Optimal control of nonlinear aeroelastic system with non-semi-simple eigenvalues at Hopf bifurcation points

This study discusses an efficient method of the Hopf bifurcation control for nonlinear aeroelastic system. The nonlinear aeroelastic system whose linear part has multiple non-semi-simple eigenvalues at critical point gives rise to Hopf bifurcations. The method of the multiple scales and the well-known linear quadratic regulator method are used to deal with the optimal control of the nonlinear system at Hopf bifurcation points. The modal optimal control equation and modal Riccati equation of the nonlinear system are developed to simplify the computations. The conventional Potter's algorithm is extended to solve modal Riccati equation for the modal Riccati matrix of the Hopf bifurcation control. The first-order approximation solutions are developed, which include the gain vectors and inputs. By the way of optimal control, the admissible control input and trajectory of the linear part of the nonlinear aeroelastic system are obtained to minimize the performance measure. Then, we set the appropriate first-order gain vector to adjust the convergence speed of this nonlinear system.