An Exact Characterization of the <i>L</i> <sub>1</sub>/<i>L</i>₋ Index of Positive Systems and Its Application to Fault Detection Filter Design
Jun Shen, Jason J. R. Liu, Yukang Cui
Abstract
In this brief, the problem of the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> /L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</sub> fault detection for positive systems is revisited. In the existing literature, the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -gain and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</sub> 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -gain and L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</sub> 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> /L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</sub> 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</sub> 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -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.