A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems

DaCunha, Jeffrey J.; Davis, John M.

doi:10.1016/j.jde.2011.07.023

Cited by 48 publications

(38 citation statements)

References 22 publications

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“…see Dacunha and Davis [2009] Corollary 2. (Floquet Transformation) The value of the fundamental matrix at t = T is called the Monodromy Matrix.…”

Section: (T) Is Known As the Fundamental Transition Matrix (Ftm) Andφmentioning

confidence: 98%

“…• A simple, necessary and sufficient criterion of stability can be extracted from periodic data, using the theory of Floquet (Dacunha and Davis [2009]). Therefore, a simple residual based on this criterion can be built • A set of time-invariant data subsequences that have the same time-varying behavior can be found for such periodic systems, as outlined in Meyer and Burrus [1975], which makes it possible to apply a detection algorithm similar to the classical time-invariant one (Mevel et al [2005]), for each of these sequences…”

Section: Introductionmentioning

confidence: 99%

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Subspace Instability Monitoring for Linear Periodically Time-Varying Systems*

Jhinaoui

Mevel²,

Morlier³

2012

IFAC Proceedings Volumes

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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. Abstract: Most subspace-based methods enabling instability monitoring are restricted to the linear time-invariant (LTI) systems. In this paper, a new subspace method of instability monitoring is proposed for the linear periodically time-varying (LPTV) case. For some LPTV systems, the system transition matrices may depend on some parameter and are also periodic in time. A certain range of values for the parameter leads to an unstable transition matrix. Early warning should be given when the system gets close to that region, taking into account the time variation of the system. Using the theory of Floquet, some symptom parameter of stability-or residual-is defined. Then, the parameter variation is tracked by performing a set of parallel cumulative sum (CUSUM) tests. Finally, the method is tested on a simulated model of a helicopter with hinged blades, for monitoring the ground resonance phenomenon.

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“…see Dacunha and Davis [2009] Corollary 2. (Floquet Transformation) The value of the fundamental matrix at t = T is called the Monodromy Matrix.…”

Section: (T) Is Known As the Fundamental Transition Matrix (Ftm) Andφmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Subspace Instability Monitoring for Linear Periodically Time-Varying Systems*

Jhinaoui

Mevel²,

Morlier³

2012

IFAC Proceedings Volumes

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“…A comprehensive Floquet theory including Lyapunov transformations was developed and their various stability preserving properties were analyzed in [8]. Colaneri [5] addresses a few theoretical aspects of LPTV systems and methodology which can be useful to characterize and extend other concepts usually exploited in the time-invariant case only.…”

Section: Systemsmentioning

confidence: 99%

A Note on Uniform Exponential Stability of Linear Periodic Time-Varying Systems

Vrabel

2020

IEEE Trans. Automat. Contr.

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In this paper we derive new criterion for uniform stability assessment of the linear periodic time-varying systemṡ x = A(t)x, A(t + T ) = A(t). As a corollary, the lower and upper bounds for the Floquet characteristic exponents are established. The approach is based on the use of logarithmic norm of the system matrix A(t). Finally we analyze the robustness of the stability property under external disturbance.Index Terms-Linear periodic time-varying system, uniform exponential stability, uniform stability, logarithmic norm.

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“…Some of its essential elements, that are related to the study hereafter, are briefly reviewed. More details can be found in [36].…”

Section: On Floquet Theorymentioning

confidence: 99%

A new SSI algorithm for LPTV systems: Application to a hinged-bladed helicopter

Jhinaoui

Mevel

Morlier

2014

Mechanical Systems and Signal Processing

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a b s t r a c tMany systems such as turbo-generators, wind turbines and helicopters show intrinsic time-periodic behaviors. Usually, these structures are considered to be faithfully modeled as linear time-invariant (LTI). In some cases where the rotor is anisotropic, this modeling does not hold and the equations of motion lead necessarily to a linear periodically timevarying (referred to as LPTV in the control and digital signal field or LTP in the mechanical and nonlinear dynamics world) model. Classical modal analysis methodologies based on the classical time-invariant eigenstructure (frequencies and damping ratios) of the system no more apply. This is the case in particular for subspace methods. For such time-periodic systems, the modal analysis can be described by characteristic exponents called Floquet multipliers. The aim of this paper is to suggest a new subspace-based algorithm that is able to extract these multipliers and the corresponding frequencies and damping ratios. The algorithm is then tested on a numerical model of a hinged-bladed helicopter on the ground. IntroductionMost existing identification techniques in mechanical and civil engineering work under the assumption that the underlying system can be modeled by a linear time-invariant (LTI) model. Unfortunately, structures that exhibit intrinsically time-varying behaviors are increasingly used in industry. For accurate analysis of such structures, that assumption is not satisfied. The time-varying aspect must be taken into account for a whole and reliable description of the system dynamics.The extension of the well-known identification techniques to the linear time-varying (LTV) systems is an ongoing active topic of research. A wide range of methods have been suggested in the literature. Among them, one can cite the frozen-time approach introduced first in [1,2] which deals with the identification of slowly time-varying systems (the frozen-time approach consists in modeling the slowly time varying system as a sequence of LTI systems, called frozen-time systems). Recursive algorithms such as recursive least squares (RLS), recursive instrumental variable and recursive predictive error [3,4] have also been widely investigated. For the class of rapidly time-varying systems, the functional expansion techniques have been suggested in manifold works [5][6][7][8].In [9][10][11][12], efforts have been undertaken to extend the subspace identification approach [13] to the LTV case by introducing the idea of repeated experiments. As pointed in [14], most subspace-based results developed thus far, even if significant, give state space realizations that are topologically equivalent from an input and output standpoint, but are not n Corresponding author. Tel.: þ33 2 99 84 73 25; fax: þ 33 2 99 84 71 71.

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A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems

Cited by 48 publications

References 22 publications

Subspace Instability Monitoring for Linear Periodically Time-Varying Systems*

Subspace Instability Monitoring for Linear Periodically Time-Varying Systems*

A Note on Uniform Exponential Stability of Linear Periodic Time-Varying Systems

A new SSI algorithm for LPTV systems: Application to a hinged-bladed helicopter

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