C. C. de Visser

Brief paper: A new approach to linear regression with multivariate splines

A new methodology for creating highly accurate, static nonlinear maps from scattered, multivariate data is presented. This new methodology uses the B-form polynomials of multivariate simplex splines in a new linear regression scheme. This allows the use of standard parameter estimation techniques for estimating the B-coefficients of the multivariate simplex splines. We present a generalized least squares estimator for the B-coefficients, and show how the estimated B-coefficient variances lead to a new model quality assessment measure in the form of the B-coefficient variance surface. The new modeling methodology is demonstrated on a nonlinear scattered bivariate dataset.

Differential constraints for bounded recursive identification with multivariate splines

The ability to perform online model identification for nonlinear systems with unknown dynamics is essential to any adaptive model-based control system. In this paper, a new differential equality constrained recursive least squares estimator for multivariate simplex splines is presented that is able to perform online model identification and bounded model extrapolation in the framework of a model-based control system. A new type of linear constraints, the differential constraints, are used as differential boundary conditions within the recursive estimator which limit polynomial divergence when extrapolating data. The differential constraints are derived with a new, one-step matrix form of the de Casteljau algorithm, which reduces their formulation into a single matrix multiplication. The recursive estimator is demonstrated on a bivariate dataset, where it is shown to provide a speedup of two orders of magnitude over an ordinary least squares batch method. Additionally, it is demonstrated that inclusion of differential constraints in the least squares optimization scheme can prevent polynomial divergence close to edges of the model domain where local data coverage may be insufficient, a situation often encountered with global recursive data approximation.

Wavefront reconstruction in adaptive optics systems using nonlinear multivariate splines

This paper presents a new method for zonal wavefront reconstruction (WFR) with application to adaptive optics systems. This new method, indicated as Spline based ABerration REconstruction (SABRE), uses bivariate simplex B-spline basis functions to reconstruct the wavefront using local wavefront slope measurements. The SABRE enables WFR on nonrectangular and partly obscured sensor grids and is not subject to the waffle mode. The performance of SABRE is compared to that of the finite difference (FD) method in numerical experiments using data from a simulated Shack-Hartmann lenslet array. The results show that SABRE offers superior reconstruction accuracy and noise rejection capabilities compared to the FD method.

A Hybrid Sensor Based Backstepping Control Approach with its Application to Fault-Tolerant Flight Control

Recently, an incremental type sensor based backstepping (SBB) control approach, based on singular perturbation theory and Tikhonov’s theorem, has been proposed. This Lyapunov function based method uses measurements of control variables and less model knowledge, and it is not susceptible to the model uncertainty caused by fault scenarios. In this paper, the SBB method has been implemented on a fixed wing aircraft with its focus on handling structural changes caused by damages. A new hybrid autopilot flight controller has been developed for a Boeing 747-200 aircraft after combining nonlinear dynamic inversion (NDI) with SBB control approach. Two benchmarks for fault tolerant flight control (FTFC), named rudder runaway and engine separation, are employed to evaluate the proposed method. The simulation results show that the proposed control approach leads to a zero tracking-error performance in nominal condition and guarantees the stability of the closed-loop system under failures as long as the reference commands are located in the safe flight envelope.

A Multidimensional Spline Based Global Nonlinear Aerodynamic Model for the Cessna Citation II

A new method is proposed for the identification of global nonlinear models of aircraft non-dimensional force and moment coefficients. The method is based ona recent type of multivariate spline, the multivariate simplex spline, which can accurately approximate very large, scattered nonlinear datasets in any number of dimensions. The new identification method is used to identify a global nonlinear aerodynamic model of high dimensionality for the Cessna Citation II laboratory aircraft operated by the Delft University of Technology and the Netherlands National Aerospace Laboratory. The data used in the identification process consisted of millions of measurements and was accumulated during more than 250 flight test maneuvers with the laboratory aircraft. The resulting models for the aerodynamic force and moment coefficients are continuous analytical functions as they consist of sets of piecewise defined, multivariate polynomials. The identified models were validated using a subset of the flight data, with validation results showing a very close match between model and reality.

Nonlinear Multivariate Spline-Based Control Allocation for High-Performance Aircraft

High performance flight control systems based on the nonlinear dynamic inversion (NDI) principle require highly accurate models of aircraft aerodynamics. In general, the accuracy of the internal model determines to what degree the system nonlinearities can be canceled; the more accurate the model, the better the cancellation, and with that, the higher the performance of the controller. In this paper a new control system is presented that combines NDI with multivariate simplex spline based control allocation. We present three control allocation strategies which use novel expressions for the analytical Jacobian and Hessian of the multivariate spline models. Multivariate simplex splines have a higher approximation power than ordinary polynomial models, and are capable of accurately modeling nonlinear aerodynamics over the entire flight envelope of an aircraft. This new method, indicated as SNDI, is applied to control a high performance aircraft (F-16) with a large flight envelope. The simulation results indicate that the SNDI controller can achieve feedback linearization throughout the entire flight envelope, leading to a significant increase in tracking performance compared to ordinary polynomial based NDI.

Global Nonlinear Aerodynamic Model Identification with Multivariate Splines

A new method for global nonlinear aerodynamic model identification is presented. This new identification method uses multivariate splines inside a linear regression framework. The linear regression framework allows the use of standard parameter estimation techniques for estimating the parameters of the multivariate splines. The new identification method is used to identify a global nonlinear aerodynamic model of the F-16 fighter aircraft based on simulated flight test data from a NASA subsonic wind tunnel model of the F-16. The high approximation power of the multivariate splines allows the pilot to fly high amplitude, long duration maneuvers resulting in a globally valid, high quality aerodynamic model. The identified aerodynamic model is compared directly with the NASA wind tunnel model showing that the multivariate splines can accurately model both small scale and large scale nonlinear aerodynamic phenomena.

Time-Varying Model Identification of Flapping-Wing Vehicle Dynamics Using Flight Data

A time-varying model for the forward flight dynamics of a flapping-wing micro aerial vehicle is identified from free-flight optical tracking data. The model is validated and used to assess the validity of the widely applied time-scale separation assumption. Based on this assumption, each aerodynamic force and moment is formulated as a linear addition of decoupled time-averaged and time-varying submodels. The resulting aerodynamic models are incorporated in a set of linearized equations of motion, yielding a simulation-capable full dynamic model. The time-averaged component includes both the longitudinal and the lateral aerodynamics and is assumed to be linear. The time-varying component is modeled as a third-order Fourier series, which approximates the flapping dynamics effectively. Combining both components yields a more complete and realistic simulation. Results suggest that while in steady flight the time-scale separation assumption applies well during maneuvers the time-varying dynamics are not fully ...

Multivariate Spline-Based Adaptive Control of High-Performance Aircraft with Aerodynamic Uncertainties

In this paper, a new modular adaptive control system is presented to compensate for aerodynamic uncertainties in high-performance flight control systems. This approach combines nonlinear dynamic inversion with multivariate spline-based adaptive control allocation. A new real-time identification routine for multivariate splines is presented to compensate for aerodynamic uncertainties in the control allocation system. This method, indicated as spline-based adaptive nonlinear dynamic inversion, is applied to control an F-16 aircraft subject to significant aerodynamics uncertainties. Simulation results indicate that the new controller can tune itself each time a model error is detected and has superior adaptability compared to an ordinary polynomial-based adaptive controller. Multivariate splines have sufficient flexibility and approximation power to accurately model nonlinear aerodynamics over the entire flight envelope. As a result, the global model remains intact. Although a part of the model is being reco...

Joint Sensor Based Backstepping for Fault-Tolerant Flight Control

The sensor based backstepping control law, based on the singular perturbation theory and Tikhonov’s theorem, is a novel nonlinear incremental control approach. This Lyapunov function based method is not susceptible to model uncertainties since it uses measured state derivatives instead of an onboard model. Considering these merits, the sensor based backstepping method is extended to handle sudden structural changes in the fault-tolerant flight control of an overactuated Boeing 747-200 aircraft with the control allocation being considered. Because of the application of the backstepping technique, this double-loop joint sensor based backstepping attitude controller allows more interaction between its outer and inner loops compared to a standard nonlinear dynamic inversion angular control approach. The benchmark with engine separation and rudder runaway failure scenarios is employed to evaluate the new controller. The simulation results show that the new joint sensor based backstepping attitude controller ca...