Low-order Nonlinearities Research Articles

Fixed-time Lyapunov criteria of stochastic nonlinear systems revisited and its applications

In all the references on stochastic fixed-time stability, the customary treatment of the stochastic noise in the worst-case sense is that it is treated as the unfavorable factor for system stability. Consequently, stochastic fixed-time Lyapunov-type conditions are rather restrictive. Realizing this limitation, we revisit stochastic fixed-time stability and present a generalized fixed-time stability theorem for stochastic differential equations. This theorem completely removes the assumption in these references that the differential operator of the Lyapunov function must be strictly negative, and reveals a positive role of the stochastic noise in stochastic fixed-time stability. As the application of this theorem and its corollary, we solve the problem of fixed-time stabilization for stochastic nonlinear systems with high-order and low-order nonlinearities.

A Low-Order Steady-State Model of Electric Springs for Conservation Voltage Reduction in Active Distribution Networks With Renewables

Electric spring (ES), as a new type of power electronic device, can flexibly regulate the voltage of noncritical load (NL) and improve the voltage quality of critical load (CL). It can further increase the energy savings of conservation voltage reduction (CVR) in active distribution networks (ADNs). However, the high-order nonlinearity of ESs steady-state model brings the difficulty of solving the CVR optimization problem Therefore, this paper proposes a low-order steady-state model of ESs for CVR in ADNs. The operation modes of ESs are extended based on analysis of voltage regulation characteristics, firstly. The equivalent circuit of the bus with an ES is also derived, and a steady-state model with low-order nonlinearity is further proposed. Then, considering significant difference in response speeds among ESs, distributed generation (DGs) and discrete control devices (DCDs), a two-stage coordinated operation model of ESs, DGs and DCDs is established. By means of convex envelope and linearization, the nonconvex nonlinear constraints are transformed into linear constraints, which can improve the convergence and solving efficiency of the optimization model. Finally, the case studies demonstrate that ESs can increase the CVR energy savings, and the proposed low-order steady-state model can improve the solving efficiency of this CVR optimization problem.

Dispersive decay bound of small data solutions to higher order KdV-type equations with low order nonlinearity

Dispersive decay bound of small data solutions to higher order KdV-type equations with low order nonlinearity

Adaptive stabilisation of output-constrained high-order nonlinear systems with high-order and low-order nonlinearities

This paper investigates the adaptive stabilisation of output-constrained high-order nonlinear systems with more general high-order and low-order nonlinearities. By skilfully introducing nonlinear mappings, a key coordinate transformation, integral Lyapunov functions and the sign function into the adding a power integrator technique, and combining a novel analysis method, a continuous adaptive state-feedback controller is constructed. When the initial value of system states lies in the constrained set, it is rigorously proved that all the closed-loop signals are uniformly bounded, the asymmetric output constraint isn't transgressed, and the equilibrium point of the closed-loop system is uniformly asymptotically stable. A simulation example shows the effectiveness of this control scheme.

Nonlinear aeroelastic stability analysis of a folding wing by using geometrically exact fully intrinsic beam equations

In this paper, the aeroelastic instability of folding wings by using the geometrically exact fully intrinsic beam equations is investigated. The important advantages of these equations in comparison with other structural beam equations are complete modeling without simplifying assumptions in large deformations, low-order nonlinearities, and thus less complexity. For the first time, folding angles have been implemented in the geometrically exact fully intrinsic beam equations and hence this is the main novelty of this study. The applied aerodynamic loads in an incompressible flow regime are determined using Peter’s unsteady aerodynamic model. In order to check the stability of the system, first the resulting non-linear partial differential equations are discretized by employing the central finite difference method, and then linearized about the nonlinear steady-state condition. By obtaining the eigenvalues of the linearized system, the stability of the wing is evaluated. Furthermore, investigation of the effects of some important parameters such as stiffness ratio and length ratio on the flutter speed of the folding wing for various folding angles, is another achievement of this study. It is observed that the geometrically exact fully intrinsic beam equations can model the folding angles for the aeroelastic analysis more accurately and the capabilities of these equations became more specific.

Low-order nonlinearities and high-order harmonics generation in Zn and ZnSe nanoparticles synthesized during femtosecond laser ablation at a 50 kHz pulse repetition rate

We synthesize Zn and ZnSe nanoparticles during laser ablation of bulk material in ethanol and analyze them using the Z-scan technique and the generation of high harmonics using a 50 kHz, 40 fs, and 1030 nm laser. The low-order nonlinear absorption and nonlinear refraction coefficients in the colloidal suspensions of the formed nanoparticles were measured using 1030 nm and 515 nm probing laser pulses, respectively. The values of the nonlinear refractive index and nonlinear absorption coefficient of Zn NPs at 1030 nm wavelength were calculated to be γ = − 1.4 × 10 − 13 c m 2 W − 1 and β = 2.7 × 10 − 11 c m W − 1 , respectively. Similarly, in the case of ZnSe NPs, γ = − 4.9 × 10 − 14 c m 2 W − 1 and β = 0.5 × 10 − 11 c m W − 1 , respectively. In the case of 515 nm probing pulses, the corresponding nonlinear optical parameters of Zn NP suspension were calculated to be γ = − 1.7 × 10 − 14 c m 2 W − 1 and β = 1.0 × 10 − 11 c m W − 1 , and for ZnSe NP suspension, the value of β was calculated to be 2.1 × 10 − 11 c m W − 1 . The nonlinear refractive index of this suspension was γ = − 2.4 × 10 − 14 c m 2 W − 1 . We also report the studies of high-order harmonics generation in plasmas containing Zn and ZnSe NPs using single- (1030 nm) and two-color (1030 nm + 515 nm) laser fields. The maximal generated extreme ultraviolet (XUV) harmonics were 33rd (Zn NPs) and 27th (ZnSe NPs) orders. The decrease of the maximal generating harmonic orders was explained using the three-step model. The high brightness of the ninth and 10th harmonics generated in the two-color laser field was obtained due to the resonance properties of Zn NPs. The maximal generated XUV harmonics were the 33rd (Zn nanoparticles) and 27th (ZnSe nanoparticles) orders.

Finite-time stabilization of output-constrained stochastic high-order nonlinear systems with high-order and low-order nonlinearities

This paper investigates finite-time stabilization of output-constrained stochastic high-order nonlinear systems with stochastic inverse dynamics and high-order and low-order nonlinearities. By characterizing the unmeasured stochastic inverse dynamics with finite-time stochastic input-to-state stability and introducing nonlinear transformations, a new design and analysis method is constructed. It is rigorously proved that all the closed-loop signals are bounded almost surely, the asymmetric output constraint isn’t violated almost surely, and the trivial solution of the closed-loop system is finite-time stable in probability.

Nonrelativistic limit of solitary waves for nonlinear Maxwell–Klein–Gordon equations

We study the nonrelativistic limit of solitary waves from Nonlinear Maxwell–Klein–Gordon equations (NMKG) to Nonlinear Schrödinger–Poisson equations (NSP). It is known that the existence or multiplicity of positive solutions depends on the choices of parameters the equations contain. In this paper, we prove that for a given positive solitary wave of NSP, which is found in Ruiz’s work (J Funct Anal 237(2):655–674, 2006), there corresponds a family of positive solitary waves of NMKG under the nonrelativistic limit. Notably, our results contain a new result of existence of positive solutions to (NMKG) with lower order nonlinearity.

Global finite-time stabilisation for a class of nonlinear systems in the p-normal form via output feedback

In this paper, a finite-time output feedback control scheme is presented for a class of nonlinear systems in the p-normal form. The nonlinear functions in the systems are assumed to be bounded by both low-order and high-order nonlinearities multiplied by a polynomial-type output-dependent growth rate, which are able to cover a more general ones. A reduced-order observer is constructed to estimate unmeasurable states, and a finite-time output feedback controller is proposed for the considered system by constructing a new Lyapunov function and using the adding a power integration technique. Two simulation examples are given to verify the effectiveness of the proposed scheme.

Parameter estimate of multi-component LFM signals based on GAPCK

In this paper, we propose a fast and robust parameter estimate method for multi-component linear frequency modulated (LFM) signals from a finite number of noisy discrete time observations. First, a new kernel called generalized adjustable parameter correlation kernel (GAPCK) is introduced to avoid the coupling terms between time and lag. Then, Matched Fourier transform (MFT) and Robust Energy accumulation (REA) are utilized to obtain the chirp rate estimate of the received LFM signals. This estimate is used to compensate the time-quadratic phase term of the GAPCK, and then Fast Fourier transform (FFT) along time-axis is performed to estimate the constant coefficient. Moreover, the asymptotic statistical properties of parameter estimates are derived. The proposed method has low computational complexity and favorite performance under low signal-to-noise ratio (SNR) due to the low-order non-linearity of the GAPCK. Finally, simulated and real data are provided to verify the robustness and effectiveness of the proposed method.

Global stabilization of a class of nonlinear systems with polynomial nonlinearities by output feedback

This article deals with the problem of global stabilization by output feedback for a class of nonlinear systems with high-order and low-order nonlinearities. By generalizing a reduced-dimensional design method, adding a power integrator technique and choosing an appropriate Lyapunov function, a continuous output feedback controller is constructed. A simulation example is given to verify the effectiveness of the proposed scheme.

Injection Locking at Fractional Frequencies of Magnetic-Tunnel-Junction-Based Read Sensors’ Ferromagnetic Resonance Modes

Being nonlinear dynamic systems, magnetic read sensors should respond to an excitation signal of a frequency considerably different from their natural ferromagnetic resonance (FMR) frequencies. Because of the magnetization dynamics' inherent nonlinear nature, the sensors' response should be measured at the DC, excitation frequency, and its multiples (harmonics). In this paper, we present results of such measurements, accomplished using a one-port nonlinear vector network analyzer (NVNA), which show distinct resonances at fractional frequencies of the free layer (FL) FMR mode. Identification of these resonances, resulting from the nonlinear nature of the spin-torque (ST)-induced magnetization dynamics, was performed using micromagnetic modeling. In particular, we show that the measured DC response at the above-mentioned fractional frequencies can be explained by a low-order nonlinearity and strong magnetodipolar feedback between magnetic layers adjacent to an MgO barrier. Additionally, we determined that the simulated harmonic response is strongly enhanced by the mutual ST effect between these layers. Finally, we demonstrate that the read sensors' nonlinear magnetization dynamics and, by extension, their harmonic response are highly sensitive to various magnetic and ST parameters. Thus, this study shows that using NVNA measurements in conjunction with micromagnetic modeling can clarify the uncertainty in the definition of these parameters.

Nonlinear Regulation of Enzyme-Free DNA Circuitry with Ultrasensitive Switches.

DNA is used to construct synthetic chemical reaction networks (CRNs), such as inorganic oscillators and gene regulatory networks. Nonlinear regulation with a simpler molecular mechanism is particularly important in large-scale CRNs with complex dynamics, such as bistability, adaptation, and oscillation of cellular functions. Here we introduce a new approach based on ultrasensitive switches as modular regulatory elements to nonlinearly regulate DNA-based CRNs. The nonlinear behavior of the systems can be finely tuned by programmable regulation of the linker length and the ligand binding sites, of which the Hill coefficients (nH) are in the range of 1.00-2.32. By integrating two different strand displacement reactions with low-order nonlinearities (nH ≈ 1.44 and 1.54), we could construct CRNs exhibiting high-order nonlinearities with Hill coefficients of up to ∼2.70. In addition, this could provide an efficient approach for designing CRNs at will with complex chemical dynamics by incorporating our design with previously developed enzyme-free DNA circuits.

Pair-cat codes: autonomous error-correction with low-order nonlinearity

We introduce a driven-dissipative two-mode bosonic system whose reservoir causes simultaneous loss of two photons in each mode and whose steady states are superpositions of pair-coherent/Barut-Girardello coherent states. We show how quantum information encoded in a steady-state subspace of this system is exponentially immune to phase drifts (cavity dephasing) in both modes. Additionally, it is possible to protect information from arbitrary photon loss in either (but not simultaneously both) of the modes by continuously monitoring the difference between the expected photon numbers of the logical states. Despite employing more resources, the two-mode scheme enjoys two advantages over its one-mode cat-qubit counterpart with regards to implementation using current circuit QED technology. First, monitoring the photon number difference can be done without turning off the currently implementable dissipative stabilizing process. Second, a lower average photon number per mode is required to enjoy a level of protection at least as good as that of the cat-codes. We discuss circuit QED proposals to stabilize the code states, perform gates, and protect against photon loss via either active syndrome measurement or an autonomous procedure. We introduce quasiprobability distributions allowing us to represent two-mode states of fixed photon number difference in a two-dimensional complex plane, instead of the full four-dimensional two-mode phase space. The two-mode codes are generalized to multiple modes in an extension of the stabilizer formalism to non-diagonalizable stabilizers. The M-mode codes can protect against either arbitrary photon losses in up to M − 1 modes or arbitrary losses and gains in any one mode.

Spectral Formulation for Geometrically Exact Beams: A Motion-Interpolation-Based Approach

This paper proposes a novel spectral formulation for geometrically exact beams based on motion interpolation schemes. Motion interpolation schemes based on matrix, quaternion, and geodesic metrics yields simple expressions for the sectional strains and linearized strain–motion relationships at the mesh nodes. Consequently, the expressions for the nodal forces and tangent stiffness matrices of spectral elements are simplified dramatically. Furthermore, the motion formalism is used to describe the kinematics of the problem, leading to equations of motion that present low-order nonlinearities. Spectral elements based on Gauss–Lobatto and Gauss quadrature rules are investigated. For both cases, the combination of the spectral formulation with the motion formalism leads to geometrically exact beam elements that are much simpler to implement than their counterparts based conventional finite element interpolation schemes using a classical description of kinematics. A global parameterization-free generalized- scheme is used to integrate the equations in time. Numerical examples demonstrate the accuracy of the proposed formulation. The elements based on Gauss–Lobatto quadrature rules are shown to suffer from axial and shear locking, whereas those based on Gauss rules are locking-free. The convergence rate of -node spectral elements based on Gauss quadrature rules is about to for all three interpolation schemes. As the number of elements increases, the proposed formulation becomes more accurate than its conventional counterpart.

Spectral collocation methods for the periodic solution of flexible multibody dynamics

Many flexible multibody systems of practical interest exhibit a periodic response. This paper focuses on the implementation of the collocation version of the Fourier spectral method to determine periodic solutions of flexible multibody systems modeled via the finite element method. To facilitate the analysis and obtain governing equations presenting low-order nonlinearities, the motion formalism is adopted. Application of Fourier spectral methods requires global interpolation schemes that approximate the unknown fields over the entire period of response with exponential convergence characteristics. The classical spectral interpolation schemes were developed for linear fields and hence do not apply to the nonlinear configuration manifolds, such as $$\mathrm {SO}(3)$$ or $$\mathrm {SE}(3)$$ , that are used to describe the kinematics of multibody systems. Furthermore, the configuration and velocity fields are related through nonlinear kinematic compatibility equations. Clearly, special procedures must be developed to adapt the Fourier spectral approach to flexible multibody systems. The spectral interpolation of motion is investigated; interpolation schemes based on the polar decomposition are proposed. Assembly of the linearized governing equations at all the grid points leads to the governing equations of the spectral method. Numerical examples illustrate the performance of the proposed approach.

A new method for parameter estimation of high-order polynomial-phase signals

Parameter estimation of a high-order polynomial phase signal (PPS) is considered in this paper. We propose a method to estimate phase parameters more efficiently and accurately. In the proposed method, we define an operator referred to as non-uniform sampled reducing-order operator (NURO) to reduce the order of a polynomial phase by half, when the order of PPS is even. By combined using NURO and phase differentiation (PD) operators, the PPS order is reduced to one, i.e., the PPS degenerates into a complex sinusoid. Then, the parameter estimation can be done by jointly using fast Fourier transform (FFT) and one-dimensional search. Compared with the traditional methods, the reducing-order procedure in the proposed method has lower-order nonlinearities. Simulation results show that the proposed method outperforms the hybrid CPF-HAF and HAF in both mean square error (MSE) and the threshold when the PPS order is higher than 5.

A region-segmentation combining surrogate model based on L-indicator and N-fold cross-validation technique

ABSTRACTFor complex engineering problems, for which the mathematical models may be linear, low-order nonlinear or even high-order nonlinear, surrogate models which have high adaptability and accuracy are required. This article develops a method for constructing a region-segmentation combining surrogate model. It is based on the idea that in the entire experimental domain, different local regions may present different characteristics (linearity, low-order nonlinearity and high-order nonlinearity), and the entire domain should be divided into several subregions to be approximated by different surrogates so as to achieve high prediction accuracy in local regions. The preferred models in each subregion then constitute a weight-average combining surrogate model. The investigations reveal that the new model not only is more adaptive to analytically unknown functions, but also gives more accurate predictions. The method has been applied to three benchmark problems and a practical engineering problem, and the results maintain validity.

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

ABSTRACTIn this paper, the problem of global stabilisation by state feedback is investigated for a class of stochastic high-order nonlinear systems with both high-order and low-order nonlinearities, to which the existing control methods are inapplicable. Based on the generalised stochastic Lyapunov theorem, and by skillfully using the method of adding a power integrator, a continuous state feedback controller is successfully constructed, which can guarantee the global asymptotic stability in probability of the resulting closed-loop system in the sense of weak solution, and also is able to lead to an interesting result of finite-time stabilisation under appropriate conditions. Finally, two simulation examples are provided to demonstrate the effectiveness of the proposed approach.

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

In this paper, we study the global output feedback stabilisation of a class of high-order nonlinear systems with more general low-order and high-order nonlinearities. By constructing the novel Lyapunov function and observer, based on the homogeneous domination theory together with adding a power integrator method, an output feedback controller is developed to guarantee the equilibrium of the closed-loop system globally uniformly asymptotically stable.