Tracking Control for $$2\times 2$$ Linear Heterodirectional Hyperbolic PDEs that Are Bidirectionally Coupled with Nonlinear ODEs

Abstract

A solution-based approach to trajectory tracking control is presented for \(2\times 2\) linear heterodirectional hyperbolic PDEs that are actuated at one boundary and bidirectionally coupled with nonlinear ODEs at the other one. The control strategy relies on the solution of the PDE subsystem, significantly simplified by a preliminary backstepping transformation, to essentially reduce the PDE-ODE system to the nonlinear boundary ODEs with a delayed input. A tracking controller, e.g. a flatness-based feedback, is assumed to exist for this reduced system in the case without an input delay. To control the delay system based on this feedback, a prediction of the ODE state is required, which follows from the solution of a general nonlinear Volterra integro-differential equation. An explicit solution is found in the special case of linear ODEs, which, with an appropriate choice of design parameters, is used for showing that the presented solution-based controller for the PDE-ODE system is equivalent to known backstepping-based stabilizing controllers. The performance of the proposed tracking controller is demonstrated in numerical simulations.