Equation Error Method Research Articles

Robust Aerodynamic Parameter Estimation of Unmanned Aircraft Based on Two-step Identification

This paper presents the estimation of stability and control derivatives of an unmanned aircraft. The aerodynamics are described using regressors composed of velocity, angular rates, flow angles and control surface deflections. The flight data is generated from numerical simulation of postulated equations of motion describing the aerodynamics model. Least squares based on the equation error method is used to estimate the parameters representing the different force and moment aerodynamic coefficients. Statistical analysis is done on the estimates to determine the accuracy and adequacy of the estimates to describe the aerodynamic model. A dynamic simulation based on the identified aerodynamic model is used to improve the parameter estimates through regression of the errors between the flight data and the model response. The aircraft under consideration is a scaled Yak-54 fixed wing unmanned aerial vehicle.

Estimation of aerodynamic parameters using neural artificial bee colony fusion algorithm for moderate angle of attack using real flight data

Aircraft system identification aims to estimate the aerodynamic force and moment coefficients utilizing intelligent modeling and parametric identification methodologies. Classical methods like output, filter, and equation error methods apply extensively as parametric approaches. In contrast, machine learning approaches like Artificial Neural Networks (ANN), Adaptive Neuro-Fuzzy Inference Systems (ANFIS), etc., are alternatives to model-based methods. This work presents a novel aerodynamic parameters estimation technique that fuses two biologically inspired optimization techniques, (i) the Artificial Bee Colony (ABC) optimization and (ii) ANN for an actual aircraft while incorporating system and measurement uncertainty. The fusion of ABC and ANN imparts the ability to address sensor noise challenges associated with system identification and parameter estimation. Comparison of the proposed method’s results with the benchmark techniques like Least Square, Filter Error, and Neural Gauss Methods using recorded flight data of the ATTAS (DLR German Aerospace Centre) and HANSA-3 (IIT Kanpur) aircrafts established its adequacy and efficacy. Furthermore, the capability of the proposed hybrid method to extract stability and control variables from the stable aircraft kinematics is shown even with insufficient information in its data history.

An adaptive law based online system identification of cropped delta unmanned aerial vehicles from flight tests

Online system identification of an aircraft has multifaceted advantages, such as real-time health monitoring, fault detection and reconfiguration of control, to name a few. Researchers have utilized variants of the equation error method, which involves non-gradient-based computation to carry out online system identification, due to its feasibility for real-time implementation. However, it was found to be underperforming while handling high angles of attack as well as time-varying bias parameters. In the current manuscript, an attempt has been made to alleviate the aforementioned limitation by proposing a novel online system identification technique based on adaptive law and its applicability is demonstrated by using compatible flight data sets, acquired from two cropped delta wing unmanned aerial vehicles, pertaining to linear, moderately nonlinear, and near stall flight regimes. It is observed that the proposed method is able to predict unknown aerodynamic derivatives, irrespective of flight envelopes, if dynamic modes are adequately excited. Confidence in the estimated aerodynamic parameters is assured with the two-sigma error bound. It is noticed that the maximum relative standard deviation in the estimates obtained with the proposed method is less than the recursive least squares method and higher than the sequential least squares-recursive Fourier transform method in the linear flight regime. Simulated responses using the estimates obtained from aforementioned methods are compared with measured flight data, and it is witnessed that the simulated nondimensional lift coefficient with the proposed method has a relative offset of less than 34.2% w.r.t measured value in near-stall flight regimes, which is least among all three methods.

Aerodynamic characterisation of delta wing unmanned aerial vehicle using non-gradient-based estimator

AbstractAerodynamic characterisation from flight testing is an integral subroutine for evaluating a new flight vehicle’s aerodynamic performance, stability and controllability. The estimation of aerodynamic parameters from flight test data has extensively been explored, in the past, using estimation methods such as the equation error method, output error method and filter error method. However, in the current era, non-gradient-based estimation techniques are gaining attention from researchers due to their inherent data-driven optimisation capability to find the global best solution. In this paper, a novel non-gradient-based estimation method is proposed for the aerodynamic characterisation of unmanned aerial vehicles from flight data, which relies on the maximum likelihood method augmented with particle swarm optimisation. Flight data sets of a wing-alone unmanned aerial vehicle are used to demonstrate the capabilities of the proposed method in estimating aerodynamic derivatives. Estimates from the proposed method are corroborated with the wind tunnel test and output error method results. It has been observed that simulated flight vehicle responses using estimated parameters are in good agreement with measured data in most of the manoeuvers considered. Confidence in the estimates of linear and nonlinear aerodynamic parameters is well established with the lower limit of Cramer-Rao bounds, which are minimal. The proposed method also demonstrates good predictability of the quasi-steady stall aerodynamic model by estimating stall characteristic parameters such as aerofoil static stall characteristics parameter, hysteresis time constant and breakpoint. The overall performance of the proposed estimation method is on par with the output error method and is validated with the proof-of-match exercise.

System Identification Approach for eVTOL Aircraft Demonstrated Using Simulated Flight Data

This paper describes a system identification method for electric vertical takeoff and landing (eVTOL) aircraft. The approach merges fixed-wing and rotary-wing modeling techniques with new strategies to develop a modeling method for eVTOL vehicles using flight-test data. The eVTOL aircraft system identification approach is demonstrated through application to the NASA Langley Aerodrome No. 8 tandem tilt-wing, distributed electric propulsion aircraft using a high-fidelity flight dynamics simulation. Orthogonal phase-optimized multisine inputs are applied to each control surface and propulsor at numerous flight conditions throughout the flight envelope to collect informative flight data. An aero-propulsive model is identified at each flight condition using the equation-error method in the frequency domain. The local model parameters are then blended to create a global model across the nominal flight envelope. The identified models are shown to provide a good fit to modeling data with good prediction capability. The methodology is developed with a discussion of unique eVTOL vehicle aerodynamic characteristics and practical strategies intended to inform future flight-based system identification efforts for eVTOL aircraft.

Estimation of longitudinal aerodynamic parameters using recurrent neural network

AbstractThe aerodynamic modelling is one of the challenging tasks that is generally established using the results of the computational fluid dynamic software and wind tunnel analysis performed either on the scaled model or the prototype. In order to improve the confidence of the estimates, the conventional parameter estimation methods such as equation error method (EEM) and output error method (OEM) are more often applied to extract the aircraft’s stability and control derivatives from its respective flight test data. The quality of the estimates gets influenced due to the presence of the measurement and process noises in the flight test data. With the advancement in the machine learning algorithms, the data driven methods have got more attention in the modelling of a system based on the input-output measurements and also, in the identification of the system/model parameters. The research article investigates the longitudinal stability and control derivatives of the aerodynamic models by using an integrated optimisation algorithm based on a recurrent neural network. The flight test data of Hansa-3 and HFB 320 aircraft were used as case studies to see the efficacy of the parameter estimation algorithm and further, the confidence of the estimates were demonstrated in terms of the standard deviations. Finally, the simulated variables obtained using the estimates demonstrate a qualitative estimation in the presence of the noise.

System identification of cropped delta UAVs from flight test methods using particle Swarm-Optimisation-based estimation

AbstractIn the era of Unmanned Aerial Systems (UAS), an onboard autopilot occupies a prominent place and is inevitable for many of their modern applications. The efficacy of autopilot heavily relies upon the accuracy of the sensors employed and the capability of the onboard flight controller. In general, aerodynamic behaviour and flight dynamic capabilities of Unmanned Aerial Vehicles (UAVs) govern the selection and the design of flight controllers. Precise modeling of linear aerodynamic characteristics from flight data can be achieved using many of the existing classical parameter estimation techniques such as Output Error Method (OEM), Equation Error Method (EEM), and Filter Error Method (FEM). However, all the classical methods may not be readily applicable for aerodynamic modeling in nonlinear flight envelopes. The current manuscript is an attempt to exploit the capabilities of the Artificial Intelligence (AI) technique, named Particle Swarm Optimisation (PSO), in combination with Least Squares (LS) cost function to perform linear as well as nonlinear aerodynamic parameter estimation. The aforementioned task is accomplished by considering flight data from manoeuvers pertaining to linear angles of attack, moderate and near stall flight envelopes of two different UAVs with cropped delta planform geometry. Parameters estimated using the proposed LS-PSO method are consistent with minimum standard deviation and are on a par with OEM estimates. The proposed LS-PSO method enhances the capabilities of LS-based EEM while estimating stall characteristic parameters, which was not possible with LS alone. The longitudinal and lateral-directional static parameters estimated from the full-scale wind tunnel testing of the two UAVs were also used to corroborate the results obtained from the flight data using the LS-PSO method.

Longitudinal aerodynamic coefficients modeling from simulated flight data of fixed wing aircraft based on three methods

The equation error method, output error method, and neural network method are used to model the aircraft longitudinal aerodynamic coefficients from simulated flight data. The equation error method and the output error method are based on postulated mathematical models of the longitudinal aerodynamic coefficients, and the least squared principle and the maximum likelihood principle are used to estimate the model parameters respectively. The neural network method uses the BP network output to fit the aerodynamic coefficients without assumptions of mathematical model of aerodynamic coefficients. Two sets of simulation tests were conducted, one set of data is used for model identification, and the other set is used for model validation.

Aerodynamic Parameter Estimation Using Reconstructed Turbulence Measurements

A classical flow-vane approach for reconstructing translational gust velocities from air data and inertial sensor measurements was improved and implemented for real-time computation. The reconstructed turbulence measurements were then included in a system identification analysis to estimate nondimensional stability and control derivatives in a longitudinal short period model using the maximum likelihood equation-error method in the frequency domain with Fourier-transform data. Flight-test results using 44 maneuvers from a subscale transport-type airplane showed that the power spectra for the reconstructed vertical gusts resembled standard Dryden or von Kármán turbulence models. Flight data in moderate and severe turbulence exhibited a decorrelation of the modeling data. In particular, pitch rate and angle-of-attack rate derivatives could both be identified from flight data about straight and level flight without special maneuvers or prior information.

A new class of fredholm integral equations of the second kind with non symmetric kernel: solving by wavelets method

In this paper, we present an ecient modication of the wavelets method to solve a new class of Fredholm integral equations of the second kind with non symmetric kernel. This -analytical method based on orthonormal wavelet basis, as a consequence three systems are obtained, a Toeplitz system and two systems with condition number close to 1. Since the preconditioned conjugate gradient normal equation residual (CGNR) and preconditioned conjugate gradient normal equation error (CGNE) methods are applicable, we can solve the systems in O(2n log(n)) operations, by using the fast wavelet transform and the fast Fourier transform.

Aircraft parameter estimation using ELM network

Purpose The purpose of this paper is to investigate the estimation methodology with a highly generalized cost-effective single hidden layer neural network. Design/methodology/approach The aerodynamic parameter estimation is a challenging research area of aircraft system identification, which finds various applications such as flight control law design and flight simulators. With the availability of the large database, the data-driven methods have gained attention, which is primarily based on the nonlinear function approximation using artificial neural networks. A novel single hidden layer feed-forward neural network (FFNN) known as extreme learning machine (ELM), which overcomes the issues such as learning rate, number of epochs, local minima, generalization performance and computational cost, as encountered in the conventional gradient learning-based FFNN has been used for the nonlinear modeling of the aerodynamic forces and moments. A mathematical formulation based on the partial differentiation is proposed to estimate the aerodynamic parameters. Findings The real flight data of longitudinal and lateral-directional motion have been considered to estimate their respective aerodynamic parameters using the proposed methodology. The efficacy of the estimates is verified with the results obtained through the conventional parameter estimation methods such as the equation-error method and filter-error method. Originality/value The present study is an outcome of the research conducted on ELM for the estimation of aerodynamic parameters from the real flight data. The proposed method is capable to estimate the parameters in the presence of noise.

Online and stable parameter estimation based on normalized brain emotional learning model (NBELM)

SummaryRecently, a computational model of Amygdala based on the brain emotional learning is presented by psychologists. This brain emotional learning model (BELM) that has a neuro‐inspired architecture is utilized to train the weights which are in Amygdala and Orbitofrontal. In this paper, unknown parameters of dynamic systems are estimated by developing the normalized BELM (NBELM). To this end, after proving the stability of the model output, the sufficient condition for weights convergence is extracted while the sensitivity analysis is applied for this model. In order to evaluate the performance of NBELM, in the first example, the matrices of a twin rotor MIMO system are estimated and compared with the equation error method (EEM). In the second example, the nonlinear model of a servomotor is utilized as a case study. In the third example, the performance of the NBELM in experimental systems is validated using a reaction wheel with a DC motor. An important feature of the brain emotional system is its fast response, leading the NBELM to have a high speed performance in estimating the parameters of dynamic systems. A few number of adjustable parameters and low computing complexity also cause the NBELM to be an appropriate method for online estimation of the unknown parameters of dynamic systems.

Higher-Order Accurate Finite Volume Discretization of the Three-Dimensional Poisson Equation Based on An Equation Error Method

Efficient higher-order accurate finite volume schemes are developed for the threedimensional Poisson’s equation based on optimizations of an equation error expansion on local control volumes. A weighted quadrature of local compact fluxes and the flux integral form of the equation are utilized to formulate the local equation error expansions. Efficient quadrature weights for the schemes are then determined through a minimization of the error expansion for higher-order accurate discretizations of the equation. Consequently, the leading numerical viscosity coefficients are more accurately and completely determined to optimize the weight parameters for uniform higher-order convergence suitable for effective numerical modeling of physical phenomena. Effectiveness of the schemes are evaluated through the solution of the associated eigenvalue problem. Numerical results and analysis of the schemes demonstrate the effectiveness of the methodology.

Aerodynamic Parameters Estimation Using Radial Basis Function Neural Partial Differentiation Method

Aerodynamic parameter estimation involves modelling of force and moment coefficients and computation of stability and control derivatives from recorded flight data. This problem is extensively studied in the past using classical approaches such as output error, filter error and equation error methods. An alternative approach to these model based methods is the machine learning such as artificial neural network. In this paper, radial basis function neural network (RBF NN) is used to model the lateral-directional force and moment coefficients. The RBF NN is trained using k-means clustering algorithm for finding the centers of radial basis function and extended Kalman filter for obtaining the weights in the output layer. Then, a new method is proposed to obtain the stability and control derivatives. The first order partial differentiation is performed analytically on the radial basis function neural network approximated output. The stability and control derivatives are computed at each training data point, thus reducing the post training time and computational efforts compared to hitherto delta method and its variants. The efficacy of the identified model and proposed neural derivative method is demonstrated using real time flight data of ATTAS aircraft. The results from the proposed approach compare well with those from the other.

Identification of spin maneuver aerodynamic nonlinear model by applying ensemble empirical mode decomposition and extended multipoint modeling

Identification of the “aircraft aerodynamic model” in some unusual flight conditions such as spin maneuver provides critical information to the flight controller to retake the “dynamic stability” after it has been disturbed by the systematic, natural or environmental oscillations. Hence, a method for identifying the appropriate aerodynamic model in spin maneuvers is presented in this paper. We present an innovative systematic method for aerodynamic modeling of spin maneuvers, which combines the ensemble empirical mode decomposition technique and extended multipoint modeling approach, using flight data. In ensemble empirical mode decomposition, in addition to having all the empirical mode decomposition features, the original signal is collected with the white noise, and by using its statistical characteristics, the ensemble empirical mode decomposition solves the mode mixing problem. By applying the ensemble empirical mode decomposition to the flight parameter data, their intrinsic mode frequencies are extracted and are used as inputs to the extended multipoint modeling model. The extended multipoint modeling structure includes some parameters describing the distribution of aerodynamic forces and moments along each of the aircraft components. Moreover, this method allows coupling between the forces and moments. Unlike conventional methods, which consider the average forces obtained by plane surfaces relative to the center of mass, in the extended multipoint modeling technique, the force generated by each plane of the aircraft is allowed to appear independently in the motion equations. For identifying the aerodynamic model with extended multipoint modeling structure, the equation error method is used with a maximum likelihood optimizer inside. The obtained algorithm has been applied to two sets of spin maneuver flight data which were recorded in actual spin flight. The results demonstrate that the proposed method is able to reproduce the aerodynamic forces and moments for the second spin flight inputs with high accuracy by using a model which is derived from the first spin data identification.

Parameter Estimation of Unstable Aircraft using Extreme Learning Machine

<p class="p1">The parameter estimation of unstable aircraft using extreme learning machine method is presented. In the past, conventional methods such as output error method, filter error method, equation error method and non-conventional method such as artificial neural-network based methods have been used for aircraft’s aerodynamic parameter estimation. Nowadays, a trend of finding an accurate nonlinear function approximation is required to represent the aircraft’s equations-of-motion. Such type of nonlinear function approximation is usually achieved using artificial neural-network which is trained with the aircraft input-output flight data using a training algorithm. The accuracy of estimated parameters, which is achieved using the trained network, is highly dependent on the generalisation capability of the network which can be improved using extreme learning machine based network in contrast to artificial neural-network. To estimate the unstable aircraft parameters from the simulated flight data, Gauss-Newton based optimisation method has been used with a predefined aerodynamic model using the trained network. Further, the confidence of the estimated parameters has been shown in comparison to that of the standard parameter estimation methods in terms of the Cramer-Rao bounds.</p>

DC motor parameter identification using equation error method

The paper deals with the identification of a direct current motor using experimentally measured transient processes of the motor starting and running-down at a no-load driving torque. The identification is based on the equation error method, which was chosen for its ability to work with noisy data. The major point of the method is that the data of the experimentally obtained measurements are reciprocally associated with the differential equations defining the analysed object. The paper discusses the process and the results of identification.

A linear regularization method for a nonlinear parameter identification problem

Abstract In order to obtain regularized approximations for the solution q of the parameter identification problem - ∇ . ( q ⁢ ∇ ⁡ u ) = f {-\nabla.(q\nabla u)=f} in Ω along with the Neumann boundary condition q ⁢ ∂ ⁡ u ∂ ⁡ ν = g {q\frac{\partial u}{\partial\nu}=g} on ∂ ⁡ Ω {\partial\Omega} , which is an ill-posed problem, we consider its weak formulation as a linear operator equation with operator as a function of the data u ∈ W 1 , ∞ ⁢ ( Ω ) {u\in W^{1,\infty}(\Omega)} , and then apply the Tikhonov regularization and a finite-dimensional approximation procedure when the data is noisy. Here, Ω is a bounded domain in ℝ d {{\mathbb{R}}^{d}} with Lipschitz boundary, f ∈ L 2 ⁢ ( Ω ) {f\in L^{2}(\Omega)} and g ∈ H - 1 / 2 ⁢ ( ∂ ⁡ Ω ) {g\in H^{-1/2}(\partial\Omega)} . This approach is akin to the equation error method of Al-Jamal and Gockenback (2012) wherein error estimates are obtained in terms of a quotient norm, whereas our procedure facilitates to obtain error estimates in terms of the regularization parameters and data errors with respect to the norms of the spaces under consideration. In order to obtain error estimates when the noisy data belongs to L 2 ⁢ ( Ω ) {L^{2}(\Omega)} instead of W 1 , ∞ ⁢ ( Ω ) {W^{1,\infty}(\Omega)} , we shall make use of a smoothing procedure using the Clement operator under additional assumptions of Ω and u.

Identification of Linear Time‐Invariant Systems under Periodic Disturbance with Its Estimation

SUMMARYIn this paper, we propose a new method for identification of linear time‐invariant systems and estimation of unknown periodic disturbances. We assume that a disturbance to be considered here is unknown except its period. In this method, by stacking not only the unknown coefficients of a mathematical model but also the unknown disturbance as a parameter vector, we modify the equation error method so as to treat the unknown disturbance in the identification scheme. In particular, we also explain how the periodicity of the disturbance is utilized. In order to show the validity of this method, we give an illustrative numerical example.

Nonlinear aerodynamic model identification using empirical mode decomposition

A conceptual method based on the empirical mode decomposition algorithm is proposed to identify a high-fidelity full-flight envelope aerodynamic model, utilising flight data. The key idea is to recognise dominant phenomena of flight containing dissimilar amplitudes and frequencies by means of the empirical mode decomposition. Being distinguished and separated from each other, flight modes can be considered in the aerodynamic model, independently. To achieve the goal, an equation error method is utilised to identify six multi-input single-output systems for aerodynamic forces and moments. The inputs of the identification systems are intrinsic mode functions of flight parameters. The nonparametric identification problem uses the Hammerstein nonlinear ARMAX structure and estimates parameters by the least squares iterative algorithm. Flight tests of an unmanned aircraft in two complex manoeuvres are employed for the modelling and simulation. Several models with different nonlinear functions are introduced and trained by the first dataset. Then, to verify the identified model, the flight simulation is performed for the second dataset. Results indicate the appropriate performance of the identification method in nonlinear aerodynamics.