## 3 Fault-Tolerant Output Regulation Scheme
#### Actuator Fault Estimation  
Let us estimate the actuator fault for system (3) with \(w(t)=0\). The faults are assumed to hold \(f_{a}^{(\sigma)}(t)\approx 0\), where \(\sigma\) is the derivative order (this assumption might seem restrictive in theory, but in practice, sinusoidal signals can also be considered; \(\sigma\) is related to the order of their Taylor polynomial approximation). Thus, by defining \(z_{1}(t)=f_{a}(t)\), \(z_{2}(t)=\dot{f}_{a}(t)\), \(\ldots\), \(z_{\sigma}(t)=f_{a}^{(\sigma-1)}(t)\), the following system can be written  
\[\dot{z}(t)=Fz(t),\text{ with }F= \begin{bmatrix}0&I_{m}&0&\ldots&0\\ 0&0&I_{m}&\ldots&0\\ 0&0&0&\ldots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\ldots&0\end{bmatrix} ,\quad z= \begin{bmatrix}z_{1}\\ z_{2}\\ z_{3}\\ \vdots\\ z_{\sigma}\end{bmatrix} , \tag{11}\]  
where \(I_{m}\) is an identity matrix of \(m\times m\); in general \(I\) and \(0\) are identity and zero matrices of proper sizes. Note that the previous expression is a chain of integrators; this idea can be traced back to [26]. Once coupled with the faulty system as in [17], an augmented system yields  
\[\dot{x}(t)=\bar{A}\bar{x}(t)+\bar{B}u(t),\quad y(t)=\bar{C}\bar{x}(t), \tag{12}\]  
where  
\[\bar{x}(t)= \begin{bmatrix}x(t)\\ z(t)\end{bmatrix} ,\quad\bar{A}=\begin{bmatrix}A&B \begin{bmatrix}I_{m}&0_{m \times m(\sigma-1)}\\ 0&F\end{bmatrix} ,\quad\bar{B}= \begin{bmatrix}B\\ 0\end{bmatrix} ,\quad\text{and}\quad\bar{C}=[C\ \ 0_{\circ\times(n+\sigma m )}].\]  
The estimation of \(x(t)\) and \(f_{a}(t)\) is done through a UI observer of the form (this observer is also named as an unknown input proportional multi-integral observer [25])  
\[\dot{\bar{x}}(t)=\bar{A}\bar{x}(t)+\bar{B}u(t)+\bar{L}(y(t)-\bar{y}(t)), \tag{13}\]with \(\hat{\mathpzc{E}}(t)=[\check{\mathpzc{E}}^{\top}(t)\ \check{\mathpzc{E}}^{\top}(t)]^{\top}\), \((\bar{A},\bar{B},\bar{C})\) are defined as in (12); while \(\bar{L}=[L_{p}^{\top}\ L_{I}^{\top}]^{\top}\), \(L_{P}\in\mathbb{R}^{o\times n}\), \(L_{I}\in\mathbb{R}^{o\times(n+\sigma m)}\) is the observer gain to be designed such that the observation error  
\[\check{e}(t)=\bar{x}(t)-\hat{\mathpzc{E}}(t)= \begin{bmatrix}x(t)-\hat{ \mathpzc{E}}(t)\\ z(t)-\hat{\mathpzc{E}}(t)\end{bmatrix} = \begin{bmatrix}e_{x}(t)\\ e_{z}(t)\end{bmatrix} , \tag{14}\]  
holds  
\[\lim_{t\to\infty}\bar{e}(t)=0.\]  
The dynamics of the augmented error system are computed from (12) and (14), that is  
\[\dot{\mathpzc{E}}(t)=(\bar{A}-\bar{L}\bar{C})\bar{e}(t). \tag{15}\]  
Like the previous section, LMI constraints can be used to find the observer gain \(\bar{L}\), guaranteeing a decay rate \(\gamma>0\). To this end, consider \(V(\bar{e}(t)){=\bar{e}(t)^{T}\bar{P}\bar{e}(t)}\), \(\bar{P}=\bar{P}^{T}\in\mathbb{R}^{(n+\sigma m)\times(n+\sigma m)}\) as Lyapunov function candidate, thus LMIs  
\[\bar{P}>0,\quad\bar{P}\bar{A}-\bar{N}\bar{C}+\bar{A}^{T}\bar{P}-\bar{C}^{T} \bar{N}^{T}+2\gamma\bar{P}<0, \tag{16}\]  
where \(P\) and \(\bar{N}\in\mathbb{R}^{p\times(n+\sigma m)}\) are decision variables. If the previous LMIs are found feasible, thus \(\dot{V}(\bar{e}(t))\leq-2\gamma V(\bar{e}(t))\) and the observer gain is \(\bar{L}=\bar{P}^{-1}\bar{N}\). Thus, both the high-gain observer and the unknown input one are implemented together, so the control law (2) is constructed. Recall that solving the output regulation problem for unmodeled signals requires finding a gain \(K\) such that \((A+BK)\) is Hurwitz; that is, the system is asymptotically stable when \(w(t)=0\); nevertheless, this analysis does not consider actuator faults. Thus, if \(f_{a}(t)\
eq 0\) in (3), then the control should take into account the fault as suggested by [27; 28]. Therefore, the control law is  
\[u(t)=Kx(t)-\hat{f}_{a}(t), \tag{17}\]  
where \(\hat{f}_{a}(t)\) comes from the UI observer (13).