## 3 Improved FFDL Model-Free Adaptive PID Controller  
### iFFDL-MFA-PID Design  
Consider the following SISO system affected by disturbances  
\[I(k+1)=f(I(k),\ \cdots,\ I(k-n_{I}),u(k),\ \cdots,\ u(k-n_{u}))\ +d(k) \tag{10}\]  
where \(d(k)\) is a bounded disturbance in the forklift system.  
For System (10) to satisfy Assumptions 1 and 2, given \(0\leq L_{I}\leq n_{I}\) and \(1\leq L_{u}\leq n_{u}\), there must be a parameter vector \(\mathbf{\Lambda}(k)\) so that System (10) can be transformed as  
\[\Delta I(k+1)=\mathbf{\Lambda}^{\text{T}}(k)\Delta\mathbf{H}(k)+q(k) \tag{11}\]  
where \(\begin{array}{c}q(k)=f(I(k-1),\cdots,I(k-n_{I}),\)\(u(k-1),\cdots,u(k-n_{u}))\\ -f(I(k-1),\cdots,I(k-n_{I}-1),\)\(u(k-1),\cdots,u(k-n_{u}-1))\!+\!\Delta d(k)\end{array}\), \(q(k)\) is the sum of the bounded disturbances of the nonlinear remainder.  
Based on (11) and the principle of equivalent feedback, we have  
\[\begin{array}{l}u(k)=u(k-1)+\frac{\rho(I_{d}(k+1)-I(k)-q(k))}{\partial+ \Lambda_{L_{v}+1}(k)}-\frac{\rho\sum\limits_{i=1}^{L_{I}}\Lambda_{i}(k) \Delta I(k-i+1)}{\partial+\Lambda_{L_{I}+1}(k)}\\ -\frac{\sum\limits_{i=L_{I}+2}^{L_{I}+L_{u}}\Lambda_{i}(k)\Delta u(k-L_{I}-i+ 1)}{\partial+\Lambda_{L_{I}+1}(k)}\end{array} \tag{12}\]  
To further enhance robustness, by drawing on the ideas of Ref. [31] and incorporating an iPID strategy into the control law (12), it can be concluded that  
\[\begin{array}{l}u(k)=u(k-1)+\frac{\rho(I_{d}(k+1)-I(k)-q(k))}{\theta+\Lambda _{I_{I}+1}(k)}-\frac{\rho\sum\limits_{i=1}^{L_{I}}\Lambda_{i}(k)\Delta I(k-i+1 )}{\theta+\Lambda_{I_{I}+1}(k)}-\frac{\sum\limits_{i=L_{I}+2}^{L_{I}+L_{u}} \Lambda_{i}(k)\Delta u(k-l_{I}-i+1)}{\theta+\Lambda_{I_{I}+1}(k)}\\ +k_{1}(e(k)-e(k-1))\!+\!k_{2}e(k)\!+\!k_{3}(e(k)-2e(k-1)+e(k-2))\end{array} \tag{13}\]  
where \(e(k)=I_{d}(k)-I(k)\) is the output error; \(k_{1},k_{2},\) and \(k_{3}\) are the control learning gains; and \(q(k)\) and \(\mathbf{\Lambda}(k)\) need to be estimated by the algorithm.  
Firstly, we use the difference algorithm in [29] to estimate \(q(k)\)  
\[\hat{q}(k)=\Delta I(k)-\mathbf{\hat{\Lambda}}(k-1)^{\text{T}}\Delta\mathbf{H}(k-1) \tag{14}\]Then, we provide the following new function:  
\[J(\mathbf{\Lambda}(k))=\mu\big{\|}\mathbf{\Lambda}(k)-\mathbf{\hat{\Lambda}}(k-1)\big{\|}^{2} +\big{|}\Delta I(k)-q(k-1)-\mathbf{\Lambda}(k)^{T}\Delta\mathbf{H}(k-1)\big{|}^{2} \tag{15}\]  
According to the minimization function (15), one has  
\[\mathbf{\hat{\Lambda}}(k)=\mathbf{\hat{\Lambda}}(k-1)+\frac{\eta\Delta\mathbf{H}(k-1) }{\mu+\|\Delta\mathbf{H}(k-1)\|^{2}}\times(\Delta I(k)-\hat{q}(k-1)-\mathbf{\hat{ \Lambda}}^{T}(k-1)\Delta\mathbf{H}(k-1)) \tag{16}\]  
Then, the following reset algorithm is proposed:  
\[\mathbf{\hat{\Lambda}}(k)=\mathbf{\hat{\Lambda}}(1)\text{ if }\big{\|}\mathbf{\hat{ \Lambda}}(k)\big{\|}\leq\varepsilon\text{ or }\big{\|}\Delta\mathbf{H}(k-1)\big{\|}^{2}\leq \varepsilon\text{ or }\text{sign}\big{(}\hat{\Lambda}_{L_{l}+1}(k)\big{)}\
eq\text{sign} \big{(}\hat{\Lambda}_{L_{l}+1}(1)\big{)} \tag{17}\]  
where \(\varepsilon>0\) is a sufficiently small positive number.  
The control law (13), PG estimation algorithm (16), reset algorithm (17), and difference estimation algorithm (14) together form the iFFDL-MFA-PID scheme, and the control block diagram is shown in Figure 1.  
**Remark 1**.: _The primary distinction between the FFDL data model (11) and the conventional model (4) lies in the expression of the form of the total system uncertainty. Unlike the linear parameter-based approach, the FFDL model considers nonlinear residual uncertainty. Incorporating the estimation of nonlinear residual uncertainty into the control law enables compensation for such uncertainties, thereby enhancing control design flexibility._