Second-order Nonlinear Systems Research Articles

Nonlinearity-induced transition in the nonlinear Su-Schrieffer-Heeger model and a nonlinear higher-order topological system

We study the topological physics in nonlinear Schr\"{o}dinger systems on lattices. We employ the quench dynamics to explore the phase diagram, where a pulse is given to a lattice point and we analyze its time evolution. There are two system parameters $\lambda $ and $\xi $, where $\lambda $ controls the hoppings between the neighboring links and $\xi $ controls the nonlinearity. The dynamics crucially depends on these system parameters. Based on analytical and numerical studies, we derive the phase diagram of the nonlinear Su-Schrieffer-Heeger (SSH) model in the ($\lambda ,\xi $) plane. It consists of four phases. The topological and trivial phases emerge when the nonlinearity $\xi $ is small. The nonlinearity-induced localization phase emerges when $\xi $ is large. We also find a dimer phase as a result of a cooperation between the hopping and nonlinear terms. A similar analysis is made of the nonlinear second-order topological system on the breathing Kagome lattice, where a trimer phase appears instead of the dimer phase.

Distributed Optimization of Second-Order Nonlinear Multiagent Systems With Event-Triggered Communication

In this article, we study the event-triggered distributed optimization problem of second-order nonlinear multiagent systems under undirected and connected communication topologies. To reduce the cost of information exchange in an optimization problem, a distributed event-triggered communication-based algorithm is proposed. In an event-triggered mechanism, the information interaction process of the network is controlled by a set of trigger conditions. The trigger condition of each agent only requires its own state information and its neighboring agents’ state information. Each agent updates its own states only at the triggering time. In order to achieve the optimal result of the optimization problem and reduce the cost of communicatio, we propose a distributed event-triggered algorithm based on gradient descent. Moreover, we analyze the convergence of the algorithm by constructing a suitable Lyapunov function. Furthermore, we prove that the Zeno behavior can be avoid. Finally, we provide examples to validate the effectiveness of the obtained result.

Leader–Follower Affine Formation Control of Second-Order Nonlinear Uncertain Multi-Agent Systems

This brief proposes two leader-follower affine formation control algorithms for a class of second-order uncertain multi-agent systems. The model of each agent is described by the second-order dynamics with nonlinear uncertainties. Due to the high robustness, flexibility and maneuverability, the affine formation approaches based on the stress matrix are adopted to achieve a target formation, in which the formation control laws are designed for followers to keep a desired formation configuration with respect to leaders. First, considering that the velocities of leaders are constant, we design an estimator to compensate for the uncertainties and propose a PI-like affine formation control law to guarantee that the followers can achieve the target formation. Then, for the case that the velocities of leaders are time-varying, we propose another control algorithm with variable gains for followers under which a time-varying formation configuration can be maintained. Finally, the corresponding simulations of a realistic example are carried out to validate the effectiveness of the proposed affine formation control algorithms.

Consensus Sliding-Mode Fault-Tolerant Control for Second-Order Multi-Agent Systems

This study investigates the consensus problem of second-order nonlinear multi-agent systems (MASs) with actuator faults via a sliding mode control approach. The consensus error dynamic is given based on the relative states of the neighbors. Then, a sliding mode surface based on consensus errors is proposed, and the asymptotic stability of the sliding mode is proved using the Lyapunov theory. Furthermore, a sliding-mode fault-tolerant consensus protocol is proposed to compensate for actuator faults. According to the sliding mode control theory, the proposed sliding-mode fault-tolerant controller ensures that the consensus of the MASs can be reached in a finite time. Finally, a simulation example of a second-order multi-robot system is presented to demonstrate the effectiveness of the proposed controller.

An adaptive fuzzy sliding mode control under model uncertainties and disturbances: second-order non-linear system

An adaptive fuzzy sliding mode control under model uncertainties and disturbances: second-order non-linear system

A Comparative Study of Nonsingular Terminal Sliding Mode and Backstepping Schemes for the Coupled Two-Tank System

This paper presents an implementation of two radically different control schemes for a state-coupled two-tank liquid-level system. This is due to the purpose of transferring theoretical studies to industrial systems. The proposed schemes to be introduced and compared are the nonsingular terminal sliding mode control (NTSMC) and the backstepping control (BC). The performances of the developed methods are experimentally tested on a particular class of second-order nonlinear systems. The main purpose of the considered control schemes is to achieve a tracking trajectory for a coupled-tank system. It is proved that the designed robust controllers guarantee the stability of the corresponding closed loop systems. The obtained results are verified with the same setup test to ensure a suitable basis for their comparison. During the experiments, we resorted to adding an integrator to the backstepping control so that we improve the results, leading to the appearance of the integrator backstepping control (IBC). To focus on the adequacy and applicability of the suggested control layout, theoretical comparisons as well as experimental results are afforded and debated.

Prediction and suppression of chaotic instability in brake squeal

Chaotic instability in a vibration phenomenon, known as brake squeal, is investigated including the combined effects of falling friction, sprag-slip and mode coupling. Brake squeal is a high-pitched noise that occurs sometimes when a vehicle is decelerated using disc brakes. The equations of motion for the two dominant coupled modes of the brake system reduce to two autonomous coupled nonlinear second-order systems. The mode coupling instability via friction causes limit cycle behaviour via a supercritical Hopf bifurcation. This limit cycle is shown to break up into chaotic motion characterised by a Poincare map with a less than one-dimensional attractor, similar to that found in a forced dry friction oscillator. For the first time, conservative analytical conditions for brake squeal chaos are developed and verified numerically over a range of sprag angles and brake pressures for fundamental and real brake systems. The predictive model is then used to identify and quantify means to suppress brake squeal chaos to unlikely, excessive friction levels. The results provide predictive insight into conditions under which brake squeal chaos occurs and its suppression.

Compensator-based periodically intermittent control for nonlinear second-order multi-agent systems

This article studies leader-following bipartite consensus control of a class of nonlinear second-order multi-agent systems. A new compensator-based periodically intermittent control algorithm is proposed for the second-order multi-agent systems. The compensator can be implemented in a fully distributed way. In other words, in order to construct the compensator for a certain agents, the values of measurement output of each agent and its neighbor agents are monitored for the construction of compensator. Considering the limited bandwidth of communication networks, the state variable of compensator is used by the controller for each follower to construct a novel periodically intermittent control algorithm, which can effectively reduce the communication frequency among agents in the control process. By combing stability theory and algebra graph theory, the criteria are obtained for the achievement of global exponential bipartite consensus under a fixed directed topology. Finally, some numerical examples are provided to illustrate the performance of the proposed control algorithm.

A novel reset control approach to leader-following consensus of second-order nonlinear multi-agent systems

This paper investigates the leader-following consensus problem of second-order nonlinear multi-agent systems with directed graph. A novel reset control approach is proposed for the aim of improving transient consensus performance, e.g., settling time. By introducing consensus error into reset conditions, the output of reset integrator will keep the same sign with the consensus error, thus, the desired states can be compensated preferentially and the system transient performance is improved accordingly. To appropriately describe the closed-loop system with reset-induced jump dynamics, a hybrid system model consisting of both flow dynamics and jump dynamics is constructed. Based on this model, and combined with backstepping method, Lyapunov-based consensus analysis is presented under hybrid system framework. Finally, a numerical example is provided to show the effectiveness of the obtained results.

Adaptive neural tracking control for stochastic nonlinear multi-agent periodic time-varying systems

This paper investigates the design of adaptive tracking neural controllers for first- and second-order nonlinear multi-agent periodic time-varying systems described based on random differential equations, respectively. First, the Fourier series expansion and neural networks are used as the main tools to describe the nonlinear periodic disturbance dynamics of the follower, which transforms the completely nonlinear problem into a partially linearizable problem. Then, the fully distributed adaptive neural controllers are developed based on adaptive estimation, in which new estimation methods are designed to address periodic time-varying perturbations. Furthermore, based on the stochastic Lyapunov stability theory, it is proved that the tracking errors converge asymptotically to a small neighborhood of zero in the sense of mean square. Finally, the efficicency of the proposed controllers is verified using simulations.

Robust flocking for non-identical second-order nonlinear multi-agent systems

This paper investigates the robust flocking problem for second-order nonlinear systems with a leader and external disturbances. In contrast with most of second-order systems in the literature, the intrinsic dynamics here are nonlinear and non-identical that depend not only on the velocity but also on the position, which is more realistic. Moreover, the interaction topology is undirected and switching. Provided that the leader’s velocity may be constant or time-varying, two distributed flocking control laws have been proposed for two cases to make the differences of the velocities between all followers and the leader approach to zero asymptotically. The proposed distributed flocking control laws are both model-independent which results in the effectiveness of the controllers to cope with the different intrinsic dynamics of the followers and the leader under some assumptions on boundedness of several states. An example is given to illustrate the validity of the theoretical results.

Formation–containment control of multi-agent systems with communication delays

This paper designs formation–containment control algorithms for a class of second-order nonlinear multi-agent systems governed by Euler–Lagrange dynamics with communication delays. The formation–containment problem consists of leader agents’ formation control and follower agents’ containment control. Firstly, to make the leaders form a desired formation and move collectively with a constant velocity, a coordinated formation control algorithm is designed and the variable-gain technique is used to eliminate the effect of communication delays on the leaders’ formation control. Secondly, considering that only the leaders have access to the desired moving velocity, we propose distributed velocity estimators for followers in which the communication delays also exist in the followers’ information interaction. By using the estimated velocity information, coordinated containment control laws are designed for the followers to drive them asymptotically converge to the convex hull spanned by all leaders. Furthermore, to increase the system robustness against uncertainties and external disturbances, the adaptive updating laws are designed for all agents. Finally, simulations are given to demonstrate these obtained results.

Iterative learning control for consensus of second-order multi-agent systems with interval uncertain topologies

Abstract This paper is devoted to the robust consensus tracking problem of second-order nonlinear multi-agent systems (MASs) with the interval uncertain topologies. For the second-order MASs including one leader agent and multiple follower agents, a control protocol is proposed by combining the iterative learning control scheme with the sliding mode control method. By analyzing the convergence of sliding mode variables, the consensus conditions including the unknown eigenvalues and the undetermined weight coefficient are obtained. In order to deal with the difficulties of weight coefficient design caused by the unknown eigenvalues of graphs, a min-max optimization problem is formulated based on the fastest convergence of the λ-norm of sliding mode variables, then the optimal weight coefficient is obtained by solving the min-max optimization problem. Moreover, for the undirected and directed interval uncertain graphs, two algorithms about the optimal weight coefficients are proposed, respectively. Finally, three numerical simulation examples are presented to demonstrate the effectiveness of the proposed methods.

Design of a non-singular fast terminal sliding mode control for second-order nonlinear systems with compound disturbance

This paper investigates a new disturbance observer based non-singular fast terminal sliding mode control technique for the path tracking and stabilization of non-linear second-order systems with compound disturbance. The compound disturbance is comprised of both parametric and non-parametric uncertainties. While warranting fast convergence rate and robustness, it also dominates the singularity and complex-value number issues associated with conventional terminal sliding mode control. Furthermore, due to the estimation properties of the observer, knowledge about the bounds of the uncertainties is not required. The simulation results of two case studies, the velocity and path tracking of an autonomous underwater vehicle and the stabilization of a chaotic Φ6-Duffing oscillator, validate the efficacy of the proposed method.

Fully Distributed Event-Triggered Consensus for a Class of Second-Order Nonlinear Multi-agent Systems

Fully Distributed Event-Triggered Consensus for a Class of Second-Order Nonlinear Multi-agent Systems

Control of Nonlinear Uncertain Systems by Extended PID

Since the classical proportional-integral-derivative (PID) controller is the most widely and successfully used ones in industrial processes, it is of vital importance to investigate theoretically the rationale of this ubiquitous controller in dealing with nonlinearity and uncertainty. Recently, we have investigated the capability of the classical PID control for second-order nonlinear uncertain systems, and provided some analytic design methods for the choices of PID parameters, where the system is assumed to be in the form of cascade integrators. In this article, we will consider the natural extension of the classical PID control for high-order affine-nonlinear uncertain systems. In contrast to most of the literature on controller design of nonlinear systems, we do not require such special system structures as normal or triangular forms, thanks to the strong robustness of the extend PID controller. To be specific, we will show that under some suitable conditions on nonlinearity, and uncertainty of the systems, the extended PID controller can semiglobally stabilize the nonlinear uncertain systems, and at the same time the regulation error converges to zero exponentially fast, as long as the control parameters are chosen from an open unbounded parameter manifold constructed in this article.

Theoretical analysis of quasi-one-dimensional compressible magnetohydrodynamic channel flow

A theoretical analysis of quasi-one-dimensional, steady, inviscid, compressible channel flow at a low magnetic Reynolds number with variable cross-section is performed in this paper. A second-order nonlinear dynamical system describing the variation of physical parameters is investigated in the phase plane. The characteristics of all possible channel flow with a constant electromagnetic field are obtained by the phase plane and the isomorphism of labeled graphs. It is revealed that the magnetic interaction number novelly derived from the variation rate of the cross-sectional area could significantly affect the phase trajectory in the phase plane of dimensionless velocity and Mach number. Meanwhile, the phase trajectory is only dependent on this magnetic interaction number. Further analysis of the second-order dynamical system discovers five critical values in the range of the magnetic interaction number. These five critical values can divide the whole range into eight subsets under the isomorphism of labeled graphs, forming eight equivalence classes of the phase plane. The study of such eight equivalence classes reveals that the variable cross-section flow in the phase plane can be viewed as a linear superposition of constant cross-section magnetohydrodynamic flow and isentropic flow, the weight ratio being exactly the magnetic interaction number. Consequently, for the divergent channel, the acceleration region in the supersonic zone is larger than that of the constant cross-section flow, while for the convergent channel, only the acceleration region in the subsonic zone is larger than that of the constant cross-section flow.

Multiple Positive Solutions of Second-Order Nonlinear Difference Systems with Repulsive Singularities

We study the existence of positive solutions for second-order nonlinear repulsive singular difference systems with periodic boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a fixed point theorem in cones and a nonlinear alternative principle of Leray-Schauder; the result is applicable to the case of a weak singularity as well as the case of a strong singularity. An example is given; some recent results in the literature are improved and generalized.

Design of Spline-Evolutionary Computing Paradigm for Nonlinear Thin Film Flow Model.

A novel design of stochastic numerical computing method is introduced for computational fluid dynamics problem governed with nonlinear thin film flow (TFF) system by exploiting the competency of polynomial splines for discretization and optimization with evolutionary computing aided with brilliance of local search. The TFF model of second grade fluid is represented with nonlinear second-order differential system. The aim of the present work is to exploit the cubic spline approach (CSA) to transform the differential equations for TFF model into an equivalent set of nonlinear equations. The approximation in mean squared error sense is introduced for the formulation of cost function for solving the nonlinear system of equations representing TFF model. The optimization of the decision variables of the cost function is carried out with global search efficacy of evolution by genetic algorithms (GAs) integrated with sequential quadratic programming (SQP) for speedy adjustments. The designed spline–evolutionary computing paradigm, CSA–GA–SQP, is evaluated for different scenarios of TFF model by variation of second grade and magnetic parameters, as well as variation in the length of splines. Results endorsed the worth of CSA–GA–SQP solver as an efficient alternative, reliable, stable, and accurate framework for the variants of nonlinear TFF systems on the basis of multiple autonomous executions. The design computing spline paradigm CSA–GA–SQP is a promising alternative numerical solver to be implemented for the solution of stiff nonlinear systems representing the complex scenarios of computational fluid dynamics problems.

Event-triggered H-infinity finite-time consensus control for nonlinear second-order multi-agent systems with disturbances

The study considers the problem of finite-time event-triggered H-infinity consensus for second-order multi-agent systems (MASs) with intrinsic nonlinear dynamics and external bounded disturbances. Based on the designed triggering function, a distributed event-triggered control strategy is presented on the basis of the designed triggering function to ensure consensus in the system, which effectively reduces the data transmission. Then, sufficient conditions for the finite-time consensus with H-infinity performance level of the resulting event-triggered MAS are derived by utilizing the Lyapunov function and finite-time stability theory. Furthermore, Zeno behavior is proven to be excluded under the proposed event-triggered scheme. Finally, the validity of the proposed results is verified by numerical simulation.