# An Improved Model-Free Adaptive Nonlinear Control and Its Automatic Application
## 2 Traditional Model-Free Adaptive Controller
### Dynamic Linearization  
For many practical systems, its order is generally multi-order, which leads to its output depending on the input and output at a certain moment. Based on the above considerations, the FFDL method is used in this paper to linearize the data. The FFDL method adds the influence of the input and output within a fixed length on the system. Theoretically, the method can both capture the complex dynamics in the original system and effectively reduce the complexity of the system [5; 18].  
Consider the following SISO discrete-time nonlinear system  
\[y(t+1)=f\big{(}y(t),\ \cdots,\ y(t-n_{y}),u(t),\ \cdots,\ u(t-n_{u})\big{)} \tag{1}\]  
where \(u(t)\), \(y(t)\) denote the input and output of the system at time \(t\); \(n_{u}\), \(n_{y}\) are the output and output orders, respectively; \(f(\cdot)\) is a nonlinear and time-varying function.  
\(\mathbf{H}(t)\) is defined as a matrix of all control inputs and all system outputs at a fixed time, as follows:  
\[\mathbf{H}(t)=[\Delta y(t),\cdots,\Delta y(t-L_{y}+1),\Delta u(t),\cdots,\Delta u( t-L_{u}+1)]^{\mathrm{T}} \tag{2}\]  
where \(0\leq L_{y}\leq n_{y},0\leq L_{u}\leq n_{u}\) are the length constants of the system output and control input linearization, respectively, also known as the pseudo-order.  
For the nonlinear system (1), the two following assumptions are given.  
**Assumption 1** ([18; 19]).: _The nonlinear time-varying function \(f(\cdot)\) has partial derivatives for all variables of the system._  
**Assumption 2** ([18; 19]).: _System (1) satisfies the following generalized Lipschitz condition, for any \(0\leq t_{1}\
eq t_{2}\) and \(\mathbf{H}(t_{1})\
eq\mathbf{H}(t_{2})\):_  
\[|y(t_{1}+1)-y(t_{2}+1)|\leq b\|\mathbf{H}(t_{1})-\mathbf{H}(t_{2})\| \tag{3}\]  
_where \(b\) is a constant greater than 0. Denoting \(\Delta\mathbf{H}(t)=\mathbf{H}(t)-\mathbf{H}(t-1)\)._  
Theorem 1 can be obtained from Assumption 1 and Assumption 2.  
**Theorem 1** ([19]).: _Consider that the discrete-time nonlinear system (5) satisfies Assumptions 1 and 2, for any \(0\leq L_{\varphi}\leq n_{\varphi},1\leq L_{u}\leq n_{u}\), and \(\|\Delta\mathbf{H}(t)\|\
eq 0\). Then, there must be a time-varying parameter vector \(\mathbf{\Phi}(t)\in\mathbf{R}^{L_{y}+L_{u}}\), which is called a pseudo gradient (PG), so that system can be transformed into the following FFDL data model:_  
\[y(t+1)=y(t)+\mathbf{\Phi}^{\mathrm{T}}(t)\Delta\mathbf{H}(t) \tag{4}\]  
_At any time \(t\), the parameter matrix \(\mathbf{\Phi}(t)=[\Phi_{1}(t),\cdots,\Phi_{L_{y}+L_{u}}(t)]^{\mathrm{T}}\) is bounded._  
**Remark 1**.: _Theorem 1 has been rigorously proven by [19]. Different FFDL data models can be obtained by selecting different linearization length constants. However, the increase in linearization length will inevitably cause difficulties in the design of the control system. The dimension of the PPJM will double with the increase in the system dimension and linearization length. On the other hand, the increase in linearization length does not necessarily improve the performance of the system because each system has a suitable linearized data model. Therefore, a reasonable choice of time-varying parameter vectors and length constants can improve the accuracy of the data model in system description._