Asymmetric Nonlinearity Research Articles

Parameter design of a multi-delayed isolator with asymmetrical nonlinearity

In this study, based on the analysis of multiple nonlinear vibration properties, the criterions of parameter design for different types of excitations and structural nonlinearities are proposed for vibration isolation. For a nonlinear vibration isolation system with Quasi-Zero-Stiffness property, the assembly process and stiffness property with different masses of isolation object are taken into account. In order to improve the isolation effectiveness in a wide frequency band, the relation between vibration performances and parameters is established. The effect of adjustable parameters on the vibration features and the effect mechanisms of multiple time delays are discussed in details based on the method of multiple scales. Because the study takes full consideration of the effect of structural nonlinearity and transition mechanism of vibration performances, the proposed criterions for parameter design could solve the conflict between the improvement of isolation effectiveness and complexity or bifurcation brought by nonlinearity. The results in this paper show that without changing the High-Static-Low-Frequency advantage brought by the nonlinear isolation structure, the isolation effectiveness could be improved in a wide frequency band.

Further results on open-loop compensation of rate-dependent hysteresis in a magnetostrictive actuator with the Prandtl-Ishlinskii model

Abstract Apart from the output-input hysteresis loops, the magnetostrictive actuators also exhibit asymmetry and saturation, particularly under moderate to large magnitude inputs and at relatively higher frequencies. Such nonlinear input-output characteristics could be effectively characterized by a rate-dependent Prandtl-Ishlinskii model in conjunction with a function of deadband operators. In this study, an inverse model is formulated to seek real-time compensation of rate-dependent and asymmetric hysteresis nonlinearities of a Terfenol-D magnetostrictive actuator. The inverse model is formulated with the inverse of the rate-dependent Prandtl-Ishlinskii model, satisfying the threshold dilation condition, with the inverse of the deadband function. The inverse model was subsequently applied to the hysteresis model as a feedforward compensator. The proposed compensator is applied as a feedforward compensator to the actuator hardware to study its potential for rate-dependent and asymmetric hysteresis loops. The experimental results are obtained under harmonic and complex harmonic inputs further revealed that the inverse compensator can substantially suppress the hysteresis and output asymmetry nonlinearities in the entire frequency range considered in the study.

Decentralized adaptive neural control for interconnected stochastic nonlinear delay-time systems with asymmetric saturation actuators and output constraints

This paper investigates the problem of decentralized adaptive backstepping control for a class of large-scale stochastic nonlinear time-delay systems with asymmetric saturation actuators and output constraints. Firstly, the Gaussian error function is employed to represent a continuous differentiable asymmetric saturation nonlinearity, and barrier Lyapunov functions are designed to ensure that the output parameters are restricted. Secondly, the appropriate Lyapunov–Krasovskii functional and the property of hyperbolic tangent functions are used to deal with the unknown unmatched time-delay interactions, and the neural networks are employed to approximate the unknown nonlinearities. At last, based on Lyapunov stability theory, a decentralized adaptive neural control method is proposed, and the designed controller decreases the number of learning parameters. It is shown that the designed controller can ensure that all the closed-loop signals are 4-Moment (or 2 Moment) semi-globally uniformly ultimately bounded (SGUUB) and the tracking error converges to a small neighborhood of the origin. Two examples are provided to show the effectiveness of the proposed method.

Adaptive output feedback tracking control for switched non‐strict‐feedback non‐linear systems with unknown control direction and asymmetric saturation actuators

The problem of adaptive tracking control is studied for a class of switched non-strict-feedback non-linear systems with unknown control direction and asymmetric saturation actuators under arbitrary switchings. The main technical difficulty comes from developing a common adaptive control scheme in the presence of unknown control direction and unknown asymmetric saturation non-linearities. A linear state transformation is exploited to deal with the unknown control coefficients and a Gaussian error function-based smooth model is used to approximate the asymmetric saturation non-linearity. Based on an input-driven observer, a common adaptive output-feedback control strategy containing only one adaptive parameter is developed for such systems via a novel combination of common Lyapunov function method, backstepping technique, variable separation approach and neural network approximation. Under the proposed control scheme, the tracking error converges to an adjustable neighbourhood of the origin. Finally, two illustrative examples are included to verify the effectiveness of the proposed control design.

Adaptive neural control for stochastic pure‐feedback non‐linear time‐delay systems with output constraint and asymmetric input saturation

In this study, the adaptive tracking control is investigated for a class of stochastic pure-feedback non-linear time-delay systems with output constraint and asymmetric input saturation non-linearity. First, the Gaussian error function is employed to represent a continuous differentiable asymmetric saturation model, and the barrier Lyapunov function is designed to cope with the output constraints. Then, the appropriate Lyapunov–Krasovskii functional and the property of hyperbolic tangent functions are used to address the effects of the unknown time-delay terms, and the neural network is employed to approximate the unknown non-linearities. At last, based on Lyapunov stability theory, a robust adaptive neural controller is proposed, which decreases the number of learning parameters and thus avoids the over-estimation problem. Under the designed neural controller, all the closed-loop signals are guaranteed to be 4-moment (or 2 moment) semi-globally uniformly ultimately bounded and the tracking error converges to a small neighbourhood of the origin for bounded initial conditions. Two simulation examples are presented to further illustrate the effectiveness of the designed method.

Trajectory tracking control for a marine surface vessel with asymmetric saturation actuators

This paper investigates the problem of trajectory tracking control for an unmanned marine surface vessel (MSV) with external disturbances and asymmetric saturation actuators. An adaptive radial basis function neural network (RBFNN) is constructed to provide an estimation of the unknown disturbances and is applied to design the trajectory tracking controller through a backstepping technique. To handle the effect of nonsmooth asymmetric saturation nonlinearity, a Gaussian error function-based continuous differentiable asymmetric saturation model is employed such that the backstepping technique can be used in the control design. It is proved that all the states in the closed-loop system are semiglobally uniformly ultimately bounded, and the tracking error converges to a small neighborhood of origin by appropriately choosing design parameters. Simulation results and comparisons illustrate the effectiveness of the proposed controller and its robustness to external disturbances and asymmetric saturation actuators.

Fuzzy Approximator Based Adaptive Dynamic Surface Control for Unknown Time Delay Nonlinear Systems With Input Asymmetric Hysteresis Nonlinearities

In this paper, a fuzzy approximator based adaptive dynamic surface control scheme is proposed for a class of unknown time delay nonlinear systems preceded by asymmetric hysteresis nonlinearities. The features of the proposed method are: 1) by combining the approximated property of the fuzzy logic systems (FLSs) with the finite covering lemma, the Krasovskii functionals are disposed of, achieving the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${L}_{{\infty }}$ </tex-math></inline-formula> norm of the tracking error by using the initializing technique; 2) the assumptions on the time-delay functions are removed due to the use of the finite covering lemma and the FLSs; and 3) the proposed adaptive fuzzy dynamic surface control scheme can also compensate the asymmetric shifted Prandtl-Ishlinskii (ASPI) hysteresis without constructing the inverse of the ASPI model with the density function of ASPI model being unknown and estimated online to compensate the hysteresis. It is proved that all the signals in the closed-loop system are ultimately uniformly bounded and can be made arbitrarily small. Simulation results show the validity of the proposed method.

Direct identification of generalized Prandtl–Ishlinskii model inversion for asymmetric hysteresis compensation

In this study, we present an identification-based direct construction of the inverse generalized Prandtl–Ishlinskii (P–I) model to facilitate inverse model-based feedforward compensation of asymmetric hysteresis nonlinearities. Compared with the derivation of the inverse model analytically from a generalized P–I model, this direct modeling approach has the following advantages. First, direct inverse model identification is formulated as a nonlinear optimization problem, which is not subject to the constraint condition on the generalized P–I model's threshold and density functions, where this is indispensable for the analytical model inversion procedure. Second, this approach may be a simple and attractive alternative when the identification precision of a generalized P–I model is limited by the constraint condition, which necessarily results in insufficient hysteresis compensation functionality for the analytically derived inverse model. Finally, direct inverse model identification can overcome the drawbacks of the analytical inversion method, including the accumulation of parameter estimation errors in an analytical inverse model because these parameters are computed from the generalized P–I model's parameters in a recursive manner. Our experimental results demonstrated that the implementation of open-loop control with the directly identified inverse generalized P–I model as a feedforward compensator achieved precise compensation for the asymmetric hysteresis nonlinearities of a piezoelectric stack actuator.

Periodic solutions of planar Hamiltonian systems with asymmetric nonlinearities

In this paper, we look for periodic solutions of planar Hamiltonian systems \t\t\t{x′=f(y)+p1(t,y),y′=−g(x)+p2(t,x).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\left \\{ \\textstyle\\begin{array}{l} x'=f(y)+p_{1}(t,y),\\\\ y'=-g(x)+p_{2}(t,x). \\end{array}\\displaystyle \\right . $$\\end{document} By using the Poincaré-Birkhoff twist theorem, we prove the existence and multiplicity of periodic solutions of the given system when f satisfies an asymmetric condition and the related time map satisfies an oscillating condition.

Adaptive tracking control for switched strict-feedback nonlinear systems with time-varying delays and asymmetric saturation actuators

This paper focuses on the problem of adaptive tracking control for a class of switched strict-feedback nonlinear systems with unknown time-varying delays and asymmetric saturation actuators under arbitrary switching. Especially, the considered time-varying delays absolutely depend on the subsystem number. The main technical difficulties lie in finding an appropriate common Lyapunov function (CLF) for all subsystems and designing a common adaptive control scheme in the presence of unknown time-varying delays and asymmetric saturation nonlinearities. Based on a novel combination of Lyapunov–Razumikhin method, dynamic surface control (DSC) technique, variable separation approach and neural network (NN) approximation, a simple quadratical CLF is constructed and a common adaptive control scheme involving only one adaptive parameter is developed. The proposed controller guarantees that all signals of closed-loop system are semi-globally uniformly ultimately bounded (SGUUB) while the tracking error converges to an adjustable neighborhood of the origin. Finally, the effectiveness of the design methodology is illustrated with two simulation examples.

Iterative solution of the dynamic responses of locally nonlinear structures with drift

Considering the operating point drift in the dynamic responses of locally nonlinear structures, a new iterative method based on describing functions is introduced to solve for the steady-state frequency response. Drift in the operating point arises in the presence of asymmetric nonlinearity, pre-deformation or static loads. In this study, the internal nonlinear forces are expressed using describing functions. The complex equations governing the responses of multiple frequency components are established and are iteratively solved using the Inverse Matrix Update Method. The nonlinear frequency responses can thus be rapidly obtained, and the validity of the method is verified by simulations. The presented method can be applied to large-scale structures with multiple nonlinear elements.

Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents

Motivated by the statistical mechanics description of stationary 2D-turbulence, for a sinh-Poisson type equation with asymmetric nonlinearity, we construct a concentrating solution sequence in the form of a tower of singular Liouville bubbles, each of which has a different degeneracy exponent. The asymmetry parameter \begin{document} $γ∈(0, 1]$ \end{document} corresponds to the ratio between the intensity of the negatively rotating vortices and the intensity of the positively rotating vortices. Our solutions correspond to a superposition of highly concentrated vortex configurations of alternating orientation; they extend in a nontrivial way some known results for \begin{document} $\gamma=1$ \end{document} . Thus, by analyzing the case \begin{document} $\gamma≠1$ \end{document} we emphasize specific properties of the physically relevant parameter \begin{document} $\gamma$ \end{document} in the vortex concentration phenomena.

Design of state and parametric estimation algorithms for nonlinear systems with hysteresis-saturation nonlinearities

The paper deals with the parameter estimation problem of Wiener state-space models with hysteresis-saturation nonlinearities. A recursive parametric and state estimation algorithm is presented for the Wiener system by combining the adjustable model idea, the least squares technique and the Kalman filter principle. The basic idea is to decompose the hysteresis-saturation nonlinearity into two asymmetric saturation nonlinearities and to estimate jointly the state variables, the parameters and the internal variable of the considered Wiener model using the available input-output data. The proposed recursive algorithm can be extended to nonlinear systems with other hard nonlinearities.

Design of state and parametric estimation algorithms for nonlinear systems with hysteresis-saturation nonlinearities

The paper deals with the parameter estimation problem of Wiener state-space models with hysteresis-saturation nonlinearities. A recursive parametric and state estimation algorithm is presented for the Wiener system by combining the adjustable model idea, the least squares technique and the Kalman filter principle. The basic idea is to decompose the hysteresis-saturation nonlinearity into two asymmetric saturation nonlinearities and to estimate jointly the state variables, the parameters and the internal variable of the considered Wiener model using the available input-output data. The proposed recursive algorithm can be extended to nonlinear systems with other hard nonlinearities.

Multiplicity of periodic solutions for systems of weakly coupled parametrized second order differential equations

We prove a multiplicity result of periodic solutions for a system of second order differential equations having asymmetric nonlinearities. The proof is based on a recent generalization of the Poincare–Birkhoff fixed point theorem provided by Fonda and Urena.

Asymmetric determinants of CDS spreads: U.S. industry-level evidence through the NARDL approach

This paper investigates the presence of asymmetries in the short- and long-run relationships between the 5-year CDS index spreads at the U.S. industry level and a set of major macroeconomic and financial variables, namely the corresponding industry stock indices, the VIX index, the 5-year Treasury bond yield and the crude oil price, using the NARDL approach. The empirical results provide significant evidence of both short-run and long-run asymmetries in the linkage between ten industry CDS spreads and the potential driving factors common for all industries, confirming the importance of asymmetric nonlinearity in this context. It is also shown that the industry equity prices, the VIX, the 5-year Treasury bond rate and, to a lesser extent, the crude oil price constitute important asymmetric determinants of these U.S. industry CDS spreads. The findings of this study have relevant implications for investors, speculators, arbitrageurs and policy makers interested in credit risk at the industry level.

Modeling, identification and compensation of hysteresis nonlinearity for a piezoelectric nano-manipulator

Hysteresis nonlinearity widely exists in piezoelectric actuated nano-positioning applications, which degrades their tracking accuracy and limits their precision positioning applications. This paper presents a novel hysteresis modeling and compensation approach to alleviate the adverse effect of the asymmetric and rate-dependent hysteresis nonlinearity for a piezoelectric transducer actuated servo stage. By integrating a generalized input function with the play operator of the classical Prandtl–Ishlinskii model, a novel polynomial-based rate-dependent Prandtl–Ishlinskii (PRPI) model is proposed to capture the hysteresis behavior of the piezoelectric positioning stage, where a polynomial function of input and a time rate function of input are introduced to formulate the generalized input function. Meanwhile, a new adaptive differential evolution optimization algorithm is developed to identify the parameters of the proposed PRPI hysteresis model. Based on the PRPI hysteresis model with the identified parameters, an inverse feedforward controller is constructed to achieve the accurate tracking motion. Furthermore, the hysteresis compensation error of the proposed PRPI model is theoretically analyzed. Finally, comparative experiments are conducted, and the experimental results provided in this paper demonstrate the effectiveness and superiority of the proposed inverse PRPI model compensation approach.

On a Robin problem with indefinite weight and asymmetric reaction superlinear at +∞

The existence of at least three nontrivial smooth solutions to a semilinear Robin problem with indefinite potential and asymmetric nonlinearity super-linear at +∞ is established. Both crossing and resonance are allowed. Proofs exploit variational methods, truncation techniques, and Morse theory.

High precision tracking of a piezoelectric positioner using a discrete-time sliding mode control

This paper proposes a new comprehensive control strategy to precisely control a piezoelectric positioner by combining discrete-time sliding mode control (DSMC) with the Prandtl-Ishlinskii hysteresis model. In order to obtain precision tracking control, a direct inverse hysteresis compensation method is firstly adopted to compensate for the asymmetric hysteresis nonlinearity. Due to the existence of hysteresis modeling error, the dynamics behavior of the piezoelectric positioner with hysteresis compensation can be treated as a linear second order plant plus an unknown lumped disturbance term. Then a discrete-time sliding mode controller with a disturbance observer is designed to stabilize the position error and improve the position accuracy. The stability of DSMC and the convergence of the disturbance observer are both carried out. It is shown that the tracking performances are robust to the parametric uncertainties and unknown disturbances. Eventually, different trajectory-tracking experiments are performed, and the comparative experimental results are presented to confirm the significantly better performance of the proposed control strategy.

Control of Gear Transmission Servo Systems With Asymmetric Deadzone Nonlinearity

This brief deals with the output control problem of gear transmission servo systems with asymmetric deadzone nonlinearity between states. To overcome the difficulty of controller design due to the nondifferentiability of the deadzone, a brand new differentiable asymmetric deadzone model is put forward, which provides an additional design degree of freedom to approximate the real deadzone with any prescribed accuracy. Based on the differentiability of the new model, a global differential homeomorphism and a simple state feedback controller are proposed to solve the output control problem. Finally, simulation studies are included to demonstrate the main results of this brief.