of the target motor are shown in Table 1. The target motor is a 24 V DC brushed gear motor and its rated torque is 2.94 Nm. Figure 2: Modified plant model with a low-pass filter. Then, the state equation of the target system is  
\[\dot{\mathbf{x}} =\left[ \begin{array}{ccc}-\frac{R_{m}}{L_{m}}&-\frac{K_{b}}{L_{m}} &0\\ \frac{R_{m}}{L_{m}}&-\frac{R_{m}}{L_{m}}&0\\ 0&\kappa_{m}&0\end{array} \right]\mathbf{x}+\left[ \begin{array}{c}\frac{1}{L_ {m}}\\ 0\\ 0\end{array} \right]u \tag{48}\] \[\approx\left[ \begin{array}{ccc}-384.62&-13.85&0\\ 1714.29&-33.33&0\\ 0&0.25&0\end{array} \right]\mathbf{x}+\left[ \begin{array}{c}153.85\\ 0\\ 0\end{array} \right]u,\]  
\[\mathbf{y}=\left[ \begin{array}{ccc}0&0&1\end{array} \right]\mathbf{x}. \tag{49}\]  
where  
\[\mathbf{x}=\left[ \begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\end{array} \right],\]  
\(x_{1}\) is the motor current, \(x_{2}\) is the angular speed of the motor, and \(x_{3}\) is the angular position of the motor. Let \(\beta=1000\), then the modified plant model is  
\[\frac{d}{dt}\left[ \begin{array}{c}\mathbf{x}\\ u\end{array} \right]=\mathbf{A}\left[ \begin{array}{c}\mathbf{x}\\ u\end{array} \right]+\mathbf{B}\overline{u}, \tag{50}\]  
\[\mathbf{y}=\mathbf{C}_{y}\left[ \begin{array}{c}\mathbf{x}\\ u\end{array} \right], \tag{51}\]  
where  
\[\mathbf{A}=\left[ \begin{array}{ccc}-384.62&-13.85&0&153.85\\ 1714.29&-33.33&0&0\\ 0&0.25&0&0\\ 0&0&0&-1000\end{array} \right]\mathbf{B}=\left[ \begin{array}{c}0\\ 0\\ 0\\ 1000\end{array} \right]\mathbf{C}_{y}=\left[ \begin{array}{cc}0&0&1&0\end{array} \right]\]  
Let \(\tau=0.001\), then the parameters for the prediction can be calculated as  
\[\mathbf{A}_{d}=\left[ \begin{array}{cccc}0.67&-0.01&0&0.09\\ 1.41&0.96&0&0.07\\ 0&0&1&0\\ 0&0&0&0.33\end{array} \right]\mathbf{B}_{d}=\left[ \begin{array}{c}0.04\\ 0.04\\ 0\\ 0.67\end{array} \right]\]  
The optimal tracking problem is  
\[\underset{\mathbf{x},u,\overline{u}}{\text{minimize}} J=\frac{1}{2}\int_{0}^{t_{f}}\left(\left[ \begin{array}{c}r-x_{3}\\ u\end{array} \right]^{T}\left[ \begin{array}{cc}100&0\\ 0&1\end{array} \right]\left[ \begin{array}{c}r-x_{3}\\ u\end{array} \right]+\overline{u}^{2}\right)dt\] (52) subject to \[\left\{\begin{array}{c}\mathbf{A}\left[ \begin{array}{c}\mathbf{x}\\ u\end{array} \right]+\mathbf{B}\overline{u}-\left[ \begin{array}{c}\dot{ \mathbf{x}}\\ \dot{u}\end{array} \right]=0\\ \mathbf{C}_{h}\left[ \begin{array}{c}\mathbf{x}\\ u\end{array} \right]+\mathbf{w}\leq 0,\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Unit** & **Value** \\ \hline Rotor inductance (\(L_{m}\)) & H & 0.0065 \\ Armature resistance (\(R_{m}\)) & \(\Omega\) & 2.3 \\ Back EMF constant (\(K_{b}\)) & V-sec/rad & 0.09 \\ Torque constant (\(K_{m}\)) & Nm/A & 0.09 \\ Friction coefficient (\(B_{m}\)) & Nm-sec & 0.00175 \\ Rotor inertia (\(J_{m}\)) & Nm-rad & 0.0000525 \\ Gear ratio (\(\kappa_{m}\)) & & 0.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the target motor. where  
\[\mathbf{C}_{h}=\left[ \begin{array}{cccc}-1&0&0&0\\ 1&0&0&0\\ 0&-1&0&0\\ 0&1&0&0\end{array} \right],\mathbf{w}=\left[ \begin{array}{c}-3\\ -3\\ -50\\ -50\end{array} \right]\]  
In this case, \(\left[ \begin{array}{c}\mathbf{C}_{y}\\ \mathbf{C}_{h}\\ \mathbf{C}_{h}\mathbf{A}_{d}\end{array} \right]\) has full rank, therefore the closed-loop system is stable.