Supplement: Linear Systems Theory

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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. The focus in on more advanced presentations of the material on linear systems, providing a more theory-based coverage of the topics. Each chapter is intended to augment the introductory topics that are introduced in Feedback Systems.

Note: These notes are in draft form and may contain errors. Permission is granted to use and modify these notes for non-commercial purposes provided that appropriate credit is given to the original author and that all modifications contain these same terms (CC-BY-NC-SA).

Quick Links


  • Contents and Preface (PDF, 27 Dec 2020)
  • Signals and Systems
    • Linear Spaces and Mappings
    • Input/Output Dynamical Systems
    • Linear Systems and Transfer Functions
    • System Norms
    • Exercises
  • Linear Input/Output Systems
    • Matrix Exponential
    • Convolution Equation
    • Linear System Subspaces
    • Input/output stability
    • Time-Varying Systems
    • Exercises
  • Reachability and Stabilization
    • Concepts and Definitions
  • Reachability for Linear State Space Systems
  • System Norms
  • Stabilization via Linear Feedback
  • Exercises
  • Ch 2 - Optimal Control (PDF, 27 Oct 2018)
    • Review: Optimization
    • Optimal Control of Systems
    • Examples
    • Linear Quadratic Regulators
    • Choosing LQR Weights
    • Advanced Topics
    • FurtherReading
  • State Estimation
    • Concepts and Definitions
    • Observability for Linear State Space Systems
    • Combining Estimation and Control
    • Exercises
  • Bibliography and Index (PDF, 27 Dec 2020)