Analysis and Nonlinear Control of Galerkin Models Using Averaging and Center Manifold Theory

In this paper, nonlinear control systems whose dynamics are quadratic with respect to state, and bilinear with respect to state and input, which exhibit an oscillation caused by a stable limit cycle for zero input are studied. The effect of linear control on this model is analyzed using modal forms and center manifold theory. It is found that the oscillation amplitude depends both on terms linear in the control and those that depend on the center manifold. To exploit the latter, a nonlinear control law is proposed. 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 an example.