## 4 Simulation Experiment
### Experiment 2: Simulation of Longitudinal Dynamic Control of Electric Forklift
\[A(z^{-1})I(k)=B(z^{-1})u(k-1)+\frac{C(z^{-1})V(k)}{1-z^{-1}} \tag{47}\]  
where \(I(k)\) and \(u(k)\) are the speed and control force of the electric forklift, respectively, and the corresponding system parameters can be defined as  
\[\left\{ \begin{array}{l}A(z^{-1})=1+a_{0}z^{-1}+\cdots+a_{n_{\rm e}}z^{-n_{ \rm e}}\\ B(z^{-1})=b_{0}+b_{1}z^{-1}+\cdots+b_{n_{\rm e}}z^{-n_{\rm b}}\\ C(z^{-1})=1+c_{1}z^{-1}+\cdots+c_{n_{\rm e}}z^{-n_{\rm c}}\end{array} \right. \tag{48}\]  
Figure 6: Diagram showing working the principle of the electric balanced forklift. We rewrite (47) in the following form  
\[ \begin{split}\Delta I(k)=[1-A(z^{-1})]\Delta I(k)+B(z^{-1})\Delta u(k -1)+V(k)\\ =\mathbf{M}^{\mathrm{T}}(k)\mathbf{Y}(k)+V(k)\end{split}  \tag{49}\]  
where \(\mathbf{M}(k)\) and \(\mathbf{Y}(k)\) are the data vectors and model parameters, respectively, defined as  
\[\left\{ \begin{array}{l}\mathbf{M}(k)=[-\Delta I(k-1),\cdots,-\Delta I(k-n_{a }),\Delta u(k-1),\cdots,\Delta u(k-n_{b}-1)]^{\mathrm{T}}\\ \mathbf{Y}(k)=[a_{1},\cdots,a_{n_{a}},b_{0},\cdots,b_{n_{b}}]^{\mathrm{T}} \end{array} \right. \tag{50}\]  
The parameter vector \(\mathbf{Y}(k)\) to be estimated is calculated using the following recursive least-squares method with a forgetting factor [38]. \[\left\{ \begin{array}{l}\mathbf{\hat{Y}}(k)=\mathbf{\hat{Y}}(k-1)+\mathbf{K} (k)[\Delta I(k)-\mathbf{M}^{\mathrm{T}}(k)\mathbf{\hat{Y}}(k-1)]\\ \mathbf{K}(k)=\frac{\mathbf{P}(k-1)\mathbf{M}(k-1)}{\mathbf{M}+\mathbf{M}^{ \mathrm{T}}(k)\mathbf{P}(k-1)\mathbf{M}(k)}\\ \mathbf{P}(k)=\frac{1}{\lambda}\left[\mathbf{I}-K(k)\mathbf{M}^{\mathrm{T}}(k )\right]\mathbf{P}(k-1)\end{array} \right. \tag{51}\]  
where \(\mathbf{\hat{Y}}(k)\) is the estimated value of the model parameters; \(\mathbf{K}(k)\) and \(\mathbf{P}(k)\) are the parameter matrices; \(\mathbf{P}(0)=(10^{4}-10^{10})\mathbf{I}\); and \(\lambda\) is the forgetting factor, with values ranging from 0.9 to 1. The four coefficients in (50) calculated using the recursive least-squares method were input into (47) to obtain the control model. To validate the effectiveness of the proposed iFFDL-MFA-PID algorithm, the operating speed, position, and other relevant information of the electric forklift were recorded. The parameters of the EFG-BC/320 electric balanced forklift, as well as the control parameters of the iFFDL-MFA-PID algorithm, are shown in Table 2. Simulation verification was conducted using the iFFDL-MFA-PID, FFDL-MFAC, and PID methods, respectively. The tracking error, control force, and acceleration were compared and analyzed to demonstrate the superiority of the iFFDL-MFA-PID method. The parameters of the other contrasting control methods were set as follows:  
1. [label=(0)]
2. FFDL-MFAC method: \(I(1)=I_{d}(1)\), \(\eta=0.85\), \(\mu=0.9\), \(\rho=0.1\), \(\partial=0.25\), \(L_{I}=1\), and \(L_{u}=1\). 3. PID method [39]: \(I(1)=I_{d}(1)\), \(K_{1}=0.65\), \(K_{2}=0.7\), and \(K_{3}=0.06\). Figure 7 shows the speed-tracking curves of the electric forklift with the iFFDL-MFA-PID, MFAC, and PID methods, and Figure 8 shows their error curves. In Figures 7 and 8, it can be seen that the speed-tracking error range of the electric forklift under the action of MFAC is in the range of \([-0.128\) km\(\cdot\)h\({}^{-1}\), \(0.194\) km\(\cdot\)h\({}^{-1}]\), whereas that of the PID is in the range of \([-0.228\) km\(\cdot\)h\({}^{-1}\), \(0.313\) km\(\cdot\)h\({}^{-1}]\). For iFFDL-MFA-PID, firstly, the MFAC law was obtained by using the principle of equivalent feedback. Secondly, to enhance the