Backstepping in Infinite Dimension for a Class of Parabolic Distributed Parameter Systems

Abstract

In this paper a family of stabilizing boundary feedback control laws for a class of linear parabolic PDEs motivated by engineering applications is presented. The design procedure presented here can handle systems with an arbitrary finite number of open-loop unstable eigenvalues and is not restricted to a particular type of boundary actuation. Stabilization is achieved through the design of coordinate transformations that have the form of recursive relationships. The fundamental di‰culty of such transformations is that the recursion has an infinite number of iterations. The problem of feedback gains growing unbounded as the grid becomes infinitely fine is resolved by a proper choice of the target system to which the original system is transformed. We show how to design coordinate transformations such that they are su‰ciently regular (not continuous but L y ). We then establish closed-loop stability, regularity of control, and regularity of solutions of the PDE. The result is accompanied by a simulation study for a linearization of a tubular chemical reactor around an unstable steady state.

Rapid stabilization of a linearized bilinear 1-D Schrödinger equation

“…Initially developed to design, in a recursive way, more effective feedback laws for globally asymptotic stable finite dimensional system for which a feedback law and a Lyapunov function are already known (we refer to [30,46,18] for a comprehensive introduction in finite dimension and [20,34] for an application of this method to partial differential equations), the backstepping approach was later used in the infinite dimensional setting to design feedback laws by mapping the system to stabilize to a target stable system. To our knowledge, this strategy was first introduced in the context of partial differential equations to design a feedback law for heat equation [1] and, later on, for a class of parabolic PDE [12]. The backstepping-like change of coordinates of the semi-discretized equation maps the discrete solution to stabilize to a stable solution.…”

Stabilization of a spatially non‐causal reaction–diffusion equation by boundary control

“…The target system (2) is stable in the sense of L 2 norm [5]. The transformation (3) introduced here is new and more complicated than the Volterra transformation, which is applied in previous papers [5][6][7][8][9]. In the Volterra transformation, there is only one kernel.…”

“…For the spatially causal system, the backstepping transformation is always the Volterra transformation, in which there is only one kernel to be determined. Generally, the kernel equation is convenient to be handled mathematically [4,6,7,[9][10][11][12]. However, for the system (1), the normal Volterra transformation cannot be applied to the PDE backstepping design because the plant is spatially non-causal.…”

Stabilization of a Heat‐ODE System Cascaded at a Boundary Point and an Intermediate Point

“…This paper considers the exponential stabilization problem of a cascaded system with a heat partial differential equation (PDE) and an ordinary differential equation (ODE): the utilization of the backstepping method for stabilizing controller designs. For example, researchers have considered the stabilizing feedback controller designs of heat equations in [3][4][5][6], the cascaded PDE-ODE systems in [7][8][9][10] and the coupled PDE-ODE systems in [11][12][13].…”