Global Exponential Convergence Research Articles

Almost automorphic behaviours of nonlocal stochastic fuzzy Cohen–Grossberg lattice neural networks

Discrete-time Cohen–Grossberg neural networks (CGNNs) play important role in feedback neural networks. The existing literatures of CGNNs only regarded integral order discrete-time models without other variables. This paper builds the lattice model for nonlocal stochastic fuzzy CGNNs with reaction diffusions by employing the finite difference and Mittag–Leffler time Euler difference techniques. Further, the existence of a unique bounded almost automorphic sequence solution in distribution and global exponential convergence in the mean-square sense to the achieved difference model are investigated. At last, by using the tool of Matlab and time Euler difference for the Brownian motions, an illustrative example with simulations is used to show the feasible of the works of the current paper. We think that this work can contribute to achieve some effective real tasks, e.g. content-addressable memory, auto-association, dynamic reconstruction of a chaotic process, etc.

Exponential Convergence of Primal-Dual Dynamics Under General Conditions and Its Application to Distributed Optimization.

In this article, we establish the local and global exponential convergence of a primal-dual dynamics (PDD) for solving equality-constrained optimization problems without strong convexity and full row rank assumption on the equality constraint matrix. Under the metric subregularity of Karush-Kuhn-Tucker (KKT) mapping, we prove the local exponential convergence of the dynamics. Moreover, we establish the global exponential convergence of the dynamics in an invariant subspace under a technically designed condition which is weaker than strong convexity. As an application, the obtained theoretical results are used to show the exponential convergence of several existing state-of-the-art primal-dual algorithms for solving distributed optimization without strong convexity. Finally, we provide some experiments to demonstrate the effectiveness of our results.

Global exponential convergence and synchronization for exponential numerical competitive neural networks with different time scales and fuzzy logic

For nonlocal fuzzy delayed competitive neural networks, a novel difference model is first developed in this article using the Mittag-Leffler Euler difference method. Secondly, the existence of a sole globally bounded solution and globally exponential stability to the discrete-time networks are addressed based on Banach contractive mapping principle and the constant variation formula for sequences. Besides, exponential synchronization of the discrete-time networks is investigated and a feedback controller is designed from the viewpoint of the theory of controls. It is the first time to consider nonlocal fuzzy delayed competitive neural networks by using Mittag-Leffler Euler differences. In contrast to nonlocal discrete approaches, such difference competitive neural networks could preferably figure the features of continuous networks. The difference method in this paper extends and improves the traditional difference approaches, for example, Adams-Bashforth Euler differences. Moreover, this article explores some avenue for the research of discrete-time fractional-order systems and develops a set of theories and methodologies for the future investigations.

Observer-based event-triggered boundary control of the one-phase Stefan problem

This paper presents an observer-based event-triggered boundary control strategy for the one-phase Stefan problem, utilising the position and velocity measurements of the moving interface. The design of the observer and controller is founded on the infinite-dimensional backstepping approach. To implement the continuous-time observer-based controller in an event-triggered framework, we propose a dynamic event triggering condition. This condition specifies the instances when the control input must be updated. Between events, the control input is maintained constant in a Zero-Order-Hold manner. We demonstrate that the dwell-time between successive triggering moments is uniformly bounded from below, thereby precluding Zeno behaviour. The proposed event-triggered boundary control strategy ensures the well-posedness of the closed-loop system and the satisfaction of certain model validity conditions. Additionally, the global exponential convergence of the closed-loop system to the setpoint is established using Lyapunov approach. A simulation example is provided to validate the theoretical findings.

Discrete-time adaptive distributed output observer for time-varying formation tracking of heterogeneous multi-agent systems

This paper presents discrete-time distributed algorithms to achieve time-varying output formation tracking of heterogeneous linear multi-agent systems subject to time-varying faults and directed graphs. To overcome the impacts caused by unknown faults and directed graphs, a discrete-time adaptive distributed output observer-based control architecture is established to guarantee time-varying output formation tracking exponentially. In particular, several distributed leader observers are firstly given to estimate and reconstruct the leader’s state over a directed graph. The new discrete-time adaptive distributed output observer is presented to remove the need of having fully access to the leader’s system matrices and to reduce the dimension and information exchange in the developed distributed observers. In terms of this estimated information from the adaptive distributed output observer, a distributed output feedback controller integrated with an observer-based scheme is developed to compensate for the impact of unknown faults such that time-varying output formation tracking is achieved with a global exponential convergence. Two examples and simulation results are provided to validate the effectiveness of the proposed designs.

Global exponential convergence of delayed inertial Cohen–Grossberg neural networks

In this paper, the exponential convergence of delayed inertial Cohen–Grossberg neural networks (CGNNs) is studied. Two methods are adopted to discuss the inertial CGNNs, one is expressed as two first-order differential equations by selecting a variable substitution, and the other does not change the order of the system based on the nonreduced-order method. By establishing appropriate Lyapunov function and using inequality techniques, sufficient conditions are obtained to ensure that the discussed model converges exponentially to a ball with the prespecified convergence rate. Finally, two simulation examples are proposed to illustrate the validity of the theorem results.

Robust and scalable distributed recursive least squares

We consider a problem of robust estimation over a network in an errors-in-variables context. Each agent measures noisy samples of a local pair of signals related by a linear regression defined by a common unknown parameter, and the agents must cooperate to find the unknown parameter in presence of uncertainty affecting both the regressor and the regressand variables. We propose a recursive least squares estimation method providing global exponential convergence to the unknown parameter in absence of uncertainty, and robust stability of the estimate, formalized in terms of input-to-state stability, in presence of uncertainty affecting all the variables. The result relies on a cooperative excitation assumption that is proved to be strictly weaker than persistency of excitation of each local data set. The proposed estimator is validated on an adaptive road pricing application.

Event-Triggered State Estimation for Networked Switched Systems: An Output Predictor Approach

In this article, we investigate the event-triggered state estimation for networked switched systems with (a)synchronous switching. A new multimode output predictor-based sampled-data observer and an event-triggered mechanism are proposed. Between event-triggered sampling times, an output predictor is utilized to compensate for the impact of sampling, and at the sampling instants, the predicted output is reset with the updated sampled output. Two scenarios are considered: the observer's switching signal is synchronous with the system's switching signal, and the switching signals among them are asynchronous, and frequent switching is allowed during interevent intervals. Average dwell time and linear matrix inequalities techniques are employed to guarantee the global uniform exponential convergence of the proposed observer. Moreover, Zeno phenomenon can be ruled out by proving the existence of a positive lower bound of interevent intervals. Finally, the effectiveness of the proposed approach is illustrated by a circuit system and an academic example.

Identifiability implies robust, globally exponentially convergent on-line parameter estimation

In this paper we propose a new parameter estimator that ensures global exponential convergence of linear regression models requiring only the necessary assumption of identifiability of the regression equation, which we show is equivalent to interval excitation of the regressor vector. An extension to – separable and monotonic – nonlinear parameterisations is also given. The estimators are shown to be robust to additive measurement noise and – not necessarily slow-parameter variations. Moreover, a version of the estimator that is robust with respect to sinusoidal disturbances with unknown internal model is given. Simulation results that illustrate the performance of the estimator compared with other algorithms are given.

Topology-based dynamic event-triggered leader-following consensus of multi-agent systems under switching topologies

This paper is devoted to the topology-based dynamic event-triggered leader-following consensus issue of general linear multi-agent systems under switching topological structures. The novel topology-based distributed dynamic event-triggered rule is formed, in which a topology-based dynamic auxiliary variable is introduced to reduce the conservatism and save resources. Under novel triggering rules, topology-based distributed event-triggered controllers are constructed. Different control gains are designed for different topological structures. Compared with existing works, conservatism is further reduced. Then the leader-following consensus is achieved with a global exponential convergence rate. Moreover, the average dwell time method is adopted to relax restrictions on the switching speed of topological structures and an original analytical thought is put forward to rule out the Zeno phenomenon concisely and clearly. Finally, two numerical examples and a realistic mass–spring system are utilized to verify the feasibility and superiority of the proposed theory.

Exponential convergence in the mean-square sense of nonlocal stochastic almost automorphic genetic regulatory lattice networks

This paper first establishes the lattice model for nonlocal stochastic genetic regulatory networks with reaction diffusions by employing a mix of the finite difference and Mittag–Leffler time Euler difference techniques. Second, the existence of a unique bounded almost automorphic sequence in distribution and global mean-square exponential convergence to the achieved difference model are investigated. An illustrative example is used to show the feasible of the works of the current paper.

Anti-periodic motion and mean-square exponential convergence of nonlocal discrete-time stochastic competitive lattice neural networks with fuzzy logic

This paper firstly establishes the discrete-time lattice networks for nonlocal stochastic competitive neural networks with reaction diffusions and fuzzy logic by employing a mix techniques of finite difference to space variables and Mittag-Leffler time Euler difference to time variable. The proposed networks consider both the effects of spatial diffusion and fuzzy logic, whereas most of the existing literatures focus only on discrete-time networks without spatial diffusion. Firstly, the existence of a unique ω-anti-periodic in distribution to the networks is addressed by employing Banach contractive mapping principle and the theory of stochastic calculus. Secondly, global exponential convergence in mean-square sense to the networks is discussed on the basis of constant variation formulas for sequences. Finally, an illustrative example is used to show the feasible of the works in the current paper with the help of MATLAB Toolbox. The work in this paper is pioneering in this regard and it has created a certain research foundations for future studies in this area.

From Zeroing Dynamics to Zeroing-Gradient Dynamics for Solving Tracking Control Problem of Robot Manipulator Dynamic System with Linear Output or Nonlinear Output

With the vigorous development of mechanical intelligence in industrial manufacturing, tracking control dynamic systems have been widely applied in many aspects of industry. In this paper, we present one theorem to discuss the validity condition of a ZD model with order-n for solving the tracking control problem of a nonlinear problem by utilizing a Lie derivative. Moreover, we also give the unified formula of the ZD model with order-n and rigorously prove it mathematically. In addition, we present three other theorems to give the global exponential convergence property of the ZD controller u(t), and the steady-state tracking error bound of the ZGD controller u(t), and the radius bound where the steady-state tracking error converges exponentially. Finally, simulations are conducted to demonstrate the validity and parameter influences of the ZD model and ZGD model for solving the tracking control problem with a single linear or nonlinear output of the single-link manipulator with flexible joints.

Attack-Resilient Distributed Convex Optimization of Cyber–Physical Systems Against Malicious Cyber-Attacks Over Random Digraphs

This article addresses a resilient exponential distributed convex optimization problem for a heterogeneous linear multiagent system under Denial-of-Service (DoS) attacks over random digraphs. The random digraphs are caused by unreliable networks and the DoS attacks, allowed to occur aperiodically, refer to an interruption of the communication channels carried out by the intelligent adversaries. In contrast to many existing distributed convex optimization works over a perfect communication network, the global optimal solution might not be sought under the adverse influences that result in performance degradations or even failures of optimization algorithms. The aforementioned setting poses certain technical challenges to optimization algorithm design and exponential convergence analysis. In this work, several resilient algorithms are presented such that a team of agents minimizes a sum of local nonquadratic cost functions in a safe and reliable manner with the global exponential convergence. Numerical simulation results are further presented to validate the effectiveness of the proposed distributed approaches.

PERIODIC DYNAMICS AND MEAN-SQUARE EXPONENTIAL CONVERGENCE OF NONLOCAL STOCHASTIC FUZZY BIDIRECTIONAL ASSOCIATIVE MEMORY LATTICE NEURAL NETWORKS

This paper firstly establishes the lattice model for nonlocal stochastic fuzzy bidirectional associative memory neural networks with reaction diffusions by employing a mix of the finite difference and Mittag-Leffler time Euler difference techniques. Secondly, the existence of a unique bounded periodic sequence solution in distribution and global mean-square exponential convergence to the achieved difference model are investigated. Some illustrative example is used to show the feasible of the works of the current paper.

Unknown piecewise constant parameters identification with exponential rate of convergence

SummaryThe scope of this research is the identification of unknown piecewise constant parameters of linear regression equation under the finite excitation condition. Compared to the known methods, to make the computational burden lower, only one model to identify all switching states of the regression is used in the developed procedure with the following two‐fold contribution. First of all, we propose a new truly online estimation algorithm based on a well‐known DREM approach to detect switching time and preserve time alertness with adjustable detection delay. Second, despite the fact that a switching signal function is unknown, the adaptive law is derived that provides global exponential convergence of the regression parameters estimates to their true values in case the regressor is finitely exciting somewhere inside the time interval between two consecutive parameters switches. The robustness of the proposed identification procedure to the influence of external disturbances is analytically proved. Its effectiveness is demonstrated via numerical experiments, in which both abstract regressions and a second‐order plant model are used.

A new least squares parameter estimator for nonlinear regression equations with relaxed excitation conditions and forgetting factor

In this note a new high performance least squares parameter estimator is proposed. The main features of the estimator are: (i) global exponential convergence is guaranteed for all identifiable linear regression equations; (ii) it incorporates a forgetting factor allowing it to preserve alertness to time-varying parameters; (iii) thanks to the addition of a mixing step it relies on a set of scalar regression equations ensuring a superior transient performance; (iv) it is applicable to nonlinearly parameterized regressions verifying a monotonicity condition and to a class of systems with switched time-varying parameters; (v) it is shown that it is bounded-input-bounded-state stable with respect to additive disturbances; (vi) continuous and discrete-time versions of the estimator are given. The superior performance of the proposed estimator is illustrated with a series of examples reported in the literature.

Target localization and multicircular circumnavigation with bearing‐only measurements

SummaryThis article studies the multicircular circumnavigation problem of a stationary target using bearing‐only measurements. The proposed estimator‐based control strategy drives the networked mobile agents to move around an unknown target along different circles with any prescribed inter‐agent angular spacing. The results presented in this article are as follows: First, a new estimator is developed by introducing a linear filter into an estimation algorithm which has been widely used in the existing literature. The novelty of the proposed estimator lies in its enhanced convergence rate. Analytical results show that the exponential convergence rate of the estimator can be adjusted to be arbitrarily high by increasing the estimation gain. Second, a distributed controller is designed such that global exponential convergence of the agents to the desired circular formation can be achieved. Besides, the proposed algorithms guarantee that each agent can keep a safe distance from the target, and high angular velocities of the agents can be avoided even if the circumnavigation radii become small. Finally, several numerical simulations are given to corroborate the theoretical results.

Trajectory tracking for nonlinear systems using extended quadratic port-Hamiltonian models without input and state coordinate transformations

In this note, an enhanced trajectory tracking (or equivalently, tracking-error) approach is developed for the control of nonlinear systems whenever the stage of feedback passivation design prior to synthesizing state feedback controllers is impossible. To achieve this purpose while using the original state vector to retain its interpretation, it is possible without the use of input and state coordinate transformations to combine the system dynamics with the so-called extended quadratic port-Hamiltonian (PH) models (including possibly the quadratic pseudo PH models) which are then divided into non-relaxing and relaxing ones for further study on control benefits. Interestingly, both cases are associated to a unifying quadratic Hamiltonian storage function similar to that of electrical, mechanical, or electromechanical systems with a specific insight. Sufficient conditions for the global asymptotic or exponential convergence of the system trajectory to the reference one are shown. In addition, a Proportional–Integral action can be added to the tracking control for improving the closed-loop performance and robustness. The proposed approach is illustrated via two case studies, including the non-minimum phase Van de Vusse reaction system and the 3-DOF SCARA robot.

A Contraction Analysis of Primal-Dual Dynamics in Distributed and Time-Varying Implementations

In this article, we provide an overarching analysis of primal-dual dynamics associated with linear equality-constrained optimization problems using contraction analysis. For the well-known standard version of the problem, we establish convergence under convexity and the contracting rate under strong convexity. Then, for a canonical distributed optimization problem, we use partial contractivity to establish global exponential convergence of its primal-dual dynamics. As an application, we propose a new distributed solver for the least-squares problem with the same convergence guarantees. Finally, for time-varying versions of both centralized and distributed primal-dual dynamics, we exploit their contractive nature to establish bounds on their tracking error. To support our analyses, we introduce novel results on contraction theory.