Uniform Global Asymptotic Stability Research Articles

Hybrid control strategy for positive switched delay systems with unstable modes

In this paper, the problem of hybrid control strategy (HCS) for time-varying delay positive switched linear systems (PSLS) with unstable modes is studied. Firstly, the HCS, which includes minimum switching strategy and discretized state feedback controller, is applied to PSLS with time-varying delay for the first time. Secondly, by using the discretized multiple linear copositive Lyapunov-Krasovskii functional, a sufficient condition of globally uniformly asymptotically stable (GUAS) under the HCS is given. Finally, the HCS is extended to discrete-time positive switched time delay systems, and a delay independent stabilization condition is obtained in the discrete system. The effectiveness of the HCS is verified by two simulation examples.

A Lyapunov-Based ISS Small-Gain Theorem for Infinite Networks of Nonlinear Systems

In this article, we show that an infinite network of input-to-state stable (ISS) subsystems, admitting ISS Lyapunov functions, itself admits an ISS Lyapunov function, provided that the couplings between the subsystems are sufficiently weak. The strength of the couplings is described in terms of the properties of an infinite-dimensional nonlinear positive operator, built from the interconnection gains. If this operator induces a uniformly globally asymptotically stable (UGAS) system, a Lyapunov function for the infinite network can be constructed. We analyze necessary and sufficient conditions for UGAS and relate them to small-gain conditions used in the stability analysis of finite networks.

Stability and Resilience of Distributed Information Spreading in Aggregate Computing

Spreading informationthrough a network of devices is a core activity for most distributed systems. Self-stabilizing algorithms for information spreading are one of the key building blocks enabling aggregate computing to provide resilient coordination in open complex distributed systems. This article improves a general spreading block in the aggregate computing literature by making it resilient to network perturbations, establishes its global uniform asymptotic stability, and proves that it is ultimately bounded under persistent disturbances. The ultimate bounds depend only on the magnitude of the largest perturbation and the network diameter, and three design parameters trading off competing aspects of performance. For example, as in many dynamical systems, values leading to greater resilience to network perturbations slow convergence and vice versa.

Robust Hierarchical Tracking Control of Vehicle-Manipulator Systems

For vehicle-manipulator systems (VMSs) to perform precise operations, a robust tracking control framework is required. Moreover, to exploit the platform's redundancy, it is desirable that the framework allows several tasks to be completed simultaneously. This work presents a robust hierarchical tracking controller for floating-base robots. In addition to providing uniform global asymptotic stability (UGAS) in free motion and allowing a total task dimension greater than the number of degrees of freedom (DoFs), the proposed controller includes a sliding mode effect and is proved to achieve UGAS even in the presence of bounded disturbances. Moreover, it is proved that uniform global ultimate boundedness (UGUB) is obtained if a continuous approximation of the sliding mode term is used. The stability results are proved mathematically, and are validated through simulations of an articulated intervention autonomous underwater vehicle (AIAUV).

Model Predictive Control for Networked Multiple Linear Motors System Under DoS Attack and Time Delay

The complex cyber layer challenges the accuracy of cooperative tracking motion for the networked multiple linear motors system. This article focuses on model predictive control (MPC) for networked control system with the denial-of-service (DoS) attack and network-induced time delay. In order to improve tracking accuracy and control performance of the system, several topologies for linear motion models are studied. The DoS attack and time delay are modeled as constraint models in the time domain. Combining dynamic prediction horizon with the buffer compensation in the controller to actuator (C-A) channel, the proposed networked control strategy based on MPC effectively solves the impact of DoS attacks and time delays on the tracking performance of multiple linear motors. In order to ensure the recursive feasibility of the MPC optimization problem and the uniform global asymptotic stability of system, the output terminal constraint and dynamic prediction horizon constraint are considered by combining four DoS attack cases of the networked control strategy, and a Lyapunov function is constructed. Finally, three linear motors with different topologies are built, and the simulation results verify the control effectiveness of the proposed control strategy on the motor tracking performance under cyber constraints.

On the Stability of Cyclic or Periodically Switched Systems Under Unstable Switching Sub-Modes

In this brief, emphasis is placed on the global uniform asymptotic stability analysis of a switched system formulated by cyclic or periodic switching laws. First, several concepts and properties of cyclic or periodically switched systems are defined. Next, by using the cycle dwell time or cyclic period time switching method, new stability criteria for cyclic or periodically switched systems with all Hurwitz stable sub-systems are obtained in the context of general solutions. Moreover, new stability conditions for cyclic or periodically switched systems with partially unstable sub-modes are derived in the framework of general solutions. Finally, two examples are provided to demonstrate the feasibility of the developed criteria.

Uniform global asymptotic stability for nonautonomous nonlinear dynamical systems

Research on uniform global asymptotic stability has the advantage of being able to predict the asymptotic velocity to the origin for all solutions of equations describing nonlinear phenomena. This study elucidates the sufficient conditions under which the zero solution of a nonautonomous nonlinear dynamical system of second order is uniformly globally asymptotically stable. From the results obtained, a certain first-order nonlinear differential equation associated with the second-order dynamical system plays a vital role in uniform global asymptotic stability. More precisely, under several reasonable assumptions, if the integral from σ to t+σ of a particular solution of the first-order differential equation diverges uniformly with respect to σ, then the zero solution of the dynamical system is guaranteed to be uniformly globally asymptotically stable. For the proof, the behavior of the solutions of the dynamical system is carefully tracked. An example that also includes a linear case is provided to illustrate the main result. Simulations are also presented to facilitate understanding of the example, and a local theorem corresponding to the main result and its application to an ecosystem model are discussed.

Robust Distributed Nash Equilibrium Seeking for Games Under Attacks and Communication Delays

In this article, a noncooperative game in which the engaged players are of double integrator-type dynamics, is investigated. The dynamics of the players are considered to be subject to unknown time-varying disturbances and unmodeled terms. Besides, the communication topology among the players suffers from attacks and time-varying communication delays. To find a Nash equilibrium for such games, a distributed robust Nash equilibrium seeking algorithm is proposed. By utilizing average dwell-time and time-ratio constraints to tackle the attacks, the closed-loop system is modeled as a hybrid system with memory. To analyze the stability of the distributed switched algorithm, a Lyapunov functional is constructed, by which we show the uniform global asymptotic stability of the equilibrium. Finally, an example is given to illustrate our results.

Uniform Global Asymptotic Stabilization of Semilinear Periodic Discrete-Time Systems

Semilinear discrete-time control systems with periodic coefficients are considered. The problem of uniform global asymptotic stabilization of the zero equilibrium of the closed-loop system by state feedback is studied. It is assumed that the free dynamic system has a Lyapunov stable zero equilibrium. The method for constructing a damping control is extended from time-invariant systems to time varying periodic semilinear discrete-time systems. By using this approach, sufficient conditions for uniform global asymptotic stabilization for those systems are obtained. Moreover, the converse Lyapunov theorem on Lyapunov (nonasymptotic) stability is proved for complex and real linear periodic discrete-time systems. Finally, examples of using the obtained results are presented.

An Adaptive Time-Varying Impedance Controller for Manipulators.

Aiming at the situation that the structural parameters of the general manipulators are uncertain, a time-varying impedance controller based on model reference adaptive control (MRAC) is proposed in this article. The proposed controller does not need to use acceleration-based feedback or to measure external loads and can tolerate considerable structure parameter errors. The global uniform asymptotic stability of the time-varying closed-loop system is analyzed, and a selection approach for control parameters is presented. It is demonstrated that, by using the proposed control parameter selection approach, the closed-loop system under the adaptive controller is equivalent to an existing result. The feasibility of the presented controller for the general manipulators is demonstrated by some numerical simulations.

Time-Varying Formation Control for Heterogeneous Planar Underactuated Multivehicle Systems

Abstract This paper presents a distributed control approach for time-varying formation of heterogeneous planar underactuated vehicle networks without global position measurements. All vehicles in the network are modeled as generic three degree-of-freedom planar rigid bodies with two control inputs, and are allowed to have nonidentical dynamics. Feasible trajectories are generated for each vehicle using the nonholonomic constraints of the vehicle dynamics. By exploiting the cascaded structure of the planar vehicle model, a transformation is introduced to define the reduced order error dynamics, and then, a sliding-mode control law is proposed. Low-level controller for each vehicle is derived such that it only requires relative position and local motion information of its neighbors in a given directed communication network. The proposed formation control law guarantees the uniform global asymptotic stability (UGAS) of the closed-loop system subject to bounded uncertainties and disturbances. The proposed approach can be applied to underactuated vehicle networks consisting of mobile robots, surface vessels, and planar aircraft. Simulations are presented to demonstrate the effectiveness of the proposed control scheme.

Sampled-data control design for systems with quantized actuators

This paper deals with the problem of designing a sampled-data state feedback control law for continuous-time linear control systems subject to uniform input quantization. The sampled-data state feedback is designed to ensure uniform global asymptotic stability (UGAS) of an attractor surrounding the origin. The closed-loop system is rewritten as a hybrid dynamical system. To do this, an auxiliary clock variable triggering the occurrence of sampling events is introduced. A numerically tractable algorithm with feasibility guarantees, based on concave-convex decomposition, is then proposed allowing to minimize the size of the attractor. Theoretical results are illustrated in a numerical example.

A nonlinear version of Halanay's inequality for the uniform convergence to the origin

<p style='text-indent:20px;'>A nonlinear version of Halanay's inequality is studied in this paper as a sufficient condition for the convergence of functions to the origin, uniformly with respect to bounded sets of initial values. The same result is provided in the case of forcing terms, for the uniform convergence to suitable neighborhoods of the origin. Related Lyapunov methods for the global uniform asymptotic stability and the input-to-state stability of systems described by retarded functional differential equations, with possibly nonconstant time delays, are provided. The relationship with the Razumikhin methodology is shown.</p>

Necessary and Sufficient Conditions for Stability of Discrete-Time Switched Linear Systems With Ranged Dwell Time

This letter deals with the stability analysis problem of discrete-time switched linear systems with ranged dwell time. A novel concept called L-switching-cycle is proposed, which contains sequences of multiple activation cycles satisfying the prescribed ranged dwell time constraint. Based on L-switching-cycle, two sufficient conditions are proposed to ensure the global uniform asymptotic stability of discrete-time switched linear systems. It is noted that two conditions are equivalent in stability analysis with the same L-switching-cycle. These two sufficient conditions can be viewed as generalizations of the clock-dependent Lyapunov and multiple Lyapunov function methods, respectively. Furthermore, it has been proven that the proposed L-switching-cycle can eventually achieve the nonconservativeness in stability analysis as long as a sufficiently long L-switching-cycle is adopted. A numerical example is provided to illustrate our theoretical results.

New results on stability of discrete-time systems with infinite delays

This paper is concerned with problems of stability of discrete-time systems with infinite delays. Several new theorems for stability analysis of the concerned delayed systems are developed. Specifically, uniform stability, uniform asymptotic stability, global uniform asymptotic stability, exponential stability and global exponential stability are included. Our results include some existing works on bounded delays as special cases. Furthermore, an example is furnished to demonstrate the effectiveness of the proposed theorems.

Practical stability in relation to a part of variables of stochastic pantograph differential equations

The main purpose of this paper is to investigate the practical stability in relation to a part of the variables of stochastic pantograph differential equations (SPDEs, in short). With the aid of Lyapunov function method, the criterions of the global practical uniform asymptotic stability and the global practical uniform p-th moment exponential stability in relation to a part of variables of SPDEs are established. An illustrative example is given to show the effectiveness and feasibility of the proposed results on the practical stability with respect to a part of variables.

Uniform global asymptotic stability for oscillators with nonlinear damping and nonlinear restoring terms

Second-order nonlinear differential equations with a time-varying coefficient describe various damped oscillators to explain the actual phenomena observed in numerous research areas. In actual phenomena, it is often important to immediately suppress the vibration of an object. For this purpose, the equilibrium of the oscillator needs to be uniformly globally asymptotically stable. This study examines uniform global asymptotic stability for a damped nonlinear oscillator. A first-order nonlinear differential equation, referred to as the characteristic equation, related to the second-order differential equation plays an important role in uniform global asymptotic stability for the oscillator. To be more precise, the property of particular solutions of this characteristic equation becomes a criterion for determining whether the equilibrium of the nonlinear oscillator is uniformly globally asymptotically stable. Careful mathematical analysis is used for the proof. In addition, a few examples are presented to show that the property of the particular solutions can be examined by drawing solution curves via a simple numerical simulation. Furthermore, detailed explanations are provided for the assumptions used to obtain the results. Our results clarify the appropriate damping coefficient required for the equilibrium to become uniformly globally asymptotically stable and have the advantage of being able to predict the speed at which a vibrating object returns to its original equilibrium state.

Asymptotic behavior of a class of perturbed differential equations

UDC 517.9 This paper deals with the problem of stability of nonlinear differential equations with perturbations. Sufficient conditions for global uniform asymptotic stability in terms of Lyapunov-like functions and integral inequality are obtained. The asymptotic behavior is studied in the sense that the trajectories converge to a small ball centered at the origin. Furthermore, an illustrative example in the plane is given to verify the effectiveness of the theoretical results.

ADVANCES IN ANALYSIS OF CAPUTO FRACTIONAL-ORDER NONAUTONOMOUS SYSTEMS: FROM STABILITY TO GLOBAL UNIFORM ASYMPTOTIC STABILITY

In this work, we analyze various stability (from stability to global uniform asymptotic stability) of Caputo fractional-order nonautonomous systems (CFONSs). With [Formula: see text] being only continuous, it is proven that the Caputo fractional derivative (CFD) of a general Lyapunov function [Formula: see text] along solutions is continuous and moreover, [Formula: see text] on the maximal interval of existence (MIE) of [Formula: see text]. This together with the continuation of solutions suffices to prove the various stability theorems for CFONSs that are as general as those for integer-order systems, and make them practically applicable. The work reduces the assumption on vector field functions [Formula: see text] for stability analysis from continuously differentiable (CD) to only continuous, which advances existing results to a large extent. Finally, some derived results are applied to real examples with numerical simulations.

Time-triggered and event-triggered control of switched affine systems via a hybrid dynamical approach

This paper focuses on the design of both periodic time- and event-triggered control laws of switched affine systems using a hybrid dynamical system approach. The novelties of this paper rely on the hybrid dynamical representation of this class of systems and on a free-matrix min-projection control, which relaxes the structure of the usual Lyapunov matrix-based min-projection control. This contribution also presents an extension of the usual periodic time-triggered implementation to the event-triggered one, where the control input updates are permitted only when a particular event is detected. Together with the definition of an appropriate optimization problem, a stabilization result is formulated to ensure the uniform global asymptotic stability of an attractor for both types of controllers, which is a neighborhood of the desired operating point. Finally, the proposed method is evaluated through a numerical example.