Robust Control of Convective-Diffusion Systems

In the presented control approach to fluid dynamics, the basal and primal motivation arises from laminar flow control. The key issues associated with this class of problems, distributed systems governed by nonlinear partial differential equations (Navier-Stokes), are identified, and a mathematical benchmark problem reflecting those properties (the Burgers equation with periodic boundary conditions and a non-homogeneous distributed forcing term) is created. A viscosity parameter κ being an analogue to the inverse Reynolds number is incorporated. In order to provide a suitable formulation for control purposes, a semi-discretization is performed using a Galerkin finite element method. The resulting state-space formulation is expanded to an unprecedented ‘real world’ control loop design, including process disturbance, measurement noise, model-error, and model-reduction. A Lyaponuv based proof for exponential stability of the origin (under certain initial conditions) can be established. For the nominal control, as well as for the required estimator, the linear quadratic regulator and the extended Kalman filter are applied. Additionally, model-error control synthesis is introduced in its one-step ahead prediction formulation for nonlinear distributed systems. This provides a computationally fast correction to cope with model-error and process disturbances. The derived and introduced techniques are subject to extensive numerical evaluation. Thereby, the combination of the linear quadratic regulator with model-error control synthesis reveals itself to be a powerful control tool, resulting in a fast attenuation of an initial distribution as well as a robust correction of process disturbance. Results hold in face of noisy measurements (additive white Gaussian noise) if the extended Kalman filter is added to the system. The problem is approached from a ‘worst case’ point of view, where the applied disturbance and noise by far exceeds ‘real world’ dimensions.