Global Uniform Exponential Stability Research Articles

Equivalent conditions for the synchronization of identical linear systems over arbitrary interconnections

We propose necessary and sufficient conditions for the synchronization of N identical single-input–single-output (SISO) systems, connected through a directed graph without imposing any assumption on the graph interconnection. We consider both the continuous-time and the discrete-time case, and we provide conditions that are equivalent to the uniform global exponential stability, with guaranteed convergence rate, of the closed and unbounded attractor that corresponds to the synchronization set.

Practical stability analysis of stochastic perturbed linear time-varying systems via integral inequality

In this paper, we investigate the problem of stability of linear time-varying stochastic perturbed systems. We present sufficient conditions ensuring the global practical uniform exponential stability of a different class of stochastic perturbed systems based on generalized integral inequalities of Gronwall type. Finally, we provide numerical examples to prove the usefulness of the main results of this paper.

Dynamic output feedback control of switched systems: A dynamic event-triggered sampling control scheme

A dynamic event-triggered sampling scheme for dynamic output feedback control for a class of switched systems is proposed, where the triggering mechanism can exclude the Zeno phenomenon even if the state of the switched systems is within the output kernel space, and subsystems of the resulting closed-loop switched systems are not necessarily required to be stable. The proposed method can guarantee the Zeno-freeness property and global uniform exponential stability of the origin simultaneously under the impacts of triggering asynchronism and sampling errors. Moreover, sufficient conditions on the globally exponential stability are also given. Finally, the effectiveness of the proposed scheme is illustrated by a numerical example.

Certainty Equivalence Control of Discrete-Time Multiagent Systems: A Framework for Uniform Global Exponential Stability

Certainty Equivalence Control of Discrete-Time Multiagent Systems: A Framework for Uniform Global Exponential Stability

An Exponential Stabilization Analysis for Switched Interval Type-2 Fuzzy Sampled-Data Control Systems

This work is focused on designing the admissible edge-dependent average dwell time (AE-DADT) based fuzzy sampled-data control problem for a class of switched nonlinear systems. The upper and lower grade of membership functions is used to characterize uncertain parameter-dependent nonlinear systems in the interval type-2 fuzzy (IT2F) model. Unlike from the existing literature, a new fuzzy dependent refined looped-Lyapunov functional (LF) is introduced which involves the new exponent terms and state information such as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\frac{e^{2\gamma _{v}(\varrho -\varrho _{\mathscr {k}})}-1}{2\gamma _{v}}, \ \frac{e^{2\gamma _{v}(\varrho _{\mathscr {k}+1}-\varrho)}-1}{2\gamma _{v}}, \ \mathscr {z}(\varrho _{\mathscr {k}})$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathscr {z}(\varrho _{\mathscr {k}+1})$</tex-math></inline-formula> with tuning parameters and few slack variables. The AE-DADT switching property has also been taken into account for nonlinear systems, and it contributes to getting more information about previous modes. Based on the proposed new refined looped-LFs, new sufficient conditions are derived in terms of linear matrix inequalities (LMI), which ensure the global uniform exponential stability (GUES) of the nominated switched IT2F systems. The feasibility of the obtained results is verified by the numerical example, and the tunnel diode circuit system model and Lorenz chaotic system are established to illustrate the feasibility and efficacy of the theoretical findings.

On Contraction Analysis of Switched Systems With Mixed Contracting-Noncontracting Modes via Mode-Dependent Average Dwell Time

This paper studies contraction analysis of switched systems that are composed of a mixture of contracting and noncontracting modes. The first result pertains to the equivalence of the contraction of a switched system and the uniform global exponential stability of its variational system. Based on this equivalence property, sufficient conditions for a mode-dependent average dwell/leave-time based switching law to be contractive are established. Correspondingly, LMI conditions are derived that allow for numerical validation of contraction property of nonlinear switched systems, which include those with all non-contracting modes.

Predictor‐based switching control for networked systems

SummaryThis paper concerns a networked predictive control (NPC) for linear systems with data time‐varying delays, packet losses and disordering. By utilizing the received time‐delayed state measurement and control input, a predictive algorithm is firstly proposed for approximating the future state prediction. Afterwards, a probability dependent switching control law is proposed. Although the large data transmission delays emerge in the forward and backward channels, the packet disordering in the backward channel can be naturally excluded thus the actuator always receives valid control command during each sampling period with correct time sequence. The proposed NPC not only guarantees the global uniform exponential stability of the overall networked system but also brings merits in tolerating locally unstable sub‐systems. Numerical examples are provided to demonstrate the effectiveness and improvements of the proposed method.

Practical stability analysis of stochastic functional differential systems with G-Brownian motion and impulsive effects

This article is concerned with the practical stability performance of nonlinear impulsive stochastic functional differential systems driven by G-Brownian motion (G-ISFDSs). Comparing with traditional Lyapunov stability theory, practical stability can portray qualitative behavior and quantitative properties of suggested systems. By employing G-Itô formula, Lyapunov-Razumikhin approach and stochastic analysis theory, some novel conditions for pth moment practical exponential stability and quasi sure global practical uniform exponential stability of G-ISFDSs are established. The obtained results show that impulses may influence dynamic behavior of the addressed system. Two numerical examples are given to verify the validity of our developed results.

Towards nonconservative conditions for equilibrium stability. Applications to switching systems with control delay

The objective of the paper is to give a unitary approach to nonconservative conditions of the equilibrium stability. The focus is on a real world system defined by a switched linear mathematical model having control delay. A control synthesis based on predictive state feedback is first applied to compensate the actuator delay. This results in a closed loop switching system free of delay. Then, a theorem giving conservative sufficient conditions of global uniform exponential stability for a switched system is proved. Next, a theorem expressing necessary and sufficient stability conditions is cited, in which the differences compared to the proved theorem are highlighted, but whose mechanism could not be applied in practice. Finally, it is shown how the problem of nonconservativeness can be solved starting from an important theorem given in the literature that uses the idea of homogeneous polynomial Lyapunov functions originating in the Hilbert’s famous 17th Problem. The paper investigates the complexity of the method in terms of numerical computation volume, and offers a way to circumvent the conflict between the method and the mathematical model. In this manner older and newer results are presented in a unitary concept and in a selfcontained approach. Numerical simulations are applied to a real-world model: an active vibration control for a physical model of an intelligent airplane wing in a wind tunnel. The substantial result of the simulations is the minimum guaranteed dwell time that characterizes the equilibrium stability of the switching system with control delay.

On stability for generalized linear differential equations and applications to impulsive systems

In this paper, we are interested in investigating notions of stability for generalized linear differential equations (GLDEs). Initially, we propose and revisit several definitions of stability and provide a complete characterization of them in terms of upper bounds and asymptotic behaviour of the transition matrix. In addition, we illustrate our stability results for GLDEs to linear periodic systems and linear impulsive differential equations. Finally, we prove that the well-known definitions of uniform asymptotic stability and variational asymptotic stability are equivalent to the global uniform exponential stability introduced in this article.

Stabilization of Switched Linear Neutral Systems With Time-Scheduled Feedback Control Strategy

This article is concerned with the stabilization of continuous-time switched linear neutral systems with tighter bounds on mode-dependent average dwell time. By introducing a novel time-scheduled control strategy, a switching-signal-based multiple discontinuous Lyapunov function (SMDLF) is constructed for continuous-time switched linear neutral systems, where the discontinuous points of the Lyapunov function during each mode-running interval are dynamically adjusted according to the actual system switching signal. First, sufficient criteria are established for global uniform exponential stability (GUES) of switched nonlinear systems. Then, based on the constructed SMDLF, we derive less conservative stability conditions for switched linear neutral systems. Furthermore, a set of time-dependent state feedback controllers are designed to guarantee the GUES of the underlying system. In the end, a simulation example with comparison results is provided to verify the effectiveness and advantages of the presented approach.

Period-Sampled Switching Event-Triggered Control of Switched Linear Systems

This brief focuses on the event-triggered control (ETC) problem for a class of switched linear systems in which the controller updates the sampled information of both the system states and the switching signal only at the moment of event triggering. When the system switching occurs in the holding interval, the controller does not update the switching signal, which leads to the asynchronous switching behavior between the controller and the switched system. To solve the problem, we develop a new switching event-triggered mechanism (ETM) with periodic sampling by introducing a relative threshold function and an asynchronous related jump function with a tuning parameter. This period-sampled switching ETM can save computing resources and avoid Zeno behavior. Then, based on the multiple Lyapunov functional technique and the average dwell time (ADT) method, some sufficient conditions for global uniform exponential stability (GUES) of the closed-loop system are proposed. Besides, the corresponding solvability conditions for the co-design of the switching signal and the event-triggered controller are established. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed method.

Stability of impulsive dynamical networks with different impulse time sequences and application to event-triggered impulsive control

This paper studies the stability for impulsive dynamical networks (IDN) where different nodes may have different impulse time sequences. By deriving the conditions of global uniform exponential stability for zero-order delayed systems, the criteria of stability for IDN are established. The stability results for IDN are then used to design event-triggered impulsive control (ETIC) for stabilization of continuous-time dynamical networks (CDN). In every node of CDN, three levels of event-triggering conditions are set for ETIC. The exponential stabilization of CDN is shown to be achieved by the designed ETIC and it also shows that the ETIC is robust w.r.t. time-delays. Moreover, a non-instantaneous ETIC (NI-ETIC) is proposed for the stabilization of CDN. Three examples with numerical simulations are given to show that the designed ETIC has lower impulse frequency and cost of control than the classic time-triggered impulsive control. It is also shown that the discontinuity and the instantanity of ETIC are improved by the NI-ETIC.

Generalized integral inequality and application on partial stability analysis

The method of Lyapunov is one of the most effective methods for the analysis of the partial stability of dynamical systems. Different authors develop the problem of partial practical stability based on Lyapunov techniques. In this paper, we investigate the partial practical stability of linear time-invariant perturbed systems based on the integral inequalities of the Gronwall type, in particular of Gamidov?s type. We derive some sufficient conditions that guarantee global practical uniform exponential stability with respect to a part of the variables of linear time-invariant perturbed systems. Also, we have developed the local partial practical stability of nonlinear systems. Further, we provide two examples to support our findings.

Distributed bearing-based localization under switching graph topologies

This paper addresses a distributed localization problem for n (n ≥ 2)-agent systems defined in d (d ≥ 2)-dimensional space under undirected switching graph topologies. It is assumed that each agent in the formation can measure the bearing to the neighboring agents and communicate with them, and at least one agent has access to its global position. Novel distributed position observers are designed under which each agent estimates its global position using its velocity, the relative bearing measurements, and the position estimates of neighboring agents received over a communication network. The main contribution of this work is that i) it relaxes the classical bearing rigidity assumption (typically required) for bearing-based localization problems addressed in the current literature and ii) it covers time-varying bearing measurements and/or time-varying graph topologies. Furthermore, the efficiency of the proposed observers is guaranteed by the rigorous mathematical analysis (i.e., uniform global exponential stability of the estimation error) along with illustrative simulation results.

Mixed [formula omitted]/passive synchronization for persistent dwell-time switched neural networks via an activation function dividing method

The mixed H∞/passive synchronization issue of discrete-time switched neural networks is studied in this paper. In order to regulate the switching between subsystems, the persistent dwell-time switching law is adopted. The paper aims to design a suitable synchronization controller to make the synchronization error system satisfy the mixed H∞/passive performance and achieve global uniform exponential stability. By employing Lyapunov stability theory, performance analysis criteria and the synchronization controller design method are given, in which an activation function dividing method is employed to reduce their conservatism. Simulation results demonstrate the superiority and effectiveness of the method.

Online optimization of switched LTI systems using continuous-time and hybrid accelerated gradient flows

This paper studies the design of feedback controllers to steer a switching linear dynamical system to the solution trajectory of a time-varying convex optimization problem. We propose two types of controllers: (i) a continuous controller inspired by the online gradient descent method, and (ii) a hybrid controller that can be interpreted as an online version of Nesterov’s accelerated gradient method with restarts of the state variables. By design, the controllers continuously steer the system toward a time-varying optimal equilibrium point without requiring knowledge of exogenous disturbances affecting the system. For cost functions that are smooth and satisfy the Polyak–Łojasiewicz inequality, we demonstrate that the online gradient-flow controller ensures uniform global exponential stability when the time scales of the system and controller are sufficiently separated and the switching signal of the system varies slowly on average. For cost functions that are strongly convex, we show that the hybrid accelerated controller can outperform the continuous gradient descent method. When the cost function is not strongly convex, we show that the hybrid accelerated method guarantees global practical asymptotic stability.

A novel criterion for global incremental stability of dynamical systems

In this paper, we establish the sufficient conditions guaranteeing global uniform exponential stability, or at least global asymptotic stability, of all solutions for nonlinear dynamical systems, also known as global incremental stability (GIS) of the systems. We provide here an alternative approach for assessment of GIS in terms of logarithmic norm under which the stability becomes a topological notion and also generalize both horizontally and vertically the well-known Demidovich criterion for GIS of dynamical systems. Convergence of all solutions to the origin x=0, which is not assumed to be an equilibrium state of system, is also analyzed. Theory is illustrated by a simulation experiment.

Event‐triggered switching‐dependent adaptive integral sliding mode control for switched systems

SummaryThis article considers the event‐triggered adaptive sliding mode control problem for switched systems with the exogenous disturbance and the unknown nonlinear disturbance subject to network‐induced delay. The system output is considered to be periodically sampled. First, a mode‐dependent event‐triggering scheme is proposed to decide whether the sampled system output is transmitted or not to the observer. Second, two new integral sliding surfaces depending on the state of the switching instants are designed for the observer and the error system respectively. Moreover, the sliding mode observer for the error system can compensate the effect of the nonlinearity and match the exogenous disturbance by its upper bound, which is more suitable in practical applications. Compared with the existing results, the proposed switching‐dependent sliding mode surface for the error system not only has the characteristics of the mode dependent sliding mode surface, but also we establish the relationship among the average dwell time, the triggering condition and the mode dependent sliding mode surface. The stability conditions of the sliding mode dynamics are presented, and two event‐triggered adaptive sliding mode controllers are derived to guarantee global uniform exponential stability of switched systems with a performance. Finally, an example is employed to show the effectiveness.

Uniform exponential stability criteria with verification for nonautonomous switched nonlinear systems with uncertainty

AbstractThis article is concerned with the sufficient and necessary conditions for the global uniform exponential stability (GUES) of a class of uncertain nonautonomous switched nonlinear systems (UNSNS), that is, systems with time‐varying uncertain information. To start with, based on a piecewise continuous scalar function and a piecewise differentiable uncertainty‐dependent Lyapunov function along the trajectories, we propose alternative sufficient and necessary conditions for the GUES of UNSNS. Afterwards, for removing the dependence on uncertain information of the required Lyapunov functions, we combine an uncertainty‐independent Lyapunov function with its right upper Dini derivative and the above piecewise continuous scalar function to further obtain other alternative sufficient and necessary conditions for the GUES of UNSNS. Note that our conditions release the requirement on negative definiteness of the time‐derivatives and Dini derivatives of Lyapunov functions respectively, which is illustrated by Example 1. Further, for the rational UNSNS, we propose a linear semidefinite programming based computable approach to mechanically verify our current theoretical results. In the end, the effectiveness of our theoretical results and mechanical approach are illustrated by two examples.