## 4 Simulation Experiment
### Experiment 1  
To enhance the applicability of the proposed method, we extended iFFDL-MFA-PID to a MIMO system and verified its effectiveness by simulating a typical controlled object. We compared and simulated the iFFDL-MFA-PID and FFDL-MFAC algorithms using time-varying signals. The first group of simulation objects consisted of a system with three inputs and three outputs, as follows:  
\[\left\{ \begin{array}{l}I_{1}(k+1)=1.5\times I_{1}(k)+0.12\times{I_{1}}^{2}(k- 1)+0.7\times u_{1}(k)\\ +0.3\times u_{2}(k)+0.3\times u_{3}(k)+0.5\times d(k)\\ I_{2}(k+1)=1.6\times I_{2}(k)+0.14\times I_{2}^{2}(k-1)+0.4\times u_{1}(k)\\ +0.6\times u_{2}(k)+0.3\times u_{3}(k)+0.5\times d(k)\\ I_{3}(k+1)=1.6\times I_{3}(k)+0.15\times{I_{3}}^{2}(k-1)+0.1\times u_{1}(k)\\ +0.7\times u_{2}(k)+0.6\times u_{3}(k)+0.5\times d(k)\end{array} \right. \tag{40}\]  
where \(d(k)\) represents a white-noise signal with a mean of zero and a variance of 0.05. The predetermined trajectory settings were as follows:  
\[I_{d}=\left\{ \begin{array}{l}\mathrm{lins}(0,50,100)\times 5/9,0<k\leq 100\\ \mathrm{lins}(250,250,50)/9,100<k\leq 150\\ \mathrm{lins}(250,290,50)/9,150<k\leq 200\\ \mathrm{lins}(290,290,600)/9,200<k\leq 800\\ \mathrm{lins}(290,0,200)/9,801<k\leq 1000\end{array} \right. \tag{41}\]  
The controller parameters were set to \(\hat{\boldsymbol{\Lambda}}(1)=\hat{\boldsymbol{\Lambda}}(2)=\mathrm{diag} \{0.55,0.6,0.55\}\), \(\eta=0.75\), \(\mu=1\), \(\rho=0.15\), \(\partial=0.2\), \(k_{1}=0.45\), \(k_{2}=0.7\), \(k_{3}=0.05\), \(L_{I}=1\), and \(L_{u}=1\). Figure 2 depicts the system output under the influence of the iFFDL-MFA-PID and FFDL-MFAC methods, whereas Figure 3 shows the corresponding output error graphs. Table 1 presents the corresponding control performance indicators. By zooming in on the details in the magnified graphs in Figure 2, it can be observed that at the 100 th and 200 th sampling points, the FFDL-MFAC method experienced phase delay and poor disturbance rejection capabilities, resulting in delayed tracking phenomena and significant oscillations. This significantly diminished the rapid response characteristics of the model-free adaptive control algorithm. On the other hand, the iFFDL-MFA-PID method exhibited no phase delays, promptly providing feedback on the output signal. It maintained the fast response characteristic of the model-free adaptive control algorithm while eliminating disturbances. At the 200 th sampling point, even when the reference signal changed, the iFFDL-MFA-PID method continued to maintain its convergence, further demonstrating the adaptability of the proposed algorithm. Additionally, by examining the local magnified graphs in the output curve in Figure 2 and the time indicators in Table 1, it can be observed that the iFFDL-MFA-PID method achieved stability within 15 s under varying reference outputs, whereas MFAC required more than 27 s. Furthermore, the iFFDL-MFA-PID method exhibited approximately half the rise time of the MFAC. Moreover, from the error change curve, it is evident that the iFFDL-MFA-PID method not only possessed fast tracking characteristics but also effectively suppressed the influence of external disturbances, consistently outperforming traditional MFAC throughout the tracking process. The error-related indicators shown in Figure 3 and Table 1 indicate that regardless of whether the reference signal was constant or changing, the tracking performance of the proposed iFFDL-MFA-PID method was superior to that of traditional MFAC. The MAE and MSE values of the iFFDL-MFA-PID method were only \(1.15\times 10^{-2}\) and \(2.32\times 10^{-4}\), respectively, whereas the MAE and MSE values of traditional MFAC were \(3.41\times 10^{-2}\) and \(7.29\times 10^{-4}\), respectively. In conclusion, the iFFDL-MFA-PID method achieves nonlinear compensation for unknown disturbances such as measurement noise without relying on a model.