Second-order Nonlinear Systems Research Articles

Inducing Tunable Conventional Photon Blockade and Two-Photon Blockade in a Second-Order Nonlinear System with Two-Level Atoms

Inducing Tunable Conventional Photon Blockade and Two-Photon Blockade in a Second-Order Nonlinear System with Two-Level Atoms

Leader-following consensus of heterogeneous nonlinear multi-agent systems within arbitrary pre-specified time

Abstract The leader-following consensus problem of second-order heterogeneous nonlinear multi-agent systems (MASs) is studied in this paper. Based on the free-will arbitrary time (FWAT) theory, a piecewise distributed controller is developed to solve this problem within an arbitrary pre-specified convergence time. It should be noted that the pre-specified convergence time is independent of any system parameters and initial conditions. Furthermore, compared with the existing FWAT methods, consensus can still be maintained after the convergence time under the protocol designed in this work. Finally, numerical examples are given to illustrate the validity of the proposed method.

Pinning Group Consensus of Multi-agent Systems Under DoS Attacks

In this paper, group consensus is investigated for a class of nonlinear multi-agent systems suffered from the DoS attacks. Firstly, a first-order nonlinear multi-agent system is constructed, which is divided into M subsystems and each subsystem has an unique leader. Then a protocol is proposed and a Lyapunov function candidate is chosen. By means of the stability theory, a sufficient criterion, which involves the duration of DoS attacks, coupling strength and control gain, is obtained for achieving group consensus in first-order system. That is, the nodes in each subsystem can track the leader of that group. Furthermore, the result is extended to nonlinear second-order multi-agent systems and the controller is also improved to obtain sufficient conditions for group consensus. Additionally, the lower bounds of the coupling strength and average interval of DoS attacks can be determined from the obtained sufficient conditions. Finally, several numerical simulations are presented to explain the effectiveness of the proposed controllers and the derived theoretical results.

Fast finite-time formation control with obstacle avoidance of second-order multi-agent systems with input saturation

This paper proposes a leader-follower formation control protocol using fast finite-time (FFT) theory, based on second-order nonlinear multi-agent systems (MASs) with input saturation constraints. The artificial potential field method is addressed to implement the formation control with obstacle avoidance of the MASs. An adaptive FFT strategy is constructed that all the agents follow required formation performance. Neural networks are considered to approximate uncertain functions, which improved convergence and ensuring safety of distributed formation control. Finally, the validity of the theoretical approach is demonstrated by FFT stability theory validated by simulation examples.

Fixed-time cooperative output regulation for second-order nonlinear multiagent systems with an unknown exosystem

This paper investigates the fixed-time cooperative output regulation (FTCOR) problem with unknown exosystem of second-order nonlinear multiagent systems. To solve this problem, it is necessary to ensure that each agent can obtain the reference trajectory originating from an unknown exosystem and reconstruct the solution of the regulator equation containing the exogenous signal and unknown parameter vectors within a fixed time, and the unknown exosystem makes it difficult to solve these two problems with existing tools. By using auxiliary filtering variables, we propose a fixed-time adaptive distributed observer (FTADO) and design a fixed-time distributed internal model (FTDIM) based on this FTADO. These two new tools ensure the solvability of the two problems just mentioned. Then, based on these two tools, we transform the problem into a fixed-time stabilization problem of a so-called augmented system. Through the backstepping technique, we propose a distributed controller that solves this fixed-time stabilization problem while relaxing the assumptions of similar references.

On the Convergence of an Approximation Scheme of Fractional-Step Type, Associated to a Nonlinear Second-Order System with Coupled In-Homogeneous Dynamic Boundary Conditions

The paper concerns a nonlinear second-order system of coupled PDEs, having the principal part in divergence form and subject to in-homogeneous dynamic boundary conditions, for both θ(t,x) and φ(t,x). Two main topics are addressed here, as follows. First, under a certain hypothesis on the input data, f1, f2, w1, w2, α, ξ, θ0, α0, φ0, and ξ0, we prove the well-posedness of a solution θ,α,φ,ξ, which is θ(t,x),α(t,x)∈Wp1,2(Q)×Wp1,2(Σ), φ(t,x),ξ(t,x)∈Wν1,2(Q)×Wp1,2(Σ), ν=min{q,μ}. According to the new formulation of the problem, we extend the previous results, allowing the new mathematical model to be even more complete to describe the diversity of physical phenomena to which it can be applied: interface problems, image analysis, epidemics, etc. The main goal of the present paper is to develop an iterative scheme of fractional-step type in order to approximate the unique solution to the nonlinear second-order system. The convergence result is established for the new numerical method, and on the basis of this approach, a conceptual algorithm, alg-frac_sec-ord_u+varphi_dbc, is elaborated. The benefit brought by such a method consists of simplifying the computations so that the time required to approximate the solutions decreases significantly. Some conclusions are given as well as new research topics for the future.

Parameter variational based adaptive bipartite formation control of second-order multi-agent systems with antagonistic interactions

This article mainly concentrates on researching the leader-following bipartite formation of second-order nonlinear multi-agent systems (MASs) with antagonistic interactions and time-varying delay. A novel distributed feedback control and impulsive control strategy is proposed for realizing the bipartite formation. By establishing an error system, the bipartite formation issue of the MASs is transformed into the stability issue of the error systems. Sufficient conditions for achieving the bipartite formation of second-order nonlinear MASs are obtained by jointly employing the extended parameters variation formula, the comparison principle and the definition of average impulsive interval. In addition, in order to obtain the optimal control gains and control effects, an adaptive control strategy is taken into account by designing the adaptive updating laws. Ultimately, the effectiveness of the theoretical results is demonstrated by two numerical simulations.

Approximate optimal and safe coordination of nonlinear second-order multirobot systems with model uncertainties

This paper investigates the approximate optimal coordination for nonlinear uncertain second-order multi-robot systems with guaranteed safety (collision avoidance) Through constructing novel local error signals, the collision-free control objective is formulated into an coordination optimization problem for nominal multi-robot systems. Based on approximate dynamic programming technique, the optimal value functions and control policies are learned by simplified critic-only neural networks (NNs). Then, the approximated optimal controllers are redesigned using adaptive law to handle the effects of robots’ uncertain dynamics. It is shown that the NN weights estimation errors are uniformly ultimately bounded under proper conditions, and safe coordination of multiple robots can be achieved regardless of model uncertainties. Numerical simulations finally illustrate the effectiveness of the proposed controller.

Novel Observer-Based Input-Constrained Control of Nonlinear Second-Order Systems with Stability Analysis: Experiment on Lever Arm

Novel Observer-Based Input-Constrained Control of Nonlinear Second-Order Systems with Stability Analysis: Experiment on Lever Arm

Practical fixed-time event-triggered output feedback containment control for second-order nonlinear multi-agent systems with switching topologies

This paper studies the fixed-time containment of the second-order nonlinear multi-agent system (MAS) accompanied by switching topologies and event-triggered communication. The agent is subjected to unknown nonlinear dynamics. A fixed-time RBF observer was constructed based on information obtained from neural networks and event-triggered communication. By combining the controller with the identification information from the observer, it is demonstrated that the system can achieve practical fixed-time output feedback containment and exclude Zeno behavior during the control process. The effectiveness of the proposed event-triggered control method for fixed-time output feedback containment control of the second-order nonlinear MAS is verified by simulation examples.

Impulsive consensus algorithms for vector second-order Lipschitz nonlinear multi-agent systems using only velocity regulation

The existing impulsive consensus algorithms for second-order Lipschitz nonlinear multi-agent systems require to apply the impulsive control to both position and velocity vectors at the same time. Such a requirement cannot be met in most of the real-world applications. To overcome the limitations of these impulsive algorithms, two kinds of new second-order impulsive consensus algorithms using only velocity regulation are proposed. Through developing a weighted discontinuous Lyapunov function-based approach that is able to leverage the spectral property of Laplacian matrix, impulse-dwell-time-dependent sufficient conditions for solving second-order impulsive consensus are derived in the form of linear matrix inequalities. Further, it is shown that if the impulsively controlled velocity subsystems are globally exponentially stable, the impulsive static consensus algorithm is able to ensure that all agents tend to an agreed position. Based on the consensus conditions, two convex optimization problems are formulated, by which the impulsive gain matrices for ensuring a prescribed exponential convergence rate can be designed. Finally, the effectiveness of the proposed distributed impulsive consensus algorithms is certified through numerical simulations.

Few- and single-molecule reservoir computing experimentally demonstrated with surface-enhanced Raman scattering and ion gating.

Molecule-based reservoir computing (RC) is promising for achieving low power consumption neuromorphic computing, although the information-processing capability of small numbers of molecules is not clear. Here, we report a few- and single-molecule RC that uses the molecular vibration dynamics in the para-mercaptobenzoic acid (pMBA) detected by surface-enhanced Raman scattering (SERS) with tungsten oxide nanorod/silver nanoparticles. The Raman signals of the pMBA molecules, adsorbed at the SERS active site of the nanorod, were reversibly perturbated by the application of voltage-induced local pH changes near the molecules, and then used to perform time-series analysis tasks. Despite the small number of molecules used, our system achieved good performance, including >95% accuracy in various nonlinear waveform transformations, 94.3% accuracy in solving a second-order nonlinear dynamic system, and a prediction error of 25.0 milligrams per deciliter in a 15-minute-ahead blood glucose level prediction. Our work provides a concept of few-molecular computing with practical computation capabilities.

Time-Varying Optimal Formation Control for Second-Order Multiagent Systems Based on Neural Network Observer and Reinforcement Learning.

This article addresses a distributed time-varying optimal formation protocol for a class of second-order uncertain nonlinear dynamic multiagent systems (MASs) based on an adaptive neural network (NN) state observer through the backstepping method and simplified reinforcement learning (RL). Each follower agent is subjected to only local information and measurable partial states due to actual sensor limitations. In view of the distributed optimized formation strategic needs, the uncertain nonlinear dynamics and undetectable states may jointly affect the stability of the time-varying cooperative formation control. Furthermore, focusing on Hamilton-Jacobi-Bellman optimization, it is almost incapable of directly dealing with unknown equations. Above uncertainty and immeasurability processed by adaptive state observer and NN simplified RL are further designed to achieve desired second-order formation configuration at the least cost. The optimization protocol can not only solve the undetectable states and realize the prescribed time-varying formation performance on the premise that all the errors are SGUUB, but also prove the stability and update the critics and actors easily. Through the above-mentioned approaches offer an optimal control scheme to address time-varying formation control. Finally, the validity of the theoretical method is proven by the Lyapunov stability theory and digital simulation.

Adaptive stabilisation of second-order strict-feedback systems based on FAS with tuning functions

This paper thoroughly investigates adaptive stabilisation for a specific class of second-order strict-feedback nonlinear systems. Drawing inspiration from the fully actuated system (FAS) approach, an adaptive control scheme incorporating FAS with tuning functions is proposed. Notably, this approach directly addresses second-order systems without the need for reducing them to first-order systems, streamlining the intricate procedures involved. To handle uncertain parameters within the system, a combined scheme of adaptive estimation and tuning functions is employed. Rigorous analysis using Lyapunov stability theory demonstrates that the designed controller ensures the system's asymptotic convergence to the equilibrium point, with all other signals in the closed-loop system bounded. Finally, two simulation experiments verify the effectiveness of the proposed strategy: numerical simulations compared with sliding mode control and backstepping control, and practical application on the single-link robotic arm with a comparative analysis against backstepping control.

Brain-Inspired Reservoir Computing Using Memristors with Tunable Dynamics and Short-Term Plasticity.

Recent advancements in reservoir computing (RC) research have created a demand for analogue devices with dynamics that can facilitate the physical implementation of reservoirs, promising faster information processing while consuming less energy and occupying a smaller area footprint. Studies have demonstrated that dynamic memristors, with nonlinear and short-term memory dynamics, are excellent candidates as information-processing devices or reservoirs for temporal classification and prediction tasks. Previous implementations relied on nominally identical memristors that applied the same nonlinear transformation to the input data, which is not enough to achieve a rich state space. To address this limitation, researchers either diversified the data encoding across multiple memristors or harnessed the stochastic device-to-device variability among the memristors. However, this approach requires additional preprocessing steps and leads to synchronization issues. Instead, it is preferable to encode the data once and pass them through a reservoir layer consisting of memristors with distinct dynamics. Here, we demonstrate that ion-channel-based memristors with voltage-dependent dynamics can be controllably and predictively tuned through the voltage or adjustment of the ion channel concentration to exhibit diverse dynamic properties. We show, through experiments and simulations, that reservoir layers constructed with a small number of distinct memristors exhibit significantly higher predictive and classification accuracies with a single data encoding. We found that for a second-order nonlinear dynamical system prediction task, the varied memristor reservoir experimentally achieved an impressive normalized mean square error of 1.5 × 10-3, using only five distinct memristors. Moreover, in a neural activity classification task, a reservoir of just three distinct memristors experimentally attained an accuracy of 96.5%. This work lays the foundation for next-generation physical RC systems that can exploit the complex dynamics of their diverse building blocks to achieve increased signal processing capabilities.

Reinforcement learning-based optimal control of uncertain nonlinear systems

This paper considers the optimal control of a second-order nonlinear system with unknown dynamics. A new reinforcement learning based approach is proposed with the aid of direct adaptive control. By the new approach, actor-critic reinforcement learning algorithms are proposed with neural network approximation. Simulation results are presented to show the effectiveness of the proposed algorithms.

Event-Triggered Fixed-Time Consensus of Second-Order Nonlinear Multi-Agent Systems with Delay and Switching Topologies

Event-Triggered Fixed-Time Consensus of Second-Order Nonlinear Multi-Agent Systems with Delay and Switching Topologies

Event-triggered control strategy based on absolute velocity and relative position measurements for second-order nonlinear multi-agent systems under DoS attacks

Consensus control in multi-agent systems (MASs) has wide-ranging applications in various domains. As the scope of applications continues to expand, ensuring the secure control of these systems to counter a diverse range of network attacks becomes highly necessary. Among these threats, Denial of Service (DoS) attacks have emerged as a major network security threat, capable of rendering targeted agents or networks unable to provide normal services. This paper presents an effective control protocol aiming to achieve leader-following consensus in a second-order MAS with event-triggered control mechanism and nonlinear dynamics under DoS attacks. The protocol utilizes absolute velocity and relative position measurements. The maximum attack frequency and duration of the DoS attacks have been obtained. Furthermore, the possibility of Zeno behavior is eliminated. Theoretical analysis demonstrates that, even under DoS attacks, consensus can be maintained among the leaders and followers within the MASs. In conclusion, the effectiveness of the proposed distributed security control protocol is validated through a case study involving a multi-vehicle cooperative system with unmanned cars.

Fixed-time convergence of second-order nonlinear systems based on nonsingular fractional sliding mode

In this paper, a fixed-time convergence scheme for fractional sliding modes is proposed for nonlinear second-order systems. Firstly, a fixed-time fractional-order sliding mode surface is designed by combining the fixed-time theory with fractional-order sliding mode control, which has a faster convergence rate and less chattering. Secondly, the proposed sliding mode controller is applied to a class of second-order nonlinear systems subject to uncertainties and external perturbations to ensure that the system is globally robust fixed-time stable. Then, a continuous fractional order approach law is designed and the proposed sliding mode controller is shown to converge in fixed time by Lyapunov function and the convergence time is related to the choice of controller parameters. Finally, the fixed-time fractional-order sliding mode control strategy is applied to a second-order nonlinear magnetic levitation system system, and the simulation results verify the effectiveness of the proposed method.

Adaptive iterative learning control for a class of parameterized fractional-order systems subjected to backlash-like hysteresis

In this article, a fractional-order adaptive iterative learning control scheme is proposed for a class of parameterized fractional-order systems with unknown control gain and backlash-like hysteresis nonlinearity under disturbance. Based on the sufficient condition for the stability of linear fractional-order systems, a sliding mode surface of tracking errors is constructed to facilitate the controller design and stability analysis. A new boundary layer function is designed by using Mittag-Leffler function to relax the restriction of the identical initial condition of iterative learning control design. The fractional-order differential-type adaptive laws and difference-type learning law are designed to estimate unknown constant parameters and time-varying parameter, respectively. To deal with the influence of backlash-like hysteresis nonlinearity reminder term and unknown bounded external disturbance, a robust control term is designed by employing the hyperbolic tangent function with a convergent series sequence which can guarantee the learning convergence along iteration axis. By constructing Lyapunov-like composite energy function, the stability analysis is presented to prove the convergence of the system output to a small neighborhood of the desired trajectory and the boundedness of all the closed-loop signals. Finally, a simulation example of second-order nonlinear fractional-order system is presented, which demonstrates the effectiveness of the proposed fractional-order adaptive iterative learning control scheme.