## 3 Improved FFDL Model-Free Adaptive PID Controller
### Stability Analysis
Then, according to Lemma 1 and Lemma 2, one has  
\[\left\|\mathbf{M}_{\Xi_{2}}(k)\right\|_{v}\parallel\mathbf{q}(k-1)-\mathbf{q} (k-2)\|_{v}+\parallel\mathbf{\Lambda}(k-1)\|_{v}+\parallel\mathbf{\Lambda}(k )\|_{v}\leq 2d_{2}B_{\alpha}+2B_{\Lambda}<\varsigma_{1} \tag{26}\]  
where \(0<\varsigma_{1}\) is a constant. In terms of (22) and (26), we can deduce that  
\[\parallel\tilde{\Lambda}(k)\|_{v}\leq d_{1}\parallel\tilde{\Lambda}(k-1)\|_{v }+\varsigma_{1}\leq\cdots\leq d_{1}^{k}\parallel\tilde{\Lambda}(0)\|_{v}+ \frac{\varsigma_{1}}{1-d_{1}} \tag{27}\]which implies that \(\mathbf{\tilde{\Lambda}}(k)\) is bounded. Obviously, \(\mathbf{\hat{\Lambda}}(k)\) is bounded since \(\mathbf{\Lambda}(k)\) is also bounded. Then, we assume that \(\big{|}\hat{\Lambda}(k)\big{|}\leq\tilde{b}_{\hat{\Lambda}}\), where \(\tilde{b}_{\hat{\Lambda}}>0\) is a constant and \(\hat{\Lambda}(k)\) are the elements of \(\mathbf{\hat{\Lambda}}(k)\). According to (13) and (14), we have  
\[ \begin{split}\Delta u(k)&=\frac{\rho\big{(}I_{d}(k +1)-2I(k)+I(k-1)+\hat{\Lambda}(k-1)\Delta u(k-1)\big{)}}{\partial+\hat{\Lambda} (k)}\\ &+k_{1}(e(k)-e(k-1))+k_{2}e(k)+k_{3}(e(k)-2e(k-1)+e(k-2))\\ &=\bigg{(}k_{1}+k_{2}+k_{3}+\frac{2\rho}{\lambda+\hat{\theta}(k) }\bigg{)}e(k)+\bigg{(}-k_{1}-2k_{3}-\frac{\rho}{\partial+\hat{\Lambda}(k)} \bigg{)}e(k-1)+k_{3}e(k-2)\\ &+\frac{\rho\hat{\Lambda}(k-1)}{\partial+\hat{\Lambda}(k)} \Delta u(k-1)+\frac{\rho}{\partial+\hat{\Lambda}(k)}(I_{d}(k+1)-2I_{d}(k)+I_{ d}(k-1))\end{split}  \tag{28}\]  
Combined with the system data model (11), the tracking error can be written as  
\[ \begin{split} e(k+1)&=I_{d}(k+1)-I(k+1)\\ &=e(k)-\Lambda(k)\Delta u(k)+I_{d}(k+1)-I_{d}(k)-q(k)\end{split}  \tag{29}\]  
Substituting (28) into (29), one has  
\[ \begin{split} e(k+1)&=e(k)\bigg{(}1-\Lambda(k) \bigg{(}k_{1}+k_{2}+k_{3}+\frac{2\rho}{\partial+\hat{\Lambda}(k)}\bigg{)} \bigg{)}\\ &-\Lambda(k)\bigg{(}-k_{1}-2k_{3}-\frac{\rho}{\partial+\hat{ \Lambda}(k)}\bigg{)}e(k-1)-\Lambda(k)k_{3}e(k-2)\\ &-\frac{\rho\hat{\Lambda}(k-1)\Lambda(k)}{\partial+\hat{\Lambda }(k)}\Delta u(k-1)+I_{d}(k+1)-I_{d}(k)-q(k)\\ &-\frac{\rho\Lambda(k)}{\partial+\hat{\Lambda}(k)}(I_{d}(k+1)-2I _{d}(k)+I_{d}(k-1))\end{split}  \tag{30}\]  
Let \(\mathbf{e}(k+1)=[e(1),\cdots,e(k),\mathbf{0}]^{\mathrm{T}}\), \(\mathbf{I}_{d}(k)=[I_{d}(0),\cdots,I_{d}(k),\mathbf{0}]^{\mathrm{T}}\), \(\mathbf{\hat{\theta}}(k)=[o(0),\cdots,\)\(q(k),\mathbf{0}]^{\mathrm{T}}\), \(\mathbf{\hat{\Sigma}}(k)=1-\Lambda(k)\Big{(}k_{1}+k_{2}+k_{3}+\frac{2\rho}{ \lambda+\Lambda(k)}\Big{)}\), and  
\[\left\{ \begin{array}{l}w(k)=\frac{e\hat{\Lambda}(k-1)\Lambda(k)}{\partial+ \hat{\Lambda}(k)}\Big{(}k_{1}+k_{2}+k_{3}+\frac{2\rho}{\partial+\hat{\Lambda} (k-1)}\Big{)}-\Lambda(k)\Big{(}k_{1}+2k_{3}+\frac{\rho}{\partial+\hat{\Lambda} (k)}\Big{)}\\ c(k)=k_{3}+\frac{p\hat{\Lambda}(k)}{\partial+\hat{\Lambda}(k+1)}\Bigg{(} \begin{array}{c}\frac{p\hat{\Lambda}(k-1)}{\partial+\hat{\Lambda}(k)}\Big{(} k_{1}+k_{2}+k_{3}+\frac{2\rho}{\partial+\hat{\Lambda}(k-1)}\Big{)}\\ -\Big{(}k_{1}+2k_{3}+\frac{\rho}{\partial+\hat{\Lambda}(k)}\Big{)}\end{array} \Bigg{)}\\ \end{array} \right. \tag{31}\]  
Then, define \(\mathbf{M}_{\Xi_{3}}(k)=\mathrm{diag}(\Xi_{3}(k)\quad\mathbf{0}),\mathbf{M}_ {\Xi_{4}}(k)=\mathrm{diag}(\Xi_{4}(k)\quad\mathbf{0})\), and  
\[m(k)=\left[\frac{\rho\Lambda(0)\hat{\Lambda}(-1)}{\partial+\hat{\Lambda}(0)}, \frac{\rho^{2}\Lambda(0)\hat{\Lambda}(0)}{\partial+\hat{\Lambda}(1)}\frac{ \rho\hat{\Lambda}(-1)}{\partial+\hat{\Lambda}(0)},\ldots,\frac{\rho^{k+1}\Lambda (k)\prod\limits_{m=-1}^{k-1}\hat{\Lambda}(n)}{\prod\limits_{m=0}^{k}( \partial+\hat{\Lambda}(m))},\mathbf{0}\right]^{\mathrm{T}} \tag{32}\]  
where  
parameters can be determined according to the convergence condition given in the following theorem if the gradient information is available. Further, \(k_{1}\), \(k_{2}\), and \(k_{3}\) can also be selected using the well-known PID control parameter-tuning methods._