Fractional Order Modeling And Nonlinear Fractional Order Pi‐Type Control For PMLSM System

Abstract

In this paper, firstly a fractional order (FO) model is proposed for the speed control of a permanent magnet linear synchronous motor (PMLSM) servo system. To identify the parameters of the FO model, a practical modeling algorithm is presented. The algorithm is based on a pattern search method and its effectiveness is verified by real experimental results. Second, a new fractional order proportional integral type controller, that is, (PIμ)λ or FO[FOPI], is introduced. Then a tuning methodology is presented for the FO[FOPI] controller. In this tuning method, the controller is designed to satisfy four design specifications: stability requirement, specified gain crossover frequency, specified phase margin, flat phase constraint, and minimum integral absolute error. Both set point tracking and load disturbance rejection cases are considered. The advantages of the tuning method are that it fully considers the stability requirement and avoids solving a complex nonlinear optimization problem. Simulations are conducted to verify the effectiveness of the proposed FO[FOPI] controller over classical FOPI and FO[PI] controllers.

New Results on Stabilization of Fractional‐Order Nonlinear Systems via an LMI Approach

“…Moreover, many systems modelled with the help of fractional calculus display rich fractional dynamical behavior, such as viscoelastic systems [10], boundary layer effects in ducts [11], electromagnetic waves [12], fractional kinetics [13,14], and electrode-electrolyte polarization [15,16]. Many stability conditions have been proposed for linear fractional-order systems [17][18][19][20][21][22][23], fractional-order nonlinear systems [24][25][26][27][28][29][30], fractional-order neural networks [31,32], fractional-order switched linear systems [33,34], fractional-order singular systems [35,36], positive fractional-order systems [37,38] and fractional chaotic complex networks systems [39]. Many stability conditions have been proposed for linear fractional-order systems [17][18][19][20][21][22][23], fractional-order nonlinear systems [24][25][26][27][28][29][30], fractional-order neural ne...…”

“…As a result, the problem of stability analysis and control of fractional-order systems is an important problem in the theory and applications of fractional calculus. Many stability conditions have been proposed for linear fractional-order systems [17][18][19][20][21][22][23], fractional-order nonlinear systems [24][25][26][27][28][29][30], fractional-order neural networks [31,32], fractional-order switched linear systems [33,34], fractional-order singular systems [35,36], positive fractional-order systems [37,38] and fractional chaotic complex networks systems [39]. Among the reported methods, the Lyapunov direct method provides an effective approach to analyze the stability of fractional nonlinear systems.…”

Modelling and simulation of nabla fractional dynamic systems with nonzero initial conditions

“…This completes the proof. 2 It is noteworthy that if the nabla Laplace transform F (s) is given, f (k) can be calculated via Theorem 1. Sometimes, it is difficult to solve such a contour integral problem.…”

“…Fractional calculus is a natural generalization of classical integer order calculus, whose inception can be traced back to 300 years ago. It is well known that fractional calculus has been widely applied to system modelling [1,2], stability analysis [3,4], controller synthesis [5,6], and optimization algorithm [7,8], etc. Due to the great efforts devoted by researchers, a large number of valuable results have been reported on fractional calculus.…”

Bifurcation Analysis and Stability Criterion for the Nonlinear Fractional‐Order Three‐Dimensional Financial System with Delay

“…However, its limitations mean that the general integer order financial system is not an adequate reflection of the overall financial world. However, the fractional differential operator has global correlation or nonlocal characteristics , and thus is more suitable to describe complex financial behavior . Moreover, fractional calculus theory has unique advantages .…”