# An Improved Model-Free Adaptive Nonlinear Control and Its Automatic Application
## 3 Model-Free Adaptive Control Design with Output Error Rate (MFAC-OER)
### Stability Analysis
 \tag{21}\]  
Due to the step factor \(\eta\in(0,2]\), and \(\mu>0\), it is not difficult to obtain  
\[\frac{\eta\parallel\Delta H(t-1)\parallel^{2}}{\mu+\parallel\Delta H(t-1) \parallel^{2}}<\frac{\eta\parallel\Delta H(t-1)\parallel^{2}}{\parallel \Delta H(t-1)\parallel^{2}}=2 \tag{22}\]  
Through Equations (21) and (22), the following can be derived  
\[\left\|\mathbf{\tilde{\Phi}}(t-1)\bigg{(}I-\frac{\eta\Delta H(t-1)\Delta H(t- 1)}{\mu+\parallel\Delta H(k-1)\parallel^{2}}\bigg{)}\right\|^{2}<\left\| \mathbf{\tilde{\Phi}}(t-1)\right\|^{2} \tag{23}\]  
From Equation (23), we can deduce that there must be a number such that the following inequality holds  
\[\left\|\mathbf{\tilde{\Phi}}(t-1)\bigg{(}I-\frac{\eta\Delta H(t-1)\Delta H(t- 1)}{\mu+\parallel\Delta H(t-1)\parallel^{2}}\bigg{)}\right\|\leq M\big{\|} \mathbf{\tilde{\Phi}}(t-1)\big{\|} \tag{24}\]  
The following inequality is obtained by substituting Equation (24) into Equation (20)  
\[ \begin{array}{l}\left\|\mathbf{\tilde{\Phi}}(t)\right\|\leq M\big{\|} \mathbf{\tilde{\Phi}}(t-1)\big{\|}+2b\leq M\big{(}M\big{\|}\mathbf{\tilde{ \Phi}}(t-2)\big{\|}+2b\big{)}+2b\\ \cdots\leq M^{t-1}\big{\|}\mathbf{\tilde{\Phi}}(1)\big{\|}+\frac{1-M^{t-1}}{ 1-M}2b\end{array}  \tag{25}\]  
From Equation (25), it can be concluded that the parameter estimation error \(\mathbf{\tilde{\Phi}}(t)\) is bounded, and because \(\mathbf{\Phi}(t)\) is bounded, the estimated value \(\mathbf{\Phi}(t)\) is bounded. The conclusion (1) of this Theorem 2 is proven. Proof. : \(\lim\limits_{l\rightarrow\infty}\lvert e(t)\rvert=0\). For the convenience of proof, let \(\theta+\frac{1-\theta}{T^{2}}=l_{1}\), \(\frac{1-\theta}{T^{2}}=l_{2}\). Substituting the data model Equation (4) into the error expression can obtain:  
\[e(t+1)=y_{d}-y(t+1)=y_{d}-y(t)-\phi_{1}\Delta y(t)-\phi_{2}\Delta u(t) \tag{26}\]  
Furthermore, substituting the control law (10) in Equation (26) to obtain:  
\[ \begin{array}{l}\frac{e(t+1)}{\mu+\parallel\Delta H(t-1)}=e(t)-\phi_{1} \Delta y(t)-\\ \phi_{2}\frac{\phi_{2}(t)\parallel_{1}(y_{d}(t+1)-y(t))-\phi_{2}(t)\phi_{1}(t )\parallel_{1}\Delta y(t)-\phi_{2}(t)\parallel_{2}e(k)}{\lambda+l_{1}\parallel \phi_{2}(t)\parallel^{2}}\\ \leq\left|1-\frac{\phi_{2}\phi_{2}(t)\theta}{\lambda+l_{1}\parallel\phi_{2}(t )\parallel^{2}}\right|e(k)+\left|\phi_{1}-\frac{\phi_{2}\phi_{2}(t)\phi_{1}l_{ 1}}{\lambda+l_{1}\parallel\phi_{2}(t)\parallel^{2}}\right|\Delta y(t)\end{array}  \tag{27}\]  
Since the time-varying parameter vector \(\mathbf{\Phi}(t)\) is bounded, enabling the upper bound of its elements to be \(\phi_{2}\leq b\), we know that:  
\[0<M\leq\left|\frac{\phi_{2}\phi_{2}(t)\theta}{\lambda+l_{1}\big{\|}\phi_{2}(t) \big{\|}^{2}}\right|\leq\left|\frac{b\big{\|}\phi_{2}(t)\big{|}\theta}{\lambda+ l_{1}\big{\|}\phi_{2}(t)\big{\|}^{2}}\right| \tag{28}\]let \(\frac{i^{2}\theta^{2}}{4l_{1}}=\lambda_{\text{min}}<\lambda\). Using the inequality \(\sqrt{ab}\leq a+b\), Equation (28) can be further expressed as:  
\[ \begin{array}{l}0<M\leq\left|\frac{\phi_{2}\hat{\phi}_{2}(t)\theta}{\lambda+l_{ 1}|\phi_{2}(t)|^{2}}\right|\leq\left|\frac{b|\hat{\phi}_{2}(t)|\theta}{\lambda+l _{1}|\phi_{2}(t)|^{2}}\right|\\ \leq\left|\frac{b|\hat{\phi}_{2}(t)|\theta}{2\left|\hat{\phi}_{2}(t)|\sqrt{ \lambda_{\text{min}}l_{1}}\right|}\right|\leq\left|\frac{b|\hat{\phi}_{2}(t)| \theta}{2\left|\hat{\phi}_{2}(t)|\sqrt{\lambda_{\text{min}}l_{1}}\right|} \right|=1\end{array}  \tag{29}\]  
Since \(\phi_{1},\phi_{2},\hat{\phi}_{2},l_{1},\lambda\) are all bounded constants, and the system satisfies Assumption 3 and \(\lim\limits_{t\rightarrow\infty}\left|\Delta y(t)\right|=0\), it can be obtained  
\[\lim\limits_{t\rightarrow\infty}\left|e(t+1)\right|\leq M|e(t)|\leq M^{2}|e(t -1)|\leq\cdots\leq M^{t-1}|e(2)| \tag{30}\]  
Equation (30) can be explained in that \(\lim\limits_{t\rightarrow\infty}\left|e(t)\right|=0\); as such, conclusion (2) of Theorem 2 is proven. Proof. : The output sequence and input sequence are bounded.