Computation of Nonlinear Feedback for Flow Control Problems

When computable, a linear feedback control guarantees the stability of a steady-state flow. However in many cases the stability region of the resulting closed-loop system might be too small to be implemented in practice, or better performance of the closed-loop system might be desired. This motivates investigations of nonlinear feedback strategies to either expand the stability region or improve the local performance. Nonlinear feedback laws can now be efficiently computed using the nonlinear systems toolbox created by Art Krener. However, for distributed parameter systems, it must be implemented on reduced-order models that are very low-dimensional (or in special cases on very coarse discretizations of the original equations). This is also the case for linear systems, but here the dimension of the model has to be much lower. In this paper, we use a model feedback control problem with quadratic and cubic nonlinear terms and compute polynomial solutions to the Hamilton-Jacobi-Bellman equations. Of particular interest is understanding the expansion of the stability regions as the degree of the control is increased for this simple problem. Then we investigate the closed-loop performance benefits on a model flow control problem described by the one-dimensional Burgers equation. This model has the same nonlinear term, but the simplicity of this distributed parameter system allows us to evaluate the performance of both linear and nonlinear feedback gains from the discretized linearized or quadratic system, respectively. We note that the setup time for these solvers can be very expensive due to the need to compute all of the required derivatives of the system and control objective functions when building the Taylor polynomials. Thus, we have developed improvements in the software by using automatic differentiation methods. We also investigate the simplification of the algorithm when the state equations have the quadratic nonlinearities that are typical of flow control problems.