Feedback control and Kalman Filtering of nonlinear wave dynamics

The paper proposes a pointwise control method for the 1D nonlinear wave equation and a filtering approach for estimating the dynamics of such a system from measurements provided by a small number of sensors. It is shown that the numerical solution of the associated partial differential equation results into a set of nonlinear ordinary differential equations. With the application of a diffeomorphism that is based on differential flatness theory it is shown that an equivalent description of the system in the linear canonical (Brunovsky) form is obtained. This transformation enables to obtain estimates about the state vector of the system through the application of the standard Kalman Filter recursion. For the local subsystems, into which the nonlinear wave equation is decomposed, it becomes possible to apply pointwise state estimation-based feedback control. The efficiency of the proposed filtering and control approach for nonlinear systems described by 1D partial differential equations of the wave type (e.g. sineGordon PDE) is confirmed through simulation experiments.