Direct Synthesis of Fractional-Order Controllers Using Only Two Design Equations with Robustness to Parametric Uncertainties
Abstract
1. Introduction
2. Theoretical Background
2.1. Second-Order Canonical Transfer Function
2.2. Fractional Calculus
2.3. Three-Term Fractional-Order Canonical Form
2.4. Controllers
2.4.1. Integer-Order Phase Lead/Lag Compensator
2.4.2. Fractional-Order PID Controller
2.4.3. Fractional-Order ID Controller
2.4.4. Fractional-Order Direct Synthesis Control
2.5. Performance Metrics and Statistical Analysis
2.5.1. Integrated Square Error (ISE)
2.5.2. Integral Square Control Error Performance Index (ISCE)
2.5.3. Monte Carlo Analysis
- % MonteCarlo Analysis
- % Parameter variations and normal distribution
- Kp = 1; % nominal gain
- N = 400; % number of cases
- des = 0.25; % 25% std deviation
- % Gain variation
- Kp_mc = normrnd(Kp, des∗Kp).
2.5.4. Lilliefors Normality Test
2.5.5. Levene’s Test
2.5.6. Kruskal–Wallis Test
3. Proposed Method
3.1. Stability Analysis
3.2. Phase Margin
3.3. Disturbance Response
3.4. Gain Margin
3.5. Implementation Steps
4. Example
4.1. Utilization of the Method
4.2. Direct Synthesis
5. Evaluation of the Proposed Method Based on Simulations Results
5.1. Scenario Without Parameter Variations
5.2. Scenario with Parameter Variations: Monte Carlo Analysis
5.3. Inferential Statistical Analysis
6. Discussion
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
FOC | Fractional-Order Controller |
IOC | Integer-Order Controller |
FOPID | Fractional-Order PID Controller |
FOID | Fractional-Order ID Controller |
DS | Direct Synthesis |
FO-DS | Fractional-order Direct Synthesis |
Integrated square error | |
Normalized ISE | |
Integrated square control and error index | |
Normalized ISCE | |
IMC | Internal Model Controller |
, | Reference signal (setup) |
, | Noise signal |
Error signal | |
Control signal | |
Controlled signal | |
System transfer function (process) | |
Closed-loop transfer function | |
Loop gain | |
Fractional-order closed-loop transfer function | |
Direct synthesis controller | |
Fractional-order ID controller | |
Lead-lag phase compensator | |
Fractional-order PID controller | |
Integer-order damping factor | |
Fractional-order damping factor | |
Overshoot | |
Settling time to 2% | |
Phase crossover frequency | |
Steady-state error | |
Integer-order gain crossover frequency | |
Fractional-order natural frequency | |
Loop gain | |
Phase margin | |
Gain margin | |
Disturbances bandwidth | |
A | Gain at in dB |
Fractional order of | |
, | Fractional orders of |
N | Number of Monte Carlo runs |
n | Number of terms in the integer-order approximation |
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64.63∘ | 64.63∘ | 64.63∘ | 60∘ | |
5% | 5% | 5% | 8.7% | |
0.0693 | 0.01 | 0.003 | 0.01 | |
1000 s | 586 s | 2000 s | 586 s | |
A | −40 dB | - | - | −40 dB |
0.6% | 3.7% | 4% | 5.5% | |
858 s | 1838 s | 1104 s | 997 s | |
0.1 | 0.1 | 0.1 | 0.1 | |
3.2 | 4.5 | 6.8 | 12.4 | |
857 s | 832 s | 650 s | 640 s | |
5.94 | 18 | 1.78 | 31.17 |
Param | ||
M = 1.4%, SD = 1.9% | M = 4.8%, SD = 3.8% | |
M = 1116.4 s, SD = 330.2 s | M = 1660.9 s, SD = 299.2 s | |
M = 0.1, SD = 0.02 | M = 0.1, SD = 0.03 | |
M = 3.9, SD = 2.2 | M = 7.6, SD = 2.8 | |
Param | ||
M = 5.4%, SD = 5.7% | M = 6.3%, SD = 6.5% | |
M = 1224 s, SD = 361.9 s | M = 1113.4 s, SD = 244.7 s | |
M = 0.1, SD = 0.02 | M = 0.1, SD = 0.02 | |
M = 8.6, SD = 5.9 | M = 40.8, SD = 2.6 |
Parameter | ||||
---|---|---|---|---|
Median | 0.49 | 3.18 | 2.80 | 3.02 |
C. Interval | (0.41, 0.68) | (2.67, 3.55) | (0.79, 4.11) | (1.05, 4.74) |
Median (s) | 1043.8 | 1684.8 | 1134.5 | 1066.2 |
C. Interval (s) | (971.2, 1116.4) | (1546.9, 1761.2) | (1071.0, 1204.6) | (1014.7, 1083.7) |
Median | 3.21 | 4.61 | 6.91 | 39.28 |
C. Interval | (3.01, 3.36) | (4.39, 4.83) | (6.27, 7.41) | (39.06, 39.56) |
Method | Analytical | Analytical | Analytical | Numerical |
(2 equations) | -Numerical | (4 equations | (FOMCON) | |
Bode plots) | ||||
No. of parameters | 4 | 3 | 3 | 5 |
Parameters | , , A, | , , | , , | , , A, , |
Difficulty | Easy | Medium | Medium | Medium |
Position in | ||||
the comparison | ||||
1st | 4th | 2nd | 3rd | |
1st | 4th | 3rd | 2nd | |
Energy efficiency | 1st | 2nd | 3rd | 4th |
No. of parameters | 2nd | 3rd | 3rd | 1st |
adjusted |
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Muñiz-Montero, C.; Munoz-Pacheco, J.M.; Sánchez-Gaspariano, L.A.; Sánchez-López, C.; Molinar-Solís, J.E.; Chavez-Portillo, M. Direct Synthesis of Fractional-Order Controllers Using Only Two Design Equations with Robustness to Parametric Uncertainties. Fractal Fract. 2025, 9, 101. https://doi.org/10.3390/fractalfract9020101
Muñiz-Montero C, Munoz-Pacheco JM, Sánchez-Gaspariano LA, Sánchez-López C, Molinar-Solís JE, Chavez-Portillo M. Direct Synthesis of Fractional-Order Controllers Using Only Two Design Equations with Robustness to Parametric Uncertainties. Fractal and Fractional. 2025; 9(2):101. https://doi.org/10.3390/fractalfract9020101
Chicago/Turabian StyleMuñiz-Montero, Carlos, Jesus M. Munoz-Pacheco, Luis A. Sánchez-Gaspariano, Carlos Sánchez-López, Jesús E. Molinar-Solís, and Melissa Chavez-Portillo. 2025. "Direct Synthesis of Fractional-Order Controllers Using Only Two Design Equations with Robustness to Parametric Uncertainties" Fractal and Fractional 9, no. 2: 101. https://doi.org/10.3390/fractalfract9020101
APA StyleMuñiz-Montero, C., Munoz-Pacheco, J. M., Sánchez-Gaspariano, L. A., Sánchez-López, C., Molinar-Solís, J. E., & Chavez-Portillo, M. (2025). Direct Synthesis of Fractional-Order Controllers Using Only Two Design Equations with Robustness to Parametric Uncertainties. Fractal and Fractional, 9(2), 101. https://doi.org/10.3390/fractalfract9020101