Robust reduced order control for nonlinear distributed systems of Burgers class
MetadataShow full item record
In the presented application of control engineering to fluid flow dynamics, the basal and primal motivation arises from laminar flow control in aerodynamics. The governing physical model is identified as the Navier-Stokes equations. These equations are reviewed and derived in a general approach so that the presented techniques and results can be related to a variety of continuity problems. The key issues associated with this class of problems, distributed systems governed by nonlinear partial differential equations, are identified, and a mathematical benchmark problem reflecting those properties (the Burgers equation with periodic boundary conditions and a non-homogeneous forcing term is created. Thereby, Burgers' equation is linked by several means to distributed nonlinear systems: as an approximation of a two-dimensional channel flow problem; as the decisive factor in the creation of turbulence (original motivation); and as a modeling equation for traffic flow. The benchmark problem is stated and classified as a continuous partial differential equation subject to periodic boundary conditions (of first and second order) and a non-homogeneous distributed forcing term (the control input). This setting has not yet been adequately addressed in previous research. A viscosity parameter κ being an analogue to the inverse Reynolds number is incorporated. In the following, the viscous problem is solved analytically for a certain class of initial conditions while the general solution to the inviscid case, as well as the steady-state solution, are derived without limitations on the initial condition. In order to provide a suitable formulation for control engineering, a semi-discretization (in space) is performed using a Galerkin finite element method. This results in a large-scale ODE system which is tested for consistency with the previously derived analytical solution. 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. Also, the benchmark problem is analyzed in control terms for stability and controllability. Thereby, a Lyaponuv based proof of exponential stability of the origin, for certain initial conditions, can be established. The exponential rate of decay is identified as being only dependent on the viscosity parameter κ. It can be shown that a feedback law with positive semi-definite gain even improves exponential stability. An argumentation toward the controllability of constant equilibria, previously identified as being the only steady-state solutions, is made. For the nominal control, as well as for the required estimator, the linear quadratic regulator and the extended Kalman filter are briefly reviewed and derived for the benchmark problem. 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 implementation in previous research is briefly presented while a modification for application in this work is suggested. The derived and introduced techniques are subject to extensive numerical evaluation, and detailed results are given. 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. Due to the differentiating character of the predictive filter, the model-error correction has to be reprocessed when a reduced-order model is applied. Then, the results prove to be as powerful as in the full-order model case. A comprehensive Matlab code is provided. The problem is approached from a 'worst case' point of view, where the applied disturbance and noise by far exceeds 'real world' dimensions. The reduced-order model is based on a coarse truncation of a linear global Galerkin finite element method; even better results are expected if refined techniques are applied. A brief review of previous research on PDE problems in control engineering, as well as a detailed reference to publications on Burgers' equation, is presented. Especially, results are compared to previous work at the Virginia Polytechnic Institute and State University. An outlook on future research and topics to be addressed concludes this thesis.