[Paper Review] A mathematical explanation via "intelligent" PID controllers of the strange ubiquity of PIDs

B. D’Andrea Novel, Michel Fliesś+3 more|arXiv (Cornell University)|May 4, 2010

Advanced Control Systems OptimizationEngineering2 references20 citations

TL;DR

This paper provides a mathematical explanation for the widespread use of classical PID controllers by showing that their sampled-time behavior approximates the structural dynamics of 'intelligent' PID controllers—model-free, adaptive regulators that estimate system dynamics in real time. The key finding is that properly tuned classic PIDs implicitly account for unknown system structure via their gains, but intelligent PIDs offer superior robustness, fault tolerance, and simpler tuning without system identification.

ABSTRACT

The ubiquity of PID controllers in the industry has remained mysterious until now. We provide here a mathematical explanation of this strange phenomenon by comparing their sampling with the the one of "intelligent" PID controllers, which were recently introduced. Some computer simulations nevertheless confirm the superiority of the new intelligent feedback design.

Motivation & Objective

• To resolve the long-standing mystery of why classical PID controllers are so widely used despite lacking theoretical justification beyond linear systems.
• To establish a mathematical link between classical PID controllers and recently developed 'intelligent' PID controllers based on model-free control theory.
• To demonstrate that the effectiveness of classic PIDs stems from their ability to approximate the structural adaptation of intelligent controllers when sampled.
• To highlight the advantages of intelligent PIDs in handling nonlinearities, parameter variations, and faults, where classical PIDs fail.
• To advocate for a paradigm shift in control engineering education and practice toward intelligent PID controllers due to their robustness and simplicity.

Proposed method

• Uses a phenomenological model of the system as $ y^{( u)} = F + \alpha u $, where $ \nu = 1 $ or $ 2 $, and $ F $ is estimated in real time using numerical differentiators.
• Introduces 'intelligent' PID (i-PID) controllers that directly use the estimated $ F $ and $ \alpha $ to compute control input: $ u = \frac{1}{\alpha}(-F + \ddot{y}^* + K_P e + K_I \int e + K_D \dot{e}) $.
• Performs a crude time-sampling of both classical and intelligent controllers to compare their discrete-time behavior.
• Derives explicit algebraic correspondences between classical PID gains ($ k_p, k_i, k_d $) and intelligent controller gains ($ K_P, K_I, K_D $) via Riemann sum approximations.
• Applies the method to a nonlinear system $ \dot{y} + y^3 = 2u $, comparing classical PI and i-PI controllers under varying conditions.
• Uses numerical simulations to validate performance under setpoint changes, parameter variations, and actuator faults.

Experimental results

Research questions

• RQ1 Why are classical PID controllers so ubiquitously used in industrial applications despite lacking theoretical grounding for nonlinear or complex systems?
• RQ2 How do the gains in classical PID controllers relate to the structural dynamics of unknown nonlinear systems?
• RQ3 To what extent does the sampled behavior of classical PIDs approximate the behavior of intelligent PID controllers?
• RQ4 What advantages do intelligent PID controllers offer over classical PIDs in terms of robustness, tuning, and fault tolerance?
• RQ5 Can the success of classical PIDs be mathematically explained as an emergent approximation of intelligent control principles?

Key findings

• The sampled behavior of classical PI and PID controllers approximates the structural adaptation of intelligent PID controllers, explaining their empirical success.
• Classical PID gains are mathematically linked to intelligent controller gains via the relations: $ k_p = K_D / (\alpha h) $, $ k_i = K_P / (\alpha h) $, $ k_{ii} = K_I / (\alpha h) $, and $ k_d = -1 / (\alpha h) $.
• Intelligent PID controllers require no system identification or complex tuning rules, unlike classical PIDs, which rely on intricate heuristic methods.
• Under large setpoint changes or system parameter variations, the classical PI controller degrades significantly, while the i-PI maintains stable performance.
• In the presence of actuator faults (e.g., 0.4% power loss per sample), the i-PI controller shows significantly better fault tolerance than the classical PI.
• The i-PI controller with $ \alpha = 1 $, $ K_P = 6 $, $ K_I = 9 $, required no calibration for different operating conditions, while the classical PI needed re-tuning after system changes.