Local Proportional-Integral Boundary Feedback Stabilization for Quasilinear Hyperbolic Systems of Balance Laws

Abstract

The proportional-integral (PI) boundary stabilization of nonlinear hyperbolic systems of balance laws is investigated for the $H^2$-norm, in which the control and output measurements are all located at the boundaries. The boundary conditions of the system are subject to unknown constant disturbances. The induced closed-loop system is proven to be locally exponentially stable with respect to the steady states. To this end, a set of matrix inequalities is given by constructing a new Lyapunov function as a weighted $H^2$-norm of the classical Cauchy solution and the integral of boundary output. Furthermore, the traffic flow dynamics of a freeway section are modeled with the Aw--Rascle--Zhang model. To stabilize the oscillations of traffic demand, a local PI boundary feedback controller is designed with integration of on-ramp metering and variable speed limit control. The exponential convergence of the nonlinear traffic flow dynamics in an $H^2$ sense is achieved and validated with simulations.