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We consider a system of linear hyperbolic PDEs where the state at one of theboundary points is controlled using the measurements of another boundary point.Because of the disturbances in the measurement, the problem of designingdynamic controllers is considered so that the closed-loop system is robust withrespect to measurement errors. Assuming that the disturbance is a locallyessentially bounded measurable function of time, we derive adisturbance-to-state estimate which provides an upper bound on the maximum normof the state (with respect to the spatial variable) at each time in terms of$\mathcal{L}^\infty$-norm of the disturbance up to that time. The analysis isbased on constructing a Lyapunov function for the closed-loop system, whichleads to controller synthesis and the conditions on system dynamics requiredfor stability. As an application of this stability notion, the problem ofquantized control for hyperbolic PDEs is considered where the measurements sentto the controller are communicated using a quantizer of finite length. Thepresence of quantizer yields practical stability only, and the ultimate boundson the norm of the state trajectory are also derived.

Disturbance-to-State Stabilization and Quantized Control for Linear Hyperbolic Systems