Let G(s) be the plant transfer function. The closed-loop transfer
Let wp be such that |G(jwp)| = 1 and ÐG(jwp) = f, then prove that
where phase margin fm = 180 + f, -180° £ f £ 180°.
; wp > wz
and wl be the
frequency at which ÐC(jwl) is maximum over all
w > 0. Show that
ÐC(jwl) = sin-1
wp/wz - 1
wp/wz + 1
20log10|C(jwl)| = 10log10(wp/wz).
The ratio wp/wz is often expressed as m. Make a
table of a few important values of m and fl.
Figure 1: Block Diagram: Problem 3
Let the true system transfer function in the above
Figure 1 be modelled using a multiplicative
uncertainty as [G\tilde](s) = G(s)(1 + L(s)).
The model we have available for design is G(s) and let the
uncertainty magnitude be, lm(jw) = |L(jw)|. The closed-loop system sensitivity function is such that
Prove that for a stable feedback system to satisfy the sensitivity function
constraint the following inequality should be satisfied: