Control Theory Of Nonlinear Airfoil Hopf Bifurcation

In the last years there has been a great interest to analyze control systems displaying complex dynamics.An emerging research field that has become very stimulating is the bifurcation control.When a nonlinear system generates dynamic instability,the system loses structural stability and gives rise to bifurcation,affecting normal operation,and some will cause disasters.In actual engineering problems,such as the linearization of flutter analysis of nonlinear systems has non-semi-simple purely imaginary eigenvalues at a critical point giving rise to multiple Hopf bifurcations.If the non-semi-simple matrix has a small perturbation,the multiple eigenvalues can be separated into distinct eigenvalues,which is known as the bifurcations of the multiple eigenvalues.In the present study,the compression technology combined with the singular value decomposition method,and the number of singular values is used to judge the characteristic of the defective system,and the generalized modal vectors of each order are solved.Connection with the perturbed eigenvalue problem of defective matrix,the small parameter fractional power expansion method is used to study the eigenvalue change law of the linear part of the nonlinear system when the system parameters pass the critical point of Hopf bifurcation,and the dynamic characteristics of the system are analyzed.The singular value decomposition method is used to discuss that the linear part of the nonlinear system has multiple Jordan blocks,and the eigenvalues of each Jordan block are repeated eigenvalues(that is,the eigenvalues of each Jordan block are equal)And isolated eigenvalues(that is,the eigenvalues of each Jordan block are not equal to each other),the controllability and observability judgment and measurement of the system.This study discusses the feedback control of nonlinear dynamic system.The method of multiple scales and center-manifold reduction are used to deal with the feedback control design of system at the critical point of the Hopf bifurcation.The presented methods are based on the Jordan form which is the simplest one.The results show that the present method is effective and valid for the control of nonlinear system with non-semi-simple eigenvalues in the center subspace.This study discusses an efficient method of the Hopf bifurcation control for nonlinear system.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 system is obtained to minimize the performance measure.