Attenuation of Oscillations in Galerkin Systems Using Center-Manifold Techniques*

Abstract

In this paper, nonlinear control systems whose dynamics are quadratic with respect to state and bilinear with respect to state and input, and which exhibit an oscillation caused by a stable limit cycle for zero input are studied. Galerkin systems which arise from reducedorder modeling of certain infinite-dimensional dynamical systems of interest in flow control belong to this category. For these models, it is customary to analyze the effect of linear control on the amplitude of the limit cycle using standard arguments involving Poincare ‘normal forms and center manifold theory. It is found that the oscillation amplitude depends both on terms linear in the control and nonlinear terms that depend on the center manifold. To exploit these latter, in this paper a nonlinear control law is proposed that aims at reducing the oscillation by shaping the center manifold. An oscillation preserving condition was defined and enforced on the system to ensure that the results are physically meaningful and practically implementable. The analysis of the closed loop system is simplified using a time varying periodic change of coordinates, time scaling, and averaging. Using center manifold theory, conditions governing the number and stability type of the limit cycles, and analytical expressions for the oscillation amplitude are derived. The results are verified using a finite-dimensional cavity flow model as a case study.