Structured spatial control of the reaction-diffusion equation with parametric uncertainties

Abstract

Feedback control problems for distributed parameter systems arise in a variety of physical, chemical, biological, and mechanical systems. This paper exploits the algebraic structure of the system of ordinary differential equations that arise from spatial discretization of the partial differential equation (PDE) to analyze and design feedback controllers that are robust to bounded perturbations in the parameters of the original PDE. As a prototypical problem, this paper investigates the spatial field control of a reaction-diffusion system whose spatial discretization has a state matrix that is circulant symmetric. Structured robust controllers are designed based on internal model control and mixed sensitivity optimization. The controllers are shown to be robust to inaccuracies in the spatial manipulation, even for arbitrarily fine spatial discretizations.

Robust Anti-Windup Compensation for Normal Systems with Application to the Reaction-Diffusion Equation

“…2a with insertion of the expressions (18)-(19) and with the saturation removed can be rearranged so that U and U * only appear in a block with U * Q(s)U and the transformed process and its model are diagonal. Following similar proofs as in Hovd et al [1997] and Kishida and Braatz [2010] a diagonal U * Q(s)U can be selected that is optimal, so an optimal Q(s) can be written as (20).…”

“…Consider the control of the distributed parameter system with reaction and diffusion (Kishida and Braatz [2010]):…”

“…Normal systems are unitarily diagonalizable systems, which include symmetric, skew-symmetric, and orthogonal systems, and can be seen in a wide variety of application domains including in electric power (Lunze [1986]), spacecraft (Wu et al [1999]), and internet congestion (Zhang and Loguinov [2009]). Circulant systems are a subclass of normal systems that arise in vehicle formations (Marshall et al [2004]), paper machines (Laughlin et al [1993]), and from the discretization of partial differential equations (Brockett and Willems [1974], Kishida and Braatz [2010]). For circulant systems, the unitary matrix is the Fourier matrix regardless of the dynamics of the system (Hovd and Skogestad [1994]).…”

Robust tracking control for a class of perturbed and uncertain reaction-diffusion equations

“…In particular in [Sira-Ramirez, 1989] the problem of distributed control for quasilinear first-order parabolic equations is addressed and a variablestructure control policy is proposed, while in [Pisano et al, 2011] the authors focus on the design of sliding-mode controllers for robust tracking in the case of unidimensional heat equation and wave equation. In the framework of reaction-diffusion equations, both boundary control [Barthel et al, 2010] and distributed control [Kishida and Braatz, 2010] have been investigated. This paper proposes the extension of some results from [Pisano et al, 2011] to the case of uncertain and perturbed reaction-diffusion equations.…”

Distributed control of parameter-varying spatially interconnected systems using parameter-dependent Lyapunov functions

“…These include systems that are inherently spatially discrete, such as vehicle Platoons [2], or cross-directional control of modern paper machines [3], but may also represent lumped approximations of systems governed by partial differential equations, such as reaction-diffusion processes [1], applications in fluid control [4], smart structures [5], etc.…”