## 3 Control Design and Stability Analysis
### Controller Design  
The controller design uses [PERSON] theory, with the adaptive backstepping strategy as framework. New states \(z_{1}\) and \(z_{2}\) are defined as function of \(x_{1}\), \(x_{2}\), and updated parameters. Furthermore, an overall Lyapunov function \(V\) is defined as the sum of \(V_{z1}\), the dead zone Lyapunov function for \(z_{1}\); \(V_{z2}\), the dead zone Lyapunov function for \(z_{2}\); and a Lyapunov function for each parameter updating error. Differential equations are defined for the new states, and the time derivatives of the Lyapunov functions are defined. The mechanisms for the updated parameters and the control input \(u\) are chosen such that \(dV/dt\) is negative semi-definite, thus implying the asymptotic convergence of \(z_{1}\). Other important features of the developed procedure area new auxiliary system of second order is proposed, whose input includes the control signal error \(\Delta u\), which is the difference between the constrained and the unconstrained control signals;
* a modified tracking error \(z_{1}\) is defined as the sum of the regular tracking error and the state of the auxiliary system;
* the definition the \(z_{i}\) states is based on the adaptive state backstepping method;
* dead zone radially unbounded quadratic forms are used instead of current quadratic forms; and
* a new treatment of the \(b\Delta u\) term is proposed, including a new parameterization of the unknown model parameters, and the formulation of a new auxiliary system. In a basic adaptive backstepping control design, the \(z_{1}\) state would be defined as the tracking error, \(z_{1}=x_{1}-y_{d}\). In contrast, we define \(z_{1}\) by adding \(-\psi_{1}\) to the tracking error:  
\[z_{1}=x_{1}-y_{d}-\psi_{1} \tag{6}\]  
where the \(\psi_{1}\) state is the output of a stable second order linear filter whose input contains the input error \(\Delta u\), the difference between the non-saturated and the saturated input signals. The controller design is aimed at driving \(z_{1}\) to \(\Omega_{z1}\), \(\Omega_{z1}=\{z_{1}:|z_{1}|\leq C_{b}\}\). The advantages of the definition of \(z_{1}\) (6) and the controller design based on \(z_{1}\) instead of the tracking error \(e\) are (i) excessive increase of the updated parameters is avoided and (ii) the convergence region of the tracking error \(e=x_{1}-y_{d}\) in presence of input saturation can be determined. The time derivative of Equation (6) is  
\[\dot{z}_{1}=\dot{x}_{1}-\dot{y}_{d}-\dot{\psi}_{1} \tag{7}\]  
Incorporating the \(\dot{x}_{1}\) expression (1) yields  
\[\dot{z}_{1}=a_{1}x_{2}-a_{2}x_{1}-r_{x1}-\dot{y}_{d}-\dot{\psi}_{1} \tag{8}\]  
Let  
\[V_{z1}=\left\{ \begin{array}{l}(1/3)(z_{1}-C_{b})^{3}\;\;\mbox{for}\;\;z_{1} \geq C_{b}\\ 0\;\;\mbox{for}\;\;z_{1}\in(-C_{b},C_{b})\\ (1/3)(-1)(z_{1}+C_{b})^{3}\;\;\mbox{for}\;\;z_{1}\leq-C_{b}\end{array} \right. \tag{9}\]  
This truncated Lyapunov function is inspired on that of [20, 21]. Early versions of dead zone Lyapunov functions are presented in [17, 22, 23]; versions for backstepping-based controllers in [21, 24], and other versions in [20, 25, 26]. The use of the dead zone Lyapunov function (9) allows designing the adaptive controller, tackling the presence of unknown varying terms or parameters, and avoiding the use of discontinuous signals. The main properties of \(V_{z1}\) (9) are  
\[V_{z1}=0\;\mbox{for}\;z_{1}\in[-C_{b},\;C_{b}]\] \[V_{z1}>0\;\mbox{for}\;z_{1}\
ot\in[-C_{b},\;C_{b}]\] \[V_{z1}\;\mbox{is continuous with respect to}\;z_{1},\;\;\mbox{and it is bounded for}\;z_{1}\;\mbox{bounded}\]  
The above properties and a stable dynamics of \(V_{z1}\) imply the convergence of \(z_{1}\) to \(\Omega_{z1}\), \(\Omega_{z1}=\{z_{1}:|z_{1}|\leq C_{b}\}\), as is shown in the convergence theorem in Section 3.2. Differentiating (9) with respect to time yields  
\[\dot{V}_{z1}=f_{z1}\dot{z}_{1} \tag{10}\] \[f_{z1}=\frac{dV_{z1}}{dz_{1}}=\left\{