PID-GM: PID Control with Gain Mapping
Bo Zhu, Wei Yu, Hugh H. T. Liu
TL;DR
This paper tackles robust PID tuning for uncertain and disturbed plants by introducing a nonlinear gain mapping from auxiliary parameters $(k_p,k_d,T)$ to PID gains $(K_P,K_I,K_D)$. It shows PID is equivalent to a nominal PD controller plus an uncertainty/disturbance estimator within a 2-DoF framework, enabling tuning to be reduced to adjusting a single parameter $T$ while maintaining stability via singular perturbation theory. The key contribution is a principled, provable relationship between the auxiliary parameter and both stability and ultimate tracking error, along with a practical three-step design procedure. The approach is validated through simulations and experiments on a 3-DOF helicopter, demonstrating expected trade-offs between robustness (smaller $T$) and control effort (larger gains).
Abstract
Proportional-Integral-Differential (PID) control is widely used in industrial control systems. However, up to now there are at least two open problems related with PID control. One is to have a comprehensive understanding of its robustness with respect to model uncertainties and disturbances. The other is to build intuitive, explicit and mathematically provable guidelines for PID gain tuning. In this paper, we introduce a simple nonlinear mapping to determine PID gains from three auxiliary parameters. By the mapping, PID control is shown to be equivalent to a new PD control (serving as a nominal control) plus an uncertainty and disturbance compensator (to recover the nominal performance). Then PID control can be understood, designed and tuned in a Two-Degree-of-Freedom (2-DoF) control framework. We discuss some basic properties of the mapping, including the existence, uniqueness and invertibility. Taking as an example the PID control applied to a general uncertain second-order plant, we prove by the singular perturbation theory that the closed-loop steady-state and transient performance depends explicitly on one auxiliary parameter which can be viewed as the virtual singular perturbation parameter (SPP) of PID control. All the three PID gains are monotonically decreasing functions of the SPP, indicating that the smaller the SPP is, the higher the PID gains are, and the better the robustness of PID control is. Simulation and experimental examples are provided to demonstrate the properties of the mapping as well as the effectiveness of the mapping based PID gain turning.