Feedback Systems: An Introduction for Scientists and Engineers

Citation

Åström, K. J., & Murray, R. M. (2008). Feedback Systems: An Introduction for Scientists and Engineers. Princeton University Press. Free electronic edition: http://www.cds.caltech.edu/~murray/amwiki

Chapter Summary

The book has twelve chapters and runs to 408 pages in the version read. The preface states its audience as scientists and engineers who want to understand and use feedback in physical, biological, information and social systems, and its aim as a concise, insightful account of current knowledge in feedback and control, with modest mathematics but without sacrificing rigor. The first half is almost entirely state space; the second half is "classical control" built on transfer functions. Every chapter ends with a short "Further Reading" section and a set of exercises.

Chapter 1: Introduction

The chapter defines feedback as the coupling of two or more dynamical systems so that each influences the other, which makes causal reasoning circular and calls for formal methods. It surveys feedback in engineering history (early technological examples such as the centrifugal governor), power, aerospace, materials and processing, instrumentation, robotics, networks and computing, economics and in nature. It then discusses properties of feedback (robustness to uncertainty, design of dynamics, higher levels of automation, and drawbacks) and simple forms of feedback: feedforward, positive feedback, on-off control and PID control.

Chapters 2–3: System Modeling; Examples

Chapter 2 presents modeling as the construction of a precise representation of a system for analysis and simulation. It draws on mechanics and electrical engineering, introduces state space models with ordinary differential equations and difference equations, and gives a modeling methodology covering simulation, block diagrams, models from experiments, normalization and scaling, and model uncertainty. Chapter 3 collects the running examples: cruise control, bicycle dynamics, operational amplifier circuits, computing systems and networks (web server control and congestion control), atomic force microscopy, drug administration and population dynamics. First-time readers are advised to concentrate on the examples closest to their own background.

Chapter 4: Dynamic Behavior

The chapter treats nonlinear differential equations: solutions, phase portraits, equilibrium points and limit cycles, definitions of stability, stability of linear systems and linear approximation, Lyapunov functions, regions of attraction, bifurcations and the design of nonlinear dynamics by feedback.

Chapter 5: Linear Systems

The matrix exponential and the convolution equation characterise the input/output behavior of linear time-invariant systems. The chapter also covers eigenvalues and modes, coordinate invariance, steady-state response, and Jacobian linearization and feedback linearization as ways to relate nonlinear systems to linear ones.

Chapters 6–7: State Feedback; Output Feedback

Chapter 6 introduces reachability and uses it to show that, under certain conditions, closed-loop eigenvalues can be placed arbitrarily by state feedback; it includes reachable canonical form, design for second-order and higher-order systems, and integral action. Chapter 7 turns to observability, observers, control using the estimated state (the separation principle is proved), Kalman filtering and a general controller structure.

Chapters 8–9: Transfer Functions; Frequency Domain Analysis

Chapter 8 introduces transfer functions through the response to exponential signals, then covers block diagrams, poles and zeros, pole/zero cancellations, algebraic loops, Bode plots and, in a closing section, Laplace transforms. Chapter 9 builds the loop transfer function, the Nyquist criterion (including conditional stability), stability margins, Bode's relations and minimum-phase systems, and generalized gain and phase, including small gain and passivity.

Chapters 10–11: PID Control; Frequency Domain Design

Chapter 10 treats PID control, simple controllers for complex systems, tuning (including relay feedback and the Ziegler-Nichols rules), integrator windup, derivative filtering, setpoint weighting, and analog and computer implementation. Chapter 11 develops sensitivity functions, feedforward design, performance specifications, loop shaping with lead and lag compensation, and fundamental limitations from right half-plane poles and zeros, time delays and Bode's integral formula. It closes with a design example: lateral control of a vectored thrust aircraft.

Chapter 12: Robust Performance

The last chapter models uncertainty (parameter variations and unmodeled dynamics), examines robust stability using the Nyquist criterion and performance under uncertainty, and presents robust pole placement. A final section previews methods for robust performance, including quantitative feedback theory and linear quadratic control.

Key Concepts

Chapters 1–5: Models and behavior

• Feedback and open versus closed loop: interconnection of systems in a cycle versus a broken loop.
• State space model: a system described by states, inputs and outputs, dx/dt = f(x, u), y = h(x, u).
• Stability and Lyapunov functions: ways to judge whether solutions stay near or converge to an equilibrium.
• Matrix exponential and convolution: complete description of linear time-invariant response.

Chapters 6–9: State feedback, observers and frequency domain

• Reachability and observability: whether the state can be steered and whether it can be inferred from outputs.
• Eigenvalue assignment and observers: designing closed-loop dynamics and estimating unmeasured states.
• Transfer function and Bode plot: input/output description from exponential response, and its frequency-response graph.
• Nyquist criterion and stability margins: closed-loop stability from the open-loop frequency response.

Chapters 10–12: Design and robustness

• PID control: proportional, integral and derivative action, tuning and windup.
• Sensitivity functions and loop shaping: shaping closed-loop response through the loop transfer function.
• Fundamental limitations: constraints set by right half-plane poles and zeros and delays.
• Robust stability and performance: stability and performance when the process is uncertain.

Critical Analysis

Strengths

• A design that serves several audiences. The preface explains that the book arose from a Caltech course attended by engineering undergraduates and by graduate students in biology, computer science and physics, and that it avoids relying on Laplace transforms, introducing transfer functions through the exponential response. Examples span mechanical, electrical, computing, biological and economic systems.
• Modern topics reached early. Nonlinear dynamics, Lyapunov stability, the matrix exponential, reachability and observability, and fundamental limits appear in a first course; the preface says these were chosen over some traditional material.
• Clear signposting. Advanced sections are marked in the book, the preface proposes concrete plans for a 10-week and a 14-to-15-week course, and each chapter ends with Further Reading pointing to textbooks and history.
• A companion web site with code, supplemental examples and exercises, and lecture material is announced in the preface.

Limitations

• Omitted classical techniques. The preface says root locus, lead and lag compensation and detailed rules for sketching Bode and Nyquist plots are deliberately de-emphasised. A course that expects these must supplement the book.
• Scalar designs. For simplicity, the state feedback and output feedback designs of Chapters 6 and 7 are developed for a single input and a single output, so multi-input design needs another source.
• Mathematical expectations. Linear algebra, differential equations and complex numbers are assumed, and no appendix reviews them. The preface describes the last chapter as an introduction to more advanced study, so it is a starting point rather than a full treatment of robust design.
• Edition. This is the 2008 first edition. A second edition appeared in 2021, and its contents differ from those described here.

Where It Fits in the Course

This is the control text in the signals, systems and control course that is written for a broad audience. Nise's Control Systems Engineering is the applied introduction that uses MATLAB and Simulink for response and controller design, and is a natural companion for students who want more worked engineering practice. Oppenheim and Willsky's Signals and Systems and Sadiku's primer provide the signals and transforms background; Åström and Murray do not require Laplace transforms, so these books are useful background but not a prerequisite. Downey's Think DSP and Craig's Introduction to Robotics are specialised neighbours: signal processing in Python and the mechanics and control of robots.

How to Use This Book

The book suits junior and senior undergraduates and beginning graduate students in engineering and the natural sciences. The preface assumes linear algebra at the sophomore level, basic ordinary differential equations and some complex numbers.

For a 10-week course the authors suggest Chapters 1–2, 4–6 and 8–11 at a chapter per week, trimming the last chapters; a 14-to-15-week course can add Chapter 3 and Chapter 12. A student working alone can follow the same plan: read Chapters 1 and 2 first, skim the examples in Chapter 3 for the ones closest to the field, then work through the state-space chapters before the frequency-domain chapters. Chapter 12 is best saved for last. Do the exercises at the end of each chapter, since the preface notes that several standard results are left to them, and run the code from the companion web site.

Edition and Sources

This guide covers the first edition (Princeton University Press, 2008) in the authors' electronic version v2.10c (4 March 2010), which is a printing with corrections and is made available for personal use from the authors' web site. It is based on the preface and contents, the opening of every chapter and its section structure, and selected sections of Chapters 1, 6, 7, 10, 11 and 12.