Second-order Systems Research Articles

Eigenvalue assignment of second-order singular systems by acceleration–velocity–displacement feedback

Eigenvalue assignment of second-order singular systems by acceleration–velocity–displacement feedback

Multiple homoclinic solutions for nonsmooth second-order differential systems

In the present paper, we obtain infinitely many pairs of homoclinic solutions for a class of nonsmooth second-order differential systems when the energy functional associated is not continuously differentiable and does not satisfy the Palais–Smale condition.

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.

Independent Control of Active and Reactive Power Flow for a Single-Phase, Unidirectional Onboard Power Converter Connecting the DC Power Bus to the AC Bus

The paper presents a proposed system that supplies a 400 Hz single-phase onboard grid from the DC onboard bus. This system enables independent compensation of reactive power in the AC grid. Independent control of active and reactive power flow requires the decomposition of current in the grid into active and reactive components. Independent control of active and reactive power requires the use of synchronizers that operate in the dq frame system. If synchronization is performed with a single-phase grid, the transformation of dq requires the virtual quadrature signals. Standard quadrature signal generation systems use a second-order generalized integrator. To improve the dynamics of the system, the paper proposes a new quadrature generator that operates on the basis of trigonometric calculations instead of a second-order integration system. The developed system was implemented in a proportional-resonant current control system. Tests carried out in steady state and in dynamic states related to typical grid disturbances proved significantly better dynamic properties than those of a standard integrator-based system.

Adaptive fuzzy finite-time PID backstepping control for chaotic systems with full states constraints and unmodeled dynamics

To solve the problem that the controlled object and system model cannot be fully mastered or the system parameters cannot be well obtained through conventional measurement methods, in this paper, a novel adaptive fuzzy finite-time PID backstepping control method is developed for a class of nonlinear chaotic systems with full state constraints and unmodeled dynamics. The unmodeled dynamics are handled by employing a dynamic signal. To avoid the violations of full state constraints (FSCs), a barrier Lyapunov function is used to implement a PID backstepping framework, whose merit consists in that three control gain constants KP, KI and KD with corresponding proportional relationships are designed to further improve control precision. Besides, the PID control method also embodies three dynamic gain components ΔKP, ΔKI and ΔKD, all of which are free functions that can be self-tuning online, and have a significant link with the adaptive law and modify accordingly. Finally, theoretical analysis confirms that all signals are ultimately uniformly bounded, the error signal converges to a nearby area near the origin in a finite-time, and the FSCs are not smashed. The practicality of the proposed method is confirmed by two simulation studies using the second-order Duffing chaotic system and third-order Chua's circuit.

Nine-Stage Runge–Kutta–Nyström Pairs Sharing Orders Eight and Six

We explore second-order systems of non-stiff initial-value problems (IVPs), particularly those cases where the first derivatives are absent. These types of problems are of significant interest and have applications in various domains, such as astronomy and physics. Runge–Kutta–Nyström (RKN) pairs stand out as highly effective methods of addressing these IVPs. In order to create a pair with eighth and sixth orders, we need to address a certain known set of equations concerning the coefficients. When constructing such pairs for use in double-precision arithmetic, we often need to meet various conditions. Primarily, we aim to maintain small coefficient magnitudes to prevent a loss of accuracy. Nevertheless, in the context of quadruple precision, we can tolerate larger coefficients. This flexibility enables us to establish pairs with eighth and sixth orders that exhibit significantly reduced truncation errors. Traditionally, these pairs are constructed to go through eight stages per step. Here, we propose using nine stages per step. Then we have available more coefficients in order to further reduce truncation errors. As a result, we construct a novel pair that, as anticipated, achieves superior performance compared to equivalent-order pairs in various significant problem scenarios.

Event-triggered global consensus of second-order multi-agent systems with asymmetric input saturation

This paper addresses the event-triggered global consensus issue of second-order multi-agent systems (MASs) with asymmetric input saturation, where the zero saturation bounds are allowed. First, for the leaderless situation, we propose an event-triggered mechanism and give the criterion for realizing global consensus. Different from the traditional event-triggered strategy in an effort to exclude the Zeno behavior, a preset minimum triggering interval is introduced. That is, the control input will not be updated within this interval after the latest trigger time, and the event-triggered condition is activated after this interval. Moreover, the above results are extended to the leader–follower case. Finally, some numerical examples illustrate the effectiveness of the developed schemes.

A Lifshitz-Slyozov type model for adipocyte size dynamics: limit from Becker-Döring system and numerical simulation.

Biological data show that the size distribution of adipose cells follows a bimodal distribution. In this work, we introduce a Lifshitz-Slyozov type model, based on a transport partial differential equation, for the dynamics of the size distribution of adipose cells. We prove a new convergence result from the related Becker-Döring model, a system composed of several ordinary differential equations, toward mild solutions of the Lifshitz-Slyozov model using distribution tail techniques. Then, this result allows us to propose a new advective-diffusive model, the second-order diffusive Lifshitz-Slyozov model, which is expected to better fit the experimental data. Numerical simulations of the solutions to the diffusive Lifshitz-Slyozov model are performed using a well-balanced scheme and compared to solutions to the transport model. Those simulations show that both bimodal and unimodal profiles can be reached asymptotically depending on several parameters. We put in evidence that the asymptotic profile for the second-order system does not depend on initial conditions, unlike for the transport Lifshitz-Slyozov model.

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

Leader–follower consensus of hybrid multiagent systems based on game

Based on game theory, this paper considers the consensus problem of second-order hybrid multiagent systems (HMASs) in a leader–follower framework, which includes continuous-time multiagents and discrete-time multiagents. Firstly, the competitive relationship between two groups is modeled as a multi-player game type. To achieve consensus, we design different cost functions according to the rules of the game and assume that each player has a global purpose of minimizing their own cost. Moreover, the players adjust their states at the next moment according to the Nash equilibrium solution. Then, using matrix theory and stability theory, we analyze the necessary and sufficient conditions for the consensus problem of the second-order HMASs with different leaders, and show that the control parameters (sampling interval and feedback gain) and the eigenvalues of the network topology have essential impact on the consensus of the system. Finally, the validity of the results is tested by simulation examples.

Improved VSG strategy of grid-forming inverters for supporting inertia and damping

A virtual synchronous generator (VSG) strategy can introduce the rotational inertia and damping characteristics of the synchronous generator to the static inverter, e.g., PV, wind generation, and ESS, which are used to enhance the system frequency support characteristics of the micro-grid. However, under various operations of the VSG, the inertia and damping support capabilities are different, and the conventional VSG strategy with a fixed control coefficient sacrifices a certain degree of dynamic regulation performance with less robustness. This research proposes an improved VSG strategy with adaptive inertia and damping coefficients to increase the flexibility of VSG. To this end, a mathematical model is first established to analyze the impact of various parameters on the characteristics of VSG, and then the root trajectory is used to explore the impacts of various rotational inertia and damping coefficients on the stability of the VSG system. Second, an improved control strategy with adaptive inertia and damping coefficients is proposed based on the characteristics of the second-order system and the system frequency deviations. Finally, a simulation system with the VSG is constructed based on MATLAB/Simulink. The effectiveness of the proposed control strategy is verified by comparing the simulation results to the conventional control strategy with the fixed control coefficients under over-frequency and under-frequency disturbances.

Asynchronous impulsive control for interval consensus of second-order multi-agent systems

This paper investigates the interval consensus problem of second-order multi-agent systems over a directed graph, where each agent is allowed to propose an interval constraint on the achievable consensus value. In order to reduce the communication among agents, a distributed impulsive controller is given. The impulsive control is implemented only at impulsive instants. Moreover, each agent has its own impulsive instants independently and the impulsive intervals are aperiodic. Sufficient conditions are established to guarantee the achievement of interval consensus for the considered systems. Finally, a numerical example is presented to demonstrate the effectiveness of the obtained result.

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.

Predefined-time sliding mode control based on exact time disturbance observer for second-order systems with matched and mismatched disturbances

This paper’s primary motivation is to present a globally predefined-time sliding mode control (PtSMC) strategy to stabilize a class of second-order systems subjected to matched and mismatched disturbances. To achieve this, the paper proposes a new exact time disturbance observer (DOB) based on a terminal time regulator, which accurately estimates the disturbances within a prescribed time, effectively preventing the system state from escaping to infinity due to high gains and overestimation. In addition, a new predefined-time sliding mode variable with the estimation of DOB is developed to ensure a predefined-time convergence on the sliding mode phase against mismatched disturbances. The proposed DOB-based technique can alleviate the chattering resulting from the use of an overestimated gain, in contrast to the controller without employing a DOB. Furthermore, a predefined-time reaching law is introduced to guarantee a global predefined-time convergence. This paper establishes the stability of the disturbed second-order system under the proposed controller through strict Lyapunov analysis. The novelty of the proposed method lies in its global predefined-time convergence, chattering-reduced properties and robustness against matched and mismatched disturbances. Finally, numerical simulations and application examples validate the proposed methodology’s effectiveness.

Prescribed-time leader-following consensus and containment control for second-order multiagent systems with only position measurements

The problems of prescribed-time leader-following consensus and prescribed-time containment control for double-integrator multiagent systems (MASs) with only position measurements are investigated in this paper. An observer-based prescribed-time control protocol is proposed, in which the two groups of observers for estimating the consensus errors of each follower both converge to zero within a specified time. Furthermore, the proposed controller only relies on its own observers and does not need to obtain the data from observers embedded in its neighbor nodes. The sufficient conditions for the MASs to achieve the defined prescribed-time consensus and to fulfill the goal of prescribed-time containment control are, respectively, given. The effectiveness of the proposed control protocol is further verified by computer simulations.

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.

Parametric study and multi-parameter optimization of a generalized second-order tri-stable stochastic resonance system

Parametric study and multi-parameter optimization of a generalized second-order tri-stable stochastic resonance system

Two strain modes and transition point of 18650 lithium-ion battery at different charging rates

The safety issues of lithium-ion batteries are one of the biggest obstacles restricting the development of lithium-ion batteries. Excessive charging rate is not conducive to extending the cycle life of lithium-ion batteries. However, there is a lack of research on the upper limit of the charging rate of lithium batteries. The strain of the battery can reflect the change of the internal material during the charging process. The purpose of this study is to explore the charging rate of the strain transition point of lithium-ion batteries and the strain characteristics of the batteries after exceeding this rate. The experiments conducted at different rates and temperatures on battery strain revealed that the transition point of strain in the battery occurs at a rate of 2C. When the charging rate is lower than 2C, the battery strain increases monotonically, which is similar to the step response of the first-order system. When the charging rate exceeds 2C, the battery strain experiences excessive strain due to severe lithium plating, showing a step response similar to a second-order system with a spike representing an overshoot phenomenon. As the charging rate continues to increase, the overshoot peak of the battery ‘s strain becomes more obvious. The experimental results confirm that the charging rate of the 18,650 ternary lithium-ion battery used in the experiment should not exceed 2C, as it would lead to excessive strain in the battery.