On slide 7, why is there almost no change in the magnitude of the perturbed transfer function, but a large change in the phase?

Richard Murray, 3 Dec 02

It's actually possible to add phase without changing the gain at all, using a "all-pass" transfer function. The simplest example is H(s) = (s-a)/(s+a), which has unity gain but phase that goes between -180 and 0. Unfortunately, the converse is not true: you can't modify the gain without changing the phase.

For the specific graph on slide 7, here's the code that was used to generate it:

m = 1000;				% mass of the car, kg
b = 50;					% damping coefficient, N sec/m
a = 0.2;				% engine lag coefficient
r = 5;					% transmission gain

% Dynamics
veh = tf([1/m], [1 b/m]);		% vehicle
eng = tf([r], [1 a]);			% engine

W2 = 0.2 * tf([1/10 1], [1]);
sysh = veh*eng*(1+W2);
sysl = veh*eng*(1-W2);

bode(veh*eng, sysh, sysl);