Spatiotemporal adaptive state feedback control of a linear parabolic partial differential equation

Abstract

This article deals with the issue of asymptotic stabilization for a linear parabolic partial differential equation (PDE) with an unknown space-varying reaction coefficient and multiple local piecewise uniform control. Clearly, the unknown reaction coefficient belongs to a function space. Hence, the fundamental difficulty for such issue lies in the lack of a conceptually simple but effective parameter identification technique in a function space. By the Lyapunov technique combined with a variant of Poincaré-Wirtinger inequality, an update law is derived for estimate of the unknown reaction coefficient in a function space.Then a spatiotemporal adaptive state feedback control law is constructed such that the estimate of the unknown coefficient is bounded and the closed-loop PDE is asymptotically stable in the sense of spatial  1 norm if a sufficient condition given in terms of space-time varying linear matrix inequalities (LMIs) is fulfilled for the estimated coefficient and the control gains. Both analytical and numerical approaches are proposed to construct a feasible solution to the space-time varying LMI problem. With the aid of the semigroup theory, the well-posedness and regularity of the closed-loop PDE is also analyzed. Moreover, two extensions of the proposed adaptive control scheme are discussed: the PDE in N-D space and the PDE with unknown diffusion and reaction coefficients. Finally, numerical simulation results are presented to support the proposed spatiotemporal adaptive control design.

The Modeling and Control of (Renewable) Energy Systems by Partial Differential Equations—An Overview

“…The use of the developed models is mostly in simulations and analytical studies of energy systems, and they are rarely used for control purposes in energy processes and systems. Distributed-parameter systems are a recent trend in control systems engineering research [175][176][177][178][179][180][181][182][183][184][185]. Since these and other recent results show the full advantage of using models described by partial differential equations, it is expected that in the near future, many more controllers will be designed and optimized for energy systems (especially renewable energy systems) on the grounds of using the partial differential equation methodology.…”

Adaptive Fault-Tolerant Control for Consensus of Nonlinear Fractional-Order Multi-Agent Systems with Diffusion

“…Actually, all processes essentially depend on both time and space [16,17]. There are many dynamic behaviors with spatio-temporal characteristics in nature, such as transport reaction processes [18], continuous casting [19], sputtering processes [20], hydraulic fracturing [21] and cracking furnaces [22].…”