## 6 Adaptive Control Law Design
#### Controller Design and Stability Analysis  
In order to achieve the ultimate control goal, we define the error vector and the generalized Hurwitz polynomial  
\[\mathbf{e}=\mathbf{z}-\mathbf{x_{d}} \tag{32}\]  
where \(\mathbf{x_{d}}=[x_{d},\cdots,x_{d}^{(n-1)}]\) are the tracking target and its  
From Theorem 1, we know that \(\lambda_{\min}(P)>\kappa\), so we have  
\[\begin{array}{l}\dot{V}\leq-B_{n}(k-\varepsilon_{1}\frac{\lambda_{\max}([0, \mathbf{\Lambda}^{\mathsf{T}}])}{\lambda_{\min}([\mathbf{\Lambda}^{\mathsf{T}},\mathsf{1}])})\dot{\
u}^{2}+(B_{\max}+B_{\min})\Big{\|}\widetilde{\mathbf{W}}^{ \mathsf{T}}\Phi\Big{\|}\dot{\
u}|\\ +B_{n}\gamma_{\max}|\dot{\
u}|-B_{\min}\beta\kappa\Big{\|}\widetilde{\mathbf{W}} \Big{\|}^{2}+B_{\min}\beta\Big{\|}\widetilde{\mathbf{W}}\Big{\|}\delta_{\max}\end{array} \tag{49}\]  
By applying [PERSON]'s inequality to \(\Big{\|}\widetilde{\mathbf{W}}^{\mathsf{T}}\Phi\Big{\|}|\dot{\
u}|\), \(B_{n}\gamma_{\max}|\dot{\
u}|\), and \(\Big{\|}\widetilde{\mathbf{W}}\Big{\|}\delta_{\max}\), and substituting the result into the above formula, we obtain  
\[\begin{array}{l}\dot{V}\leq-\min\biggl{\{}B_{n}(k-\varepsilon_{1}\frac{ \lambda_{\max}([0,\mathbf{\Lambda}^{\mathsf{T}}])}{\lambda_{\min}([\mathbf{ \Lambda}^{\mathsf{T}},\mathsf{1}])})-\frac{2(B_{\max}+B_{\min})\|\Phi\|^{2}}{ \kappa B_{\min}\beta},\frac{\beta\kappa}{\lambda_{\max}([0,\mathbf{\Lambda}^ {\mathsf{T}}])}\biggr{\}}V\\ +\frac{B_{\max}^{2}\gamma_{\max}^{2}}{2B_{\min}(k-\varepsilon_{1}\frac{ \lambda_{\max}([0,\mathbf{\Lambda}^{\mathsf{T}}])}{\lambda_{\min}([\mathbf{ \Lambda}^{\mathsf{T}},\mathsf{1}])})}+\frac{B_{\min}\beta_{\max}^{2}}{\kappa} \end{array} \tag{50}\]  
Equation (50) reveals that by appropriately increasing the gain parameter \(k\) and the correction parameter \(\beta\), it can be ensured that  
\[\min\biggl{\{}B_{n}(k-\varepsilon_{1}\frac{\lambda_{\max}([0, \mathbf{\Lambda}^{\mathsf{T}}])}{\lambda_{\min}([\mathbf{\Lambda}^{\mathsf{ T}},\mathsf{1}])})-\frac{2(B_{\max}+B_{\min})\|\Phi\|^{2}}{\kappa B_{\min} \beta},\frac{\beta\kappa}{\lambda_{\max}([0,\mathbf{\Lambda}^{\mathsf{T}}])} \biggr{\}}>0 \tag{51}\] \[\frac{B_{\max}^{2}\gamma_{\max}^{2}}{2B_{\min}(k-\varepsilon_{1} \frac{\lambda_{\max}([0,\mathbf{\Lambda}^{\mathsf{T}}])}{\lambda_{\min}([ \mathbf{\Lambda}^{\mathsf{T}},\mathsf{1}])})}+\frac{B_{\min}\beta_{\max}^{2}}{ \kappa}>0 \tag{52}\]  
From the [PERSON] theorem, we know that the errors \(\hat{v}\) and \(\widetilde{\mathbf{W}}\) are bounded and converge to a compact set near the 0 point. At the same time, from Equations (35) and (43), and since \(\tilde{\mathbf{z}}\) is bounded, we conclude that \(\
u\), \(\mathbf{e}\), and \(\tilde{\mathbf{e}}\) are bounded and that the weight vector \(\hat{\mathbf{W}}\) is bounded. From Equation (41), the control signal \(u\) is bounded.  
The proof is completed. \(\square\)  
In this paper, the adaptive law based on an MTN ensures that the estimated weight vector approaches the true weight vector in a direction with infinitely small errors. At the same time, compared with the traditional dual-neural-network identification method, the number of calculations is reduced, and the identification process of the unknown dynamics of the entire system is completed.