Analysis and Control of Linear Time Periodic System using Normal Forms

Multiple techniques have been developed in the past towards stability and control of linear time periodic systems. Though the method of normal forms was predominantly applied to nonlinear equations, in this work, it is utilized to transform a linear time varying system with periodic coefficients to a time-invariant system (similar to a Lyapunov-Floquet transformation). The direct application of time independent normal forms is facilitated by a combination of an intuitive state augmentation technique and modal transformation. Additionally, this approach yields a closed form analytical expression for the Lyapunov–Floquet (L-F) transformation and state transition matrix. The transition curves and stability bounds are identified and multiple feedback control strategies are also discussed in this work. Furthermore, the authors demonstrate the control of an unstable periodic system to a stable point, desired periodic orbit and optimally controlled system states using the normal forms approach. The theoretical framework and controller implementation are illustrated using numerical simulations for the case of a linear Mathieu equation.