An Exact Characterization of the L 1/L₋ Index of Positive Systems and Its Application to Fault Detection Filter Design

Abstract

In this brief, the problem of the $L_{1}/L_{-}$ fault detection for positive systems is revisited. In the existing literature, the $L_{1}$ -gain and $L_{-}$ index for positive systems are often characterized separately, and thus their linear programming descriptions involve different Lyapunov vectors. This casts the fault detection filter design as a bilinear optimization problem. To circumvent this obstacle, we first show that, for an externally positive system, the $L_{1}$ -gain and $L_{-}$ index are determined, respectively, by the largest and smallest column sums of the static gain matrices. Based on this fact, an exact characterization is given for the $L_{1}/L_{-}$ index for positive systems in terms of a linear program with equality constraints. The new characterization only involves one single Lyapunov vector, and thus renders the fault detection filter design problem convex. In addition, we find that the maximum fault sensitivity (characterized by the $L_{-}$ index from the fault to the residual) that can be achieved by the filter design approach is proportional to the required upper bound on the $L_{1}$ -gain from the disturbance to the residual. Finally, an illustrative example of a positive electric circuit is presented to show the effectiveness of the theoretical results.