Mixed ℓ₁/ℓ_Fault Detection for Positive 2-D Systems With Distributed Delays
Zhaoxia Duan, Jinna Fu, Choon Ki Ahn, Zhengrong Xiang, Imran Ghous
Abstract
This work discusses the fault detection problem for two-dimensional (2-D) positive systems (PSs) with distributed delays. First, a systematic method is proposed for the computation of exact values of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula> -gain and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{-}$ </tex-math></inline-formula> index for delay-free systems, and the dimension expansion technique is used to transfer the computation of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1}/\ell _{-}$ </tex-math></inline-formula> index for 2-D delayed PSs to one of the delay-free systems. Second, necessary and sufficient conditions are established such that the 2-D delayed PSs are asymptotically stable with a desired <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1}/\ell _{-}$ </tex-math></inline-formula> performance level (PL) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\gamma /\beta $ </tex-math></inline-formula> . Third, based on the above results, some NCSs are established such that the disturbance and the fault impact on the output signal are minimized and maximized, respectively. An algorithm (iterative) is formulated for the solution of the convex optimization problem. Finally, the potency and accuracy of the developed theoretical results are exhibited by an example.