## 3 Fault-Tolerant Output Regulation Scheme  
The output regulation for a nonlinear system (1) or its linear approximation (3) with \(f_{a}(t)=0\) requires the existence of an exosystem for the generation of references/perturbations \(w(t)\); traditionally, such system is known [4]. An interesting case arises when this exosystem is unknown; according to [24], the tracking signals can be represented by \(\Psi(t)\in\mathbb{R}^{p+d}\), as long as they are available for measurement, smooth and continuously differentiable, i.e.,  
\[\Psi(t)= \begin{bmatrix}\psi_{1}&\psi_{2}&\ldots&\psi_{p}&\ldots&\psi_{p+d} \end{bmatrix} ^{\top}, \tag{4}\]where \(\psi_{1}\), \(\psi_{2},\ldots,\psi_{p}\) are reference signals to be tracked and \(\psi_{p+1},\ldots,\psi_{p+d}\) are the disturbance signals to be rejected. Thus, following [24], the solution to the output regulation problem for unknown reference/disturbance signals for (3) consists in constructing the exosystem by estimating the states and outputs of \(\Psi(t)\) through the following HGO  
\[w(t)=Sw(t)+S_{H}\Psi(t),\quad y_{w}= \begin{bmatrix}y_{ref}(t)\\ y_{dis}(t)\end{bmatrix} =Q_{w}w(t), \tag{5}\]  
where \(Q_{w}= \begin{bmatrix}Q_{ref}\\ Q_{dis}\end{bmatrix} ,y_{ref}\in\mathbb{R}^{p}\) are the \(p\) references to track and \(y_{dis}\in\mathbb{R}^{d}\) are the \(d\) disturbances to reject; matrices \(S\in\mathbb{R}^{\rho(p+d)\times\rho(p+d)}\), \(S_{H}\in\mathbb{R}^{\rho(p+d)\times(p+d)}\), and \(Q_{ref}\in\mathbb{R}^{p\times\rho(p+d)}\), \(Q_{dis}\in\mathbb{R}^{d\times\rho(p+d)}\) have the following structure  
\[S= \begin{bmatrix}S_{1}&0&0&\ldots&\ldots&0\\ 0&S_{2}&0&\ldots&\ldots&0\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ 0&0&\ldots&S_{p}&\ldots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&\ldots&\ldots&S_{p+d}\end{bmatrix} ,\quad S_{H}= \begin{bmatrix}H_{1 }&0&0&\ldots&\ldots&0\\ 0&H_{2}&0&\ldots&\ldots&0\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ 0&0&\ldots&H_{p}&\ldots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&\ldots&\ldots&H_{p+d}\end{bmatrix} ,\]  
\[Q_{ref}= \begin{bmatrix}Q_{1}&0&0&\ldots&\ldots&0\\ 0&Q_{2}&0&\ldots&\ldots&0\\ \vdots&\vdots&\ddots&\ldots&\ldots&0\\ 0&0&\ldots&Q_{p}&\ldots&0\end{bmatrix} ,\quad Q_{dis}= \begin{bmatrix}0&\ldots&Q_ {p+1}&0&\ldots&0\\ 0&\ldots&0&Q_{p+2}&\ldots&0\\ \vdots&\vdots&\vdots&\vdots&\ldots&\vdots\\ 0&\ldots&0&0&\ldots&Q_{p+d}\end{bmatrix} ,\]  
with \(S_{i}\in\mathbb{R}^{\rho\times\rho}\), \(H_{i}\in\mathbb{R}^{\rho\times 1}\) and \(Q_{i}\in\mathbb{R}^{1\times\rho}\) are block matrices defined as:  
\[S_{i}= \begin{bmatrix}-\frac{\alpha_{1}}{\epsilon}&1&0&\ldots&0\\ -\frac{\alpha_{2}}{\epsilon}&0&1&\ldots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ -\frac{\alpha_{p-1}}{\epsilon^{p-1}}&0&0&\ldots&1\\ -\frac{\alpha_{p}}{\epsilon^{p}}&0&0&\ldots&0\end{bmatrix} ,\quad H_{i}= \begin{bmatrix}\alpha_{1}/\epsilon&\alpha_{2}/\epsilon^{2}&\ldots&\alpha_{ \rho}/\epsilon^{\rho}\end{bmatrix} ^{\top},\]  
\[Q_{i}= \begin{bmatrix}1&0&0&\ldots&0\\ 1&0&0&\ldots&0\\ 0&0&\ldots&0\end{bmatrix} ,\]  
Additionally, \(\epsilon\) is related with the convergence rate and \(\alpha_{i}\) is chosen such as \(s^{\rho}+\alpha_{1}s^{\rho-1}+\ldots+\alpha_{\rho-1}s+\alpha_{\rho}\) be Hurwitz. One of the inconveniences of using HGOs is the high gain required when the dimension of \(\rho\) is high, which can cause a peaking phenomenon during the transient states and potentially can destabilize the system. To address this issue, in this work, the order of \(\rho\) is set to 2, resulting in a maximum power of \(\epsilon^{2}\) that reduces the peak effect. Additionally, the selection of \(\epsilon\) to be small and an adequate \(\alpha_{\rho}\) can improve the convergence of the unknown signal.