State Estimation of a Spatial 2-D Linear Diffusion Process With Mobile Sensors

This paper studies the state observer design of a spatial two-dimensional (2-D) linear diffusion process described by a linear parabolic partial differential equation (PDE) under mobile sensors. Firstly, we analyze the well-posedness of the PDE system and give the structure form of the state observer with mobile sensors. Subsequently, according to the number of mobile sensors, the 2-D space domain is divided into multiple sub-domains, and the mobile sensors are guided by the projection operator method, which can guarantee that the mobile sensors can only move in their respective 2-D sub-domains. Then, in the light of Lyapunov theory, Poincaré-Wirtinger inequality and Barbalat lemma, we propose an observation-plus-guidance design method to ensure the asymptotic stability of the state estimation error system. In the designed mobile strategy, the actual guidance of mobile sensors is essentially a physical synthesis of two direction guidance laws, where two dimensional guidance laws are designed separately. Moreover, the existence condition of the observer is given by linear matrix inequalities. At last, a numerical example is provided to demonstrate the efficacy of the proposed design scheme. Note to Practitioners—For the actual physical temporal-space process, the spatial 2-D case makes more sense. With the increase of space dimension, the difficulty of system design increases sharply. The observer design methodology and mobile sensor guidance approach developed for spatial 1-D systems are not directly extendable to 2-D systems. However, there have also been few reports on the observer design of 2-D physical temporal-space process using mobile sensors, and it is still a challenging problem. In this article, this study addresses the problem of designing state observer for 2-D linear parabolic PDE systems utilizing mobile sensors. To prevent collisions among mobile sensors, the 2-D spatial domain is partitioned into several subregions, and the projection operator approach is employed to develop the motion strategy. Then, a Lyapunov-based observation-plus-guidance design method is provided to achieve the desired design goals. At last, the finite difference method is used to verify the design method.