# An Improved Model-Free Adaptive Nonlinear Control and Its Automatic Application
## 2 Traditional Model-Free Adaptive Controller
### Traditional Model-Free Adaptive Control Scheme  
Traditional MFAC considered the following performance index function:  
\[J(u(t))=|y_{d}(t+1)-y(t+1)|^{2}+\lambda|u(t)-u(t-1)|^{2} \tag{5}\]  
where \(y_{d}\) is the reference output; \(\lambda>0\) is a weight factor, which limits the change of control input. By solving the first-order partial differential of \(u(t)\) in Equation (5) and making it 0, the following control law expression can be obtained:  
\[ \begin{split} u(t)&=u(t-1)+\frac{\rho_{L_{y}+1}\phi _{L_{y}+1}(t)(y_{d}(t+1)-y(t))}{\lambda+\left|\phi_{L_{y}+1}(t)\right|^{2}}\\ &-\frac{\phi_{L_{y}+1}(t)\left(\sum\limits_{i=1}^{L_{y}}\rho_{i} \phi_{\bar{i}}(t)\Delta y(t-i+1)+\sum\limits_{i=L_{y}+2}^{L_{y}+L_{u}}\rho_{i }\phi_{\bar{i}}(t)\Delta u(t-L_{y}-i+1)\right)}{\lambda+\left|\phi_{L_{y}+1}( t)\right|^{2}}.\end{split}  \tag{6}\]  
where the step factor \(\rho\in(0,1]\) is added to make the algorithm FFDL-MFAC design process more flexible. Next, we need to estimate the time-varying parameter vector \(\mathbf{\Phi}(t)\) in real time, and introduce the parameter estimation criterion function  
\[J(\mathbf{\Phi}(t))=\left|\Delta y(t)-\mathbf{\Phi}^{T}(t)\Delta\mathbf{H}(t-1)\right|^{2 }+\mu\big{\|}\mathbf{\Phi}(t)-\mathbf{\hat{\Phi}}(t-1)\big{\|}^{2} \tag{7}\]  
where the weight factor \(\mu>0\) constrains the rate of change of adjacent parameters. On the other hand, the choice of parameter \(\mu\) affects the accuracy of the estimated value of the time-varying parameter matrix. We continuously adjust the size of the parameter\(\mu\) until the system control performance is optimal (there is no specific parameter value for the estimation of the time-varying parameter matrix; however, the estimation results are for control purposes, so we judge the parameter selection problem by observing the control performance). Minimizing the parameter estimation criterion function (7), the following parameter estimation algorithms can be obtained  
\[\mathbf{\Phi}(t)=\mathbf{\Phi}(t-1)+\frac{\eta\Delta\mathbf{H}(t-1)\big{(}\Delta y(t)-\mathbf{ \Phi}^{T}(t-1)\Delta\mathbf{H}(t-1)\big{)}}{\mu+\left\|\Delta\mathbf{H}(t-1)\right\|^{2}} \tag{8}\]  
where \(\mathbf{\hat{\Phi}}(t)\) is the estimated value of \(\mathbf{\Phi}(t)\); and \(\eta\in(0,2]\) is the step factor. In order to enhance the robustness of the system, the parameter reset algorithm is given. \[ \begin{array}{c}\mathbf{\Phi}(t)=\mathbf{\Phi}(1)\;\;\text{if}\;\;\left\|\mathbf{\Phi}(t) \right\|\leq\varepsilon\;\;\text{or}\;\;\left\|\Delta\mathbf{H}(t-1)\right\|^{2} \leq\varepsilon\\ \;\text{or}\;\;\text{sign}\left(\hat{\phi}_{t_{y}+1}(t)\right)\
eq\text{sign} \left(\hat{\phi}_{t_{y}+1}(1)\right)\end{array}  \tag{9}\]  
Therefore, the FFDL-MFAC control scheme can be organized as follows:  
\[ \begin{array}{c}u(t)=u(t-1)+\frac{\rho_{t_{y}+1}\hat{\[PERSON]}_{y+1}(t)(y_{d}(t+ 1)-y(t))}{\lambda+\left\|\hat{\[PERSON]}_{t_{y}+1}(t)\right\|^{2}}-\\ \frac{\hat{\phi}_{t_{y}+1}(t)(\sum\limits_{i=1}^{L_{y}}\rho_{i}\hat{\[PERSON]}_{i}(t )\Delta y(t-i+1)+\sum\limits_{i=1}^{L_{y}}\rho_{i}\hat{\[PERSON]}_{i}(t)\Delta u(t- L_{y}-i+1))}{\lambda+\left\|\hat{\[PERSON]}_{t_{y}+1}(t)\right\|^{2}}\\ \mathbf{\hat{\[PERSON]}}(t)=\mathbf{\hat{\[PERSON]}}(t-1)+\frac{\eta\Delta H(t-1)(\Delta y(t)- \mathbf{\hat{\[PERSON]}}^{T}(t-1)\Delta H(t-1))}{\mu+\left\|\Delta H(t-1)\right\|^{2}} \\ \mathbf{\hat{\[PERSON]}}(t)=\mathbf{\hat{\Phi}}(1)\;\;\text{if}\;\left\|\mathbf{\hat{\Phi}}(t) \right\|\leq\varepsilon\;\;\text{or}\;\;\left\|\Delta H(t-1)\right\|^{2}\leq \varepsilon\\ \;\text{or}\;\;\text{sign}(\hat{\phi}_{t_{y}+1}(t))\
eq\text{sign}(\hat{\phi }_{t_{y}+1}(1))\end{array}  \tag{10}\]  
**Remark 2**. : _When the sampling period and \(\Delta u(t)\) are small, the interaction between the time-varying parameter vector \(\mathbf{\Phi}(t)\) and \(\Delta u(t)\) can be ignored [26]._  
**Remark 3**. : _Algorithms (8) and (10) are independent of the mathematical model of the controlled system. The update of the pseudo partial derivative estimate is only related to the online 1/O data of the control system.