Asymptotic Stability Research Articles

Dynamic event-based cooperative control of virtually coupled high-speed trains: an ADP-based optimal control scheme

This paper is concerned with the cooperative control issue of virtually coupled high-speed trains (VCHSTs) under an adaptive dynamic programming (ADP) framework, where a dynamic event-triggered mechanism is employed to govern the communication between sensors and observers. A learning-based observer, equipped with a robust term, is exploited to mitigate the effect of the unknown nonlinear dynamics of the VCHSTs. Furthermore, to realise the distributed design, an auxiliary control signal is employed to simply the closed-loop system and also integrated into the cost function to handle the coupled nature of the investigated systems. The Hamilton–Jacobi–Bellman (HJB) equation is approximated by actor and critic networks in the ADP framework, which iteratively acquires the desired control strategy. By resorting to the common Lyapunov stability, some sufficient conditions are derived to achieve the desired observer gain and learning rates of NNs. Ultimately, the control scheme's efficiency is confirmed by simulations of VCHSTs.

Adaptive second-order backstepping control for a class of 2DoF underactuated systems with input saturation and uncertain disturbances

An adaptive second-order backstepping control algorithm is proposed for a kind of two degrees of freedom (2DoF) underactuated systems. The system dynamics is transformed into a nonlinear feedback cascade system with an improved global change of coordinates. Fully taking the cascade structure into consideration and in order to simplify the design process, each step in the backstepping process is designed for a second-order subsystem. Two neural networks are applied to approximate system unknown functions and two adaptive laws are designed to estimate the upper bound of the sum of approximation error and external disturbances. To overcome the explosion problem of complexity, a second-order filter is applied to produce the virtual control and its second-order derivative that is needed in the next backstepping step. Two auxiliary dynamic systems are proposed and integrated into the backstepping process to eliminate the effects of filtering error and input saturation. The system stability is analyzed by the Lyapunov stability theory and verified by numerical simulations with two 2DoF benchmark underactuated systems: the translational oscillator with a rotational actuator (TORA) and the inertial wheel pendulum (IWP).

Neural Network Energy Management-Based Nonlinear Control of a DC Micro-Grid with Integrating Renewable Energies

The broad acceptance of sustainable and renewable energy sources as a means of integrating them into electrical power networks is essential to promote sustainable development. Microgrids using direct currents (DCs) are becoming more and more popular because of their great energy efficiency and straightforward design. In this work, we discuss the control of a PV-based renewable energy system and a battery- and supercapacitor-based energy storage system in a DC microgrid. We describe a hierarchical control approach based on sliding-mode controllers and the Lyapunov stability theory. To balance the load and generation, a fuzzy logic-based energy management system has been created. Using a neural network, maximum power defects for the PV system were determined. The global asymptotic stability of the framework has been verified using Lyapunov stability analysis. In order to simulate the proposed DC microgrid and controllers, MATLAB/SimulinkR (2019a) was utilized. The outcomes show that the system operates effectively with changing production and consumption.

Lyapunov stability of suspension bridges in turbulent flow

AbstractIn the era of sleek, super slender suspension bridges, facing the issue of stability against dynamic wind actions represents an increasingly complex challenge. Despite the significant progress over the last decades, the impact of atmospheric turbulence on bridge stability remains partially not understood, evoking the need for innovative research approaches. This study aims to address a gap in current research by investigating the random flutter stability associated with variations in the angle of attack due to turbulence, which has not formally been addressed yet. The present investigation employs the 2D rational function approximation model to express self-excited forces in a turbulent flow. The application of this type of models to bridge dynamics yields a viscoelastic coupled dynamic system characterized by memory effects and driven by broad-band long-time-scale noise, described here by a linear homogeneous time-variant differential equation, which shows apparent nonlinear features, and which has rarely been matter of research. Utilizing a Monte Carlo methodology, this work innovates in applying the largest Lyapunov exponent (LE) and the moment Lyapunov exponents (MLE) to study bridge random flutter stability. The calculation of LE and MLE under diverse turbulent wind conditions uncovers lower flutter stability than without turbulence effects. In most cases, sample and low-order p-th moment stability thresholds closely align with the bridge dynamic response pattern; therefore, the flutter critical wind speed is unequivocal. However, under certain turbulence scenarios, it is necessary to resort to MLE for a complete description of stability, evoking some additional consideration of which statistical moments should be considered for the engineering assessment of the flutter limit. Finally, this work provides a qualitative insight into the instability mechanisms by approximating the random parametric excitation with a sinusoidal gust and evaluating the time-periodic system stability via Floquet theory.

CONTRÔLE SANS CAPTEUR DE TENSION D'UN ONDULEUR À CELLULES U À CINQ NIVEAUX BASÉ SUR L'APPROCHE LYAPUNOV POUR UN SYSTÈME PHOTOVOLTAÏQUE CONNECTÉ AU RÉSEAU

This paper deals with the nonlinear control of a five-level packed U-Cell (PUC5) inverter used in a grid-on photovoltaic (PV) system. The sensorless modulation strategy is adopted to control the switching of the six power switches. This topology generates a five-level output voltage with fewer active and passive components than traditional multilevel inverters. A nonlinear controller was designed to control the grid current using the Lyapunov theorem to ensure the asymptotic stability of the overall system. The proposed control scheme addresses two key challenges: minimizing the injected grid current's total harmonic distortion (THD) while balancing the capacitor voltage at their reference under various operating conditions. The efficiency of the proposed approach was comprehensively evaluated using theoretical analysis, mathematical modeling, and simulation results obtained using MATLAB/Simulink software.

COMMANDE HYBRIDE DE MODE COULISSANT FUZZY TYPE-1 ET 2 DU MOTEUR À INDUCTION

This paper focuses on developing two innovative induction motor (IM) control techniques. These techniques are based on the hybridization of Lyapunov theory (sliding mode) and artificial intelligence (type 1 and type 2 fuzzy logic). We will then compare these two control techniques to determine which is more robust. This comparative analysis will be based on a series of tests that we have carried out, covering the system's transient and steady-state operations under identical conditions. The first test involves observing the simulation results obtained by applying these control techniques to the motor to control the generated mechanical power. This qualitative comparison enables these controls to be evaluated for and without the application of external variations. The second test quantifies the different control laws based on quantified measurements, highlighting the performance of each technique in terms of error and time. This test is called a quantitative comparison. Finally, the last examination involves altering the machine parameters, as these values naturally experience fluctuations caused by diverse physical phenomena like inductance saturation and heating of the resistors. This comparison enables the robustness of the control techniques to be assessed.

Trajectory Tracking of Stochastic Open Quantum Systems Based on Online Estimated State Feedback Control

AbstractAn online estimated state feedback control for trajectory tracking in stochastic open quantum systems is proposed in this paper, which is based on the Lyapunov‐based control method. By inducing the error between the controlled state, and the target state as the error state, the trajectory tracking problem of the quantum system is transformed into the error state transition control problem. The quantum state online estimation method QST‐OADM is applied to estimate the state of the error state system online, and the tracking control laws are designed by using the quantum Lyapunov stability theorem for driving the stochastic open quantum system from an arbitrary initial state to an arbitrary trajectory. The numerical simulation experiments and results analyses are given.

Neuro-Adaptive-Based Fixed-Time Composite Learning Control for Manipulators With Given Transient Performance.

This article investigates an adaptive neural network (NN) control technique with fixed-time tracking capabilities, employing composite learning, for manipulators under constrained position error. The first step involves integrating the composite learning method into the NN to address the dynamic uncertainties that inevitably arise in manipulators. A composite adaptive updating law of NN weights is formulated, requiring adherence solely to the relaxed interval excitation (IE) conditions. In addition, for the output error, instead of knowing the initial conditions, this article integrates the error transfer function and asymmetric barrier function to achieve the specific performance for position error in both steady and transient states. Furthermore, the fixed-time control methodology and Lyapunov stability criterion are synergistically employed in order to guarantee the convergence of all signals in the manipulators to a compact neighborhood around the origin within a fixed-time. Finally, numerical simulation and experiments with the Baxter robot results both determine the capability of the NN composite learning technique and fixed-time control strategy.

Induced ℒ∞ control for continuous-time nonlinear systems

In this paper, the problem of induced L ∞ control in the presence of disturbance signals in the control output of continuous-time nonlinear systems is considered. Under the condition that the system control output has a disturbance signal, the expected design conditions of the controller that satisfy the asymptotic stability and the specified performance are given. The sufficient conditions for such an induced L ∞ controller are expressed in terms of linear matrix inequalities (LMIs). Finally, the effectiveness and feasibility of the proposed induced L ∞ control design strategy is illustrated by a tunnelling diode circuit.

Neural network-based distributed intelligent supervisory control for multi-agent systems with unknown input powers

In this paper, a synchronisation problem is studied for a class of nonlinear leader-following systems. Based on adaptive control with some algebra lemmas, we propose a distributed control scheme. This scheme ensures each follower to asymptotically synchronise with the leader. Compared with the existing works where system input powers are assumed one, the input powers considered in this paper are unknown but larger than one. Based on Graph theory, Lyapunov theory and radial basis function (RBF) networks, we design a distributed adaptive supervisory control method. It is proven that the consensus among the leader and followers are achieved and all the signals including tracking errors asymptotically converge to a small neighbourhood of the origin. Finally, simulation results are displayed to demonstrate the effectiveness of control scheme.

Design Principles for Perfect Adaptation in Biological Networks with Nonlinear Dynamics.

Establishing a mapping between the emergent biological properties and the repository of network structures has been of great relevance in systems and synthetic biology. Adaptation is one such biological property of paramount importance that promotes regulation in the presence of environmental disturbances. This paper presents a nonlinear systems theory-driven framework to identify the design principles for perfect adaptation with respect to external disturbances of arbitrary magnitude. Based on the prior information about the network, we frame precise mathematical conditions for adaptation using nonlinear systems theory. We first deduce the mathematical conditions for perfect adaptation for constant input disturbances. Subsequently, we translate these conditions to specific necessary structural requirements for adaptation in networks of small size and then extend to argue that there exist only two classes of architectures for a network of any size that can provide local adaptation in the entire state space, namely, incoherent feed-forward (IFF) structure and negative feedback loop with buffer node (NFB). The additional positiveness constraints further narrow the admissible set of network structures. This also aids in establishing the global asymptotic stability for the steady state given a constant input disturbance. The proposed method does not assume any explicit knowledge of the underlying rate kinetics, barring some minimal assumptions. Finally, we also discuss the infeasibility of certain IFF networks in providing adaptation in the presence of downstream connections. Moreover, we propose a generic and novel algorithm based on non-linear systems theory to unravel the design principles for global adaptation. Detailed and extensive simulation studies corroborate the theoretical findings.

Asymptotic stability and bifurcations of a perturbed McMillan map

This paper presents various bifurcations of the McMillan map under perturbations of its coefficients, such as period-doubling, pitchfork, and hysteresis bifurcation. The associated existence regions are located. Using the quasi-Lyapunov function method, the existence of asymptotically stable fixed point is also demonstrated.

A novel robust MPC scheme established on LMI formulation for surge instability of uncertain compressor system with actuator constraint and piping acoustic

This paper presents a new control method for surge instability of the compressor system. Due to the importance of the active control approach in improving the efficiency of the compressor system, a new scheme based on the model predictive control (MPC) technique has been considered to control surge which gives the benefits of control signal optimization by considering the constraints on states and input. Because of designing a novel MPC by using of linear matrix inequality scheme, the optimization problem is solved in less time and with less complexity. The proposed method is able to consider the limitation of the close coupled valve actuator and also covers the uncertainty on the model. In addition, the developed model describes the nonlinear behaviour of the compressor system and handles the effects of the pipe on surge instability. Using Lyapunov theory, the stability of the closed-loop system under the proposed robust active controller is shown. The simulation results show the high pressure and high efficiency operation of the system under study in different operating conditions.

Distributed event‐triggered fixed‐time formation tracking control for multi‐spacecraft systems based on adaptive immersion and invariance technique

AbstractIn this study, the problem of fixed‐time formation tracking for multi‐spacecraft systems without internal collisions was investigated. Aiming to ensure that the formation members can accurately realize and maintain the required configuration within the user‐given time, we designed a novel adaptive immersion and invariance (I&I)‐based control protocol. The novelty here lies in two aspects. First and foremost, this study was based on the adaptive I&I technique, combined with a new artificial potential function, to achieve the desired formation tracking without internal collision. Second, unlike the asymptotic convergence of the traditional I&I‐related works, to guarantee the fixed‐time stability, the proposed protocol introduced the prescribed performance control, which can also alleviate the possible system performance degradation caused by the non‐periodic control signal update of the event‐triggered mechanism. Further, the introduction of the event‐triggered mechanism can reduce the unnecessary information interaction. Lyapunov stability analysis shows that this proposed protocol can enable the defined implicit manifold to converge to the origin for the most initial conditions. Also, benefiting from the prescribed performance techniques, the convergence time can eliminate the dependence of the system on designed controller parameters or initial system conditions, relying only on the actual mission requirements. In addition, we adopted a linear extended state observer to deal with the parameter uncertainties and external disturbances, and used the I&I adaption to estimate the observer errors, thus further improving the system performance. Moreover, a new exponential‐type artificial potential function was designed to avoid close proximity between formation members and prevent internal collisions. Finally, numerical simulations were conducted to verify the theoretical results.

Fault-Tolerant Tracking Control of Hypersonic Vehicle Based on a Universal Prescribe Time Architecture

An adaptive tracking control strategy with a prescribe tracking error and the convergence time is proposed for hypersonic vehicles with state constraints and actuator failures. The peculiarity is that constructing a new time scale coordinate translation mapping method, which maps the prescribe time on the finite field to the time variable on the infinite field, and the convergence problem of the prescribe time is transformed into the conventional system convergence problem. The improved Lyapunov function, the improved tuning function, and the adaptive fault-tolerant mechanism are further constructed. Combined with the neural network, the prescribe time tracking control of the speed subsystem and the height subsystem are realized respectively. Combined with the Barbalat lemma and Lyapunov stability theory, the boundedness of the closed-loop system is proved. The simulation results have proven that, compared with other control strategies, it can ensure that the tracking error converges to the prescribe interval in the prescribe time and meets the constraints of the whole state of the system.

A novel saturation degree‐dependent adaptive event‐triggered control for systems with asymmetric saturation

AbstractThis paper investigates the control problem of systems with asymmetric saturation by the formulated saturation degree‐dependent event‐triggered mechanism. The subregion partition method is adopted to represent the asymmetric saturation as the symmetric saturation in each subregion. In order to further reduce the triggering number when the system is saturated and save the communication resources of the system, this paper establishes the saturation degree‐dependent adaptive event‐triggered condition. Then, the subregion‐related control law under the presented event‐triggered scheme is designed, and the asymptotic stabilization of the underlying system is realized. Moreover, an optimization algorithm is proposed to estimate the maximal domain of attraction. Finally, simulation examples are used to validate the theoretical results.

Two Schemes of Impulsive Runge–Kutta Methods for Linear Differential Equations with Delayed Impulses

In this paper, two different schemes of impulsive Runge–Kutta methods are constructed for a class of linear differential equations with delayed impulses. One scheme is convergent of order p if the corresponding Runge–Kutta method is p order. Another one in the general case is only convergent of order 1, but it is more concise and may suit for more complex differential equations with delayed impulses. Moreover, asymptotical stability conditions for the exact solution and numerical solutions are obtained, respectively. Finally, some numerical examples are provided to confirm the theoretical results.

Nonlinear Robust Stabilization of Ship Roll by Convex Optimization

For ship roll stabilization, this paper presents a nonlinear robust controller design method using the sum of squares (SOS) technique combined with the dual of Lyapunov's stability theorem. Varying ship speed and initial metacentric height are seen as uncertainties, sea waves are regarded as external disturbance, and then the robust nonlinear controller is also designed based on the suggested method. Simulations are performed by taking container ship model as an example. It is shown that the designed system guarantees robust performance with respect to the uncertainties and disturbance.

Optimal control of rotavirus infection in breastfed and non-breastfed children

According to the World Health Organization’s guidelines, a two-year mandatory breastfeeding period is strongly recommended, supported by research demonstrating its significant benefits in reducing childhood illnesses and mortality rates. The present study is designed to address this recommendation by focusing on a specific infection, namely rotavirus. Acknowledging this evidence, researchers recognize the need for a deeper understanding of the dynamics of rotavirus disease. In response, our study focuses on constructing and analyzing an SIR epidemic model uniquely designed for transmitting rotavirus in children. The model partitions the children population into three compartments: susceptible, infected, and recovered. To enhance precision, we further categorize susceptible children based on breastfeeding status and infected individuals based on infectious or non-infectious nature. This detailed categorization enables a thorough examination of rotavirus transmission dynamics. The proposed model analyzes two equilibria, disease-free and endemic, revealing local and global stability conditions determined by the basic reproduction number (R0). Our exploration extends to local stability analysis for the endemic equilibrium when R0>1 and global stability assessment using the Lyapunov theory under specific conditions. Subsequently, we employ optimal control theory through Pontryagin’s maximum principle to minimize the cost burden associated with disease control in children. Then, our work provides a sensitivity analysis of different parameters for the basic reproduction number to enhance our understanding of model dynamics. Finally, numerical simulations conducted with MATLAB and Python software validate our analytical findings, offering a comprehensive and practical assessment of the proposed model’s implications for controlling rotavirus transmission in children.

Learning from NN-based extended PID control for a class of high-order uncertain nonlinear systems

As is well known, traditional PID controllers are the most widely used type of controllers in industrial control due to their simplicity and other advantages. However, the existing research results on PID controllers lack a general theory for addressing the nonlinearity and uncertainty present in practical systems. In the era of intelligent control, it is highly valuable to design an intelligent PID controller with learning capability for complex unknown nonlinear systems. In this paper, we consider an extended form of PID controller based on deterministic learning (DL) for high-order nonlinear systems with unknown dynamics. The introduction of neural networks (NNs) provides an effective tool to compensate for the composite unknown nonlinearities in controlled systems, overcoming the limitations of traditional PID controllers relying solely on feedback mechanisms to combat unknown nonlinearities. Based on the Lyapunov theory, it not only ensures the stability and tracking control of the system but also simplifies the selection of controller parameters. Furthermore, based on the DL theory, the composite unknown nonlinear dynamics can be accurately compensated and approximated, and the acquired knowledge can be stored in a constant-value neural network. When encountering similar control tasks, the acquired knowledge can be quickly accessed to construct a knowledge-based extended PID controller, thereby improving the overall control performance. Simulation results demonstrate the effectiveness of the proposed algorithm.