Concept Of Finite-time Stability Research Articles

Optimal Finite-Time State Estimation for Discrete-Time Switched Systems under Switching Frequency Constraint

The state estimation problem for a class of switched linear systems which only switches in some short interval is addressed. Besides the asymptotic stability of error dynamics, the boundness of error state is a significant issue for short-time switched systems. By introducing the concept of finite-time stability, the state estimation procedure is formulated to determine appropriate observer gains ensuring the error dynamics is finite-time stable in the short-time switching intervals of interest. Optimal finite-time observers are designed through iterative algorithms to minimize the bound of error state, in the cases with and without disturbances. Particularly, when the total activation time is known, a less conservative result can be derived and an optimization problem can be solved with the help of the genetic algorithm. A numerical example is provided to illustrate the theoretical findings in this paper.

Finite-time stability for switched singular systems

In this paper, finite-time stability and stabilization of switched singular systems are studied. Firstly, we discuss the solvability condition of the switched singular system and introduce the concepts of finite-time stability and finite-time boundness. Secondly, using the mode-dependent average dwell time method and the Lyapunov function method, we provide sufficient conditions to guarantee that the switched singular system is regular, impulse free, and finite-time stable or finite-time bounded. Then, we design the state feedback controllers to ensure that a closed-loop system is finite-time stable and finite-time bounded with a present H disturbance attenuation level . Finally, numerical examples are given to verify the efficiency of the proposed theory.

Finite-time stability of nonlinear switched impulsive systems

In this article, the problem of finite-time stability (FTS) of nonlinear switched impulsive systems is first considered. A new concept of FTS is introduced and sufficient conditions for a switched impulsive system to be finite-time stable are presented by using switched Lyapunov functions. Some results of FTS of impulsive systems and switched systems can be obtained from this article as special cases, respectively. Two numerical examples are given to illustrate the effectiveness of the proposed results.

Finite time H∞ filtering for uncertain discrete-time switching systems

The concept of finite-time stability (FTS) has gained much attention from researchers in recent years, after the emergent utilization of linear matrix inequalities (LMIs) in the field of control systems. In this paper, FTS and finite-time boundedness (FTB) analysis and synthesis procedure for a ‘finite-time H∞ filter’ for discrete-time switching systems are presented. Assuming the external disturbance to be energy bounded, sufficient conditions for the designated H∞ filter-based system to be finite-time stable are proposed. Considering dwell-time switching, average and minimum dwell time in terms of ‘finite-time’ system parameters has been formulated. Furthermore, these results are extended to uncertain switching system, by assuming the switching system subject to norm-bounded parameter uncertainties. The intended filter design steps have been demonstrated for both certain and uncertain system models. A cost function is introduced and then a parametric optimization algorithm is devised to calculate optimal filter parameters and present the derived results in LMI form. Numerical design examples are presented at the end to demonstrate the results.

Stability of singular time-delay systems in the sense of non-Lyapunov: Classical and modern approach

This paper provides sufficient conditions for both practical stability and finite-time stability of linear singular continuous time-delay systems which can be mathematically described as Ex(t)=Aox(t)+A1x(t-t). Considering a finite-time stability concept, new delay independent and delay dependent conditions have been derived using the approach based on the Lyapunov-like functions and their properties on the subspace of consistent initial conditions. These functions do not need to have the properties of positivity in the whole state space and negative derivatives along the system trajectories. When the practical stability has been analyzed the above mentioned approach was combined and supported by the classical Lyapunov technique to guarantee the attractivity property of the system behavior. Moreover an linear matrix inequality (LMI) approach has been applied in order to get less conservative conditions.

Finite-time control for discrete-time switched systems with time delay

The problem of finite-time state feedback control is considered in this paper for a class of discrete-time switched systems with time delay. Firstly, the concepts of finite-time stability and finitetime boundedness are extended to discrete-time switched systems with time delay, respectively. Then, by resorting to the average dwell time approach and Lyapunov-Krasovskii functional technology, some new delay-dependent criteria guaranteeing finite-time boundedness and stability are developed. The state feedback controller is also obtained by virtue of a cone complement linearization (CCL) method. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed results.

State constrained control of impulsive quadratic systems in integrated pest management

In this paper, an approach based on the concept of finite-time stability is applied to the problem of designing optimal control strategies for an integrated pest management model with disease in the pest, which is subject to time-dependent impulsive control. The goal is to design a resetting law such that, given a set of admissible initial conditions, the state trajectories of the resulting impulsive quadratic system remain in a bounded region of the state space which is prescribed on the basis of some constraints on ecological balance and resource saving. The determination of an optimal resetting period, which is related to the release period of chemical pesticides, is reduced to the solution of a feasibility problem with linear matrix inequalities constraints.

Uniform finite-time stability of discrete-time switched descriptor systems

The problem of uniform finite-time stability for a class of discrete-time switched descriptor systems is considered. Firstly, the concept of finite-time stability for continuous descriptor systems is extended to discrete-time switched descriptor systems. Secondly, based on the Lyapunov-like function method, and under arbitrary switching signal, sufficient conditions under which discrete-time switched descriptor systems are regular and causal, the uniform finite-time is bounded and uniform finite-time is stable, are derived. Then, the state feedback controllers are designed to guarantee the discrete-time switched descriptor system uniform finite-time stablility. Finally, some numerical examples show that the results obtained in this paper are effective.

Fixed-Time Stabilization via Second Order Sliding Mode Control

The paper addresses the finite-time stabilization problem with possible predefining of the convergence time independently on the initial conditions. The corresponding extension of finite-time stability concept is called by fixed-time stability. The second order sliding mode control algorithm provided fixed-time stabilization of the multi-input linear control system is presented. Theoretical results are supported by numerical simulations.

State Constrained Control of Impulsive Quadratic Systems in Integrated Pest Management

AbstractIn this paper, an approach based on the concept of Finite-Time Stability is applied to the problem of designing optimal control strategies for an integrated pest management model with disease in the pest, which is subject to time-dependent impulsive control. The goal is to design a resetting law such that, given a set of admissible initial conditions, the state trajectories of the resulting impulsive quadratic system remain in a bounded region of the state space which is prescribed on the basis of some constraints on ecological balance and resource saving. The determination of an optimal resetting period, which is related to the release period of chemical pesticides, is reduced to the solution of a feasibility problem with Linear Matrix Inequalities constraints.

Finite-time stability theorem of stochastic nonlinear systems

A new concept of finite-time stability, called stochastically finite-time attractiveness, is defined for a class of stochastic nonlinear systems described by the Itô differential equation. The settling time function is a stochastic variable and its expectation is finite. A theorem and a corollary are given to verify the finite-time attractiveness of stochastic systems based on Lyapunov functions. Two simulation examples are provided to illustrate the applications of the theorem and the corollary established in this paper.

Finite-time stability of linear time-varying singular systems with impulsive effects

In this paper, we introduce a new concept of finite-time stability of linear time-varying singular systems with impulses at fixed times, and present new results for the above-mentioned class of system in the form of sufficient conditions. A sufficient condition is given in terms of matrix inequalities, which gives the opportunity of fitting the finite-time control problem in the general framework via impulsive control. A numerical example is presented to illustrate the efficiency of the proposed result.

Finite-time stabilization of linear systems via impulsive control

This paper introduces a new concept of finite-time stability of linear time-invariant impulsive systems. The sufficient condition of finite-time stability of the system via impulsive control at fixed times and variable times is obtained, respectively. Such sufficient conditions of finite-time stabilization are given in terms of matrix inequalities. A numerical example is presented to illustrate the efficiency of the proposed results.

On the Stability of Perfect Viscoelastic Columns

The stability problem associated with an Euler-Bernoulli beam made of an arbitrary linear viscoelastic material is formulated. The three parameter and the Kelvin-Voigt models are analyzed in the presence of constant as well as periodic loads. The application of a finite time stability concept is shown for the constant loading case when the traditional stability criterion fails to make sense. For the case of a periodic loading, the stability diagrams are obtained through an application of Floquet theory. It is found that the addition of periodic loads may significantly alter the stability behavior of a column which is originally subjected to a constant load only.

Numerical solution of an optimal control problem with a probability criterion

An optimal stochastic control problem is considered with a probability criterion. A stochastic differential equation model is assumed. The optimization problem is to maximize the probability that the state trajectory remain in a given bounded region D over a given finite time interval. This type of criterion is especially relevant to certain technical problems where it is essential that certain state variables not exceed given values, and is closely related to the concepts of finite time stability. For the class of dynamical systems considered, the optimal control is bang-bang. A numerical method developed by Samarskii is used to solve the optimization equation, a nonlinear partial differential equation of the parabolic type. A second-order system is computed to illustrate numerical results. Time-varying switching curves for the optimal bang-bang solution are plotted.

Finite time stability under perturbing forces and on product spaces

This paper continues the development of a qualitative theory of stability, recently initiated by the authors, for systems operating over finite time intervals. The theory is motivated by 1) the need for a more practical concept of stability than is provided by the classical theory; and 2) the search for methods for investigating stability of a system trajectory (either analytically or numerically given) without the necessity of performing complicated transformations of the differential equations involved. The systems studied in this paper are nonautonomous, i.e., they are under the influence of external forces, and the concept of finite time stability (precisely defined in the paper) in this case involves the bounding of trajectories within specified regions of the state space during a given finite time interval. (The input is assumed to be bounded by a known quantity during this time interval.) Sufficient conditions are given for various types of finite time stability of a system under the influence of perturbing forces which enter the system equations linearly. These conditions take the form of existence of "Liapunov-like" functions whose properties differ significantly from those of classical Liapunov functions. In particular, there is no requirement of definiteness on such functions or their derivative. The remainder of the paper deals with the problem of determining finite time stability properties of a system from knowledge of the finite time stability properties of lower-order subsystems which, when appropriately coupled, form the original system. An example is given which illustrates some of the concepts discussed in the paper.