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Time-Periodic and High-Order Time-Invariant Linearized Models of Rotorcraft: A Survey
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.
... Recently, LTI reformulations of LTP systems were employed in the design of load alleviation control (LAC) laws [6,7]. A comprehensive survey of the methods used to extract harmonically decomposed LTI models of rotorcraft can be 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 number of studies have sought to identify the LTP dynamics of rotorcraft or flapping-wing vehicles. For rotorcraft LTP system identification, see Ref. [8] and the references therein. For flapping-wing vehicles, see Refs. ...
... Consider now a discrete-time representation of the harmonic decomposition system in Equation (8) with unknown coefficient matrices: ...
This paper presents a novel step in the extension of subspace identification toward the direct identification of harmonic decomposition linear time-invariant models from nonlinear time-periodic system responses. The proposed methodology is demonstrated through examples involving the nonlinear time-periodic dynamics of a flapping-wing micro aerial vehicle. These examples focus on the identification of the vertical dynamics from various types of input–output data, including linear time-invariant, linear time-periodic, and nonlinear time-periodic input–output data. A harmonic analyzer is used to decompose the linear time-periodic and nonlinear time-periodic responses into harmonic components and introduce spurious dynamics into the identification, which make 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 in the frequency and time domains.
... Considering the forecasted increase in the number of wind farms in the coming years, it is important to implement reliable fault detection methods based on data collected during operation. The methods based on Operational Modal Analysis (OMA) system matrices and the EMP signal [17,18]. Theoretically, with this representation, the dimension of the new system matrices is infinite, as the Fourier series decomposition of the periodic matrices is also infinite. ...
... The generated data are sampled at 25 Hz during 600 s. For each run of the Monte Carlo simulation, the identification of the wind turbine parameters is performed at model order (18) and with the same lag parameters (q + 1 = p = 100). Finally, the empirical uncertainties from the Monte Carlo simulation are compared with the mean estimated uncertainties from 100 other identifications (referred as the estimation in the following (est)), where denotes the standard deviation of the frequency and denotes the standard deviation of the damping. ...
... 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. ...
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.
... The current investigation adopts the method described in [25][26][27][28], which is summarized as follows: Consider an NLTP system in firstorder form representative of the coupled rotor and free-vortex wake dynamics [22][23][24][25][26][27][28][29][30][31]: ...
This article describes the implementation, linearization, and model-order reduction of a free-vortex wake model in state-variable form and its use in the real-time prediction of complex aerodynamic interactions during shipboard launch and recovery operations. Following a detailed mathematical description, the wake model is implemented for a UH-60 rotor and tested in different flight conditions and for simple control inputs. Interference effects due to the sea surface, ship deck, and ship superstructures are modeled using the method of images. A periodic solution to the wake model is found by time marching the coupled rotor and vortex wake dynamics. Next, linearized harmonic decomposition models are obtained and validated against nonlinear simulations. Order reduction methods are explored to guide the development of linearized wake models with real-time performance that still guarantee satisfactory prediction of the periodic response of the wake.
... Within the context of rotorcraft, harmonic decomposition models have been used to: (1) study the interference effects between higher-harmonic control (HHC) and the aircraft flight control system (AFCS) [35,[38][39][40]; (2) design load alleviation control (LAC) laws (the PI's efforts in [41][42][43]; and (3) prediction and avoidance of flight envelope limits [43][44][45]. A survey by the PI on the use of harmonic decomposition models in the rotorcraft field can be found in [46]. When coupled with a harmonic balance scheme, harmonic decomposition can also be used to solve for (stable and unstable) periodic solutions [47] and compute open-loop higher-harmonic control (HHC) inputs that attenuate arbitrary state/output harmonics [9]. ...
This article investigates vibrational stabilization effects in rotorcraft flight dynamics. This study is motivated by recent results in flapping-wing flight, which showed that the time-varying aerodynamic and inertial loads due to the insect wing periodic motion induce a vibrational stabilization mechanism in hover. The dynamics of flapping-wing flyers and rotary- wing vehicles are both described by time-periodic systems so vibrational stabilization mechanisms can also have an effect on stability characteristics of rotary-wing vehicles. The article extends the use of the harmonic decomposition method to vibrational stability analysis of rotorcraft. Two cases are considered: vibrational stability due blade imbalance at hover, and vibrational stability due to number-of-blades-per-rotor-revolution (Nb/rev) in high-speed forward flight. Results show that while vibrations induced by rotor blade imbalance do not stabilize the hovering dynamics of a helicopter, these vibrations can still have a significant effect on the hovering dynamics. Rotor blade imbalance results in a symmetric effect on the roll and pitch axes, in that it tends to decrease the frequency of the subsidence modes of the hovering cubic, while the unstable oscillatory modes tend to increase in frequency and decrease in damping (destabilizing effect). On the other hand, the yaw/ heave dynamics are relatively unaffected compared to the lateral and longitudinal axes. Moreover, Nb/rev rotor loads in forward flight are shown to reduce the damping of the coupled roll/pitch oscillation mode.
... 28) was adopted to represent LTP systems with higher order LTI models for use in prediction of loads and vibration in a control design framework (Ref. [30][31][32]. While harmonic decomposition originated from within the rotorcraft community, it was also recently extended to flapping-wing flight (Ref. ...
This article describes the implementation and linearization of free-vortex wake models in state-variable form as applied to rotary- and flapping-wing vehicles. More specifically, the wake models are implemented and tested for a UH-60 rotor in forward flight and for a hovering insect representative of a hawk moth. A periodic solution to each wake model is found by time marching the coupled rotor/wing and vortex wake dynamics. Next, linearized harmonic decomposition models are obtained and validated against nonlinear simulations. Order reduction methods are explored to guide the development of linearized wake models that provide increased runtime performance compared to the nonlinear and linearized harmonic decomposition wake models while guaranteeing satisfactory prediction of the periodic response of the wake. This constitutes a first attempt to extend free-vortex wake methods in state-variable form, originally developed for rotary-wing applications, to flapping-wing flight.
... 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. ...
This paper discusses the development of a numerical method for the approximation of the nonlinear time-periodic rotorcraft flight dynamics with higher order linear time-invariant (LTI) models. The method relies on a per-rotor revolution perturbation scheme, which is of particular importance for the linearization of simulation models that do not allow for per-time-step perturbations, and for those output measures that necessitate the solution of partial differential equations and thus require several time steps to be computed. The paper demonstrates the application of the proposed methodology to obtain high-order LTI models capable of predicting vibrations for a generic utility helicopter. Simulations are used to validate the response of the linearized models against those from nonlinear simulations and from competing approaches in the literature. The proposed method is shown to predict accurately the nonlinear response for the case shown and for small amplitude maneuvers. Frequency-domain validation is also performed to compare the linear models derived with the proposed method with those obtained with harmonic decomposition, a competing approach based on a per-time-step perturbation scheme. Interestingly, the proposed algorithm yields nearly identical numerical results compared to harmonic decomposition, suggesting that the two methods are in fact equivalent but rely on different formulations.
This work employs the principles of time-variant systems theory to investigate the unsteady aerodynamics of rotary-wing configurations under periodic equilibrium conditions. Their application enables an extension of the pulse technique for system identification, as well as the adaptation of the linear-frequency-domain formulation commonly utilized in fixed-wing to rotary-wing scenarios. These methodologies effectively incorporate the aerodynamic nonlinearities associated with the equilibrium state into an efficient time-variant linearized representation of the unsteady aerodynamics. To promote its application in the context of rotary-wing aeroelasticity, a state-space realization based on a periodic autoregressive model with exogenous input is subsequently employed. Upon transformation from discrete to continuous time, the resulting aerodynamic model adopts a linear continuous-time periodic state-space formulation, offering compatibility for its coupling with a wide range of structural models. The proposed aerodynamic framework tailored to rotary-wing aeroelasticity holds applicability across a spectrum of aerodynamic models of arbitrary complexity, spanning from incompressible potential flow approximations to potentially more sophisticated methods. Showcasing the potential of this framework, the widely studied lossy Mathieu equation and the aerodynamic response to a flap perturbation about the periodic equilibrium condition of a prototypical rotor blade section, incorporating nonlinearities through an analytical dynamic stall model, are considered.
This paper investigates vibrational stabilization effects in rotorcraft flight dynamics. This study is motivated by the fact that eigenvalues of the rotorcraft flight dynamics identified from flight test often differ from those computed with physics-based simulations, and that some commonly observed mismatches may be ascribed to vibrational stability effects due to rotor blade imbalance or other periodic disturbance on the rotorcraft. Starting from a simple example involving an inverted pendulum, the paper demonstrates the use of the harmonic decomposition method for the study of vibrational stabilization effects. The concept is then extended to analyze the effect of blade imbalance on the flight dynamics of a helicopter. Additionally, vibrational stablization of a slung load in forward flight is investigated using small-amplitude and disturbances on an active cargo hook. Results show that while vibrations induced by rotor blade imbalance do not stabilize the hovering dynamics of a helicopter, these vibrations still have a significant effect on the hovering dynamics. Rotor blade imbalance results in a symmetric effect on the roll and pitch axes, in that it tends to decrease the frequency of the subsidence modes of the hovering cubic, while the unstable oscillatory modes tend to increase in frequency and decrease in damping. On the other hand, the yaw/heave dynamics are relatively unaffected compared to the lateral and longitudinal axes. Moreover, small-amplitude oscillations of an active cargo hook were shown to significantly decrease the amplitude of the limit cycle oscillation of a slung load in forward flight and to stabilize its dynamics. This constitutes a practical solution to the semi-active control of a suspended load.
Accurate and real-time load monitoring of vital components located in the rotor system is important for not only
inferring usage and estimating fatigue in those components, but also for developing load alleviation /limiting
control schemes. An approach previously developed for online estimation of rotor component loads is evaluated
in this paper in which a linear time invariant (LTI) model of helicopter coupled body/rotor dynamics is combined
with a Linear Quadratic Estimator (LQE), that is designed to correct LTI model state response using fixed system
measurements. The estimation fidelity of the LTI/LQE scheme is evaluated in simulations using a nonlinear
model of a generic helicopter for online prediction of rotor blade pitch link loads arising from vehicle maneuvers.
This paper explores the use of Model Predictive Control (MPC) as a viable solution for real time load limiting for critical helicopter components that are subjected to significant fatigue loading. The development of the (structural) load limiting controller is posed as an optimal control problem where given a desired user defined max limit, the estimate of control margin associated with the component load limit is found and is provided as a cue to the pilot. The proposed load limiting via MPC, which makes use of an on-board reduced order model derived from a higher order Linear Time Invariant (LTI) model of helicopter coupled body/rotor/inflow dynamics, is described in detail. The proposed scheme is evaluated using nonlinear model simulations for its ability to limit harmonic pitch link loads and indirectly limit the peak-to-peak of the total pitch link load
In this paper we show how higher-order averaging can be used to remedy serious technical issues with the direct application of the averaging theorem. While doing so, we reconcile two higher-order averaging methodologies that were developed independently using different tools and within different communities: (i) perturbation theory using a near-identity transformation and (ii) chronological calculus using Lie algebraic tools. We provide the underpinning concepts behind each averaging approach and provide a mathematical proof for their equivalence up to the fourth order. Moreover, we provide a higher-order averaging study and analysis for two applications: the classical problem of the Kapitza pendulum and the modern application of flapping flight dynamics of micro-air-vehicles and/or insects.
A matrix-based computational scheme is developed based on the Generalized Harmonic Balance method for periodic solutions of nonlinear dynamical systems. The nonlinear external loading is expanded into a Taylor's series as a function of displacement and velocity, and is then expressed as a combination of Fourier harmonics through the Generalized Harmonic Balance method. Using the Newton-Raphson's approach, an iteration scheme is formulated to obtain the solution of harmonic coefficients for the displacement. The present scheme is a general purpose realization of the Generalized Harmonic Balance method in the sense that it does not need an analytical Fourier expansion of loadings, and all of the coefficient matrices involved with the scheme are created in a standard way. An example of a periodically forced Duffing oscillator is provided to demonstrate the performance of the present scheme. Numerical solutions of period-1 motion from the present scheme are compared with numerical results given by the Runge-Kutta method. The numerical results agree well with analytical predictions by Luo et al.
In this paper, the generalized harmonic balance method is presented for approximate, analytical solutions of periodic motions in nonlinear dynamical systems. The nonlinear damping, periodically forced, Duffing oscillator is studied as a sample problem. The approximate, analytical solution of period-1 periodic motion of such an oscillator is obtained by the generalized harmonic balance method. The stability and bifurcation analysis of the HB2 approximate solution of period-1 motions in the forced Duffing oscillator is carried out, and the parameter map for such HB2 solutions is achieved. Numerical illustrations of period-1 motions are presented. Similarly, the same ideas can be extended to period-k motions in such an oscillator. The methodology presented in this paper can be applied to other nonlinear vibration systems, which are independent of small parameters.
System identification methodology is developed for a linear time-periodic (LTP) system and applied to an experimental setup of an integrally twist-actuated helicopter rotor blade. Identification is conducted for a controller design, which alleviates vibratory loads induced in forward flight. Since a rotor in forward flight is a time-periodic system due to the aerodynamic environment varying once per rotation, the adopted methodology requires determination of the multicomponent harmonic transfer functions. A simplified identification formula is also derived for a linear time-invariant (LTI) system, such as a rotor system in hover. The latter approach gives another estimate of the primary component among the harmonic transfer functions. The identification experiment is conducted at NASA Langley Transonic Dynamics Tunnel. The magnitude of the higher-order harmonic transfer functions is observed to be small in the frequency range of interest when compared with that of the primary component. This indicates that the present active rotor system may be regarded as a LTI system under the level flight conditions considered. Results obtained in system identification are interpreted in terms of the closed-loop controller design.
Linear and nonlinear time-varying controller synthesis for systems represented by nonlinear dif ferential equations with periodic coefficients is addressed. A recently developed technique, based on the Liapunov-Floquet (L-F) theorem, is employed so that time-varying control gains can be obtained via time- invariant techniques. Furthermore, a simple time-varying pole-placement approach for the design of linear control has also been devised for linear time-periodic systems. The robustness of the above control designs under structured perturbations of the nominal system matrices is studied. In many cases, the linear control de sign alone may not meet the desired performance specifications of the nonlinear periodic systems due to the time-varying nature of the problem. Therefore, to improve the controlled response of the nonlinear system, a nonlinear time-varying controller is designed and incorporated. The linear control is used to stabilize and the nonlinear controller is employed to improve the response specifications of the system. The linear control de signs are based on the L-F transformation approach and the time-varying pole-placement approach, whereas the nonlinear controller is obtained using the Liapunov direct method. The responses obtained through the above approaches are compared and the advantages and disadvantages of the methods are discussed. Notice ably, the combination of linear and nonlinear controllers based on the L-F transformation approach has been found to have better performance and robustness characteristics than the other approach.
This paper presents an aeromechanical closed-loop stability and response analysis of a hingeless rotor helicopter with a higher harmonic control system for vibration reduction. The analysis includes the rigid body dynamics of the helicopter and blade flexibility. The gain matrix is assumed to be fixed and computed offline. The discrete elements of the higher harmonic control loop are rigorously modeled, including the presence of two different time scales in the loop. By also formulating the coupled rotor-fuselage dynamics in discrete form, the entire coupled helicopter higher harmonic control system could be rigorously modeled as a discrete system. The effect of the periodicity of the equations of motion is rigorously taken into account by converting the system into an equivalent system with constant coefficients and identical stability properties using a time-lifting technique. The most important conclusion of the present study is that the discrete elements in the higher harmonic control loop must be modeled in any higher harmonic control analysis. Not doing so is unconservative. For the helicopter configuration and higher harmonic control structure used in this study, an approximate continuous modeling of the higher harmonic control system indicates that the closed-loop, coupled helicopter higher harmonic control system is always stable, whereas the more rigorous discrete analysis shows that closed-loop instabilities can occur. The higher harmonic control gains must be reduced to account for the loss of gain margin brought about by the discrete elements. Other conclusions of the study are 1) the higher harmonic control is effective in quickly reducing vibrations, at least at its design condition; 2) a linearized model of helicopter dynamics is adequate for higher harmonic control design, as long as the periodicity of the system is correctly taken into account, that is, periodicity is more important than nonlinearity, at least for the mathematical model used in this study; and 3) when discrete and continuous systems are both stable, the predicted higher harmonic control control harmonies are in good agreement, although the initial transient behavior can be considerably different.
This paper presents a concise review of the state of the art for vibration reduction in rotorcraft using active controls, The principal approaches to vibration reduction in helicopters described in the paper are 1) higher harmonic control, 2) individual blade control, 3) vibration reduction using an actively controlled flap located on the blade, and 4) active control of structural response, The special attributes of the coupled rotor/flexible fuselage vibration reduction problem are also briefly discussed to emphasize that vibration reduction at the hub is not equivalent to acceleration reduction at specific fuselage locations, Based on the comparison of the various approaches, it appears that the actively controlled flap has remarkable potential for vibration reduction.
Dynamic derivatives are used to represent the influence of the aircraft rates on the aerodynamic forces and moments needed for flight dynamics studies. These values have traditionally been estimated by processing measurements made from periodic forced motions applied to wind tunnel models. The use of Computational Fluid Dynamics has potential to supplement this approach. This paper considers the problem of the fast computation of forced periodic motions using the Euler equations. Three methods are evaluated. The first is computation in the time domain, and this provides the benchmark solution in the sense that the time accurate solution is obtained. Two acceleration techniques in the frequency domain are compared. The first uses an harmonic solution of the linearised problem (referred to as the linear frequency domain approach). The second uses the Harmonic Balance method, which approximates the nonlinear problem using a number of Fourier modes. These approaches are compared in the sense of their ability to predict dynamic derivatives and their computational cost. The standard NACA aerofoil CT cases, the SDM fighter model geometry and the DLR F12 passenger jet wind tunnel model are used as test cases. Compared to time accurate simulations an order-of-magnitude reduction in CPU costs is achieved for flows with a narrow frequency spectrum and moderate amplitudes, as the solution does not evolve through transients to reach periodicity.
The development of a state-space formulation for a multi-input/multi-output (MIMO) higher-harmonic-control (HHC) system is described. Results are also presented of a numerical investigation into closed-loop performance and stability of an HHC system, implemented in the rotating system, based oil the simulation of a hingeless rotor helicopter. The results show that the HHC controller reduces the 4/rev accelerations at the center of gravity. The percentage reductions obtained in the simulations are in excess of 80-90%. The vibration attenuation occurs within 5-7 s after the HHC system is turned on. This is equivalent to a frequency of around 1 rad/s, where flight control systems and human pilots tend to operate. Therefore, interactions and potential adverse effects on the stability and control characteristics of the helicopter should be explored. The HHC problem is intrinsically time periodic if the HHC inputs include frequencies other than the frequency one wishes to attenuate. This is true even if the rest of the model is assumed to be time invariant. In these cases, the closed-loop stability results obtained using constant coefficient approximations can be incorrect even at lower values of the advance ratio mu, where constant coefficient approximations of the open-loop dynamics are accurate.
Higher harmonic control (HHC) has been shown to be capable of reducing vibration and noise in rotorcraft. The majority of the previous work has concentrated on the HHC itself, and not the interaction between the automatic flight control system (AFCS) and HHC. Limited studies have been done on HHC/AFCS interaction using a linear time invariant (LTI) model and offline inputs including the use of dynamic crossfeeds to improve HHC performance while maneuvering. Most recently, results on HHC/AFCS interaction using nonlinear piloted simulations have been obtained. It has been observed that the nominal T-matrix controller approach, while effective in steady state, does not reduce vibration during a maneuver. Increasing the feedback gain can slightly improve RMS and settling time but magnifies peak response. The aim of the current work is to optimize HHC controller performance in a realistic piloted maneuver, and validate the design on a nonlinear simulation model. Achieving this objective required the following accomplishments: 1) Extraction, reduction, and validation of a linear time invariant model from a nonlinear model. 2) Baseline higher harmonic controller design and validation based on the linear model applied to the nonlinear model. 3) Optimization of the higher harmonic controller design to improve performance during a maneuver while maintaining stability. 4) Fitting ideal crossfeeds with a low-order approximation. 5) Implementation of crossfeeds to further improve vibration controller performance during a maneuver.
This paper addresses the use of dynamic inversion with direct load feedback to provide combined load alleviation and flight control of rotorocraft. The method is applied to a compound utility rotorcraft with similar airframe properties as a UH-60A along with a lifting wing. The controller makes use of flaperons and horizontal stabilizer in addition to the conventional main rotor / tail rotor blade pitch controls to track pilot commands while also minimizing pitch link loads. The nonlinear simulation is developed in FLIGHTLAB ® with structural models of the rotor blades and control system. This model must be linearized to a linear time-invariant (LTI) system to support linear Dynamic Inversion control design. The vehicle dynamics and critical fatigue load are modeled with a linear time-periodic (LTP) model which is converted via harmonic decomposition into a high-order LTI model. This model is then reduced to design controllers across a range of airspeeds. The controllers are tested both in linear model simulations and using the full nonlinear FLIGHTLAB ® model. The results show that the load alleviating controller achieves significant reduction in the pitch link peak-to-peak loads with minimal change in response characteristics, indicating that load alleviation can be achieved with no degradation in handling qualities.
This paper proposes a new methodology in linear time-periodic (LTP) system identification. In contrast to previous methods that totally separate dynamics at different tag times for identification, the method focuses on imposing appropriate structural constraints on the linear time-invariant (LTI) reformulation of LTP systems. This method adopts a periodically-switched truncated infinite impulse response model for LTP systems, where the structural constraints are interpreted as the requirement to place the poles of the non-truncated models at the same locations for all sub-models. This constraint is imposed by combining the atomic norm regularization framework for LTI systems with the group lasso technique in regression. As a result, the estimated system is both uniform and low-order, which is hard to achieve with other existing estimators. Monte Carlo simulation shows that the grouped atomic norm method does not only show better results compared to other regularized methods, but also outperforms the subspace identification method under high noise levels in terms of model fitting.
In this paper, a numerical method is proposed for determining the periodic state and control solutions of nonlinear time-periodic systems. Starting from an initial guess at the solution, the algorithm uses a harmonic balance technique to refine the solution through a gradient-based optimization approach. The algorithm introduces three major innovations when compared to previous techniques: it does not rely on state transition matrices, it simultaneously solves for the approximate higher-order linear time-invariant dynamics about the periodic solution, and it can be used to compute the harmonic control inputs that attenuate arbitrary state harmonics. Following a description of the algorithm, it is applied to three different aerospace vehicles with nonlinear time-periodic dynamics: a flapping-wing micro aerial vehicle, a helicopter, and a flapping-tail airplane. In all cases, the algorithm derives periodic solutions for the states and controls even when starting from very poor initial guesses. The algorithm has clear application in the development of advanced flight control laws that attenuate certain state harmonics and in the prediction of loads and vibrations.
The objective of this research effort is to assess the impact of load alleviation control on the quantitative handling qualities specifications (also known as predicted handling qualities) of both conventional helicopters and compound rotorcraft in forward flight. First, an overview on how the harmonic decomposition methodology is used toward load alleviation control is presented. Next, flight control laws are developed based on an explicit model-following architecture. Parametric studies are performed to provide insights on how both the feedforward and feedback paths of the model-following control laws can be used to alleviate the rotor loads. The impact of load alleviation on predicted handling qualities is studied. It is shown that, for the standard helicopter configuration considered, load alleviation comes at the cost of a degradation in handling qualities. However, for the compound rotorcraft considered, allocation of the control signal to the redundant control surfaces provides load alleviation without degradation in predicted handling qualities. The flight control laws are subsequently optimized using the Control Designer’s Unified Interface (CONDUIT) to meet a comprehensive set of stability, handling-qualities, and performance specifications for specific mission task elements while minimizing the unsteady rotor loads.
The present study considers two notional rotorcraft models: a conventional utility helicopter, representative of an H-60, and a wing-only compound utility rotorcraft, representative of an H-60 with a wing similar to the X-49A wing. An explicit model following (EMF) control scheme is designed to achieve stability and desired rate command / attitude hold response around the roll, pitch, and yaw axes, while alleviating vibratory loads through both feed-forward and feedback compensation. The harmonic decomposition methodology is extended to enable optimization of primary flight control laws that mitigate vibratory loads. Specifically, linear time-periodic systems representative of the periodic rotorcraft dynamics are approximated by linear time-invariant (LTI) models. The LTI models are subsequently reduced and used in linear quadratic regulator (LQR) design to constrain the harmonics of the vibratory loads. Both fuselage state feedback and rotor state feedback are considered. A pseudo-inverse strategy is incorporated into the EMF scheme for redundant control allocation on the compound configuration. Simulations of the load alleviating controllers are compared to results from a baseline controller. Finally, an analysis is performed to assess the impact that load alleviating control action, rotor state feedback, and pseudo-inverse have on handling qualities in terms of ADS-33E specifications.
This note proposes a new methodology for subspace-based state-space identification for linear time-periodic (LTP) systems. Since LTP systems can be lifted to equivalent linear time-invariant (LTI) systems, we first lift input-output data from the unknown LTP system as if it was collected from an equivalent LTI system. Then, we use frequency-domain subspace identification methods to find an LTI system estimate. Subsequently, we propose a novel method to obtain a time-periodic realization for the estimated lifted LTI system by exploiting the specific parametric structure of Fourier series coefficients of the frequency-domain lifting method. Our method can be used to both obtain state-space estimates for unknown LTP systems as well as to obtain Floquet transforms for known LTP systems.
Rotor control techniques for active control of vibrations require the availability of dynamic models of the response of the rotor to control inputs. Such models have to be time-varying ones. The results of black-box identification trials of discrete-time periodic models for the response of rotor loads to perturbations of the collective control input are presented and discussed.
Several methods for analysis of linear time periodic (LTP) systems have successfully been demonstrated using harmonic decompositions. Onemethod recently examined is to create a linear time invariant (LTI) model approximation by expansion of the LTP system states into various harmonic state representations, and formulating corresponding LTI models. Although this method has shown success, it relies on a second-order formulation of the original LTP system. This second-order formulation can prove problematic for degrees of freedom not explicitly represented in second-order form. Specifically, difficulties arise when performing the harmonic decomposition of body and inflow states as well as interpretation of LTI velocities. Instead this paper presents a more generalized LTI formulation using a first-order formulation for harmonic decomposition. The new first-order approach is evaluated for a UH-60A rotorcraft model and is used to show the significance of particular harmonic terms, specifically that the coupling of harmonic components of body and inflow states with the rotor states makes a significant contribution to LTI model fidelity in the prediction of vibratory hub loads.
This paper presents a new frequency domain identification technique to estimate multivariate Linear Parameter-Varying (LPV) continuous-time state space models, where a periodic variation of the parameters is assumed or imposed. The main goal is to obtain an LPV state space model suitable for control, from a single parameter-varying experiment. Although most LPV controller synthesis tools require continuous time state space models, the identification of such models is new. The proposed identification method designs a periodic input signal, taking the periodicity of the parameter variation into account. We show that when an integer number of periods is observed for both the input and the scheduling, the state space model representation has a specific, sparse structure in the frequency domain, which is exploited to speed up the estimation procedure. A weighted non-linear least squares algorithm then minimizes the output error. Two initialization methods are explored to generate starting values. The first approach uses a Linear Time-Invariant (LTI) approximation. The second estimates a Linear Time-Variant (LTV) input–output differential equation, from which a corresponding state space realization is computed.
The linearized stability characteristics of rotor blades in forward flight are examined. Equations of motion are derived for the case of a centrally hinged spring-restrained rigid blade. Various commonly used approximations for the equations of motion are examined in order to determine their effect on stability calculations. These comparisons show that periodic coefficients are important for flap-lag stability even at low advance ratios. The effect of rotor equilibrium on blade stability is studied. It is found that the variation in rotor trim and inflow with forward speed has a significant effect of blade stability. Finally, rotor stability boundaries are presented for a variety of rotor parameters, showing that forward flight can often change the qualitative effects of certain parameters on blade stability.
Formulation of linear time invariant (LTI) models of a nonlinear system about a periodic equilibrium using the harmonic domain representation of LTI model states is well established in the literature. This paper presents a computationally efficient scheme for implementation of a previously developed method for extraction of linear time invariant (LTI) models from a helicopter nonlinear model in forward flight. The fidelity of the extracted LTI models is evaluated using response comparisons between the extracted LTI models and the nonlinear model in both time and frequency domains. For time domain evaluations, individual blade control (IBC) inputs that have been tried in the literature for vibration and noise control studies are used. For frequency domain evaluations, frequency sweep inputs are used to obtain frequency responses of fixed system hub loads to a single blade IBC input. The evaluation results demonstrate the fidelity of the extracted LTI models, and thus, establish the validity of the LTI model extraction process for its use in integrated flight and rotor control studies.
Quasi-periodic motions and their stability are addressed from the point of view of different harmonic balance-based approaches. Two numerical methods are used: a generalized multidimensional version of harmonic balance and a modification of a classical solution by harmonic balance. The application to the case of a nonlinear response of a Duffing oscillator under a bi-periodic excitation has allowed a comparison of computational costs and stability evaluation results. The solutions issued from both methods are close to one another and time marching tests showing a good agreement with the harmonic balance results confirm these nonlinear responses. Besides the overall adequacy verification, the observation comparisons would underline the fact that while the 2D approach features better performance in resolution cost, the stability computation turns out to be of more interest to be conducted by the modified 1D approach. [DOI: 10.1115/1.4005823]
This paper describes a novel technique for predicting limit parameter values and calculating the corresponding control margins of an aircraft. This new approach utilizes an observer type adaptive neural network loop for the estimation of the correct aircraft model. The constructed aircraft model is then used to predict the quasi-steady response behavior of the limit parameters and the corresponding control margins using a second adaptive neural network loop. Though the approach does not require any off-line training of the neural networks, existing off-line trained neural network data maps can be accommodated in the procedure. Only standard sensor measurements are used for adaptation. A detailed development of the method along with simulation evaluation of the method using a linear helicopter model and a nonlinear tiltrotor model are included. © 2001 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
This paper describes carefree maneuvering techniques for aircraft under multiple limit constraints on multiple control axes. Adaptive multi-layered neural networks are employed for on-line learning and modelling error compensation. The approach utilizes an observer type adaptive neural network loop for an estimation of the correct aircraft model. The identified aircraft model is then used to predict the dynamic trim response behavior of the limit parameters. The corresponding control margins are calculated for each control corresponding to each different limiting value. A penalty vector for each control channel can than be calculated based on the control margins. Only standard sensor measurements are used for adaptation. The effectiveness of the proposed adaptive limit detection technique is evaluated through a series of simulations using the nonlinear Generic Tiltrotor Simulation (GTRSIM) program. © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
In this study, the authors present an overview of closed-loop subspace identification methods found in the recent literature. Since a significant number of algorithms has appeared over the last decade, the authors highlight some of the key algorithms that can be shown to have a common origin in autoregressive modelling. Many of the algorithms found in the literature are variants on the algorithms that are discussed here. In this study, the aim is to give a clear overview of some of the more successful methods presented throughout the last decade. Furthermore, the authors retrace these methods to a common origin and show how they differ. The methods are compared both on the basis of simulation examples and real data. Although the main focus in the literature has been on the identification of discrete-time models, identification of continuous-time models is also of practical interest. Hence, the authors also provide an overview of the continuous-time formulation of the identification framework.
In this paper, the optimized predictor-based subspace identification (PBSIDopt) method is applied to identify linear models of DLR's research helicopter ACT/FHS and to evaluate its usage to enhance existing physics based models in the future. For this effort, dedicated identification flight test data is used. This paper first describes the well known Maximum Likelihood frequency domain output error method and the applied physical model briefly. Then, the PBSIDopt method is presented and parameters, which influence the identification process, are discussed. Results from both methods using the same flight test data of the ACT/FHS are compared; model accuracy, order and missing dynamics are investigated. Advantages and disadvantages of both methods are evaluated and the applicability of the PBSIDopt method to rotorcraft system identification and its usage to improve the existing physical model structure is discussed.
In this paper, four characterizations of stabilizability and detectability of linear periodic systems are considered. Two of them look as natural extensions of the classical definitions given for time-invariant systems. The remaining two are modal characterizations which turn out to be useful in the analysis of the periodic Lyapunov and Riccati equations. It is shown that all these notions of stabilizability (and detectability) are in fact equivalent to each other.One of the various definitions calls for the existence of the Kalman canonical decomposition of periodic systems. This issue is addressed in the Appendix.
A discussion is presented of sampled-data systems with only one sampler ; which has a periodically time varying sampling rate, i.e., the sampling pattern ; is repetitive. The z-transform is used in this analysis although matrix methods ; could be just as easily applied. The method can be used in problems of inventory ; control, production control, and in the operation research field. (W.D.M.);
In this paper, a nonparametric estimation procedure is presented in order to track the evolution of the dynamics of continuous (discrete)-time (non)-linear periodically time-varying (PTV) systems. Multisine excitations are applied to a PTV system since this kind of excitation signals allows us to discriminate between the noise and the nonlinear distortion from a single experiment. The key idea is that a linear PTV system can be decomposed into an (in)finite series of transfer functions, the so-called harmonic transfer functions (HTFs). Moreover, a systematic methodology to determine the number of significant branches is provided in this paper as well. Making use of the local polynomial approximation, a method that was recently developed for multivariable (non)-linear time invariant systems, the HTFs, together with their uncertainties embedded in an output-error framework, are then obtained from only one single experiment. From these nonparametric estimates, the evolution of the dynamics, described by the instantaneous transfer function (ITF), can then be achieved in a simple way. The effectiveness of the identification scheme will be first illustrated through simulations before a real system will be identified. Eventually, the methodology is applied to a weakly nonlinear PTV electronic circuit.
This dissertation presents an aeromechanical closed loop stability and response analysis of a hingeless rotor helicopter with a Higher Harmonic Control (HHC) system for vibration reduction. The analysis includes the rigid body dynamics of the helicopter and blade flexibility. The gain matrix is assumed to be fixed and computed off-line. The discrete elements of the HHC control loop are rigorously modeled, including the presence of two different time scales in the loop. By also formulating the coupled rotor-fuselage dynamics in discrete form, the entire coupled helicopter-HHC system could be rigorously modeled as a discrete system. The effect of the periodicity of the equations of motion is rigorously taken into account by converting the system into an equivalent system with constant coefficients and identical stability properties using a time lifting technique. The most important conclusion of the present study is that the discrete elements in the HHC loop must be modeled in any HHC analysis. Not doing so is unconservative. For the helicopter configuration and HHC structure used in this study, an approximate continuous modeling of the HHC system indicates that the closed loop, coupled helicopter-HHC system remains stable for optimal feedback control configurations which the more rigorous discrete analysis shows can result in closed loop instabilities. The HHC gains must be reduced to account for the loss of gain margin brought about by the discrete elements. Other conclusions of the study are: (i) the HHC is effective in quickly reducing vibrations, at least at its design condition, although the time constants associated with the closed loop transient response indicate closed loop bandwidth to be 1 rad/sec on average, thus overlapping with FCS or pilot bandwidths, and raising the issue of potential interactions; (ii) a linearized model of helicopter dynamics is adequate for HHC design, as long as the periodicity of the system is correctly taken into account, i.e., periodicity is more important than nonlinearity, at least for the mathematical model used in this study; and (iii) when discrete and continuous systems are both stable, and quasisteady conditions can be guaranteed, the predicted HHC control harmonics are in good agreement.
Dynamic analysis of rotorcraft usually involves a nonlinear trim solution followed by a linearized Floquet analysis. This paper utilizes results by McNulty and by McVicar and Bradley to show that, when the rotor is composed of Q identical blades, both the Floquet analysis and the trim can be obtained in 1/Q of the normal computing times. This paper also generalizes the earlier work to show that these savings can be obtained for most Floquet algorithms and for either individual-blade or multi-blade descriptions. Finally, the general result leads to a new formulation of multi-blade coordinates.
An automatic feedback system, based on continuous monitoring of hub loads, is used to find the control settings that are required to obtain a given flight condition for a helicopter rotor. Optimum values of gains and time constants are determined, and the limitations of the controller are examined for flap-pitch dynamics. It is found that the present method shows good convergence and is superior to other trim techniques for systems with moderate damping or with many degrees of freedom.
From the Publisher:Aeronautical engineers concerned with the analysis of aircraft dynamics and the synthesis of aircraft flight control systems will find an indispensable tool in this analytical treatment of the subject. Approaching these two fields with the conviction that an understanding of either one can illuminate the other, the authors have summarized selected, interconnected techniques that facilitate a high level of insight into the essence of complex systems problems. These techniques are suitable for establishing nominal system designs, for forecasting off-nominal problems, and for diagnosing the root causes of problems that almost inevitably occur in the design process. A complete and self-contained work, the text discusses the early history of aircraft dynamics and control, mathematical models of linear system elements, feedback system analysis, vehicle equations of motion, longitudinal and lateral dynamics, and elementary longitudinal and lateral feedback control. The discussion concludes with such topics as the system design process, inputs and system performance assessment, and multi-loop flight control systems.
A method for using neural networks to provide predictive flight envelope limit information was developed. The method was applied to provide a tactile cueing system for normal load factor and angle-of-attack buffet limits on the V-22 tilt-rotor aircraft. Results from a real-time piloted simulation showed that the system enabled the pilot to maneuver along the flight envelope boundaries without exceeding the limits. Results indicated that the approach has the potential to expand the effective safe maneuvering flight envelope of aircraft with structural load limits.
The application of Floquet analysis to rotorcraft has demonstrated that there is often misunderstanding as to how to interpret the integer-multiple arbitrariness in the imaginary part of the characteristic exponents of the system (i. e., in the frequency). Although some papers have offered various methodologies for choosing the "correct" frequency (such as choice of the strongest harmonic in the periodic eigenvector), confusion still persist and none of the proposed methods completely solves the difficulty. The purpose of this present paper is to (1) offer a history of analysis of periodic systems with respect to the treatment of the integer multiples, (2) propose a practical tool for fundamental insight into system frequencies, and (3) show demonstratively for simple cases why no single integer can adequately describe system dynamics. It is hoped that this approach will provide clarity to a problem that, although long understood in principle, still leads to confusion in published papers.
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.
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.
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.