The authors consider a discrete-time bilinear system described by \[ \begin{aligned} x_{k+1} &= A^0x_k+ Bu_k+ \sum^q_{i=1} A^iu_k(i)x_k+ Fd_k+ Gf_{a(k)},\tag{1}\\ y_k &= Cx_k+Qf_{b(k)},\end{aligned} \] where \(x_k\in\mathbb{R}^n\), \(u_k\in\mathbb{R}^m\) and \(y_k\in\mathbb{R}^p\) are the state input and output vectors respectively; \(u_k(i)\) are the components of \(u_k\) \((i=1,\dots,q)\) \((q\leq m)\), \(d_k\in\mathbb{R}^l\), \(f_{a(k)}\in\mathbb{R}^a\) and \(f_{b(k)}\in\mathbb{R}^b\) are the unknown input, the component and actuator fault and sensor fault vectors, respectively; \(A^0\), \(A^i\), \(B\), \(C\), \(F\), \(G\) and \(Q\) are matrices of appropriate dimensions.
The authors propose for system (1) a bilinear fault detection observer. They give sufficient conditions for the existence of the observer and results for the explicit calculation of the observer design matrices. An application to a hydraulic drive system is presented to illustrate the obtained results.