Stability and stabilization for singularly perturbed systems with Markovian jumps

This article focuses on the stability and stabilization problems of singularly perturbed jump systems. Here, the singularly perturbed parameter (SPP) is also with Markov switching and satisfies any with positive bound predefined. First, stability conditions expressed ?_i‐free but involving its bound are developed by constructing an ?_i‐dependent Lyapunov function. Then, a method for state feedback stabilization controller depending on SPP is proposed, whose conditions are given in terms of linear matrix inequalities. Moreover, some special cases about deterministic SPP are considered too. Finally, two practical examples are used to demonstrate the effectiveness and superiorities of the proposed methods.     

Robust predictive extended state observer for a class of nonlinear systems with time-varying input delay

ABSTRACT

This paper deals with asymptotic stabilisation of a class of nonlinear input-delayed systems via dynamic output feedback in the presence of disturbances. The proposed strategy has the structure of an observer-based control law, in which the observer estimates and predicts both the plant state and the external disturbance. A nominal delay value is assumed to be known and stability conditions in terms of linear matrix inequalities are derived for fast-varying delay uncertainties. Asymptotic stability is achieved if the disturbance or the time delay is constant. The controller design problem is also addressed and a numerical example with an unstable system is provided to illustrate the usefulness of the proposed strategy.     

Full and reduced-order H∞ filtering of Takagi–Sugeno fuzzy time-varying delay systems: Input–output approach

This article addresses the issue of delay-dependent H_∞ filtering design for TakagiŮSugeno fuzzy time-varying delay systems using the input–output approach. A three-term approximation model has been used to transform the original system into two interconnected subsystems. Since the nonquadratic Lyapunov–Krasovskii functional requires to deal with the membership function's (MF) time derivative, upper-bound inequalities have been added to the obtained conditions. Based on the scaled small gain theorem, nonquadratic Lyapunov–Krasovskii functional approach and considering the bounds of the MF time-derivative, the H_∞ full- and reduced-order filters are designed and then formulated in terms of linear matrix inequalities. Finally, illustrative examples are presented to demonstrate the validity of the proposed methods.     

Numerical analysis of history-dependent hemivariational inequalities and applications to viscoelastic contact problems with normal penetration

In this paper numerical approximation of history-dependent hemivariational inequalities with constraint is considered, and corresponding Céa’s type inequality is derived for error estimate. For a viscoelastic contact problem with normal penetration, an optimal order error estimate is obtained for the linear element method. A numerical experiment for the contact problem is reported which provides numerical evidence of the convergence order predicted by the theoretical analysis.     

泰勒公式在不等式中的应用

泰勒公式在高等数学中应用广泛.在简单介绍泰勒公式的基础上，通过具体实例，将泰勒公式（包含佩亚诺型余项、拉格朗日型余项）应用于高阶不等式的证明中，最后对证明步骤及注意事项进行相应分析，以便为学生学习泰勒公式提供参考。     

The Marginal Density of a TMA(1) Process

This note reconsiders the marginal density of a threshold moving average process and proposes a simple yet effective numerical algorithm to implement that by solving an associated integral equation. This algorithm can also be applied to calculate stationary probability density or distribution functions of a few other types of nonlinear stationary stochastic processes numerically.