L2 disturbance attenuation for highly dissipative nonlinear spatially distributed processes via HJI approach

Abstract

For many practical industrial spatially distributed processes (SDPs), their dynamics are usually described by highly dissipative nonlinear partial differential equations (PDEs). In this paper, we address the L2 disturbance attenuation problem of nonlinear SDPs using the Hamilton–Jacobi–Isaacs (HJI) approach. Firstly, by collecting an ensemble of PDE states, Karhunen–Loève decomposition (KLD) is employed to compute empirical eigenfunctions (EEFs) of the SDP based on the method of snapshots. Subsequently, these EEFs together with singular perturbation (SP) technique are used to obtain a finite-dimensional slow subsystem of ordinary differential equation (ODE) that accurately describes the dominant dynamics of the PDE system. Secondly, based on the slow subsystem, the L2 disturbance attenuation problem is reformulated and a finite-dimensional H∞ controller is synthesized in terms of the HJI equation. Moreover, the stability and L2-gain performance of the closed-loop PDE system are analyzed. Thirdly, since the HJI equation is a nonlinear PDE that has proven to be impossible to solve analytically, we combine the method of weighted residuals (MWR) and simultaneous policy update algorithm (SPUA) to obtain its approximate solution. Finally, the simulation studies are conducted on a nonlinear diffusion-reaction process and a temperature cooling fin of high-speed aerospace vehicle, and the achieved results demonstrate the effectiveness of the developed control method.