Note the similarity with Chapter 5. This is important material, but
again try not to get lost in the details. Try to make a table of the main
results in the two chapters - also to highlight the difference between
MIMO and SISO. It is recommended that each student makes such a
table (I think probably the learn the most from doing it themselves).
Here are some additional points to note:
The basis of many of the results in this chapter are the
"interpolation constraints" (they are called so because they
give restrictions on S(s) and T(s) in terms on some fixed values
such as S(z) and S(p)), see p. 215.
+ RHP-zero z. T(s) must have a RHP-zero at z, i.e., T(z)
has a zero gain in the direction of output direction yz of
the zero, and we get: yz^H T(z) = 0 and yz^H S(z) =
yz^H.
+ RHP-pole p. S(s) must have a RHP-zero (!) at p, i.e.
S(p) has a zero gain in the input direction of the output
direction yp of the RHP-pole, and we get: S(p) yp = 0
and T(p) yp = yp.
+ Based on this we can generalize most of the SISO
results, but directions must be taken into account.
+ We define the angle between the pole and zero direction
as phi = arccos | yz^H yp|. If phi = 90 degrees, then the
pole and zero are in completely different directions and
there is no interaction (they maybe considered
separately). If phi = 0 degrees, then they interact as in a
SISO system.
p. 218-219: A plant is Functional controllanle if its normal
rank is equal to the number of outputs.
For example, a plant with fewer inputs than outputs is not
functional controllable. For exampl, this applies to the
"bath-tub" example in Chapter 4, see p. 124-125. In that
example, we have only input (u=T0 - the inlet temperature)
and four outputs (the four tank temperatures). These four
outputs can not be controlled independently as functions of
time, although we can achieve a given "point value" (so it is
state controllable).
p. 220: Note that a time delay in a single element may help (!) if
it reduces the interactions. Obviously, the magnitude of the
time delay which can be factored out from a given output is
always bad for control of that output.
p. 221: Note that the "bad" effect of a RHP-zero may be
moved to particular output, unless the yz has a zero for that
output. For example, if we have a RHP-zero with yz = [0.03
-0.04 0.9 0.43]^T, then one may in theory move the bad effect
of the RHP-zero to any of the outputs (with the other outputs
perfectly controlled). However, in practice, in will be difficult to
avoid the effect of the RHP-zero on output 3, because the zero
direction is mainly in that output - and trying to move it
somewhere else will give large interactions and poor
performance - see Example 6.2 on page 222.
p.222 and 223: Generally, it is a bad idea to try to get a
deceoupled response for a plant with a RHP-zero. You have to
pay for the decoupling by having poor response for all outputs
(i.e., for a nxn plant with 1 RHP-zero in G(s), you get n
RHP-zeros in T by requiring decoupling).
Note that the RGA is a very usefull tool to indicate sensitivity
to uncertainty (and in particular to diagonal input uncertainty
which is always present. We have: A plant G(s) with large
RGA-elements (say larger than 5-10) in the
frequency-range imporrtant for feedback control is
fundamentally difficult to control.
Generally, a diagonal controller (decentralized control) is
insensitive to (multivariable) uncertainty - the problem is that it
will not always give good performance even nominally - for
example, for a plant with large RGA-elements.
p. 228: For a single disturbance with model gd we may have
the performance objective that the H-infinity norm of Sgd is
less than 1. However, if G(s) has a RHP-zero z, then we must
as a prerequisite require that |yz^H gd(z)| is less than 1. That is,
gd(z) must be less than 1 in the output direction of the
RHP-zero.
p. 229: For a single disturbance input saturation poses no
problem if all elements in the vector G^-1 gd are less than 1 at
all frequencies (then we may even achieve perfect control, ie.e
e=0).
If this is violated, then it may still be possible to achieve
acceptable control (|e|<1) if (6.46) is satisfied for all singular
values of G.
For some good exercises to test your understanding of the material in
this chapter, check out Problem 1 in the Trondheim-Exam from 1996
(and its solution).
-Sigurd