Fractional Order Models of Dynamic Systems

2020 ◽  
pp. 115-152
Author(s):  
Andrzej Dzieliński ◽  
Grzegorz Sarwas ◽  
Dominik Sierociuk
Keyword(s):  
Fractional Order ◽  
Dynamic Systems

scholarly journals The Microcontroller Implementation of the Basic Fractional-Order Element

2020 ◽  
Vol 24 (4) ◽  
pp. 19-26
Author(s):  
Krzysztof Oprzędkiewicz ◽  
Maciej Rosół ◽  
Jakub Żegleń-Włodarczyk
Keyword(s):  
Control Systems ◽  
Real Time ◽  
Fractional Order ◽  
Dynamic Systems ◽  
Broad Class ◽  
Fractional Order Control ◽  
Linear And Nonlinear ◽  
Hard Real Time ◽  
Order Element ◽  
Accuracy Of Approximation

The paper presents the implementation of the basic fractional order element sγ on the STM32 microcontroller platform. The implementation employs the typical CFE and FOBD approximations, the accuracy of approximation as well as duration of calculations are experimentally tested. Microcontroller implementation of fractional order elements is known; however, real-time tests of such implementations have been not presented yet. Results of experiments show that both methods can be implemented at the considered platform. The FOBD approximation is more accurate, but the CFE one is faster. The presented experimental results prove that the STM32F7 family processor could be used to develop the embedded fractional-order control systems for a broad class of linear and nonlinear dynamic systems. This is crucial during the implementation of the fractional-order control in the hard real-time or embedded systems.

Mittag–Leffler stability of nabla discrete fractional-order dynamic systems

2020 ◽  
Vol 101 (1) ◽  
pp. 407-417
Author(s):  
Yingdong Wei ◽  
Yiheng Wei ◽  
Yuquan Chen ◽  
Yong Wang
Keyword(s):  
Fractional Order ◽  
Dynamic Systems

H ∞ and Sliding Mode Observers for Linear Time-Invariant Fractional-Order Dynamic Systems With Initial Memory Effect

2014 ◽  
Vol 136 (5) ◽  
Author(s):  
Sang-Chul Lee ◽  
Yan Li ◽  
YangQuan Chen ◽  
Hyo-Sung Ahn
Keyword(s):  
Memory Effect ◽  
Fractional Order ◽  
Dynamic Systems ◽  
Sliding Mode ◽  
Linear Time ◽  
Unknown Input ◽  
Unknown Input Observer ◽  
Linear Time Invariant ◽  
Sliding Mode Observers ◽  
Time Invariant

The H∞ and sliding mode observers are important in integer-order dynamic systems. However, these observers are not well explored in the field of fractional-order dynamic systems. In this paper, the H∞ filter and the fractional-order sliding mode unknown input observer are developed to estimate state of the linear time-invariant fractional-order dynamic systems with consideration of proper initial memory effect. As the first result, the fractional-order H∞ filter is introduced, and it is shown that the gain from the noise to the estimation error is bounded in the sense of the H∞ norm. Based on the extended bounded real lemma, the H∞ filter design is formulated in a linear matrix inequality form, and it will be seen that numerical methods to solve convex optimization problems are feasible in fractional-order systems (FOSs). As the second result of this paper, not only state but also unknown input disturbance are estimated by fractional-order sliding-mode unknown input observer, simultaneously. In this paper, it is shown that the design and stability analysis of the two estimation techniques are not related with the initial history. Through two numerical examples, the performance of the fractional-order H∞ filter and the fractional-order sliding-mode observer is illustrated with consideration of the initialization functions.

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