Domain Of Attraction Research Articles

Advanced Optimal Tracking Control With Stability Guarantee via Novel Value Learning Formulation.

In this article, to solve the optimal tracking control problem (OTCP) for discrete-time (DT) nonlinear systems, general value iteration (GVI) scheme and online value iteration (VI) algorithms with novel value function are discussed. First, the disadvantage of the traditional value function for the OTCP is presented and the novel value function is introduced. Second, we analyze the monotonicity and convergence of GVI and establish the admissibility condition of GVI to evaluate the admissibility of the current iterative control. Note that a novel approach is introduced to analyze the admissibility. Third, based on the attraction domain, improved control policies with online VI can be obtained by judging the location of the current tracking error and reference point. Finally, the stability of the online VI-based control system is guaranteed. Besides, we provide two simulation examples to show the performance of the proposed methods.

Estimation of the Domain of Attraction on Controlled Nonlinear Neutral Complex Networks via Razumikhin Approach

This paper is devoted to dealing with the issue of the estimation of the domain of attraction (DOA) for highly nonlinear neutral complex networks (HNNCNs) with time delays. Firstly, by the Razumikhin approach, we establish several novel lemmas on the estimation of DOA for highly nonlinear neutral differential systems. The cases of bounded non-differentiable delays and unbounded proportional delays are discussed, respectively. Subsequently, by utilizing the proposed lemmas, combining the Lyapunov stability theory and inequality technique, the estimation of DOA on HNNCNs with bounded delays or proportional delays is derived when the chosen control gain is sufficiently large. If initial values start from DOA, then the states of systems will exponentially or polynomially converge to the equilibrium point, which means that the local exponential or polynomial synchronization of HNNCNs is realized. Additionally, the weighted outer-coupling matrix of complex networks is not required to be symmetric, which means that the derived results can be applied to both the undirected networks and directed networks. Finally, several numerical examples are provided to illustrate the feasibility of theoretical findings.

Learning Lyapunov terminal costs from data for complexity reduction in nonlinear model predictive control

AbstractA classic way to design a nonlinear model predictive control (NMPC) scheme with guaranteed stability is to incorporate a terminal cost and a terminal constraint into the problem formulation. While a long prediction horizon is often desirable to obtain a large domain of attraction and good closed‐loop performance, the related computational burden can hinder its real‐time deployment. In this article, we propose an NMPC scheme with prediction horizon and no terminal constraint to drastically decrease the numerical complexity without significantly impacting closed‐loop stability and performance. This is attained by constructing a suitable terminal cost from data that estimates the cost‐to‐go of a given NMPC scheme with long prediction horizon. We demonstrate the advantages of the proposed control scheme in two benchmark control problems.

Systematic Controller Design for Inverter-Based Microgrids With Certified Large-Signal Stability and Domain of Attraction

Systematic Controller Design for Inverter-Based Microgrids With Certified Large-Signal Stability and Domain of Attraction

Composite quadratic Lyapunov function event-triggered control of vertical take-off and landing aircraft systems with actuator saturation

This paper deals with the problem of event-triggered control (ETC) for vertical take-off and landing (VTOL) aircraft systems under actuator saturation. By introducing the VTOL system with saturated actuators, a composite quadratic Lyapunov function (CQLF) approach is first proposed to describe the problem of the event triggering mechanism. The proposed transfer mechanism not only further saves communication resources, but also further suppresses the conservatism of the attraction domain estimation. Then, sufficient conditions are established via linear matrix inequality (LMI) by incorporating new event triggering mechanism condition and prescribed passive performance metrics under the closed-loop system that is asymptotically stable and strictly passive, respectively. Finally, the method proposed in this paper can effectively reduce the number of triggers and expand the domain of attraction. The validity and feasibility of the obtained results are demonstrated by two example simulations.

An Extension of the Feedback Linearization Method in the Control Problem of an Inverted Pendulum on a Wheel

This paper continues previous studies on designing stabilizing control laws for a mechanical system consisting of a wheel and a pendulum suspended on its axis. The control objective is to simultaneously stabilize the vertical position of the pendulum and a given position of the wheel. The difficulty of this problem is that the same control is used to achieve two targets, i.e., stabilize the pendulum angle and the wheel rotation angle. Previously, the output feedback linearization method was applied to this problem. The sum of the pendulum angle and the wheel rotation angle was taken as the output. For the closed loop system to be not only asymptotically stable in the output but also to have asymptotically stable zero dynamics, a dissipative term was added to the output-stabilizing control law. Below, a two-parameter modification of this law is described. Along with the dissipative term, we introduce a positive factor. The more general parameterization allows stabilizing this system in the cases where the control law proposed previously appeared ineffective. The properties of the new control law are investigated, and the attraction domain is estimated. The estimation procedure is reduced to checking the feasibility of linear matrix inequalities.

Quasi-synchronization of heterogeneous neural networks with hybrid time delays via sampled-data saturating impulsive control

This paper focuses on the quasi-synchronization matter of directed coupled heterogeneous neural networks with actuator saturation and hybrid time-varying delays under the sampled-data impulsive control. By utilizing the extended impulsive comparison principle and a dead-zone function representation of saturation nonlinearity function, multiple sufficient criteria are established for attaining the quasi-synchronization of heterogeneous dynamic neural networks. Meanwhile, the maximum estimating domain of attraction is acquired by solving an optimization problem which is confined by the linear matrix inequalities converted from the bilinear matrix inequalities. In addition, through utilizing the extended formula for the variation of parameters with hybrid delays, precise exponential convergence rate and theoretical error bound are calculated. Ultimately, the theoretical results are illustrated by means of a numerical simulation.

Asynchronous dissipative control of time-delay singular Markovian jump systems with actuator saturation

This paper focuses on the problem of asynchronous dissipative control for time-delay singular Markovian jump systems (SMJSs) with actuator saturation. In order to describe the asynchronous phenomenon between the system and the controller, a hidden Markov model (HMM) is adopted. In addition, the problem of actuator saturation is considered. Firstly, by the mode-dependent Lyapunov function, some novel sufficient criteria for guaranteeing the stochastic admissibility with dissipative performance of the system are proposed. Secondly, according to the derived criteria, a novel approach is established to design the controller gains. Thirdly, the domain of attraction of the system is given by the optimal method. Finally, two examples are given to verify the correctness of the proposed approach.

Methods for Studying Asymptotics of Solutions to Singularly Perturbed Equations in Complex Domains

When studying the asymptotic behavior of solutions to singularly perturbed equations with analytic functions, various methods are used. The main goal of this work is to conduct a comprehensive analysis of the methods used and divide them into groups. To achieve this goal, a scalar, nonlinear first-order equation with an initial condition is considered. The equation is considered in a certain circle, the complex plane of the independent variable, and the real part of the function — the coefficient, for a linear unknown function, changes its sign from negative to positive on part of the segment of the real axis contained in the circle. The concept of the domain of attraction of the solution of a singularly perturbed equation to the solution of a nonlinear equation (degenerate equation) is introduced. The task is set to prove the existence of a region of attraction. In the process of solving the problem, the methods used are divided into three groups.

Accelerated Gradient Descent Driven by Lévy Perturbations

In this paper, we mainly consider two kinds of perturbed accelerated gradient descents driven by Lévy perturbations, which is of great importance for enhancing the global search ability. By using Lévy representation, Lévy perturbations can be divided into two parts: small jumps and large jumps, whose properties are then carefully discussed. By introducing the concept of attraction domain for local minima, Makovian transition properties are proven for the proposed two perturbed accelerated gradient descents with different infinitesimal matrices. Finally, all the results are extended to the vector case and two simulation examples are provided to validate all the conclusions.

Evolution of trust in the N-player trust game with the margin system

Investigations into trust behavior between investors and trustees using trust games have gained significant attention in recent years. However, understanding the process of trust evolution remains challenging. The margin system, widely employed in the financial industry as an economic and behavioral incentive mechanism, aims to maintain order in financial markets and protect participants' rights and interests. Despite its extensive use, the precise influence of the margin system on trust evolution dynamics remains unclear. To address this gap, we propose an N-player trust game model that incorporates a margin system within an infinite well-mixed population. In our model, a third-party institution collects a proportionate amount of margin from trustees, which is returned upon fulfilling their obligations but forfeited otherwise. By utilizing replicated dynamic equations, we find that implementing the margin system can establish a stable coexistence state between investors and trustworthy trustees, representing a significant advancement in trust evolution. Additionally, an intriguing finding arises from our observation of an inverted U-shaped relationship between the magnitude of the attraction domain for coexistence and the coefficient of the margin. This suggests that excessively high margin proportions can discourage trustees' participation, while excessively low proportions may fail to effectively deter dishonest behavior. Optimal margin proportions can effectively promote trust evolution.

The positivity and event-triggered stabilization of Takagi-Sugeno fuzzy systems with actuator saturation

The positivity and event-triggered stabilization of Takagi-Sugeno fuzzy systems with actuator saturation are investigated in this paper. Different from the existing quadratic event-triggered mechanism (ETM), a linear ETM is constructed, which can be reduced to a time-triggered mechanism (TTM). Based on parallel distribution compensation, an event-triggered fuzzy controller is designed, and sufficient conditions under the estimation of a certain domain of attraction are proposed to ensure the ETM-based and TTM-based closed-loop systems are positive and stable. The controller gain is solved directly by the proposed conditions. And then these sufficient conditions are extended to the conditions under the estimation of the largest domain of attraction by establishing an optimization problem. Furthermore, the linear ETM is proved to exclude the Zeno behavior of the system. Finally, two simulation examples are utilized to verify the effectiveness of the proposed event-triggered control scheme.

The Fast Recurrent Subspace on an N-Level Quantum Energy Transport Model

The fast recurrent subspace (the biggest support of all invariant states) of a weak coupling limit type quantum Markov semigroup modeling a quantum transport open system of [Formula: see text]-energy levels is determined. This is achieved by characterizing the structure of all the invariant states and their spectra in terms of a natural generalization of the discrete Fourier transform operator. Finally, the attraction domains and long-time behaviour of the evolution are studied on hereditary subalgebras where faithful invariant states exist.

Data-Driven Computational Methods for the Domain of Attraction and Zubov's Equation

Data-Driven Computational Methods for the Domain of Attraction and Zubov's Equation

Pattern-Moving-Modelling and Analysis Based on Clustered Generalized Cell Mapping for a Class of Complex Systems

Considering a class of complex nonlinear systems whose dynamics are mostly governed by statistical regulations, the pattern-moving theory was developed to characterise such systems and successfully estimate the outputs or states. However, since the pattern class variable is not computable directly, this study establishes a clustered generalized cell mapping (C-GCM) to reveal system characteristics. C-GCM is a two-stage approach consisting of a pattern-moving-based description and analysis method. First, a density algorithm, named density-based spatial clustering of applications with noise (DBSCAN), is designed to obtain cell space Ω and the corresponding classification guidelines; this algorithm is initiated after the initial pre-image cells, and the total number of entity cells amounts to Ns. Then, the GCM provides several image cells based on a cell mapping function that refers to the multivariate ARMAX model. The global dynamic analysis employing both searching and storing algorithms depend on the attractor, domain of attraction, and periodic cell groups. At last, simulation results of two examples emphasise the practicality as well as efficacy of the technique suggested. The chief aim of this study was to offer a new perspective for a class of complex systems that could inspire research into nonmechanistic principles modelling and application to nonlinear systems.

Exponential synchronization of nonlinear dynamical systems via an aperiodic hybrid control with actuator saturation and impulse time windows

AbstractConsidering impulse time windows and the actuator saturation, we put forward a novel aperiodic hybrid control scheme to achieve the exponential synchronization of two nonlinear dynamical systems (NDSs). Different from the existing research, the actuator saturation of two types of control signals and impulse time windows are both considered in the exponential synchronization analysis. Some sufficient conditions are found with designing the hybrid controller, and the attraction domain (AD) of the error system is also estimated. On the other hand, the conditions which can ensure the exponential synchronization between two NDSs with this hybrid control without saturation limit are also given. Finally, experimental results for the synchronization of three dimensional Hopfield‐type neural networks and the image reproduction based on the synchronization of neural networks demonstrate the feasibility of our theoretical results.

Semiglobal exponential stability of the discrete-time Arrow-Hurwicz-Uzawa primal-dual algorithm for constrained optimization

We consider the discrete-time Arrow-Hurwicz-Uzawa primal-dual algorithm, also known as the first-order Lagrangian method, for constrained optimization problems involving a smooth strongly convex cost and smooth convex constraints. We prove that, for every given compact set of initial conditions, there always exists a sufficiently small stepsize guaranteeing exponential stability of the optimal primal-dual solution of the problem with a domain of attraction including the initialization set. Inspired by the analysis of nonlinear oscillators, the stability proof is based on a non-quadratic Lyapunov function including a nonlinear cross term.

Distributed Saturated Impulsive Control for Local Consensus of Nonlinear Time-Delay Multiagent Systems With Switching Topologies

In this paper, the local consensus problem of nonlinear time-delay multi-agent systems with switching topologies via distributed saturated impulsive control is discussed and the maximum domain of attraction is well estimated. Specifically, we develop a new estimation approach that is quite distinct from the contractive invariant set to estimate the domain of attraction. Moreover, a novel composite impulsive-instant-dependent Lyapunov function is constructed and an improved convex hull representation with more slack variables is adopted to minimize the conservativeness. And then, some delay-independent sufficient criteria in the form of bilinear matrix inequalities that guarantee local consensus of nonlinear time-delay multi-agent systems are obtained by means of the Lyapunov-Razumikhin technique and proof by contradiction. Through appropriate transformation, some linear matrix inequalities-based optimization problems constrained by the proposed local consensus criteria are formulated, and the corresponding numerical solutions are solved by the Yalmip. Finally, a simulation example is given to confirm the validity and superiority of the proposed theoretical results.

Dynamics analysis of helical gear considering de-meshing and reverse impact with EHL lubrication condition.

The purpose of this paper was to study the evolution and dynamic characteristics of the helical gear system co-existence response considering the occurrence of de-meshing and reverse impact under lubrication conditions. The double-three tooth alternate meshing model of a helical gear was established, and the time-varying geometric parameters, motion parameters, and load distribution of positive meshing and reverse impact were analyzed, respectively. The variation of oil film parameters and the coupling between oil film stiffness and meshing stiffness of a gear system were studied according to the equivalent lubrication model of a helical gear. Considering the phenomenon that the meshing state changed from positive meshing to de-meshing and reverse impact, the dynamics model was established. The global bifurcation diagram, the attraction domain, the phase portrait, and the Poincaré section were used to analyze the evolution of a coexistence response and dynamic characteristics with helical gear system parameters.

Chemical oscillation in ultracold chemistry

We demonstrate the occurrence of oscillatory reactions in the ultra-cold chemistry of atom-molecular Bose-Einstein condensate. Nonlinear oscillations in the mean-field dynamics occur for a specific range of elliptic modulus, giving rise to both in- and out-phase modulations in the atom-molecule population density. The reaction front velocity is found to be controlled by photoassociation, which also regulates the condensate density. Two distinct pairs of in-phase bright localized gap solitons are found as exact solutions, the existence of one of which necessarily requires a background. Cnoidal atomic density waves along with a plane-wave molecular motion are observed in both attractive and repulsive domains. The role of intra- and inter-species interactions on both existence and stability is explicated in the presence of photoassociation.