Delayed point control of a reaction–diffusion PDE under discrete-time point measurements

• Published in: Automatica (Oxford) Vol. 96; pp. 224 - 233
• Main Authors: Selivanov, Anton, Fridman, Emilia
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.10.2018
• Subjects: Boundary control; Distributed parameter systems; Input delay; Networked control systems; Point control; Boundary control; Input delay; Networked control systems; Distributed parameter systems; Point control
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2018.06.050

We consider stabilization problem for reaction–diffusion PDEs with point actuations subject to a known constant delay. The point measurements are sampled in time and transmitted through a communication network with a time-varying delay. To compensate the input delay, we construct an observer for the future value of the state. Using a time-varying observer gain, we ensure that the estimation error vanishes exponentially with a desired decay rate if the delays and sampling intervals are small enough while the number of sensors is large enough. The convergence conditions are obtained using a Lyapunov–Krasovskii functional, which leads to linear matrix inequalities (LMIs). We design output-feedback point controllers in the presence of input delays using the above observer. The boundary controller is constructed using the backstepping transformation, which leads to a target system containing the exponentially decaying estimation error. The in-domain point controller is designed and analysed using an improved Wirtinger-based inequality. We show that both controllers can guarantee the exponential stability of the closed-loop system with an arbitrary decay rate smaller than that of the observer’s estimation error.