Stability and robustness analysis of a linear time-periodic system subjected to random perturbations

Abstract As per Floquet theory, a transformation matrix (Lyapunov Floquet transformation matrix) converts a linear time periodic system to a linear time-invariant one. Though a closed form expression for such a matrix was missing in the literature, this method has been widely used for studying the dynamical stability of a time periodic system. In this paper, the authors have derived a closed form expression for the Lyapunov Floquet (L-F) transformation matrix analytically using intuitive state augmentation, Modal Transformation and Normal Forms techniques. The results are tested and validated with the numerical methods on a Mathieu equation with and without damping. This approach could be applied to any linear time periodic systems.

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A Simple Approach for the Computation of Lyapunov-Floquet Transformations for the Mathieu Equation

10.1115/detc2021-71028 ◽

2021 ◽

Author(s):

Ashu Sharma

Keyword(s):

Periodic System ◽

Linear Time ◽

Mathieu Equation ◽

Adjoint System ◽

Simple Approach ◽

Periodic Systems ◽

Linear Ordinary Differential Equations ◽

Constant Coefficients ◽

Time Invariant ◽

Time Periodic

Abstract Lyapunov-Floquet (L-F) transformations reduce linear ordinary differential equations with time-periodic coefficients (so-called linear time-periodic systems) to equations with constant coefficients. The present work proposes a simple approach to construct L-F transformations. The solution of a linear time-periodic system can be expressed as a product of an exponential term and a periodic term. Using this Floquet form of a solution, the ordinary differential equation corresponding to a linear time-periodic system reduces to an eigenvalue problem. Next, eigenanalysis is performed to obtain the general solution and subsequently, the state transition matrix of the time-periodic system is constructed. Then, the Lyapunov-Floquet theorem is used to compute L-F transformation. The inverse of L-F transformation is determined by defining the adjoint system to the time-periodic system. Mathieu equation is investigated in this work and L-F transformations and their inverse are generated for stable and unstable cases. These transformations are very useful in the design of controllers using time-invariant methods and in the bifurcation studies of nonlinear time-periodic systems.

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Computation of Lyapunov–Perron transformation for linear quasi-periodic systems

Journal of Vibration and Control ◽

10.1177/1077546321993568 ◽

2021 ◽

pp. 107754632199356 ◽

Cited By ~ 1

Author(s):

Susheelkumar C Subramanian ◽

Peter MB Waswa ◽

Sangram Redkar

Keyword(s):

Closed Form ◽

Periodic System ◽

Linear Time ◽

Closed Form Expression ◽

Periodic Systems ◽

Form Expression ◽

State Augmentation ◽

Time Invariant ◽

Time Periodic ◽

Time Invariant System

The transformation of a linear time periodic system to a time-invariant system is achieved using the Floquet theory. In this work, the authors attempt to extend the same toward the quasi-periodic systems, using a Lyapunov–Perron transformation. Though a technique to obtain the closed-form expression for the Lyapunov–Perron transformation matrix is missing in the literature, the application of unification of multiple theories would aid in identifying such a transformation. In this work, the authors demonstrate a methodology to obtain the closed-form expression for the Lyapunov–Perron transformation analytically for the case of a commutative quasi-periodic system. In addition, for the case of a noncommutative quasi-periodic system, an intuitive state augmentation and normal form techniques are used to reduce the system to a time-invariant form and obtain Lyapunov–Perron transformation. The results are compared with the numerical techniques for validation.

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