2017/03/01 by Aneel Tanwani, Christophe Prieur, Tanwani, Aneel +3 · 2 citations
Engineering · #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Electrical engineering #FOS: Mathematics #Stability and Controllability of Differential Equations #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.1703.00302
openalex publication_date 2017/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a system of linear hyperbolic PDEs where the state at one of the boundary points is controlled using the measurements of another boundary point. Because of the disturbances in the measurement, the problem of designing dynamic controllers is considered so that the closed-loop system is robust with respect to measurement errors. Assuming that the disturbance is a locally essentially bounded measurable function of time, we derive a disturbance-to-state estimate which provides an upper bound on the maximum norm of the state (with respect to the spatial variable) at each time in terms of L^∞-norm of the disturbance up to that time. The analysis is based on constructing a Lyapunov function for the closed-loop system, which leads to controller synthesis and the conditions on system dynamics required for stability. As an application of this stability notion, the problem of quantized control for hyperbolic PDEs is considered where the measurements sent to the controller are communicated using a quantizer of finite length. The presence of quantizer yields practical stability only, and the ultimate bounds on the norm of the state trajectory are also derived.