Sampled-data relay control of diffusion PDEs

Abstract

We consider a vector reaction-advection-diffusion equation on a hypercube. The measurements are weighted averages of the state over different subdomains. These measurements are asynchronously sampled in time. Subject to matched disturbances, the discrete control signals are applied through shape functions and zero-order holds. The feature of this work is that we consider generalized relay control: the control signals take their values in a finite set. This allows for networked control through low capacity communication channels. First, we derive linear matrix inequalities (LMIs) whose feasibility guarantees the ultimate boundedness with a limit bound proportional to the sampling period. Then we construct a switching procedure for the controller parameters that ensures semi-global practical stability: for an arbitrarily large domain of initial conditions the trajectories converge to a set whose size does not depend on the domain size. For the disturbance-free system this procedure guarantees exponential convergence to the origin. The results are demonstrated by two examples: 2D catalytic slab and a chemical reactor.

Spatially Piecewise Fuzzy Control Design for Sampled-Data Exponential Stabilization of Semilinear Parabolic PDE Systems

“…where ρ 1 min{λ min (P 1 ), λ min (P 2 )}. For t = 0, the following inequality can be obtained from (22)…”

“…Although the problem of sampled-data fuzzy control design of nonlinear parabolic DPSs has been addressed in [39] and [40], the proposed Lyapunov-Krasovskii functional candidate is constructed based on the low-dimensional ODE approximations obtained from the singular perturbation formulation of Galerkin's method. On the other hand, the Lyapunov-Krasovskii functional candidate of the form (18)-(21) is also different from the ones used in [20]- [22] for sampled-data control design of semi-linear parabolic PDE systems, where an exponential term exp(2α(s − t)) (α is a given positive constant) is introduced in construction of Lyapunov-Krasovskii functional candidates.…”

Global Stabilization of a Class of Nonlinear Reaction-Diffusion Partial Differential Equations by Boundary Feedback

“…Unstable nonlinear PDEs are usually treated by performing a feedback design for the linearized model (see for example [36]). Very few feedback design methodologies have been proposed for unstable nonlinear parabolic PDEs: see the extension of the backstepping boundary feedback design in [37,38] as well as feedback designs for distributed inputs in [9,16,31,32]. In many cases the stabilization results are local, guaranteeing exponential stability in specific spatial norms.…”

Lyapunov-based boundary feedback design for parabolic PDEs

“…When studying systems described by Partial Differential Equations (PDEs), the Control Lyapunov Function becomes a Control Lyapunov Functional (CLF). The use of CLFs for the solution of global feedback stabilization problems for systems with PDEs has been presented in detail in [5] and has been used for instance in [1,7,12,14,20,21].…”

“…The complicated nature of the CLF obtained by backstepping has not allowed the use of the CLF methodology for nonlinear parabolic PDEs. Very few feedback design methodologies have been proposed for unstable nonlinear parabolic PDEs: see the extension of the backstepping boundary feedback design in [24,25] as well as feedback designs for distributed inputs in [4,7,20,21]. In many cases the stabilization results are local, guaranteeing exponential stability in specific spatial norms.…”