scholarly journals Generalized convergence analysis of the fractional order systems

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 404-411
Author(s):  
Ahmad Ruzitalab ◽  
Mohammad Hadi Farahi ◽  
Gholamhossien Erjaee
Keyword(s):  
Fractional Order ◽  
Fractional Order Systems ◽  
Necessary And Sufficient Condition ◽  
Lyapunov Matrix Equation ◽  
Discrete Time Systems ◽  
Lyapunov Matrix ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Contraction Theory ◽  
Time Systems

Abstract The aim of the present work is to generalize the contraction theory for the analysis of the convergence of fractional order systems for both continuous-time and discrete-time systems. Contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another. The result of this study is a generalization of the Lyapunov matrix equation and linear eigenvalue analysis. The proposed approach gives a necessary and sufficient condition for exponential and global convergence of nonlinear fractional order systems. The examples elucidate that the theory is very straightforward and exact.

scholarly journals New Stability Tests for Discretized Fractional-Order Systems Using the Al-Alaoui and Tustin Operators

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Rafał Stanisławski ◽  
Marek Rydel ◽  
Krzysztof J. Latawiec
Keyword(s):  
Fractional Order ◽  
Discrete Time ◽  
Fractional Order Systems ◽  
Simulation Experiments ◽  
Stability Tests ◽  
Analytical Stability ◽  
Discrete Time Systems ◽  
The Stability ◽  
Time Systems ◽  
Order Continuous

This paper provides new results on a stable discretization of commensurate fractional-order continuous-time LTI systems using the Al-Alaoui and Tustin discretization methods. New, graphical, and analytical stability/instability conditions are given for discrete-time systems obtained by means of the Al-Alaoui discretization scheme. On this basis, an analytically driven stability condition for discrete-time systems using the Tustin-based approach is presented. Finally, the stability of discrete-time systems obtained by finite-length approximation of the Al-Alaoui and Tustin operators are discussed. Simulation experiments confirm the effectiveness of the introduced stability tests.

scholarly journals Controllability of Linear Discrete-Time Systems with Both Delayed States and Delayed Inputs

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Hong Shi ◽  
Guangming Xie ◽  
Wenguang Luo
Keyword(s):  
Linear Systems ◽  
Discrete Time ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Control Delay ◽  
Discrete Time Systems ◽  
Necessary And Sufficient ◽  
And Control ◽  
Time Systems ◽  
Time Linear

The controllability issues for discrete-time linear systems with delay in state and control are addressed. By introducing a new concept, the controllability realization index (CRI), the characteristic of controllability is revealed. An easily testable necessary and sufficient condition for the controllability of discrete-time linear systems with state and control delay is established.

scholarly journals Controllability Analysis of Linear Discrete Time Systems with Time Delay in State

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Hong Shi ◽  
Guangming Xie ◽  
Wenguang Luo
Keyword(s):  
Time Delay ◽  
Linear Systems ◽  
Discrete Time ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Delay ◽  
Discrete Time Systems ◽  
Necessary And Sufficient ◽  
Time Systems ◽  
Time Linear

The controllability issues for linear discrete-time systems with delay in state are addressed. By introducing a new concept, the minimum controllability realization index (MinCRI), the characteristic of controllability is revealed. It is proved that the MinCRI of a system with state delay exists and is finite. Based on this result, a necessary and sufficient condition for the controllability of discrete-time linear systems with state delay is established.

scholarly journals Modified Function Projective Synchronization for a Partially Linear and Fractional-Order Financial Chaotic System with Uncertain Parameters

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Yehong Yang ◽  
Guohua Cao
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Projective Synchronization ◽  
Uncertain Parameters ◽  
Lyapunov Matrix Equation ◽  
Function Projective Synchronization ◽  
Partially Linear ◽  
Modified Function Projective Synchronization ◽  
Lyapunov Matrix ◽  
The Stability

This paper investigates the modified function projective synchronization between fractional-order chaotic systems, which are partially linear financial systems with uncertain parameters. Based on the stability theory of fractional-order systems and the Lyapunov matrix equation, a controller is obtained for the synchronization between fractional-order financial chaotic systems. Using the controller, the error systems converged to zero as time tends to infinity, and the uncertain parameters were also estimated so that the phenomenon of parameter distortion was effectively avoided. Numerical simulations demonstrate the validity and feasibility of the proposed method.

scholarly journals Losslessness of Nonlinear Stochastic Discrete-Time Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Xikui Liu ◽  
Yan Li ◽  
Ning Gao
Keyword(s):  
Discrete Time ◽  
State Feedback ◽  
Zero Dynamics ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Discrete Time System ◽  
Time System ◽  
Discrete Time Systems ◽  
Necessary And Sufficient ◽  
Time Systems

This paper will study stochastic losslessness theory for nonlinear stochastic discrete-time systems, which are expressed by the Itô-type difference equations. A necessary and sufficient condition is developed for a nonlinear stochastic discrete-time system to be lossless. By the stochastic lossless theory, we show that a nonlinear stochastic discrete-time system can be lossless via state feedback if and only if it has relative degree0,…,0and lossless zero dynamics. The effectiveness of the proposed results is illustrated by a numerical example.

scholarly journals H-Index for Stochastic Linear Discrete-Time Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Yan Li ◽  
Weihai Zhang ◽  
Xikui Liu
Keyword(s):  
Discrete Time ◽  
Observer Design ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
H Index ◽  
Discrete Time Systems ◽  
Necessary And Sufficient ◽  
Index Problem ◽  
Time Systems ◽  
Theoretical Results

This paper discusses theH-index problem for stochastic linear discrete-time systems. A necessary and sufficient condition ofH-index is given for such systems in finite horizon. It is proved that when theH-index is greater than a given value, the feasibility ofH-index is equivalent to the solvability of a constrained difference equation. The above result can be applied to the fault detection observer design. Finally, some examples are presented to illustrate the proposed theoretical results.

scholarly journals On the Q–S Chaos Synchronization of Fractional-Order Discrete-Time Systems: General Method and Examples

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Adel Ouannas ◽  
Amina-Aicha Khennaoui ◽  
Giuseppe Grassi ◽  
Samir Bendoukha
Keyword(s):  
Fractional Order ◽  
Discrete Time ◽  
Chaos Synchronization ◽  
Control Strategies ◽  
Linearization Method ◽  
Discrete Time Systems ◽  
Control Scheme ◽  
The Stability ◽  
Time Systems ◽  
General Method

In this paper, we propose two control strategies for the Q–S synchronization of fractional-order discrete-time chaotic systems. Assuming that the dimension of the response system m is higher than that of the drive system n, the first control scheme achieves n-dimensional synchronization whereas the second deals with the m-dimensional case. The stability of the proposed schemes is established by means of the linearization method. Numerical results are presented to confirm the findings of the study.

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