A unified approach for realization of discrete-time fractional-order proportional derivative controller for two classes of plant models

Bhanita Adhikary; Jaydeep Swarnakar

doi:10.3934/mcrf.2024027

Article Contents

2025, Volume 15, Issue 2: 590-614. Doi: 10.3934/mcrf.2024027

This issue Previous Article Local Halanay's inequality with application to feedback stabilization Next Article Null controllability for backward stochastic parabolic convection-diffusion equations with dynamic boundary conditions

A unified approach for realization of discrete-time fractional-order proportional derivative controller for two classes of plant models

Bhanita Adhikary^1, and
Jaydeep Swarnakar^1, ,

1.
North-Eastern Hill University, India

^*Corresponding author: Jaydeep Swarnakar
^*Corresponding author: Jaydeep Swarnakar

Received: June 2023

Revised: April 2024

Early access: June 20, 2024

Published online: June 20, 2024

Abstract / Introduction Full Text(HTML) Figure(22) / Table(5) Related Papers Cited by

Abstract

In this work, a method has been proposed for the realization of a discrete-time fractional-order proportional derivative (FOPD) controller for two different classes of plant models. The methodology has been developed in two stages. In the first stage, the continuous-time FOPD controller has been designed from some of the standard frequency domain specifications employing a graphical approach, namely a vector model method. In the second stage, a generating function has been framed in a delta domain using alpha approximation or parameterized Al-Alaoui approximation. The proposed generating function is named the Modified Al Alaoui-Delta (MALD) approximation, which has been expanded using the continued fraction expansion (CFE) method to realize the discrete-time FOPD controller. The discrete-time realization of the controller has been executed in a delta domain instead of a traditional $ z $-domain, as the delta operator offers unification between a discrete-time system and its continuous-time counterpart at the low sampling time limit. The efficacy of the proposed controller realization method, has been justified over some of the existing methods taking suitable examples from the literature.

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Mathematics Subject Classification: Primary: 93C55, 93C62; Secondary: 47N70.

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Figure 1. Vector model of the FOPD controller

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Figure 2. Plot of $ k_d $ versus $ \beta $ (Example 1)

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Figure 3. Frequency response comparison between $ C_1(s) $ and $ C_{\delta1}(\gamma) $ at $ \Delta = 0.001 $ sec

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Figure 4. Frequency response comparison between $ G_{OL1}(s) $ and $ G_{OL\delta1}(\gamma) $ at $ \Delta = 0.001 $ sec

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Figure 5. Comparison of frequency responses obtained from ideal open loop system and its two approximants at $ \Delta = 0.5 $ sec (Example 1)

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Figure 6. Comparison of frequency responses obtained from ideal open loop system and its two approximants at $ \Delta = 0.01 $ sec (Example 1)

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Figure 7. Comparison of frequency responses obtained from ideal open loop system and its two approximants at $ \Delta = 0.001 $sec (Example 1)

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Figure 8. Comparison of frequency responses obtained from ideal open loop system and its three approximants at $ \Delta = 0.005 $ sec (Example 1)

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Figure 9. Frequency responses of $ G_{OL\delta1}(\gamma) $ for different values of plant gain at $ \Delta = 0.001 $ sec

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Figure 10. (a) Comparison of step responses obtained from an ideal closed loop system and its three approximants at $ \Delta = 0.005 $ sec (Example 1). (b) Step responses of the closed loop system for different values of plant gain at $ \Delta = 0.001 $ sec (Example 1)

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Figure 11. Plot of $ k_d $ versus $ \beta $ (Example 2)

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Figure 12. Frequency response comparison between $ C_{2}(s) $ and $ C_{\delta2}(\gamma) $ at $ \Delta = 0.001 $ sec

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Figure 13. Frequency response comparison between $ G_{OL2}(s) $ and $ G_{OL\delta2}(\gamma) $ at $ \Delta = 0.001 $ sec

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Figure 14. Comparison of frequency responses obtained from ideal open loop system and its two approximants at $ \Delta = 0.5 $ sec (Example 2)

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Figure 15. Comparison of frequency responses obtained from ideal open loop system and its two approximants at $ \Delta = 0.01 $sec (Example 2)

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Figure 16. Comparison of frequency responses obtained from the ideal open loop system and its two approximants at $ \Delta = 0.001 $ sec (Example 2)

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Figure 17. Comparison of frequency responses obtained from the ideal open loop system and its three approximants at $ \Delta = 0.005 $ sec (Example 2)

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Figure 18. Frequency responses of $ G_{OL\delta2}(\gamma) $ for different values of plant gain at $ \Delta = 0.001 $ sec (Example 2)

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Figure 19. (a) Comparison of step responses obtained from an ideal closed loop system and its three approximants at $ \Delta = 0.01 $ sec (Example 2.) (b) Step responses of the closed loop system for different values of plant gain at $ \Delta = 0.001 $ sec (Example 2)

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Figure 20. Frequency response comparison between $ C_{3}(s) $ and $ C_{\delta 3}(\gamma) $ at $ \Delta = 0.001 $ sec (Example 3)

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Figure 21. Frequency response comparison between $ G_{OL3}(s) $ and $ G_{OL\delta 3}(\gamma) $ at $ \Delta = 0.001 $ sec (Example 3)

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Figure 22. Step response plot for the closed loop system of Luo et.al. [13] at $ \Delta = 0.01 $ sec (Example 3)

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Table 1. Numerator and Denominator coefficients for fifth-order approximation of $ s^\beta $

$ M_0=32768 C_0+ 28672\Delta C_1 +25088 \Delta^2 C_2+ 21952 \Delta^3 C_3+ 19208 \Delta^4 C_4+ 16807 \Delta^5 C_5 $
$ M_1=32768 C_1+ 57344\Delta C_2 + 75264 \Delta^2 C_3+ 87808 \Delta^3 C_4+ 96040 \Delta^4 C_5 $
$ M_2=32768 C_2+ 86016\Delta C_3 + 150528 \Delta^2 C_4+ 219520 \Delta^3 C_5 $
$ M_3=32768 C_3+ 114688\Delta C_4 + 250880 \Delta^2 C_5 $
$ M_4=32768 C_4+ 143360\Delta C_5 $
$ M_5=32768 C_5 $
$ N_0=32768 C_5+ 28672\Delta C_4 +25088 \Delta^2 C_3+ 21952 \Delta^3 C_2+ 19208 \Delta^4 C_1+ 16807 \Delta^5 C_0 $
$ N_1=32768 C_4+ 57344\Delta C_3 + 75264 \Delta^2 C_2+ 87808 \Delta^3 C_1+ 96040 \Delta^4 C_0 $
$ N_2=32768 C_3+ 86016\Delta C_2+ 150528 \Delta^2 C_1+ 219520 \Delta^3 C_0 $
$ N_3=32768 C_2+ 114688\Delta C_1 + 250880 \Delta^2 C_0 $
$ N_4=32768 C_1+ 143360\Delta C_0 $
$ N_5=32768 C_0 $
$ C_0=- \left(\beta^5+ 15\beta^4+ 85\beta^3+ 225\beta^2 +274\beta+ 120\right) $
$ C_1= 5\beta^5+ 45\beta^4+ 5\beta^3 -1005\beta^2 -3250\beta -3000 $
$ C_2=- \left(10\beta^5+ 30\beta^4- 410\beta^3- 1230\beta^2 +4000\beta+ 12000\right) $
$ C_3=10\beta^5- 30\beta^4 -410\beta^3+ 1230\beta^2 +4000\beta- 12000 $
$ C_4=- \left(5\beta^5- 45\beta^4+ 5\beta^3+ 1005\beta^2 -3250\beta+ 3000\right) $
$ C_5= \beta^5- 15\beta^4+ 85\beta^3- 225\beta^2 +274\beta- 120 $

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Table 3. Discrete-time OLTFs at different sampling time limits (Example 1)

Sampling Time $ (\Delta) $	$ G_{OL\delta 1}(\gamma) $
$ \Delta=0.5 $ sec	$ \frac{\begin{array}{c} 2.902 \times 10^{10} \gamma^7 +2.71 \times 10^{11} \gamma^6+ 1.015 \times 10^{12}\gamma^5\\ + 1.942 \times10^{12} \gamma^4 +1.99 \times10^{12} \gamma^3 + 1.044 \times10^{12}\gamma^2\\ + 2.395 \times 10^{11}\gamma+ 1.584 \times 10^{10} \end{array}}{\begin{array}{c} 6.063 \times 10^{9}\gamma^7 + 3.983 \times 10^{10}\gamma^6+ 9.951 \times 10^{10} \gamma^5\\ +1.183 \times 10^{11} \gamma^4 + 6.795 \times 10^{10} \gamma^3\\ + 1.651 \times 10^{10} \gamma^2+ 1.128 \times 10^9\gamma \end{array}} $
$ \Delta=0.01 $ sec	$ \frac{\begin{array}{c} 3.769 \times 10^{5} \gamma^7 +9.25 \times 10^{7} \gamma^6+ 6.401 \times 10^{9}\gamma^5\\ + 8.921 \times10^{10} \gamma^4 + 3.685 \times10^{11} \gamma^3 + 4.891 \times10^{11}\gamma^2\\ + 1.92 \times 10^{11}\gamma+ 1.584 \times 10^{10} \end{array}}{\begin{array}{c} 1.406 \times 10^{7}\gamma^7 + 1.034 \times 10^{9}\gamma^6+ 9.802 \times 10^{9} \gamma^5\\ +3.038 \times 10^{10} \gamma^4 + 3.567 \times 10^{10} \gamma^3\\ + 1.361 \times 10^{10} \gamma^2+ 1.128 \times 10^{9}\gamma \end{array}} $
$ \Delta=0.001 $ sec	$ \frac{\begin{array}{c} 3286 \gamma^7 +7.571 \times 10^{6} \gamma^6+ 4.428 \times 10^{9}\gamma^5\\ + 7.791 \times10^{10} \gamma^4+ 3.494 \times10^{12} \gamma^3 +4.8 \times10^{11}\gamma^2\\ + 1.911 \times 10^{11}\gamma+ 1.584 \times 10^{10} \end{array}}{\begin{array}{c} 6.503 \times 10^{6}\gamma^7 + 8.856 \times 10^{8}\gamma^6+ 9.097 \times 10^{9} \gamma^5\\ +2.926 \times 10^{10} \gamma^4+ 3.513 \times 10^{10} \gamma^3\\ + 1.356 \times 10^{10} \gamma^2+ 1.128 \times 10^9\gamma \end{array}} $

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Table 2. Frequency domain specifications resulted from continuous-time FOPD and discrete-time FOPD (Example1)

Design Criterions	Continuous-time domain	Discrete-time domain
Gain crossover frequency	$ \omega_g=8 $ rad/sec	$ \omega_g=7.975 $ rad/sec
Phase margin	$ \phi_m=60^\circ $	$ \phi_m=60.235^\circ $
Flat phase around $ \omega_g $	Satisfied	Satisfied

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Table 4. Frequency domain specifications resulting from continuous-time FOPD and discrete-time FOPD (Example 2)

Design Criterions	Continuous-time domain	Discrete-time domain
Gain crossover frequency	$ \omega_g=2 $ rad/sec	$ \omega_g=1.9 $ rad/sec
Phase margin	$ \phi_m=65^\circ $	$ \phi_m=65.8^\circ $
Flat phase around $ \omega_g $	Satisfied	Satisfied

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Table 5. Discrete-time OLTFs at different sampling times (Example 2)

Sampling Time $ (\Delta) $	$ G_{OL\delta 1}(\gamma) $
$ \Delta=0.5 $ sec	$ \frac{\begin{array}{c} 9.297 \times 10^{18} \gamma^{13} + 1.428 \times 10^{20} \gamma^{12}+ 9.72 \times 10^{20}\gamma^{11}\\ + 3.861 \times10^{21} \gamma^{10} + 9.922 \times10^{21} \gamma^9 + 1.728 \times10^{22}\gamma^8\\ + 2.079 \times 10^{22}\gamma^7 +1.727 \times 10^{22} \gamma^6 + 9.734 \times 10^{21}\gamma^5\\ + 3.6 \times10^{21} \gamma^4 + 8.244 \times10^{20} \gamma^3 + 1.063 \times10^{20}\gamma^2\\ + 6.551 \times 10^{18}\gamma+ 1.45 \times 10^{17} \end{array}}{\begin{array}{c} 1.163 \times 10^{19} \gamma^{13} + 1.447 \times 10^{20} \gamma^{12}+ 7.839 \times 10^{20}\gamma^{11}\\ + 2.424 \times10^{21} \gamma^{10} + 4.722 \times10^{21} \gamma^9 + 6.039 \times10^{21}\gamma^8\\ + 5.123 \times 10^{21}\gamma^7 + 2.852 \times 10^{21}\gamma^6 + 1.01 \times 10^{21} \gamma^5\\ +2.152 \times 10^{20} \gamma^4 + 2.515 \times 10^{19} \gamma^3\\ + 1.367 \times 10^{18} \gamma^2+ 2.69 \times 10^{16}\gamma \end{array}} $
$ \Delta=0.01 $ sec	$ \frac{\begin{array}{c} 5.56 \times 10^{11} \gamma^{13} + 1.63 \times 10^{14} \gamma^{12}+ 1.623 \times 10^{16}\gamma^{11}\\ + 6.485 \times10^{17} \gamma^{10} + 1.141 \times10^{19} \gamma^9 + 9.5 \times10^{19}\gamma^8\\ + 4.174 \times 10^{20}\gamma^7 +1.018 \times 10^{21} \gamma^6 + 1.385 \times 10^{21}\gamma^5\\ + 1.031 \times10^{21} \gamma^4 + 3.992 \times10^{20} \gamma^3 + 7.469 \times10^{19}\gamma^2\\ + 5.742 \times 10^{18}\gamma+ 1.45 \times 10^{17} \end{array}}{\begin{array}{c} 1.093 \times 10^{14} \gamma^{13} + 1.715 \times 10^{16} \gamma^{12}+ 6.047 \times 10^{17}\gamma^{11}\\ + 8.809 \times10^{18} \gamma^{10} + 6.295 \times10^{19} \gamma^9 + 2.404\times10^{20}\gamma^8\\ + 5.069 \times 10^{20}\gamma^7 + 5.886 \times 10^{20}\gamma^6 + 3.704 \times 10^{20} \gamma^5\\ +1.208 \times 10^{20} \gamma^4 + 1.902\times 10^{19} \gamma^3\\ + 1.229 \times 10^{18} \gamma^2+ 2.69 \times 10^{16}\gamma \end{array}} $
$ \Delta=0.001 $ sec	$ \frac{\begin{array}{c} 3.164 \times 10^{9} \gamma^{13} + 7.485 \times 10^{12} \gamma^{12}+ 4.717 \times 10^{15}\gamma^{11}\\ + 3.442 \times10^{17} \gamma^{10}+ 8.038 \times10^{18} \gamma^9+ 7.656 \times10^{19}\gamma^8\\ + 3.636 \times 10^{20}\gamma^7 +9.331 \times 10^{20} \gamma^6+ 1.313 \times 10^{21}\gamma^5\\ + 9.994 \times10^{20} \gamma^4+ 3.928 \times10^{20} \gamma^3+ 7.415 \times10^{19}\gamma^2\\ + 5.728 \times 10^{18}\gamma+ 1.45 \times 10^{17} \end{array}}{\begin{array}{c} 7.685 \times 10^{12} \gamma^{13} + 9.144 \times 10^{15} \gamma^{12}+ 4.189 \times 10^{17}\gamma^{11}\\ + 6.971 \times10^{18} \gamma^{10}+ 5.394 \times10^{19} \gamma^9 + 2.172 \times10^{20}\gamma^8\\ + 4.751 \times 10^{20}\gamma^7 + 5.656 \times 10^{20}\gamma^6+ 3.619 \times 10^{20} \gamma^5\\ +1.193 \times 10^{20} \gamma^4+ 1.892 \times 10^{19} \gamma^3\\ + 1.226 \times 10^{18} \gamma^2+ 2.69 \times 10^{16}\gamma \end{array}} $

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