## 3 Adaptive-Dynamic-Programming-Based Robust Control Design
### Online-Uncertainty-Compensation-Based Fixed-Time NN-Based Observers
According to [PERSON]'s inequality, one has  
\[h_{1}=\max\{u_{\hat{d}},\alpha_{2},\alpha_{3},\alpha_{4}\},h_{2} =\max\{\tau_{\hat{d}},\alpha_{5},\alpha_{6},\alpha_{7}\},\]2  
Substituting (15) into (14), one can obtain that  
\[h_{1}=\max\{u_{\hat{d}},\alpha_{2},\alpha_{3},\alpha_{4}\},h_{2} =\max\{\tau_{\hat{d}},\alpha_{5},\alpha_{6},\alpha_{7}\},\]3 \[h_{1}=\max\{u_{\hat{d}},\alpha_{2},\alpha_{3},\alpha_{4}\},h_{2} =\max\{\tau_{\hat{d}},\alpha_{5},\alpha_{6},\alpha_{7}\},\]4  
If \(\dot{I}_{i}^{2}\geq 1\), one can obtain that  
\[h_{1}=\max\{u_{\hat{d}},\alpha_{2},\alpha_{3},\alpha_{4}\},h_{2} =\max\{\tau_{\hat{d}},\alpha_{5},\alpha_{6},\alpha_{7}\},\]5  
If \(\dot{I}_{i}^{2}<1\), one can obtain that  
\[h_{1}=\max\{u_{\hat{d}},\alpha_{2},\alpha_{3},\alpha_{4}\},h_{2} =\max\{\tau_{\hat{d}},\alpha_{5},\alpha_{6},\alpha_{7}\},\]6  
On account of the inequalities \(\left|\dot{I}_{i}\right|=\left|\dot{I}_{i}-\dot{I}_{i}\right|\leq\left|\dot{I}_ {i}\right|\), \(0<\dot{I}_{i}<\dot{I}_{i}\) and according to Lemmas 2 and 3, one can further obtain that  
\[h_{1}=\max\{u_{\hat{d}},\alpha_{2},\alpha_{3},\alpha_{4}\},h_{2} =\max\{\tau_{\hat{d}},\alpha_{5},\alpha_{6},\alpha_{7}\},\]7  
where \(\pi_{i1}=\min\{r_{i3}2^{a_{i2}},\ \varrho_{i1}2^{a_{i2}}\}\), \(\pi_{i2}=\min\{r_{i4}2^{a_{i4}},\ \varrho_{i1}2^{a_{i4}}\}\), \(a_{i3}=\frac{q_{i}-p_{i}}{2\dot{q}_{i1}}\), and \(\vartheta_{i1}=\frac{1}{4c_{i}}+a_{i1}+\varrho_{i1}(\frac{p_{i}}{2}+l_{i}^{2a _{i4}})\). Based on Lemma 1, \(L_{i2}\) will reduce to a residual set in fixed time. The settling time can be achieved by \(t_{if}\leq\frac{1}{\pi_{i1}\omega_{i}(1-a_{i2})}+\frac{1}{\pi_{i2}2^{a_{i3}} \omega_{i}(a_{i4}-1)}\) with \(0<\omega_{i}<1\). Defining an augmented vector \(\hat{\mathbf{X}}_{i}=[\mathbf{\Xi}_{i}^{\mathrm{T}},\dot{I}_{i}]^{\mathrm{T}}\), \(L_{i2}\) is also represented as \(L_{i2}=\frac{1}{2}\hat{\mathbf{X}}_{i}^{\mathrm{T}}\hat{\mathbf{X}}_{i}\), and it is bounded by  
\[h_{1}=\max\{u_{\hat{d}},\alpha_{2},\alpha_{3},\alpha_{4}\},h_{2} =\max\{\tau_{\hat{d}},\alpha_{5},\alpha_{6},\alpha_{7}\},\]8  
where \(\vartheta_{i2}=\Big{(}\frac{\varrho_{i0}}{\pi_{i1}}\Big{)}^{\frac{1}{4\beta_{i 2}}}\) and \(\vartheta_{i3}=\Big{(}\frac{\varrho_{i0}}{\pi_{i2}2^{a_{i3}}}\Big{)}^{\frac{1}{ \beta_{i}4}}\) with \(\varrho_{i0}=\frac{\vartheta_{i1}}{1-\omega_{i}}\). Then, one can conclude that  
\[h_{1}=\max\{u_{\hat{d}},\alpha_{2},\alpha_{3},\alpha_{4}\},h_{2} =\max\{\tau_{\hat{d}},\alpha_{5},\alpha_{6},\alpha_{7}\},\]9  
where \(\tilde{\vartheta}_{i1}=\min\{\vartheta_{i2},\vartheta_{i3}\}\). Consider the definition of \(\Xi_{i}\); one has  
\[H_{1}(\mathbf{S}_{1})=1+\|\mathbf{\zeta}\|+\left\|\mathbf{\zeta}\right\|^{2},H_{2}(\mathbf{S}_{2})=1+\|\mathbf{\Theta}\|+\left\|\hat{\mathbf{\Theta}}\right\|^{2},\]0Then, according to the result (16), the upper norm bounds of \(\mathbf{\tilde{d}}_{i}\) are given by  
\[H_{1}(\mathbf{S}_{1})=1+\|\mathbf{\zeta}\|+\left\|\mathbf{\zeta}\right\|^{2},H_{2}(\mathbf{S}_{2})=1+\|\mathbf{\Theta}\|+\left\|\hat{\mathbf{\Theta}}\right\|^{2},\]1 \[H_{1}(\mathbf{S}_{1})=1+\|\mathbf{\zeta}\|+\left\|\mathbf{\zeta}\right\|^{2},H_{2}(\mathbf{S}_{2})=1+\|\mathbf{\Theta}\|+\left\|\hat{\mathbf{\Theta}}\right\|^{2},\]2  
Based on the above analysis and the selection pf appropriate parameter values, the uncertainties' observation errors \(\mathbf{\tilde{d}}_{i}\) will converge to a bounded region of origin in a fixed time. **Remark 1**. : _Compared with previous NN-based observer methods, it is obvious that the proposed FTNNOBs only need to estimate a scalar parameter online instead of a weight vector or matrix, which reduces the computational complexities. Moreover, the fixed-time convergence of the estimation error is guaranteed._