A New Fractional Integration Operational Matrix of Chebyshev Wavelets in Fractional Delay Systems
Abstract
:1. Introduction
2. Preliminaries
Useful Properties of CWs
- can be expressed in terms of by the integration operational matrix of CWs denoted by as
- the product operational matrix of CWs for denoted by the symbol simplifies to
- and , where is a time-delay and is a piecewise delay, are expressed by the delay and the piecewise delay operational matrices of CWs denoted by the symbols, in turn, and , where
- is expressed by the inverse (reverse) time operational matrix of CWs denoted by as
- is obtained as the integration matrix of the product of two CWs vectors on denoted by , that is,
3. The Fractional Integration Operational Matrix of CWs
4. Chebyshev Wavelet Methods for Fractional Delay Systems
4.1. Optimal Control of Linear-Quadratic Fractional Time-Delay Systems
4.2. Analysis of Linear Fractional Time-Delay Systems
5. Illustrative Examples
5.1. Example 1
- Case 1:
- Case 2:
5.2. Example 2
5.3. Example 3
5.4. Example 4
5.5. Example 5
5.6. Example 6
- (1)
- We use the proposed method directly on this system;
- (2)
- By the technique used in [27], we first select the first derivative of as a new state which is , and set , then we solve the new problem as
- Case 1:
- and .
- Case 2:
- and .
5.7. Example 7
- Case 1:
- and the system is unconstrained.
- Case 2:
- and the path constraint is .
- Case 3:
- and the path constraint is .
6. Conclusions
Funding
Acknowledgments
Conflicts of Interest
References
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This Work; | [28]; | [34]; | [18] | [19] | [20] | |
---|---|---|---|---|---|---|
1 | 0.37311293528 | 0.373112935096 | 0.373112935279 | 0.01451 | 0.04553 | 0.37311264 |
0.999 | 0.37302124305 | 0.01450 | ||||
0.99 | 0.37219761493 | 0.01436 | 0.3721964 | |||
0.95 | 0.36856850562 | 0.3685506 | ||||
0.9 | 0.36409192174 | 0.01336 | 0.3640344 | |||
0.89 | 0.36320335165 | |||||
0.8 | 0.35528976948 | 0.01314 | 0.3551193 | |||
0.7 | 0.34662823700 | 0.3463065 | ||||
0.5 | 0.32938391796 |
, Strategy 1 | , Strategy 2 | |
---|---|---|
1.001 | 0.34564307806 | 0.34564144998 |
1.01 | 0.34638051376 | 0.34636577766 |
1.05 | 0.34965701531 | 0.34961784656 |
1.1 | 0.35375919815 | 0.35376402419 |
1.11 | 0.35458119323 | 0.35460259170 |
Case 1 | Case 2 | |||||
---|---|---|---|---|---|---|
1 | 0.38129264275 | 0.0499999578 | 1 | 0.37958583678 | 0.0499999519 | 0.0553443427 |
0.999 | 0.38121931108 | 0.0499999578 | 0.999 | 0.37950598319 | 0.0499999519 | 0.0553602137 |
0.99 | 0.38056110670 | 0.0499999583 | 0.99 | 0.37878828547 | 0.0499999523 | 0.0555045828 |
0.9 | 0.37412772028 | 0.0499999624 | 0.9 | 0.37167971572 | 0.0499999560 | 0.0570967526 |
0.8 | 0.36721704588 | 0.0499999659 | 0.8 | 0.36383702347 | 0.0499999589 | 0.0591798411 |
This Work, | [28] | [34] | [20] | [37] | [38] | ||
---|---|---|---|---|---|---|---|
0 | 1 | 4.79679791916 | 4.79679791913 | 4.79679870920 | 4.79679868 | 4.7968 | 4.796817 |
0 | 0.999 | 4.79697117915 | |||||
0 | 0.99 | 4.79853220259 | 4.7766443 | ||||
0 | 0.95 | 4.80544625813 | 4.6907801 | ||||
0 | 0.9 | 4.81377758646 | 4.5728139 | ||||
0 | 0.8 | 4.82825064845 | 4.3096610 | ||||
0 | 0.7 | 4.83888850210 | 4.0256671 | ||||
0 | 0.5 | 4.84814116845 | |||||
1 | 0.9 | 5.27371052428 | |||||
1 | 0.99 | 5.24118253395 | |||||
1 | 0.999 | 5.23791614199 | |||||
1 | 1 | 5.23755370619 | 5.23755370619 | 5.23755466744 | |||
1 | 1.001 | 5.49646031081 | |||||
1 | 1.01 | 5.49452438269 | |||||
1 | 1.1 | 5.46836520817 | |||||
1 | 1.2 | 5.42892272878 | |||||
1 | 1.3 | 5.38170076246 |
This Work; , | [20] | [18] | [19] | [17] | |
---|---|---|---|---|---|
1 | 1.647874 | 1.64787419 | 0.4727464 | 0.00002674 | 0.3048 |
0.99 | 1.648911 | 1.6459912 | 0.4778890 | ||
0.9 | 1.658451 | 1.6248785 | 0.5021900 | ||
0.8 | 1.669404 | 1.5926486 | 0.4985242 | ||
0.7 | 1.680951 | 1.5519859 | |||
0.5 | 1.708245 | 0.00186172 |
This Work, | [39] | |
---|---|---|
1 | 74.1065868949 | 74.1173 |
0.99 | 75.4717293676 | |
0.98 | 76.8343875120 | |
0.97 | 78.1913320078 | |
0.96 | 79.5404056247 | |
0.95 | 80.8802694101 |
This Work | [31] | |||
---|---|---|---|---|
0 | 1 | 1.56224137355 | 1.56224137354 | |
1 | 1 | 0.999 | 1.41013747158 | |
1 | 1 | 0.99 | 1.41106062244 | |
1 | 1 | 0.95 | 1.41378641071 | |
1 | 1 | 0.91 | 1.41668866306 | |
1 | 1 | 0.9 | 1.41740062973 | |
1 | 1 | 0.8 | 1.42442691113 |
t | This Work; | [30] |
---|---|---|
0.1 | 0.00238 | 0.00230 |
0.3 | 0.00194 | 0.00342 |
0.00231 | 0.00795 | |
0.00027 | 0.00182 | |
0.7 | 0.00015 | 0.00053 |
0.9 | 0.00024 | 0.00092 |
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Malmir, I. A New Fractional Integration Operational Matrix of Chebyshev Wavelets in Fractional Delay Systems. Fractal Fract. 2019, 3, 46. https://doi.org/10.3390/fractalfract3030046
Malmir I. A New Fractional Integration Operational Matrix of Chebyshev Wavelets in Fractional Delay Systems. Fractal and Fractional. 2019; 3(3):46. https://doi.org/10.3390/fractalfract3030046
Chicago/Turabian StyleMalmir, Iman. 2019. "A New Fractional Integration Operational Matrix of Chebyshev Wavelets in Fractional Delay Systems" Fractal and Fractional 3, no. 3: 46. https://doi.org/10.3390/fractalfract3030046