Some Ideas on the Local Control of Nonlinear Systems With Time-Periodic Coefficients

In this paper ideas on local control of linear and nonlinear time-periodic systems are presented. Our first goal is to stabilize the system far away from bifurcation points. In this case, the classical linear state feedback stabilization based on pole placement is generalized such that it is applicable to time-periodic systems. The linear state-feedback controller design involves computation of the fundamental solution matrix of the system in a symbolic form as function of the control parameters. Next, we focus on the bifurcation control of time-periodic systems. When the linearized system is in a critical case of stability (i.e. when it has Floquet multipliers on the unit circle of the complex plane) and the critical modes are uncontrollable in the linear sense, then a purely nonlinear state-feedback controller is designed to stabilize the equilibrium at the bifurcation point and ensure the stability of the bifurcated nontrivial solution. When the linearized system is linearly controllable, then it is shown that an appropriately chosen linear state-feedback control can also modify the nonlinear features of the bifurcations, such as stability or size of the limit cycles or quasi-periodic limit sets. The control techniques are based on a series of transformations that convert the system into a time-invariant form. First, the Lyapunov-Floquet transformation is used to make the linear part of the periodic system time-invariant. Then, time-periodic center manifold reduction and time-dependent normal form theory are applied to obtain the simplest nonlinear form of a system undergoing bifurcation. For most codimension one bifurcations the normal form is completely time-invariant and therefore, it is a rather simple task to choose the appropriate control gains. These ideas are illustrated by an example of a parametrically excited simple pendulum undergoing symmetry breaking bifurcation.