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This page contains information about version 2.4a of ''Feedback Systems'' by Karl J. {{Astrom}} and Richard M. Murray.

Version 2.4a is the first complete copy of the book that is being used for [[RMM:CDS 101|CDS 101/110]] at Caltech.  For Caltech students, this version is available in the Caltech bookstore as a paperback book for $35.  The only difference between the paperback version and the online version is that citations in the online version are alphanumeric.  Since this book will be used for a course, this page was created to collect any errata associated with the printed version.

Additinal info:
* Revision date: 16 September 2006
* {{am05pdf|am06-complete|16Sep06|Complete book|}} - this a PDF of the entire book, as a single PDF file (10 MB).

== Errata ==

The errata listed below are referenced according to their location in the ''printed'' copy of this text (available in the Caltech bookstore).  Line and page numbers for the PDF version posted here may vary slightly. Small errors (typos, grammar fixes) are generally not listed.

Conventions: each line of text and each line of a displayed equation are counted as a line. Negative 
line numbers mean number of lines from the bottom of the page. First line on page = 1, last line 
on page = -1. Entries in boldface indicate errors which might change the interpretation of the text.

==== Chapter 2 - System Modeling ====
<ul>
<li><p>Page 44, line -1: The sign on the <math>mgl sin\theta</math> term is incorrect.  The equation should read
<center><amsmath>
 \begin{bmatrix}
    (M+m) & - ml\cos\theta \\
    - m l \cos\theta & (J+ml^2)
   \end{bmatrix} \begin{bmatrix} \ddot p \\ \ddot\theta  \end{bmatrix}
  + \begin{bmatrix} c \dot p + ml \sin\theta\, \dot\theta^2 \\
    \underline{-mgl\sin\theta} + \gamma \dot\theta
  \end{bmatrix} = \begin{bmatrix} F \\ 0 \end{bmatrix},
</amsmath></center>
(F. Zabala, 2 Oct 06)</p></li>

<li><p> Page 45, line -4: The equations of motion are missing a term and have some sign and coefficient errors.  The correct equations are
<center><amsmath>
  \aligned[c]
    \frac{d}{dt} \begin{bmatrix} p \\ \theta \\ \dot p \\ \dot \theta \end{bmatrix} &=
    \bmatrix \dot p \\ \dot \theta \\
      \dfrac{
        - m l \sin\theta \dot\theta^2 
        + m g (m l^2 / J_t) \sin\theta \cos\theta 
	- c \dot p \underline{- \gamma l m \cos\theta \dot\theta}
        + u}
        {M_t - m (m l^2 / J_t) \cos^2\theta  } \\[2ex]
      \dfrac{
	- m l^2 \sin\theta \cos\theta \dot\theta^2 
        + M_t g l \sin\theta 
        \underline{- c l \cos\theta \dot p - \gamma (M_t/m) \dot \theta}
	+ l \cos\theta u}
        {J_t (M_t/m) - m (l \cos\theta)^2}
    \endbmatrix \\
    y &= \begin{bmatrix}p \\ \theta \end{bmatrix}.
  \endaligned
</amsmath></center>
(F. Zabala, 2 Oct 06)</p></li>

<li><p>Page 46, line 2: Errors from the previous derivation carry through to the linearization.  The correct equation is
<center><amsmath>
  \aligned 
    \frac{d}{dt}\bmatrix p \\ \theta \\ \dot p \\ \dot \theta \endbmatrix &=
    {\bmatrix
      0 & 0 & 1 & 0 \\
      0 & 0 & 0 & 1 \\[1ex]
      0 & \frac{m^2 l^2 g}{\mu}
        & \frac{-c J_t}{\mu} 
	& \underline{\frac{-\gamma J_t l m}{\mu}} \\[2ex]
      0 & \frac{M_t m g l}{\mu}
        & \underline{\frac{-c l m}{\mu}}
        & \underline{\frac{-\gamma M_t}{\mu}}
    \endbmatrix} \bmatrix p \\ \theta \\ \dot p \\ \dot \theta \endbmatrix + 
    {\bmatrix
      0 \\ 0 \\[1ex]
        \frac{J_t}{\mu} \\[2ex] 
        \frac{l m}{\mu} 
    \endbmatrix} u \\[1ex]
    y &= \bmatrix 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \endbmatrix x,
  \endaligned
</amsmath></center>
(F. Zabala, 2 Oct 06)</p></li>

<li><p>Page 46, line 4: <math>\mu</math> should be 
<math>\mu = M_t J_t - m^2 l^2</math>.
(J. Carson, 16 Oct 06)</p></li>

<li><p>Page 47, equation (2.10): The equation should read
<center><amsmath>
  \aligned 
    x_k &= A^k x_0 + \sum_{i=0}^{k-1} A^{k-i-1} B u_i \\
    y_k &= C A^k x_0 + \sum_{i=0}^{k-1} C A^{k-i-1} B u_i + D u_k \\
  \endaligned \qquad k > 0.
</amsmath></center>
(D. Rizzo, 8 Oct 06; F. Zabala, 19 Oct 06)
</li></p>

<li><p>Page 60, line -8: the equations for <math>b</math> and <math>a</math> should read <math>b = r_\alpha \tan \delta</math> and <math>a = r_\alpha \tan \alpha</math>.  (M. Tao, 4 Oct 06)</p></li>
</ul>

==== Chapter 3 - Examples ====

<ul>
<li><p> Page 80, equation (3.3): <math>C_v</math> should be <math>C_d</math>, the aerodynamic drag coefficient. [K. Watts, 8 Oct 06] </p></li>
</ul>

==== Chapter 4 - Dynamic Behavior ====

<ul>
<li><p> Page 115, line 2 (first displayed equation): equation should read
<center><amsmath>
  \aligned[c]
    \frac{d}{dt} \begin{bmatrix} p \\ \theta \\ \dot p \\ \dot \theta \end{bmatrix} &=
    \bmatrix \dot p \\ \dot \theta \\
      \dfrac{
        - m l \sin\theta \dot\theta^2 
        + m g (m l^2 / J_t) \sin\theta \cos\theta 
	- c \dot p \underline{- \gamma l m \cos\theta \dot\theta}
        + u}
        {M_t - m (m l^2 / J_t) \cos^2\theta  } \\[2ex]
      \dfrac{
	- m l^2 \sin\theta \cos\theta \dot\theta^2 
        + M_t g l \sin\theta 
        \underline{- c l \cos\theta \dot p - \gamma (M_t/m) \dot \theta}
	+ l \cos\theta u}
        {J_t (M_t/m) - m (l \cos\theta)^2}
    \endbmatrix \\
    y &= \begin{bmatrix}p \\ \theta \end{bmatrix}.
  \endaligned
</amsmath></center>
[F. Zabala, 19 Oct 06]
</li></p>

<li><p> Page 132, line 5: the matrix at the end of the displayed equation should be the negative of the identity matrix (minus sign missing). [D. Hurtado, 16 Oct 06]</p></li>

<li><p> Page 133, lines 11 and 12: the last row in the nonlinear portion of <math>F(z)</math> should be zero.  The term <math>z_1 + z_2</math> is already included in the linear term in the sum.  [D. Hurtado, 15 Oct 06] </p></li>
</ul>

==== Chapter 5 - Linear Systems ====

<ul>
<li><p> Page 146, equation (5.2): <math>x(0)</math> should be <math>x_0</math>. [K. Galloway, 19 Oct 06]</p></li>

<li><p> Page 149, second line of <math>y(t)</math> expression: <math>+ \cdots</math> missing from end of line. [K. Galloway, 19 Oct 06]</p></li>

<li><p> Page 151, equation (5.7): second term in expansion should be <math>A^2 t</math> (missing factor of <math>A</math>). [K. Galloway, 19 Oct 06]</p></li>

<li><p> Page 154, line 10 (second displayed equation): third term has an additional <math>v</math> after the end of the parentheses. [K. Galloway, 19 Oct 06]</p></li>

<li><p> Page 154, line 15 (third displayed equation): denominator should be <math>v_j</math>. [K. Galloway, 19 Oct 06]</p></li>

<li><p> Page 185, exercise 7(a): the displayed equation should read
<center><amsmath>
    y_k = C A^k x_0 + \sum_{i=0}^{k-1} C A^{k-i-1} B u_i + D u_k \\
</amsmath></center>
[C. Lepine, 16 Oct 06]</p></li>
</ul>

==== Chapter 6 - State Feedback ====

<ul>
<li><p> Page 177, equations (5.27) and (5.29): <math>C_v</math> should be <math>C_d</math>, the aerodynamic drag coefficient.  [C. Lepine, 11 Oct 06] </p></li>

<li><p> Page 191, equation (6.4): the right hand sides have multiple errors.  The correct equations are
<center><amsmath>
  \aligned 
    (M+m) \ddot p - ml\cos\theta\, \ddot\theta &= -c \dot p -
      ml \sin\theta\, \dot\theta^2 + F \\
    (J+ml^2)\ddot\theta - m l \cos\theta\, \ddot p &= -\gamma +
    \dot\theta mgl\sin\theta,
  \endaligned
</amsmath></center>
[K. Galloway, 27 Oct 06]
</p></li>

<li><p> Page 193, line 9 (second displayed equation): spaces missing in the matrix.  Should read
<center><amsmath>
  H W_r = H \begin{bmatrix} B & AB & \cdots & A^{n-1}B \end{bmatrix} = 0.
</amsmath></center>
[K. Galloway, 27 Oct 06]
</p></li>

<li><p> Page 195, line -1: There is a missing equals sign after <amsmath>W_r^{-1}</amsmath>.
[K. Watts, 26 Oct 06]</p></li>

<li><p> Page 202, line -4: <math>d_1</math> should be <math>a_1</math>.  [K. Galloway, 27 Oct 06]</p></li>

<li><p> Page 203, line -14: extra <math>d</math>'s in displayed equation
<center><amsmath>
\frac{d}{dt} \begin{bmatrix} z_1 \\ z_2 \end{bmatrix} = \cdots
</amsmath></center>
[K. Galloway, 27 Oct 06]</p></li>

<li><p> Page 210, line -2: extra factor of <math>g</math> in the equation for <amsmath>\dot {\tilde v}</amsmath>.  Should read:
<center><amsmath>
    \dot{\tilde v} = a \tilde v - b_g \theta + b \tilde u
</amsmath></center>
[K. Galloway, 27 Oct 06]</p></li>

<li><p> Page 211, line 6 and line 9: <math>g</math> should be <math>b_g</math>. </p></li>

<li><p> Page 202, line -4: <math>d_1</math> should be <math>a_1</math>.  [K. Galloway, 27 Oct 06]</p></li>

</ul>

==== Chapter 7 - State Estimation ====

<ul>
<li><p> Page 225, line -1: The first part of the equation should read
<center><amsmath>
\frac{d\hat x}{dt} = A \hat x + B u + L (y \underline{ - C \hat x})
</amsmath></center>
</p></li>

<li><p> Page 221, paragraph just above heading for Section 7.2.  There is an error in both sentences of this paragraph where the word reachable is used instead of observable.
First sentence: "... converts the system into [observable] canonical form (Exercise 3)."
Second sentence: "... system is in [observable] canoncial form without...".
(J. Carson, 7 Jan 07)
</p></li>
</ul>

==== Chapter 9 - Loop Analysis ====

<ul>
<li><p> Page 281, line 8: <math>g</math> should be <math>b_g</math>.</p></li>
<li><p> Page 281, line 14: <math>\alpha</math> in denominator of <math>G_{yu}(s)</math> should be <math>a</math>. [C. Lepine, 5 Nov 06]</p></li>
<li><p> Page 286, line -10: "real axis" should be "imaginary axis".  [S. Han, 11 Nov 06]</p></li>
</ul>

==== Chapter 10 - Loop Analysis ====
<ul>
<li><p> Page 306, last paragraph: the wording regarding the scaling of the system response is confusing.  For the Ziegler Nichols method, a unit step should be applied as the input and the output should be measured without any additional scaling.   [J. Carson, 17 Nov 06]
</p></li>
<li><p>Page 308, Table 10.2, last line: <math>T_{del}/2</math> should be <math>1/2</math> [J. Carson, 17 Nov 06]
</p></li>
</ul>

== Links to individual chapters ==

The links below are to the PDF files for the individual chapters in version 2.4a.  For the most recent version of the chapters, use the contents menu (usually on the left side of the browser window).

* {{am05pdf|am06-preface|16Sep06|Preface and Table of Contents|}}
* {{am05pdf|am06-intro|16Sep06|Chapter 1 - Introduction|}}
* {{am05pdf|am06-modeling|16Sep06|Chapter 2 - System Modeling|}}
* {{am05pdf|am06-examples|16Sep06|Chapter 3 - Examples|}}
* {{am05pdf|am06-dynamics|16Sep06|Chapter 4 - Dynamic Behavior|}}
* {{am05pdf|am06-linsys|16Sep06|Chapter 5 - Linear Systems|}}
* {{am05pdf|am06-statefbk|16Sep06|Chapter 6 - State Feedback|}}
* {{am05pdf|am06-outputfbk|16Sep06|Chapter 7 - Output Feedback|}}
* {{am05pdf|am06-xferfcns|16Sep06|Chapter 8 - Transfer Functions|}}
* {{am05pdf|am06-analysis|16Sep06|Chapter 9 - Loop Analysis|}}
* {{am05pdf|am06-pid|16Sep06|Chapter 10 - PID Control|}}
* {{am05pdf|am06-loopsyn|16Sep06|Chapter 11 - Loop Shaping|}}
* {{am05pdf|am06-robperf|16Sep06|Chapter 12 - Robust Performance|}}
* {{am05pdf|am06-bibliography|16Sep06|Bibliography|}}