Analysis Of Fractional-order Systems Research Articles

Exploring nonlinearity in quarter car models with an experimental approach to formulating fractional order form and its dynamic analysis.

This study explores the inherent nonlinearity of quarter car models by employing an experimental and numerical approach. The dynamics of vehicular suspension systems are pivotal for ensuring passenger comfort, vehicle stability, and overall ride quality. In this paper we assessed the impact of various parameters and components on suspension performance, enabled the optimization of ride comfort, stability, and handling characteristics. Firstly, experimental analysis allowed for the investigation of factors that are challenging to model theoretically, such as stiffness nonlinearity and damping characteristics, which may vary under different operating conditions. Time domain and frequency response diagram of the model has been obtained. Secondly, a quarter-car with single degree-of-freedom presented and investigated in fractional order form. Fractional order dynamics emphasize nonlinearities in quarter car models, capturing real-world dynamics effectively. The proposed fractional-order nonlinear quarter car model employed Caputo derivative. For numerical analysis of fractional order system, the Adam-Bashforth-Moulton method is used and the disturbance of road assumed to be stochastic. Results show that the dynamic response of the vehicle can be chaotic. Influence of road roughness amplitude and frequency on vehicle vibration is investigated.

Stability analysis of fractional difference equations with delay.

Long-term memory is a feature observed in systems ranging from neural networks to epidemiological models. The memory in such systems is usually modeled by the time delay. Furthermore, the nonlocal operators, such as the "fractional order difference," can also have a long-time memory. Therefore, the fractional difference equations with delay are an appropriate model in a range of systems. Even so, there are not many detailed studies available related to the stability analysis of fractional order systems with delay. In this work, we derive the stability conditions for linear fractional difference equations with an arbitrary delay τ and even for systems with distributed delay. We carry out a detailed stability analysis for the cases of single delay with τ=1 and τ=2. The results are extended to nonlinear maps. The formalism can be easily extended to multiple time delays.

Analysis of fractional order systems using newton iteration-based approximation technique

<span lang="EN-US">Fractional differential equations play a major role in expressing mathematically the real-world problems as they help attain good fit to the experimental data. It is also known that fractional order controllers are more flexible than integer order controllers. But when it comes to the numerical approximation of fractional order functions inaccuracies arise if the conversion technique is not chosen properly. So, when a fractional order plant model is approximated to an integer order system, it is required that the approximated model be accurate, as the overall system performance is based on the estimated integer order model. Nitisha-Pragya-Carlson (NPC) is a recent approximation technique proposed in 2018 to derive the rational approximation of fractional order differ-integrators. In this paper, three fractional order plant models having fractional powers 3.1, 1.25 and 1.3 is analyzed in frequency domain in terms of magnitude and phase response. The performance of approximated third and second order NPC based integer model is studied and compared with the integer models developed using other existing technique. The approximation error is calculated by comparing the frequency response of the developed models with the ideal response. It has been found that in all the three examples NPC based models are very much close to the ideal values. Hence proving the efficacy of NPC technique in approximation of fractional order systems.</span>

LMI-based stability analysis of fractional order systems of neutral type with time varying delays under actuator saturation

LMI-based stability analysis of fractional order systems of neutral type with time varying delays under actuator saturation

Stability analysis of fractional-order systems with randomly time-varying parameters

This paper is concerned with the stability of fractional-order systems with randomly timevarying parameters. Two approaches are provided to check the stability of such systems in mean sense. The first approach is based on suitable Lyapunov functionals to assess the stability, which is of vital importance in the theory of stability. By an example one finds that the stability conditions obtained by the first approach can be tabulated for some special cases. For some complicated linear and nonlinear systems, the stability conditions present computational difficulties. The second alternative approach is based on integral inequalities and ingenious mathematical method. Finally, we also give two examples to demonstrate the feasibility and advantage of the second approach. Compared with the stability conditions obtained by the first approach, the stability conditions obtained by the second one are easily verified by simple computation rather than complicated functional construction. The derived criteria improve the existing related results.

A Novel Stability Criteria of a Class Nonlinear Fractional-order HIV-1 System with Multiple Delay

This paper mainly deduces a new stability criteria of the fractional-order HIV-1 system with delay on the basis of Wirtinger inequality, fractional-order Lyapunov method and integral mean value theorem. The Wirtinger inequality is rarely applied to stability analysis of fractional-order system. Nevertheless, this paper extends the general form of the Lyapunov-krasovskii function to a novel fractional expression form by applying definition of Caputo fractional derivative. Via the the integral mean value theorem, fractional-order Lyapunov method and Wirtinger inequality, the novel stability criteria is deduced. It is the integral mean value theorem that reduces the conservatism of the stability criteria. The simulation results show that the proposed criteria can satisfy different fractional-order operators.In addition, it can not only solve the stability problem of fractional-order HIV-1 system with the constant time delay, but also of the fractional-order HIV-1 system with time-varying time delay. Thus, the new stability criteria has generality and universality. So as to verify our theoretical results, many numerical simulations are provided.

Some Remarks on Estimate of Mittag-Leffler Function

The estimate of Mittag-Leffler function has been widely applied in the dynamic analysis of fractional-order systems in some recently published papers. In this paper, we show that the estimate for Mittag-Leffler function is not correct. First, we point out the mistakes made in the estimation process of Mittag-Leffler function and provide a counterexample. Then, we propose some sufficient conditions to guarantee that part of the estimate for Mittag-Leffler function is correct. Meanwhile, numerical examples are given to illustrate the validity of the two newly established estimates.

Fractional Order Simple Chaotic Oscillator with Saturable Reactors and Its Engineering Applications

Chaos is often observed in forced nonlinear electrical circuits with saturable reactors. Simplest of such a circuit is presented and investigated in both integer order and non-integer (fractional) order form. For numerical analysis of fractional order system we used Adam-Bashforth-Moulton method and investigated the various dynamic properties of the system. For implementing the system in field programmable gate arrays (FPGA) we used the Adomian decomposition method. Power and resource utilization of the fractional order system for implementing in FPGA are presented. Also, in this work, a new fractional based sound encryption application has been implemented as a unique application.

The Role of FGATool in Fractional Order System Analysis Education

The scenario for engineering education is changing by the effect of theoretical developments in recent years. Appropriate usage of computer technology gives great opportunity to follow these changes and provides student achievement. This paper presents FGATool, a Matlab based interface developed to take role in fractional order system analysis education. Fractional order calculus can be an unfamiliar concept for students, thus it is an indisputable fact that some analysis tools will be useful in the education process. The tool is experienced in classes of master students and the outcomes are recorded. Two main modules of FGATool are presented with their mathematical background all over the paper. Analysis of fractional order systems with fixed and uncertain parameters separately can be realized without complicated technical background. FGATool is also thought to be useful with its user-friendly interface.

Hurwitz stability analysis of fractional order LTI systems according to principal characteristic equations

With power mapping (conformal mapping), stability analyses of fractional order linear time invariant (LTI) systems are carried out by consideration of the root locus of expanded degree integer order polynomials in the principal Riemann sheet. However, it is essential to show the left half plane (LHP) stability analysis of fractional order characteristic polynomials in the s plane in order to close the gap emerging in stability analyses of fractional order and integer order systems. In this study, after briefly discussing the relation between the characteristic root orientations and the system stability, the author presents a methodology to establish principal characteristic polynomials to perform the LHP stability analysis of fractional order systems. The principal characteristic polynomials are formed by factorizing principal characteristic roots. Then, the LHP stability analysis of fractional order systems can be carried out by using the root equivalency of fractional order principal characteristic polynomials. Illustrative examples are presented to explain how to find equivalent roots of fractional order principal characteristic polynomials in order to carry out the LHP stability analyses of fractional order nominal and interval systems.

Convex Lyapunov functions for stability analysis of fractional order systems

This study presents an inequality which can be used to analyse the stability of fractional order systems by constructing Lyapunov functions. By using the presented inequality, it is shown that the fractional order system is Mittag-Leffler stable if there is a convex and positive definite function such that its fractional order derivative is negative definite. This result generalises the existing works and gives a useful method to construct the Lyapunov function for the stability analysis of the fractional order systems. Finally, the authors illustrate the advantages of the proposed method by two examples and their numerical simulation.

New power law inequalities for fractional derivative and stability analysis of fractional order systems

Three new power law inequalities for fractional derivative are proposed in this paper. We generalize the original useful power law inequality, which plays an important role in the stability analysis of pseudo state of fractional order systems. Moreover, three stability theorems of fractional order systems are given in this paper. The stability problem of fractional order linear systems can be converted into the stability problem of the corresponding integer order systems. For the fractional order nonlinear systems, a sufficient condition is obtained to guarantee the stability of the true state. The stability relation between pseudo state and true state is given in the last theorem by the final value theorem of Laplace transform. Finally, two examples and numerical simulations are presented to demonstrate the validity and feasibility of the proposed theorems.

Controllability and observability analysis of continuous-time multi-order fractional systems

This paper proposes some analytical criteria for controllability and observability analysis of fractional order systems describing by multi-order state space equations. The controllability and observability gramians are presented in which their non-singularity is equivalent to controllability and observability of the mentioned systems. Moreover, to overcome the gramian calculation complexities, the controllability and observability matrices are introduced. It is proved that the sufficient condition for controllability or observability of a multi-order fractional system is that its controllability or observability matrix is nonsingular. Some illustrative examples are given to show the efficiency of the proposed method.

Stability Analysis of Fractional Order Systems with Time Delay

In this paper, we mainly study the stability of linear and interval linear fractional systems with time delay. By applying the characteristic equations, a necessary and sufficient stability condition is obtained firstly, and then some sufficient conditions are deserved. In addition, according to the equivalent relationship of fractional order systems with order 0 < α ≤ 1 and with order 1 ≤ β < 2, one may get more relevant theorems. Finally, two examples are provided to demonstrate the effectiveness of our results. Keywords—Fractional order systems, Time delay, Characteristic

Fractional order control of unstable processes: the magnetic levitation study case

Although a considerable amount of research has been carried out in the field of fractional order controllers, the majority of the results deal with stable processes. Very little research has been reported regarding the design, analysis, and tuning of fractional order controllers for unstable processes. This paper proposes a methodology for designing and tuning fractional order controllers for a class of unstable second-order processes. The design is carried out using the stability analysis of fractional order systems, by means of Riemann surfaces and a proper mapping in the \(w{\text {-}}\hbox {plane}\). The resulting fractional order controllers are implemented using graphical programming on industrial equipment and are validated experimentally using a laboratory scale magnetic levitation unit.

Stability Analysis of Fractional Order Systems with Time Delay

Stability Analysis of Fractional Order Systems with Time Delay

Stability Analysis of Fractional Order Systems with Time Delay

Stability Analysis of Fractional Order Systems with Time Delay

Stability Analysis of Fractional Order Systems with Time Delay

Stability Analysis of Fractional Order Systems with Time Delay

Stability Analysis of Fractional Order Systems with Time Delay

Stability Analysis of Fractional Order Systems with Time Delay

Stability Analysis of Fractional Order Systems with Time Delay

Stability Analysis of Fractional Order Systems with Time Delay