Global Nonlinear Model Identification with Multivariate Splines

At present, model based control systems play an essential role in many aspects of modern society. Application areas of model based control systems range from food processing to medical imaging, and from process control in oil refineries to the flight control systems of modern aircraft. Central to a model based control system is a mathematical model of the physical system or process that is being controlled. The field of science concerned with the identification of models of physical systems is called system identification. In this thesis, a new methodology is proposed for the identification of models of nonlinear systems using multivariate simplex splines. This new methodology has the potential to increase the performance of any model based control system by improving the quality of system models. Multivariate simplex splines consist of polynomial basis functions, called B-form polynomials, which are defined on geometric structures called simplices. Every simplex supports a single B-form polynomial which itself consists of a linear combination of Bernstein basis polynomials. Each individual Bernstein basis polynomial is scaled by a single coefficient called a B-coefficient. The B-coefficients have a special property in the sense that they have a unique spatial location inside their supporting simplex. This spatial structure, also known as the B-net, provides a number of unique capabilities that add to the desirability of the simplex splines as a tool for data approximation. For example, the B-net simplifies local model modification by directly relating specific model regions to subsets of B-coefficients involved in shaping the model in those regions. This particular capability has the potential to play an important role in future adaptive model based control systems. In such a control system, an on-board simplex spline model can be locally adapted in real time to reflect changes in system dynamics. The approximation power of the multivariate simplex splines can be increased by joining any number of simplices together into a geometric structure called a triangulation. Triangulations come in many shapes and sizes, ranging from configurations consisting of just two simplices to configurations containing millions of simplices. Triangulations can be optimized by locally increasing or decreasing the density of simplices to reflect local system complexity. The new methodology was applied in the identification of a complete set of aerodynamic models for the Cessna Citation II laboratory using flight data obtained during seven test flights. In total, 247 flight test maneuvers were flown which together provided a significant coverage of the flight envelope of the Citation II. The complete identification dataset consisted of millions of measurements on more than sixty flight parameters. More than 2000 prototype spline based aerodynamic models were identified using a newly developed, highly optimized software implementation of the simplex spline identification algorithm. Using the developed methods for simplex spline model validation it was proved that the models are both accurate and of guaranteed numerical stability inside the spline domain. The identification and validation results of the simplex spline models were compared with those of ordinary polynomial models identified using standard identification methods. These results showed that the multivariate simplex spline based aerodynamic models were of significantly higher quality than the aerodynamic models based on ordinary polynomials.Control & OperationsAerospace Engineerin

Horizontal and Vertical Wind Measurements from GOCE Angular Accelerations

Because of the highly accurate accelerometers, the GOCE mission has proven to be a unique source of thermosphere neutral density and cross-wind data. In the current methods, in which only the horizontal linear accelerations are used, the vertical winds cannot be obtained. In the algorithm proposed in this paper, angular accelerations derived from the individual gradiometer accelerations are used to obtain the vertical wind speeds as well. To do so, the measured angular rate and acceleration are combined to find a measurement of the torque acting on the spacecraft. This measurement is then corrected for modeled control torque applied by the magnetic torquers, aerodynamic torque, gravity gradient torque, solar radiation pressure torque, the torque caused by the misalignment of the thrust with respect to the center of gravity, and magnetic torque caused by the operation of several different subsystems of the spacecraft bus. Since the proper documentation of the magnetic properties of the payload were not available, a least squares estimate is made of one hard- and one soft-magnetic dipole pertaining to the payload, on an aerodynamically quiet day. The model for aerodynamic torque uses moment coefficients from Monte-Carlo Test Particle software ANGARA. Finally the neutral density, horizontal cross-wind, and vertical wind are obtained from an iterative process, in which the residual forces and torques are minimized. It is found that, like horizontal wind, the vertical wind responds strongly to geomagnetic storms. This response is observed over the whole latitude range, and shows seasonal variations.Astrodynamics & Space MissionsControl & Simulatio

Attitude Estimation of a Quadcopter with one fully damaged rotor using on-board MARG Sensors

Quadcopters are becoming increasingly popular across diverse sectors. Since rotor damages occur frequently, it is essential to improve the attitude estimation and thus ultimately the ability to control a damaged quadcopter. This research is based on a state-of-the-art method that makes it possible to control the quadcopter despite the total failure of a single rotor, where the attitude and position of the quadcopter are provided by an external system. In the present research, a novel attitude estimator called Adaptive Fuzzy Complementary Kalman Filter (AFCKF) has been developed and validated that works independently of any external systems. It is able to estimate the attitude of a quadcopter with one fully damaged rotor while only relying on the on-board MARG (Magnetometer, Accelerometer, Rate Gyroscope) sensors. The AFCKF provides significantly better attitude estimates for flights with a damaged rotor than mainstream filters, estimating the roll and pitch of the quadcopter with an RMS error of less than 1.7 degrees and a variance of less than 2 degrees. The proposed filter also provides accurate yaw estimates despite the fast spinning motion of the damaged quadcopter, and thus outperforms existing methods at the cost of only a small increase in computation.Control & Simulatio

GOCE Aerodynamic Torque Modeling

In recent studies thermospheric densities and cross-winds have been derived from linear acceleration measurements of the gradiometer on board the GOCE satellite. Our current work is aimed at analyzing also the angular accelerations, in order to improve the thermosphere density and wind data by allowing for the estimation of more unknown parameters. On this poster an overview is provided of the modeling efforts involved in isolating the aerodynamic torque. The intermediate result is a comparison of modeled and measured torques. Each box contains a plot of the torque from a specific source, compared to the measured torque, on October 16th, 2013. A short description of the model for each torque is also provided.Astrodynamics & Space MissionsControl & Simulatio

System Identification using the Multivariate Simplotope B-Spline

In recent research efforts the multivariate simplex spline has shown great promise in system identification applications. It has high approximation power, while its linearity in the parameters allows for computationally efficient estimation of the coefficients. In this paper the multivariate simplotope spline is derived from this spline, and compared to its simplex counterpart in a system identification setting. Contrary to the simplex spline, the simplotope spline allows the user to incorporate expert knowledge of the system in his models. Whereas in the first spline all variables are included in a complete polynomial, in the latter the user can split the variables in decoupled subsets. By fitting models to specifically designed test functions it is shown that this can indeed improve the approximation performance in terms of both the error metrics and the number of B-coefficients required. This comes at the price of a higher total degree, and therefore an increased sensitivity to Runge's phenomenon in case of poor data distribution. Finally an attempt is made to apply the proposed methods to a set of flight data of the DelFly II, a flapping wing micro aerial vehicle. It is found that the used data set is not suitable for global system identification, as the data in concentrated in low-dimensional clusters in the five-dimensional state space. Therefore it is advised that a more suitable data set is obtained to validate the simplotope spline in a system identification setting.Control & SimulationAstrodynamics & Space Mission

Torque model verication for the GOCE satellite

Astrodynamics & Space MissionsControl & Simulatio

Characterization of Thermospheric Vertical Wind Activity at 225- to 295-km Altitude Using GOCE Data and Validation Against Explorer Missions

Recently, the horizontal and vertical cross wind at 225- to 295-km altitude were derived from linear acceleration measurements of the Gravity field and steady-state Ocean Circulation Explorer satellite. The vertical component of these wind data is compared to wind data derived from the mass spectrometers of the Atmosphere Explorer C and E and Dynamics Explorer 2 satellites. From a statistical analysis of the 120-s moving-window standard deviation of the vertical wind (σ(Vz)), no consistent discrepancy is found between the accelerometer-derived and the mass spectrometer-derived data. The validated Gravity field and steady-state Ocean Circulation Explorer data are then used to investigate the influence of several parameters and indices on the vertical wind activity. To this end, the probability distribution of σ(Vz) is plotted after distributing the data over bins of the parameter under investigation. The vertical wind is found to respond strongly to geomagnetic activity at high latitudes, although the response settles around a maximum standard deviation of 50 m/s at an Auroral Electrojet index of 800. The dependence on magnetic local time changes with magnetic latitude, peaking around 4:30 over the polar cap and around 01:30 and 13:30 in the auroral oval. Seasonal effects only become visible at low to middle latitudes, revealing a peak wind variability in both local summer and winter. The vertical wind is not affected by the solar activity level.Astrodynamics & Space MissionsControl & Simulatio

Horizontal and vertical thermospheric cross-wind from GOCE linear and angular accelerations

Thermospheric wind measurements obtained from linear non-gravitational accelerations of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite show discrepancies when compared to ground-based measurements. In this paper the cross-wind is derived from both the linear and the angular accelerations using a newly developed iterative algorithm. The two resulting data sets are compared to test the validity of wind derived from angular accelerations and quantify the uncertainty in accelerometer-derived wind data. In general the difference is found to be less than 50 m/s vertically after high-pass filtering, and 100 m/s horizontally. A sensitivity analysis reveals that continuous thrusting is a major source of uncertainty in the torque-derived wind, as are the magnetic properties of the satellite. The energy accommodation coefficient is identified as a particularly promising parameter for improving the consistency of thermospheric cross-wind data sets in the future. The algorithm may be applied to obtain density and cross-wind from other satellite missions that lack accelerometer data, provided the attitude and orbit are known with sufficient accuracy.Astrodynamics & Space MissionsControl & Simulatio

Parallel Real-Time Tracking and 3D Reconstruction with TBB for Intelligent Control and Smart Sensing Framework

Recent advancements in aircraft controllers paired with increasingly flexible aircraft designs create the need for adaptive and intelligent control systems. To correctly capture the motion of a flexible aircraft wing and provide feedback to the controller, a large number of states (nodes along the span) must be monitored in real-time. Visual sensing methods carry the promise of flexibility needed for this type of smart sensing and control. However, visual sensing requires capturing and tracking keypoint features (marker tracking), while detecting thereof from a feature-rich image can be a computationally intensive task. The computational effort significantly increases with image size or when an image stereo pair is used to find matching keypoints. In this study, a parallel approach is presented with Threading Building Blocks (TBB), using sub-matrix computations, for extraction of corresponding keypoints from an image-stereo pair, and triangulation with the Direct Linear Transform (DLT) method to reconstruct the 3D position of the object in space. Additional robustness is investigated by implementing a Kalman filter for tracking prediction during the domain transition between the sub-matrices. Furthermore, a flexible simulation framework is set up for smart sensing with a coupled unsteady aeroservoelastic model of a 3D wing and a visual model to test the method for intelligent control feedback in a simulation environment. The methodology is tested in a laboratory environment with a stereo camera setup, and in a virtual environment, where the virtual camera parameters are reconstructed to meet a stereo setup. The proposed approach aims to advance the state-of-the-art in smart sensing, particularly in the context of real-time state estimation of aeroelastic structures and control feedback. The parallel approach shows a significant improvement of speed and efficiency, allowing real-time computation from a live image stream at 50 fps.Aerospace Structures & Computational MechanicsOLD SnC CultureControl & Simulatio

Peaking into the Black-box: Prediction Intervals Give Insight into Data-driven Quadrotor Model Reliability

Ensuring the reliability and validity of data-driven quadrotor model predictions is essential for their accepted and practical use. This is especially true for grey- and black-box models wherein the mapping of inputs to predictions is not transparent and subsequent reliability notoriously difficult to ascertain. Nonetheless, such techniques are frequently and successfully used to identify quadrotor models. Prediction intervals (PIs) may be employed to provide insight into the consistency and accuracy of model predictions. This paper estimates such PIs for polynomial and Artificial Neural Network (ANN) quadrotor aerodynamic models. Two existing ANN PI estimation techniques - the bootstrap method and the quality driven method – are validated numerically for quadrotor aerodynamic models using an existing high-fidelity quadrotor simulation. Quadrotor aerodynamic models are then identified on real quadrotor flight data to demonstrate their utility and explore their sensitivity to model interpolation and extrapolation. It is found that the ANN-based PIs widen considerably when extrapolating and remain constant, or shrink, when interpolating. While this behaviour also occurs for the polynomial PIs, it is of lower magnitude. The estimated PIs establish probabilistic bounds within which the quadrotor model outputs will likely lie, subject to modelling and measurement uncertainties that are reflected through the PI widths.Control & Simulatio