Adaptive compensation of diffusion-advection actuator dynamics using boundary measurements

Abstract

Abstract-For (potentially unstable) Ordinary Differential Equation (ODE) systems with actuator delay, delay compensation can be obtained with a prediction-based control law. In this paper, we consider another class of PDE-ODE cascade, in which the Partial Differential Equation (PDE) accounts for diffusive effects. We investigate compensation of both convection and diffusion and extend a previously proposed control design to handle both uncertainty in the ODE parameters and boundary measurements. Robustness to small perturbations in the diffusion and convection coefficients is also proved.

Uniform stabilization of a wave equation with partial Dirichlet delayed control

“…x ∈ Ω, t > 0, w(ξ, t) = 0, ξ ∈ Γ 1 , t > 0, w(ξ, t) = u(ξ, t), ξ ∈ Γ 0 , t > 0, w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x), x ∈ Ω. (3) This is a classical problem of the partial Dirichlet boundary control with control space U = L 2 (Γ 0 ).…”

“…Due to the challenge in mathematical analysis, the boundary control problem of systems has been a hot topic in mathematical control field since 1970. For instance, Smyshlyaev et al [30] considered the boundary stabilization of a 1-D wave equation with in-domain antidamping based on the backstepping method(similar method also appears in [3]). Guo et al [11] solved the error feedback regulator problem for 1-D wave equation by using the adaptive control approach.…”

Adaptive boundary control of constant-parameter reaction–diffusion PDEs using regulation-triggered finite-time identification

“…Three different methodologies of adaptive boundary control design for parabolic PDEs are available in the literature (see [19,21,23,32,33,34]): i) Lyapunov-based design, ii) design with passive identifiers, and iii) design with swapping identifiers. Recently, the scope of adaptive controllers has been extended to more complicated cases: parabolic PDEs with input delays (see [11]) and parabolic PDEs with distributed parameters and inputs (see [27]). Moreover, adaptive controllers have been used extensively for hyperbolic PDEs: see [1][2][3][4][5][6][7][8][9][10]20].…”

“…An illustrative example allows the comparison with other adaptive control design methodologies.Keywords: parabolic PDEs, boundary feedback, backstepping, adaptive control. Recently, the scope of adaptive controllers has been extended to more complicated cases: parabolic PDEs with input delays (see [11]) and parabolic PDEs with distributed parameters and inputs (see [27]). Moreover, adaptive controllers have been used extensively for hyperbolic PDEs: see [1][2][3][4][5][6][7][8][9][10]20].The purpose of the present work is the development of a novel adaptive boundary control scheme for parabolic PDEs.…”

Robustness to diffusion of prediction-based control for convection processes

“…When the actuation path is subject to both convection and diffusion, this technique has been recently extended in [12], [21] (see also [8] for an adaptive version) to compensate for both effects. However, while delay dynamics can be finitely stabilized, diffusion compensation should then be understood differently as it introduces an infinite relative degree.…”