⚡ electrical engineering

PID Parameters Optimization by Using Genetic Algorithm

This paper proposes using a Genetic Algorithm to optimize PID controller parameters for First Order Lag plus Time Delay systems to compensate for time delays, demonstrating its effectiveness by comparing the results against Iterative Method and Ziegler-Nichols rule approaches.

Original authors: Andri Mirzal, Shinichiro Yoshii, Masashi Furukawa

Published 2026-06-03

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CC BY 3.0

Original authors: Andri Mirzal, Shinichiro Yoshii, Masashi Furukawa

Original paper licensed under CC BY 3.0 (http://creativecommons.org/licenses/by/3.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to drive a car, but there is a strange delay between when you turn the steering wheel and when the car actually turns. Maybe it's a heavy truck, or maybe there's a slow computer processing your commands. In the world of engineering, this is called a "time delay." It makes systems sluggish and unstable, like trying to steer a boat with a 10-second lag.

To fix this, engineers use a "PID controller," which is essentially a smart autopilot that constantly adjusts the steering to keep the car on track. The big question is: How do you tune this autopilot to work perfectly?

This paper is about a team of researchers who tried three different ways to find the perfect settings for this autopilot. Here is the breakdown of their journey:

1. The Three Competitors

The researchers set up a race between three different methods to tune the controller:

The Rulebook (Ziegler-Nichols): This is like following a cookbook recipe. You measure the delay, plug the number into a standard formula, and get your settings. It's fast and easy, but it's a "one-size-fits-all" approach.
The Trial-and-Error (Iterative Method): This is like a mechanic tweaking the engine by hand, testing it, adjusting it, and testing it again until it feels right. It's more careful than the rulebook but still requires the system to be stopped and tested offline.
The Evolutionary Solver (Genetic Algorithm - GA): This is the star of the show. Imagine a virtual breeding ground where you create 80 to 100 different "champions" (sets of settings). You let them "race" (simulate the car driving). The ones that drive the smoothest get to "reproduce," mixing their best traits to create a new generation. Over 300 generations, the population "evolves" to find the absolute best settings.

2. The Goal: What Are We Measuring?

The researchers wanted to see which method made the car drive the best. They looked at two main things:

The "Ride Quality" (Standard Measures):
- Overshoot: Did the car swing wildly past the turn before correcting? (The goal is to swing as little as possible).
- Settling Time: How long did it take to stop wobbling and drive straight?
- Rise Time: How fast did it react to the turn?
- Stability Margin: How close did it get to crashing before it became unstable?
The "Error Score" (Performance Indices):
- Think of this as a scorecard for how much the car deviated from the perfect path. The lower the score, the better. They used five different ways to calculate this score (like MSE, IAE, ITAE, etc.).

3. The Results: Who Won?

The Big Win: Smoother Rides
The Genetic Algorithm was the clear winner for making the ride smooth.

It reduced the "swing" (overshoot) by more than 70% compared to the Rulebook and 30% compared to the Trial-and-Error method.
It also helped the car stop wobbling faster (settling time).
The Catch: To get this smooth ride, the car became slightly less "stable" in terms of how much it could handle before tipping over. It's a trade-off: a smoother ride, but a slightly narrower margin for error.

The Scorecard: Lower Errors
When looking at the "Error Scores," the Genetic Algorithm consistently scored lower than the other two methods. It found settings that minimized the deviation from the path better than the Rulebook or the Trial-and-Error method.

The Surprise: Some Things Stayed the Same
Interestingly, the Genetic Algorithm didn't make the car react faster (Rise Time) or reach its peak speed quicker (Peak Time) than the other methods. It just made the ride smoother and more controlled.

4. The Hiccups (Limitations)

The researchers noted two main problems with using the Genetic Algorithm:

It's not always perfect: Sometimes, the "evolution" gets stuck and doesn't find the best solution, even after many tries. In their tests, it failed to find a solution about 4% of the time.
It's limited by its starting point: To make sure the algorithm worked, they had to tell it to only look for settings within a specific range (based on the other two methods). This means they might have missed a "super-optimal" solution that existed outside of that range.

The Bottom Line

The paper concludes that using a Genetic Algorithm is a powerful way to tune controllers for systems with time delays. It produces a much smoother, more controlled response with fewer errors than traditional methods. However, it requires more computing power, takes longer to run, and needs careful setup to ensure it actually finds a solution. It's a high-tech, evolutionary approach that beats the old-school recipes, but it's not a magic wand that solves every problem instantly.

Technical Summary: PID Parameters Optimization by Using Genetic Algorithm

Problem Statement
Time delays are inherent components in physical, chemical, biological, and economic systems, as well as in measurement and computation processes. These delays introduce time-lags that can significantly reduce the stability of minimum phase systems. While previous work by the authors demonstrated that the Iterative Method and Ziegler-Nichols rule could effectively tune PID controllers for First Order Lag plus Time Delay (FOLPD) systems, these methods have limitations. Specifically, they require the system to be taken offline for retuning, making them unsuitable for dynamic plants where parameters change constantly and real-time retuning is necessary. This paper addresses the challenge of optimizing PID controller parameters to compensate for time delays in FOLPD systems using a method capable of handling complex, non-linear search spaces without requiring offline system shutdowns.

Methodology
The authors propose the implementation of a Genetic Algorithm (GA) to determine optimal PID controller parameters ( $K_p, K_i, K_d$ ). The study utilizes the Matlab Genetic Algorithm Toolbox and follows a structured approach:

System Modeling: The time delay component is approximated using a second-order Direct Frequency Response (DFR) series. The authors selected DFR over the standard Pade approximation (used in Matlab's Control Systems Toolbox) because DFR demonstrated the smallest average error in previous comparisons and allows for a polynomial series representation suitable for the GA's transfer function construction.
Objective Functions: To evaluate the fitness of each chromosome, the study employs five performance indices: Mean of the Squared Error (MSE), Integral of Absolute Magnitude of the Error (IAE), Integral of the Squared Error (ISE), Integral of Time multiplied by Absolute Error (ITAE), and Integral of Time multiplied by the Squared Error (ITSE). The fitness value is defined as the reciprocal of the performance index value.
GA Configuration:
- Encoding: Real number encoding is used for the three PID parameters to avoid precision loss associated with binary encoding.
- Population: A population size of 80 to 100 chromosomes is used, with each chromosome containing $K_d, K_p, K_i$ values and a fitness value.
- Bounds: Parameter value bounds are not set arbitrarily but are constrained based on the ranges found in the Iterative Method and Ziegler-Nichols rule results to ensure convergence.
- Operators: Normalized Geometric Selection is used for parent selection. Arithmetic crossover is applied with four crossover points to accommodate the three alleles. Mutation probability is set low (0.1%) to prevent the search from devolving into a random process.
- Termination: The algorithm runs for a maximum of 300 generations, a limit determined by observing that no visible improvement occurs beyond this point.

Key Results
The GA-optimized PID controllers were compared against the Iterative Method and the Ziegler-Nichols rule across various time delays (0.01 to 1 second). The results were analyzed using standard performance measures and performance indices.

Standard Performance Measures:
- Percent Overshoot: GA methods significantly reduced percent overshoot compared to the other two methods. The average overshoot for GA was approximately 10%, compared to 38% for Ziegler-Nichols and 15% for the Iterative Method. Optimization by the IAE criterion yielded the best results (6%).
- Settling Time: GA optimization by ITAE and IAE showed improvements (0.745s and 1.01s, respectively) compared to the other methods (approx. 1.5s). However, other GA variations did not show significant improvement over the tuning rules.
- Rise Time and Peak Time: GA methods produced results similar to the Ziegler-Nichols rule and were generally worse than the Iterative Method in terms of rise time. The Iterative Method showed the worst peak time (3.43s), while GA and Ziegler-Nichols were comparable (approx. 0.57s–0.83s).
- Stability Margin: GA methods resulted in a slightly reduced stability margin compared to the tuning methods, indicating a potential trade-off where improved transient response comes at the cost of reduced stability margins.
Performance Indices:
- As expected, the GA methods produced the smallest average values for all five performance indices (MSE, IAE, ISE, ITAE, ITSE) across the tested delay range.
- The Ziegler-Nichols rule generally produced smaller performance index values than the Iterative Method.
- While GA consistently minimized the error criteria, the magnitude of improvement over the tuning methods was not always "impressive," except in the case of the ITAE objective function where the difference increased with time delay.

Significance and Claims
The paper concludes that applying Genetic Algorithms to PID controller tuning for FOLPD systems offers distinct advantages in transient response, specifically achieving a reduction in percent overshoot of more than 70% compared to the Ziegler-Nichols rule and over 30% compared to the Iterative Method. The GA approach maintains rise time and peak time comparable to the best traditional methods while improving settling time in specific configurations.

However, the authors maintain a modest stance regarding the overall superiority of GA. They note that while GA minimizes error criteria effectively, the differences in performance indices compared to traditional tuning rules are not always substantial enough to definitively claim GA is "much better" in all aspects. Furthermore, the study highlights a critical limitation: convergence issues. In the experiments, 2 out of 45 cases failed to reach convergence. The authors acknowledge that while setting value bounds based on existing tuning methods improves convergence probability, it inherently restricts the search space, potentially discarding the possibility of finding optimal results outside those established ranges. Thus, while GA is a viable tool for real-time retuning in dynamic systems where offline methods fail, its application requires careful consideration of convergence reliability and the trade-off between transient performance and stability margins.

1. The Three Competitors

2. The Goal: What Are We Measuring?

3. The Results: Who Won?

4. The Hiccups (Limitations)

The Bottom Line

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