PID Controller Parameter Calculation for First-Order Systems with Delay

In this article, we present a streamlined computational tool that implements three established PID tuning methodologies – Ziegler-Nichols, Cohen-Coon, and Internal Model Control (IMC) – for first-order systems with time delays. The program demonstrates practical implementations of these algorithms through systematic parameter calculations, enabling users to compare different tuning approaches and select optimal gains for improved control system performance and stability. The code structure includes dedicated functions for each method, with clear mathematical formulations for proportional, integral, and derivative gain computations based on system time constant (τ) and delay time (L) inputs. Whether you're beginning your control engineering journey or are an experienced practitioner, this tool provides an efficient framework for PID parameter optimization with immediate practical applications. The implementation includes error handling for invalid system parameters and visualization options for comparing closed-loop responses across different tuning methods. We encourage you to explore this computational assistant to enhance your control system design workflow and observe measurable improvements in your control applications.