Control of 2×2 linear hyperbolic systems: Backstepping-based trajectory generation and PI-based tracking

We consider the problems of trajectory generation and tracking for general 2×2 systems of first-order linear hyperbolic PDEs with anti-collocated boundary input and output. We solve the trajectory generation problem via backstepping. The reference input, which generates the desired output, incorporates integral operators acting on advanced and delayed versions of the reference output with kernels which were derived by Vazquez, Krstic, and Coron for the backstepping stabilization of 2×2 linear hyperbolic systems. We apply our approach to a wave PDE with indefinite in-domain and boundary damping. For tracking the desired trajectory we employ a PI control law on the tracking error of the output. We prove exponential stability of the closed-loop system, under the proposed PI control law, when the parameters of the plant and the controller satisfy certain conditions, by constructing a novel “non-diagonal” Lyapunov functional. We demonstrate that the proposed PI control law compensates in the output the effect of in-domain and boundary disturbances. We illustrate our results with numerical examples.