Optimal state space solution to the fault detection problem at DC

In this work, we consider an observer-based fault detection (FD) problem for linear time invariant systems at DC (ω = 0). A simple parameterization of all optimal solutions is derived such that the FD filter generates a residual signal which minimizes the sensitivity of the residual to disturbances while maintaining a minimum level of sensitivity to faults. Linear matrix inequality (LMI) techniques are used to obtain the optimal solution. Furthermore, we show that, at DC, the resulting degrees of freedom allow the FD observer to be used simultaneously to achieve other control specifications. Conversely, our work demonstrates that observers which are used for control system design can, with minor modifications, be also used to implement fault detection schemes at DC. Finally, the effectiveness of the proposed scheme is illustrated using a numerical example.