View source for Errata: sign errors in Example 5.18 (noise cancellation)

{{Errata
|Chapter=Dynamic Behavior
|Page number=5-34
|Line=8
|Version=3.1.5
|Date=26 Nov 2024
}}
In Example 5.18 (noise cancellation), there are two sign errors in equation (5.26) that are propagated through the next several lines.  The corrected text should read (with changes in red):

Assuming for simplicity that <math>S=0</math>, introduce <math>x_1=e=z-w</math>, <math>x_2 = a - a_0</math>, and <math>x_3 = b - b_0</math>.  Then
{| width=100%
|-
|
<center><math>
\frac{dx_1}{dt} = a_0 (z - w) + (a - a_0) w + (b - b_0) n
    = a_0 x_1 {\color{red} \boldsymbol{-}} x_2 w {\color{red} \boldsymbol{-}} x_3 n.
</math></center>
| (5.26)
|}
We will achieve noise cancellation if we can find a feedback law for changing the parameters <math>a</math> and <math>b</math> so that the error <math>e</math> goes to zero. To do this we choose
<center><math>
  V(x_1,x_2,x_3)=\frac{1}{2} \bigl( \alpha x_1^2+x_2^2+x_3^2 \bigr)
</math></center>
as a candidate Lyapunov function for
equation (5.26). The derivative of <math>V</math> is
<center><math>
  \dot V = \alpha x_1 \dot x_1 + x_2 \dot x_2 + x_3 \dot x_3
  = \alpha a_0 x_1^2 + x_2 (\dot x_2 {\color{red} \boldsymbol{-}} \alpha w x_1) + x_3 (\dot x_3 {\color{red} \boldsymbol{-}}
    \alpha n x_1).
</math></center>
Choosing
{| width=100%
|-
|
<center><math>
  \dot a = \dot x_2 = {\color{red}+} \alpha w x_1 = {\color{red}+} \alpha w e,\qquad 
  \dot b =\dot x_3 = {\color{red}+} \alpha n x_1 = {\color{red}+} \alpha n e,
</math></center>
| (5.27)
|}
we find that <math>\dot V = \alpha a_0 x_1^2 < 0</math>, and it follows that the
quadratic function will decrease as long as <math>e = x_1 = w - z \neq 0</math>.