Errata: sign errors in Example 5.18 (noise cancellation)

Chapter	Dynamic Behavior
Page	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 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 S=0} , introduce 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 x_1=e=z-w} , 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 x_2 = a - a_0} , and 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 x_3 = b - b_0} . Then

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 \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. }

(5.26)

We will achieve noise cancellation if we can find a feedback law for changing the parameters $a$ and $b$ so that the error $e$ goes to zero. To do this we choose

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 V(x_1,x_2,x_3)=\frac{1}{2} \bigl( \alpha x_1^2+x_2^2+x_3^2 \bigr) }

as a candidate Lyapunov function for equation (5.26). The derivative of 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 V} 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} \boldsymbol{-}} \alpha w x_1) + x_3 (\dot x_3 {\color{red} \boldsymbol{-}} \alpha n x_1). }

Choosing

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, }

(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} .