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Time-Periodic and High-Order Time-Invariant Linearized Models of Rotorcraft: A Survey

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Abstract

The objective of this paper is to summarize the relevant published research studies on the extraction of linear time-periodic (LTP) systems and their higher order linear time-invariant (LTI) reformulations from rotorcraft physics-based models and on the identification of LTP systems from rotorcraft experimental data. The paper begins with an introductory overview of LTP system theory. Next, the relevant methods for the extraction of LTP and high-order LTI systems from physics-based models are presented. The paper continues with an overview of LTP model identification methods, followed by a discussion on the application of these methods toward the identification of the rotor dynamics alone and the coupled rigid-body/rotor dynamics. Final remarks summarize the overall findings of the study and identify areas for future work including, but not limited to, the context of the Future Vertical Lift (FVL) program.

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... 6,7). A comprehensive survey of the methods used to extract harmonically-decomposed LTI models of rotorcraft is found in Ref. 8. In flapping-wing applications, LTI reformulations of LTP systems are important for the study of dynamic stability. ...
... In fact, only a limited amount of studies sought the identification of the LTP dynamics of rotorcraft or flapping-wing vehicles. For rotorcraft LTP system identification, see Ref. 8 and the references therein. For flapping-wing, see Refs. ...
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This paper presents a first step in the extension of subspace identification toward the direct identification of harmonic decomposition linear time-invariant (LTI) models from nonlinear time-periodic (NLTP) system responses. The proposed methodology is demonstrated through examples involving the NLTP dynamics of a flapping-wing micro aerial vehicle (FWMAV). These examples focus on the identification of the vertical dynamics from various types of input-output data, including LTI, LTP, and NLTP input-output data. The use of a harmonic analyzer to decompose the LTP and NLTP responses into harmonic components is shown to introduce spurious dynamics in the identification, which makes the identified model order selection challenging. A similar effect is introduced by measurement noise. The use of model-order reduction and model-matching methods in the identification process is studied to recover the harmonic decomposition structure of the known system. The identified models are validated both in the frequency and time domains.
... A detailed summary of the literature on the extraction of LTP systems and their higher order LTI reformulations from rotorcraft physics-based models can be found in Ref. 16. To frame the current study, it is sufficient to mention the current state of the art, which consists of two major approaches. ...
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A considerable amount of work has been dedicated in the past to the problem of the system identification of helicopter flight dynamics, while much less activity has been oriented to the goal of developing suitable identification procedures for rotor dynamics, mainly because of the difficulties associated with the task. This paper shows that subspace and optimization based identification techniques can be used to determine discrete-time linear parameter-varying models that have the potential to provide accurate descriptions for the (intrinsically time-varying) dynamics of a rotor blade. The identification techniques are presented and applied to simulated data generated by a physical model that describes the out-of-plane bending dynamics of a helicopter rotor blade.
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This paper is concerned with the identification of linear time-varying systems. The discrete-time state space model of freely vibrating systems is used as an identification model. The focus is placed on identifying successive discrete transition matrices that have the same eigenvalues as the original transition matrices. First a typical subspace-based method is presented to illustrate the extraction of the observability range space using the singular value decomposition (SVD) of a general Hankel matrix. Then, the identification of varying transition matrices is approached by using an ensemble of response sequences. For arbitrarily varying systems, a series of the Hankel matrices are formed by an ensemble set of responses which are obtained through multiple experiments on the system with the same time-varying behavior. The varying transition matrix at each moment is estimated through the SVD of two successive Hankel matrices. The proposed algorithm is applied to two special cases that require only a single response series, i.e., periodically varying systems and slowly varying systems. The use of the eigenvalues of the transition matrices is discussed and the pseudomodal parameters are defined. Finally, a two-link manipulator subjected to a varying end force is used as an example to illustrate the tracking capability and performance of the proposed algorithm. (C) 1997 Academic Press Limited.
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A non-stochastic state-space identification algorithm for linear time- invariant systems is modified for use with periodic linear systems. Like its predecessor, the modified algorithm forms block Hankel matrices from input-output data and uses the singular value decomposition of these Hankel matrices to compute state vector sequences. The state vector sequences are then used to compute system matrices associated with the periodic linear system by solving an (overdetermined) linear system. The modification to the original algorithm are as follows. First, the periodic system of period p is viewed as p separate time-invariant period-mapped systems. This technique allows the structure of a periodic Hankel matrix to be deduced, which in turn allows a state vector subsequence for the periodic system to be computed. When a complete state vector sequence is computed, it is used directly to construct the periodic state-space models. Second, similarities in the structure of each of the periodic Hankel matrices associated with the p time- invariant period-mapped systems is exploited in such a way that not only reduces the amount of computation necessary, but also improves the accuracy of the computation of the system matrices.
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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 rotor-bearing systems, wind turbines and nonlinear systems linearized about a periodic trajectory; all of these have been treated analytically in the literature. However, few methods exist for experimentally characterizing LTP systems. This paper presents a set of tools that can be used to experimentally characterize an LTP system, using a frequency domain approach and utilizing existing algorithms to perform parameter identification. One of the approaches is based on lifting the response to obtain an equivalent Linear Time-Invariant (LTI) form and the other based on Fourier series expansion. The development focuses on the pre-processing steps needed to apply LTI identification to the measurements, the post-processing needed to reconstruct the LTP model from the identification results and the interpretation of the measurements. This approach elucidates the similarities between LTP and LTI identification, allowing the experimentalist to transfer insight from time-invariant systems to the LTP identification problem. The approach determines the model order of the system, and post processing reveals the shapes of the time-periodic functions comprising the LTP model. Further post-processing is also presented that allows one to generate the full state transition matrix and the time-varying state matrix of the system from the parametric model if the measurement set is adequate. The experimental techniques are demonstrated on simulated measurements from a Jeffcott rotor mounted on an anisotropic, flexible shaft, supported by anisotropic bearings.