Feedback Control Theory by John C. Doyle Bruce A. Francis Allen R. Tannenbaum

Author:John C. Doyle, Bruce A. Francis, Allen R. Tannenbaum
Language: eng
Format: epub, pdf
Publisher: Dover Publications

Proof In Lemma 3 take ω0 = 0 to get

or equivalently,

Multiply by σ0:

It can be shown that the left-hand side converges to

as σ0 → ∞ [The idea is that for very large σ0 the function

equals nearly 1 up to large values of ω. On the other hand, log |S(jω)| tends to zero as ω tends to ∞.] So it remains to show that

We can write

where

It is claimed that

To see this, note that since L has relative degree at least 2 we can write

for some constant c and some integer k ≥ 2. Thus as a σ → ∞

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