Uniform Global Asymptotic Stability Research Articles

Improving extensions for the global uniform asymptotic stability of continuous-time nonlinear systems

This paper considers the classical problems of global asymptotic stability (GAS) of nonlinear autonomous (time-invariant) systems and global uniform asymptotic stability (GUAS) of nonlinear non-autonomous (time-varying) systems. By imposing new conditions on the Lyapunov function and its derivative, the state convergence to the origin from any initial condition with bounded norm in the classical setting is extended to the initial conditions with unbounded norms in the present work. The proposed new notions of stability are called extended global asymptotic stability (EGAS) and extended global uniform asymptotic stability (EGUAS) in this paper. The proposed Lyapunov-based criteria yield the global asymptotic stability of the origin and fixed-time/predefined-time attractiveness of any selected neighbourhood of the origin. The result is that the dynamical behaviour of the state trajectories from the convergence time point of view is improved considerably. The relationship between the smoothness and input-to-state stability (ISS) properties is also studied from a quantitative point of view. Under some new sufficient conditions, it is shown that dynamical systems with smaller Lipschitz constants represent smaller ultimate bounds, and as a result, are more robust. The desirable dynamical behaviour and robustness property of the proposed stability notion makes it comparable with the finite/fixed/predefined/prescribed (exact) stable systems that are naturally non-Lipschitz at the origin. Numerical results illustrate the effectiveness of the proposed theoretical results.

Event‐triggered anti‐disturbance control for switched singular systems based on the uncertainty and disturbance estimator

AbstractThis article studies the event‐triggered anti‐disturbance control problem for switched singular systems (SSSs) based on the uncertainty and disturbance estimator (UDE). First, a composite controller consisting of feedback control and anti‐disturbance control based on UDE is proposed. By constructing the multiple Lyapunov functions and using the average dwell time (ADT) method, sufficient conditions are given to ensure that the closed‐loop system is regular, impulse‐free and globally uniformly asymptotically stable (GUAS). Further, in order to save resources, the event‐triggered control method is introduced into the feedback control, and a novel hybrid event‐triggered mechanism (ETM) is proposed, which guarantees that asynchronous switching and Zeno behavior are excluded. And the solvable conditions in terms of linear matrix inequalities (LMIs) are given. Finally, the validity of the proposed results is verified by the simulations.

Initialization-free Lie-bracket Extremum Seeking

Stability results for extremum seeking control in Rn have predominantly been restricted to local or, at best, semi-global practical stability. Extending semi-global stability results of extremum-seeking systems to unbounded sets of initial conditions often demands a stringent global Lipschitz condition on the cost function, which is rarely satisfied by practical applications. In this paper, we address this challenge by leveraging tools from higher-order averaging theory. In particular, we establish a novel second-order averaging result with global (practical) stability implications. By leveraging this result, we characterize sufficient conditions on cost functions under which uniform global (i.e., under any initialization) practical asymptotic stability can be established for a class of extremum-seeking systems acting on static maps. Our sufficient conditions include the case when the gradient of the cost function, rather than the cost function itself, satisfies a global Lipschitz condition, which covers quadratic cost functions. Our results are also applicable to vector fields that are not necessarily Lipschitz continuous at the origin, opening the door to non-smooth Lie-bracket ES dynamics. We illustrate all the results via different analytical and/or numerical examples.

Forward completeness does not imply bounded reachability sets and global asymptotic stability is not necessarily uniform for time-delay systems

An example of a time-invariant time-delay system that is uniformly globally attractive and exponentially stable, hence forward complete, but whose reachability sets from bounded initial conditions are not bounded over compact time intervals is provided. This gives a negative answer to two current conjectures by showing that (i) forward completeness is not equivalent to robust forward completeness (i.e. boundedness of reachability sets) and (ii) global asymptotic stability is not equivalent to uniform global asymptotic stability. In addition, a novel characterization of robust forward completeness for systems having a finite number of discrete delays is provided. This characterization relates robust forward completeness of the time-delay system with the forward completeness of an associated nondelayed finite-dimensional system.

Adaptive cyber-tolerant finite-time frequency control framework for renewable-integrated power system under deception and periodic denial-of-service attacks

This work concentrates on designing and applying an adaptive cyber-tolerant finite-time frequency control framework for smart power systems under state-dependent sensor deception and periodic denial-of-service (DoS) attacks. An adaptive radial-basis function neural network (RBFNN)-based disturbance observer (DO) is designed to estimate undeniable plant disturbances, which includes structural uncertainties and unmodelled dynamics. Then, the DO-based resilient secondary control law based on the adaptive nonlinear sliding mode strategy for the cyber-physical power system (CPPS) is derived to guarantee global uniform asymptotic stability of the parametric error and the selected nonlinear sliding manifold. Furthermore, an in-depth analysis of uniformly ultimately boundedness (UUB) of the estimation error of RBFNN has been demonstrated. Numerical simulation and qualitative assessment have revealed the proficiency of the applied feedback control framework for the undertaken CPPS and a high-degree of cyber-tolerance against malicious attacks.

Quantized filtering for switched memristive neural networks against deception attacks

This paper deals with the quantized mixed H∞/L2−L∞ filtering issue for switched memristive neural networks in the presence of deception attacks, in which the switching of parameters is not only subject to neural states, but also to the persistent dwell-time switching rule in the continuous-time case. First, a switched dynamic quantizer is utilized to quantize the measurement output before being transmitted to the filter. Second, deception attacks governed by a Bernoulli distribution are considered during the quantized output transmission. Then, in virtue of the Lyapunov stability theory together with stochastic analysis theory, a sufficient condition is given to ensure the global uniform asymptotic stability in the mean-square sense for the filtering error system with a mixed H∞/L2−L∞ performance. Furthermore, a co-design scheme for the desired mode-dependent filter gain and dynamic parameter is provided. Ultimately, the effectiveness of the proposed method is verified by a numerical example.

Emulation-based stabilization for networked control systems with stochastic channels

This paper studies the stabilization problem of networked control systems (NCSs) with random packet dropouts caused by stochastic channels. To describe the effects of stochastic channels on the information transmission, the transmission times are assumed to be deterministic, whereas the packet transmission is assumed to be random. We first propose a stochastic scheduling protocol to model random packet dropouts, and address the properties of the proposed stochastic scheduling protocol. The proposed scheduling protocol provides a unified modelling framework for a general class of random packet dropouts due to different stochastic channels. Next, the proposed scheduling protocol is embedded into the closed-loop system, which leads to a stochastic hybrid model for NCSs with random packet dropouts. Based on this stochastic hybrid model, we follow the emulation approach to establish sufficient conditions to guarantee uniform global asymptotical stability in probability. In particular, an upper bound on the maximally allowable transmission interval is derived explicitly for all stochastic protocols satisfying Lyapunov conditions that guarantee uniform global asymptotic stability in probability. Finally, two numerical examples are presented to demonstrate the derived results.

A Stochastic Hybrid Framework for Event-Triggered Nonlinear Control Systems With Stochastic Denial of Service Attacks

In this paper, the stability problem for event-triggered nonlinear control systems is addressed in the presence of stochastic denial of service (DoS) attacks on the communication network. Dynamic event-triggered control (DETC) and periodic event-triggered control (PETC) are considered under the stochastic attacks. The stochastic attacks are modeled by a stochastic hybrid timer and a logic variable which is used to identify whether the network is attacked or not. Then the entire closed-loop system can be modeled into the framework of stochastic hybrid systems. Uniform global asymptotic stability in probability is concluded for the resulting stochastic system by proving that the equilibrium set is uniformly globally stable in probability and each neighborhood of the equilibrium set is uniformly globally recurrent. A nonlinear system example is provided to illustrate the effectiveness of the proposed results.

Adaptive Regulation With Global KL Guarantees

In the absence of persistency of excitation (PE), referring to adaptive control systems as “uniformly asymptotically stable” typically indicates insufficient understanding of stability concepts. While the state is indeed regulated to zero and the parameter estimate has some limit, namely, the overall state converges to some equilibrium, the equilibrium reached is not unique (and not even necessarily stable) but is dependent on the initial condition. The equilibrium set in the absence of PE is not uniformly attractive (from an open set containing the equilibrium set); hence, uniform asymptotic stability does not hold and KL-estimates are unavailable for the full state. In this paper we pursue adaptive control design with KL guarantees on the regulated state, something that is possible but previously unachieved with smooth, time-invariant and non-hybrid adaptive controllers. This property is referred to as Uniform Global Asymptotic Output Stability, where the regulated state is thought of as a system output. We provide designs for (i) systems with a matched uncertainty and (ii) systems in the parametric strict feedback form. We assume no a priori known bound for the parameters. To guarantee KL-estimates in the absence of PE, our designs employ time-invariant nonlinear damping terms, which depend both on the state and the parameter estimate. With an example, we illustrate the theory.

Practical asymptotic stability for time-varying nonlinear systems

This paper is concerned with the global uniform practical asymptotic and exponential stability for a large class of time-varying systems by using the concept of practical scalar functions. A new converse stability theorem is established which is used to give some sufficient conditions that guarantee the global uniform practical asymptotic and exponential stability of a class of perturbed systems. Furthermore, some examples are presented.

Improved results of asynchronous mixed [formula omitted] and passive control for discrete-time linear switched system with mode-dependent average dwell time

This paper primarily discusses the hybrid H∞ and passive control problems of asynchronous discrete-time switching systems based on mode-dependent average dwell time. In the process of analysis, we divide the running time of the subsystem into two cases: matching and not matching the controller. Firstly, we employ the method of mode-dependent average dwell time to model the system and derive a suitable switching signal, ensuring the system is global uniform asymptotic stability. Meanwhile, the weighted hybrid H∞ and passivity are achieved. Then, according to the different performance parameters of each subsystem, the appropriate controller is designed using the form of linear matrix inequalities to improve the stability of the system. Finally, the effectiveness of the proposed method is demonstrated by a numerical example and a population ecosystem.

Uniting Nesterov and heavy ball methods for uniform global asymptotic stability of the set of minimizers

We propose a hybrid control algorithm that guarantees fast convergence and uniform global asymptotic stability of the unique minimizer of a C1, convex objective function. The algorithm, developed using hybrid system tools, employs a uniting control strategy, in which Nesterov’s accelerated gradient descent is used “globally” and the heavy ball method is used “locally”, relative to the minimizer. Without knowledge of its location, the proposed hybrid control strategy switches between these accelerated methods to ensure convergence to the minimizer without oscillations, with a (hybrid) convergence rate that preserves the convergence rates of the individual optimization algorithms. We analyze key properties of the resulting closed-loop system including existence of solutions, uniform global asymptotic stability, and convergence rate. Additionally, stability properties of Nesterov’s method are analyzed, and extensions on convergence rate results in the existing literature are presented. Numerical results validate the findings.

A Distributed Robust Two-Time-Scale Switched Algorithm for Constrained Aggregative Games

Robust Nash equilibrium (NE) seeking for a constrained aggregative game is explored. Our setting has the following three features: 1. Local feasibility sets are considered. 2. Each player has first-order dynamics that are influenced by unknown time-varying disturbances and unmodeled terms. 3. The communication topology among players may be influenced by attacks. The existence of unknown disturbances results in the trajectories of the players going out of the feasible sets even though the initial values start from inside. To make the ideas clear, we first use an example that exhibits the three features to introduce the problem formulation to be studied. Subsequently, a two-time-scale distributed, projected algorithm is proposed. The simulation results are also shown. To further illustrate the effectiveness of the proposed algorithm through theoretical analysis, such an algorithm is generalized into a two-time-scale hybrid system by using an average dwell-time automaton and a time-ratio monitor to impose restrictions on the frequency and relative activation time of the attacks. To analyze stability of such a hybrid system, the reduced system and the boundary layer system are presented and corresponding Lyapunov functions are constructed. Correspondingly, uniform (uniform global) asymptotic stability is obtained under some mild assumptions. Also, some results are given for systems without attacks. Finally, two aggregative games are used to show the effectiveness of the proposed algorithm and theoretical analysis.

Robust I&I adaptive tracking control of systems with nonlinear parameterization: An ISS perspective

This paper studies the immersion and invariance (I&I) adaptive tracking problem for a class of nonlinear systems with nonlinear parameterization in the ISS framework. Under some mild assumptions, a novel I&I adaptive control algorithm is proposed, leading to an interconnection of an ISS estimation error subsystem and an ISS tracking error subsystem. Using an ISS small-gain condition, the desired uniform global asymptotic stability of the resulting interconnected “error” system can be achieved and a sum-type strict Lyapunov function can be explicitly constructed. Taking advantage of this ISS-based design framework, it is shown that the corresponding robustness with respect to the input perturbation can be rendered ISS. To remove the need to solve the immersion manifold shaping PDE, a new filter-based approach is proposed, which preserves the ISS-based design framework. Finally, we demonstrate the validness of the proposed framework on a tracking problem for series elastic actuators.

Sufficient conditions for uniform global asymptotic stabilization of affine discrete-time systems with periodic coefficients

Sufficient conditions for uniform global asymptotic stabilization of affine discrete-time systems with periodic coefficients

Distributed observer-based finite-time control of moving target tracking for UAV formation

This paper disseminated two formation control strategies for multiple unmanned aerial vehicles (multi-UAV) system of moving target tracking in a windy environment. The communication among UAVs is modeled by a directed graph. The first control strategy proposes a distributed dynamic error observer and a guidance law to make the system global uniform asymptotic stability when the wind disturbance is a known constant. The second control strategy employs a distributed fixed-time observer and a finite-time stable guidance law to make the system globally finite-time stable with unknown wind disturbances. The stability of both formation control strategies is rigorous demonstrated mathematically. Finally, the excellent performance and reliability of the proposed guidance law for target tracking in a windy environment are verified through several simulation examples.

A Lyapunov-Based Small-Gain Theory for Infinite Networks via Infinite-Dimensional Gain Operators

In this paper, we develop a new approach to study gain operators built from the interconnection gains of infinite networks of dynamical systems. Our focus is on the construction of paths of strict decay which are used for building Lyapunov functions for the network and thus proving various stability properties, including input-to-state stability. Our approach is based on the study of an augmented gain operator whose fixed points are precisely the points of decay for the original gain operator. We show that plenty of such fixed points exist under a uniform version of the no-joint-increase condition. Using these fixed points to construct a path of strict decay, in general, requires specific dynamical properties of associated monotone operators. For particular types of gain operators such as max-type operators and subadditive operators, these properties follow from uniform global asymptotic stability of the system induced by the original gain operator. This is consistent with former results in the literature which can readily be recovered from our results.

Analysis of the Relation Between L<sub>2</sub> Gain and Dwell Time in Discrete Time Switching System

This paper focuses on the design of <i>l</i>₂ gain controller for discrete time switching systems with dwell time (DT) switching. So far, the quantitative relationship between <i>l</i>₂ gain stability and dwell time is one of the unsolved problems in the field of switching systems. Different from previous studies, a novel dwell time dependent Lyapunov function (DTDLF) is proposed in this paper for the first time to obtain a less conservative stability criterion. The innovation of this Lyapunov function lies in that it is no longer built by the traditional two-step controller design method, but combines the two design tasks of controller and switching law into one framework, in which solving linear matrix inequalities (LMIs) involves only one step. Compared with the two-step design scheme, this method greatly simplifies the design process. Based on the resident time dependent Lyapunov function and the resident time switching strategy, a dwell time dependent (DTD) H_∞ controller is designed such that the system is globally uniformly asymptotically stable (GUAS) and has a guaranteed H_∞ performance index. Finally, a numerical example is given to verify the validity of the proposed design method. The simulation results show that the proposed design method can guarantee a shorter resident time limit and a smaller error result.

Composite event‐triggered H∞ control for nonlinear switched singular systems with a mode‐dependent triggering strategy

SummaryThis paper considers the event‐triggered control problem of nonlinear switched singular systems (NSSSs) with incremental quadratic constraints (IQCs) via a mode‐dependent hybrid triggering strategy. First, impulsive control is introduced to reduce the effect of impulse caused by switching, and the closed‐loop system is modeled as an impulsive NSSS. Then, a novel mode‐dependent hybrid event‐triggering strategy (ETS) consisting of disturbance‐influenced ETS and Lyapunov‐function‐dependent ETS is constructed to avoid Zeno behavior and asynchronous phenomena. Based on this ETS and admissible edge‐dependent average dwell time (AED‐ADT) method, some new sufficient conditions in terms of linear matrix inequalities (LMIs) are derived to ensure that the closed‐loop system is regular, impulse‐free, has a unique solution, and is globally uniformly asymptotically stable (GUAS) with a weighted performance. Further, Zeno behavior is excluded by calculating the minimum lower bound of inter‐event time. Finally, a numerical example and a practical application are proposed to demonstrate the validity of the results.

A Switching Strategy for Stabilization of Discrete-Time Switched Positive Time-Varying Delay Systems with All Modes Being Unstable and Application to Uncertain Data

The stability problem of switched systems plays an essential role in the study of long-term behavior. In fact, systems containing both time delay and uncertainty terms may lead to performance degradation of those systems. Therefore, we are interested in the robust stability for discrete-time switched positive time-varying delay systems with interval uncertainties in the case of all modes being unstable. Based on the proposed time-scheduled multiple co-positive Lyapunov–Krasovskii functional of each mode, new sufficient conditions for the global uniform asymptotic stability of the systems are derived. An effective time-dependent switching law utilized in this work is mode-dependent dwell time. In addition, the robust stability criteria in an asymptotic sense are formulated for the systems without time-varying delay. Compared with the existing related works, our results are less conservative and more general than some previous research. Finally, two numerical examples are provided to illustrate the effectiveness and correctness of the developed theoretical results.