Supplement: Optimization-Based Control

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Richard M. Murray (Caltech)

These notes serve as a supplement to Feedback Systems by Åström and Murray and expand on some of the topics introduced there. Our focus is on the use of optimization-based methods for control, including optimal control theory, receding horizon control and Kalman filtering. Each chapter is intended to be a standalone reference for advanced topics that are introduced in Feedback Systems.

Note: These notes are in draft form and may contain errors. Permission is granted to download and print a copy for individual use, but this material may not be reproduced, in whole or in part, without written consent from the author.

Quick Links
News (archive)
  • 15 Feb 10: updated version of Chapter 4 (stochastic systems) is posted; multiple fixes + some new material
  • 3 Jan 10: posted updated version of Chapters 1 and 2 (mainly small fixes) + working errata link
  • 22 Dec 09: updated versions of all chapters posted; working through small fixes over the next few weeks
  • 22 Dec 09: updated supplement main page to a new format
  • 4 Jan 09: Version 2.0b posted - fixes and error corrections for chapters 1-4


  • Contents and Preface (PDF, 03 Jan 2010)
  • Ch 1 - Trajectory Generation and Tracking (PDF, 03 Jan 2010)
    • Two Degree of Freedom Design
    • Trajectory Tracking and Gain Scheduling
    • Trajectory Generation and Differential Flatness
  • Ch 2 - Optimal Control (PDF, 27 Oct 2018)
    • Review: Optimization
    • Optimal Control of Systems
    • Examples
    • Linear Quadratic Regulators
    • Choosing LQR Weights
    • Advanced Topics
  • Ch 3 - Receding Horizon Control (PDF, 22 Dec 2009)
    • Optimization-Based Control
    • Receding Horizon Control with CLF Terminal Cost
    • Receding Horizon Control Using Differential Flatness
    • Implementation on the Caltech Ducted Fan
  • Ch 4 - Stochastic Systems (PDF, 15 Feb 2010)
    • Review of Random Variables
    • Introduction to Random Processes
    • Continuous-Time, Vector-Valued Random Processes
    • Linear Stochastic Systems
    • Random Processes in the Frequency Domain
  • Ch 5 - Kalman Filtering (PDF, 22 Dec 2009)
    • Linear Quadratic Estimators
    • Extensions of the Kalman Filter
    • LQG Control
    • Application to a Vectored Thrust Aircraft
  • Ch 6 - Sensor Fusion (PDF, 22 Dec 2009)
    • Discrete-Time Stochastic Systems
    • Kalman Filters in Discrete Time
    • Predictor-Corrector Form
    • Sensor Fusion
    • Information Filters
    • Additional Topics
  • Bibliography and Index (PDF, 15 Feb 2010)