## 3 Control Algorithm Synthesis
### Adaptive Prescribed Performance Control
\begin{array}{rl}\dot{V}_{3}&=e_{2}\Big{(}-\theta_{2}^{T}\chi+\theta_{f}+\theta_ {u}u-\dot{x}_{2,d}\Big{)}+\gamma_{f}\overset{\sim}{\theta}\dot{\theta}\dot{ \theta}+\gamma_{u}\overset{\sim}{\theta}_{u}\dot{\theta}_{u}+\overset{\sim}{ \theta}_{2}^{T}\mathbf{\Gamma}_{2}\dot{\theta}_{2}\\ &=-k_{2}e_{2}^{2}+e_{2}\Big{(}-\overset{\sim}{\theta}_{2}^{T}\chi+\overset{ \sim}{\theta}_{f}+\overset{\sim}{\theta}_{u}u\Big{)}+\gamma_{f}\overset{\sim }{\theta}_{f}\dot{\theta}_{f}+\gamma_{u}\overset{\sim}{\theta}_{u}\dot{\theta}_ {u}+\overset{\sim}{\theta}_{2}^{T}\mathbf{\Gamma}_{2}\dot{\theta}_{2}.\end{array}  \tag{37}\]  
To ensure \(\dot{V}_{3}\leq-k_{2}e_{2}^{2}\) with \(k_{2}>0\), the following adaptive law for updating the estimators is employed:  
\[ \begin{array}{rl}\dot{\theta}_{2}&=-\mathbf{\Gamma}_{2}^{-1}\big{(}\mathbf{x}\mathbf{e}_ {2}+\mathbf{\eta}_{2}\dot{\theta}_{2}\big{)}\\ \dot{\theta}_{f}&=-\frac{1}{\gamma_{f}}\Big{(}e_{2}+\eta_{f}\dot{\theta}_{f} \Big{)}\\ \dot{\theta}_{u}&=-\frac{1}{\gamma_{u}}\big{(}ue_{2}+\eta_{u}\ddot{\theta}_{u} \big{)},\end{array}  \tag{38}\]  
where \(\gamma_{f}\), \(\gamma_{u}\), \(\eta_{z}\), \(\eta_{f}\) and \(\eta_{u}\) are positive design parameters, \(\mathbf{\Gamma}_{2}\) is a positive definite matrix. The structured uncertainties are characterized by a linear separation of unknown parameters from known basis functions. This property is leveraged to design adaptive controllers with stability guarantees. The adaptive law (38) exploits the linear parametric structure of \(\mathbf{\theta}_{2}\) in Equation (17). By estimating \(\mathbf{\theta}_{2}\). via gradient-based update driven by regressor \(\chi\) and error \(e_{2}\), the controller compensates structured uncertainties directly. By comparing the steps of the control strategy with the content in Figure 3, it can be seen that Step 1 corresponds to the \"State monitor\"; Step 2 is responsible for computing the ideal control strategy, i.e., Equation (34), corresponding to the \"Adaptive backstepping control\"; and Step 3 corresponds to the dashed arrow, which compensates for the system's parametric uncertainties through parameter adaptation. Hence, the development of the adaptive prescribed performance controller can be summarized as the following theorem:  
**Theorem 1**. : _For the controlled system in Equation (17), the proposed controller (35) with the adaptive law (38) can guarantee that all state variables \(x_{i}\)\((i=1,2)\) are bounded. Furthermore, \(x_{1}\) is retained within the prescribed bound as defined in Equation (26)._  
Proof of Theorem 1.: Considering the following Lyapunov function:  
\[V=\frac{1}{2}\varepsilon_{1}^{2}+\frac{1}{2}\varepsilon_{2}^{2}+\frac{1}{2} \gamma_{f}\overset{\sim}{\theta}_{f}^{2}+\frac{1}{2}\gamma_{u}\overset{\sim}{ \theta}_{u}^{2}+\frac{1}{2}\overset{\sim}{\theta}_{2}^{T}\mathbf{\Gamma}_{2}\overset {\sim}{\theta}_{2}. \tag{39}\]  
The time derivative of this function is given as  
\[ \begin{array}{rl}\dot{V}=\varepsilon_{1}\dot{\varepsilon}_{1}+e_{2}\dot{ \varepsilon}_{2}+\gamma_{f}\overset{\sim}{\theta}_{f}\dot{\theta}_{f}+\gamma_ {u}\overset{\sim}{\theta}_{u}\dot{\theta}_{u}+\overset{\sim}{\theta}_{2}^{T} \mathbf{\Gamma}_{2}\dot{\theta}_{2}\\ =\varepsilon_{1}r_{1}\Big{(}x_{2}-\dot{x}_{1,d}-\frac{\dot{\varphi}_{1}}{ \varphi_{1}}e_{1}\Big{)}+\varepsilon_{2}\Big{(}-\theta_{2}^{T}\chi+\theta_{f} +\theta_{zu}u-\dot{x}_{2,d}\Big{)}+\gamma_{f}\overset{\sim}{\theta}_{f}\dot{ \theta}_{f}+\gamma_{u}\overset{\sim}{\theta}_{u}\dot{\theta}_{u}+\overset{ \sim}{\theta}_{2}^{T}\mathbf{\Gamma}_{2}\dot{\theta}_{2}.\end{array}