Linear approximation

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

Analytic $Q$-ball solutions and their stability in a piecewise parabolic potential

Abstract: Explicit solutions for extended objects of a $Q$-ball type were found analytically in a model describing complex scalar field with piecewise parabolic potential in ($3+1$)- and ($1+1$)-dimensional space-times. Such a potential provides a variety of solutions which were thoroughly examined. It was shown that, depending on the values of the parameters of the model and according to the known stability criteria, there exist stable and unstable solutions. The classical stability of solutions in ($1+1$)-dimensional space-time was examined in the linear approximation and it was shown explicitly that the spectrum of linear perturbations around some solutions contains exponentially growing modes while it is not so for other solutions.

Quadratic Basis Pursuit

Abstract: In many compressive sensing problems today, the relationship between the measurements and the unknowns could be nonlinear. Traditional treatment of such nonlinear relationships have been to approximate the nonlinearity via a linear model and the subsequent un-modeled dynamics as noise. The ability to more accurately characterize nonlinear models has the potential to improve the results in both existing compressive sensing applications and those where a linear approximation does not suffice, e.g., phase retrieval. In this paper, we extend the classical compressive sensing framework to a second-order Taylor expansion of the nonlinearity. Using a lifting technique and a method we call quadratic basis pursuit, we show that the sparse signal can be recovered exactly when the sampling rate is sufficiently high. We further present efficient numerical algorithms to recover sparse signals in second-order nonlinear systems, which are considerably more difficult to solve than their linear counterparts in sparse optimization.

Nonlinear predictive control for Hammerstein-Wiener systems.

Abstract: This paper discusses a nonlinear Model Predictive Control (MPC) algorithm for multiple-input multiple-output dynamic systems represented by cascade Hammerstein–Wiener models. The block-oriented Hammerstein–Wiener model, which consists of a linear dynamic block embedded between two nonlinear steady-state blocks, may be successfully used to describe numerous processes. A direct application of such a model for prediction in MPC results in a nonlinear optimisation problem which must be solved at each sampling instant on-line. To reduce the computational burden, a linear approximation of the predicted system trajectory linearised along the future control scenario is successively found on-line and used for prediction. Thanks to linearisation, the presented algorithm needs only quadratic optimisation, time-consuming and difficult on-line nonlinear optimisation is not necessary. In contrast to some control approaches for cascade models, the presented algorithm does not need inverse of the steady-state blocks of the model. For two benchmark systems, it is demonstrated that the algorithm gives control accuracy very similar to that obtained in the MPC approach with nonlinear optimisation while performance of linear MPC and MPC with simplified linearisation is much worse.