Symmetric differential operators of fractional order and their extensions

2018 ◽  
Vol 79 ◽  
pp. 177-185
Author(s):  
N. E. Tokmagambetov ◽  
B. T. Torebek
Keyword(s):  
Fractional Order ◽  
Differential Operators ◽  
Symmetric Differential Operators

scholarly journals Characterization of Self-Adjoint Domains for Two-Interval Even Order Singular C-Symmetric Differential Operators in Direct Sum Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Qinglan Bao ◽  
Xiaoling Hao ◽  
Jiong Sun
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Direct Sum ◽  
Even Order ◽  
Special Case ◽  
Symmetric Differential Operators

This paper is concerned with the characterization of all self-adjoint domains associated with two-interval even order singular C-symmetric differential operators in terms of boundary conditions. The previously known characterizations of Lagrange symmetric differential operators are a special case of this one.

scholarly journals Balance Adjustment of Power-Line Inspection Robot Using General Type-2 Fractional Order Fuzzy PID Controller

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 479
Author(s):  
Yao Chen ◽  
Tao Zhao ◽  
Songyi Dian ◽  
Xiaodong Zeng ◽  
Haipeng Wang
Keyword(s):  
Fractional Order ◽  
Pid Controller ◽  
General Type ◽  
Fuzzy Logic Controller ◽  
Differential Operators ◽  
Power Line ◽  
State Variables ◽  
Fuzzy Pid ◽  
Robot System

In this study, a general type-2 fractional order fuzzy PID (GT2FO-FPID) controller is proposed to fulfil the balance adjustment of the Power-line Inspection (PLI) robot system. It is a combination of Mamdani general type-2 fuzzy logic controller (GT2-FLC) and fractional PID controller. Since the PLI robot system is an under-actuated system, it’s necessary to get complete information of the system. However, when all state variables are treated as input to the controller, there is a problem with the rule explosion. Because of this, the information fusion method is adopt to solve the problem and simplify the controller design. At the same time, fractional-order integral-differential operators and input-output scaling factors, which are taken as design variables and optimized by genetic algorithm (GA). To assess the performance of proposed controller based on symmetry criterion, we compared it against existing controllers, i.e., interval type-2 fractional order fuzzy PID (IT2FO-FPID), type-1 fractional order fuzzy PID (T1FO-FPID), and conventional fractional order (FOPID) controllers. Furthermore, to show the anti-inference ability of the proposed controller, three common perturbed process are tested. Finally, simulation results show that the GT2FO-FPID controller outperforms other controllers in the presence of external perturbations on the PLI robot system.

scholarly journals New Improved Fractional Order Differentiator Models Based on Optimized Digital Differentiators

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Maneesha Gupta ◽  
Richa Yadav
Keyword(s):  
Fractional Order ◽  
Differential Operators ◽  
Fitness Function ◽  
Hybrid Optimization ◽  
Nyquist Frequency ◽  
Swarm Optimization ◽  
Optimization Simulation ◽  
Fractional Order Differentiator ◽  
Simulation Results ◽  
The Ideal

Different evolutionary algorithms (EAs), namely, particle swarm optimization (PSO), genetic algorithm (GA), and PSO-GA hybrid optimization, have been used to optimize digital differential operators so that these can be better fitted to exemplify their new improved fractional order differentiator counterparts. First, the paper aims to provide efficient 2nd and 3rd order operators in connection with process of minimization of error fitness function by registering mean, median, and standard deviation values in different random iterations to ascertain the best results among them, using all the abovementioned EAs. Later, these optimized operators are discretized for half differentiator models for utilizing their restored qualities inhibited from their optimization. Simulation results present the comparisons of the proposed half differentiators with the existing and amongst different models based on 2nd and 3rd order optimized operators. Proposed half differentiators have been observed to approximate the ideal half differentiator and also outperform the existing ones reasonably well in complete range of Nyquist frequency.

Limit Point and Limit Circle Criteria for a Class of Singular Symmetric Differential Operators

1976 ◽  
Vol 28 (5) ◽  
pp. 905-914 ◽  
Author(s):  
Robert L. Anderson
Keyword(s):  
Limit Point ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Limit Circle ◽  
Image Position ◽  
Point Type ◽  
Symmetric Differential Operators

For certain classes of singular symmetric differential operators L of order 2n, this paper considers the problem of determining sufficient conditions for L to be of limit point type or of limit circle type. The operator discussed here is defined by

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