Lyapunov Asymptotic Stability Research Articles

The proof of Lyapunov asymptotic stability theorems for Caputo fractional order systems

A recent comment paper (Wu, 2021) pointed out that the proof for the existing asymptotic stability theorems of Caputo fractional order systems is questionable. Since these theorems are fundamental and widely adopted in many scenarios, such a comment indeed causes panic in fractional community and doubt from other academic communities. Under this background, the mentioned theorems are revisited and confirmed.

Stabilisation of nonlinear hybrid stochastic systems with time-varying delay by discrete-time feedback controls with a time delay

In this paper, we investigate stabilisation of hybrid stochastic systems with time-varying delay by feedback control based on discrete-time state and mode observations with a time delay. By constructing a proper Lyapunov functional, the , asymptotic and exponential stability of the controlled systems are studied in the pth moment sense for p>1. Meanwhile, we obtain the upper bound on the duration τ between two consecutive state and mode observations. At last, two numerical examples are illustrated to support our theoretical results.

Bipartite consensus-based formation control of high-order multi-robot systems with time-varying delays

This paper considers a multi-robot formation control scheme for high-order bipartite consensus under time-varying delays. First, a third-order bipartite consensus protocol under time-varying delays is presented, and then a high-order bipartite consensus protocol is extended from it. Then, necessary and sufficient conditions for third-order and high-order multi-agent systems without time-delays are given to attain bipartite consensus following the protocols. Communication time-delays are added to the protocols, and the theory of Lyapunov asymptotic stability is used to obtain sufficient conditions for the multi-agent systems with communication delays to ultimately attain a bipartite consensus. Furthermore, the given protocol is employed for the formation control of multi-robot systems, and a formation control scheme for multi-robot systems with time-varying delays is proposed. Finally, a few simulations are implemented to demonstrate the rationality and availability of the method.

Combined Input–Output Finite-time Stability with H∞ Static Output-feedback Control Approach for Active Suspension

Combined input–output finite-time stability condition with static output-feedback H ∞ control is proposed in this work to effectively attenuate the short-duration road disturbance. The control problem is formulated based on the pre-defined condition of input–output finite-time stability with H ∞ bound along with pre-established quarter-car active suspension model. Then, a feasible static output-feedback controller is designed via variable substitution approach and linear matrix inequalities. Output-feedback control’s initial infeasibility issue is solved by state-feedback technique. The controller is specifically designed for the system which exposed with short-duration exogenous disturbances hence, this work prefers finite-time approach instead Lyapunov asymptotic stability condition. To enumerate the significance of the proposed controller numerical examples are presented and its response is effectively analyzed in distinct domains (time and frequency). The theoretical results denote that the combined input–output finite-time stability condition with static output-feedback H ∞ control can enhance the vehicle performance better when compared with passive and conventional static output-feedback control scheme. Additionally, the controller robustness is verified.

FEA-based tracking control of flexible body switching dynamic structure

AbstractIn dynamically switched systems with unknown switching signal, the control system is conventionally designed based on the worst switching scenario to ensure system stability. Such conservative design demands excessive control effort in less critical switching configurations. In the case of continuum mechanics systems, such excessive control inputs result in increased structural deformations and resultant modeling uncertainties. These deformations alter differential equations of motion which cripple the task of control. In this paper, a new approach for tracking control of uncertain continuum mechanics multivariable systems undergoing switching dynamics and unknown time delay has been proposed. Control algorithm is constructed based on the mathematical rigid model of the plant and a Common Lyapunov Function (CLF) is proposed upon sliding hyperplane regarding all switching configurations. Considering the model-based nature of sliding mode control (SMC) and inevitable uncertainties induced from modeling simplifications of continuum system or parameter evaluation errors, Finite Element Analysis (FEA) is utilized to approximate total model uncertainties. To obtain robust stability, instead of conventional switching functions in the construction of control law, the control inputs are selected such that system dynamics reside within stability bounds which are calculated based on the Lyapunov asymptotic stability criterion. Therefore, the unwanted chattering issue caused by continuous switching is not observed in control input signals. Eventually, the accuracy of the proposed method has been verified through the student version of ANSYS® mechanical APDL-based simulations and its effectiveness has been demonstrated in multiple operating conditions.

A Takagi-Sugeno fuzzy-model-based finite-time H-infinity control for a hydraulic turbine governing system with time delay

Hydraulic turbine governing systems (HTGSs) are complicated nonlinear systems with time delays, and stability control is an important factor for the safe operation of hydropower units. The control of an HTGS that focuses on Lyapunov asymptotic stability may have poor transient performance, and external disturbance will also make the control effect undesirable. Therefore, a robust Takagi-Sugeno (T-S) fuzzy finite-time H-infinity H∞ (FFTH) control method for a nonlinear time-delay HTGS is proposed in this paper. First, a nonlinear time-delay HTGS model under a rigid water hammer is established, and its T-S fuzzy model is developed on the basis of fuzzy rules. Second, based on a selected Lyapunov candidate function, a sufficient condition where the fuzzy HTGS achieves finite-time stability and satisfies H∞ performance along with the existence of a fuzzy controller is given in the form of linear matrix inequalities (LMIs). Finally, the numerical simulation results show that the designed control scheme resists random disturbances well and has better robustness compared to the conventional methods. It provides a new idea for the stability control of the HTGS and a reference for the practical engineering application.

Finite-Time Fuzzy Control for Nonlinear Singularly Perturbed Systems With Input Constraints

Singularly perturbed systems have found widespread applications in practice. The existing results on singularly perturbed systems mainly focused on the Lyapunov asymptotic stability, which are unable to deal with the cases that the system states cannot exceed a given threshold during a fixed time interval. This article addresses the finite-time fuzzy control issue for discrete-time nonlinear singularly perturbed systems with input constraints. The aim is to guarantee the boundedness of the states of singularly perturbed systems during a finite-time interval. Based on the matrix inequality technique, some conditions are established to guarantee the finite-time boundedness of the fuzzy singularly perturbed systems, where the singularly perturbed parameter is independent so as to avoid the ill-conditioned problem caused by the small singularly perturbed parameter. The gains of the finite-time fuzzy controller can be obtained by solving some singularly perturbed parameter independent linear matrix inequalities. Finally, the proposed finite-time fuzzy controller design approach for nonlinear singularly perturbed systems is illustrated via the Van der Pol circuit.

Finite-time Takagi–Sugeno fuzzy controller design for hydraulic turbine governing systems with mechanical time delays

The grid has increasingly higher reliability requirements for hydropower supply and regulation, imposing higher requirements for the stability of hydraulic turbine governing system (HTGS). The mechanical structure of the unit is complex, and the adjustment of the active guide vane needs the hydraulic servo system with multistage amplifier to provide force, which makes the governor response output lag. The control of an HTGS that focuses on Lyapunov asymptotic stability may have poor transient performance. Therefore, a brand-new nonlinear finite-time Takagi–Sugeno fuzzy control scheme for HTGSs with mechanical time delay is proposed. First, based on the Takagi–Sugeno fuzzy model with time delay, a time-delay HTGS that considers the inherent delay effect of the HTGS relay component as well as its fuzzy model is presented. Second, based on the finite-time stability theorem, a nonlinear finite-time Takagi–Sugeno fuzzy controller for the HTGS is proposed and a practical stability condition is given in the form of matrix inequalities. Furthermore, the matrix inequalities are transformed into a set of linear matrix inequalities, which are easy to solve. Finally, numerical experiments on HTGS under various operating conditions demonstrate the proposed scheme’s effectiveness and superiority compared with the traditional proportional integral derivative control.

Nonlinear Control of a Multilink Aerial System and ASEKF-Based Disturbances Compensation

The actual article presents the modeling and control of a multilink unmanned aerial system whose dynamics is computed by means of the Euler-Lagrange approach. The aforementioned system is subjected to lumped disturbances, which comprise external disturbances and parametric uncertainties. An augmented-state extended Kalman filter intended to estimate endogenous and exogenous uncertainties is conceived and a trajectory-tracking controller fulfilling Lyapunov asymptotic stability is synthesized. A simulation stage is conducted to validate the effectiveness of the proposal.

Lyapunov-Stable Orientation Estimator for Humanoid Robots

In this paper, we present an observation scheme, with proven Lyapunov stability, for estimating a humanoid's floating base orientation. The idea is to use velocity aided attitude estimation, which requires to know the velocity of the system. This velocity can be obtained by taking into account the kinematic data provided by contact information with the environment and using the IMU and joint encoders. We demonstrate how this operation can be used in the case of a fixed or a moving contact, allowing it to be employed for locomotion. We show how to use this velocity estimation within a selected two-stage state tilt estimator: (i) the first which has a global and quick convergence (ii) and the second which has smooth and robust dynamics. We provide new specific proofs of almost global Lyapunov asymptotic stability and local exponential convergence for this observer. Finally, we assess its performance by employing a comparative simulation and by using it within a closed-loop stabilization scheme for HRP-5P and HRP-2KAI robots performing whole-body kinematic tasks and locomotion.

Randomized probabilistic approach for parametric uncertainties in unmanned helicopters

SummaryParametric uncertainties and the problems associated with the design of robust state feedback control tends to be a major concern for achieving stable operation of nonlinear systems. To overcome these drawbacks, this article provides a perspective on a randomized algorithm based probabilistic control for nonlinear systems. The control approach aims at development of randomized algorithms for control of unmanned helicopters which are subjected to uncertainties. A controller gain is synthesized by developing a robust probabilistic approach using sequential randomized algorithm with the two degree of freedom helicopter system. Subsequently, the stability and reachability for the action of developed controller is validated with the help of Lyapunov asymptotic stability and reach tube analysis, and the corresponding results were analyzed. A comparative assessment of simulation and real‐time control results of projected approaches on helicopter model are presented for validation of the controller response.

On a Periodic Inner Layer in the Reaction–Diffusion Problem with a Modular Cubic Source

The article studies a singularly perturbed periodic problem for the parabolic reaction–diffusion equation in the case of a discontinuous source: a nonlinearity describing the reaction (interaction). The case of the existence of an inner transition layer under conditions of an unbalanced and a balanced reaction is considered. An asymptotic approximation is constructed, and the asymptotic Lyapunov stability of periodic solutions in each of the cases is investigated. To prove the existence of a solution and its asymptotic stability, the asymptotic method of differential inequalities is used. The theoretical result is illustrated by an example and numerical calculations.

Model-Predictive-Control-Based Path Tracking Controller of Autonomous Vehicle Considering Parametric Uncertainties and Velocity-Varying

The automated steering control technology is crucial for an autonomous vehicle, but due to parametric uncertainties and time varying, the performance of automated steering control can be degraded. Therefore, a vehicle automated steering controller based on a model predictive control (MPC) approach is proposed in this article. First, considering tire nonlinear characteristics, the state and control matrices are modified, then the time-varying vehicle speed is considered and a linear parameter varying lateral model is established through utilizing a polytope with finite vertices to describe vehicle longitudinal velocity. Then, the MPC-based vehicle path tracking controller, which is robust against parameter uncertainties, is designed; the proposed controller can be solved via a set of linear matrix inequalities (LMI), which are derived from Lyapunov asymptotic stability and the minimization of the worst case infinite horizon quadratic objective function. The proposed control system is evaluated by both cosimulations of MATLAB/Simulink & CarSim and real-bus tests. Results show the effectiveness of the proposed controller, and it can ensure the control accuracy and strong robustness.

Robust Stabilization And Control Of Takagi-Sugeno Fuzzy Systems With Parameter Uncertainties And Disturbances Via State Feedback And Output Feedback

Most of the systems in the industry contain extreme non-linearity and uncertainties, which are hard to design and control utilizing general nonlinear systems. To conquer this sort of troubles, different plans have been produced in the most recent two decades, among which a popular methodology is Takagi-Sugeno fuzzy control. In this article, we present robust stabilization and control of Takagi-Sugeno (T-S) fuzzy systems with parameter uncertainties and disturbances. Initially, Takagi and Sugeno (TS) fuzzy model is used to represent a nonlinear system. Based on this T-S fuzzy model, fuzzy controller design schemes for state feedback and output feedback is also developed. Then, necessary conditions are derived for robust stabilization in the intelligence of Lyapunov asymptotic stability and are expressed in the arrangement of linear matrix inequalities (LMIs). The proposed system is implemented in the working platform of MATLAB and the simulation results are provided to illustrate the effectiveness of the proposed methods.

Fixed-Time Convergent Guidance Law with Impact Angle Control

The existing convergence control guidance laws are designed via the Lyapunov asymptotic stability theory or finite-time stability theory. However, guidance law based on the Lyapunov asymptotic stability theory would lead the states to zero only as time approaches infinity, which is imperfect theory. The convergence time for guidance laws based on finite-time stable theory is dependent on the initial states. A fixed-time convergent guidance law with impact angle control is proposed in this paper. The proposed guidance law consists of two parts. One is the heading error angle shaping term, and the other is the bias term to achieve the desired impact angle. The guidance command is continuous during the engagement without utilizing the switching logics. Unlike the existing guidance law in the literature, the fixed-time stability theory is utilized to ensure the impact angle error to converge to zero before the interception. Furthermore, the convergence rate is merely related to control parameters. Simulations are carried out to illustrate the effectiveness of the proposed guidance law.

Asymptotically Stable Stationary Solutions of the Reaction–Diffusion–Advection Equation with Discontinuous Reaction and Advection Terms

We study the Lyapunov asymptotic stability of the stationary solution of the spatially one-dimensional initial–boundary value problem for a nonlinear singularly perturbed differential equation of the reaction–diffusion–advection (RDA) type in the case where the advection and reaction terms undergo a discontinuity of the first kind at some interior point of an interval. Sufficient conditions are derived for the existence of a stable stationary solution with a large gradient near the point of discontinuity. An asymptotic method of differential inequalities is used to prove the existence and stability theorems. The resulting stability conditions can be employed to create mathematical models and develop numerical methods for solving “stiff” problems arising in various applications, for example, when simulating combustion processes and in nonlinear wave theory.

The Existence and Stability of Periodic Solutions with a Boundary Layer in a Two-Dimensional Reaction-Diffusion Problem in the Case of Singularly Perturbed Boundary Conditions of the Second Kind

The existence of time-periodic solutions of the boundary-layer type to a two-dimensional reaction–diffusion problem with a small-parameter coefficient of a parabolic operator is proved in the case of singularly perturbed boundary conditions of the second kind. An asymptotic approximation with respect to the small parameter is constructed for these solutions. The set of boundary conditions for which these solutions exist is studied and the local uniqueness and asymptotic Lyapunov stability are established for them. It is shown that, unlike the analogous Dirichlet problem, for which such a solution is unique, there can be several solutions of this kind for the problem under consideration, each of which has its domains of stability and local uniqueness. To prove these facts, results based on the asymptotic principle of differential inequalities are used. Keywords: , , , , , , .

Asymptotically Stable Periodic Solutions in One Problem of Atmospheric Diffusion of Impurities: Asymptotics, Existence, and Uniqueness

The basis of this work is the use of modern methods of asymptotic analysis in reaction–diffusion–advection problems in order to describe the classical boundary-layer periodic solution of one singularly perturbed problem for the nonlinear diffusion–advection equation. An asymptotic approximation of an arbitrary order of such a solution is constructed, and the formal construction is justified. The uniqueness theorem is proved, the asymptotic Lyapunov stability is established, and the local domain of attraction of the boundary-layer periodic solution is found. One of the applications of this result to atmospheric diffusion problems is discussed, namely, mathematical modeling of the processes of transport and chemical transformation of anthropogenic impurities in the atmospheric boundary layer with allowance for periodic, e.g., daily or seasonal changes. The analytical algorithms developed for this problem as well will form the basis for a new method for calculating daily corrected emission fluxes of anthropogenic impurities from urban sources, which will make it possible to develop improved methods for determining daily integral emissions from the entire territory of a city or a urban agglomeration, based on the use of analytical solutions of model problems in combination with information obtained on a network of atmospheric monitoring stations.

Multiple Lagrange stability and Lyapunov asymptotical stability of delayed fractional-order Cohen–Grossberg neural networks**Project supported by the Natural Science Foundation of Zhejiang Province of China (Grant Nos. LY18F030023, LY17F030016, LQ18F030015, and LY18F020028) and the National Natural Science Foundation of China (Grant Nos. 61503338, 61773348, and 61972354).

This paper addresses the coexistence and local stability of multiple equilibrium points for fractional-order Cohen–Grossberg neural networks (FOCGNNs) with time delays. Based on Brouwer’s fixed point theorem, sufficient conditions are established to ensure the existence of equilibrium points for FOCGNNs. Through the use of Hardy inequality, fractional Halanay inequality, and Lyapunov theory, some criteria are established to ensure the local Lagrange stability and the local Lyapunov asymptotical stability of equilibrium points for FOCGNNs. The obtained results encompass those of integer-order Hopfield neural networks with or without delay as special cases. The activation functions are nonlinear and nonmonotonic. There could be many corner points in this general class of activation functions. The structure of activation functions makes FOCGNNs could have a lot of stable equilibrium points. Coexistence of multiple stable equilibrium points is necessary when neural networks come to pattern recognition and associative memories. Finally, two numerical examples are provided to illustrate the effectiveness of the obtained results.

Existence and Asymptotic Stability of Periodic Two-Dimensional Contrast Structures in the Problem with Weak Linear Advection

We consider the boundary-value singularly perturbed time-periodic problem for the parabolic reaction-advection-diffusion equation in the case of a weak linear advection in a two-dimensional domain. The main result of the present paper is the justification, under certain sufficient assumptions, of the existence of a periodic solution with internal transition layer near some closed curve and the study of the Lyapunov asymptotic stability of such a solution. For this purpose, an asymptotic expansion of the solution is constructed; the justification of the existence of the solution with the constructed asymptotics is carried out by using the method of differential inequalities. The proof of Lyapunov asymptotic stability is based on the application of the so-called method of contraction barriers.