scholarly journals Linear Time-Periodic System Identification with Grouped Atomic Norm Regularization

2020 ◽  
Vol 53 (2) ◽  
pp. 1237-1242
Author(s):  
Mingzhou Yin ◽  
Andrea Iannelli ◽  
Mohammad Khosravi ◽  
Anilkumar Parsi ◽  
Roy S. Smith
Keyword(s):  
System Identification ◽  
Periodic System ◽  
Linear Time ◽  
Atomic Norm ◽  
Time Periodic

Floquet Experimental Modal Analysis for System Identification of Linear Time-Periodic Systems

2007 ◽  
Author(s):  
Matthew S. Allen
Keyword(s):  
System Identification ◽  
Linear Time ◽  
Time Varying ◽  
Response Model ◽  
State Matrix ◽  
Linear Time Invariant ◽  
Response Data ◽  
Time Varying Systems ◽  
Time Invariant ◽  
Time Periodic

A variety of systems can be faithfully modeled as linear with coefficients that vary periodically with time or Linear Time-Periodic (LTP). Examples include anisotropic rotorbearing systems, wind turbines, satellite systems, etc… A number of powerful techniques have been presented in the past few decades, so that one might expect to model or control an LTP system with relative ease compared to time varying systems in general. However, few, if any, methods exist for experimentally characterizing LTP systems. This work seeks to produce a set of tools that can be used to characterize LTP systems completely through experiment. While such an approach is commonplace for LTI systems, all current methods for time varying systems require either that the system parameters vary slowly with time or else simply identify a few parameters of a pre-defined model to response data. A previous work presented two methods by which system identification techniques for linear time invariant (LTI) systems could be used to identify a response model for an LTP system from free response data. One of these allows the system’s model order to be determined exactly as if the system were linear time-invariant. This work presents a means whereby the response model identified in the previous work can be used to generate the full state transition matrix and the underlying time varying state matrix from an identified LTP response model and illustrates the entire system-identification process using simulated response data for a Jeffcott rotor in anisotropic bearings.

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.

System Identification of Dynamic Systems With Cubic Nonlinearities Using Linear Time-Periodic Approximations

2009 ◽  
Author(s):  
Matthew S. Allen ◽  
Michael W. Sracic
Keyword(s):  
System Identification ◽  
Limit Cycle ◽  
Dynamic Systems ◽  
Nonlinear Dynamic ◽  
Linear Time ◽  
Stable Limit Cycle ◽  
Parametric Models ◽  
Stable Limit ◽  
Nonlinear Dynamic Model ◽  
Time Periodic

This work develops methods to identify parametric models of nonlinear dynamic systems from response measurements using tools for Linear Time Periodic (LTP) systems. The basic approach is to drive the system periodically in a stable limit cycle and then measure deviations of the response from that limit cycle. Under certain conditions, the resulting response can be well approximated as that of a linear-time periodic system. In the analytical realm it is common to linearize a system about a periodic trajectory and then use Floquet analysis to assess the stability of the limit cycle. This work is concerned with the inverse problem, using a measured time-periodic response to derive a nonlinear dynamic model for the system. Recently, a few new methods were developed that facilitate the experimental identification of linear time periodic systems, and those methods are exploited in this work. The proposed system identification methodology is evaluated by applying it to a Duffing oscillator, demonstrating that the nonlinear force-displacement relationship can be identified without a priori knowledge of its functional form. The proposed methods are also applied to simulated measurements from a cantilever beam with a cubic nonlinear spring on its tip, revealing that the model order of the system and the displacement dependent stiffness can be readily identified.

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.

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