Non-commensurate Fractional-order Systems Research Articles

FOMCON Toolbox-Based Direct Approximation of Fractional Order Systems Using Gaze Cues Learning-Based Grey Wolf Optimizer

This study discusses a new method for the fractional-order system reduction. It offers an adaptable framework for approximating various fractional-order systems (FOSs), including commensurate and non-commensurate. The fractional-order modeling and control (FOMCON) toolbox in MATLAB and the gaze cues learning-based grey wolf optimizer (GGWO) technique form the basis of the recommended method. The fundamental advantage of the offered method is that it does not need intermediate steps, a mathematical substitution, or an operator-based approximation for the order reduction of a commensurate and non-commensurate FOS. The cost function is set up so that the sum of the integral squared differences in step responses and the root mean squared differences in Bode magnitude plots between the original FOS and the reduced models is as tiny as possible. Two case studies support the suggested method. The simulation results show that the reduced approximations constructed using the methodology under consideration have step and Bode responses more in line with the actual FOS. The effectiveness of the advocated strategy is further shown by contrasting several performance metrics with some of the contemporary approaches disseminated in academic journals.

State-space model realization for non-commensurate fractional-order systems based on Gleason’s problem

This paper presents a novel state-space model (SSM) realization approach for non-commensurate fractional-order (NCFO) systems. A new notion of the resolvent invariant backward shift space of Gleason’s problem associated with the NCFO rational transfer function matrix is introduced firstly, based on which a necessary and sufficient condition for the existence of the NCFO-SSM realization is given. Then, a sufficient condition for the construction of the resolvent invariant backward shift space of NCFO Gleason’s problem is presented, and the corresponding realization algorithms are developed. In order to obtain a lower-dimensional NCFO-SSM, further improvement based on a polynomial description is studied. Finally, symbolic and numerical examples are provided to illustrate the details as well as the effectiveness of the proposed NCFO-SSM realization approach.

A Fractional Atmospheric Circulation System under the Influence of a Sliding Mode Controller

The earth’s surface is heated by the large-scale movement of air known as atmospheric circulation, which works in conjunction with ocean circulation. More than 105 variables are involved in the complexity of the weather system. In this work, we analyze the dynamical behavior and chaos control of an atmospheric circulation model known as the Hadley circulation model, in the frame of Caputo and Caputo–Fabrizio fractional derivatives. The fundamental novelty of this paper is the application of the Caputo derivative with equal dimensionality to models that includes memory. A sliding mode controller (SMC) is developed to control chaos in this fractional-order atmospheric circulation system with uncertain dynamics. The proposed controller is applied to both commensurate and non-commensurate fractional-order systems. To demonstrate the intricacy of the models, we plot some graphs of various fractional orders with appropriate parameter values. We have observed the influence of thermal forcing on the dynamics of the system. The outcome of the analytical exercises is validated using numerical simulations.

Fractional-order circuit design with hybrid controlled memristors and FPGA implementation

A hybrid memristor circuit consisting of a magnetic-controlled and a charge-controlled is designed. In order to construct a relatively complex dynamical system, an absolute value and a square root algorithm are embedded in the magnetic-controlled model specially. This novel five-dimensional system is first analyzed with integer order for the existence of chaotic phenomena, and then implemented the fractional-order memristor system by the Grunwald–Letnikov algorithm. Five fractional order values of qi(i=1,2,3,4,5) are taken as identical and different in the numerical simulation, respectively. The bifurcation diagram, 0-1 test, SALI algorithm, and complexity analysis reveal the existence of nonlinear dynamic characteristics to the hybrid fractional-order memristor system. Finally, the commensurate and non-commensurate fractional-order systems based on the frequency domain method are implemented by FPGA technology, their accuracy can reach to 14 decimal places. The results of numerical calculation and circuit simulation have the same phase trajectory, which can verify that the hybrid memristors have practical values in the future.

Modified Mikhailov stability criterion for continuous-time noncommensurate fractional-order systems

This paper presents the modified Mikhailov stability criterion, which can be effectively used in stability analysis for continuous-time noncommensurate fractional-order systems. The main advantage of the proposed methodology is that the stability analysis of noncommensurate fractional-order systems leads to the same computational complexity as for the commensurate order ones. Simulation examples confirm the usefulness of the introduced methodology.

Matlab Code for Lyapunov Exponents of Fractional-Order Systems, Part II: The Noncommensurate Case

In this paper, the Benettin–Wolf algorithm for determining all Lyapunov exponents of noncommensurate fractional-order systems modeled by Caputo’s derivative and the corresponding Matlab code are presented. The paper continues the work started in [ Danca & Kuznetsov, 2018 ], where the Matlab code of commensurate fractional-order systems is given. To integrate the extended systems, the Adams–Bashforth–Moulton scheme for fractional differential equations is utilized. Like the Matlab program for commensurate-order systems, the program presented in this paper prints and plots all Lyapunov exponents as function of time. The program can be simply adapted to plot the evolution of the Lyapunov exponents as a function of orders, or a function of a bifurcation parameter. Special attention is paid to the periodicity of fractional-order systems and its influences. The case of noncommensurate Lorenz system is demonstrated.

A modified Mikhailov stability criterion for a class of discrete-time noncommensurate fractional-order systems

This paper introduces an extension of the Mikhailov stability criterion to a class of discrete-time noncommensurate fractional-order systems using the nabla fractional-order Grünwald-Letnikov difference. The new stability analysis methods proposed in the paper are computationally simple and can be effectively used both for commensurate and noncommensurate fractional-order systems. The main advantage of the proposed methodology is the fact that the stability analysis of noncommensurate fractional-order systems leads to exactly the same computational complexity as for the commensurate-order ones. Simulation examples confirm usefulness of the proposed methodology.

Fractional-Order Adaptive Backstepping Control of a Noncommensurate Fractional-Order Ferroresonance System

In this paper, fractional calculus is applied to establish a novel fractional-order ferroresonance model with fractional-order magnetizing inductance and capacitance. Some basic dynamic behaviors of this fractional-order ferroresonance system are investigated. And then, considering noncommensurate orders of inductance and capacitance and unknown parameters in an actual ferroresonance system, this paper presents a novel fractional-order adaptive backstepping control strategy for a class of noncommensurate fractional-order systems with multiple unknown parameters. The virtual control laws and parameter update laws are designed in each step. Thereafter, a novel fractional-order adaptive controller is designed in terms of the fractional Lyapunov stability theorem. The proposed control strategy requires only one control input and can force the output of the chaotic system to track the reference signal asymptotically. Finally, the proposed method is applied to a noncommensurate fractional-order ferroresonance system with multiple unknown parameters. Numerical simulation confirms the effectiveness of the proposed method. In addition, the proposed control strategy also applies to commensurate fractional-order systems with unknown parameters.

New integer-order approximations of discrete-time non-commensurate fractional-order systems using the cross Gramian

This paper presents a new approach to modeling of linear time-invariant discrete-time non-commensurate fractional-order single-input single-output state space systems by means of the Balanced Truncation and Frequency Weighted model order reduction methods based on the cross Gramian. These reduction methods are applied to the specific rational (integer-order) FIR-based approximation to the fractional-order system, which enables to introduce simple, analytical formulae for determination of the cross Gramian of the system. This leads to significant decrease of computational burden in the reduction algorithm. As a result, a rational and relatively low-order state space approximator for the fractional-order system is obtained. A simulation experiment illustrates an efficiency of the introduced methodology in terms of high approximation accuracy and low time complexity of the proposed method.

Stability Analysis for a Class of Non-Commensurate Fractional-Order Systems

Stability Analysis for a Class of Non-Commensurate Fractional-Order Systems

Lyapunov Stability of Noncommensurate Fractional Order Systems: An Energy Balance Approach

Lyapunov stability of linear noncommensurate order fractional systems is treated in this paper. The proposed methodology is based on the concept of fractional energy stored in inductor and capacitor components, where natural decrease of the stored energy is caused by internal Joule losses. The Lyapunov function is expressed as the sum of the different reversible fractional energies, whereas its derivative is interpreted in terms of internal and external Joule losses. Stability conditions are derived from the energy balance principle, adapted to the fractional case. Examples are taken from electrical systems, but this methodology applies also directly to mechanical and electromechanical systems.

Observer-Based Sliding Mode Control for a Class of Noncommensurate Fractional-Order Systems

This paper investigates an application of fractional theory for control and state estimation for a class of noncommensurate systems. A single-link flexible manipulator is considered as a representative example of this class of second-order underactuated systems. A fractional model is proposed and validated experimentally. The fractional-order sliding mode control is synthesized using a novel stable fractional surface. A fractional sliding mode observer is designed for state estimation necessary for implementing the control. The proposed method is validated through simulation and experimentation both.

A graphic stability criterion for non-commensurate fractional-order time-delay systems

This study focuses on a graphical approach to determine the stability of non-commensurate fractional-order systems with time-delay. By the iterative decomposition technique, the fractional-order systems described by the transfer functions are represented as a series of closed loop systems. The innermost closed loop system is straightforward to test the stability by its coefficient and the fractional-order. A function with respect to each open loop system is defined, and the stability of a non-commensurate fractional-order system is determined by the unstable poles of the innermost system and the times of the curve depending on the defined function encircling the origin in the clockwise direction. Finally, three illustrative examples are provided to demonstrate the effectiveness of the proposed criterion.