Let 1 ... n be the unstable poles of 0 (i.e., the poles of 0 that have nonneg-
p
p
P
P
ative real part) and let 1 ... m be the unstable zeros of 0 (i.e., the zeros of 0
z
z
P
P
that have nonnegative real part) we will assume for simplicity that they are distinct
and have multiplicity one. Let denote the relative degree of 0, i.e., the di erence
r
P
between the degrees of its numerator and its denominator. Then a transfer matrix
is achievable with a stabilizing controller if and only if it has the form (7.23),
H
where satis es:
T
1. is stable,
T
2. ( 1) = = ( n) = 1,
T
p
T
p
3. ( 1) = = ( m) = 0, and
T
z
T
z
4. the relative degree of is at least .
T
r
These conditions are known as the interpolation conditions (on they can also
T
be expressed in terms of or other closed-loop transfer functions). The interpolation
S
conditions can be easily understood in classical control terms. Condition 2 re ects
the fact that the loop gain 0 is in nite at the unstable plant poles, and so we
P
K
have perfect tracking ( = 1) at these frequencies. Conditions 3 and 4 re ect the
T
fact that there is no transmission through 0 at a frequency where 0 has a zero,
P
P
and thus = 0 at such a frequency. (Internal stability prohibits an unstable pole
T
or zero of 0 from being canceled by a zero or pole of .)
P
K
The interpolation conditions are also readily understood in terms of our descrip-
tion of stable given in (7.20). Substituting the plant transfer matrix for the 1-DOF
H
control system (2.10) into (7.20) and using =
0 we get:
R
T
=P
stable =
of form (7 23)
0 (1
) 0 are stable
(7.24)
H
fH
:
j
T
T
=P
;
T
P
g
:
Assuming is stable,
0 will be stable if vanishes at 1 ... m and in addition
T
T
=P
T
z
z
has relative degree at least that of 0 in other words,
0 is stable if conditions
T
P
T
=P
1, 3, and 4 of the interpolation conditions hold. Similarly, (1
) 0 will be stable
;
T
P
if is stable and 1
vanishes at 1 ... n (i.e., conditions 1 and 2 of the
T
;
T
p
p
interpolation conditions hold).
The interpolation conditions are the earliest description of stable, and date
H
back at least to 1955 (see the Notes and References at the end of this chapter for
details).
7.2.6
General Free Parameter Representation
In the general case there is a free parameter description of the set of closed-loop
transfer matrices achievable with stabilizing controllers:
stable = 1 + 2 3
is a stable u y transfer matrix
(7.25)
H
fT
T
QT
j
Q
n
n
g
7.3 MODIFIED CONTROLLER PARADIGM
157
where 1 2 and 3 are certain stable transfer matrices that depend on the plant.
T
T
T
is referred to as the parameter in (7.25), not in the sense of a real number that is
Q
to be designed (e.g., the integrator time constant in a PI controller), but rather in
the sense that it is the free parameter in the description (7.25). We saw a special
case of this form already in the example of the stable plant|in that case, zw, zu
P
P
and yw are possible choices for 1, 2 and 3, respectively.
P
T
T
T
The controller that stabilizes the plant and yields closed-loop transfer matrix
= 1 + 2 3 has the linear fractional form
H
T
T
QT
Q = ( +
) 1( +
)
(7.26)
;
K
A
B
Q
C
D
Q
where
are certain stable transfer matrices related to 1, 2, and 3.
A
B
C
D
T
T
T
Thus the dependence of Q on is bilinear (c.f. equation (7.10)).
K
Q
It is not hard to understand the basic idea behind the free parameter represen-
tation (7.25) of the set of achievable closed-loop transfer matrices (7.20), although
a complete derivation is fairly involved (see the Notes and References at the end of
this chapter).
We consider the subspace of u y transfer matrices given by
n
n
=
yu
yu yu yu are stable
S
fS
j
S
P
S
S
P
P
S
P
g
:
The basic idea is that an
must have the appropriate zeros that cancel the
S
2
S
unstable poles of yu. These zeros can be arranged by multiplying a stable transfer
P
matrix on the left and right by appropriate stable transfer matrices and ~:
D
D
= n
~
is stableo
S
D
QD
Q
:
and ~ are not unique, but any suitable choice has the property that if is stable,
D
D
Q
then each of
~, yu ~,
~ yu, and yu ~ yu are stable. We shall not
D
QD
P
D
QD
D
QD
P
P
D
QD
P
derive the form of and ~.
D
D
By comparing (7.20) and (7.25) we see that one possible choice for 2 and 3
T
T
in (7.25) is
2 = zu
T
P
D
3 = ~ yw
T
D
P
:
1 can be taken to be any closed-loop transfer matrix achieved by some stabilizing
T
controller. The references cited at the end of this chapter contain the complete
details.
7.3
Modified Controller Paradigm
The descriptions of stable given in the previous sections can be given an interpre-
H
tation in terms of modifying a given nominal controller that stabilizes the plant.
158