Linear approximation

About: Linear approximation is a research topic. Over the lifetime, 3901 publications have been published within this topic receiving 74764 citations.

On the optimality of neural-network approximation using incremental algorithms

Abstract: The problem of approximating functions by neural networks using incremental algorithms is studied. For functions belonging to a rather general class, characterized by certain smoothness properties with respect to the L/sub 2/ norm, we compute upper bounds on the approximation error where error is measured by the L/sub q/ norm, 1/spl les/q/spl les//spl infin/. These results extend previous work, applicable in the case q=2, and provide an explicit algorithm to achieve the derived approximation error rate. In the range q/spl les/2 near-optimal rates of convergence are demonstrated. A gap remains, however, with respect to a recently established lower bound in the case q>2, although the rates achieved are provably better than those obtained by optimal linear approximation. Extensions of the results from the L/sub 2/ norm to L/sub p/ are also discussed. A further interesting conclusion from our results is that no loss of generality is suffered using networks with positive hidden-to-output weights. Moreover, explicit bounds on the size of the hidden-to-output weights are established, which are sufficient to guarantee the established convergence rates.

On the linear control of underactuated systems: The flywheel inverted pendulum

Abstract: Underactuated systems are usually represented by nonlinear models and their control requires the design of nonlinear controllers. The flywheel inverted pendulum is a special case, also nonlinear, but admitting a linear approximation around the unstable equilibrium point in the upper position. In this paper, a linear controller is designed in two steps. First, the output is controlled with a simple PID controller, leading to an internally unstable plant. Then, a second linear controller in an outer loop provides the global stability, showing good regulation and disturbance rejection performance. Experimental results show the effectiveness of the design.

Using MILP for UAVs Trajectory Optimization under Radar Detection Risk

Abstract: This paper presents an approach to trajectories optimization for unmanned aerial vehicle (UAV) in presence of obstacles, waypoints, and threat zones such as radar detection regions, using mixed integer linear programming (MILP). The main result is the linear approximation of a nonlinear radar detection risk function with integer constraints and indicator 0-1 variables. Several results are presented to show that the approach can yields trajectories depending on the acceptable risk of detection.

Modeling of nonlinear medical diagnostic ultrasound

Abstract: In this PhD Thesis, a numerical method is described that accurately predicts the pulsed acoustic pressure field generated by a medical diagnostic phased array transducer in a nonlinear acoustic medium. The method is called the Iterative Nonlinear Contrast Source (INCS) method, and it is capable of handling a large-scale, threedimensional domain of interest, in the order of 100 wavelengths in each spatial dimension and 100 periods in the temporal dimension. Unlike many existing methods, the method is based on a full-wave approach and it does not employ an implicit or explicit plane wave approximation. Starting from a set of two nonlinear first-order field equations and a two nonlinear constitutive equations, we show that the nonlinear acoustic field may be approximated with a second-order wave equation which is a lossless form of the Westervelt equation including source terms. The Westervelt equation may be solved efficiently by means of the INCS method. In this method, the nonlinear wave problem is formally solved by a Neumann iterative solution, in which the nonlinear term in the Westervelt equation acts as a nonlinear contrast source and provides iterative corrections to the linear approximation of the nonlinear wave problem. The linear step in the Neumann scheme is solved by a spatiotemporal convolution integral of the (primary or contrast) source with the Green's function of the linear background medium. For the evaluation of the convolution integral as a discrete convolution sum on a spatiotemporal grid, the Green's function and the (primary or contrast) source are filtered and windowed in all spatiotemporal dimensions, allowing for their coarse discretization at the Nyquist limit of two points per wavelength/period for the maximum frequency of interest. This approach is referred to as the Filtered Convolution (FC) method. The resulting discretized convolution sum is efficiently evaluated using a Fast Fourier Transform (FFT) method. Results for various one-dimensional and three-dimensional wave problems show that in all cases the INCS method produces accurate results. A validation experiment with a rectangular transducer also shows that the measured nonlinear acoustic field is reproduced very well with the INCS method. Because of its accuracy, reliability and the general validity of its solution, we conclude that the INCS method can be used as a benchmark model for weak to moderate nonlinear distortion, as it occurs in medical diagnostic ultrasound.

Accurate Modeling of Nonlinear Systems using Volterra Series Submodels

Abstract: We investigate the problem of accurately modeling nonlinear systems (such as aircraft flight in high angle-of-attack/sideslip flight) using simple low-order Volterra submodels. First, we apply this technique to a simplified nonlinear stall/post-stall aircraft model for the case of a longitudinal limit cycle. Our simulation study demonstrates that the responses of the Volterra submodels accurately match the responses of the original nonlinear model, whereas the responses of a piecewise-linear model do not. Next, we apply the technique to a simplified high a nonlinear model of wing rock. Our simulation study demonstrates that the second-order Volterra approximation predicts the wing rock limit cycle, while a linear approximation does not. Third-, fourth- and fifth-order Volterra approximations are observed to give wing rock amplitudes that converge quadratically to the nonlinear value.