Conditions For Exponential Convergence Research Articles

Adaptive observer design for coupled ODE–hyperbolic PDE systems with application to traffic flow estimation

This paper studies the state estimation problem for a class of coupled linear ODE–hyperbolic PDE systems with unknown in-domain parameters. Based on the swapping transformation and the least squares parameter estimation method, a Luenberger-type adaptive boundary observer is designed to estimate the ODE–PDE states under the unknown parameters. The exponential convergence conditions w.r.t. the observer feedback gains are given by employing the Lyapunov technique. Finally, an application to traffic state estimation of the Aw–Rascle–Zhang (ARZ) model involving the unknown relaxation time and the boundary input speed deviation is provided to validate the theoretical result.

Design of infinite horizon LQR controller for discrete delay systems in satellite orbit control: A predictive controller and reduction method approach

In the realm of satellite orbit control, powerful controller design plays a pivotal role in minimizing fuel consumption and ensuring orbit stability. This article introduces an advanced approach to the design of a Linear Quadratic Regulator (LQR) controller with an infinite horizon, tailored for discrete delay systems. The proposed methodology integrates predictive control with a reduction method, aiming for optimality while addressing performance and system constraints. Formulating the control problem as a quadratic program, the predictive control method generates a sequence of control inputs using a reducing horizon strategy. Stability analysis, employing Lyapunov-Krasovsky functions and linear matrix inequalities, yields delay-independent conditions for exponential convergence. A numerical example showcases the controller's effectiveness in maintaining orbit and reducing fuel consumption, underlining its capacity to achieve control objectives despite uncertainties and time delays. This research contributes to robust control strategies in satellite orbit systems, enhancing control performance and operational efficiency.

Unknown input observer for Takagi-Sugeno implicit models with unmeasurable premise variables

<p>Recent years have seen a great deal of interest in implicit nonlinear systems, which are used in many different engineering applications. This study is dedicated to presenting a new method of fuzzy unknown inputs observer design to estimate simultaneously both non-measurable states and unknown inputs of continuous-time nonlinear implicit systems defined by Takagi-Sugeno (T-S) models with unmeasurable premise variables. The suggested observer is based on the singular value decomposition approach and rewritten the continuous-time T-S implicit models into an augmented fuzzy system, which gathers the unknown inputs and the state vector. The exponential convergence condition of the observer is established by using the Lyapunov theory and linear matrix inequalities are solved to determine the gains of the observer. Finally, the effectiveness of the suggested method is then assessed using a numerical application. It demonstrates that the estimated variables and the unknown input converge to the real variables accurately and quickly (less than 0.5 s).</p>

Non-Euclidean Contraction Theory for Monotone and Positive Systems

In this note we study contractivity of monotone systems and exponential convergence of positive systems using non-Euclidean norms. We first introduce the notion of conic matrix measure as a framework to study stability of monotone and positive systems. We study properties of the conic matrix measures and investigate their connection with weak pairings and standard matrix measures. Using conic matrix measures and weak pairings, we characterize contractivity and incremental stability of monotone systems with respect to non-Euclidean norms. Moreover, we use conic matrix measures to provide sufficient conditions for exponential convergence of positive systems to their equilibria. We show that our framework leads to novel results on (i) the contractivity of excitatory Hopfield neural networks, and (ii) the stability of interconnected systems using non-monotone positive comparison systems.

Exponential stabilization of stochastic complex networks with Markovian switching topologies via intermittent discrete-time state observation control

This paper is concerned with the exponential stability of the stochastic complex networks with Markovian switching topologies. It is worth emphasizing that the topological structure of the stochastic complex networks is Markovian switching. Not all switching subnetworks must contain a spanning tree or be strongly connected. Moreover, a new type of aperiodically intermittent discrete-time state observation control is proposed, which is a significant extension of discrete-time state observation control and intermittent control. Based on the M-matrix, the Lyapunov method, and the graph theory, some sufficient conditions for exponential convergence. Moreover, we obtain that the average control rate in this paper is greater than the control rate proposed in the existing literature, which is less conservative. In addition, the theoretical results are used to discuss the exponential stability of stochastic coupled oscillators and a communication network model, respectively. Finally, two numerical examples are given to verify the effectiveness of the results.

Fault diagnosis for a class of nonlinear uncertain systems using deterministic learning approach

In this paper, we propose a fault diagnosis (FD) approach for a class of nonlinear uncertain systems based on the deterministic learning approach (DLA). Specifically, an adaptive learning observer is constructed, in which the adaptive neural networks (NNs) are constructed to approximate the unknown system dynamics under normal and fault modes. Based on the strictly positive real (SPR) condition, the convergence of the state estimation can be guaranteed. When the system is undergoing a periodic or periodic-like (recurrent) motion, the states of the observer will also become recurrent. Thus through DLA, the partial persistent excitation (PE) condition of the associated subvectors of NNs is satisfied. By utilizing the partial (PE) condition, the uniformly completely observable (UCO) property of the identification system is analyzed and the exponential convergence condition of the identification system is derived. Under this condition, the unknown dynamics under normal and fault modes can be accurately identified along the system trajectory. And by utilizing the knowledge obtained in the identification phase, the fault can be detected in the diagnosis phase. The main attraction of this paper lies in the analytical result, which shows that the exponential convergence condition of the learning observer not only depends on the observer gain matrix, but also depends on the PE level of the regressor subvector of NN. Simulation results are included to illustrate the effectiveness of the proposed scheme.

Research on a Cooperative Adaptive Cruise Control (CACC) Algorithm Based on Frenet Frame with Lateral and Longitudinal Directions.

Research on the cooperative adaptive cruise control (CACC) algorithm is primarily concerned with the longitudinal control of straight scenes. In contrast, the lateral control involved in certain traffic scenes such as lane changing or turning has rarely been studied. In this paper, we propose an adaptive cooperative cruise control (CACC) algorithm that is based on the Frenet frame. The algorithm decouples vehicle motion from complex motion in two dimensions to simple motion in one dimension, which can simplify the controller design and improve solution efficiency. First, the vehicle dynamics model is established based on the Frenet frame. Through a projection transformation of the vehicles in the platoon, the movement state of the vehicles is decomposed into the longitudinal direction along the reference trajectory and the lateral direction away from the reference trajectory. The second is the design of the longitudinal control law and the lateral control law. In the longitudinal control, vehicles are guaranteed to track the front vehicle and leader by satisfying the exponential convergence condition, and the tracking weight is balanced by a sigmoid function. Laterally, the nonlinear group dynamics equation is converted to a standard chain equation, and the Lyapunov method is used in the design of the control algorithm to ensure that the vehicles in the platoon follow the reference trajectory. The proposed control algorithm is finally verified through simulation, and validation results prove the effectiveness of the proposed algorithm.

Adaptive boundary observer for ARZ traffic flow model with domain and boundary uncertainties

This paper studies the problem of the adaptive boundary observer design for the Aw-Rascle-Zhang (ARZ) traffic flow model, which is subject to both relaxation time uncertainty in the domain and boundary input disturbance. The boundary input disturbance comes from merging vehicles’ velocities at the downstream on-ramp, and we scale the disturbance by a low-pass filter based on the ordinary differential equation (ODE). Then, the ARZ model with the domain uncertainty and boundary input disturbance can be linearized to a coupled ODE-PDE system. Based on the swapping transformation, an adaptive boundary observer with least-squares type parameter estimation is designed to estimate the traffic states, the domain uncertainty, and the boundary input disturbance, simultaneously. The exponential convergence conditions w.r.t. the observer feedback gains are given by employing the Lyapunov technique. Finally, the simulation results are presented to illustrate the effectiveness of the designed adaptive boundary observer.

Unique quasi-stationary distribution, with a possibly stabilizing extinction

We establish sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the Q-process, the process conditioned upon never being absorbed. The technique relies on a coupling procedure that is related to Harris recurrence (for Markov Chains). It applies to general continuous-time and continuous-space Markov processes. The main novelty is that we modulate each coupling step depending both on a final horizon of time (for survival) and on the initial distribution. By this way, we could notably include in the convergence a dependency on the initial condition. As an illustration, we consider a continuous-time birth–death process with catastrophes and a diffusion process describing a (localized) population adapting to its environment.

Cooperative control of double-integrator agents with heterogeneous control gains: Exponential consensus conditions and the heterogeneity-metric

This paper considers cooperative control of multiple inertial (i.e., double-integrator) agents, with the couplings of the agents represented as a digraph. Particularly, the control inputs can be a general setting of heterogeneous control gains for the agents’ couplings. The main contributions in this paper are as follows: First, for the digraph being any unidirectional digraph (i.e., the corresponding Laplacian of the digraph can be expressed as an upper or lower triangular matrix if with proper numbering sequence of the agents) that contains a directed spanning-tree, we show that the necessary and sufficient condition for exponential convergence to consensus is that these gains can be merely arbitrarily positive. Second, for the digraph being generally weighted and strongly-connected (this type of the digraph includes all undirected and connected graphs), we provide the analytical lower-bounds of the heterogeneous gains for exponential convergence to consensus, whose bounds are less conservative than existing analytical results. Further, we propose the notion of the heterogeneity-metric to characterize the heterogeneity-degree of the heterogeneous gains and the notion of the maximum-heterogeneity-degree (MHD), and for the heterogeneous gains with the constraint of a designated MHD, we provide the conditions for exponential convergence to consensus, whose results are novel. Finally, we show the relation between the MHD constraint and the lower-bounds of the gains for exponential consensus.

Hybrid Control for Robust and Global Tracking on Smooth Manifolds

In this paper, we present a hybrid control strategy that allows for global asymptotic tracking of reference trajectories evolving on smooth manifolds, with nominal robustness. Two different versions of the hybrid controller are presented: one which allows for discontinuities of the plant input and a second one that removes the discontinuities via dynamic extension. By taking an exosystem approach, we provide a general construction of a hybrid controller that guarantees global asymptotic stability of the zero tracking error set. The proposed construction relies on the existence of proper indicators and a transport map-like function for the given manifold. We provide a construction of these functions for the case where each chart in a smooth atlas for the manifold maps its domain onto the Euclidean space. We also provide conditions for exponential convergence to the zero tracking error set. To illustrate these properties, the proposed controller is exercised on three different compact manifolds—the two-dimensional sphere, the unit-quaternion group, and the special orthogonal group of order three—and further applied to the problems of obstacle avoidance in the plane and global synchronization on the circle.

Exponential Convergence of the Discrete- and Continuous-Time Altafini Models

This paper considers the discrete-time version of Altafini's model for opinion dynamics in which the interaction among a group of agents is described by a time-varying signed digraph. Prompted by an idea from [3] , exponential convergence of the system is studied using a graphical approach. Necessary and sufficient conditions for exponential convergence with respect to each possible type of limit states are provided. Specifically, under the assumption of repeatedly jointly strong connectivity, it is shown that 1) a certain type of two-clustering will be reached exponentially fast for almost all initial conditions if, and only if, the sequence of signed digraphs is repeatedly jointly structurally balanced corresponding to that type of two-clustering; 2) the system will converge to zero exponentially fast for all initial conditions if, and only if, the sequence of signed digraphs is repeatedly jointly structurally unbalanced. An upper bound on the convergence rate is provided. The results are also extended to the continuous-time Altafini model.

Adaptation, coordination, and local interactions via distributed approachability

This paper investigates the relation between cooperation, competition, and local interactions in large distributed multi-agent systems. The main contribution is the game-theoretic problem formulation and solution approach based on the new framework of distributed approachability, and the study of the convergence properties of the resulting game model. Approachability theory is the theory of two-player repeated games with vector payoffs, and distributed approachability is here presented for the first time as an extension to the case where we have a team of agents cooperating against a team of adversaries under local information and interaction structure. The game model turns into a nonlinear differential inclusion, which after a proper design of the control and disturbance policies, presents a consensus term and an exogenous adversarial input. Local interactions enter into the model through a graph topology and the corresponding graph-Laplacian matrix. Given the above model, we turn the original questions on cooperation, competition, and local interactions, into convergence properties of the differential inclusion. In particular, we prove convergence and exponential convergence conditions around zero under general Markovian strategies. We illustrate our results in the case of decentralized organizations with multiple decision-makers.

P-th moment exponential convergence analysis for stochastic networked systems driven by fractional Brownian motion

In this paper, the existence, uniqueness and asymptotic behavior of mild solutions of stochastic neural network systems driven by fractional Brownian motion are investigated. By applying the Banach fixed point theorem, the existence and uniqueness of mild solution are analytically proved in a Hilbert space. Based on the moment inequality of wick-type integral analysis technique, the p-th moment exponential convergence condition of the mild solution is presented. Finally, two numerical examples are presented to demonstrate the validity of the theoretical results.

Robust cooperative learning control for directed networks with nonlinear dynamics

This paper studies a class of robust cooperative learning control problems for directed networks of agents (a) with nonidentical nonlinear dynamics that do not satisfy a global Lipschitz condition and (b) in the presence of switching topologies, initial state shifts and external disturbances. All uncertainties are not only time-varying but also iteration-varying. It is shown that the relative formation of nonlinear agents achieved via cooperative learning can be guaranteed to converge to the desired formation exponentially fast as the number of iterations increases. A necessary and sufficient condition for exponential convergence of the cooperative learning process is that at each time step, the network topology graph of nonlinear agents can be rendered quasi-strongly connected through switching along the iteration axis. Simulation tests illustrate the effectiveness of our proposed cooperative learning results in refining arbitrary high precision relative formation of nonlinear agents.

Dissipativity-based observer design for a class of coupled 1-D semi-linear parabolic PDE systems

In the present paper the dissipativity-based observer design approach is extended to a class of 1-D semi-linear parabolic PDE systems with pointwise in-domain measurement. On the basis of a modal innovation over the set of slow eigenvalues of the linear operator and the consideration of a dissipativity (sector) condition for the nonlinearity an exponential convergence condition for the estimation error is derived in form of a linear operator inequality, which is solved on the basis of an eigenfunction expansion of the linear operator. The approach is illustrated by numerical simulations for a coupled two-state semi-linear PDE system.

Exponential convergence-tractability of general linear problems in the average case setting

We study d-variate general linear problems defined over Hilbert spaces in the average case setting. We consider algorithms that use finitely many evaluations of arbitrary linear functionals. We obtain the necessary and sufficient conditions for Exponential Convergence-(Strong) Polynomial Tractability, Quasi-Polynomial Tractability and (Uniform) Weak Tractability. All results are in terms of the eigenvalues of the corresponding covariance operators.

Exponential convergence to quasi-stationary distribution and $$Q$$-process

For general, almost surely absorbed Markov processes, we obtain necessary and sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the $$Q$$ -process (the process conditioned to never be absorbed). We apply these results to one-dimensional birth and death processes with catastrophes, multi-dimensional birth and death processes, infinite-dimensional population models with Brownian mutations and neutron transport dynamics absorbed at the boundary of a bounded domain.

Mathematical analysis on an extended Rosenzweig-MacArthur model of tri-trophic food chain

In this paper, we study a new model as an extension of theRosenzweig-MacArthur tritrophic food chain model in which thesuper-predator consumes both the predator and the prey. We first obtainthe ultimate bounds and conditions for exponential convergence for thesepopulations. We also find all possible equilibria and investigate their stability orinstability in relation with all the ecological parameters. Our main focus is on theconditions for the existence, uniqueness and stability of a coexistence equilibrium.The complexity of the dynamics in this model is theoretically discussed and graphicallydemonstrated through various examples and numerical simulations.

Regularized Nonlinear Moving-Horizon Observer With Robustness to Delayed and Lost Data

Moving-horizon estimation provides a general method for state estimation with strong theoretical convergence properties under the critical assumption that global solutions are found to the associated nonlinear programming problem at each sampling instant. A particular benefit of the approach is the use of a moving window of data that is used to update the estimate at each sampling instant. This provides robustness to temporary data deficiencies such as lack of excitation and measurement noise, and the inherent robustness can be further enhanced by introducing regularization mechanisms. In this paper, we study moving-horizon estimation in cases when output measurements are lost or delayed, which is a common situation when digitally coded data are received over low-quality communication channels or random access networks. Modifications to a basic moving-horizon state estimation algorithm and conditions for exponential convergence of the estimation errors are given, and the method is illustrated by using a simulation example and experimental data from an offshore oil drilling operation.