CHAOTIFYING CONTINUOUS-TIME NONLINEAR AUTONOMOUS SYSTEMS

Based on the principle of chaotification for continuous-time autonomous systems, which relies on two basic properties of chaos, i.e. being globally bounded with necessary positive-zero-negative Lyapunov exponents, this paper derives a feasible and unified chaotification method for designing a general chaotic continuous-time autonomous nonlinear system. For a system consisting of a linear and a nonlinear subsystems, chaotification is achieved using separation of state variables, which decomposes the system into two open-loop subsystems interacting through mutual feedback resulting in an overall closed-loop nonlinear feedback system. Under the condition that the nonlinear feedback control output is uniformly bounded where the nonlinear function is of bounded-input/bounded-output, it is proved that the resulting system is chaotic in the sense of being globally bounded with a required placement of Lyapunov exponents. Several numerical examples are given to verify the effectiveness of the theoretical design. Since linear systems are special cases of nonlinear systems, the new method is also applicable to linear systems in general.

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Positive fractional continuous-time linear systems with singular pencils

Bulletin of the Polish Academy of Sciences Technical Sciences ◽

10.2478/v10175-012-0002-0 ◽

2012 ◽

Vol 60 (1) ◽

pp. 9-12 ◽

Cited By ~ 11

Author(s):

T. Kaczorek

Keyword(s):

Linear Systems ◽

Continuous Time ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Descriptor Systems ◽

State Variables ◽

Algebraic Constraints ◽

Necessary And Sufficient ◽

Time Linear

Positive fractional continuous-time linear systems with singular pencils A method for checking the positivity and finding the solution to the positive fractional descriptor continuous-time linear systems with singular pencils is proposed. The method is based on elementary row and column operations of the fractional descriptor systems to equivalent standard systems with some algebraic constraints on state variables and inputs. Necessary and sufficient conditions for the positivity of the fractional descriptor systems are established.

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Stability analysis of continuous-time linear systems consisting of n subsystems with different fractional orders

Bulletin of the Polish Academy of Sciences Technical Sciences ◽

10.2478/v10175-012-0037-2 ◽

2012 ◽

Vol 60 (2) ◽

pp. 279-284 ◽

Cited By ~ 8

Author(s):

M. Busłowicz

Keyword(s):

Linear Systems ◽

Continuous Time ◽

State Equation ◽

The State ◽

Natural Order ◽

State Variables ◽

The Stability ◽

Derivatives Of ◽

Fractional Orders ◽

Time Linear

Abstract. The stability problem of continuous-time linear systems described by the state equation consisting of n subsystems with different fractional orders of derivatives of the state variables has been considered. The methods for asymptotic stability checking have been given. The method proposed in the general case is based on the Argument Principle and it is similar to the modified Mikhailov stability criterion known from the stability theory of natural order systems. The considerations are illustrated by numerical examples.

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Chaotification of a Class of Linear Switching Systems by Hybrid Driven Methods

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414500333 ◽

2014 ◽

Vol 24 (03) ◽

pp. 1450033 ◽

Cited By ~ 11

Author(s):

Yuping Zhang ◽

Xinzhi Liu ◽

Hong Zhu ◽

Yong Zeng

Keyword(s):

Lyapunov Exponents ◽

Continuous Time ◽

Autonomous Systems ◽

Phase Portraits ◽

Computational Formula ◽

Switching Systems ◽

Time Switching ◽

Switching Rule ◽

Lyapunov Spectra ◽

Linear Switching Systems

This paper investigates a class of linear continuous-time switching systems and proposes a new approach to generate chaos by designing a hybrid switching rule. First, a computational formula for Lyapunov exponents is derived by extending the definition of Lyapunov exponent for continuous-time autonomous systems to that of a class of linear continuous-time switching systems. Then, a novel switching rule is proposed to gain global boundedness property as well as the required placement of Lyapunov exponents for chaos. A numerical example is given to illustrate the chaotic dynamic behavior of the generated system. The Lyapunov dimension of the system in the example is calculated and the corresponding bifurcation diagram and Lyapunov spectra are sketched, which, together with other phase portraits, clearly verify the validity of the main result.

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Selection of nonlinear function in integral sliding mode-based composite nonlinear feedback method for transient improvement of uncertain linear systems

The 2nd International Conference on Control, Instrumentation and Automation ◽

10.1109/icciautom.2011.6356711 ◽

2011 ◽

Cited By ~ 4

Author(s):

Saleh Mobayen ◽

Vahid J. Majd ◽

Mohammad Hassan Asemani

Keyword(s):

Linear Systems ◽

Sliding Mode ◽

Nonlinear Function ◽

Nonlinear Feedback ◽

Composite Nonlinear Feedback ◽

Integral Sliding Mode ◽

Transient Improvement ◽

Feedback Method ◽

Uncertain Linear Systems ◽

Selection Of

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A geometric characterisation of persistently exciting signals generated by continuous-time autonomous systems

IFAC-PapersOnLine ◽

10.1016/j.ifacol.2016.10.268 ◽

2016 ◽

Vol 49 (18) ◽

pp. 826-831 ◽

Cited By ~ 6

Author(s):

Alberto Padoan ◽

Giordano Scarciotti ◽

Alessandro Astolfi

Keyword(s):

Continuous Time ◽

Autonomous Systems ◽

Time Autonomous

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A Geometric Criterion for the Existence of Chaos Based on Periodic Orbits in Continuous-Time Autonomous Systems

Journal of Dynamical and Control Systems ◽

10.1007/s10883-021-09582-x ◽

2022 ◽

Author(s):

Xu Zhang ◽

Guanrong Chen

Keyword(s):

Periodic Orbits ◽

Continuous Time ◽

Autonomous Systems ◽

Geometric Criterion ◽

Time Autonomous

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Estimation of Nonlinear Functions of State Vector for Linear Systems with Time-Delays and Uncertainties

Mathematical Problems in Engineering ◽

10.1155/2015/217253 ◽

2015 ◽

Vol 2015 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Il Young Song ◽

Georgy Shevlyakov ◽

Vladimir Shin

Keyword(s):

Linear Systems ◽

State Vector ◽

Time Delays ◽

Nonlinear Function ◽

Computational Procedure ◽

Target System ◽

State Variables ◽

Receding Horizon ◽

Unscented Transformation ◽

Optimal Estimator

This paper focuses on estimation of a nonlinear function of state vector (NFS) in discrete-time linear systems with time-delays and model uncertainties. The NFS represents a multivariate nonlinear function of state variables, which can indicate useful information of a target system for control. The optimal nonlinear estimator of an NFS (in mean square sense) represents a function of the receding horizon estimate and its error covariance. The proposed receding horizon filter represents the standard Kalman filter with time-delays and special initial horizon conditions described by the Lyapunov-like equations. In general case to calculate an optimal estimator of an NFS we propose using the unscented transformation. Important class of polynomial NFS is considered in detail. In the case of polynomial NFS an optimal estimator has a closed-form computational procedure. The subsequent application of the proposed receding horizon filter and nonlinear estimator to a linear stochastic system with time-delays and uncertainties demonstrates their effectiveness.

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