Interval estimation and control for linear reaction–diffusion systems

Interval estimation and control synthesis are studied for linear reaction–diffusion systems with bounded disturbances and measurement noise. Unlike the existing methods treated this problem based on Luenberger observers and Galerkin projection, we proposed an interval estimation method based on spatial central difference. In detail, the reaction–diffusion system is transformed into an approximated ordinary differential equations (AODEs) by central difference method. Then, we design interval observers based on the decoupling technique for AODEs, so that the states of AODEs are contained between the ones of two sub-observers. Furthermore, by integrating the estimates with truncation error, the state estimation is derived for reaction–diffusion systems. Additionally, these observations are leveraged to design a controller aimed at achieving practical stability and input-to-state stability. Ultimately, the effectiveness of the proposed scheme is validated through numerical simulations.