Lyapunov Asymptotic Stability Research Articles

On the Global Existence of Solution and Lyapunov Asymptotic Practical Stability for Nonlinear Impulsive Caputo Fractional Derivative via Comparison Principle

This paper examines the existence of maximal solution of the comparison differential system and also establishes sufficient conditions for the asymptotic practical stability of the trivial solution of a nonlinear impulsive Caputo fractional differential equations with fixed moments of impulse using the vector Lyapunov functions. First, it was discovered that the vector form of the Lyapunov function was majorized by the maximal solution of the comparison system. From the results obtained, it was also established that the main system is asymptotically practically stable in the sense of Lyapunov.

Active impedance control based adaptive locomotion for a bionic hexapod robot

AbstractIn recent years, with the continuous development of human exploration of the natural world, there has been a growing demand across various fields for robots capable of free movement in diverse environments. In this study, we address the issue of compliant control for a hexapod robot in diverse environments and propose a novel control method based on an adaptive impedance model for position control. Our approach enables the hexapod robot to stabilize foot force on complex terrains while preserving balance and body height. Specifically, we analyze the algorithm's parameters and stability by establishing the hexapod robot's structural and impedance control models. To tackle this challenge, we introduce an adaptive impedance control algorithm that estimates environmental parameters using Lyapunov's asymptotic stability theorem and achieves tracking of actual foot‐end forces to desired foot forces. Furthermore, to ensure body stability and height, we incorporate attitude feedback and body feedback. Experimental results from foot force control experiments conducted on a multilegged robot demonstrate that our proposed algorithm enhances the adaptability and robustness of the robot. This research holds significant implications for the stable control of hexapod robots in complex environments and has promising practical applications.

Existence and Stability of a Stationary Solution in a Two-Dimensional Reaction-Diffusion System with Slow and Fast Components

In the paper, the existence of a stable stationary solution in a reaction-diffusion system with slow and fast components in a two-dimensional spatial variable case is investigated. The theorem of the existence of a stationary solution with boundary layers in the case of Dirichlet boundary conditions is proven, its asymptotic approximation is constructed, and conditions for Lyapunov asymptotic stability of this solution are obtained. The research is based on the asymptotic method of differential inequalities, applied to a new class of problems. This result is practically important both for various applications described by similar systems and for the application of numerical stationing methods when solving elliptical boundary value problems.

Stochastic response and stability analysis of nonlinear vehicle energy-regenerative suspension system with time delay

With the rapid development of new energy vehicles, a novel vehicle energy-regenerative suspension for simultaneous vibration suppression and energy harvesting has become one of the key development directions of the next generation of vehicle design. In order to improve the performance of suspension system, the time-delayed feedback control is applied to the novel suspension system subjected to different external excitation. At the same time, there are few studies related to the stochastic dynamics of nonlinear vehicle energy-regenerative suspension system with time delay. This paper studies stochastic response and stability of nonlinear vehicle energy-regenerative suspension with time-delayed feedback control under the excitation of Gaussian white noise. First of all, a stochastic dynamic model of the nonlinear vehicle energy-regenerative suspension with time delay system is established. Secondly, using the stochastic averaging method based on generalized harmonic function, the steady-state probability density function (PDF) of amplitude is obtained. The accuracy of the presented approach is verified through the comparison of analytical and numerical solutions. In addition, through combining the stochastic averaging method with the Khasminskii method, the top Lyapunov exponent (TLE) of the system is obtained, and then the influence of time delay and system parameters on the asymptotic Lyapunov stability is discussed. Finally, the effects of time delay and system parameters on system performance are also studied.

A novel impact angle control guidance law design via predefined time convergence theory

A predefined time guidance law with impact angle control is proposed for intercepting the stationary target. With such a guidance law, the convergence time of the impact angle error for the missile is independent of the initial conditions, and it can be selected at the will of the designer. First, a heading error shaping method based on the Lyapunov asymptotic stability theory will be introduced, which ensures the missile hits the target. Then, a bias term will be designed and introduced into this method, which can make sure the impact angle error converges to zero in the predefined time. The Lyapunov stability analysis is discussed in detail for the resulting predefined-time convergent guidance system. Finally, simulation examples are carried out to illustrate the effectiveness of the proposed guidance law.

Control systems with Thyristor converters

A system is considered, in the control loop of which there are thyristor converters or a special type of indirect control system. An example of constructing self-oscillations in a system describing the behavior of electrical devices using such elements in the control loop is given. Solutions for self-oscillations of a given period are constructed. Conditions for the existence of orbital asymptotic stability and Lyapunov stability for periodic solutions with a given period are presented, and a control loop is synthesized that ensures the existence of such solutions.

Системы управления с тиристорными преобразователями

The article discuses systems whose control circuit contains thyristor converters or a special type of indirect control system. An example of constructing self-oscillations in a system describing the behavior of electrical devices using such elements in a control loop is given. The criterion of orbital stability and the method of synthesis of stabilizing controls are proposed. The conditions for the existence of orbital asymptotic stability and Lyapunov stability for periodic solutions with a given period are presented. A control loop has been synthesized that ensures the existence of such solutions. Examples of self-oscillations are given for structurally linear systems. Solutions for self-oscillations of a given period are constructed.

Existence and Stability of Solutions with Internal Transition Layer for the Reaction–Diffusion–Advection Equation with a KPZ-Nonlinearity

We study a boundary value problem for a quasilinear reaction–diffusion–advection ordinary differential equation with a KPZ-nonlinearity containing the squared gradient of the unknown function. The noncritical and critical cases of existence of an internal transition layer are considered. An asymptotic approximation to the solution is constructed, and the asymptotics of the transition layer point is determined. Existence theorems are proved using the asymptotic method of differential inequalities, the Lyapunov asymptotic stability of solutions is proved by the narrowing barrier method, and instability theorems are proved with the use of unordered upper and lower solutions.

Stabilizing traffic flow by autonomous vehicles: Stability analysis and implementation considerations

The emergence of autonomous vehicles (AVs) is expected to revolutionize road transportation and it has been demonstrated that AVs can be employed as mobile actuators to effectively dissipate stop and go waves. This paper proposes a comprehensive theoretical framework including stability and implementation considerations for analyzing traffic platoon control via sparse AVs and discussing the number of HDVs that can be stabilized by one AV. Considering multiple disturbance sources, we first discuss the output error composition and analyze traffic dynamics in a platoon system. Then we discuss the Lyapunov asymptotic stability (LAS) of the system based on eigenvalue analysis and propose a novel ring string stability (RSS), which ensures the boundedness of disturbance error and guarantees that the effect of disturbance will not be amplified over a lap. The proposed RSS is flexible and applicable to mixed traffic, since the string instability for predecessor-follower pairs are allowed, provided they can be compensated by AVs. In addition, the practical constraints including driving safety and vehicle physical limitations are modeled. The worst-case evaluation of constraints is conducted to reproduce the critical scenarios in real life. Finally, we conduct numerical experiments and analyze the traffic platoon control from the aspects of both stability and practical implementations. New findings include: (1) The introduction of AVs is beneficial to dampen the waves and stabilize the HDV flow. (2) Considering the implementation issues, there is an upper bound of HDVs that can be stabilized by one single AV. (3) As the number and amplitudes of exogenous disturbances increase, it will be more challenging for sparse AVs to ensure stability and meanwhile satisfy the practical constraints.

Substability and Substabilization: Control of Subfully Actuated Systems.

The region of attraction of the Lyapunov asymptotic stability at the origin is defined to be a ball centered at the origin, which is clearly simply connected and also bounded in the local case. In this article, the concept of substability is proposed, which allows "gaps" and "holes" in the region of attraction of the Lyapunov exponential stability, and also allows the origin to be a boundary point of the region of attraction. The concept is meaningful and useful in many practical applications, but is particularly made so with the control of single- and multi-order subfully actuated systems. Specifically, the singular set of a sub-FAS is first defined, and a substabilizing controller is then designed such that the closed-loop system is a constant linear one with an arbitrarily assignable eigen-polynomial, but with its initial values restricted within a so-called region of exponential attraction (ROEA). Consequently, the substabilizing controller drives all the state trajectories starting from the ROEA exponentially to the origin. The introduced concept of substabilization is of great importance because, on the one side, it is often practically useful since the designed ROEA is often large enough for certain applications, while on the other side, Lyapunov asymptotically stabilizing controllers can be further easily established based on substabilization. Several examples are given to demonstrate the proposed theories.

Model-Free Sliding Mode Enhanced Proportional, Integral, and Derivative (SMPID) Control

This study proposes a type of Sliding Mode-based Proportional, Integral, and Derivative (SMPID) controllers to establish a model-free (treat dynamic plants as a whole uncertainty) sliding model control (MFSMC) platform for Bounded-Input and Bounded-Output (BIBO) dynamic systems. The SMPID design (1) proposes a sliding mode error (rather than error) as the PID input, (2) directly links to Lyapunov asymptotic stability to provide total robust nonlinear dynamic inversion (NDI), and (3) reduces the chattering effects in terms of Lyapunov definite positive stability. Further, the study proposes a general SMC framework to accommodate asymptotic time stabilisation and finite-time stabilisation for both model-based and model-free designs. A U-control framework is presented to integrate the SMPID control (for NDI) and an invariant control (IC) (for specifying the whole control system’s dynamic and static responses), which significantly relaxes the PID tunings and generates the specified performance. To provide assurance and guidance for applications and expansions, this study presents the relevant fundamental analyses and transparent simulated bench tests. It should be noted that the new SMPID in forms of u=SMPID(σ(e))=PID(sliding-mode) is different from that studied u=sliding-mode(PID(e)) in expression and functionality.

Bifurcation and stability analysis of a hybrid energy harvester with fractional-order proportional–integral–derivative controller and Gaussian white noise excitations

Hybrid energy harvester (HEH) has received much attention in the field of energy harvesting. The HEH inevitably suffers from external stochastic disturbances in the working environment such as winds, waves, and ocean currents. Only few works deal with the stochastic dynamics of a hybrid energy harvester with fractional-order proportional–integral–derivative (PID) controller. This paper aims to investigate bifurcation control and stability analysis of a HEH with fractional-order PID controller subjected to Gaussian white noise excitations. An approximately equivalent dimensionally reduced system is formulated via the variables transformation method, and its approximate stationary solutions are derived through the stochastic averaging method. One example is worked out in detail to verify the effectiveness of the proposed procedure. It is shown that the analytical results provide a good approximation to the numerical simulation results. The influences of system parameters on the stochastic bifurcation and the asymptotic Lyapunov stability are also discussed.

Dynamic surface sliding mode control of chaos in the fourth-order power system

As a typical complex nonlinear model, the power system is influenced by the parameters and initial values, and chaotic oscillations can easily occur in the actual operation process, which seriously endangers the stability of its operation and may even lead to the collapse of the whole power system. To suppress chaotic oscillations in power systems, a method of dynamic surface sliding mode control is first proposed, which overcomes the “term explosion” problem in the traditional backstepping design process by introducing first-order low-pass filters, and reduces the design steps of dynamic surface controllers and the complexity of stability analysis by combining sliding mode control. And then, the steadiness of the closed-loop controlled system is evaluated according to Lyapunov asymptotic stability theory. Lastly, the simulation and experiment are implemented to show that the designed control method can effectively inhibit the power system's chaotic oscillations and stabilize it.

Asymptotic stability of non‐instantaneous impulsive systems and T‐S fuzzy non‐instantaneous impulsive control for nonlinear systems

AbstractThis paper studies the asymptotic stability for a class of non‐instantaneous impulsive systems and Takagi‐Sugeno (T‐S) fuzzy non‐instantaneous impulsive control for linear and nonlinear systems. First, a class of concrete comparison system for a more accurate and universal nonlinear non‐instantaneous impulsive model is constructed. Then, the less conservative sufficient conditions of Lyapunov asymptotic stability are studied by impulsive differential inequalities and generalized comparison principle. On the other hand, an improved non‐instantaneous impulsive control scheme with T‐S fuzzy model is also presented for the asymptotic stability of the nonlinear systems. Finally, simulation results on controlling a linear and a nonlinear system are presented to illustrate the rationality and effectiveness of the obtained results.

Feedback stabilization of quasi nonintegrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A procedure for designing a feedback control to asymptotic Lyapunov stability with probability one of quasi nonintegrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations is proposed. First, a one dimensional partially averaged Itô stochastic differential equation for controlled Hamiltonian is derived from the motion equations of the system by using the stochastic averaging method. Second, the dynamical programming equation for the ergodic control problem of the averaged system with undetermined cost function is set up based on the dynamical programming principle and the jump–diffusion chain stochastic differential rules. The optimal control law is obtained by solving the dynamical programming equation. Third, the analytical expression for the largest Lyapunov exponent of the averaged system is derived. Finally, the asymptotic Lyapunov stability with probability one of the originally controlled system is analyzed approximately by using the largest Lyapunov exponent. The cost function and optimal control forces are determined by the requirements of stabilizing the system. An example is worked out in detail to illustrate the effectiveness of the proposed method for stabilization control, and the control effect of the proposed feedback stabilization varies with the change of parameters is also studied in this paper, such as, the greater the excitation intensity of Gaussian and Poisson white noise, the better the stabilization control effect.

Economic MPC of Markov Decision Processes: Dissipativity in undiscounted infinite-horizon optimal control

Economic Model Predictive Control (MPC) dissipativity theory is central to discussing the stability of policies resulting from minimizing economic stage costs. In its current form, the dissipativity theory for economic MPC applies to problems based on deterministic dynamics or to very specific classes of stochastic problems, and does not readily extend to generic Markov decision processes. In this paper, we clarify the core reason for this difficulty, and propose a generalization of the economic MPC dissipativity theory that circumvents it. This generalization focuses on undiscounted infinite-horizon problems and is based on nonlinear stage cost functionals, allowing one to discuss the Lyapunov asymptotic stability of policies for Markov decision processes in terms of the probability measures underlying their stochastic dynamics. This theory is illustrated for the stochastic linear quadratic regulator with Gaussian process noise, for which a storage functional can be provided explicitly. For the sake of brevity, we limit our discussion to undiscounted Markov decision processes.

Stability analysis for fractional‐order neural networks with time‐varying delay

SummaryThis paper focuses on the stability analysis for fractional‐order neural networks with time‐varying delay. A novel Lyapunov's asymptotic stability determination theorem is proved, which can be used for fractional‐order systems directly. Different from the classical Lyapunov stability theorem, constraint condition on the derivative of Lyapunov function is revised as an uniformly continuous class‐K function in the fractional‐order case. Based on this novel Lyapunov stability theorem and free weight matrix method, a new sufficient condition on Lyapunov asymptotic stability of fractional‐order Hopfield neural networks is derived by constructing a suitable Lyapunov function. Moreover, two numerical examples are provided to illustrate the effectiveness of these criteria.

Quantized Output Feedback Control of AFS for Electric Vehicles With Transmission Delay and Data Dropouts

This article proposes a robust <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_\infty $ </tex-math></inline-formula> output feedback control strategy to improve vehicle lateral stability through active front wheel steering with quantization error, transmission delay and data dropouts. Since vehicle lateral dynamics presents inherently uncertain challenges, the polytopic technique is used and the variation of tire cornering stiffness is compensated via norm-bounded uncertainty. Meanwhile, considering that the measurement outputs are inevitably suffer from network-induced constrains, i.e. the quantization error, transmission delay and data dropouts, which degrade the stability and handling performance of vehicle, a robust gain-scheduling <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_\infty $ </tex-math></inline-formula> output feedback controller design strategy is proposed and the stability conditions are presented in the form of linear matrix inequalities that are derived from Lyapunov asymptotic stability and quadratic <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_\infty $ </tex-math></inline-formula> performance. Finally, two simulation cases are carried out with MATLAB/Simulink-Carsim to validate the effectiveness of the proposed method.

Dynamical analysis of an SDOF quasi‐linear system with jump noises and multitime‐delayed feedback forces

AbstractIn this paper, stochastic dynamics of a single degree‐of‐freedom quasi‐linear system with multitime delays and Poisson white noises are investigated using an approximate procedure based on the stochastic averaging method. The simplified equations, including the averaged stochastic differential equation and the averaged generalized Fokker–Planck–Kolmogorov equation, are obtained to calculate the probability density functions (PDFs) to explore stationary responses. The expression of the Lyapunov exponent is presented to examine the asymptotic stochastic Lyapunov stability. An illustrative example of a quasi‐linear oscillator with two Poisson white noises controlled by two time‐delayed feedback forces is worked out to demonstrate the validity of the proposed method. The approximate stationary PDFs of stochastic responses and asymptotic stochastic stability are demonstrated numerically and theoretically. The results show that the Gaussian white noise has a stronger influence on the dynamics than the Poisson white noise with a small mean arrival rate. Moreover, the influence of the time delay and noise parameters on stochastic dynamics is investigated. It is found that the PDFs under the Poisson white noise approach those under Gaussian white noise as the mean arrival rate increases. The time delay can induce stochastic P‐bifurcation of the system. It is also found that the increase of time delay and the mean arrival rates of the Poisson white noises will broaden the unstable parameter region. The comparison between numerical and theoretical results shows the effectiveness of the proposed method.

Family of Bounded Regulators with Variable Gains and Lyapunov Asymptotic Stability for Robot Manipulators

In this paper, the regulation problem for robot manipulators in joint space through the proposal of a new family of bounded regulators with variable gains is presented. The proposed regulators have bounded functions that replace the classical position error and the velocity; moreover, the variable gains are formed by a family of Lipchitz functions with the position error and the velocity as their arguments. This structure avoids exceeding the physical limits of the servomotors. A strict Lyapunov function is proposed to demonstrate the global and asymptotic stability. Finally, the functionality and performance of the proposal are examined by experimental results on a direct-drive-robot of 3-degrees-of-freedom against the PD regulator