On Mixed ℓ1/ℓ− Fault Detection Observer Design for Positive 2D Roesser Systems: Necessary and Sufficient Conditions

Abstract This paper addresses the problem of fault detection observer design for positive two-dimensional (2D) systems in the Roesser model. Firstly, an analytical method is presented to compute the exact value of l − index for positive 2D systems, where l − index measures the sensitivity to faults. Necessary and sufficient conditions are developed under which the l 1 -gain from noise to residuals is not larger than a prescribed level γ and the l 1 -gain from faults to residuals is not less than a prescribed level β . Secondly, based on above results, necessary and sufficient conditions for the existence of mixed l 1 / l − fault detection observer are derived such that the effect of disturbance on the residual output is minimized and the effect of faults on the residual output is maximized. Thirdly, an iterative algorithm is proposed to solve the convex optimization conditions. Finally, an example is provided to show the effectiveness and correctness of the theoretical results.