All classes PID & FOPID Metaheuristics Syllabus · Episode 4 of 6

Class PID & FOPID Metaheuristics #4: Objective Functions ITAE/ISE and New Metaheuristic Algorithms (Whale, Harris Hawks)

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Tutorial on ITAE objective function and Whale Optimization and Harris Hawks Optimization algorithms from scratch using Python. Episode 4 of the PID and FOPID metaheuristic optimization series.

Software: Python (NumPy, SciPy, Matplotlib)
Level: Advanced
Duration: ± 11 min
Series total: 52 min · 6 videos
Published: October 25, 2026

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Class PID & FOPID Metaheuristics #4: Objective Functions ITAE/ISE and New Metaheuristic Algorithms (Whale, Harris Hawks): Cover: PID & FOPID Metaheuristic Class #4: ITAE/ISE Objective Functions and New Metaheuristic Algorithms (Whale, Harris Hawks)

Class PID & FOPID Metaheuristics #4: Objective Functions ITAE/ISE and New Metaheuristic Algorithms (Whale, Harris Hawks): Before you start, steps list, and Step 1

Class PID & FOPID Metaheuristics #4: Objective Functions ITAE/ISE and New Metaheuristic Algorithms (Whale, Harris Hawks): Steps 2–3: Whale Optimization Algorithm

Class PID & FOPID Metaheuristics #4: Objective Functions ITAE/ISE and New Metaheuristic Algorithms (Whale, Harris Hawks): Step 4: Benchmark test Sphere & Rastrigin

Class PID & FOPID Metaheuristics #4: Objective Functions ITAE/ISE and New Metaheuristic Algorithms (Whale, Harris Hawks): Step 5: Convergence curve

Class PID & FOPID Metaheuristics #4: Objective Functions ITAE/ISE and New Metaheuristic Algorithms (Whale, Harris Hawks): Done: Final Result and Further Practice

Cover: PID & FOPID Metaheuristic Class #4: ITAE/ISE Objective Functions and New Metaheuristic Algorithms (Whale, Harris Hawks)

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Introduction

This is the final result, two new generations of metaheuristic algorithms, Whale Optimization and Harris-Hawks, which we wrote from scratch and tested on benchmark functions. Harris-Hawks even found the global minimum of the highly deceptive Rastrigin function. Let's get started. Hello, welcome to episode four of the PID and Fractional Order PID optimization series. To automatically tune PID, we need two things. Numbers that evaluate the good or bad of the response, which are the objective function and the algorithm that searches for the best parameters. Today we are making both of them, with two metaheuristics inspired by animal hunting behavior.

Step 1: ITAE & ISE Objective Function

Create metaheuristik.py file The objective function calculates the step response of the closed-loop system for 3 seconds then evaluates the error. ITAE, integral of time multiplied by absolute error, gives a heavy penalty for long-lasting errors, thereby producing a fast and quiet response. ISE, integral of squared error, punishes large errors at the beginning. If the system is unstable, we give a value of one million so that this parameter is avoided.

Step 2: Whale Optimization Algorithm

The WOA function is the Whale Optimization Algorithm. Whales hunt by encircling prey and spitting bubble nets in spiral shapes. Parameter A decreases from 2 to 0. When A is small, whales approach the best whale. When large, it explores following a random whale. Half the time, whales move in spiral around the best position with an exponential multiplied by cosine formula.

Step 3: Harris Hawks Optimization

The HO function is Harris-Hawks Optimization. A swarm of Harris hawks surrounds a rabbit whose energy E is continuously decreasing. When the rabbit's energy is still high, the hawks explore. When the energy decreases, the hawks encircle softly or aggressively. And if the rabbit still tries to run, the hawks pounce with a Levy jump, a random step that can sometimes be very far, to escape from local minima.

Step 4: Benchmark Test Sphere & Rastrigin

Create the test underscore metaheuristik.py file and run it. The sfir function has one valley, whereas rastrigin has hundreds of trap valleys. On sfir 10 dimensions, WIOA reaches 10 to the power of minus 22 and HHO 10 to the power of minus 49. On rastrigin, WIOA gets trapped at 13.99, whereas HHO finds the global minimum 0. As a reference, PI in Ziegler-Nichols on AVR has ITAI 0.4331 and ISI 0.2776.

Step 5: Convergence Curve

Let's look at the graphs. The convergence curve of HHO on sfir decreases gradually with several sharp jumps, a result of the encircling and Levy jump strategies. On rastrigin, WIOA remains flat because it is trapped, whereas HHO continues to decrease until 0. But benchmark results do not always apply to real-world problems. So we will test both on PID and FOPID in the next episode. Done.

Done!

We have formulated the ITAI and ISI objective functions, written Well Optimization Algorithm and Harris-Hox Optimization from scratch, and tested them on benchmark functions. In the next episode, we will optimize PID and FOPID for AVR and DC motor. 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, to tuning PI using Ziegler-Nichols, fractional order PID, up to meta-heuristic optimization such as Whale Optimization and Harris Hawks, as well as opening a Python programming course for electrical engineering. Want to master the entire series, from function design to statistical analysis and robustness testing? All 6 episodes, complete with step-by-step PDF guides and source code, are available for 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 the single class 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

0:46Step 1: ITAE & ISE Objective Function

2:20Step 2: Whale Optimization Algorithm

4:06Step 3: Harris Hawks Optimization

7:22Step 4: Benchmark Test Sphere & Rastrigin

9:32Step 5: Convergence Curve

10:01Done!

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▶ Video Lesson

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Transcript

0:03Introduction

0:46Step 1: ITAE & ISE Objective Function

2:20Step 2: Whale Optimization Algorithm

4:06Step 3: Harris Hawks Optimization

7:22Step 4: Benchmark Test Sphere & Rastrigin

9:32Step 5: Convergence Curve

10:01Done!

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Questions & answers

Your instructor

Introduction

Step 1: ITAE & ISE Objective Function

Step 2: Whale Optimization Algorithm

Step 3: Harris Hawks Optimization

Step 4: Benchmark Test Sphere & Rastrigin

Step 5: Convergence Curve

Done!

Introduction

Step 1: ITAE & ISE Objective Function

Step 2: Whale Optimization Algorithm

Step 3: Harris Hawks Optimization

Step 4: Benchmark Test Sphere & Rastrigin

Step 5: Convergence Curve

Done!