Note: If the phase is -90deg, we refer to it as a 90 deg phase lag. So it doesnt make sense to say a "phase lag of -90deg" unless one purposely wishes to say "phase lead of 90deg" using two negatives (not recommended).
[Here is the answer assuming that the question relates to the 90 deg phase lag in PI controllers.
Please send me email if this is not what you meant to ask.]
An integral control action is of the form Ki/s. This means that the
contribution of this term to the magnitude of the open loop transfer
function is Ki/|jw| =Ki/|w|. Thus for low frequencies, the integral action
results in a large gain and helps to eliminate offset (or steady state error)
by beating down any disturbances. Integral action is therefore desirable to
combat steady state error. Now, the integral action also has a phase component
of -90deg. So if the original system did not have any integrators (i.e.,
no 1/s terms), then at low frequencies, the original transfer function has a
phase of 0deg. But the introduction of the integral action has now
introduced a phase of -90deg at low frequencies (in general, it introduces a
phase lag of 90deg at low frequencies). Thus the price we pay for eliminating
steady state error by introducing an integral action is the additional phase
lag of 90deg.
A similar reasoning (look at the magnitude and phase of Kp + Ki/(jw) as w->0)
yields the same conclusions (at low frequencies) for PI controller.
The reason why addition of phase lag (at low frequencies) is not a good thing
is that it can reduce
the phase margin of the system: pm=180+phi, where phi is the phase of the open
loop transfer function when its magnitude=1. So if phi was initially -100 and
then becomes say -160, then the pm has reduced from 80deg to 20deg, making the
system less robust. An addition of a -90 phase at low frequencies has exactly
this effect of reducing phi (typically by an amount less than 90deg).
In fact, for the Caltech ducted fan example,
it turns out that the phase margin becomes negative (!) when
a PI controller is used.
[The instability caused by the integral action can be combated using a
derivative action - hence the use of PID controllers].