## 3 Improved FFDL Model-Free Adaptive PID Controller
### Stability Analysis  
The stability analysis includes the boundedness of the parameter vector estimation and the convergence proof for the system error. For ease of description, let \(L_{I}=L_{u}=1\); other situations follow similarly. Firstly, the following lemmas are introduced. **Lemma 1**. : _For any matrix \(\mathbf{B}_{m}\in R^{m\times m}\), \(b_{1},\cdots,b_{m}\) are the eigenvalues of \(\mathbf{B}_{m}\). Then, the spectral radius of \(\mathbf{B}_{m}\) is \(s(\mathbf{B}_{m})=\max\limits_{j}\big{|}b_{j}\big{|}\cdot j\in\{1,\cdots,m\}\)._  
**Lemma 2**. : _For any matrix \(\mathbf{B}_{n}\in R^{n\times n}\) and a constant \(\sigma>0\), there exists a matrix norm \(\|\cdot\|\) that satisfies_  
\[\|\mathbf{B}_{n}\|\leq s(\mathbf{B}_{n})+\sigma \tag{18}\]  
**Theorem 2**. : _If System (11) adopts the iFFDL-MFA-PID scheme, the system has the following properties:_  
1. _The parameter vector estimation_ \(\tilde{\Lambda}(k)=\tilde{\Lambda}(k)-\Lambda(k)\) _is bounded;_
2. \(e(k+1)\) _converges to a small bound if_ \(\big{|}1-\tilde{\Lambda}(k)\Big{(}k_{1}+k_{2}+k_{3}+\frac{2\rho}{\lambda+ \Lambda(k)}\Big{)}\big{|}<1\)_._  
Figure 1: The iFFDL-MFA-PID control block diagram. **Proof 1**. : Subtracting \(\Lambda(k)\) from both sides of (16) simultaneously yields  
\[ \begin{split}\tilde{\Lambda}(k)&=\left(1-\frac{\eta \Delta u^{2}(k-1)}{\mu+|\Delta u(k)|^{2}}\right)\tilde{\Lambda}(k-1)+\frac{\eta \Delta u(k-1)\Delta u(k-2)}{\mu+|\Delta u(k)|^{2}}\tilde{\Lambda}(k-2)\\ &+\frac{\eta\Delta u(k-1)}{\mu+|\Delta u(k)|^{2}}(q(k-1)-q(k-2))+ \Lambda(k-1)-\Lambda(k)\end{split}  \tag{19}\]  
Define \(\mathbf{M}_{\Xi_{1}}(k)=\text{diag}(\Xi_{1}(k),\mathbf{0})\), \(\mathbf{M}_{\Xi_{2}}(k)=\text{diag}(\Xi_{2}(k),\mathbf{0})\), and  
\[\Xi_{1}(k)=\left( \begin{array}{cccc}1-\frac{\eta\Delta u^{2}(0)}{\mu+|\Delta u (0)|^{2}}&0&\cdots&0&0\\ \frac{\eta\Delta u(1)\Delta u(0)}{\mu+|\Delta u(1)|^{2}}&\frac{\eta\Delta u^{ 2}(1)}{\mu+|\Delta u(1)|^{2}}&\cdots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&1-\frac{\eta\Delta u^{2}(k-2)}{\mu+|\Delta u(k-1)|^{2}}&0\\ 0&0&\cdots&\frac{\eta\Delta u(k-1)\Delta u(k-2)}{\mu+|\Delta u(k-1)|^{2}}&1- \frac{\eta\Delta u^{2}(k-1)}{\mu+|\Delta u(k-1)|^{2}}\end{array} \right) \tag{20}\]  
\[\Xi_{2}(k)=\text{diag}\Big{(}\frac{\eta\Delta u^{2}(0)}{\mu+|\Delta u(0)|^{2}} \quad\frac{\eta\Delta u^{2}(1)}{\mu+|\Delta u(1)|^{2}}\quad\cdots\quad\frac{ \eta\Delta u^{2}(k-1)}{\mu+|\Delta u(k-1)|^{2}}\Big{)} \tag{21}\]  
Then, (19) can be rewritten as follows  
\[\tilde{\Lambda}(k)=\mathbf{M}_{\Xi_{1}}(k)\tilde{\Lambda}(k-1)+\mathbf{M}_{ \Xi 2}(k)(\mathbf{q}(k-1)-\mathbf{q}(k-2))+\mathbf{\Lambda}(k-1)-\mathbf{\Lambda}(k) \tag{22}\]  
where \(\tilde{\Lambda}(k)=[\tilde{\Lambda}(1),\cdots,\tilde{\Lambda}(k),\mathbf{0}] ^{\text{T}}\), \(\mathbf{\Lambda}(k)=[\Lambda(1),\cdots,\Lambda(k),\mathbf{0}]^{\text{T}}\), \(\mathbf{q}(k)=[q(1),\cdots,\)\(q(k),\mathbf{0}]^{\text{T}}\). Taking the norm on both sides of (22), we obtain  
\[ \begin{split}&\parallel\tilde{\Lambda}(k)\|_{v}\leq\parallel \mathbf{M}_{\Xi_{1}}(k)\|_{v}\parallel\tilde{\Lambda}(k-1)\|_{v}+\left\| \mathbf{M}_{\Xi_{2}}(k)\right\|_{v}\parallel\mathbf{q}(k-1)-\mathbf{q}(k-2) \|_{v}\\ &+\parallel\mathbf{\Lambda}(k-1)\|_{v}+\parallel\mathbf{\Lambda}(k )\|_{v}\end{split}  \tag{23}\]  
Since \(|\Lambda(k)|\leq\bar{b}_{\alpha}\), \(|q(k)|\leq\bar{b}_{\alpha}\), so that both \(\mathbf{\Lambda}(k)\) and \(\mathbf{q}(k)\) are bounded. Next, we assume that \(\left\|\mathbf{\Lambda}(k)\right\|\leq B_{\Lambda}\) and \(\left\|\mathbf{q}(k)\right\|\leq B_{\alpha}\), where \(B_{\Lambda}>0\) and \(B_{\alpha}>0\) are two constants. Since \(0<\eta<2\) and \(\mu>0\), one has  
\[\left|1-\frac{\eta\Delta u^{2}(k)}{\mu+|\Delta u(k)|^{2}}\right|<1 \tag{24}\]  
According to Lemma 1, one deduces that \(s(\mathbf{M}_{\Xi_{1}}(k))<1\). From Lemma 2, we have  
\[\left\|\mathbf{M}_{\Xi_{1}}(k)\right\|_{v}\leq s(\mathbf{M}_{\Xi_{2}}(k))+ \sigma_{1}\leq d_{1}<1 \tag{25}\]  
where \(\sigma_{1}>0\) and \(d_{1}>0\) are small constants.