Errata: sign errors in Example 5.18 (noise cancellation)

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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 , introduce , , and . Then

(5.26)

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

Choosing

(5.27)

we find that , and it follows that the quadratic function will decrease as long as .