Mixed l₁/l₋ fault detection observer design for delayed 2D positive systems in FM LSS Models

This study is concerned with the problem of the mixed l₁/l₋ fault detection (FD) observer for delayed two-dimensional (2D) positive systems (PSs). The necessary and sufficient conditions (NSCs) are derived under which the residual error system is asymptotically stable (AS) and has the prescribed performance level. The conservatism is greatly reduced compared to the existing results. A new performance analysis method and the mixed l₁/l₋ FD observer design are presented. Firstly, the calculation of l₁/l₋ index for delayed 2D PSs is proposed by establishing the equivalence between the delayed system and the higher dimensional delay-free system in sense of l₁ and l₋ indexes. Secondly, NSCs are developed such that the delayed 2D PS is AS with a desired mixed l₁/l₋ performance. Thirdly, the sufficient conditions of the observer design are further developed based on linear programming. An iterative algorithm is formulated to minimize and maximize the impact of system disturbances and faults on the output signal, respectively. Finally, we present two examples to verify the superiority of the obtained results.