Uniform Asymptotic Stability Research Articles

On the Partial Stability of Nonlinear Discrete-Time Systems with Delay

A system of nonlinear discrete (finite-difference) of a general form with a bounded delay is considered. Interest in the tasks of qualitative analysis of such systems has increased significantly in recent years. At the same time, the problem of stability with respect to all variables of the zero equilibrium position, which has a great generality, is mainly analyzed in domestic and foreign literature. The main research method is a discrete-functional analogue of the direct Lyapunov method. In this article, it is assumed that the system under consideration admits a “partial” (in some part of the state variables) zero equilibrium position. The problem of stability of a given equilibrium position is posed, and stability is considered not in all, but only in relation to a part of the variables that determine this equilibrium position. Such a problem belongs to the class of problems of partial stability, which are actively studied for systems of various forms of mathematical description. The proposed statement of the problem complements the scope of the indicated studies in relation to the system under consideration. To solve this problem, a discrete version of the Lyapunov– Krasovskii functionals method is used in the space of discrete functions with appropriate specification of the functional requirements. To expand the capabilities of this method, it is proposed to use two types of additional auxiliary (vector, generally speaking) discrete functions in order to: 1) adjustments of the phase space region of the system in which the Lyapunov–Krasovskii functional is constructed; 2) finding the necessary estimates of the functionals and their differences (increment) due to the system under consideration, on the basis of which conclusions about partial stability are made. The expediency of this approach lies in the fact that as a result, the Lyapunov-Krasovskii functional, as well as its difference due to the system under consideration, can be alternating in the domain that is usually considered when analyzing partial stability. Sufficient conditions of partial stability, partial uniform stability, and partial uniform asymptotic stability of the specified type are obtained. The features of the proposed approach are shown on the example of two classes of nonlinear systems of a given structure, for which partial stability is analyzed in parameter space. Attention is drawn to the expediency of using a one-parameter family of functionals.

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.

Practical Input-to-State Stability of Nonlinear Systems with Time Delay

An investigation of the input-to-state practical stability (ISpS) and the integral input-to-state practical stability (iISpS) of nonlinear systems with time delays (NSWTDs) is presented in this paper. The ISpS and iISpS of the systems are obtained by using a continuously differentiable Lyapunov–Krasovskii functional (LKF) within-definite derivative, which generalizes the classical LKF with a positive definite derivative. As a result, the uniform practical asymptotic stability (UPAS) criteria of NSWTD were also given by using the proposed LKF.

Some New Results on the Uniform Asymptotic Stability for Volterra Integro-differential Equations with Delays

In this work, we establish sufficient conditions of the uniform asymptotic stability (UAS) of solutions to second-order and third-order of Volterra integro-differential equations (VIDE) with delay. Here, we prove two new theorems on the UAS of the solutions of the considered VIDEs. Our approach is based on Lyapunov’s second method. Our results improve and form a complement to some known recent results in the literature. Two illustrative examples are considered to support the results and two graphs are drawn to illustrate the asymptotic stability of the zero solution for the considered numerical equations. The obtained results are new and original.

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

Underactuated Source Seeking by Surge Force Tuning: Theory and Boat Experiments

We extend source seeking algorithms, in the absence of position and velocity measurements, and with the tuning of the surge input, from velocity-actuated (unicycle) kinematic models to force-actuated generic Euler–Lagrange dynamic underactuated models. In the design and analysis, we employ a symmetric product approximation, averaging, passivity, and partial-state stability theory. The proposed control law requires only real-time measurement of the source signal at the current position of the vehicle and ensures semi-global practical uniform asymptotic stability (SPUAS) with respect to the linear motion coordinates for the closed-loop system. The performance of our source seeker with surge force tuning is illustrated with both numerical simulations and experiments of an underactuated boat.

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.

Uniform asymptotic stability for convection–reaction–diffusion equations in the inviscid limit towards Riemann shocks

The present contribution proves the asymptotic orbital stability of viscous regularizations of stable Riemann shocks of scalar balance laws, uniformly with respect to the viscosity/diffusion parameter \varepsilon . The uniformity is understood in the sense that all constants involved in the stability statements are uniform and that the corresponding multiscale \varepsilon -dependent topology reduces to the classical W^{1, \infty} -topology when restricted to functions supported away from the shock location. Main difficulties include that uniformity precludes any use of parabolic regularization to close regularity estimates, that the global-in-time analysis is also spatially multiscale due to the coexistence of nontrivial slow parts with fast shock-layer parts, that the limiting smooth spectral problem (in fast variables) has no spectral gap and that uniformity requires a very precise and unusual design of the phase shift encoding orbital stability. In particular, our analysis builds a phase that somehow interpolates between the hyperbolic shock location prescribed by the Rankine–Hugoniot conditions and the nonuniform shift arising merely from phasing out the nondecaying 0-mode, as in the classical stability analysis for fronts of reaction–diffusion equations.

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.

A hybrid trigger strategy based mixed impulsive control scheme on uncertain local field neural networks with leakage delay

Solved in this paper is the stabilization of uncertain local field neural networks (ULFNNs) with leakage delay by dint of a mixed impulsive control scheme. The impulsive control instants are decided by a Lyapunov functional-based event triggered scheme and a periodic impulse triggered scheme. Based on the proposed control scheme, sufficient conditions are deduced for eliminating the Zeno behavior and guaranteeing the uniform asymptotic stability (UAS) of the delayed ULFNNs in the framework of Lyapunov functional analysis method. Different from individual event triggered impulse control strategy where the triggered instants may be unexpected, the mixed impulsive control scheme can release the impulse control instants in line with the distance between any two adjacent successful control points, which improves the control performance and also achieves the goal of saving the communication resource. Furthermore, the decay behavior of impulse control signal is considered to make mathematical derivation more practical, and based on this behavior, a criterion is derived to make ensure the exponential stability of delayed ULFNNs. Finally, numerical examples are given to illustrate the effectiveness of the designed controller for the ULFNNs with leakage 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.

A fractional-order model to study the dynamics of the spread of crime

Numerous crucial factors and parameters influence the dynamic process of the spread of crime. Various integer-order differential models have been proposed to capture crime spread. Most of these introduced dynamic systems have not considered the history of the criminal and the impact of crime on society. To address these shortcomings, a fractional-order crime transmission model is proposed in this manuscript considering five different classes viz law-abiding citizens, non-incarcerated criminals, incarcerated criminals, prison-released and recidivists. The primary focus of the proposed model is to study the effect of recidivism in society and decide the adequate imprisonment for repeat offenders. The existence, uniqueness, non-negativity and boundedness of the solution of the proposed model are examined. The local stability of the equilibrium points is also analysed using Routh–Hurwitz Criteria with Matignon conditions. Further, the threshold condition for the uniform asymptotic stability of the system is evaluated by the Lyapunov stability method. Moreover, the long-term impact of the imprisonment of criminals on society is also examined in the current study. The numerical simulations of the model for a range of fractional orders are obtained using power series expansion method to strengthen the theoretical results.

Uniform convergence for directed time-varying signed networks under generalized Laplacian-class model

This paper addresses the uniform convergence problems for signed networks under directed time-varying topologies. A generalized Laplacian-class model is proposed by admitting the time-varying relative self-effect of each agent, which can be reflected by a newly introduced notion of the diagonal dominance degree. With a state transition matrix-based approach, sufficient conditions on uniform asymptotic stability and uniform bipartite consensus subject to any initial time are obtained for signed networks, respectively. It is shown that these uniform convergence behaviors depend on not only the topology structure but also the diagonal dominance degree. The uniform convergence results for time-varying signed networks are further applied to cope with the uniform asymptotic stability problem of general linear time-varying systems.

Sliding Mode Control of Active Magnetic Bearings - A Cascaded Architecture

Accurate and robust positioning of rotor axle is essential for efficient and safe operation of high-speed rotational machines with active magnetic bearings. This study presents a cascaded nonlinear control strategy for vertical axial positioning of an active magnetic bearing system. The proposed scheme employs two sliding mode controllers for regulating rotor vertical position and current and an adaptive estimator to invert the nonlinear input mapping. Uniform asymptotic stability is proven for the closed-loop system and the efficacy and performance of the proposed design is evaluated in simulation.

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.

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).

A High-Order Sliding-Mode Adaptive Observer for Uncertain Nonlinear Systems

A high-order sliding-mode adaptive observer is proposed to solve the problem of an adaptive estimation, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.,</i> the simultaneous estimation of the state and parameters, for a class of uncertain nonlinear systems in the presence of external disturbances, which do not need to satisfy a relative degree condition equal to one. This approach is based on a high-order sliding-mode observer and a nonlinear parameter identification algorithm. The practical, global, and uniform asymptotic stability of the adaptive estimation error, despite the external disturbances, is guaranteed through the small-gain theorem. The convergence proofs are developed based on the Lyapunov and input-to-state stability theories. Some simulation results illustrate the performance of the proposed high-order sliding-mode adaptive observer.

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.

Passivity-Based Motion and Force Tracking Control for Constrained Elastic Joint Robots

In the past, several motion and force controls were successfully implemented on rigid-joint robots with constraints. With the invention of mechanically compliant robots, the focus on designing controllers for elastic joint robots with constraints is increasing, especially involving the complexity of the joint elasticity in control. Aiming to bridge the gap between the control schemes of rigid- and elastic-joint robots, this letter presents a controller consisting of a PD+ task-space tracking and integral force control, while the intrinsic inertial and elastic properties of the system are fully preserved. We provide a passivity analysis and prove uniform asymptotic stability of the equilibrium. Simulations on a planar two-armed benchmark system with constraints validate the proposed control law.