## 2 Description of the Mathematical Model
### Design of the FSB-ASMC with \(H_{\infty}\) Tracking Performance  
Because the control voltage for the PM-ASS is specified and bounded, all of the states, which are real-world physical signals, are bounded. This implies that \(\overline{F}(\mathbf{z}(t))\) and \(\overline{G}(\mathbf{z}(t))\) are bounded and meet the Dirichlet conditions. If time is the input for the FNN, i.e., \(\chi=t\) in (28), and two Fourier-NNs are used to respectively approximate \(\overline{F}(\mathbf{z}(t))\) and \(\overline{G}(\mathbf{z}(t))\), Equation (21) is rewritten as:  
\[ \begin{array}{rl}\dot{z}_{3}&=(\mathbf{W}_{\overline{F}}^{T}\mathbf{q}_{ \overline{F}}(t)+\varepsilon_{\overline{G}}(t))+(\mathbf{W}_{\overline{G}}^{ T}\mathbf{q}_{\overline{G}}(t)+\varepsilon_{\overline{G}}(t))u\\ &=\mathbf{W}_{\overline{F}}^{T}\mathbf{q}_{\overline{F}}(t)+\mathbf{W}_{ \overline{G}}^{T}\mathbf{q}_{\overline{G}}(t)u+w_{t}\end{array}  \tag{32}\]  
where \(\mathbf{q}_{\overline{F}}(t)\) and \(\mathbf{q}_{\overline{G}}(t)\) are the family of orthogonal Fourier activation function vector, \(\mathbf{W}_{\overline{F}}\) and \(\mathbf{W}_{\overline{G}}\) are the weight vectors, and \(w_{t}=\varepsilon_{\overline{F}}(t)+\varepsilon_{\overline{G}}(t)u\) is the lumped uncertainty subject to \(|w_{t}|\leq w_{t}{}^{u}\), where \(w_{t}{}^{u}\) is a positive constant. The terms \(\mathbf{W}_{\overline{F}}^{T}\mathbf{q}_{\overline{F}}\) and \(\mathbf{W}_{\overline{G}}^{T}\mathbf{q}_{\overline{G}}\) produce an optimal mean-square approximation to the uncertain time-varying functions \(\overline{F}(\mathbf{z}(t))\) and \(\overline{G}(\mathbf{z}(t))\). **Assumption 1**. : _The lumped uncertainty in (19) is assumed such that \(w\in L_{2}[0,T]\), \(\forall T\in[0,\,\infty)\)._  
**Assumption 2**. : _The internal dynamics of the system (19) that uses adaptive sliding-mode control law are stable._  
**Theorem 1**. : _If a system can be expressed as a form of Equation (21) with unknown nonlinear time-varying functions \(\overline{F}(\mathbf{z}(t))\) and \(\overline{G}(\mathbf{z}(t))\) and has finite bandwidth. The sliding-mode control input is:_  
\[u=\frac{-\hat{\mathbf{W}}_{\overline{F}}^{T}\mathbf{q}_{\overline{F}}(t)- \sum\limits_{i=1}^{2}c_{i}\varepsilon_{i+1}-\sum\limits_{i=1}^{2}p_{2i} \varepsilon_{i}+y_{d}^{(3)}-\frac{s}{2\omega^{2}}}{\hat{\mathbf{W}}_{ \overline{G}}^{T}\mathbf{q}_{\overline{G}}(t)} \tag{33}\]  
_where \(\hat{\mathbf{W}}_{\overline{F}}^{T}\) and \(\hat{\mathbf{W}}_{\overline{G}}^{T}\) are the estimates of \(\mathbf{W}_{\overline{F}}^{T}\) and \(\mathbf{W}_{\overline{G}}^{T}\), respectively. According to Lemma 1, \(\mathbf{P}\succ 0\), \(\mathbf{P}\in R^{2\times 2}\) is selected to satisfy the Lyapunov matrix equation:_  
\[\mathbf{A}_{1}^{T}\mathbf{P}+\mathbf{P}\mathbf{A}_{1}=-\mathbf{Q}_{1} \tag{34}\]  
_where \(s\) is the sliding surface that is defined in Equation (23), \(p_{2i}\) are elements of \(\mathbf{P}\), \(\sigma\) is a designed constant, \(\mathbf{Q}_{t}\succ 0\) is given and the adaptive laws are:_  
\[\dot{\mathbf{W}}_{\overline{F}}=\mathbf{\eta}_{1}s\mathbf{q}_{\overline{F}}(t) \tag{35}\]  
_and:_  
\[\dot{\mathbf{W}}_{\overline{G}}=\mathbf{\eta}_{2}s\mathbf{q}_{\overline{G}}(t)u \tag{36}\]_where \(\mathbf{\eta}_{1}\) and \(\mathbf{\eta}_{2}\) (\(\mathbf{\eta}_{1}>0\) and \(\mathbf{\eta}_{2}>0\)) are the adaptation gain matrices.