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 
, introduce 
, 
, and 
. Then
|
(5.26)
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We will achieve noise cancellation if we can find a feedback law for changing the parameters 
and 
so that the error 
goes to zero. To do this we choose
as a candidate Lyapunov function for
equation (5.26). The derivative of 
is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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}-} \alpha w x_1) + x_3 (\dot x_3 {\color{red}-} \alpha n x_1). }
Choosing
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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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, }
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(5.27)
|
we find that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \dot V = \alpha a_0 x_1^2 < 0}
, and it follows that the
quadratic function will decrease as long as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e = x_1 = w - z \neq 0}
.