Low-order Nonlinearities Research Articles

Finite‐time state‐feedback stabilisation of non‐holonomic systems with low‐order non‐linearities

This study investigates the problem of finite-time stabilisation by state feedback for a class of non-holonomic systems in chained form with low-order non-linearities, to which the existing control methods are inapplicable. By introducing sign function and necessarily modifying the method of adding a power integrator, a state-feedback controller is successfully constructed. Together with a novel switching control strategy, the designed controller guarantees that the states of closed-loop system are regulated to zero in a finite time. A simulation example is provided to illustrate the effectiveness of the proposed approach.

Dynamic and stability analysis of the linear guide with time-varying, piecewise-nonlinear stiffness by multi-term incremental harmonic balance method

The dynamic behaviors and stability of the linear guide considering contact actions are studied by multi-term incremental harmonic balance method (IHBM). Based on fully considering the parameters of the linear guide, a static model is developed and the contact stiffness is calculated according to Hertz contact theory. A generalized time-varying and piecewise-nonlinear dynamic model of the linear guide is formulated to perform an accurate investigation on its dynamic behaviors and stability. The numerical simulation is used to confirm the feasibility of the approach. The effects of excitation force and mean load on the system are analyzed in low-order nonlinearity. Multi-term IHBM and numerical simulation are employed to the effect of high-order nonlinearity and show the transition to chaos. Additionally, the effects of preload, initial contact angle, the number and diameter of balls are discussed.

Femtosecond stimulated Raman spectroscopy by six-wave mixing.

Femtosecond Stimulated Raman Spectroscopy (FSRS) is motivated by the knowledge of the molecular geometry changes that accompany sub-picosecond chemical reactions. The detection of vibrational resonances throughout the entire fingerprint region of the spectrum with sub-100-fs delay precision is fairly straightforward to accomplish with the FSRS technique. Despite its utility, FSRS must contend with substantial technical challenges that stem from a large background of residual laser light and lower-order nonlinearities when all laser pulses are electronically resonant with the equilibrium system. In this work, a geometry based on five incident laser beams is used to eliminate much of this undesired background in experiments conducted on metmyoglobin. Compared to a three-beam FSRS geometry with all electronically resonant laser pulses, the five-beam approach described here offers major improvements in the data acquisition rate, sensitivity, and background suppression. The susceptibility of the five-beam geometry to experimental artifacts is investigated using control experiments and model calculations. Of particular concern are undesired cascades of third-order nonlinearities, which are known to challenge FSRS measurements carried out on electronically off-resonant systems. It is generally understood that "forbidden" steps in the desired nonlinear optical processes are the origin of the problems encountered under off-resonant conditions. In contrast, the present experiments are carried out under electronically resonant conditions, where such unfortunate selection rules do not apply. Nonetheless, control experiments based on spectroscopic line shapes, signal phases, and sample concentrations are conducted to rule out significant contributions from cascades of third-order processes. Theoretical calculations are further used to estimate the relative intensities of the direct and cascaded responses. Overall, the control experiments and model calculations presented in this work suggest promise for multidimensional resonance Raman investigations of heme proteins.

Finite-time stabilisation for high-order nonlinear systems with low-order and high-order nonlinearities

This paper considers the finite-time stabilisation for a class of high-order nonlinear systems with both low-order and high-order nonlinearities. Based on the finite-time Lyapunov stability theorem together with the methods of dynamic gain control and adding one power integrator, a state feedback controller with gains being tuned online by two dynamic equations is proposed to guarantee the global finite-time stabilisation of the closed-loop system.

Global stabilisation for a class of more general high-order time-delay nonlinear systems by output feedback

This paper addresses the problem of global stabilisation by output feedback for a class of high-order time-delay nonlinear systems with more general uncertainties, which cover both high-order and low-order nonlinearities. By introducing sign function and necessarily modifying the homogeneous domination approach, an output feedback controller is successfully constructed to guarantee the global asymptotic stability of the resulting closed-loop system. Due to the versatility of the homogeneous domination approach and homogeneous properties, the proposed method is also utilised to the control design of upper-triangular systems.

Semi-global finite-time stabilization of uncertain nonholonomic systems via output feedback

This paper investigates the problem of semi-global finite-time stabilization by output feedback for a class of nonholonomic systems with both high-order and low-order nonlinearities. By using the homogeneous domination approach, a constructive design procedure for output feedback control is given. Together with a switching control scheme, the designed controller renders that the states of a closed-loop system are semi-globally regulated to zero in a finite time. A simulation example is provided to illustrate the effectiveness of the proposed approach.

Global state feedback stabilisation of nonlinear systems with high-order and low-order nonlinearities

This paper studies the state feedback control problem for a class of nonlinear systems with high-order and low-order nonlinearities. The introduction of the sign function together with the method of adding a power integrator and Lyapunov stability theorem makes the closed-loop system globally asymptotically stable. Exploiting the idea of how to deal with growth nonlinearities with both high order and low order being relaxed to some intervals is the focus of this work.

An Improved Algorithm for Estimating the Parameters of Polynomial-Phase Signals

This paper presents an algorithm for estimating the parameters of polynomial-phase signals (PPSs). This algorithm combines the CPF-HAF method and Radon transform, and can be referred to as the Radon-CPF-HAF method. In the proposed algorithm, the HAF is first applied on the original PPS to produce a cubic phase signal, whose parameters are then estimated by the Radon transform of the cubic phase function for the derived cubic phase signal. As involving in lower order nonlinearity, the proposed algorithm outperforms the HAF in terms of the accuracy and signal-to-noise-ratio (SNR) threshold. When multicomponent PPSs are considered, the product version of the proposed algorithm removes the identifiability problem which the product version of the CPF-HAF (PCPF-HAF) suffers from. Computer simulations have been carried out to support the theoretical results.

Low- and high-order nonlinear optical characterization of C60-containing media

Third- and higher-order nonlinear optical processes in fullerenes were studies to define the influence of low-order nonlinearities on the high-order harmonic generation in these media. We measured the nonlinear absorption coefficients of the C60:toluene solution using the 532 and 1064 nm, 50 ps pulses. The high-order harmonic generation was studied during propagation of the 790 nm, 150 fs pulses through the plasmas produced on surfaces containing fullerene powder. These studies have shown that the low-order nonlinearities of fullerenes have no impact on the generation of harmonics in such mediums in the vacuum ultraviolet range at optimal intensity of laser radiation.

Global Stabilization of a Class of Feedforward Systems with Lower-Order Nonlinearities

This note considers the problem of global stabilization for a class of feedforward systems whose nonlinearities are allowed to be lower-order growing, instead of higher-order or linear growing as required by many existing results. To solve the problem, a domination approach is developed to design the stabilizer with a new structure, which is also integrated with the adding a power integrator and nested saturation methods. The use of a locally homogeneous stabilizer enables us to relax the growth condition imposed on the nonlinearities and hence enlarges the class of feedforward systems which are globally stabilizable.

Observation of a Microscopic Cascaded Contribution to the Fifth-Order Nonlinear Susceptibility

Typically, low-order nonlinearities are much stronger than high-order nonlinearities. In this Letter, we demonstrate a procedure by which strong high-order nonlinearities can be synthesized out of low-order nonlinearities. Our procedure involves the use of the previously largely overlooked process of microscopic cascading, which results from local-field effects. We have performed an experiment that allows us to distinguish the influence of microscopic cascading from the more-well-known process of macroscopic cascading, and we find conditions under which microscopic cascading can be the dominant effect. The ability to create a large high-order nonlinear response could prove useful for applications in quantum-information science that require the detection of the simultaneous presence of N entangled photons.

Existence and Liouville-type theorems for some indefinite quasilinear elliptic problems with potentials vanishing at infinity

We study the existence versus absence of nontrivial weak solutions for a class of indefinite quasilinear elliptic problems on unbounded domains with noncompact boundary, in the presence of competing lower order nonlinearities with potentials decaying to zero at infinity.

TAKING ADVANTAGE OF HOMOGENEITY: A UNIFIED FRAMEWORK FOR OUTPUT FEEDBACK CONTROL OF NONLINEAR SYSTEMS

A unified framework is presented for global output feedback stabilization of a family of uncertain systems with low- and high-order nonlinearities. By taking advantage of homogeneity, together with a domination-based observer/controller design, this framework not only incorporates a number of results proposed recently in the literature, but also yields new results for more general classes of non-uniformly observable and non-smoothly stabilizable systems, whose global stabilization problem cannot be addressed by existing output feedback control schemes. The novelty of this paper lies in the development of a reduced order observer-based output feedback design method, making it possible to compensate simultaneously low-order nonlinearities that dominate near the origin, and high-order nonlinearities that dominate far away from the origin.

Indefinite quasilinear elliptic problems with subcritical and supercritical nonlinearities on unbounded domains

By using the fibering method, we study the existence of non-negative solutions for a class of indefinite quasilinear elliptic problems on unbounded domains with noncompact boundary, in the presence of competing subcritical and supercritical lower order nonlinearities.

Branches of positive solutions of quasilinear elliptic equations on non-smooth domains

We prove existence of an unbounded global branch (i.e. connected set) of weak solutions of a second order quasilinear equation depending on a real parameter λ on an arbitrary (possibly non-smooth) bounded domain in R N , with a Leray–Lions operator as the leading part. Here, we can allow lower order nonlinearities which depend on first derivatives, satisfying appropriate growth conditions including the critical case. Furthermore, we give sufficient conditions for the existence of a branch consisting entirely of nonnegative solutions for positive λ . Our approach also yields a new existence result in the case of critical growth in derivatives of lower order.

On Finite Fractal Dimension of the Global Attractor for the Wave Equation with Nonlinear Damping

We prove that the global attractor to a semilinear damped wave equation has finite fractal dimension provided that the damping function and the lower order nonlinearity are smooth with certain polynomial growth.

Effect on soft error rate of kinks and open loops in the read-head transfer curve

In this paper we study the effects of nonlinearities in the GMR sensor on soft error rate. Previous studies have focused on the effects of nonhysteretic, low order nonlinearities. Here we focus mainly on transfer curves with "kinks" and open loops.

Generalized Quasilinearization for Quasilinear Parabolic Equations with Nonlinearities of DC Type

In this paper, we consider an initial boundary-value problem for a class of quasilinear parabolic equations whose lower-order nonlinearity is of d.c. function type with respecxt to the dependent variable. Assuming the existence of an ordered pair of weak upper and lower solutions, we establish a generalized quasilinearization method for the problem under consideration. A characteristic feature of this generalized quasilinearization method consists in the construction of monotone sequences converging to the unique solution within the interval of upper and lower solutions, and whose convergence rate is quadratic.

Diffractive optics based two-color six-wave mixing: phase contrast heterodyne detection of the fifth order Raman response of liquids

Fifth order Raman scattering is extremely sensitive to anharmonicities and nonlinear polarizabilities and thus provides a unique probe of the many-body potential of liquids. The weak signals from this high order, χ (5), process are complicated by cascaded lower-order χ (3) processes. A new approach has reduced the beam delivery to a single diffractive optic that generates the beam pattern for phase-matched six-wave mixing, and passively phase locks all six fields. Heterodyne detection introduces signal amplification and phase contrast against lower-order nonlinearities. The findings provide new insight into the nature of the direct χ (5) nonlinear response function and two-time correlation of liquids.

A novel network for nonlinear modeling of neural systems with arbitrary point-process inputs

This paper address the issue of nonlinear model estimation for neural systems with arbitrary point-process inputs using a novel network that is composed of a pre-processing stage of a Laguerre filter bank followed by a single hidden layer with polynomial activation functions. The nonlinear modeling problem for neural systems has been attempted thus far only with Poisson point-process inputs and using cross-correlation methods to estimate low-order nonlinearities. The specific contribution of this paper is the use of the described novel network to achieve practical estimation of the requisite nonlinear model in the case of arbitrary (i.e. non-Poisson) point-process inputs and high-order nonlinearities. The success of this approach has critical implications for the study of neuronal ensembles, for which nonlinear modeling has been hindered by the requirement of Poisson process inputs and by the presence of high-order nonlinearities. The proposed methodology yields accurate models even for short input–output data records and in the presence of considerable noise. The efficacy of this approach is demonstrated with computer-simulated examples having continuous output and point-process output, and with real data from the dentate gyrus of the hippocampus.