## 4 Data-Based Model-Free Adaptive Controller
: To verify the convergence of the closed-loop tracking error and the convergence of the system's internal signals, the Lyapunov function is selected:  
\[u(k)=\varphi_{c}^{T}(k)\beta_{c}(k), \tag{36}\]4  
Therefore, the Lyapunov function differentiation is calculated as  
\[u(k)=\varphi_{c}^{T}(k)\beta_{c}(k), \tag{36}\]5  
Defining \(\tilde{\beta_{c}}(k+1)\) as the difference in the ideal weight vector \(\beta^{*}\) and the current iteration \(\beta(k+1)\), it is also established that  
\[u(k)=\varphi_{c}^{T}(k)\beta_{c}(k), \tag{36}\]6  
Substituting the update law (40) in the last equation  
\[u(k)=\varphi_{c}^{T}(k)\beta_{c}(k), \tag{36}\]7  
the weight vector error is also described as  
\[u(k)=\varphi_{c}^{T}(k)\beta_{c}(k), \tag{36}\]8  
In a similar sense, considering that the ideal weight produces the ideal controller and no tracking error, it is inferred that  
\[u(k)=\varphi_{c}^{T}(k)\beta_{c}(k), \tag{36}\]9  
hence,  
\[\beta_{c}(k+1)=\beta_{c}(k)-\eta\left[\frac{\partial E(k+1)}{ \partial\beta_{c}(k)}\right], \tag{37}\]0Substituting the last equation in the tracking error (35),  
\[\beta_{c}(k+1)=\beta_{c}(k)-\eta\left[\frac{\partial E(k+1)}{ \partial\beta_{c}(k)}\right], \tag{37}\]1  
is rearranged as  
\[\beta_{c}(k+1)=\beta_{c}(k)-\eta\left[\frac{\partial E(k+1)}{ \partial\beta_{c}(k)}\right], \tag{37}\]2  
With Equations (44) and (45), the Lyapunov function differentiation (43) is rewritten as  
\[\beta_{c}(k+1)=\beta_{c}(k)-\eta\left[\frac{\partial E(k+1)}{ \partial\beta_{c}(k)}\right], \tag{37}\]3  
hence, if we set  
\[\beta_{c}(k+1)=\beta_{c}(k)-\eta\left[\frac{\partial E(k+1)}{ \partial\beta_{c}(k)}\right], \tag{37}\]4  
and substitute it into the upgrade law (40), we obtain  
\[\beta_{c}(k+1)=\beta_{c}(k)-\eta\left[\frac{\partial E(k+1)}{ \partial\beta_{c}(k)}\right], \tag{37}\]5  
and according to (39), it is rearranged as  
\[\beta_{c}(k+1)=\beta_{c}(k)-\eta\left[\frac{\partial E(k+1)}{ \partial\beta_{c}(k)}\right], \tag{37}\]6  
If we define  
\[\beta_{c}(k+1)=\beta_{c}(k)-\eta\left[\frac{\partial E(k+1)}{ \partial\beta_{c}(k)}\right], \tag{37}\]7  
and substitute it in (50), we obtain the upgrade law in (41). From the previous boundary, it is derived that the Lyapunov differentiation (46) can be rewritten as  
\[\beta_{c}(k+1)=\beta_{c}(k)-\eta\left[\frac{\partial E(k+1)}{ \partial\beta_{c}(k)}\right], \tag{37}\]8 \[\beta_{c}(k+1)=\beta_{c}(k)-\eta\left[\frac{\partial E(k+1)}{ \partial\beta_{c}(k)}\right], \tag{37}\]9  
With the knowledge of the boundaries of function \(h_{e}(k+1)\), it is also deduced that \(|h_{e}(k+1)|\leq 1.2\). Replacing the boundaries, the last equation is rearranged as  
\[E(k+1)=\tanh\big{(}e(k+1)\big{)}e(k+1). \tag{38}\]0 \[E(k+1)=\tanh\big{(}e(k+1)\big{)}e(k+1). \tag{38}\]1  
For negative or positive constants \(a\) and \(b\), the inequality \((a+b)^{2}\leq 2a^{2}+2b^{2}\) is always met. With that property in mind, the last equation can be rewritten as  
\[E(k+1)=\tanh\big{(}e(k+1)\big{)}e(k+1). \tag{38}\]2 \[E(k+1)=\tanh\big{(}e(k+1)\big{)}e(k+1). \tag{38}\]3  
To ensure stability as in the previous equation, we must analyze the different term boundaries. The boundary of the tracking error \(e(k+1)\) is established from the equation  
\[E(k+1)=\tanh\big{(}e(k+1)\big{)}e(k+1). \tag{38}\]4  
and is defined as  
\[E(k+1)=\tanh\big{(}e(k+1)\big{)}e(k+1). \tag{38}\]5  
with \(\Omega_{e}\triangleq 1.44\eta^{2}g^{2}(k)\varphi_{c}^{2}(k)+2\varepsilon^{2}(k)\).