Polynomial Fuzzy Observer-Based Feedback Control for Nonlinear Hyperbolic PDEs Systems
This article is organized as follows. In Section II, preliminaries and problem formulation are introduced. Section III brings the observer-based polynomial fuzzy controller design and the stabilization analysis for the polynomial fuzzy PDEs system without estimation of Bi . The segmental algorithm for the SD-SOS condition is developed and the experiment results for the proposed results are demonstrated in Section IV. In Section V, the observer-based polynomial fuzzy controller design and the SD-SOS condition for stability analysis with estimation of Bi and the corresponding experiment results are demonstrated. Finally, Section VI gives the conclusions.
Fig. 1. the states are divergent over time for the open-loop nonlinear systems.
Technology Overview
This article introduces an observer-based feedback control method for nonlinear hyperbolic Partial Differential Equation (PDE) systems using a polynomial fuzzy PDE model. It proposes a novel SD-SOS (Sum-of-Squares) exponential stability condition, built upon the LKFPM (Lyapunov-Krasovskii Functional Polynomial Matrix) and SOS approach, leveraging Euler's homogeneous theorem. To find feasible solutions for this condition, a segmental algorithm has been developed.
Applications & Benefits
This research offers a robust solution for stabilizing complex nonlinear PDE systems, which are prevalent in various engineering and scientific domains. The proposed control method, demonstrated through three experiments, provides exponential stability, ensuring predictable and reliable system behavior. While current computations can be intensive, future work aims to reduce this burden, paving the way for its application in real-world models and advancing control strategies for systems involving phenomena like wave propagation and fluid dynamics.
Abstract：
This article explores the observer-based feedback control problem for a nonlinear hyperbolic partial differential equations (PDEs) system. Initially, the polynomial fuzzy hyperbolic PDEs (PFHPDEs) model is established through the utilization of the fuzzy identification approach, derived from the nonlinear hyperbolic PDEs model. Various types of state estimation and controller design problems for the polynomial fuzzy PDEs system are discussed concerning the state estimation problem. To investigate the relaxed stability problem, Euler’s homogeneous theorem, Lyapunov–Krasovskii functional with polynomial matrices (LKFPM), and the sum-of-squares (SOSs) approach are adopted. The exponential stabilization condition is formulated in terms of the spatial-derivative-SOSs (SD-SOSs). Additionally, a segmental algorithm is developed to find the feasible solution for the SD-SOS condition. Finally, a hyperbolic PDEs system and several numerical examples are provided to illustrate the validity and effectiveness of the proposed results.
Polynomial Fuzzy Observer-Based Feedback Control for Nonlinear Hyperbolic PDEs Systems
Author： Shun-Hung Tsai, Wen-Hsin Lee, Kazuo Tanaka, Ying-Jen Chen, Hak-Keung Lam
Year：2024
Source publication： IEEE Transactions on Cybernetics Volume: 54, Issue: 9
Subfield Highest percentage： 99% Control and Systems Engineering #4 / 375