# 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  
The following only shows the proof process of the SISO scheme, and the process of the MIMO scheme is similar. The stability proof of this paper is divided into three parts, namely pseudo partial derivative estimate \(\hat{\mathbf{\Phi}}(t)\) boundedness, algorithm convergence, and output sequence and input sequence boundedness proof. In order to make the proof more rigorous, the following assumptions are given. **Assumption 3** ([20]). : _For any time \(t\), when \(\Delta u(t)\
eq 0\), the sign of PPD remains unchanged and satisfies the conditions \(\phi_{1}(t),\phi_{2}(t)>\varepsilon>0\), where \(\varepsilon\) is a very small positive real number._  
Assumption 3 is similar to the assumption that the control direction is known in model-based control theory. This assumption indicates that, as the control input becomes larger, the system output does not decrease, and vice versa. Many actual controlled objects meet this limitation. **Assumption 4** ([5]). : _System (1) will not be affected by severe external factors during operation. In other words, when Assumption 3 is satisfied and the sampling time is small enough, the output variation tends to be infinitesimal when the system tends to be stable._  
**Theorem 2**. : _For a nonlinear system (1) satisfying Assumptions 1-3, using the control algorithm proposed in this paper, the system satisfies:_  
1. _The pseudo-partial derivative estimate_ \(\hat{\mathbf{\Phi}}(t)\) _is bounded;_
2. \(\lim\limits_{t\to\infty}\left|e(t)\right|=0\)_;_
3. _The output sequence and the input sequence are bounded._  
Proof. : The pseudo-partial derivative estimate \(\hat{\mathbf{\Phi}}(t)\) is bounded. \[\hat{\mathbf{\Phi}}(t)=\hat{\mathbf{\Phi}}(t-1)+\frac{\eta\Delta\mathbf{H}(t-1)\big{(} \Delta y(t)-\hat{\mathbf{\Phi}}^{T}(t-1)\Delta\mathbf{H}(t-1)\big{)}}{\mu+\left\| \Delta\mathbf{H}(t-1)\right\|^{2}} \tag{17}\]  
Let \(\hat{\mathbf{\Phi}}(t)=\mathbf{\Phi}(t)+\tilde{\mathbf{\Phi}}(t)\). Subtracting \(\mathbf{\Phi}(t)\) on both sides of formula (17)  
\[\tilde{\mathbf{\Phi}}(t)=\tilde{\mathbf{\Phi}}(t-1)+\mathbf{\Phi}(t-1)-\mathbf{\Phi}(t)+\frac {\eta\Delta\mathbf{H}(t-1)\big{(}\Delta y(t)-\hat{\mathbf{\Phi}}^{T}(t-1)\Delta\mathbf{H} (t-1)\big{)}}{\mu+\left\|\Delta\mathbf{H}(t-1)\right\|^{2}} \tag{18}\]  
Substituting the dynamic linearized data model (4) into formula (18), we can obtain  
\[\hat{\mathbf{\Phi}}(t)=\tilde{\mathbf{\Phi}}(t-1)\bigg{(}I-\frac{\eta\Delta\mathbf{H}(t-1 )\Delta\mathbf{H}(t-1)}{\mu+\left\|\;\Delta\mathbf{H}(t-1)\right\|^{2}}\bigg{)}+\mathbf{ \Phi}(t-1)-\mathbf{\Phi}(t) \tag{19}\]Since the time-varying parameter vector \(\mathbf{\Phi}(t)\) is bounded, let its upper bound be \(\|\mathbf{\Phi}(t)\|\leq b\). The formula (19) takes the norm on both sides and can be derived. \[\|\mathbf{\tilde{\Phi}}(t)\|\leq\left\|\mathbf{\tilde{\Phi}}(t-1)\bigg{(}I- \frac{\eta\Delta H(t-1)\Delta H(t-1)}{\mu+\parallel\Delta H(k-1)\parallel^{2}} \bigg{)}\right\|+2b \tag{20}\]  
Squaring and expanding the first term on the right side of Equation (20)  
\[ \begin{array}{l}\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\|^{2 }=\\ \left\|\mathbf{\tilde{\Phi}}(t-1)\right\|^{2}+\left(\frac{\eta\parallel\Delta H (t-1)\parallel^{2}}{\mu+\parallel\Delta H(t-1)\parallel^{2}}-2\right)\frac{ \eta\left\|\mathbf{\tilde{\Phi}}(t-1)\right\|^{2}\left\|\Delta H(t-1) \right\|^{2}}{\mu+\parallel\Delta H(t-1)\parallel^{2}}\end{array}