L1/L– Observer-Based Fault Detection for 2-D Delayed Positive Continuous-Time Systems

This brief focuses on the design of fault detection observer for continuous 2D delayed positive systems. Firstly, the
$L_{1}$
-gain and
$L_{-}$
-index of the 2D positive delayed systems are calculated by 2D Laplace transform; secondly, necessary and sufficient conditions for the system to satisfy the
$L_{1}/L_{-}$
performance are derived; finally, sufficient conditions for the existence of a mixed
$L_{1}/L_{-}$
observer are derived. The solution to the observer gain is given by an iterative linear programming algorithm. An example is given to demonstrate the validity of the proposed theoretical results.