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.
In this paper, an extension of the output-only subspace identification, to the class of linear periodically time-varying (LPTV) systems, is proposed. The goal is to identify a useful information about the system's stability using the Floquet theory which gives a necessary and sufficient condition for stability analysis. This information is retrieved from a matrix called the monodromy matrix, which is extracted by some simultaneous singular value decompositions (SVD) and from a resolution of a least squares criterion. The method is, finally, illustrated by a simulation of a model of a helicopter with a hinged-blades rotor.
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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