Asymptotic Stability Properties Research Articles

Robust dissipativity of discrete systems and its investigation with the help of a sequence of sets of Lyapunov functions

A generalized concept of robust dissipativity is introduced and theorems on the analysis of this property are formulated and proved with the help of the Lyapunov functions method. It is proposed to use a specially constructed sequence of sets of Lyapunov functions that can allow one to improve the initial estimate of the boundary set of a dissipative system up to the establishment of the asymptotic robust stability property in the limiting case. An example of investigating the dissipativity of a linear discrete system with unknown parameters under additive perturbation is given.

Stability Results for Switched Linear Systems with Constant Discrete Delays

This paper investigates the stability properties of switched systems possessing several parameterizations (or configurations) while being subject to internal constant point delays. Some of the stability results are formulated based on Gronwall′s lemma for global exponential stability, and they are either dependent on or independent of the delay size but they depend on the switching law through the requirement of a minimum residence time. Another set of results concerned with the weaker property of global asymptotic stability is also obtained as being independent of the switching law, but still either dependent on or independent of the delay size, since they are based on the existence of a common Krasovsky‐Lyapunov functional for all the above‐mentioned configurations. Extensions to a class of polytopic systems and to a class of regular time‐varying systems are also discussed.

Asymptotic Stability of Hierarchical Inner-Outer Loop-Based Flight Controllers

In this paper, we present a hierarchical controller for autonomous aerial vehicles control and navigation. For autonomous rotorcraft flight, it is common to separate the flight control problem into an inner loop that controls attitude and an outer loop that controls the translational trajectory of the rotorcraft. The resulted nonlinear controller is thus, easy to implement and to tune. Although satisfactory results have been reported in the literature and various navigational tasks have been achieved using this hierarchical control technique, this control scheme suffers from the lack of the stability proof and analysis of the closed-loop system. Here, we propose a 3D flight controller which is based on the inner- and outer-loop control scheme, and we prove the asymptotic stability property for the connected closed-loop system.

On the boundedness and local stability of

We show that every positive solution of the difference equation in the title is bounded whenever and . This confirms a conjecture posed by Amleh et al.. We also establish local asymptotic stability properties of the difference equation.

Nonlinear stability and asymptotic stability of implicit Euler method for stiff Volterra functional differential equations in Banach spaces

This paper is concerned with the numerical solution of Volterra functional differential equations. We focus on the stability and asymptotic stability properties of implicit Euler method when applied to nonlinear stiff Volterra functional differential equations in Banach spaces. A series of new stability and asymptotic stability results are obtained. Numerical example is given to confirm the theoretical results.

Comparison of the asymptotic stability properties for two multirate strategies

This paper contains a comparison of the asymptotic stability properties for two multirate strategies. For each strategy, the asymptotic stability regions are presented for a 2 × 2 test problem and the differences between the results are discussed. The considered multirate schemes use Rosenbrock type methods as the main time integration method and have one level of temporal local refinement. Some remarks on the relevance of the results for 2 × 2 test problems are presented.

A Numerical Study of the Stability of Solitary Waves of the Bona–Smith Family of Boussinesq Systems

In this paper we study, from a numerical point of view, some aspects of stability of solitary-wave solutions of the Bona–Smith systems of equations. These systems are a family of Boussinesq-type equations and were originally proposed for modelling the two-way propagation of one-dimensional long waves of small amplitude in an open channel of water of constant depth. We study numerically the behavior of solitary waves of these systems under small and large perturbations with the aim of illuminating their long-time asymptotic stability properties and, in the case of large perturbations, examining, among other, phenomena of possible blow-up of the perturbed solutions in finite time.

Stabilizing Inputs for Implicit Functional Differential Equations of the Neutral Type with Unknown Time-Varying Delays

A robust stabilization problem is investigated for a class of imperfectly known implicit systems with discrete and distributed time-varying delays. The delays are assumed to be unknown but bounded, with known bounds. The imperfections acting on the systems, which may be time-, state-, delayed state-, and/ or inputdependent, are modelled as additive nonlinear perturbations influencing a known set of implicit nonlinear functional differential equations of the neutral type. Sufficient conditions, which include delaydependent matrix inequalities and a delay-dependent stability criterion involving some bounding parameters for the uncertainty and delays in the system, are presented, and a class of robust feedback memoryless control inputs are designed to guarantee a prescribed asymptotic stability property for the class of implicit systems.

Nonlinear feedback controllers and compensators: a state-dependent Riccati equation approach

State-dependent Riccati equation (SDRE) techniques are rapidly emerging as general design and synthesis methods of nonlinear feedback controllers and estimators for a broad class of nonlinear regulator problems. In essence, the SDRE approach involves mimicking standard linear quadratic regulator (LQR) formulation for linear systems. In particular, the technique consists of using direct parameterization to bring the nonlinear system to a linear structure having state-dependent coefficient matrices. Theoretical advances have been made regarding the nonlinear regulator problem and the asymptotic stability properties of the system with full state feedback. However, there have not been any attempts at the theory regarding the asymptotic convergence of the estimator and the compensated system. This paper addresses these two issues as well as discussing numerical methods for approximating the solution to the SDRE. The Taylor series numerical methods works only for a certain class of systems, namely with constant control coefficient matrices, and only in small regions. The interpolation numerical method can be applied globally to a much larger class of systems. Examples will be provided to illustrate the effectiveness and potential of the SDRE technique for the design of nonlinear compensator-based feedback controllers.

Lyapunov Stability of Complementarity and Extended Systems

A linear complementarity system (LCS) is a piecewise linear dynamical system consisting of a linear time-invariant ordinary differential equation (ODE) parameterized by an algebraic variable that is required to be a solution to a finite-dimensional linear complementarity problem (LCP), whose constant vector is a linear function of the differential variable. Continuing the authors’ recent investigation of the LCS from the combined point of view of system theory and mathematical programming, this paper addresses the important system-theoretic properties of exponential and asymptotic stability for an LCS with a C$^1$ state trajectory. The novelty of our approach lies in our employment of a quadratic Lyapunov function that involves the auxiliary algebraic variable of the LCS; when expressed in the state variable alone, the Lyapunov function is piecewise quadratic, and thus nonsmooth. The nonsmoothness feature invalidates standard stability analysis that is based on smooth Lyapunov functions. In addition to providing sufficient conditions for exponential stability, we establish a generalization of the well-known LaSalle invariance theorem for the asymptotic stability of a smooth dynamical system to the LCS, which is intrinsically a nonsmooth system. Sufficient matrix-theoretic copositivity conditions are introduced to facilitate the verification of the stability properties. Properly specialized, the latter conditions are satisfied by a passive-like LCS and certain hybrid linear systems having common quadratic Lyapunov functions. We provide numerical examples to illustrate the stability results. We also develop an extended local exponential stability theory for nonlinear complementarity systems and differential variational inequalities, based on a new converse theorem for ODEs with B-differentiable right-hand sides. The latter theorem asserts that the existence of a “B-differentiable Lyapunov function” is a necessary and sufficient condition for the exponential stability of an equilibrium of such a differential system.

Design of adaptive sliding surfaces for systems with mismatched perturbations to achieve asymptotical stability

On the basis of the Lyapunov stability theorem, a design of adaptive sliding surfaces for a class of MIMO systems with matched and mismatched perturbations is proposed to solve regulation problems. By utilising some adaptive gains designed both in the sliding surface function and in the controllers, not only are the mismatched perturbations automatically overcome during the sliding mode, but the property of asymptotical stability of controlled systems is also achieved at the same time. Furthermore, the knowledge of the upperbound of partial perturbations is not required. A numerical example is given to demonstrate the feasibility of the proposed control scheme.

Stability of sequential Markov Chain Monte Carlo methods

Sequential Monte Carlo Samplers are a class of stochastic algorithms for Monte Carlo integral estimation w.r.t. probability distributions, which combine elements of Markov chain Monte Carlo methods and importance sampling/resampling schemes. We develop a stability analysis by funtional inequalities for a nonlinear flow of probability measures describing the limit behavior of the methods as the number of particles tends to infinity. Stability results are derived both under global and local assumptions on the generator of the underlying Metropolis dynamics. This allows us to prove that the combined methods sometimes have good asymptotic stability properties in multimodal setups where traditional MCMC methods mix extremely slowly. For example, this holds for the mean field Ising model at all temperatures.

Stabilization of complex network with hybrid impulsive and switching control

This paper studies the asymptotic stability properties of a class of complex dynamical networks under a hybrid impulsive and switching control. By utilizing the concept of impulsive control and the stability results for impulsive systems, some new criteria for global and local stability are established for this model. Some numerical examples and simulations are included to illustrate the effectiveness of the theoretical results.

Injectivity of differentiable maps ℝ2 → ℝ2 at infinity

The main result given in Theorem 1.1 is a condition for a map X, defined on the complement of a disk D in ℝ2 with values in ℝ2, to be extended to a topological embedding of ℝ2, not necessarily surjective. The map X is supposed to be just differentiable with the condition that, for some e > 0, at each point the eigenvalues of the differential do not belong to the real interval (-e,∞). The extension is obtained by restricting X to the complement of some larger disc. The result has important connections with the property of asymptotic stability at infinity for differentiable vector fields.

Adaptive sliding mode controller design based on T–S fuzzy system models

An adaptive sliding mode control (ASMC) technique based on T-S fuzzy system models is proposed in this paper for a class of perturbed nonlinear MIMO dynamic systems in order to solve tracking problems. A T-S fuzzy model is firstly formed by utilizing fuzzy theorem to amalgamate a set of linearized dynamic equations. The adaptive sliding mode controller is then designed based on this fuzzy model with perturbations. The proposed control scheme can drive the dynamics of controlled system into a designated sliding surface in finite time, and guarantee the property of asymptotical stability. It is also shown that the information of upper bound of modeling errors as well as perturbations, except the information of upper bound of input uncertainty, is not required when using the proposed controller.

Asymptotic stability and disturbance attenuation properties for a class of networked control systems

In this paper, stability and disturbance attenuation issues for a class of Networked Control Systems (NCSs) under uncertain access delay and packet dropout effects are considered. Our aim is to find conditions on the delay and packet dropout rate, under which the system stability and H∞ disturbance attenuation properties are preserved to a desired level. The basic idea in this paper is to formulate such Networked Control System as a discrete-time switched system. Then the NCSs’ stability and performance problems can be reduced to the corresponding problems for switched systems, which have been studied for decades and for which a number of results are available in the literature. The techniques in this paper are based on recent progress in the discrete-time switched systems and piecewise Lyapunov functions.

On a nonlinear Volterra integrodifferential equation in Banach spaces

The objective of the present paper is to study the local existence, global existence, uniqueness, continuous dependence, asymptotic stability and other properties of solutions of a nonlinear Volterra integrodifferential equation in Banach spaces of more general type. The technique employed in our analysis is based on treating the equation in the domain of the infinitesimal generator of semigroups of linear operators in a Banach space with graph norm and using results from linear semigroup theory. Mathematics subject classification (2000): 45N05, 45J05.

SWITCHING ADAPTIVE CONTROL AND SYNCHRONIZATION OF AFFINE NONLINEAR SYSTEM

new switching adaptive control scheme is presented, which provides global asymptotic stability property with respect to state space vector of the plant or for the state error of synchronization system. In the presence of disturbances the scheme ensures boundedness of all trajectories of the system.

Some new results on an allelopathic competition model with quorum sensing and delayed toxicant production

A 4-equation delay differential system representing a bacterial allelopathic competition is analyzed. A distributed delay term models a linear quorum-sensing mechanism which regulates the delayed allelochemicals’ production process. The proved qualitative properties of the solutions are positivity, boundedness, global existence in the future, and uniqueness. Sufficient conditions for local asymptotic stability properties of biologically meaningful steady-state solutions are given in terms of the parameters of the system. The global asymptotic stability of a biologically meaningful steady-state solution is proved by constructing a suitable Lyapunov functional.

An allelopathic competition model with quorum sensing and delayed toxicant production

The dynamics of a differential functional equation system representing an allelopathic competition is analyzed. The delayed allelochemical production process is represented by means of a distributed delay term in a linear quorum-sensing model. Sufficient conditions for local asymptotic stability properties of biologically meaningful steady-state solutions are given in terms of the parameters of the system. A global asymptotic stability result is also proved by constructing a suitable Lyapunov functional. Some simulations confirm the analytical results.