FEEDBACK CONTROL OF LYAPUNOV EXPONENTS FOR DISCRETE-TIME DYNAMICAL SYSTEMS

1996 ◽  
Vol 06 (07) ◽  
pp. 1341-1349 ◽  
Author(s):  
GUANRONG CHEN ◽  
DEJIAN LAI
Keyword(s):  
Dynamical Systems ◽  
Feedback Control ◽  
Lyapunov Exponents ◽  
Design Method ◽  
Design Procedure ◽  
Asymptotically Stable ◽  
Controlled System ◽  
Dimensional Dynamical System ◽  
Convenient Technique ◽  
Discrete Time Dynamical Systems

A simple, yet mathematically rigorous feedback control design method is proposed in this paper, which can make all the Lyapunov exponents of the controlled system strictly positive, for any given n-dimensional dynamical system that could be originally nonchaotic or even asymptotically stable. The argument used is purely algebraic and the design procedure is completely schematic, with no approximations used throughout the derivation. This is a rigorous and convenient technique suggested as an attempt for anticontrol of chaotic dynamical systems, with explicit computational formulas derived for applications.

CHAOTIFICATION OF DISCRETE-TIME DYNAMICAL SYSTEMS: AN EXTENSION OF THE CHEN–LAI ALGORITHM

2005 ◽  
Vol 15 (01) ◽  
pp. 109-117 ◽  
Author(s):  
DEJIAN LAI ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Time ◽  
Design Method ◽  
Control Design ◽  
Asymptotically Stable ◽  
Controlled System ◽  
Dimensional Dynamical System ◽  
Discrete Time Dynamical Systems ◽  
Uniformly Bounded

In this article, we propose and study an extension of the Chen–Lai algorithm for chaotification of discrete-time dynamical systems. The proposed method is a simple but mathematically rigorous feedback control design method that can gradually make all the Lyapunov exponents of the controlled system strictly positive for any given n-dimensional dynamical system that has a uniformly bounded Jacobian but otherwise could be originally nonchaotic or even asymptotically stable.

A Design Procedure for Asymptotically Optimal Dynamic Compensators

1990 ◽  
Vol 112 (1) ◽  
pp. 10-16 ◽  
Author(s):  
Kiyotaka Shimizu ◽  
Masakazu Suzuki ◽  
Misao Kato
Keyword(s):  
Optimal Parameter ◽  
Design Method ◽  
Design Procedure ◽  
Pole Placement ◽  
Feedback Gain ◽  
Controlled System ◽  
Parameter Matching ◽  
One Step ◽  
Optimal Dynamic ◽  
Optimal Regulator

This paper is concerned with a design method for optimizing dynamic compensators of Pearson’s type. Optimal parameter matrices are obtained by use of a parameter matching technique and an arbitrary pole placement technique. The controlled system has the optimal LQ modes and the modes with arbitrarily quick damping. The presented compensator works as the optimal regulator with observer and performs about the same control as the optimal regulator. And it is designed not in two steps; observer, regulator, but in one step; optimization of output feedback gain without considering any state estimation.

CHAOTIFICATION OF DISCRETE DYNAMICAL SYSTEMS IN BANACH SPACES

2006 ◽  
Vol 16 (09) ◽  
pp. 2615-2636 ◽  
Author(s):  
YUMING SHI ◽  
PEI YU ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical Systems ◽  
Feedback Control ◽  
Banach Spaces ◽  
Original System ◽  
Real Space ◽  
Discrete Dynamical Systems ◽  
State Feedback Control ◽  
Controlled System ◽  
Partial Difference ◽  
Finite Dimensional

This paper is concerned with chaotification of discrete dynamical systems in Banach spaces via feedback control techniques. A criterion of chaos in Banach spaces is first established. This criterion extends and improves the Marotto theorem. Discussions are carried out in general and some special Banach spaces. All the controlled systems are proved to be chaotic in the sense of both Devaney and Li–Yorke. As a consequence, a controlled system described in a finite-dimensional real space studied by Wang and Chen is shown chaotic not only in the sense of Li–Yorke but also in the sense of Devaney. The original system can be driven to be chaotic by using an arbitrarily small-amplitude state feedback control in a certain space. In addition, the Chen–Lai anti-control algorithm via feedback control with mod-operation in a finite-dimensional real space is extended to a certain infinite-dimensional Banach space, and the controlled system is shown chaotic in the sense of Devaney as well as in the sense of both Li–Yorke and Wiggins. Differing from many existing results, it is not here required that the map corresponding to the original system has a fixed point in some cases. An application of the theoretical results to a class of first-order partial difference equations is given with some numerical simulations.

Necessary and sufficient conditions for stable synchronization in random dynamical systems

2017 ◽  
Vol 38 (5) ◽  
pp. 1857-1875 ◽  
Author(s):  
JULIAN NEWMAN
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Compact Space ◽  
Sufficient Conditions ◽  
Invariant Set ◽  
Stochastic Flow ◽  
Asymptotically Stable ◽  
Random Maps ◽  
Necessary And Sufficient ◽  
Being Moved

For a composition of independent and identically distributed random maps or a memoryless stochastic flow on a compact space$X$, we find conditions under which the presence of locally asymptotically stable trajectories (e.g. as given by negative Lyapunov exponents) implies almost-sure mutual convergence of any given pair of trajectories (‘synchronization’). Namely, we find that synchronization occurs and is ‘stable’ if and only if the system exhibits the following properties: (i) there is asmallestnon-empty invariant set$K\subset X$; (ii) any two points in$K$are capable of being moved closer together; and (iii) $K$admits asymptotically stable trajectories.

scholarly journals A generalized PID controller for high-order dynamical systems

2021 ◽  
Vol 72 (2) ◽  
pp. 119-124
Author(s):  
Günyaz Ablay
Keyword(s):  
Dynamical Systems ◽  
Control Method ◽  
Controller Design ◽  
Design Method ◽  
Output Feedback Control ◽  
Design Procedure ◽  
Control Design ◽  
Polynomial Equation ◽  
High Order ◽  
Mechatronic Systems

Abstract This paper introduces a generalized PID type controller for controlling high-order dynamical systems. An optimal generalized PID control design method is developed to provide a simplified high-order output feedback control design procedure and tunable response characteristics. The controller design procedure is reduced to the specification of the desired natural frequency and the solution of a polynomial equation. The control method is capable of providing a desired control performance under set-point and disturbance variations. The performance of the proposed control method is implemented on some unstable and nonlinear mechatronic systems to illustrate the robustness, e ectiveness and feasibility of the method.

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