## 8 Verification of the EPS Control Algorithm
**Step 2:** Given a shaped plant \(G_{\mathrm{s}}\) and \(A\), \(B\), \(C\), \(D\) represent the shaped plant in the state-space form. To determine \(\gamma_{\mathrm{min}}\), there is a unique method, as follows [33]. \[\gamma_{\mathrm{min}}=\left(1+\lambda_{\mathrm{max}}(Z_{\mathrm{s}}X_{\mathrm{ s}})\right)^{1/2} \tag{38}\]  
where \(\lambda_{\mathrm{max}}\) is the maximum eigenvalue and \(X_{\mathrm{s}}\) and \(Z_{\mathrm{s}}\) are the solutions of two Riccati, as:  
\[\left(A-BS^{-1}D^{T}C\right)Z_{\mathrm{s}}+Z_{\mathrm{s}}\left(A-BS^{-1}D^{T}C \right)^{T}-Z_{\mathrm{s}}C^{T}R^{-1}CZ+BS^{-1}B^{T}=0 \tag{39}\]  
\[\left(A-BS^{-1}D^{T}C\right)^{T}X_{\mathrm{s}}+X_{\mathrm{s}}\left(A-BS^{-1}D^{ T}C\right)-X_{\mathrm{s}}BS^{-1}B^{T}X+C^{T}R^{-1}C=0 \tag{40}\]  
where  
\[S=I+D^{T}D\]Figure 27:0  
To ensure the robust stability of the nominal plant, the weighting function is selected so that \(1<\gamma_{\text{min}}<5\)[33]. If \(\gamma_{\text{min}}\) is too large, then return to step 1 and change \(W_{1},\ W_{2}\). **Step 3:** Choose \(\gamma>\gamma_{\text{min}}\); the \(\overset{\wedge}{K_{\infty}}\) controller must satisfy the following equation [13]:  
\[\|\left[ \begin{array}{c}\overset{\wedge}{K_{\infty}}\\ I\end{array} \right](I-G_{\text{s}}\overset{\wedge}{K_{\infty}}^{-1}M_{\text{s}} ^{-1}\|\leq\gamma \tag{41}\]  
**Step 4:** The controller is synthesized as follows:  
\[\overset{\wedge}{K}=W_{1}\overset{\wedge}{K_{\infty}}W_{2}=\frac{372.3s^{4}+67 14s^{3}+67960s^{2}+55790s+11950}{s^{5}+40s^{4}+709.3s^{3}+6310s^{2}+2648s+2.641} \tag{42}\]  
Moreover, we established the control algorithms of the EPS based on MATLAB/Simulink and the control panel of Control Desk. The results of the reference assist torque step response with an amplitude of 1 Nm are shown in Figure 29. The obtained values for rise time \(t_{\text{ris}}\), maximum overshoot \(o_{\text{max}}\), and settling time \(t_{\text{set}}\) enable an objective comparison of the different controllers in Table 11.  
Figure 29 shows a comparison of the assist torque response under the PID controller and loop-shaping controller. The results obtained with the proposed loop-shaping controller are very good, because the settling time and the overshoot time are very small. Although the PI controller shows a fast response, its behavior is not satisfied, because it oscillates strongly. Consequently, the proposed loop-shaping controller reduces the disturbances for the EPS system, and it provides a good driver feel during the steering process. Figure 30 is an actual assist torque at 40 km/h speed under the PID controller. From the figure it can be seen that the actual torque tracks the desired torque, but the assist torque characteristic does not have enough stability. Its behavior is not satisfied, because it oscillates strongly. Figure 31 is an actual assist torque at 20 km/h speed under the loop-shaping controller. It shows that the actual assist torque is good tracking the desired assist torque. Furthermore, the assist torque characteristic is high stability without overshoot. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Controller** & \(\mathbf{O}_{\text{max}}\) **(\%)** & \(\mathbf{t}_{\text{ris}}\) **(s)** & \(\mathbf{t}_{\text{set}}\) **(s)** \\ \hline
**PID controller** & 23.4 & 0.056 & 0.178 \\
**Loop-shaping controller** & 8.7 & 0.064 & 0.075 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Results of the 1 Nm reference assist torque response. Figure 29: Comparison of the assist torque response using the PID controller and loop-shaping controller. Figure 32 shows actual assist characteristic curve under the PID control algorithm with different velocities. Although the actual assist torque characteristic tracks the desired assist torque, the desired assist torque characteristic oscillates strongly. Figure 33 shows actual assist characteristic curve using the loop-shaping controller with different velocities.