Chapters 5 and 6 discuss inherent control limitations imposed by the plan G(s)
which cannot be overcome by any controller. Try not to get swamped with all
the details in this chapter. Look quickly through the material and try to
understand the significance of the controllability rules.
Some key points:
The definition of (input-output) controllability used in this chapter is
as follows:
A plant is controllable if we at each frequency can keep the control
error e = y- r less than 1, for any disturbance d with magnitude less
than 1 (and in particular for |d|=1), and for any reference change r with
magnitude less than R (and in particulat for |r|=R), using manipulated
inputs u which are less than 1 in magnitude.
This definition assumes that the variables have been scaled as outlined
in Chapter 1, page 6. SCALING IS IMPORTANT !
- A lot of insight into the inherent limitation imposed by the properties of
the plant (its "controllability") can be obtained by considering the
idealized case of perfect control, where we must have u = G^-1 r -
G^-1 G_d d; see p. 163-164.
- First waterbed formula: If you push |S| down somewhere to get good
performance, it will peak up somewhere else
- Second waterbed formula (for plants with RHP-zeros): Peak is even
higher
- RHP-zeros are bad! (we show this in many ways...). We must have
T(z)=0, so we cannot have tight control close to the RHP-zero.
- For RHP-poles we must have T(p)=1, so we must have tight control
close to the RHP-pole (the opposite of a RHP-zero!) Thus, RHP-poles
may be bad, especially if combined with RHP-zeros, see (5.49), or if
combined with input saturation, see (5.61).
- Fast control is required to reject large disturbances (with a large |Gd|).
This may be inconsistent with the presence of RHP-zeros and time
delays, see (5.53) and (5.75).
- Large input signals is required to reject large disturbances, which may
be inconsistent with input saturation, see (5.59).
- Feedforward control is sensitive to disturbances, especially if |Gd| is
large (say, larger than 5-10) at some frequency.
- Make sure you understand everything in the first application example on
p. 291-293.