## 6 Adaptive Control Law Design  
By adjusting \(\hat{\mathbf{W}}\) to make it infinitely close to \(\mathbf{W}^{*}\), we finally achieve the control purpose. To this end, we designed an error-based adaptive adjustment rate that is \(\widehat{\mathbf{W}}=\mathbf{W}^{*}-\hat{\mathbf{W}}\).  
Parameter definition:  
\[F=\frac{\beta\Psi_{\theta}}{\Psi_{\theta}^{\mathrm{T}}\Psi_{\theta}+\gamma} \tag{20}\]  
where \(\beta\) and \(\gamma\) are positive constants.  
We design two auxiliary variables, \(P\in R^{2N\times 2N}\) and \(Q\in R^{2N\times 1}\), based on \(F\):  
\[\left\{\begin{array}{ll}\dot{P}=-\beta P+F\Psi_{\theta}^{\mathrm{T}}&P(0)=0 \\ \dot{Q}=-\beta Q+Fu_{\theta}^{\mathrm{T}}&Q(0)=0\end{array}\right. \tag{21}\]  
Since both \(\beta\) and \(\gamma\) are greater than 0, it can be guaranteed that \(P\) and \(Q\) are both bounded.  
We calculate the above formula to obtain  
\[Q=P\mathbf{W}^{*}-\int_{0}^{t}e^{\beta(\tau-t)}F\lambda d\tau \tag{22}\]  
By setting \(\delta=\int_{0}^{t}e^{\beta(\tau-t)}F\lambda d\tau\), we obtain  
\[Q=P\mathbf{W}^{*}-\delta \tag{23}\]  
where the norm of \(\delta\) is a bounded function, that is, \(\|\delta\|\leq\delta_{\text{max}}\).  
The error vector is defined as  
\[S=P\mathbf{\hat{W}}-Q \tag{24}\]  
By subtracting the above two formulas, we obtain  
\[S=\delta-P\widetilde{\mathbf{W}} \tag{25}\]  
Then, after the low-pass filter, the auxiliary variables \(F\), \(P\), and \(Q\) are calculated and \(S\) is obtained. An adaptive rate based on \(S\) can be designed as  
\[\Delta\mathbf{\hat{W}}=-\lambda\cdot S \tag{26}\]  
where \(\lambda>0\) is an adaptive adjustment step.  
**Theorem 1**.: _If the above-mentioned adaptive rate is used, the weight error vector finally converges near the 0 point under the condition of a continuous excitation of \(\Psi_{\theta}\)._  
Proof.: We define the Lyapunov function as  
\[V=\frac{1}{2}\widetilde{\mathbf{W}}^{\mathrm{T}}Step^{-1}\widetilde{\mathbf{W}} \tag{27}\]  
Deriving the above formula provides  
\[\dot{V}=\widetilde{\mathbf{W}}^{\mathrm{T}}Step^{-1}\dot{\widetilde{\mathbf{W}}}= \widetilde{\mathbf{W}}^{\mathrm{T}}\delta-\widetilde{\mathbf{W}}^{\mathrm{T}}P \widetilde{\mathbf{W}} \tag{28}\]  
Since \(\Psi_{\theta}\) continues to excite, there is a normal number, \(\kappa\), for \(\forall t>0\):  
\[\int_{t}^{t+\Delta t}F\Psi_{\theta}^{\mathrm{T}}d\tau\geq\kappa I \tag{29}\]  
It can be seen from the auxiliary variable \(P\) that  
\[P(t)\geq e^{-\beta\Delta t}\int_{t-\Delta t}^{t}F\Psi_{\theta}^{\mathrm{T}}d \tau\geq e^{-\beta\Delta t}\kappa \tag{30}\]For simplicity, we define \(\varsigma=e^{-\beta\Delta t}\kappa\) and obtain  
\[\dot{V}=\widetilde{\mathbf{W}}^{\mathsf{T}}\delta-\widetilde{\mathbf{W}}^{\mathsf{T}}P \widetilde{\mathbf{W}}\leq\frac{\delta_{\max}^{2}}{2\varsigma}-\frac{\varsigma}{ \lambda_{\max}(Step^{-1})}V \tag{31}\]  
Since both \(\frac{\delta_{\max}^{2}}{2\varsigma}\) and \(\frac{\varsigma}{\lambda_{\max}(Step^{-1})}\) are greater than 0, it can be known from the Lyapunov theorem that \(V\) and the error vector \(\widetilde{\mathbf{W}}\) converge to around 0 according to the exponential law.  
The proof is complete. \(\square\)  
Compared with the traditional gradient method, the auxiliary variables \(F\), \(P\), and \(Q\) are constructed in this paper, and then an adaptive rate based on \(S\) is designed, ensuring that the error vector \(\widetilde{\mathbf{W}}\) converges to near 0, i.e., \(\hat{\mathbf{W}}\) is infinitely close to \(\mathbf{W}^{*}\). Thus, the system estimation affords guaranteed accuracy.