All classes PID & FOPID Metaheuristics Syllabus · Episode 3 of 6

Class PID & FOPID Metaheuristics #3: Fractional-Order PID: Fractional Calculus and Oustaloup Approximation

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Tutorial on fractional-order PID and Oustaloup approximation with Python: fractional calculus, Bode diagram, PI lambda D mu. Episode 3 of the PID and FOPID metaheuristic optimization series.

Software: Python (NumPy, SciPy, Matplotlib)
Level: Advanced
Duration: ± 8 min
Series total: 52 min · 6 videos
Published: October 25, 2026

Episode 3 of 6 · view syllabus

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Introduction

This is the final result, the half-derivative operator, S to the power of 0.5, which we built with the Oustaloup approximation and matches the theory. Rise of 10 decibels per decade with a phase of 45 degrees. With this operator, we create a fractional-order PID. Let's start. Hello, welcome to episode three of the PID and fractional-order PID optimization series. In a standard PID, the integral and derivative are of order 1. Fractional calculus allows fractional orders, for example, a derivative of 0.7. Fractional-order PID, or FOPID, uses an integral of order lambda and a derivative of order mu, so it has five parameters and is more flexible in design. Today we are building it.

Step 1: Oustaloup Approximation

Create a file named FOPID.py The operator S to the power of alpha does not have an exact transfer function of integer order, so we approximate it using the Oustaloup method. In the frequency range from 0.001 to 1000 radians per second, we place nine pairs of zeros and poles spaced evenly on a logarithmic scale. The alternating combination of zeros and poles makes the magnitude curve rise on average alpha times 2,0 decibels per decade, with a phase of approximately alpha times 90 degrees. The FOPID function constructs the controller as KP plus KI times S to the power of minus lambda plus KD times S to the power of mu.

Step 2: PI-lambda D-mu Controller

Fractional integration comes from the Oustaloup with a negative alpha. The three terms are summed with a common denominator. If lambda and mu are equal to one, FOPID returns to the standard PI.

Step 3: Validation & Effect of lambda-mu

Create a test_oustaloup.py file and run it. At a frequency of one radian per second, the magnitude of S to the power of 0.5 approaches 0 decibels according to theory. The phase is 58.2 degrees with N1, 48.2 with N2, and 45.1 degrees with N4, almost exactly 45. On AVR, lambda 0.8 accelerates the settling time to 0.69 seconds, mu 0.6 increases the overshoot to 51.9 percent, while mu 1.3 reduces the overshoot to 2.6 percent.

Step 4: Bode Diagram & FOPID Response

Let's look at the graph. Within the gray frequency band, the magnitude of the Oustaloup approximation coincides with the ideal line, and the phase becomes flatter at 45 degrees as N increases. Outside the band, the approximation is flat, so the band must cover the system's working frequency. The right graph shows that lambda and mu indeed shape the response, providing FOPI with two degrees of freedom additional.

Done!

Done. We have understood the fractional operator, built the Oustaloup approximation, validated it with the BODE diagram, and assembled the fractional-order PID. In the next episode, we will create the objective function and metaheuristic algorithm to automatically find the five FOPI parameters. Currently working on a thesis, dissertation, or research in the field of control systems and optimization? Cerdas Inovasi Digital helps implement control systems, from modeling DC motors and AVR generators, tuning PI via Ziegler-Nichols, fractional-order PID, up to metaheuristic optimization such as whale optimization and Harris-Hawkes, as well as offering Python programming courses for electrical engineering. Want to master the entire series, from alias functions to statistical analysis and robustness testing? All 6 episodes, complete with PDF step-by-step guides and source code, are available to premium tutorial subscribers. Subscribe starting at Rp69.000 per month, save 28% with the 12-month package, or pay once Rp999.000 for lifetime access. Want just one class? Buy single classes with lifetime access. Subscribe now at the premium page cerdasinovasidigital.com Contact our WhatsApp or visit cerdasinovasidigital.com Thank you for watching, see you in the next episode.

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Transcript

0:03Introduction

0:54Step 1: Oustaloup Approximation

2:20Step 2: PI-lambda D-mu Controller

2:54Step 3: Validation & Effect of lambda-mu

6:03Step 4: Bode Diagram & FOPID Response

6:31Done!

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Transcript

0:03Introduction

0:54Step 1: Oustaloup Approximation

2:20Step 2: PI-lambda D-mu Controller

2:54Step 3: Validation & Effect of lambda-mu

6:03Step 4: Bode Diagram & FOPID Response

6:31Done!

Ask the class material

My notes

Questions & answers

Your instructor

Introduction

Step 1: Oustaloup Approximation

Step 2: PI-lambda D-mu Controller

Step 3: Validation & Effect of lambda-mu

Step 4: Bode Diagram & FOPID Response

Done!

Introduction

Step 1: Oustaloup Approximation

Step 2: PI-lambda D-mu Controller

Step 3: Validation & Effect of lambda-mu

Step 4: Bode Diagram & FOPID Response

Done!