## 3 Adaptive-Dynamic-Programming-Based Robust Control Design  
### Online-Uncertainty-Compensation-Based Fixed-Time NN-Based Observers  
Consider the systems (7) with uncertainties; the auxiliary systems are introduced as  
\[\hat{\mathbf{e}}_{i}=\mathbf{f}(\mathbf{e}_{i})+\mathbf{g}_{i}(\mathbf{x}_{i}+\hat{\mathbf{d}}_{i}), \tag{9}\]  
where \(\hat{\mathbf{d}}_{i}\) are the estimated values of the lumped uncertainties. Define the auxiliary variables as \(\Xi_{i}=\mathbf{g}_{i}^{\intercal}(\mathbf{e}_{i}-\hat{e}_{i})\) with \(\mathbf{g}_{i}^{\intercal}\) being pseudo inverse matrices of \(\mathbf{g}_{i}\). Consider the Assumptions 1 and 2; this yields  
\[\|\mathbf{d}_{i}\|\leq h_{i}H_{i}(\mathbf{S}_{i})=\check{\varsigma}_{i}(\mathbf{S}_{i})\]  
with  
\[h_{1}=\max\{u_{\hat{d}},\alpha_{2},\alpha_{3},\alpha_{4}\},h_{2} =\max\{\tau_{\hat{d}},\alpha_{5},\alpha_{6},\alpha_{7}\},\] \[H_{1}(\mathbf{S}_{1})=1+\|\mathbf{\zeta}\|+\left\|\mathbf{\zeta}\right\|^{2},H_{2}(\mathbf{S}_{2})=1+\|\mathbf{\Theta}\|+\left\|\hat{\mathbf{\Theta}}\right\|^{2},\]  
where \(H_{i}(\mathbf{S}_{i})\) are defined as the core functions [44]. Furthermore, they are dependent on \(\mathbf{S}_{1}=[\mathbf{\zeta},\hat{\mathbf{\zeta}},\hat{\mathbf{\zeta}}^{2}]^{\mathrm{T}}\) and \(\mathbf{S}_{2}=[\mathbf{\Theta},\hat{\mathbf{\Theta}},\hat{\mathbf{\Theta}}^{2}]^{\mathrm{T}}\). As pointed out in [45; 46], the continuous nonlinear scalar functions \(\check{\varsigma}_{i}(\mathbf{S}_{i})\) can be approximated using NNs with Gaussian basis functions:  
\[\check{\varsigma}_{i}(\mathbf{S}_{i})=\mathbf{W}_{fi}^{\mathrm{T}}\mathbf{Z}(\mathbf{S}_{i})+ \varepsilon_{fi},\]  
where \(\mathbf{Z}(\mathbf{S}_{i})=[Z_{1}(\mathbf{S}_{i}),Z_{2}(\mathbf{S}_{i}),\cdots,Z_{n}(\mathbf{S}_{ i})]^{\mathrm{T}}\) are activation function vectors. Furthermore, \(Z_{n}(\mathbf{S}_{i})\) are expressed in detail as  
\[Z_{n}(\mathbf{S}_{i})=\exp\left(-\frac{\|\mathbf{S}_{i}-\mathbf{\iota}_{in}\|^{2}}{h_{in }^{2}}\right),\]  
where \(\mathbf{\iota}_{in}\) and \(h_{in}\) are the center vectors and width and \(n\) is the sum of the number of the Gaussian basis functions. It is straightforward to show that \(\mathbf{Z}(\mathbf{S}_{i})\) have norm upper bounds \(\varepsilon_{i}\), that is \(\|\mathbf{Z}(\mathbf{S}_{i})\|<\varepsilon_{i}<\infty\) with \(\varepsilon_{i}\) are positive constants. \(\mathbf{W}_{fi}\) denote the optimal weight vectors, and \(\varepsilon_{fi}\) are the approximation errors, in which \(\mathbf{W}_{fi}\) and \(\varepsilon_{fi}\) are bounded, i.e., \(\left\|\mathbf{W}_{fi}\right\|\leq\tilde{W}_{fi}\) and \(\left\|\varepsilon_{fi}\right\|\leq\tilde{\varepsilon}_{fi}\). In order to reduce the number of parameters for online calculation, the following procedure is designed. Firstly, we have  
\[\|\mathbf{d}_{i}\|\leq\check{\varsigma}_{i}(\mathbf{S}_{i})\leq\left\|\mathbf{W}_{fi}^{ \mathrm{T}}\right\|\|\mathbf{Z}(\mathbf{S}_{i})\|+\left\|\varepsilon_{fi}\right\|\leq b _{i}Z_{fi}\]  
with \(b_{i}=\max\{\left\|\mathbf{W}_{fi}\right\|,\left\|\varepsilon_{fi}\right\|\},\ Z_{fi}=1+\|\mathbf{Z}(\mathbf{S}_{i})\|\). Then, the FTNNOBs are established as  
\[\hat{\mathbf{d}}_{i}=r_{i3}\mathrm{sig}^{\frac{\pi_{i}}{\pi_{i}}}(\Xi_{i})+r_{i4} \mathrm{sig}^{\frac{\pi_{i}}{q_{i}}}(\Xi_{i})+c_{i}\check{l}_{i}\Xi_{i}Z_{fi}^ {2}, \tag{10}\]  
where \(m_{i},n_{i},p_{i}\) and \(q_{i}\) are positive odd integers satisfying \(m_{i}<n_{i},p_{i}>q_{i}\). \(r_{i3},r_{i4}\), and \(c_{i}\) are positive constants. \(\check{l}_{i}\) are the estimated values of \(l_{i}=b_{i}^{2}\), which are tuned by the following adaptive laws:  
\[\dot{l}_{i}=-\varrho_{i1}\dot{l}_{i}+c_{i}\|\Xi_{i}\|^{2}Z_{fi}^{2}\left(\dot{ l}_{i}(0)\geq 0\right) \tag{11}\]  
with \(\varrho_{i1}\) being positive constants. The estimation errors of \(l_{i}\) are defined as \(\check{l}_{i}=l_{i}-\check{l}_{i}\). We have the following result for the proposed procedure.