Initialization of Fractional-Order Systems Using the Hankel Operator

2011 ◽  
Author(s):  
Jay L. Adams ◽  
Robert J. Veillette ◽  
Tom T. Hartley
Keyword(s):  
Fractional Order ◽  
Hankel Operator ◽  
Order System ◽  
Approximation Methods ◽  
Initial Condition ◽  
Fractional Order Systems ◽  
General Characterization ◽  
System A ◽  
General Initial Condition

This paper demonstrates the use of the Hankel operator to characterize the initial-condition response of a fractional-order system. A general initial-condition response can be determined for any input applied before t = 0. Two techniques for approximating the Hankel operator are discussed. These approximation methods are applied to illustrative examples, demonstrating a general characterization of natural responses.

Fractional-order Kalman filters for continuous-time fractional-order systems involving correlated and uncorrelated process and measurement noises

2018 ◽  
Vol 41 (7) ◽  
pp. 1933-1947 ◽  
Author(s):  
Fanghui Liu ◽  
Zhe Gao ◽  
Chao Yang ◽  
Ruicheng Ma
Keyword(s):  
Fractional Order ◽  
Continuous Time ◽  
Kalman Filters ◽  
Equation Model ◽  
Order System ◽  
Fractional Order Systems ◽  
Derivative Method ◽  
Measurement Noises ◽  
Computation Load ◽  
Average Derivative

This paper presents fractional-order Kalman filters using the fractional-order average derivative method for linear fractional-order systems involving process and measurement noises. By using the fractional-order average derivative method, a difference equation model is obtained by discretizing the investigated continuous-time fractional-order system, and the accuracy of state estimation is improved. Meanwhile, compared with the Tustin generating function, the fractional-order average derivative method proposed in this paper can reduce computation load and save calculation time. Two kinds of fractional-order Kalman filters are given, for the correlated and uncorrelated cases, in terms of the process and measurement noises for linear fractional-order systems, respectively. Finally, simulation results are illustrated to verify the effectiveness of the proposed Kalman filters using the fractional-order average derivative method.

Model Predictive Control of General Fractional Order Systems Using a General Purpose Optimal Control Problem Solver: RIOTS

2019 ◽  
Author(s):  
Sina Dehghan ◽  
Tiebiao Zhao ◽  
YangQuan Chen ◽  
Taymaz Homayouni
Keyword(s):  
Optimal Control ◽  
Model Predictive Control ◽  
Fractional Order ◽  
Predictive Control ◽  
Order System ◽  
Problem Solver ◽  
Fractional Order Systems ◽  
Application Process ◽  
Order Model ◽  
Integer Order

Abstract RIOTS is a Matlab toolbox capable of solving a very general form of integer order optimal control problems. In this paper, we present an approach for implementing Model Predictive Control (MPC) to control a general form of fractional order systems using RIOTS toolbox. This approach is based on time-response-invariant approximation of fractional order system with an integer order model to be used as the internal model in MPC. The implementation of this approach is demonstrated to control a coupled MIMO commensurate fractional order model. Moreover, the performance and its application process is compared to examples reported in the literature.

scholarly journals Stabilization of the FO-BLDCM Chaotic System in the Sense of Lyapunov

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Ping Zhou ◽  
Rongji Bai ◽  
Hao Cai
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Direct Method ◽  
Order System ◽  
Lyapunov Direct Method ◽  
Brushless Dc Motors ◽  
Brushless Dc ◽  
Dc Motors ◽  
System A ◽  
Control Scheme

Based on an integer-order Brushless DC motors (IO-BLDCM) system, we give a fractional-order Brushless DC motors (FO-BLDCM) system in this paper. There exists a chaotic attractor for fractional-order0.95<q≤1in the FO-BLDCM system. Furthermore, using the Lyapunov direct method for fractional-order system, a control scheme is proposed to stabilize the FO-BLDCM chaotic system in the sense of Lyapunov. Numerical simulation shows that the control scheme in this paper is valid for the FO-BLDCM chaotic system.

Stability analysis of fractional-order systems with double noncommensurate orders for matrix case

2011 ◽  
Vol 14 (3) ◽  
Author(s):  
Zhuang Jiao ◽  
YangQuan Chen
Keyword(s):  
Fractional Order ◽  
Linear Time ◽  
Simple Algorithm ◽  
Inverse Laplace Transform ◽  
Order System ◽  
Necessary Condition ◽  
Fractional Order Systems ◽  
Numerical Inverse Laplace Transform ◽  
Laplace Transform Technique ◽  
Matrix Case

AbstractBounded-input bounded-output stability issues for fractional-order linear time invariant (LTI) system with double noncommensurate orders for the matrix case have been established in this paper. Sufficient and necessary condition of stability is given, and a simple algorithm to test the stability for this kind of fractional-order systems is presented. Based on the numerical inverse Laplace transform technique, time-domain responses for fractional-order system with double noncommensurate orders are shown in numerical examples to illustrate the proposed results.

scholarly journals Modified Projective Synchronization between Different Fractional-Order Systems Based on Open-Plus-Closed-Loop Control and Its Application in Image Encryption

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Hongjuan Liu ◽  
Zhiliang Zhu ◽  
Hai Yu ◽  
Qian Zhu
Keyword(s):  
Fractional Order ◽  
Closed Loop ◽  
Sufficient Conditions ◽  
Projective Synchronization ◽  
Order System ◽  
Coupling Scheme ◽  
Fractional Order Systems ◽  
Parameter Mismatch ◽  
Loop Control ◽  
Modified Projective Synchronization

A new general and systematic coupling scheme is developed to achieve the modified projective synchronization (MPS) of different fractional-order systems under parameter mismatch via the Open-Plus-Closed-Loop (OPCL) control. Based on the stability theorem of linear fractional-order systems, some sufficient conditions for MPS are proposed. Two groups of numerical simulations on the incommensurate fraction-order system and commensurate fraction-order system are presented to justify the theoretical analysis. Due to the unpredictability of the scale factors and the use of fractional-order systems, the chaotic data from the MPS is selected to encrypt a plain image to obtain higher security. Simulation results show that our method is efficient with a large key space, high sensitivity to encryption keys, resistance to attack of differential attacks, and statistical analysis.

LAG SYNCHRONIZATION OF THE FRACTIONAL-ORDER SYSTEM VIA NONLINEAR OBSERVER

2011 ◽  
Vol 25 (29) ◽  
pp. 3951-3964 ◽  
Author(s):  
HAO ZHU ◽  
ZHONGSHI HE ◽  
SHANGBO ZHOU
Keyword(s):  
Numerical Simulation ◽  
Fractional Order ◽  
Stability Theorem ◽  
Nonlinear Observer ◽  
Order System ◽  
Fractional Order Systems ◽  
Lag Synchronization ◽  
Systematic Method ◽  
Fractional System ◽  
The Stability

In this paper, based on the idea of nonlinear observer, lag synchronization of chaotic fractional system with commensurate and incommensurate order is studied by the stability theorem of linear fractional-order systems. The theoretical analysis of fractional-order systems in this paper is a systematic method. This technique is applied to achieve the lag synchronization of fractional-order Rössler's system, verified by numerical simulation.

scholarly journals Some Fundamental Properties on the Sampling Free Nabla Laplace Transform

2019 ◽  
Author(s):  
Yiheng Wei ◽  
Yuquan Chen ◽  
Yong Wang ◽  
YangQuan Chen
Keyword(s):  
System Theory ◽  
Laplace Transform ◽  
Fractional Order ◽  
Fractional Order System ◽  
The Other ◽  
Order System ◽  
Fractional Order Systems ◽  
Fundamental Properties ◽  
Other Hand

Abstract Discrete fractional order systems have attracted more and more attention in recent years. Nabla Laplace transform is an important tool to deal with the problem of nabla discrete fractional order systems, but there is still much room for its development. In this paper, 14 lemmas are listed to conclude the existing properties and 14 theorems are developed to describe the innovative features. On one hand, these properties make the Ntransform more effective and efficient. On the other hand, they enrich the discrete fractional order system theory.

Application of Incomplete Gamma Functions to the Initialization of Fractional-Order Systems

2007 ◽  
Author(s):  
Tom T. Hartley ◽  
Carl F. Lorenzo
Keyword(s):  
System Theory ◽  
Fractional Order ◽  
Gamma Function ◽  
Incomplete Gamma Function ◽  
Fractional Order System ◽  
Order System ◽  
Fractional Order Systems ◽  
Gamma Functions ◽  
Incomplete Gamma Functions

This paper reviews some properties of the gamma function, particularly the incomplete gamma function and its complement, as a function of the Laplace variable s. The utility of these functions in the solution of initialization problems in fractional-order system theory is demonstrated.

Research on identification of irrational fractional-order systems in frequency domain based on optimization algorithm

2019 ◽  
Vol 41 (15) ◽  
pp. 4351-4357
Author(s):  
Chen Lanfeng ◽  
Xue Dingyu
Keyword(s):  
Frequency Domain ◽  
Fractional Order ◽  
Optimization Algorithm ◽  
Controller Design ◽  
Order System ◽  
Model Parameters ◽  
Fractional Order Systems ◽  
Convolution Integral ◽  
Fractional Order Calculus ◽  
Frequency Domain Response

Fractional-order calculus can obtain better results than the integer-order in control theory, so it has become a research hotspot in recent years. However, the structure of the irrational fractional-order system is complex, so its theoretical analysis and controller design are more difficult. In this paper, a method based on convolution integral is proposed to obtain the frequency domain response of the irrational model. Combined with the optimization algorithm, the model parameters are identified. Moreover, the rationalization of the irrational model is realized, which facilitates the analysis and application design of this kind models. Finally, two examples are given to illustrate the effectiveness and feasibility of the method by identifying parameters and rationalization.

Sign in / Sign up

Export Citation Format

Share Document