Finite-Dimensional Backstepping Controller Design

• 成果类型: Article
• 署名作者: Kalantarov, Varga K.; Ozsari, Turker; Yilmaz, Kemal Cem
• 署名单位: Koc University; Ihsan Dogramaci Bilkent University; Izmir Institute of Technology
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2024.3521806
• 发表日期: 2025
• 页码: 3816-3829
• 关键词: Backstepping; mathematical models; Eigenvalues and eigenfunctions; Boundary conditions; STANDARDS; Adaptive control; tail; stability analysis; Navier-Stokes equations; Hands; boundary feedback; reaction-diffusion equation; stabilization

摘要

In this article, we introduce a finite-dimensional version of backstepping controller design for stabilizing solutions of partial differential equations (PDEs) from boundary. Our controller uses only a finite number of Fourier modes of the state of solution, as opposed to the classical backstepping controller which uses all (infinitely many) modes. We apply our method to the reaction-diffusion equation, which serves only as a canonical example but the method is applicable also to other PDEs whose solutions can be decomposed into a slow finite-dimensional part and a fast tail, where the former dominates the evolution in large time. One of the main goals is to estimate the sufficient number of modes needed to stabilize the plant at a prescribed rate. In addition, we find the minimal number of modes that guarantee the stabilization at a certain (unprescribed) decay rate. Theoretical findings are supported with numerical solutions.