Non-smooth Nonlinear Systems Research Articles

Performance Improvement of Three-Spring Quasi-Zero-Stiffness Isolator by Shape Memory Alloy Damper

Quasi-zero stiffness vibration isolators (QZSI) can effectively resolve the contradiction between the load bearing and the vibration isolation for low-frequency vibration systems. The three-spring QZSI is a classical QZSI. However, constrained by its geometrical relationship, the working stroke of the QZSI is limited, and the isolation performance will deteriorate when the displacement response is significant. To solve this question, the shape memory alloy (SMA) damper, characterized by lower secondary stiffness, partially replaces the positive-stiffness spring of the three-spring QZS isolator to widen its effective working stroke, which forms a novel SMA-QZSI. Based on a piecewise linear constitutive model of SMA, a theoretical model of the force–displacement relationship of SMA-QZSI is established. Subsequently, the approximate analytic amplitude–frequency response equation of the non-smooth nonlinear SMA-QZSI system under harmonic excitation is obtained by the average method and verified numerically. In addition, the effects of mechanical parameters of SMA springs and external excitation parameters on mechanical behaviors and low-frequency isolation performance of the SMA-QZSI isolator are discussed. Finally, the static and dynamic experiments of SMA-QZSI are conducted to verify its force–displacement relationship and isolation performance. The results show that the proposed SMA-QZSI has remarkable low-frequency vibration isolation performance.

Comparison of Genetic Algorithm and Particle Swarm Optimization in Determining the Solution of Nonlinear System of Equations

Nonlinear systems of equations often appear in various fields of science and engineering, but their analytical solutions are difficult to find, so numerical methods are needed to solve them. Optimization algorithms are very effective in finding solutions to nonlinear systems of equations especially when traditional analytical and numerical methods are difficult to apply. Two popular optimization methods used for this purpose are Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). This study aims to compare the effectiveness of GA and PSO in finding solutions to nonlinear systems of equations. The criteria used for comparison include accuracy and speed of convergence. This research uses several examples of nonsmooth nonlinear systems of equations for experimentation and comparison. The results provide insight into when and how to effectively use these two algorithms to solve nonlinear systems of equations as well as their potential combinations

Nonlinearly preconditioned semismooth Newton algorithms for nonlinear nonsmooth systems

We aim to develop efficient and robust algorithms for nonsmooth nonlinear systems arising from complementarity problems. The semismooth Newton algorithm is popular due to its reliability and efficiency. However, it struggles with issues with imbalanced nonlinearities of the problems, leading to degraded convergence rates or failure despite help from the globalization techniques like linesearch or trust region. We introduce a right nonlinearly preconditioned semismooth Newton algorithm to address this difficulty. The critical success ingredient is that before each global Newton update, a nonlinear preconditioning step implicitly removes the so-called ‘bad components’ causing trouble via nonlinear subspace correction, inspired by Gaussian elimination but adapted nonlinearly to balance system nonlinearities. Additionally, our method integrates with a domain decomposition framework, enhancing parallelism. Numerical results on two classes of problems demonstrate significantly improved convergence over standard semismooth Newton methods.

Output-Feedback Adaptive Neural Network Control for Uncertain Nonsmooth Nonlinear Systems With Input Deadzone and Saturation.

Nonsmooth nonlinear systems can model many practical processes with discontinuous property and are difficult to be stabilized by classical control methods like smooth nonlinear systems. This article considers the output-feedback adaptive neural network (NN) control problem for nonsmooth nonlinear systems with input deadzone and saturation. First, the nonsmooth input deadzone and saturation is converted to a smooth function of affine form with bounded estimation error by means of the mean-value theorem. Second, with the help of approximation theorem and Filippov's differential inclusion theory, the given nonsmooth system is converted to an equivalent smooth system model. Then, by introducing a proper logarithmic barrier Lyapunov function (BLF), an output-feedback adaptive NN strategy is set up by constructing an appropriate observer and adopting the adaptive backstepping technique. A new stability criterion is established to guarantee that all the signals in the closed-loop system are semiglobally uniformly ultimately bounded (SGUUB). Finally, comparative simulations through Chua's oscillator are offered to verify the effectiveness of the proposed control algorithm.

Improving the monodromy matrix computation in pathfollowing schemes for nonsmooth dynamics

An enhanced pathfollowing strategy was recently proposed employing a modified Newton–Raphson solver accelerated by a low-cost routine via a Krylov subspace iterator. One of the key points of the numerical strategy is the computation of the Jacobian of the Poincaré map (i.e., the monodromy matrix), which, by serving as iteration matrix, governs both the accuracy and robustness of the whole strategy. In the context of nonsmooth nonlinear dynamic systems, this matrix can only be computed via numerical differentiation, which leads to subtractive cancellation errors.A new technique is here proposed by performing a central finite difference of the vector field associated with the ODE problems within the time-step integration to assemble the monodromy matrix. This technique allows us to bound the errors induced by the standard numerical differentiation.By means of a wide numerical campaign for meaningful multi-dof dynamical systems, the enhanced pathfollowing scheme is shown to yield a higher level of accuracy than the standard one, as well as higher robustness, with improved convergence properties for the whole numerical strategy.The C++ code implementing the proposed methodology is freely available at https://zenodo.org/record/7245478, and includes code for both the classic method as well as the new approaches for the computation of the Jacobian matrix.

Dynamical analysis and numerical verification of a non-smooth nonlinear energy sink

The performance of the vibration reduction of a lightweight non-smooth nonlinear energy sink (NES) attached to a two-story linear damped primary structure (PS) is studied through dynamical analysis and numerical verification. First, a three-degree-of-freedom non-smooth nonlinear system is formulated to describe the coupled dynamics involving the PS and non-smooth NES, in which nonlinearity is induced by the piecewise linear restoring force of the NES and described by a non-smooth function. Second, when a 1:1:1 internal resonance case of the coupled system is considered, a modified complex-average technique and the method of multiple scales are employed to obtain a set of algebraic equations between the vibration amplitudes of the PS and attached non-smooth NES. The stability conditions and initial critical energy for activating targeted energy transfer (TET) can be analytically obtained. Finally, the linear stiffness, equivalent viscous damping coefficients and impact clearance of the non-smooth NES are numerically optimized by the peak estimates. The initial critical energy for the TET of NES is numerically verified to show the excellent performance of vibration reduction for the non-smooth NES by comparing with other attached dissipation devices under a wide range of impulsive loads.

Command filter‐based adaptive neural network controller design for uncertain nonsmooth nonlinear systems with output constraint

SummaryThis paper investigates the command filter‐based adaptive neural network tracking control problem for uncertain nonsmooth nonlinear systems. First, an integral barrier Lyapunov function is introduced to deal with the symmetric output constraint and make the output comply with prescribed restrictions. Second, by the Filippov's differential inclusion theory and approximation theorem, the considered nonsmooth nonlinear system is converted to an equivalent smooth nonlinear system. Third, the Levant's differentiator is used to deal with the “explosion of complexity” problem. An error compensation mechanism is established to attenuate the effect of the filtering error on control performance. Then, an adaptive neural network controller is set up by resorting to the backstepping technique. It is strictly mathematically proved that the tracking error can converge to an arbitrarily small neighborhood of the origin and all the signals in the closed‐loop system are semi‐globally uniformly ultimately bounded. Finally, a numerical example and an application example of the robotic manipulator system are provided to demonstrate the availability of the proposed control strategy.

Design of nonsmooth Kalman filter for compound sandwich systems with backlash and dead zone

AbstractIn this article, a new nonsmooth Kalman filtering (KF) method is proposed for noise suppression of the compound sandwich with backlash and dead zone. Based on the characteristics of the system, a nonsmooth stochastic state‐space equation is constructed. Then, nonsmooth KF is developed to adapt the nonsmooth stochastic state space model. The filter can switch automatically among different operating zones according to the operating conditions. Moreover, the convergence of the switchable compound nonsmooth KF is discussed. Subsequently, a motor‐driving mechanical transmission system which can be described as a compound nonsmooth nonlinear sandwich system with backlash and dead zone is studied as an application case. Finally, the comparison between the proposed filtering scheme and conventional filter are presented. It is demonstrated that the nonsmooth KF has a better performance than the conventional one in terms of convergence and accuracy in state estimation under noisy environment.

A 6M digital twin for modeling and simulation in subsurface reservoirs

A 6M digital twin for modeling and simulation in subsurface reservoirs

Bifurcation of multi-stable behaviors in a two-parameter plane for a non-smooth nonlinear system with time-varying parameters

To obtain the correlation between multiple parameters, multi-initial values and multi-stable behaviors for a non-smooth nonlinear system with time-varying parameters, a new method of calculating the bifurcation of multi-stable behaviors in the parametric plane is first proposed based on Poincare mapping theory, Lyapunov theory and Floquot theory. The bifurcation and distribution of multi-stable behaviors of a nonlinear gear system with time-varying meshing stiffness in a two-parameter plane are studied by using the proposed method. Various multi-stable behaviors and potential hidden bifurcation curves are fully revealed. Double-bifurcation points formed by the intersection of two different bifurcation curves are further investigated. The probability of occurrence of hidden bifurcation curve is calculated and analyzed based on statistical theory. Results indicate that saddle-node bifurcation curves are sensitive to the initial value and change both the type of multi-stable behaviors and the topology of the attraction basin. However, period-doubling bifurcation curves are not sensitive to the initial value, and only change the type of multi-stable behavior, but do not greatly change the topology of the attraction basin. Four different types of multi-stable behaviors are observed around double-bifurcation points. Multi-stable behaviors and bifurcation curves are easily hidden in the parametric plane due to their small occurring probabilities.

A modulus-based nonmonotone line search method for nonlinear complementarity problems

A modulus-based nonmonotone line search method is proposed for nonlinear complementarity problem. The considered problem is first reformulated to a nonsmooth nonlinear system based on the modulus-based decomposition. Then a nonmonotone line search method using simulated annealing rule is generalized to solve the resulting system. The global convergence of the proposed method is established under some suitable assumptions. Preliminary numerical experiments show that, compared with some existing methods, the proposed method is feasible and effective.

Adaptive Neural Backstepping Control Design for A Class of Nonsmooth Nonlinear Systems

This paper investigates the problems of backstepping control for a class of nonsmooth nonlinear systems. With the help of set-valued functions (or maps) and set-valued derivatives, a new Lyapunov criterion is developed for nonsmooth systems. The concept of semi-globally uniformly ultimately bounded (SGUUB) stability that is widely used for smooth nonlinear systems in lower triangular form is, for the first time, applied to the nonsmooth case, and such a concept provides a theory foundation for the subsequent backstepping control design. Then, two types of continuous controllers are designed based on the backstepping technique and adaptive neural network (NN) algorithm. In the light of Cellina approximate selection theorem and smooth approximation theorem for Lipschitz functions, the system under investigation is first transformed into an equivalent model. Then, the first type of controller can be efficiently designed by using the traditional backstepping technique. On the other hand, when the unknown uncertainty is taken into account in the equivalent systems, another type of controller is designed based on adaptive NN control strategy. It is proved that in both cases, all the closed-loop signals are SGUUB by using our proposed controllers. Finally, a numerical example with two types of controllers is supplied to show the availability of the provided control schemes.

Parallel reservoir simulators for fully implicit complementarity formulation of multicomponent compressible flows

The numerical simulation of multicomponent compressible flow in porous media is an important research topic in reservoir modeling. Traditional semi-implicit methods for such problems are usually conditionally stable, suffer from large splitting errors, and may accompany with violations of the boundedness requirement of the numerical solution. In this study we reformulate the original multicomponent equations into a nonlinear complementarity problem and discretize it using a fully implicit finite element method. We solve the resultant nonsmooth nonlinear system of equations arising at each time step by a parallel, scalable, and nonlinearly preconditioned semismooth Newton algorithm, which is able to preserve the boundedness of the solution and meanwhile treats the possibly imbalanced nonlinearity of the system. Some numerical results are presented to demonstrate the robustness and efficiency of the proposed algorithm on the Tianhe-2 supercomputer for both standard benchmarks as well as realistic problems in highly heterogeneous media.

Adaptive Backstepping Control Design for Uncertain Non-smooth Strictfeedback Nonlinear Systems with Time-varying Delays

This paper is concerned with the problem of adaptive neural tracking control for uncertain non-smooth nonlinear time-delay systems with a class of lower triangular form. Based on Filippov’s theory, the bounded stability and asymptotic stability are extended to the ones for the considered systems, which provides the theory foundation for the subsequent adaptive control design. In the light of Cellina approximate selection theorem and smooth approximation theorem for Lipschitz functions, the system under investigation is first transformed into an equivalent system model, based on which, two types of controllers are designed by using adaptive neural network (NN) algorithm. The first designed controller can guarantee the system output to track a target signal with bounded error. In order to achieve asymptotic tracking performance, the other type of controller with proportional-integral(PI) compensator is then proposed. It is also noted that by exploring a novel Lyapunov-Krasovskii functional and designing proper controllers, the singularity problem frequently encountered in adaptive backstepping control methods developed for time-delay nonlinear systems with lower triangular form is avoided in our design approach. Finally, a numerical example is given to show the effectiveness of our proposed control schemes.

Estimating the robust domain of attraction for non-smooth systems using an interval Lyapunov equation

The Lyapunov equation allows finding a quadratic Lyapunov function for an asymptotically stable fixed point of a linear system. Applying this equation to the linearization of a nonlinear system can also prove the exponential stability of its fixed points. This paper proposes an interval version of the Lyapunov equation, which allows investigating a given Lyapunov candidate function for non-smooth nonlinear systems inside an explicitly given neighborhood, leading to rigorous estimates of the domain of attraction (EDA) of exponentially stable fixed points. These results are developed in the context of uncertain systems. Experiments are presented, which show the interest of the approach including with respect to usual approaches based on sum-of-squares for the computation of EDA.

Fuzzy-Approximation-Based Adaptive Output-Feedback Control for Uncertain Nonsmooth Nonlinear Systems

This paper proposes a solution to adaptive output-feedback control for a class of nonsmooth nonlinear systems. First, the concept of semiglobally uniformly ultimately bounded (SGUUB) stability that has been widely used for smooth nonlinear systems with lower triangular systems is extended to the nonsmooth systems. Then by resorting to set-valued maps and set-valued derivatives, a new Lyapunov criterion ensuring the SGUUB stability is developed for nonsmooth nonlinear systems, which establishes the theory foundation for the subsequent backstepping control design. With the help of Cellina approximate selection theorem and smooth approximation theorem for Lipschitz functions, the system under investigation is first transformed into an equivalent model. In the sequel, exploring some efficient techniques, an adaptive fuzzy output-feedback controller is constructed for the systems under consideration by utilizing an appropriate observer and the approximation ability of fuzzy systems. Finally, a numerical example is given to show the effectiveness of the proposed control method.

Midwives in the Byron Shire: Managing challenging conversations about vaccination

The Lyapunov equation allows finding a quadratic Lyapunov function for an asymptotically stable fixed point of a linear system. Applying this equation to the linearization of a nonlinear system can also prove the exponential stability of its fixed points. This paper proposes an interval version of the Lyapunov equation, which allows investigating a given Lyapunov candidate function for non-smooth nonlinear systems inside an explicitly given neighborhood, leading to rigorous estimates of the domain of attraction (EDA) of exponentially stable fixed points. These results are developed in the context of uncertain systems. Experiments are presented, which show the interest of the approach including with respect to usual approaches based on sum-of-squares for the computation of EDA.

Probabilistic response of nonsmooth nonlinear systems under Gaussian white noise excitations

This paper proposes an approach to obtain the steady state probability density function (PDF) of nonsmooth nonlinear dynamical systems driven by the Gaussian white noise. The steady state PDF solution is assumed to contain two probability flows obtained by the detailed balance method. The weighted mean squared residuals of the error of the FPK equation are minimized with respect to the unknown coefficients in the assumed solution. The integrability of the weighted mean squared residuals is found to provide a means to deal with the singularities in the steady state PDF that arise from the nonsmooth dynamics of the nonlinear stochastic system. The accuracy of the proposed approach is estimated by the Monte Carlo simulations via two examples.

Redefinición de la función de complementariedad de Kanzow

En los ´ultimos a˜nos, ha aumentado la investigaci´on relacionada con la b´usqueda de algoritmos eficientes que resuelvan el problema de complementariedad no lineal mediante su reformulaci´on como un sistema de ecuaciones no lineales, no diferenciable, usando las llamadas funciones de complementariedad. En este art´ıculo, consideramos la funci´on de complementariedad uniparam´etrica propuesta en [11] y proponemos una nueva forma de definirla mediante una forma cuadr´atica, sim´etrica y definida positiva. Con esta nueva definici´on, demostramos que la funci´on est´a bien definida y algunas de sus propiedades. Adem´as, encontramos cotas de gran utilidad en el desarrollo de teor´ıas de convergencia (local y global) de algoritmos que resuelven el problema de complementariedad no lineal.

Experimental feedback linearisation of a non-smooth nonlinear system by the method of receptances

Input–output partial feedback linearisation is experimentally implemented on a non-smooth nonlinear system without the necessity of a conventional system matrix model for the first time. The experimental rig consists of three lumped masses connected and supported by springs with low damping. The input and output are at the first degree of freedom with a non-smooth clearance-type nonlinearity at the third degree of freedom. Feedback linearisation has the effect of separating the system into two parts: one linear and controllable and the other nonlinear and uncontrollable. When control is applied to the former, the latter must be shown to be stable if the complete system is to be stable with the desired dynamic behaviour.