Uniform exponential stability of periodic discrete switched linear system

Let {Π_τ(m, n): m?≥?n?≥?0} be the family of periodic discrete transition matrices generated by bounded valued square matrices Λ_τ(n), where $\tau :[0,1,2,?)\to \Omega$ is an arbitrary switching signal. We prove that the family {Π_τ(m, n): m?≥?n?≥?0} of bounded linear operator is uniformly exponentially stable if and only if the sequence $n?{\sum }_{k=0}^n{e}^{i\alpha k}{\Pi }_{\tau }(n,k)w\left(k\right):{\mathbb{Z}}_{+}\to \mathbb{R}$ is bounded.     

Chaotic behavior of discrete-time linear inclusion dynamical systems

Given any finite family of real d-by-d nonsingular matrices $\{S_1,\dots ,S_l\},$ by extending the well-known Li–Yorke chaos of a deterministic nonlinear dynamical system to a discrete-time linear inclusion or hybrid or switched system:

$\begin{array}{c}\hfill x_n\in \{S_k{x}_{n?1};1\le k\le l\},x_0\in {\mathbb{R}}^d\mathrm{and}n\ge 1,\end{array}$

we study the chaotic dynamics of the state trajectory (x_n(x₀, σ))_{n ≥ 1} with initial state $x_0\in {\mathbb{R}}^d,$ governed by a switching law $\sigma :\mathbb{N}\to \{1,\dots ,l\}$. Two sufficient conditions are given so that for a “large” set of switching laws σ, there exhibits the scrambled dynamics as follows: for all $x_0,y_0\in {\mathbb{R}}^d,x_0\ne y_0,$

$\begin{array}{c}\hfill \underset{n\to +\infty }{lim\; inf}\parallel x_n(x_0,\sigma )?x_n(y_0,\sigma )\parallel =0\mathrm{and}\underset{n\to +\infty }{lim\; sup}\parallel x_n(x_0,\sigma )?x_n(y_0,\sigma )\parallel =\infty .\end{array}$

This implies that there coexist positive, zero and negative Lyapunov exponents and that the trajectories (x_n(x₀, σ))_{n ≥ 1} are extremely sensitive to the initial states $x_0\in {\mathbb{R}}^d$. We also show that a periodically stable linear inclusion system, which may be product unbounded, does not exhibit any such chaotic behavior. An explicit simple example shows the discontinuity of Lyapunov exponents with respect to the switching laws.     

Generalization of all stabilizing compensators for finite-dimensional linear systems

For finite-dimensional linear systems, the Youla-Kucera parameterization (YKP) with a Q parameter over RH_∞ is assumed to satisfy the Diophantine identity. However, the stability is guaranteed if the Diophantine equation is the “U(RH_∞)“ equality, but not if it is the “identity” equality. However, Vidyasagar's structure with an H parameter over U(RH_∞) is an observer-controller configuration that satisfies the Diophantine equation. This study discusses the deficiency of the Diophantine identity; expands the YKP using an H parameter over U(RH_∞), and expands the Vidyasagar's structure using a Q_v parameter over RH_∞ so that both of the expanded parameterizations satisfy the Diophantine equation and are equivalent for all stabilizing compensators. Moreover, an equation that relates to Q, Q_v, and H will be introduced to establish relationships among the YKP, Vidyasagar's structure and both expanded parameterizations.     

Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback

This paper deals with the problem of the dissipative control for a class of nonlinear Markovian jump systems through Takagi–Sugeno fuzzy model approach. The transition rates of Markovian process under consideration are assumed to be partly known. We aim to design retarded feedback controllers such that the resulting closed-loop system is stochastically stable and strictly (Q,S,R)-θ-dissipative$(Q,S,R)\mathrm{-}\theta \mathrm{-}\mathrm{dissipative}$. By introducing a novel augmented Lyapunov functional and some free Markovian switching matrices, some sufficient conditions for the solvability of the above problem are given in terms of linear matrix inequalities. Finally, two numerical examples are given to demonstrate the effectiveness of our proposed approach.     

Non-fragile robust H∞ control for uncertain discrete-time singular systems with time-varying delays

In this paper, the problem of delay-dependent non-fragile robust H∞$H\infty$ control for a class of discrete-time singular systems with state-delay and parameter uncertainties is investigated. Based on singular value decomposition approach, a delay-dependent sufficient condition for the H∞$H\infty$ control problem for a class of discrete-time singular systems is proposed by constructing generalized Lyapunov–Krasovskii function and a new difference inequality. A memoryless state feedback controller under controller gain perturbations is designed, which guarantees that, for all admissible uncertainties, the resultant closed-loop system is regular, causal, and stable with an H∞$H\infty$ norm bound constraint. Numerical examples in the last will show that our results have the better performance in conservativeness than some results reported in the literature.     

On the stability and boundedness of a class of higher order delay differential equations

We establish some sufficient conditions which guarantee asymptotic stability of the null solution and boundedness of all the solutions of the following nonlinear differential equation of third order with the variable delay, r(t)

     

一个浮游生物的数学模型的全局稳定性

考虑如下用来描述浮游动物-营养物相互作用时滞微分方程模型 其中 是营养物浓度, 是浮游动物种群的度量,并且参数 为正常数. 如果 成立, 则该系统的正平衡点是全局吸引的. 也给出了正平衡点局部稳定的充分条件. （编辑：摘要中有公式,但未能正常显示）     

Non-fragile finite-time extended dissipative control for a class of uncertain discrete time switched linear systems

In this paper, the issues of finite-time extended dissipative analysis and non-fragile control are investigated for a class of uncertain discrete time switched linear systems. Based on average dwell-time approach, sufficient conditions for the finite-time boundedness and finite-time extended dissipative performance of the considered systems are proposed by solving some linear matrix inequalities, where using the concept of extended dissipative, we can solve the H_∞, $L_2?L_{\infty },$ Passivity and (Q, S, R)-dissipativity performance in a unified framework. Furthermore, two form of non-fragile state feedback controllers are designed to guarantee that the closed-loop systems satisfy the finite-time extended dissipative performance. Finally, simulation example is given to show the efficiency of the proposed methods.     

Privacy: an institutional fact

Let us show how property is grasped as an institutional fact. If Jones steals a computer, he does not own it in the sense of property, but only exercises control towards it. If he buys the computer, he controls it too, and moreover owns it in the sense of property. In other words, simply exercising control towards something is a brute fact. This control counts as property only in a certain context: the computer counts as Jones’s property only if he got it through a licit transfer. This is why property is not a brute fact, and is therefore an institutional fact. The same kind of reasoning applies to privacy. When a personal information P about Jones is openly diffused, it seems that P becomes public. From this point of view, a violation of privacy equates to a publication. The problem about this account is the following: who would call “publication of a book” the hacking of it on its author’s computer? No one, because the word “publication” is an institutional word that only refers to a licit diffusion. Considering this answer, we may conclude as follows: if the diffusion of P is illicit, P still counts as private, even if everyone knows about it. If that conclusion is true, privacy is an institutional fact.     

Improved stability criteria for uncertain neutral-type Lur׳e systems with time-varying delays

This paper is concerned with the robust stability of uncertain neutral-type Lur?e system with time-varying delays and sector bounded nonlinearities. By constructing a modified Lyapunov–Krasovskii functional (LKF), some new robust stability criteria are proposed in terms of linear matrix inequalities (LMIs) without using the general free-weighting matrix method. The criteria are less conservative and more efficiently calculative than some previous ones. Four numerical examples are presented to show the effectiveness and merits of the proposed approach.     

Non-fragile H∞ filter design with sparse structure for linear discrete-time systems

This paper presents a study on the problem of designing non-fragile _H∞$H_{\infty }$ filters with sparse structure for linear discrete-time systems. The filters to be designed with sparse structure are assumed to be with additive gain variations, which are resulted from filter implementations. Firstly, a class of sparse structures is specified from a given fully parameterized _H∞$H_{\infty }$ filter. Then, an LMI-based procedure for designing non-fragile _H∞$H_{\infty }$ filters with the sparse structure is provided. The resulting design guarantees the augmented system asymptotically stable and the _H∞$H_{\infty }$ attenuation level less than a prescribed level. A numerical example is given to illustrate the proposed method.     

Empirical studies on the impact of lexical resources on CLIR performance

In this paper, we compile and review several experiments measuring cross-lingual information retrieval (CLIR) performance as a function of the following resources: bilingual term lists, parallel corpora, machine translation (MT), and stemmers. Our CLIR system uses a simple probabilistic language model; the studies used TREC test corpora over Chinese, Spanish and Arabic. Our findings include:

• •One can achieve an acceptable CLIR performance using only a bilingual term list (70–80% on Chinese and Arabic corpora).
• •However, if a bilingual term list and parallel corpora are available, CLIR performance can rival monolingual performance.
• •If no parallel corpus is available, pseudo-parallel texts produced by an MT system can partially overcome the lack of parallel text.
• •While stemming is useful normally, with a very large parallel corpus for Arabic–English, stemming hurt performance in our empirical studies with Arabic, a highly inflected language.

     

Modification of Mikhailov stability criterion for fractional commensurate order systems

In this paper we present the modification of the Mikhailov stability criterion for linear fractional commensurate order systems. The modification consists in determining the appropriate measure for the total argument change depending on the highest fractional order ${\alpha }_n=n\alpha$ of the system and not only on the integer n as stated in the literature. The validity of the result is illustrated by means of several examples.