1st Edition

PID Tuning A Modern Approach via the Weighted Sensitivity Problem

    154 Pages 74 B/W Illustrations
    by CRC Press

    154 Pages 74 B/W Illustrations
    by CRC Press

    The PID controller is the most common option in the realm of control applications and is dominant in the process control industry. Among the related analytical methods, Internal Model Control (IMC) has gained remarkable industrial acceptance due to its robust nature and good set-point responses. However, the traditional application of IMC results in poor load disturbance rejection for lag-dominant and integrating plants. This book presents an IMC-like design method which avoids this common pitfall and is devised to work well for plants of modest complexity, for which analytical PID tuning is plausible. For simplicity, the design only focuses on the closed-loop sensitivity function, including formulations for the H and H2 norms. Aimed at graduate students and researchers in control engineering, this book:

    • Considers both the robustness/performance and the servo/regulation trade-offs
    • Presents a systematic, optimization-based approach, ultimately leading to well-motivated, model-based, and analytically derived tuning rules
    • Shows how to tune PID controllers in a unified way, encompassing stable, integrating, and unstable processes
    • Finds in the Weighted Sensitivity Problem the sweet spot of robust, optimal, and PID control
    • Provides a common analytical framework that generalizes existing tuning proposals




    1 Introduction
    1.1 Servo, regulation, and stability
    1.2 Industrial (PID) control
    1.3 Internal model and Hcontrol
    1.3.1 Internal model control
    1.3.2 H control
    1.3.3 Blending internal model and H control
    1.3.4 Vilanova’s (2008) design for robust PID tuning revisited
    1.4 Outline of the book


    2 Simple Model-Matching Approach to Robust PID Control
    2.1 Problem statement
    2.1.1 The control framework
    2.1.2 The model-matching problem
    2.1.3 The model-matching problem within H control
    2.2 Analytical solution
    2.2.1 Initial formulation for set-point response
    2.2.2 Alternative formulation
    2.3 Stability analysis
    2.3.1 Nominal stability
    2.3.2 Robust stability
    2.4 Automatic PID tuning derivation
    2.4.1 Control effort constraints
    2.5 Simulation examples

    3 Alternative Design for Load Disturbance Improvement
    3.1 Problem statement
    3.1.1 The control framework
    3.1.2 The model-matching problem formulation
    3.2 Model-matching solution for PID design
    3.3 Trade-off tuning interval considering load disturbances
    3.3.1 Nominal stability
    3.4 Tuning guidelines
    3.5 Simulation examples

    4 Analysis of the Smooth/TightServo/Regulation Tuning Approaches
    4.1 Revisiting the model-matching designs
    4.2 Smooth/tight tuning
    4.3 Servo/regulation tuning
    4.4 Implementation aspects
    4.5 Simulation examples
    4.6 Summary


    5 H Design with Application to PI Tuning
    5.1 Problem scenario
    5.2 Analytical solution
    5.3 Weight selection
    5.4 Stability and robustness analysis
    5.5 Application to PI tuning
    5.5.1 Stable/unstable plants
    5.5.2 Integrating plant case (τ → ∞)
    5.6 Simulation examples


    6 Generalized IMC Design and H₂ Approach
    6.1 Motivation for the input/output disturbance trade-off
    6.2 Problem statement
    6.3 Weight selection
    6.4 Analytical solution
    6.4.1 Interpretation in terms of alternative IMC filters
    6.4.2 Extension to plants with integrators or complex poles
    6.5 Performance and robustness analysis
    6.6 Tuning guidelines
    6.7 Simulation examples


    7 PID Design as a Weighted Sensitivity Problem

    7.1 Context, motivation, and objective
    7.2 Servo/regulation and robustness/performance trade-offs
    7.3 Unifying tuning rules
    7.4 Special cases and tuning-rule simplifications
    7.4.1 First-order cases (τ2 = 0)
    7.4.2 Second-order cases
    7.5 Applicability: normalized dead time range

    8 PID Tuning Guidelines for Balanced Operation
    8.1 Robustness and comparable servo/regulation designs
    8.2 Servo/regulation performance evaluation: Jmax and Javg indices
    8.3 PI control using first-order models
    8.3.1 Stable and integrating cases Tuning based on Jmax Tuning based on Javg
    8.3.2 Unstable case Tuning based on Jmax and Javg
    8.4 PID control using second-order models
    8.4.1 Stable and integrating cases Tuning based on Jmax Tuning based on Javg
    8.4.2 Unstable case Tuning based on Jmax and Javg

    Appendix A




    Salvador Alcántara Cano graduated in Computer Science & Engineering and then obtained the MSc and PhD degrees in Systems Engineering & Automation, all from Universitat Autònoma de Barcelona, in 2005, 2008, and 2011, respectively. During his short-lived research career, he focused on PID control and the analytical derivation of simple tuning rules guided by robust and optimal principles. He also made two research appointments with Professors Weidong Zhang and Sigurd Skogestad, almost completed a degree in Mathematics, and held a Marie Curie postdoctoral position in the Netherlands. Back in Barcelona, "Salva" worked as an automation & control practitioner for one more year, before definitively shifting his career into software development. Apart from programming and DevOps in general, his current interests include Stream Processing, Machine Learning, and Functional Programming & Category Theory.

    Ramon Vilanova Arbós graduated from the Universitat Autònoma de Barcelona (1991), obtaining the title of Doctor through the same university (1996). At present, he's Full Professor of Automatic Control and Systems Engineering at the School of Engineering of the Universitat Autònoma de Barcelona where he develops educational task-teaching subjects of Signals and Systems, Automatic Control, and Technology of Automated Systems. His research interests include methods of tuning of PID regulators, systems with uncertainty, analysis of control systems with several degrees of freedom, applications to environmental systems, and development of methodologies for the design of machine-man interfaces. He is an author of several book chapters and has more than 100 publications in international congresses/journals. He is a member of IFAC and IEEE-IES. He's also  a member of the Technical Committee on Factory Automation.

    Carles Pedret i Ferré was born in Tarragona, Spain, on January 29, 1972. He received the BSc degree in Electronic Engineering and the PhD degree in System Engineering and Automation from Universitat Autònoma de Barcelona, in 1997 and 2003, respectively. He is Associate Professor at the Department of Telecommunications and System Engineering of Universitat Autònoma de Barcelona. His research interests are in uncertain systems, time-delay systems, and PID control.