## 3 Adaptive-Dynamic-Programming-Based Robust Control Design
### Adaptive-Dynamic-Programming-Based Robust Control Law Design
Based on Assumptions 2 and 3, it is reasonable to assume that \(\frac{1}{2}\
abla\sigma_{i}(\mathbf{e}_{i})\mathbf{g}_{i}\mathbf{R}_{i}^{-1}\mathbf{g}_{i}^{ \mathrm{T}}\
abla^{\mathrm{T}}\mathbf{\sigma}_{i}(\mathbf{e}_{i})\), \(\frac{1}{2}\
abla^{\mathrm{T}}\mathbf{\varepsilon}_{i}\mathbf{g}_{i}\mathbf{R}_{i}^{-1} \mathbf{g}_{i}^{\mathrm{T}}\
abla\mathbf{\varepsilon}_{i}\), and \((\
abla^{\mathrm{T}}\sigma_{i}(\mathbf{e}_{i})\mathbf{W}_{i}+\
abla^{\mathrm{T}}\mathbf{ \varepsilon}_{i})\mathbf{g}_{i}\) are norm-bounded. From Theorem 1, since the disturbance error bounds are fixed-time stable, we can assume that \(\tilde{\mathbf{d}}_{i}\) are also norm-bounded, that is \(\left\|\tilde{\mathbf{d}}_{i}\right\|\leq d_{iM}\) with \(d_{iM}\) being positive constants. Then, one has  
\[\left\|\frac{1}{2}\
abla\sigma_{i}(\mathbf{e}_{i})\mathbf{g}_{i}\mathbf{R}_{i}^{-1}\mathbf{g}_{ i}^{\mathrm{T}}\
abla^{\mathrm{T}}\mathbf{\sigma}_{i}(\mathbf{e}_{i})\right\|\leq b_{\mathbf{D}_{i}}, \left\|\frac{1}{2}\
abla^{\mathrm{T}}\mathbf{\varepsilon}_{i}\mathbf{g}_{i}\mathbf{R}_{i}^{-1 }\mathbf{g}_{i}^{\mathrm{T}}\
abla\mathbf{\varepsilon}_{i}+\mathbf{\phi}_{i4}\right\|\leq\alpha _{i8}\]  
where \(b_{\mathbf{D}_{i}}\) are positive constants. Consider the inequality \(\|\mathbf{\beta}_{1}\|\leq\
abla\sigma_{i}\|\mathbf{e}_{i}\|\); the following inequalities can be obtained by using [PERSON]'s inequality:  
\[-k_{i3}\rho_{i2}\tilde{\mathbf{W}}_{ic}^{\mathrm{T}}\mathbf{\beta}_{i1} \leq\frac{\
abla\sigma_{i}^{2}}{2}\|\tilde{\mathbf{W}}_{ic}\|^{2}+\frac{k_{i3} \rho_{i2}^{2}}{2}\|\mathbf{e}_{i}\|^{2}, \tag{30}\] \[L_{i3}=V^{*}(\mathbf{e}_{i})+\frac{\rho_{i1}}{2}\tilde{\mathbf{W}}_{ia}^{\mathrm{T}} \tilde{\mathbf{W}}_{ia}+\frac{\rho_{i2}}{2}\tilde{\mathbf{W}}_{ic}^{\mathrm{T}}\tilde {\mathbf{W}}_{ic},\]0In addition, there exist constants \(b_{iQ}\) satisfying \(b_{iQ}<\lambda_{\min}(\mathbf{Q}_{i})\). By utilizing the above inequalities (30) and (31), one can further obtain that  
\[ \begin{split}\dot{L}_{i3}\leq&-(b_{iQ}-\frac{k_{i3}^{2 }\rho_{i1}^{2}}{2}-\frac{k_{i3}^{2}\rho_{i2}^{2}}{2})\|\mathbf{e}_{i}\|^{2}-\frac{k_ {i2}\rho_{i2}-k_{i1}\rho_{i2}\gamma_{i}-b_{\mathbf{D}_{i}}-\
abla\sigma_{i}^{2}}{2 }\|\hat{\mathbf{W}}_{ia}\|^{2}+\alpha_{i8}\\ &-\frac{k_{i1}\rho_{i1}r_{i2}-k_{i2}\rho_{i2}-\
abla\bar{\sigma}_ {i}^{2}}{2}\|\hat{\mathbf{W}}_{ic}\|^{2}+\alpha_{i9},\end{split}  \tag{32}\]  
where \(\gamma_{i}=\Lambda_{\max}=r_{i1}\|\mathbf{\beta}_{i1}\|_{\max}+r_{i2}\) and \(\alpha_{i9}=\frac{k_{i1}\rho_{i1}\gamma_{i}}{2}\tilde{W}_{i}^{2}\). Then, we can select the parameters \(k_{i1},~{}k_{i2}\), \(k_{i3}\), such that \(2b_{iQ}-k_{i3}^{2}\rho_{i1}^{2}-k_{i3}^{2}\rho_{i2}^{2}>0\), \(k_{i2}\rho_{i2}-k_{i1}\rho_{i2}\gamma_{i}-b_{\mathbf{D}_{i}}-\
abla\sigma_{i}^{2}>0\), and \(k_{i1}\rho_{i1}r_{i2}-k_{i2}\rho_{i2}-\
abla\sigma_{i}^{2}>0\) hold. Furthermore, one can conclude that \(\dot{L}_{i3}\leq 0\) when the condition:  
\[\left\{ \begin{aligned} \|\mathbf{e}_{i}\|>\sqrt{\frac{2\alpha_{i8}}{2b_{iQ} -k_{i2}^{2}\rho_{i1}^{2}-k_{i2}^{2}\rho_{i2}^{2}}}\\ \|\hat{\mathbf{W}}_{ia}\|>\sqrt{\frac{2\alpha_{i9}}{k_{i2}\rho_{i2}-k_{i1} \rho_{i2}\gamma_{i}-b_{\mathbf{D}_{i}}-\
abla\sigma_{i}^{2}}}\text{ or }\|\hat{\mathbf{W}}_{ic}\|>\sqrt{\frac{2\alpha_{i9}}{k_{i1}\rho_{i1}r_{i2}-k_{i2} \rho_{i2}-\
abla\sigma_{i}^{2}}}\end{aligned} \right.\]  
is satisfied. Therefore, (32) indicates that the error states of the systems (7) and the AC NNs' weight errors are UUB stable. **Remark 2**. : _Different from [16; 25], we designed weight update laws of the AC NNs that avoid the use of the PE condition by using an adaptive method, where \(\Lambda_{i}\) are designed to guarantee stable convergence performances of \(\hat{\mathbf{W}}_{ic}\) and \(\hat{\mathbf{W}}_{ia}\). Moreover, the positive constants \(r_{i2}\) are introduced in \(\Lambda_{i}\) to guarantee \(\hat{\mathbf{W}}_{ic}\) and \(\hat{\mathbf{W}}_{ia}\) can converge to a neighborhood of the origin when \(\|\hat{\mathbf{P}}_{i1}\|=0\)._  
**Remark 3**.