## 3 Design of Parameter Adaptive Controllers
### Simulation with LQ SOF and Parameter Adaptive Controllers on CarSim  
Two LQSOFCs, LQSOF1 and LQSOF2, and two PACs, PAC1 and PAC2, were simulated on CarSim connected to MATLAB/Simulink. The calculation procedure on \(v_{zc}\) and \(\omega_{yc}\) given in Figure 3 was implemented in MATLAB/Simulink. When implementing it in MATLAB/Simulink, band-limited white noises were added into the sensor signals measured with \(Az_{S}\), \(Az_{S}\), \(Az_{F}\), \(Azw_{f}\) and \(Azw_{r}\). Figures 6 and 7 show the simulation responses of four controllers on the LHSB and SWR, respectively. Tables 2 and 3 show the maximums of the absolute values of \(a_{zc}\) and \(\omega_{yc}\) (MAVHA and MAVPR) over the simulation period from 1 s to 5 s and from 5 s to 10 s on the LHSB and SWR, respectively. In Tables 2 and 3, the numbers in the parenthesis stand for the percentage reduction to the no control case. An increased percentage reduction corresponds to improved performance. Figure 6: Simulation responses of LQSOFCs and PACs on the LHSB. **(a)**\(a_{zc}\); **(b)**\(\omega_{yc}\); **(c)**\(u_{f}\) and \(u_{r}\). As shown in Figures 6 and 7, LQSOF1 shows poor performance in respect of RC and MS. This was not expected from the frequency responses given in Figure 5. This is caused by the fact that there are nonlinearities in springs and dampers in \(S_{f}\) and \(S_{r}\), and that the magnitudes of signals used for LQSOF1 in CarSim are much smaller than linear model, i.e., SSEQN, when nonlinear spring and dampers are used. For those reasons, the magnitude of the control input for LQSOF1 is much smaller than those of LQSOF2, PAC1 and PAC2. On the other hand, LQSOF2, PAC1 and PAC2 show good performance in respect of RC and MS. Especially, PACs proposed in this paper show much better performance than LQSOFCs. The difference between them originated from the fact that LQSOFCs have  
Figure 7: Simulation responses of LQSOFCs and PACs on the SWR. (**a**) \(a_{\infty}\); (**b**) \(\omega_{\text{yc}}\); (**c**) \(u_{f}\) and \(u_{r}\). constant gains, while PACs adaptively update the gains of SOFCs. From the simulation responses given in Figures 6 and 7, PAC2 is the best controller in respect of RC and MS.  
As presented in Table 2, PAC1 and PAC2 provided the reductions of 61% and 61% in relation to \(a_{zc}\), and the reductions of 85% and 84% in relation to \(\omega_{yc}\), on LHSB and SWR, respectively. Moreover, PAC1 and PAC2 provided the reductions of 63% and 57% in relation to \(a_{zc}\), and the reductions of 90% and 89% in relation to \(\omega_{yc}\), on LHSB and SWR as presented in Table 3, respectively. These results mean that the parameter adaptive controllers, PAC1 and PAC2, proposed in this paper are quite effective for RC enhancement and MS reduction. In simulation, the matrices, **X** and **\(\boldsymbol{\Sigma}\)**, of EKF were set to _diag_(\(\alpha\),\(\alpha\),\(\alpha\),\(\alpha\)) or _diag_(\(\alpha\),\(\alpha\)) and _diag_(\(\beta\),\(\beta\)) where \(\alpha\) and \(\beta\) were set to \(10^{2}\) and \(1\), respectively. If \(\alpha\) and \(\beta\) are set to \(0.1\) and \(1\), it was identified from the simulation result that there are little differences between the previous and current sets of \(\alpha\) and \(\beta\). This is caused by the fact that the system matrix of EKF is the identity, as shown in (34). Figure 8 shows the simulation responses of LQSOFCs and PACs on the BSSR. Table 4 shows the MAVHAs and MAVPRs over the simulation period from 1 s to 7 s. Figure 9 shows the frequency responses of LQSOFCs and PACs drawn from the simulation results on the BSSR. Figure 8: Simulation responses of LQSOFCs and PACs on the BSSR. (**a**) \(a_{zc}\); (**b**) \(\omega_{yc}\); (**c**) \(u_{f}\) and \(u_{r}\). Figure 9: Frequency responses of LQSOFCs and PACs on the BSSR. (**a**) \(a_{zc}\); (**b**) \(\omega_{yc}\). As shown in Figure 8 and Table 4, PAC1 shows the best performance on BSSR. Moreover, PAC1 is better than PAC2 because the former uses four signals and the latter does two signals for control.