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.

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.

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.

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.

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.

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