Control of Transport PDE/Nonlinear ODE Cascades With State-Dependent Propagation Speed

In this paper, we deal with the control of a transport partial differential equation/nonlinear ordinary differential equation (PDE/nonlinear ODE) cascade system in which the transport coefficient depends on the ODE state. We develop a PDE-based predictor-feedback boundary control law, which compensates the transport dynamics of the actuator and guarantees global asymptotic stability of the closed-loop system. The stability proof is based on an infinite-dimensional backstepping transformation and a Lyapunov-like argument. The relation of the PDE-ODE cascade with a state-dependent propagation speed to an ODE system with a state-dependent input delay, which is defined implicitly via an integral of past values of the ODE state, is also highlighted and the corresponding equivalent predictor-feedback design is presented together with an alternative proof of global asymptotic stability of the closed-loop system based on the construction of a Lyapunov functional. The practical relevance of our control framework is illustrated in an example that is concerned with the control of a metal rolling process.