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

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.

Unification of Poincaré and Floquet Theory for Time Periodic Systems

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.

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

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.

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

1996 ◽  
Vol 2 (2) ◽  
pp. 219-249 ◽  
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.

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

2008 ◽  
Vol 2008 ◽  
pp. 1-18 ◽  
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.

Lyapunov Perron Transformation for Linear Quasi-Periodic Systems

2020 ◽  
Author(s):  
Susheelkumar C. Subramanian ◽  
Sangram Redkar ◽  
Peter Waswa
Keyword(s):  
Periodic System ◽  
Floquet Theory ◽  
Normal Forms ◽  
Closed Form Expression ◽  
Periodic Systems ◽  
Integration Technique ◽  
Form Expression ◽  
P Transformation ◽  
State Augmentation ◽  
Time Invariant

Abstract It is known that a Lyapunov Perron (L-P) transformation converts a quasi-periodic system into a reduced system with a time-invariant coefficient. Though a closed form expression for L-P transformation matrix is missing in the literature, the application of combination of multiple theories would aid in such transformation. In this work, the authors have worked on extending the Floquet theory to find L-P transformation. As an example, a commutative system with linear quasi-periodic coefficients is transformed into a system with time-invariant coefficient analytically. Furthermore, for non-commutative systems, similar results are obtained in this work, with the help of an intuitive state augmentation and Normal Forms technique. The results of the reduced system are compared with the numerical integration technique for validation.

Reducibility and Analysis of Linear Quasi-Periodic Systems Via Normal Forms

2020 ◽  
Vol 15 (9) ◽  
Author(s):  
Peter M. B. Waswa ◽  
Sangram Redkar
Keyword(s):  
Periodic System ◽  
Constant Coefficient ◽  
Floquet Theory ◽  
Normal Forms ◽  
Time History ◽  
Mathieu Equation ◽  
Periodic Systems ◽  
State Augmentation ◽  
The Stability ◽  
Averaging Techniques

Abstract This article introduces a technique to accomplish reducibility of linear quasi-periodic systems into constant-coefficient linear systems. This is consistent with congruous proofs common in literature. Our methodology is based on Lyapunov–Floquet transformation, normal forms, and enabled by an intuitive state augmentation technique that annihilates the periodicity in a system. Unlike common approaches, the presented approach does not employ perturbation or averaging techniques and does not require a periodic system to be approximated from the quasi-periodic system. By considering the undamped and damped linear quasi-periodic Hill-Mathieu equation, we validate the accuracy of our approach by comparing the time-history behavior of the reduced linear constant-coefficient system with the numerically integrated results of the initial quasi-periodic system. The two outcomes are shown to be in exact agreement. Consequently, the approach presented here is demonstrated to be accurate and reliable. Moreover, we employ Floquet theory as part of our analysis to scrutinize the stability and bifurcation properties of the undamped and damped linear quasi-periodic system.

Cramer–Rao Bound Development for Linear Time Periodic Systems

2010 ◽  
Vol 133 (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.

Stability Analysis and Controller Design for Linear Time Periodic Systems Using Normal Forms

2020 ◽  
Author(s):  
Susheelkumar C. Subramanian ◽  
Sangram Redkar
Keyword(s):  
Stability Analysis ◽  
Normal Forms ◽  
Controller Design ◽  
Linear Time ◽  
Identity Transformation ◽  
Feedback Controller ◽  
Linear Feedback ◽  
Periodic Systems ◽  
Linear Feedback Controller ◽  
Time Periodic

Abstract The investigation of stability bounds for linear time periodic systems have been performed using various methods in the past. The Normal Forms technique has been predominantly used for analysis of nonlinear equations. In this work, the authors draw comparisons between the Floquet theory and Normal Forms technique for a linear system with time periodic coefficients. Moreover, the authors utilize the Normal Forms technique to transform a linear time periodic system to a time-invariant system by using near identity transformation, similar to the Lyapunov Floquet (L-F) transformation. The authors employ an intuitive state augmentation technique, modal transformation and near identity transformations to enable the application of time independent Normal Forms directly without the use of detuning or book-keeping parameter. This method provides a closed form analytical expression for the state transition matrix with the elements as a function of time. Additionally, stability analysis is performed on the transformed system and the resulting transitions curves are compared with that of numerical simulation results. Furthermore, a linear feedback controller design is discussed based on the stability bounds and the implementation of an effective feedback controller for an unstable case is discussed. The theory is validated and verified using numerical simulations of temporal variation of a simple linear Mathieu equation.

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