On the stability of linear systems with fractional-order elements

2009 ◽  
Vol 40 (5) ◽  
pp. 2317-2328 ◽  
Author(s):  
A.G. Radwan ◽  
A.M. Soliman ◽  
A.S. Elwakil ◽  
A. Sedeek
Keyword(s):  
Linear Systems ◽  
Fractional Order ◽  
The Stability

scholarly journals Stability of Systems of Fractional-Order Differential Equations with Caputo Derivatives

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 914
Author(s):  
Oana Brandibur ◽  
Roberto Garrappa ◽  
Eva Kaslik
Keyword(s):  
Differential Equations ◽  
Detailed Analysis ◽  
Linear Systems ◽  
Fractional Order ◽  
Numerical Experiments ◽  
Limit Case ◽  
Fractional Order Differential Equations ◽  
Fractional Differential ◽  
The Stability

Systems of fractional-order differential equations present stability properties which differ in a substantial way from those of systems of integer order. In this paper, a detailed analysis of the stability of linear systems of fractional differential equations with Caputo derivative is proposed. Starting from the well-known Matignon’s results on stability of single-order systems, for which a different proof is provided together with a clarification of a limit case, the investigation is moved towards multi-order systems as well. Due to the key role of the Mittag–Leffler function played in representing the solution of linear systems of FDEs, a detailed analysis of the asymptotic behavior of this function and of its derivatives is also proposed. Some numerical experiments are presented to illustrate the main results.

scholarly journals Adaptive Backstepping Method for Stabilizing Systems of Fractional Order Ordinary Differential Equations

2021 ◽  
Vol 24 (4) ◽  
pp. 46-51
Author(s):  
Asad J. Taher ◽  
◽  
Fadhel S. Fadhel ◽  
Nabaa N. Hasan ◽  
◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Systems ◽  
Fractional Order ◽  
Lyapunov Functions ◽  
Backstepping Method ◽  
Adaptive Backstepping ◽  
Quadratic Lyapunov Functions ◽  
The Stability

In this paper the method of adaptive backstepping for stabilizing and solving system of ordinary and partial differential equations will be used and applied to investigate and study the stability linear systems of Caputo fractional order ordinary differential equations. The basic idea of this approach is to find a quadratic Lyapunov functions for stabilizing the subsystems.

scholarly journals Control Techniques for a Class of Fractional Order Systems

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2357
Author(s):  
Mircea Ivanescu ◽  
Ioan Dumitrache ◽  
Nirvana Popescu ◽  
Decebal Popescu
Keyword(s):  
Time Delay ◽  
Linear Systems ◽  
Fractional Order ◽  
Closed Loop ◽  
Fractional Order Systems ◽  
Loop Control ◽  
Order Linear ◽  
Control Techniques ◽  
The Stability ◽  
Nonlinear Components

The paper discusses several control techniques for a class of systems described by fractional order equations. The paper presents the unit frequency criteria that ensure the closed loop control for: Fractional Order Linear Systems, Fractional Order Linear Systems with nonlinear components, Time Delay Fractional Order Linear Systems, Time Delay Fractional Order Linear Systems with nonlinear components. The stability criterion is proposed for the systems composed of fractional order subsystems. These techniques are used in two applications: Soft Exoskeleton Glove Control, studied as a nonlinear model with time delay and Disabled Man-Wheelchair model, analysed as a fractional-order multi-system.

CONTINUOUS-TIME FRACTIONAL ORDER LINEAR SYSTEMS IDENTIFICATION USING CHEBYSHEV WAVELET

2020 ◽  
Vol 6 (8(77)) ◽  
pp. 23-28
Author(s):  
Shuen Wang ◽  
Ying Wang ◽  
Yinggan Tang
Keyword(s):  
Linear Systems ◽  
Fractional Order ◽  
Continuous Time ◽  
Fractional Integration ◽  
Operational Matrices ◽  
Computation Complexity ◽  
Order Linear ◽  
Fractional Integration Operator ◽  
Systems Identification ◽  
Chebyshev Wavelet

In this paper, the identification of continuous-time fractional order linear systems (FOLS) is investigated. In order to identify the differentiation or- ders as well as parameters and reduce the computation complexity, a novel identification method based on Chebyshev wavelet is proposed. Firstly, the Chebyshev wavelet operational matrices for fractional integration operator is derived. Then, the FOLS is converted to an algebraic equation by using the the Chebyshev wavelet operational matrices. Finally, the parameters and differentiation orders are estimated by minimizing the error between the output of real system and that of identified systems. Experimental results show the effectiveness of the proposed method.

scholarly journals Observer-based adaptive neural network backstepping sliding mode control for switched fractional order uncertain nonlinear systems with unmeasured states

2021 ◽  
pp. 002029402110211
Author(s):  
Tao Chen ◽  
Damin Cao ◽  
Jiaxin Yuan ◽  
Hui Yang
Keyword(s):  
Neural Network ◽  
Nonlinear Systems ◽  
Sliding Mode Control ◽  
Fractional Order ◽  
Sliding Mode ◽  
Tracking Error ◽  
Adaptive Neural Network ◽  
Dynamic Surface ◽  
Mode Control ◽  
The Stability

This paper proposes an observer-based adaptive neural network backstepping sliding mode controller to ensure the stability of switched fractional order strict-feedback nonlinear systems in the presence of arbitrary switchings and unmeasured states. To avoid “explosion of complexity” and obtain fractional derivatives for virtual control functions continuously, the fractional order dynamic surface control (DSC) technology is introduced into the controller. An observer is used for states estimation of the fractional order systems. The sliding mode control technology is introduced to enhance robustness. The unknown nonlinear functions and uncertain disturbances are approximated by the radial basis function neural networks (RBFNNs). The stability of system is ensured by the constructed Lyapunov functions. The fractional adaptive laws are proposed to update uncertain parameters. The proposed controller can ensure convergence of the tracking error and all the states remain bounded in the closed-loop systems. Lastly, the feasibility of the proposed control method is proved by giving two examples.

Global stability analysis of fractional-order gene regulatory networks with time delay

2019 ◽  
Vol 12 (06) ◽  
pp. 1950067 ◽  
Author(s):  
Zhaohua Wu ◽  
Zhiming Wang ◽  
Tiejun Zhou
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Gene Regulatory Networks ◽  
Existence And Uniqueness ◽  
Regulatory Networks ◽  
Lyapunov Method ◽  
Regulation Mechanism ◽  
Global Stability Analysis ◽  
Gene Regulatory ◽  
The Stability

Fractional-order gene regulatory networks with time delay (DFGRNs) have proven that they are more suitable to model gene regulation mechanism than integer-order. In this paper, a novel DFGRN is proposed. The existence and uniqueness of the equilibrium point for the DFGRN are proved under certain conditions. On this basis, the conditions on the global asymptotic stability are established by using the Lyapunov method and comparison theorem for the DFGRN, and the stability conditions are dependent on the fractional-order [Formula: see text]. Finally, numerical simulations show that the obtained results are reasonable.

A computationally efficient adaptive robust control scheme for a quad-rotor transporting cable-suspended payloads

2021 ◽  
pp. 095441002110136
Author(s):  
Nasim Ullah ◽  
Irfan Sami ◽  
Wang Shaoping ◽  
Hamid Mukhtar ◽  
Xingjian Wang ◽  
...  
Keyword(s):  
Robust Control ◽  
Fractional Order ◽  
Sliding Mode ◽  
Adaptive Robust Control ◽  
Computationally Efficient ◽  
Control Scheme ◽  
Control Schemes ◽  
Adaptive Laws ◽  
Unknown Disturbances ◽  
The Stability

This article proposes a computationally efficient adaptive robust control scheme for a quad-rotor with cable-suspended payloads. Motion of payload introduces unknown disturbances that affect the performance of the quad-rotor controlled with conventional schemes, thus novel adaptive robust controllers with both integer- and fractional-order dynamics are proposed for the trajectory tracking of quad-rotor with cable-suspended payload. The disturbances acting on quad-rotor due to the payload motion are estimated by utilizing adaptive laws derived from integer- and fractional-order Lyapunov functions. The stability of the proposed control systems is guaranteed using integer- and fractional-order Lyapunov theorems. Overall, three variants of the control schemes, namely adaptive fractional-order sliding mode (AFSMC), adaptive sliding mode (ASMC), and classical Sliding mode controllers (SMC)s) are tested using processor in the loop experiments, and based on the two performance indicators, namely robustness and computational resource utilization, the best control scheme is evaluated. From the results presented, it is verified that ASMC scheme exhibits comparable robustness as of SMC and AFSMC, while it utilizes less sources as compared to AFSMC.

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