Fault Detection for Switched Positive Systems With Application to Traffic Signal Systems

This article is concerned with $L_{-}$ fault detection functional observer design for the switched positive systems (SPSs) with unknown external input under mode-dependent minimum dwell time (MDMDT) constraint. First, a necessary and sufficient condition is presented, such that the designed p-order functional observer provides a positive estimation with an $L_{-}$ fault sensitivity performance. Then, a new discretized linear co-positive Lyapunov function method is proposed to derive the asymptotic tracking condition of the $L_{-}$ fault detection functional observer under the MDMDT constraint. Further, the matrix decomposition technology is applied to solve the bilinear inequality problem in the observer gain matrix design, and an effective mode-dependent piece-wise design scheme of the $L_{-}$ fault detection functional observer in the linear programming form is proposed. The designed $L_{-}$ functional observer is not only sensitive to fault but also robust against any form of unknown input. The results are also extended to the cases of mode-dependent constant dwell time (MDCDT) constraint, mode-independent dwell time constraint, and non-SPSs. Finally, the proposed method is illustrated in detail by constructing a fault detection observer for a traffic signal system. Simulation results verify the effectiveness of the theoretical results.