Stabilization of a class of nonlinear uncertain ordinary differential equation by parabolic partial differential equation controller

Abstract

This article is concerned with stabilization for a class of uncertain nonlinear ordinary differential equation (ODE) with dynamic controller governed by linear 1 − d heat partial differential equation (PDE). The control input acts at the one boundary of the heat's controller domain and the second boundary injects a Dirichlet term in ODE plant. The main contribution of this article is the use of the recent infinite-dimensional backstepping design for state feedback stabilization design of coupled PDE-ODE systems, to stabilize exponentially the nonlinear uncertain systems, under the restrictions that (a) the right-hand side of the ODE equation has the classical particular form: linear controllable part with an additive nonlinear uncertain function satisfying lower triangular linear growth condition, and (b) the length of the PDE domain has to be restricted. We solve the stabilization problem despite the fact that all known backstepping transformation in the literature cannot decouple the PDE and the ODE subsystems. Such difficulty is due to the presence of a nonlinear uncertain term in the ODE system. This is done by introducing a new globally exponentially stable target system for which the PDE and ODE subsystems are strongly coupled. Finally, an example is given to illustrate the design procedure of the proposed method. K E Y W O R D Sboundary stabilization, coupled partial differential equation-ordinary differential equation, infinite-dimensional backstepping, uncertain nonlinear system INTRODUCTIONIn the last decades, coupled partial differential equation (PDE)-ordinary differential equation (ODE) systems have attracted much attention in research communities because such systems can be used to model various processes such as road traffic, 1 gas flow pipeline, 2 power converters connected to transmission lines, 3 oil drilling, 4 and many other systems. The coupled PDE-ODE system considered in this article can be viewed as a perturbation of a simplified version that models the solid-gas interaction of heat diffusion and chemical reaction, where the interaction occurs at the interface. 5Abbreviations: ODE, ordinary differential equation; PDE, partial differential equation. Int J Robust Nonlinear Control. 2020;30:3023-3038. wileyonlinelibrary.com/journal/rnc

Boundary Stabilization of Heat Equation with Multi-Point Heat Source

“…Actually, the proposed finite-dimension feedback linearization method was infinite dimensional extension of the backstepping approach. It is well known that backstepping transformation is mainstream method to deal with boundary stabilization problems of parabolic PDEs, such as, linear parabolic PDEs [15][16][17][18][19], nonlinear parabolic PDEs [20][21][22], quasi-linear parabolic PDEs [23], coupled parabolic PDEs and ODE [24][25][26][27].…”

Rapid exponential stabilization by boundary state feedback for a class of coupled nonlinear ODE and $ 1-d $ heat diffusion equation

“…Recently, in [3] and under some restriction on the length of the heat domain, a stabilization result is obtained for a class of nonlinear ODE coupled with stable heat equation. This work is a continuation in the same direction which aims at stabilizing a general coupled nonlinear ODE with heat equation (stable or unstable) such as the one investigated by [2] for the coupled nonlinear ODE and wave equation.…”

“…To the best of our knowledge, the class of system considered is not addressed in the works dealing with ODE-PDE coupled systems. To be more precise, the coupled system combines the following three features, i) the ODE subsystem (1) contains a nonlinear part, ii) the PDE subsystem (2) is unstable, and, iii) a mixed boundary condition (3). Notably, the presence of a nonlinear term in the ODE subsystem (1) adds complexity to the boundary feedback design.…”

“…However, in the case of the nonlinear ODE-heat coupling, the problem is completely different and an original idea must be implemented to overcome the effects of the heat equation. This paper and [3] can be viewed as initiatives to solve the general case by adopting the backstepping design method for coupled ODE-PDE under the assumption that nonlinear ODE has a stabilizing linear state feedback. To address this stabilization problem, it is natural to start by considering the well-known special form g(X, v) = AX + Bv + f (X) where f (X) satisfies the linear growth condition (6).…”

“…In this setting, there exists a linear change of coordinates rendering the matrices A and B in the forms (5). For more details see [3].…”