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

Mapping Intimacies ◽

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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Comparison of Poincaré Normal Forms and Floquet Theory for Analysis of Linear Time Periodic Systems

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4048715 ◽

2020 ◽

Vol 16 (1) ◽

Author(s):

Susheelkumar C. Subramanian ◽

Sangram Redkar

Keyword(s):

Floquet Theory ◽

Normal Forms ◽

Linear Time ◽

Mathieu Equation ◽

Temporal Variations ◽

Periodic Systems ◽

Comparative Results ◽

Time Invariant ◽

Identity Transformations ◽

Time Periodic

Abstract In this work, the authors draw comparisons between the Floquet theory and Normal Forms technique and apply them towards the investigation of stability bounds for linear time periodic systems. Though the Normal Forms technique has been predominantly used for the analysis of nonlinear equations, in this work, the authors utilize it to transform a linear time periodic system to a time-invariant system, similar to the Lyapunov–Floquet (L–F) transformation. The authors employ an intuitive state augmentation technique, modal transformation, and near identity transformations to facilitate the application of time-independent Normal Forms. This method provides a closed form analytical expression for the state transition matrix (STM). Additionally, stability analysis is performed on the transformed system and the comparative results of dynamical characteristics and temporal variations of a simple linear Mathieu equation are also presented in this work.

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Unification of Poincaré and Floquet Theory for Time Periodic Systems

Volume 2: 16th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC) ◽

10.1115/detc2020-22524 ◽

2020 ◽

Author(s):

Susheelkumar C. Subramanian ◽

Sangram Redkar

Keyword(s):

Closed Form ◽

Periodic System ◽

Floquet Theory ◽

Linear Time ◽

Mathieu Equation ◽

Transformation Matrix ◽

Closed Form Expression ◽

Periodic Systems ◽

Form Expression ◽

Time Periodic

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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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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On The Response of Linear Time-Periodic Systems Subjected to Deterministic and Stochastic Excitations

Journal of Vibration and Control ◽

10.1177/107754639600200205 ◽

1996 ◽

Vol 2 (2) ◽

pp. 219-249 ◽

Cited By ~ 1

Author(s):

J.M. Spires ◽

S.C. Sinha

Keyword(s):

Chebyshev Polynomial ◽

Linear Time ◽

Original System ◽

Invariant Form ◽

Periodic Systems ◽

Solution Technique ◽

Stiffness Matrices ◽

Original Time ◽

Time Invariant ◽

Time Periodic

In many situations, engineering systems modeled by a set of linear, second-order differential equations, with periodic damping and stiffness matrices, are subjected to external excitations. It has been shown that the fundamental solution matrix for such systems can be efficiently computed using a Chebyshev polynomial series solution technique. Further, it is shown that the Liapunov-Floquet transformation matrix associated with the system can be computed, and the original time-periodic system can be put into a time- invariant form. In this paper, these techniques are applied in finding the transient response of periodic systems subjected to deterministic and stochastic forces. Two formulations are presented. In the first formulation, the response of the original system is computed directly. In the second formulation, first the original system is transformed to a time-invariant form, and then the response is found by determining the response of the time-invariant system. Both formulations use the convolution integral to form an expression for the response. This expression can be evaluated numerically, symbolically, or through a Chebyshev polynomial expansion technique. Results for some time-invariant and periodic systems are included, as illustrative examples.

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On Fault Detection in Linear Discrete-Time, Periodic, and Sampled-Data Systems

Journal of Control Science and Engineering ◽

10.1155/2008/849546 ◽

2008 ◽

Vol 2008 ◽

pp. 1-18 ◽

Cited By ~ 31

Author(s):

P. Zhang ◽

S. X. Ding

Keyword(s):

Fault Detection ◽

Linear Time ◽

Periodic Systems ◽

Data Systems ◽

Sampled Data ◽

Linear Time Invariant ◽

Linear Time Invariant Systems ◽

Problem Formulations ◽

Time Invariant ◽

Time Periodic

This paper gives a review of some standard fault-detection (FD) problem formulations in discrete linear time-invariant systems and the available solutions. Based on it, recent development of FD in periodic systems and sampled-data systems is reviewed and presented. The focus in this paper is on the robustness and sensitivity issues in designing model-based FD systems.

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Linear Time Invariant Approximations of Linear Time Periodic Systems

Journal of the American Helicopter Society ◽

10.4050/jahs.62.012006 ◽

2017 ◽

Vol 62 (1) ◽

pp. 1-10 ◽

Cited By ~ 4

Author(s):

MarkJ. S. Lopez ◽

J. V. R. Prasad

Keyword(s):

Linear Time ◽

Periodic Systems ◽

Linear Time Invariant ◽

Time Invariant ◽

Time Periodic

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Cramer–Rao Bound Development for Linear Time Periodic Systems

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.4002104 ◽

2010 ◽

Vol 133 (1) ◽

Cited By ~ 1

Author(s):

Chris S. Schulz ◽

Donald L. Kunz ◽

Norman M. Wereley

Keyword(s):

Linear Time ◽

System Model ◽

Periodic Systems ◽

Accurate Representation ◽

Linear Time Invariant ◽

Lti System ◽

Identification Techniques ◽

Cramer Rao Bound ◽

Time Invariant ◽

Time Periodic

System identification techniques are often used to determine the parameters required to define a model of a linear time invariant (LTI) system. The Cramer–Rao bound can be used to validate those parameters in order to ensure that the system model is an accurate representation of the system. Unfortunately, the Cramer–Rao bound is only valid for LTI systems and is not valid for linear time periodic (LTP) systems such as a helicopter rotor in forward flight. This paper describes an extension of the Cramer–Rao bound to LTP systems and demonstrates the methodology for a simple LTP system.

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