## 2 System Model  
Consider the following strictly non-linear feedback system:  
\[\left\{\begin{array}{l}\dot{x}_{1}=f_{1}(x_{1})+h_{1}(x_{1})x_{2}\\ \vdots\\ \dot{x}_{i}=f_{i}(\textbf{x}_{i})+h_{i}(\textbf{x}_{i})x_{i+1}\\ \vdots\\ \dot{x}_{n}=f_{n}(\textbf{x}_{n})+h_{n}(\textbf{x}_{n})u\\ y=x_{1}\end{array}\right. \tag{1}\]  
where \(\textbf{x}_{i}=[x_{1},x_{2},\cdots,x_{i}]\in R^{i}\), \(i=1,2,\cdots,n\) is the system state variable, \(u\in R\) is the system control input, \(y\in R\) is the system output, \(f_{i}(\cdot)\), \(i=1,2,\cdots,n\) is the non-linear system mapping, \(h_{i}(\cdot)\), \(i=1,2,\cdots,n\) is the system's non-linear control gain function, and \(h_{i}(\cdot)\), \(\ldots\), \(h_{i}(\cdot)\) are not equal to 0.  
The main task of this paper is designing an MTN-based output feedback controller for the above-mentioned strictly non-linear feedback system, affording the output of the system \(y\) to track a given signal \(y_{d}\).  
Traditional control methods usually require all state variable information for such problems, that is, \(x_{1},x_{2},\cdots,x_{n}\). At the same time, the multi-step backstepping controller design suffers from error accumulation, and the process is complicated and cumbersome. Thus, this paper proposes a feedback algorithm based only on the output to simplify the control algorithm and reduce the calculation burden. In order to realize the control algorithm, the original strict feedback system needs to be transformed.  
We define the state variables as  
\[\left\{\begin{array}{l}z_{1}=x_{1}\\ \vdots\\ z_{i}=\dot{z}_{i-1}\ \ \ i=2,\cdots,n\\ y=x_{1}=z_{1}\end{array}\right. \tag{2}\]  
Then, there is  
\[\begin{array}{l}z_{2}=\dot{z}_{1}=\dot{x}_{1}\end{array} \tag{3}\]  
\[\begin{array}{l}z_{3}=\dot{z}_{2}=\frac{\partial f_{1}}{\partial x_{1}}\dot {x}_{1}+\frac{\partial h_{1}}{\partial x_{1}}\dot{x}_{1}x_{2}+h_{1}\dot{x}_{2 }\\ =\frac{\partial f_{1}}{\partial x_{1}}\dot{x}_{1}+\frac{\partial h_{1}}{\partial x _{1}}x_{1}x_{2}+h_{1}(f_{2}+h_{2}x_{3})\\ =(\frac{\partial f_{1}}{\partial x_{1}}+\frac{\partial h_{1}}{\partial x_{1}}x _{2})(f_{1}+h_{1}x_{2})+h_{1}f_{2}+h_{1}h_{2}x_{3}\end{array} \tag{4}\]  
Setting \(A_{2}=(\frac{\partial f_{1}}{\partial x_{1}}+\frac{\partial h_{1}}{\partial x _{1}}x_{2})(f_{1}+h_{1}x_{2})+h_{1}f_{2}\) and \(B_{2}=h_{1}h_{2}\), we obtain  
\[z_{3}=\dot{z}_{2}=A_{2}+B_{2}x_{3} \tag{5}\]  
By analogy, Equation (5) can be expressed as  
\[\dot{z}_{i}=A_{i}+B_{i}x_{i+1} \tag{6}\]\[\begin{array}{l}\text{where}\;\;A_{i}\;=\;\sum\limits_{k=1}^{i-1}{(\frac{ \partial A_{i-1}}{\partial x_{k}}+\frac{\partial B_{i-1}}{\partial x_{k}}x_{i})( f_{k}+h_{k}x_{k+1})+B_{i-1}f_{i}}\;\text{and}\;\;B_{i}\;=\;B_{i-1}h_{i}\;=\;B_{i-2}h_{i-1}h_{i} \;\text{,}\\ =\;\prod\limits_{k=1}^{i}{h_{k}}\text{.}\end{array}\]  
After the above changes, the original strict-feedback non-linear system can be expressed as  
\[\left\{\begin{array}{l}\dot{z}_{1}=z_{2}\\ \dot{z}_{2}=z_{3}\\ \vdots\\ \dot{z}_{i-1}=z_{i}i=2,\cdots,n-1\\ \vdots\\ \dot{z}_{n}=A_{n}+B_{n}u\\ y=z_{1}\end{array}\right. \tag{7}\]  
In addition, \(A_{n}\) and \(B_{n}\) include the unknown non-linear mappings \(f_{i}\) and \(h_{i}\) of the original system. Since \(h_{i}\) in the original hypothesis is not equal to 0, it is assumed that the gain function \(B_{n}\) is a bounded function greater than 0 and that \(0<B_{\min}\leq B_{n}\leq B_{\max}\), where \(B_{\min}\) and \(B_{\max}\) are constants greater than 0.  
After the transformation, the original strict feedback system has a general standard shape. Since \(z_{1}=x_{1}\), after the transformation, the system output is unchanged, and the original control target is consistent. However, \(A_{n}\) and \(B_{n}\) are unknown, and except for \(z_{1}\), the higher-order state \(z_{i}\) is unavailable, so a state observer needs to be designed.