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

2021 ◽  
pp. 107754632199356 ◽  
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.

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.

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.

Lyapunov Perron Transformation for Quasi-Periodic Systems and its Applications

2021 ◽  
pp. 1-26
Author(s):  
Susheelkumar Cherangara Subramanian ◽  
Sangram Redkar
Keyword(s):  
Periodic System ◽  
Normal Forms ◽  
Identity Transformation ◽  
Periodic Systems ◽  
Invariant System ◽  
External Excitation ◽  
P Transformation ◽  
State Augmentation ◽  
Time Invariant ◽  
Time Invariant System

Abstract This paper depicts the application of symbolically computed Lyapunov Perron (L-P) Transformation to solve linear and nonlinear quasi-periodic systems. The L-P transformation converts a linear quasi-periodic system into a time-invariant one. State augmentation and the method of Normal Forms are used to compute the L-P transformation analytically. The state augmentation approach converts a linear quasi-periodic system into a nonlinear time invariant system as the quasi-periodic parametric excitation terms are replaced by ‘fictitious’ states. This nonlinear system can be reduced to a linear system via Normal Forms in the absence of resonances. In this process, one obtains near identity transformation that contains fictitious states. Once the quasi-periodic terms replace the fictitious states they represent, the near identity transformation is converted to the L-P transformation. The L-P transformation can be used to solve linear quasi-periodic systems with external excitation and nonlinear quasi-periodic systems. Two examples are included in this work, a commutative quasi-periodic system and a non-commutative Mathieu-Hill type quasi-periodic system. The results obtained via the L-P transformation approach match very well with the numerical integration and analytical results.

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.

scholarly journals Optimal Preview Control for Linear Discrete-Time Periodic Systems

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Fucheng Liao ◽  
Mengyuan Sun ◽  
Usman
Keyword(s):  
Discrete Time ◽  
Original Problem ◽  
Control Method ◽  
Original System ◽  
Periodic Systems ◽  
Invariant System ◽  
Preview Control ◽  
Time Invariant ◽  
Time Periodic ◽  
Time Invariant System

In this paper, the optimal preview tracking control problem for a class of linear discrete-time periodic systems is investigated and the method to design the optimal preview controller for such systems is given. Initially, by fully considering the characteristic that the coefficient matrices are periodic functions, the system can be converted into a time-invariant system through lifting method. Then, the original problem is also transformed into the scenario of time-invariant system. Later on, the augmented system is constructed and the preview controller of the original system is obtained with the help of existing preview control method. The controller comprises integrator, state feedback, and preview feedforward. Finally, the simulation example shows the effectiveness of the proposed preview controller in improving the tracking performance of the close-loop system.

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.

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 Terminal Wiener Index of Fibonacci trees

2020 ◽  
Vol 9 (3) ◽  
pp. 2533-2535
Keyword(s):  
Closed Form ◽  
Linear Time ◽  
Wiener Index ◽  
Time Algorithm ◽  
Closed Form Expression ◽  
Linear Time Algorithm ◽  
Form Expression ◽  
Sum Of Distances

The terminal Wiener index of a tree is defined as the sum of distances between all leaf pairs of T. We derive closed form expression for the terminal Wiener index of fibonacci trees. We also describe a linear time algorithm to compute terminal Wiener index of a tree.

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.

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