Lyapunov Method Research Articles

Adaptive NNs asymptotic tracking control for high‐order nonlinear systems under prescribed performance and asymmetric output constraints

SummaryThis article studies the adaptive neural networks (NNs) asymptotic tracking control of high‐order nonlinear systems subject to prescribed performance, non‐strict‐feedback structure, and output constraints. To address the output constraint issue while guaranteeing that the tracking error stays within the specified area, a variable fused with the time‐varying constraint functions is introduced. Then, a pivotal form of coordinate transformation is developed, which plays a key role in achieving asymptotic tracking performance. Based on the backstepping and Lyapunov method, the designed control scheme assures that all system variables are semi‐globally uniformly ultimately bounded, the output constraints are never broken, and the tracking error always stays within the predefined function and asymptotically converges to zero. Finally, the effectiveness of theoretical findings is verified via simulation studies.

Exponential stability analysis of delayed partial differential equation systems: Applying the Lyapunov method and delay-dependent techniques

This paper presents an investigation into the stability and control aspects of delayed partial differential equation (PDE) systems utilizing the Lyapunov method. PDEs serve as powerful mathematical tools for modeling diverse and intricate systems such as heat transfer processes, chemical reactors, flexible arms, and population dynamics. However, the presence of delays within the feedback loop of such systems can introduce significant challenges, as even minor delays can potentially trigger system instability. To address this issue, the Lyapunov method, renowned for its efficacy in stability analysis, is employed to assess the exponential stability of a specific cohort of delayed PDE systems. By adopting Dirichlet boundary conditions and incorporating delay-dependent techniques such as the Galerkin method and Halanay inequality, the inherent stability properties of these systems are rigorously examined. Notably, the utilization of Dirichlet boundary conditions in this study allows for simplified analysis, and it is worth mentioning that the stability analysis outcomes under Neumann conditions and combined boundary conditions align with those of the Dirichlet boundary conditions discussed herein. Furthermore, this research endeavor delves into the implications of the obtained results in terms of control considerations and convergence rates. The integration of the Galerkin method aids in approximating the behavior of dominant modes within the system, thereby enabling a more comprehensive understanding of stability and control. The exploration of convergence rates provides valuable insights into the speed at which stability is achieved in practice, thus enhancing the practical applicability of the findings. The outcomes of this study contribute significantly to the broader comprehension and effective control of delayed PDE systems. The elucidation of stability behaviors not only provides a comprehensive understanding of the impact of delays but also offers practical insights for the design and implementation of control strategies in various domains. Ultimately, this research strives to enhance the stability and reliability of complex systems represented by PDEs, thereby facilitating their effective utilization across numerous scientific and engineering applications.

Cooperative Control and Performance Evaluation of a Linear MIMO Parabolic Spatiotemporal Dynamic System on Directed and Switching Topologies.

The current work concerns on cooperative control of exponential stabilization and control performance improvement in the spatial domain for a linear spatiotemporal dynamic system associated with multiple control actuators and multiple collaborative measurement sensors. By assuming that the system dynamics is modeled by a MIMO parabolic partial differential equation (PPDE) and each sensor can share measurement information with its topological neighbors in directed and switching topological networks, a cooperative control protocol is proposed to achieve the control aim of this article. With the help of a combination of multiagent consensus theory, Lyapunov's method, and integral inequality technique, sufficient conditions are presented for the closed-loop exponential stability of the PPDE in the norm |·|2 . Moreover, some performance indexes are defined to evaluate the closed-loop control performance improvement in the spatial domain. Extensive simulation results are finally presented for a simple numerical example and a practical heat treatment process to verify the effectiveness and performance improvement capability of the proposed cooperative control protocol.

Model Predictive Collision Avoidance Control for Object Transport of Unmanned Underwater Vehicle-Dual-Manipulator Systems

Unmanned underwater vehicle-dual-manipulator systems (UVDMSs) have attracted much research due to their humanoid operation capabilities, which have the advantage of cooperative manipulations and transporting underwater objects. Meanwhile, collision avoidance of UVDMSs is more challenging than that of unmanned underwater vehicle-dual manipulator systems (UVMSs). In this work, a model predictive control (MPC) approach is proposed for collision avoidance in objects transporting tasks of UVDMSs. The minimum distances of mutual manipulators and frame obstacles are handled as velocity constraints in the optimization of the UVDMS’s object tracking control. The command velocity generated by the model predictive kinematic controller is tracked by a dynamic inversion control scheme while model uncertainties are compensated by a neural network. Moreover, the tracking errors of the proposed dynamic controller are proved to be convergent by the Lyapunov method. At last, a three-dimensional (3D) UVDMS simulation platform is developed to verify the effectiveness of the proposed control strategy in the tasks of collision avoidance and object transport.

Intermittent event‐triggered control for input‐to‐state stability of stochastic systems

AbstractIn this article, we investigate the input‐to‐state stability (ISS) of stochastic systems via intermittent event‐triggered control. The control update sequence during the control intervals is determined through an event‐triggered mechanism (ETM), where the periodic ETM and continuous ETM are considered separately. For the continuous ETM, a positive minimum inter‐execution time is ensured by adding waiting time, which avoids Zeno behavior. For the periodic ETM, with the help of Halanay‐like inequality, the maximum allowable bound of the sampling period is given. The number of control updates is further reduced by adding a dynamic term. In addition, sufficient conditions for ISS in stochastic systems are proposed by designing an auxiliary timer and applying the Lyapunov method. Finally, two numerical examples are presented to verify the validity of the results.

Composite control based on FNTSMC and adaptive neural network for PMSM system

In this paper, a novel fixed-time non-singular terminal sliding mode control (NFNTSMC) method with an adaptive neural network (ANN) is proposed for permanent magnet synchronous motor (PMSM) system to improve PMSM performance. For nominal PMSM system without disturbance, a novel fixed-time non-singular terminal sliding mode control is designed to achieve fixed-time convergence property to improve the dynamic performance of the system. However, parameters mismatch and external load disturbances generally exist in PMSM system, the controller designed by NFNTSMC requires a large switching gain to ensure the robustness of the system, which will cause high-frequency sliding mode chattering. Therefore, an adaptive radial basis function (RBF) neural network is designed to approximate the unknown nonlinear lumped disturbance including parameters mismatch and external load disturbances online, and then the output of the neural network can be compensated to the NFNTSMC controller to reduce the switching gain and sliding mode chattering. Finally, the fixed-time convergence property and stability of the system are proved by Lyapunov method. The simulation and experimental results show that the presented strategy possesses satisfactory dynamic performance and strong robustness for PMSM system. And the proposed control scheme also provides an effective and systematic idea of the controller design for PMSM.

Persistence and Stochastic Extinction in a Lotka–Volterra Predator–Prey Stochastically Perturbed Model

The classical Lotka–Volterra predator–prey model is globally stable and uniformly persistent. However, in real-life biosystems, the extinction of species due to stochastic effects is possible and may occur if the magnitudes of the stochastic effects are large enough. In this paper, we consider the classical Lotka–Volterra predator–prey model under stochastic perturbations. For this model, using an analytical technique based on the direct Lyapunov method and a development of the ideas of R.Z. Khasminskii, we find the precise sufficient conditions for the stochastic extinction of one and both species and, thus, the precise necessary conditions for the stochastic system’s persistence. The stochastic extinction occurs via a process known as the stabilization by noise of the Khasminskii type. Therefore, in order to establish the sufficient conditions for extinction, we found the conditions for this stabilization. The analytical results are illustrated by numerical simulations.

Nonlinear Attitude and Altitude Trajectory Tracking Control of a Bicopter UAV in Geometric Framework

In this paper, we deal with a Bicopter drone that has two thrusters and two tilting servos. Both the position and attitude dynamics of Bicopter are globally expressed on the Special Euclidean group SE3. A simple control allocation method is proposed to map between the control wrench and actuator inputs for the Bicopter. A geometric nonlinear attitude and altitude tracking controller is developed for the Bicopter and the asymptotic stability analysis is performed using the Lyapunov method for the closed-loop nonlinear system. The performance of the proposed altitude and attitude stabilization controller is validated through experimental hardware developed in-house. The attitude controller performance is validated through simulations and shown to be comparable against an linear matrix inequality-based control law.

Fixed‐time stability of nonlinear impulsive systems: Sufficient criteria and necessary condition

AbstractThis article explores the criteria for fixed‐time stability (FxTS) within nonlinear impulsive dynamical systems, encompassing both stabilizing and destabilizing impulses. Initially, employing the Lyapunov method, the article delineates a criterion for FxTS, particularly when impulsive sequences are delineated by an average impulsive interval. Following this, the article innovatively applies two integral inequalities to introduce an optimized criterion for uniform impulsive sequences, evidencing its lesser conservativeness relative to the initial criterion. Moreover, this article derives a necessary condition for dynamical systems subjected to destabilizing impulses. Intriguingly, the investigation reveals the existence of a minimal threshold for the maximum impulsive interval, below which the system cannot achieve FxTS. This result elucidates why existing literature uniformly imposes a specific condition on the impulsive interval for destabilizing impulses. Finally, the efficacy of the proposed theory is underscored through numerical examples.

Multi-USV cooperative oil spill source seeking via extremum seeking combined with information fusion

This paper addresses the issue of cooperative source seeking for oil spill pollution by multiple unmanned surface vehicles (USVs), where the oil spill plume propagation is modeled by a linear advection–diffusion equation in two-dimensional space. The main objective is to construct a cooperative source seeking scheme for a group of USVs in the framework of extremum seeking combined with information fusion to solve such cooperative source seeking problem. To do this, both a gradient-based control law and a formation control law are included in the cooperative source seeking scheme. The gradient-based control law that is constructed by extremum-seeking control combined with information fusion drives the multiple USVs to approach the oil spill source, where the gradient is estimated online via the extremum-seeking control and the fused concentration value that is derived via a linear H∞ filter from the local concentration measurements at the USV’s current position. In the source seeking process, the USVs driven by the formation control law based on the virtual structure method maintain a preset circular formation. The convergence performance of the cooperative source seeking scheme is analyzed rigorously by the Lyapunov method. Finally, extensive numerical simulations are included to validate the theoretical algorithm.

Trajectory tracking of a wheeled mobile robot based on the predefined‐time sliding mode control scheme

AbstractIn this paper, a double‐loop control method is proposed to improve the trajectory tracking performance of a wheeled mobile robot (WMR) with slippage properties and external disturbances. Considering the strong robustness and the ability to enable the system to converge in a short time, the predefined‐time sliding mode control (PTSMC) strategy is applied to the design of a double‐loop controller. Furthermore, a nonlinear disturbance observer (NDO) is adopted to comply with the feedforward compensation of the external disturbances. In turn, the controller only needs to deal with the internal disturbance of the system under the premise of using the NDO, thereby reducing the burden of the sliding mode controller. Based on Lyapunov methods, the stability of the double‐loop controller and the NDO is analyzed. Finally, the feasibility of the double‐loop control strategy is proven by simulations of the WMR operating along circular, sin‐shaped, and eight‐shaped roads.

Adaptive Neural Control for a Class of Random Fractional-Order Multi-Agent Systems with Markov Jump Parameters and Full State Constraints

Based on an adaptive neural control scheme, this paper investigates the consensus problem of random Markov jump multi-agent systems with full state constraints. Each agent is described by the fractional-order random nonlinear uncertain system driven by random differential equations, where the random noise is the second-order stationary stochastic process. First, in order to deal with the unknown functions with Markov jump parameters, a radial basis function neural network (RBFNN) structure is introduced to achieve approximation. Second, for the purpose of keeping the agents’ states from violating the constraint boundary, the tan-type barrier Lyapunov function is employed. By using the stochastic stability theory and adopting the backstepping technique, a novel adaptive neural control design method is presented. Furthermore, to cope with the differential explosion problem in the design course, the extended state observer (ESO) is developed instead of neural network (NN) approximation or command filtering techniques. Finally, the exponentially noise-to-state stability in the mean square is analyzed rigorously by the Lyapunov method, which guarantees the consensus of the considered multi-agent systems and all the agents’ outputs are bounded in probability. Two simulation examples are provided to verify the effectiveness of the suggested control strategy.

Finite-time fault-tolerant control of a spacecraft using composite learning in the presence of moment of inertia uncertainty and external disturbances

In this paper, a neuro-adaptive finite-time fault-tolerant control for a class of nonlinear systems is proposed based on the backstepping method, considering model uncertainties and external disturbances. To compensate for model uncertainties, a neural network is employed, while external disturbances and the estimation error of the neural network are handled using a disturbance observer. Additionally, actuator faults are estimated and compensated for distinctly using two adaptive elements in the proposed control system. Moreover, to enhance the learning of the neural network, disturbance observer, and other adaptive elements, particularly in the lack of persistently exciting signals, a composite learning approach is utilized. To this end, a state-observer is introduced, and its estimation error along with the tracking error is used in the updating rules of all the adaptive elements in the design. Additionally, to prevent the explosion of terms in the backstepping method, a command filter with practical finite-time stable auxiliary variables is incorporated into the proposed controller. Finite-time convergence of the tracking error to zero is rigorously proved using the Lyapunov method. To demonstrate the effectiveness of the proposed control method, the problem of attitude control of a rigid spacecraft with moment of inertia uncertainty, external disturbances, and multiple actuator faults is considered since precise attitude control of a spacecraft is of great importance in different maneuvers. The results show that using the introduced control system, the spacecraft is able to track the desired trajectory under different types of uncertain dynamics.

Effects of stochastic perturbations on a phytoplankton allelopathy competitive system

In this paper, we investigate stochastic models of two species competitive phytoplankton system with allelopathic interaction. We construct stochastic models from deterministic models by introducing three different stochastic perturbations to the equations. For the first model, we prove that this model is asymptotically stable in probability at three different equilibrium points. It is also shown that the system is positively recurrent under some conditions. For the second model, we study the extinction, existence of positive recurrence and global asymptotic stability under some sufficient conditions. For the last system, almost sufficient and necessary conditions for global asymptotic stability, the existence of positive recurrence and extinction of each population are established. In addition, using a suitable stochastic Lyapunov method, we prove that there is a unique solution to three models, we also discuss the ultimate boundedness of the three stochastic systems. Moreover, some numerical simulations are introduced to illustrate our mathematical results. We show that stochastic models still retain the desirable stability of their deterministic counterparts if stochastic perturbations are relatively small.

Kaczmarz Projection Algorithms with Rank Two Gain Update

New Kaczmarz algorithms with rank two gain update, extended orthogonality property and forgetting mechanism which includes both exponential and instantaneous forgetting (implemented via a proper choice of the forgetting factor and the window size) are introduced and systematically associated in this paper with well-known Kaczmarz algorithms with rank one update. The parameter convergence was proved using Lyapunov method and convergence of the inverse of the information matrix can be used for further performance improvement. The performance of new algorithms is examined in the problem of estimation of the grid events in the presence of significant harmonic emissions.

Safe tracking in games: Achieving optimal control with unknown dynamics and constraints

AbstractThis paper introduces mix‐zero‐sum differential (MZSD) game theory to address multi‐player tracking systems, offering a better understanding of the coexistence of cooperation and competition among players. Within this framework, we present an optimal safety tracking control (OSTC) method, which incorporates a control barrier function (CBF) into the value function to ensure that the tracking error remains within a specified range, thus guaranteeing safety while achieving optimization. Simultaneously, to eliminate the need for system dynamics, we propose a novel approach leveraging off‐policy integral reinforcement learning (IRL) technology to obtain the Nash equilibrium solution of the MZSD games. We establish a unique critics–actors neural network (NN) structure that updates concurrently. Furthermore, we analyze stability and convergence using the Lyapunov method. We conduct two simulations to demonstrate the effectiveness of the proposed algorithm.

Almost Sure Stability of Complex-Valued Complex Networks: A Noise-Based Delayed Coupling Under Random Denial-of-Service Attacks.

This article is concerned with stability for stochastic complex-valued delayed complex networks under random denial-of-service (RDoS) attacks. Different from the existing literature on the stability of stochastic complex-valued systems that concentrate on moment stability, we investigate almost sure stability (ASS), where noise plays a stabilizing role. It is noted that, besides the vertex systems influenced by noise, the interactions among vertices are also at the mercy of noise. As a consequence, an innovative noise-based delayed coupling (NDC) in the presence of RDoS attacks is proposed first to accomplish the stability of complex-valued networks, where the RDoS attacks have a certain probability of triumphantly interfering with communications at active intervals of attackers. Namely, RDoS attacks considered are randomly launched at active periods, which is more realistic. In terms of the Lyapunov method and stochastic analysis theory, an almost sure exponential stability (ASES) criterion of the system discussed straightforwardly is developed by constructing a delay-free auxiliary system, while removing the traditional assumption of moment stability. The criterion strongly linked with topological structure, RDoS frequency, attack successful probability, and noise intensity reveals that the higher the noise intensity, the faster the convergence rate is for the stability of the network. In light of the criterion established, we present an algorithm that can be employed to analyze the tolerable attack parameters and the upper bound of the coupling delays, under the prerequisite of guaranteeing the stability of the network. Eventually, the theoretical results are applied to inertial complex-valued neural networks (ICNNs) and an illustrative example is presented to substantiate the efficiency of the theoretical works.

Adaptive Multi-Input Super Twisting Control for a Quadrotor: Singular Perturbation Approach

Singular perturbations easily result in the problem of high gain. Instead of high gain feedbacks, this work proposes an adaptive two-time-scale (TTS) super twisting control (STWC) algorithm for the attitude tracking of a singularly perturbed quadrotor system (SPQS). The bounds of uncertainties are assumed to be perturbed around their nominal values. First, a nominal STWC with slow and fast control gains is designed to achieve the finite-time attitude tracking for the nominal SPQS. The finite-time stability of the close-loop system is proven by utilizing the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -dependent Lyapunov method. Then, dynamically adapted slow and fast control gains initialized at their nominal values are used for the accurate finite-time convergence of tracking errors to the region around the origin, which avoids overestimating the values of control gains. Experimental results demonstrate the performance of the proposed controller in terms of attitude stabilization and tracking, and robustness to external disturbances.

Application of the Second Lyapunov Method for Getting the Conditions of Stability in Systems with Quadratic Right-Hand Side

Application of the Second Lyapunov Method for Getting the Conditions of Stability in Systems with Quadratic Right-Hand Side

An Adaptive Robust Control Method for Chaotic State of Permanent Magnet Synchronous Motor

In this paper, a chaotic control method based on adaptive robust controller is proposed to solve the chaos phenomenon of the permanent magnet synchronous motor system by mathematically analyzing the dynamic model of the permanent magnet synchronous motor system to eliminate or suppress chaos in the permanent magnet synchronous motors. This method also takes into account the parameter uncertainties (structural uncertainties) of the system and the external load torque interference (unstructured uncertainty).The former is compensated by adaptive control, and the latter by robust term. Feedforward elimination technology can integrate them together, and effectively reduce the influence of chaos on system performance and realize the high precision position tracking of the permanent magnet synchronous motor system. The Lyapunov method is used to prove the stability of the controller. The simulation results show that the method is better in robustness and control precision.