Accelerated disturbance damping of an unknown distributed system by nonlinear feedback

We propose a new control design methodology for distributed systems with unknown nonlinear dynamics. The approach is appropriate for systems whose linearized differential operator possesses an eigenspectrum that can be partitioned into a low‐dimensional slow spectrum and an infinite‐dimensional fast complement. This separation of time scales allows us to utilize center manifold and normal form techniques of modern geometric theories for dynamical systems. It is shown that the convergence to an asymptotically stable equilibrium point after a small‐amplitude transient disturbance quickly (exponentially fast) approaches an invariant manifold W which is locally tangent to the eigenspace of the slow modes and is hence of the same dimension. The nonlinear dynamics on this invariant manifold is much slower than the fast approach and is dominated by the slow modes with the fast modes coupled “adiabatically” to them. The convergence can hence be best accelerated using a slow control with smooth nonlinear feedback involving only the slow modes. Nonlinear feedback is shown to drastically improve the performance of linear feedback. The only required information about the system in our approach is the slow adjoint eigenfunctions which can be easily estimated with a Karhunen‐Loeve scheme for distributed systems. This identification scheme is quite robust to changes in process dynamics and can hence be carried out on‐line in parallel with feedback control. The overall approach is verified by numerical experiments.