Realizability
Freudenberg and Looze FL88] refer to some of the realizability constraints (e.g., +
S
= 1 in the classical 1-DOF control system) as algebraic constraints on closed-loop
T
transfer functions. In contrast, they refer to constraints imposed by stability of and the
T
interpolation conditions as analytic.
Internal Stability
Arguments forbidding unstable pole-zero cancellations can be found in any text on classical
control, e.g. Oga90, p606{607]. For MAMS plants, nding a suitable denition of a zero
is itself a dicult task, so this classical unstable cancellation rule was not easily extended.
An extensive discussion of internal stability and state-space representations can be found
in Kailath Kai80, p175] or Callier and Desoer CD82a].
Desoer and Chan's denition appears in DC75]. Their denition has been widely used
since, e.g., in Fra87, p15{17] and Vid85, p99{108].
Parametrization for Stable Plants
The parametrization given in (7.21) appears for example in the articles Zam81], DC81a],
BD86], and chapter 8 of Callier and Desoer CD82a]. In process control, the parametriza-
tion is called the internal model principle, since the controller in gure 7.9 contains a
K
model of
see Morari and Zarou MZ89, ch.3].
P
y
u
Parametrization via Interpolation Conditions
An early version of the interpolation conditions appears in Truxal's 1955 book Tru55,
p308{309]. There he states that if 0 has an unstable zero, so should the closed-loop I/O
P
transfer function . He does not mention unstable plant poles, and his reasoning is not
T
quite right (see BBN90]).
The rst essentially correct and explicit statement of the interpolation conditions appears
in a 1956 paper by Bertram on discrete-time feedback control Ber56]. He states:
In summary, for the classical SASS 1-DOF control system], the following
design restrictions must be considered.
Any zeros of the plant on or outside the unit circle in the -plane must
z
be contained in ( )].
T
z
Any poles of the plant on or outside the unit circle in the -plane must
z
be contained in 1
( )].
;
T
z
He does not explicitly state that these conditions are not only necessary, but also sucient
for closed-loop stability but it is implicit in his design procedure.
Another early exposition of the interpolation conditions can be found in chapter 7 of
Ragazzini and Franklin's 1958 book RF58, p157{158]. The equivalent interpolation
conditions for continuous-time systems rst appear in a 1958 paper by Bigelow Big58].
A recent paper that uses the interpolation conditions is Zames and Francis ZF83]. Inter-
polation conditions for MAMS plants appear in AS84].
NOTES AND REFERENCES
169
Parametrization for MAMS Plants
The results on parametrization of achievable closed-loop transfer matrices in the multiple-
actuator, multiple-sensor case depend on factorizations of transfer matrices. Early treat-
ments use factorization of transfer matrices in terms of matrices of polynomials see
e.g., Ros70] and Wol74] extensive discussion appears in Kai80]. The rst parametriza-
tion of closed-loop transfer matrices that can be achieved with stabilizing controllers ap-
pears in Youla, Jabr, and Bongiorno's articles on Wiener-Hopf design YJB76, YBJ76].
For discrete-time systems, the parametrization appears in the book by Kucera Kuc79].
A more recent version of the parametrization uses factorization in terms of stable trans-
fer matrices, and appears rst in Desoer, Liu, Murray and Saeks DLM80]. The book
by Vidyasagar Vid85, ch.3,5] contains a complete treatment of the parametrization of
achievable closed-loop transfer matrices in terms of stable factorizations. A state-space
parametrization can be found in Francis Fra87, ch.4] or Vidyasagar Vid85].
Parametrization Using Observer-Based Controller
The observer-based controller parametrization was rst pointed out by Doyle Doy84] it
also appears in Anderson and Moore AM90, 9.2], Maciejowski Mac89, 6.4], and a
x
x
recent article by Moore, Glover, and Telford MGT90].
Why We Hear So Much About Stability
We mentioned in section 7.2.2 that any sensible set of design specications will constrain
the four critical transfer matrices more tightly than merely requiring stability. For example,
the specications may include specic nite limits on some norm of the four critical transfer
matrices, such as
11
12
21
22
(7.42)
kH
k
a
kH
k
a
kH
k
a
kH
k
a
u
1
u
1
y
1
y
1
1
2
1
2
whereas internal stability only requires that these norms be nite:
(7.43)
kH
k
<
1
kH
k
<
1
kH
k
<
1
kH
k
<
1:
u
1
u
1
y
1
y
1
1
2
1
2
In any particular problem, a design in which these transfer matrices are extremely large but
stable is just as unacceptable as a design in which one or more of these transfer matrices
is actually unstable. So in any particular problem, the \qualitative" (ane) specication
of internal stability (7.43) will need to be replaced by a stronger \quantitative" (convex
but not ane) specication such as (7.42).
We see so much discussion about the qualitative specication of internal stability for
historical reasons. In Newton, Gould, and Kaiser NGK57, p21] we nd
In the classical view, a feedback control problem could be identied almost
always as a stability problem. To the early workers in the eld, the problem
of assuring stability was nearly always the foremost consideration. ... A bad
controller] caused the system to exhibit sustained oscillations of the output
even though the input was quiescent. This phenomenon, often called hunting,
so plagued the control engineer that even to the present time 1957] it has all
but dwarfed the many other aspects of the feedback control problem.
In Black Bla34] we nd
170