Another sufficient condition for the stability of grey discrete-time systems

In this paper, the stability of grey discrete-time systems is discussed whose state matrices are interval matrices. A new approach is obtained which guarantee the stability of grey discrete-time systems. The sufficient condition for robust stability of grey time delay systems subjected to interval systems is also derived. By mathematical analysis, the stability criterion is less conservative than those in previous results. Examples are given to compare the proposed method with reported recently.     

Positivity and exponential stability of discrete-time coupled homogeneous systems with time-varying delays

This paper investigates the positivity and stability of discrete-time coupled homogeneous systems with time-varying delays. First, an explicit criterion is given for the positivity of discrete-time coupled homogeneous delay systems. Then, by using the properties of homogeneous functions, a sufficient condition is presented for ensuring the stability of the considered systems. Moreover, the obtained result is applied to study the stability of positive singular systems with time-varying delay. It should be noted that it is the first time that the stability result is given for discrete-time coupled homogeneous positive systems with time-varying delays. Two numerical examples are presented to demonstrate the effectiveness of the derived results.     

基于区间时变时滞模糊系统的稳定性分析

考虑了区间时变时滞模糊系统的稳定性问题.利用T-S模糊模型对模糊系统进行了研究,利用线性矩阵不等式的形式给出了此类模糊系统在时滞相关意义下保守性更小的稳定性判据.由于加入了自由矩阵,所得结果保守性更小.并且给出了一个数值例子说明了所得稳定性判据的有效性.     

New finite-time stability for fractional-order time-varying time-delay linear systems: A Lyapunov approach

The primary goal of this paper is to examine the finite-time stability and finite-time contractive stability of the linear systems in fractional domain with time-varying delays. We develop some sufficient criteria for finite-time contractive stability and finite-time stability utilizing fractional-order Lyapunov-Razumikhin technique. To validate the proposed conditions, two different types of dynamical systems are taken into account, one is general time-delay fractional-order system and another one is fractional-order linear time-varying time-delay system, furthermore the efficacy of the stability conditions is demonstrated numerically.     

Stability equation method for control systems with lags and distributed-parameters

In this paper, control systems with lags and with distributed-parameters are considered. First, the relation between the stability equation method and the theorem of Pontryagin for testing stability of the zeros of exponential polynomials is considered, then the distributions of roots of double-valued functions are analyzed, and finally the applications of the stability equation method for stability analysis of process control systems are presented.     

A neutral system approach to stability of singular time-delay systems

This paper is concerned with the problem of delay-dependent stability for a class of singular time-delay systems. By representing the singular system as a neutral form, using an augmented Lyapunov–Krasovskii functional and the Wirtinger-based integral inequality method, we obtain a new stability criterion in terms of a linear matrix inequality (LMI). The criterion is applicable for the stability test of both singular time-delay systems and neutral systems with constant time delays. Illustrative examples show the effectiveness and merits of the method.     

New fractional-order integral inequalities: Application to fractional-order systems with time-varying delay

In this paper, the problem of delay-dependent stability analysis of fractional-order systems with time-varying delay is investigated. First, a class of novel fractional-order integral inequalities for quadratic functions by constructing appropriate auxiliary functions is proposed, which has been proven to be useful in analyzing fractional-order systems with time-varying delay. Based on these proposed inequalities, the Lyapunov–Krasovskii functions are designed to deal with the time-varying delay terms, reducing the conservatism of the stability criteria. Furthermore, delay-dependent criteria are derived to achieve asymptotic stability of fractional-order systems with time-varying delay. Finally, two examples are provided to illustrate the effectiveness and feasibility of the proposed stability criteria.     

On input-to-state stability of impulsive stochastic systems

This paper is concerned with the input-to-state stability (ISS) of impulsive stochastic systems. First, appropriate concepts of stochastic input-to-state stability (SISS) and pth moment input-to-state stability (p-ISS) for the mentioned systems are introduced. Then, we prove that impulsive stochastic systems possessing SISS-Lyapunov functions are uniformly SISS and p-ISS over a certain class of impulse sequences. As a byproduct, a criterion on the uniform global asymptotic stability in probability for the system in isolation (without inputs) is also derived. Finally, we provide a numerical example to illustrate our results.     

Delay-independent stability conditions for a class of nonlinear difference systems

The paper addresses the asymptotic stability problem for a class of difference systems with nonlinearities of a sector type and time-delay. A new approach to Lyapunov–Krasovskii functionals constructing for considered systems is proposed. On the basis of the approach, delay-independent asymptotic stability conditions and estimates of the convergence rate of solutions are derived. In addition, stability of perturbed systems is investigated in the case where nonstationary perturbations admit zero mean values. Some examples are given to illustrate the obtained results.     

On design of continuous Lyapunov's feedback control

A common approach to Lyapunov's stability control is to design a controller such that a Lyapunov function can be derived for the control system to ensure stability. This procedure often leads to a discontinuous controller. When the controller is implemented, the discontinuous terms are replaced with continuous functions to avoid chattering of the control signal. Two associated problems have been overlooked during this procedure. One is that discontinuous control systems are non-smooth, which violates the fundamental assumptions of solution theories and the applicability of Lyapunov's stability theory is questionable. Another problem is that the replacement of discontinuous terms may weaken stability, which can be critical. In this paper, we discuss proper stability analysis of discontinuous control systems using the extended Lyapunov's second method based on Filippov's solution concept for non-smooth systems. We further propose to utilize the concept of Lyapunov exponents to quantitatively analyze the stability of continuous control systems obtained by replacing the discontinuous terms in the discontinuous controllers. An example involving the stabilization of a two-link non-fixed-base robotic manipulator is presented for demonstration. This research fills the gap in designing continuous Lyapunov's stability controllers regarding limited available Lyapunov functions.     

A necessary and sufficient condition for stability of linear discrete systems with parameter-variation

A simple sufficient stability criterion for linear discrete systems obtained previously is proved to be necessary and sufficient for the stability of a class of such systems with parameter-variation.     

Robust stability analysis for a class of fractional order systems with uncertain parameters

The research of robust stability for fractional order linear time-invariant (FO-LTI) interval systems with uncertain parameters has become a hot issue. In this paper, it is the first time to consider robust stability of uncertain parameters FO-LTI interval systems, which have deterministic linear coupling relationship between fractional order and other model parameters. Linear matrix inequalities (LMI) methods are used, and a criterion for checking asymptotical stability of this class of systems is presented. One numerical illustrative example is given to verify the correctness of the conclusions.     

Improved stability theorems of random nonlinear time delay systems and their application

This paper focuses on stability theorems of random nonlinear time delay systems and their application. On the one hand, some noise-to-state practical stability theorems are improved for random nonlinear systems with time delay in this paper. Compared with the existing stability theorems, some strict constraints have been removed and the existing stability theorems can be viewed as special cases of the new established stability theorems in this paper. On the other hand, the state feedback tracking control problem is investigated to show the applicability of improved stability theorems in this paper.     

Design of optimal PID controller for multivariable time-varying delay discrete-time systems using non-monotonic Lyapunov-Krasovskii approach

This paper is concerned with stability analysis and stabilization of time-varying delay discrete-time systems in Lyapunov-Krasovskii stability analysis framework. In this regard, a less conservative approach is introduced based on non-monotonic Lyapunov-Krasovskii (NMLK) technique. The proposed method derives time-varying delay dependent stability conditions based on Lyapunov-Krasovskii functional (LKF), which are in the form of linear matrix inequalities (LMI). Also, a PID controller designing algorithm is extracted based on obtained NMLK stability condition. The stability of the closed loop system is guaranteed using the designed controller. Another property that is important along with the stability, is the optimality of the controller. Thus, an optimal PID designing technique is introduced in this article. The proposed method can be used to design optimal PID controller for unstable multi-input multi-output time-varying delay discrete-time systems. The proposed stability and stabilization conditions are less conservative due to the use of non-monotonic decreasing technique. The novelty of the paper comes from the consideration of non-monotonic approach for stability analysis of time-varying delay discrete-time systems and using obtained stability conditions for designing PID controller. Numerical examples and simulations are given to evaluate the theoretical results and illustrate its effectiveness compared to the existing methods.     

基于Lyapunov方法的鲁棒稳定性条件

分别针对具有非结构不确定性、强结构不确定性线性离散系统,利用lyapunov方法进行讨论,给出了相应系统的鲁棒稳定性判别条件,并通过算例进行了验证。     

Finite-time stabilization of nonlinear dynamical systems via control vector Lyapunov functions

Finite-time stability involves dynamical systems whose trajectories converge to an equilibrium state in finite time. Since finite-time convergence implies nonuniqueness of system solutions in reverse time, such systems possess non-Lipschitzian dynamics. Sufficient conditions for finite-time stability have been developed in the literature using Hölder continuous Lyapunov functions. In this paper, we develop a general framework for finite-time stability analysis based on vector Lyapunov functions. Specifically, we construct a vector comparison system whose solution is finite-time stable and relate this finite-time stability property to the stability properties of a nonlinear dynamical system using a vector comparison principle. Furthermore, we design a universal decentralized finite-time stabilizer for large-scale dynamical systems that is robust against full modeling uncertainty. Finally, we present two numerical examples for finite-time stabilization involving a large-scale dynamical system and a combustion control system.     

Modified Mikhailov stability criterion for continuous-time noncommensurate fractional-order systems

This paper presents the modified Mikhailov stability criterion, which can be effectively used in stability analysis for continuous-time noncommensurate fractional-order systems. The main advantage of the proposed methodology is that the stability analysis of noncommensurate fractional-order systems leads to the same computational complexity as for the commensurate order ones. Simulation examples confirm the usefulness of the introduced methodology.     

Robust stability testing function for a complex interval family of fractional-order polynomials

The robust stability of a family of interval fractional-order systems with complex coefficients is investigated in this study. The concept of “a family of interval fractional-order systems with complex coefficients” means that the characteristic function of a control system can be of both commensurate and non-commensurate orders, the coefficients of the characteristic function can be uncertain parameters, and may be complex numbers. At first, a simple graphical procedure is presented for robust stability analysis. The “robust stability testing function” is then extended to look at the robust conditions. To the best of the authors’ knowledge, no auxiliary function for analyzing the robust stability of the systems under investigation has been introduced until now. Moreover, lower and upper frequency bounds are provided which are useful to improve the computational efficiency of testing the robust stability conditions. Eventually, to verify the results, analytical examples and numerical simulations are provided.     

Novel stability results for aperiodic sampled-data systems

This paper is concerned with stability for aperiodic sampled-data systems. Firstly, for aperiodic sampled-data systems without uncertainties, a new Lyapunov-like functional is constructed by introducing the double integral of the derivative of the state, the integral of the state, and the integral of the cross term of the state and the sampled state. When estimating the derivative of the Lyapunov-like functional, superior integral inequalities to Jensen inequality are employed to get a tighter upper bound. By the Lyapunov-like functional principle, sampling-interval-dependent stability results are derived. Then, the stability results are extended to aperiodic sampled-data systems with polytopic uncertainties. Finally, some examples are listed to show the stability results have less conservatism than some existing ones.     

Robust bounds for fractional-order systems with uncertain order and structured perturbations via Cylindrical Algebraic Decomposition method

This paper considers the robust stability problem of fractional-order systems with uncertain order and structured perturbations. A stability check procedure is proposed for determining the robust bounds of uncertain order and other uncertain parameters for fractional-order systems.The results are obtained in terms of Cylindrical Algebraic Decomposition which is first used for analyzing the robust stability problem of fractional-order systems with uncertain order. The method is non-conservative for fractional-order systems with the uncertain order α satisfying 0?<?α?<?2. Examples are given to demonstrate the effectiveness of proposed approach.