View source for Errata: Closed loop dynamics matrix is incorrect in Example 4.6

{{errata page
| chapter = Dynamic Behavior
| page = 107
| line = 12-23
| contributor = S. Han
| date = 12 Oct 08
| version = First printing
| text =
The dynamics matrix of the closed loop system (on line 12) is incorrect.  The displayed equation should read
<center><amsmath>
  \aligned 
    \frac{dx}{dt} &= \begin{bmatrix}
      -k_0 -k_1 & k_1 - b_0 k \\ k_2 & - k_2
    \end{bmatrix} x +
    \begin{bmatrix} b_0 \\ 0 \end{bmatrix} u_d =: Ax + B u_d.
  \endaligned
</amsmath></center>
This error propagates into the computation of the equilibrium point <math>y_e</math> in line 15, which should be
<center><amsmath>
y_e = -C A^{-1} B u_d =\frac{b_0 k_2}{k_0 k_2 + b_0 k_2 k}u_d. 
</amsmath></center>
Following this, the dynamics matrix in the next displayed equation (line 19) is again incorrect.  The shifted, closed loop dynamics should be
<center><amsmath>
  \frac{dz}{dt} = \begin{bmatrix}-k_0 - k_1 & k_1 - b_0 k \\ k_2 & -k_2 \end{bmatrix} z.
</amsmath></center>
Finally, the characteristic equation on line 23 is incorrect and should instead be
<center><amsmath>
  \lambda(s) = s^2 + (k_0 + k_1 + k_2) s + (k_0 k_2 + b_0 k_2 k).
</amsmath></center>
}}