# Sensor Fusion Architecture for Fault Diagnosis with a Predefined-Time Observer
## 2 Sensor Fusion Architecture
### Diagnoser  
The naive approach of solving the unknown faults \(m(t)\) from Equation (1) and the estimated state \(\bar{x}\) (output from the proposed architecture) is not possible, since it involves the computation of \(\hat{x}\), increasing the noise and leading to false faults detection. This inconvenience is avoided by using a fault diagnoser; in particular, this work proposes a diagnoser based on a predefined-time observer to swiftly diagnose the fault. First, we assume that the estimated system state \(\bar{x}\) is partitioned as \(\bar{x}=[\hat{x}_{m}\,\hat{x}_{m}]\), where the states in \(\hat{x}_{m}\) are directly affected by the faults, i.e., \(l_{m}(\bar{x})m(t)\
eq 0\) when \(m(t)\
eq 0\), and the states in \(\hat{x}_{m}\) are not affected by the faults, i.e., \(l_{m}(\bar{x})m(t)=0\) when \(m(t)\
eq 0\); then, from Equation (1) we obtain  
\[ \begin{bmatrix}\hat{x}_{m}\\ \hat{x}_{m}\end{bmatrix} = \begin{bmatrix}f_{m}(\bar{x})\\ f_{m}(\bar{x})\end{bmatrix} + \begin{bmatrix}g_{m}(\bar{x})\\ g_{m}(\bar{x})\end{bmatrix} u(t)+ \begin{bmatrix}l_{m}(\bar{x})\\ l_{m}(\bar{x})\end{bmatrix} m(t) \tag{7}\]  
The diagnoser is proposed as the following predefined-time observer:  
\[\hat{x}_{m}=f_{m}(\bar{x})+g_{m}(\bar{x})u(t)+\Theta \tag{8}\]  
where \(\Theta\) (that is related to the fault value) is the observer feedback law that must be selected to ensure that \(\hat{x}_{m}\) converges to \(\hat{x}_{m}\) at a predefined-time \(T_{c}\). The estimation error is  
\[e=\hat{x}_{m}-\hat{x}_{m} \tag{9}\]  
where the measurements of \(x_{m}\) are in \(\tilde{\Sigma}\). The time derivative of the error in Equation (9) is  
\[\dot{e}=l_{m}(\bar{x})m(t)-\Theta \tag{10}\]  
Proposition 1 shows how to compute \(\Theta\). The proof of this proposition is based on the next theorem, presented in [21]. **Theorem 1** ([21]). : _Consider the nonlinear system_  
\[\dot{x}(t)=f(x,t;\rho) \tag{11}\]_where the system state is \(x\in\mathbb{R}^{n}\), the system parameters are \(\rho\in\mathbb{R}^{b}\), which are assumed to be constant, and \(f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) is assumed to be a continuous nonlinear function. If there exists a continuous definite-positive radially unbounded function \(V:R^{n}\to R\) such that_  
\[\dot{V}(x)\leq-\frac{\gamma}{T_{c}}(\alpha V(x)^{p}+\beta V(x)^{q})^{\kappa}, \ \forall x\in R^{n}\setminus\{0\} \tag{12}\]  
_with constants \(\beta,\alpha,p,q,\kappa>0\), \(\kappa p<1\), and \(\kappa q>1\), and the constant \(\gamma\) is given by_  
\[\gamma=\frac{\Gamma(m_{p})\Gamma(m_{q})}{\alpha^{\kappa}\Gamma(\kappa)(q-p)} \left(\frac{\alpha}{\beta}\right)^{m_{p}} \tag{13}\]  
_where \(\Gamma\) is the gamma function [22], \(m_{p}=\frac{1-\kappa p}{q-p}\), and \(m_{q}=\frac{\kappa q-1}{q-p}\). Then, the origin of (11) is predefined-time stable with \(T_{c}\) as its predefined time._  
**Proposition 1**. : _Let \(\Delta=l_{m}(\tilde{x})m(t)\), where its \(i\)-th component is bounded by a positive constant, i.e., \(|\Delta_{i}|\leq\xi_{i}<\infty\), \(\xi_{i}\in\mathbb{R}^{+}\). If the \(i\)-th entry \(\Theta_{i}\) of the observer feedback law \(\Theta\) is chosen as_  
\[\Theta_{i}=[\frac{\gamma}{T_{c}}(\alpha|e_{i}|^{p}+\beta|e_{i}|^{q})^{k}+\xi_{ i}]sign(e_{i}),\quad i=1,\ldots,d \tag{14}\]  
_then, the origin of the error system (10) is predefined-time stable with \(T_{c}\) as its predefined time._  
Proof. : For every estimation error \(e_{i}\) consider the continuous radially unbounded Lyapunov function \(V(e_{i})=|e_{i}|\), \(i=1,\ldots,d\). The time derivative of each Lyapunov function \(V(e_{i})\) yields  
\[ \begin{split}\dot{V}(e_{i})&=\text{sign}(e_{i})[ \Delta_{i}-\Theta_{i}]\\ &=-\frac{\gamma}{T_{c}}(\alpha|e_{i}|^{p}+\beta|e_{i}|^{q})^{k}- \xi_{i}+\Delta_{i}\text{sign}(e_{i})\\ &\leq-\frac{\gamma}{T_{c}}(\alpha V(e_{i})^{p}+\beta V(e_{i})^{q })^{k}-\xi_{i}+|\Delta_{i}\text{sign}(e_{i})|\\ &=-\frac{\gamma}{T_{c}}(\alpha V(e_{i})^{p}+\beta V(e_{i})^{q})^{ k}-(\xi_{i}-|\Delta_{i}\text{sign}(e_{i})|)\\ &\leq-\frac{\gamma}{T_{c}}(\alpha V(e_{i})^{p}+\beta V(e_{i})^{q })^{k}\end{split}