An Approach, Via Entropy, to the Stability of Random Large-Scale Sampled-Data Systems Under Structural Perturbations

1982 ◽  
Vol 104 (1) ◽  
pp. 49-57 ◽  
Author(s):  
Guy Jumarie
Keyword(s):  
Stochastic Stability ◽  
Large Scale ◽  
Sufficient Conditions ◽  
Direct Application ◽  
Jacobian Determinant ◽  
Random Perturbations ◽  
Data Systems ◽  
Sampled Data ◽  
Large Scale Systems ◽  
The Stability

The concept of entropy in information theory is used to investigate the sensitivity and the stability of sampled-data systems in the presence of random perturbations. After a brief background on the definition, the practical meaning and the main properties of the entropy, its relations with asymptotic insensitiveness are exhibited and then some new results on the sensitivity and the stochastic stability of linear and nonlinear multivariable sampled data systems are derived. A new concept of stochastic conditional asymptotic stability is obtained which seems to be of direct application in the analysis of large-scale systems. Sufficient conditions for stability are stated. This approach provides a new look over stochastic stability. In addition, variable transformations act additively on entropy, via Jacobian determinant, and as a result the corresponding calculus is very simple.

scholarly journals Strong stochastic stability for non-uniformly expanding maps

2012 ◽  
Vol 33 (3) ◽  
pp. 647-692 ◽  
Author(s):  
JOSÉ F. ALVES ◽  
HELDER VILARINHO
Keyword(s):  
Stochastic Stability ◽  
Hyperbolic Systems ◽  
Sufficient Conditions ◽  
Unperturbed System ◽  
Random Perturbations ◽  
Expanding Maps ◽  
Srb Measure ◽  
Central Direction ◽  
Open Class ◽  
Discrete Time Dynamical Systems

AbstractWe consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in Alves and Araújo [Random perturbations of non-uniformly expanding maps. Astérisque 286 (2003), 25–62], where the stochastic stability in the $\mathrm {weak}^*$ topology was proved. Here, under slightly weaker assumptions on the random perturbations, we obtain a stronger version of stochastic stability: convergence of the density of the stationary measure to the density of the Sinai–Ruelle–Bowen (SRB) measure of the unperturbed system in the $L^1$-norm. As an application of our results, we obtain strong stochastic stability for two classes of non-uniformly expanding maps. The first one is an open class of local diffeomorphisms introduced in Alves, Bonatti and Viana [SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140 (2000), 351–398] and the second one is the class of Viana maps.

scholarly journals Optimal preview control for a class of linear continuous-time large-scale systems

2017 ◽  
Vol 40 (14) ◽  
pp. 4004-4013 ◽  
Author(s):  
Fucheng Liao ◽  
Yu Wang ◽  
Yanrong Lu ◽  
Jiamei Deng
Keyword(s):  
Continuous Time ◽  
Large Scale ◽  
Closed Loop ◽  
Reference Signal ◽  
Sufficient Conditions ◽  
Preview Control ◽  
Correlation Matrices ◽  
Large Scale Systems ◽  
Augmented System ◽  
Limiting Condition

In this paper, the problem of optimal preview control is studied for a class of linear continuous-time large-scale systems. We first construct an augmented system including the error signal and the reference signal to transform the tracking problem into the regulator problem. Then, the controllers are designed for isolated augmented subsystems, which also constitute the controller of large-scale systems. On the basis of proving the asymptotic stability of closed-loop large-scale systems and the existence of the controller, sufficient conditions for reaching optimal preview control are given. In particular, the limiting condition of the correlation matrices is determined by the fact that the total derivative of a positive definite Lyapunov function is negative definite. The numerical simulation indicates that the controller can drive the large-scale systems to track the reference signal without steady-state error, and the tracking effect is improved with the increasing preview length.

Analysis of Stability for Generalized Discrete Large-Scale Systems

2013 ◽  
Vol 467 ◽  
pp. 627-632
Author(s):  
Chen Fang ◽  
Jiang Hong Shi ◽  
Shuang Yu ◽  
Jian Fang Mao
Keyword(s):  
Lyapunov Function ◽  
Large Scale ◽  
Lyapunov Equation ◽  
Large Scale Systems ◽  
Stability And Instability ◽  
Analysis Of Stability ◽  
The Stability

Based on the condition that all independent subsystems of the generalized large-scale systems are regular and causal, this thesis studied both stability and instability of discrete linear generalized large-scale systems through Lyapunov equation and Lyapunov function, and proposed the criterion theorem for the stability or instability of discrete linear generalized large-scale systems.

Robust Decentralized Stabilization of Large-Scale Delay Systems Via Sliding Mode Control

1997 ◽  
Vol 119 (2) ◽  
pp. 307-312 ◽  
Author(s):  
Jun-Juh Yan ◽  
Jason Sheng-Hong Tsai ◽  
Fan-Chu Kung
Keyword(s):  
Sliding Mode Control ◽  
Large Scale ◽  
Sliding Mode ◽  
Delay Systems ◽  
Sliding Mode Controller ◽  
Stabilization Problem ◽  
Large Scale Systems ◽  
Decentralized Stabilization ◽  
Mode Control ◽  
The Stability

The present paper is concerned with the decentralized stabilization problem of large-scale systems with delays in the intercon-nections using sliding mode control. A robust stability condition of the sliding mode and a robust decentralized sliding mode controller are newly derived for large-scale delay systems. Also a proportional-integral sliding mode is designed to make it easy to assure the stability of dynamics in the sliding mode.

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