An Exact Characterization of the L1/L₋ Index of Positive Systems and Its Application to Fault Detection Filter Design

In this brief, the problem of the <inline-formula> <tex-math notation="LaTeX">$L_{1}/L_{-}$ </tex-math></inline-formula> fault detection for positive systems is revisited. In the existing literature, the <inline-formula> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula>-gain and <inline-formula> <tex-math notation="LaTeX">$L_{-}$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula>-gain and <inline-formula> <tex-math notation="LaTeX">$L_{-}$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$L_{1}/L_{-}$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$L_{-}$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula>-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.

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