Piecewise Linear Approximation and PID Control Optimization for Nonlinear Systems

TL;DR

This paper presents a method combining piecewise linear approximation with PID control optimization via Particle Swarm Optimization to effectively control nonlinear systems, enabling frequency-domain analysis and improved stability.

Contribution

It introduces a novel approach that integrates piecewise linear models with PID tuning using PSO, enhancing control of nonlinear systems with analytical tractability.

Findings

The method achieves stable control of nonlinear systems.

Frequency-domain analysis becomes feasible through linear approximation.

Optimized PID parameters improve system performance and accuracy.

Abstract

This paper investigates the control of nonlinear systems using a piecewise linear approximation framework. The proposed approach combines a PID controller with locally linearized models obtained by partitioning the nonlinear function into subregions over a compact domain. This approximation yields an analytically tractable representation of the system dynamics, enabling the application of transfer-based and frequency-domain analysis tools that are not directly applicable to nonlinear systems. As the number of linear segments increases, the approximated system progressively approaches the behavior of the original nonlinear system, allowing for a meaningful frequency-domain interpretation of the dynamics. The PID controller parameters are optimized using the Particle Swarm Optimization method with performance criteria based on ITAE (Integral of Time-weighted Absolute Error) and ISO (Integral of Squared Overshoot). Numerical simulations confirm the effectiveness of the proposed method, demonstrating that controller parameters obtained from the piecewise linear model ensure stable and accurate control when applied to the original nonlinear system, while maintaining a balance between computational effort and approximation accuracy.