Norms of Signals and Systems for Control System Analysis
The use of norms in feedback system analysis was popularized in the 1960's by researchers
such as Zames Zam66b], Sandberg San64], Narendra NG64], and Willems Wil69],
although some norms had been used in control systems analysis before these papers. For
example, the square of the 2 norm is referred to as
in the 1957 book NGK57]
H
I
y
chapter 7 of the 1947 book JNP47], written by Philips, is entitled RMS-Error Criterion
in Servomechanism Design.
A thorough reference on norms for signals and systems in the context of control systems
is Desoer and Vidyasagar DV75]. This book contains general and precise denitions of
many of the norms in this and the previous chapter. Mathematics texts covering many of
the norms we have seen include Kolmogorov and Fomin KF75] and Aubin Aub79]. The
bold in the symbols 2 and
stands for the mathematician G. H. Hardy.
H
H
H
1
The observation that the total variation of the step response is the peak gain of a transfer
function appears in Lunze Lun89].
Singular Value Plots
Singular value plots are discussed in, for example, Callier and Desoer CD82a], Ma-
ciejowski Mac89], and Lunze Lun89]. Analytical properties of singular values and nu-
merical algorithms for computing them are covered in Golub and Van Loan GL89].
The Entropy Interpretation
The simple interpretation of the entropy, as the average square of the 2 norm when a
H
random feedback is connected around a transfer function, has not appeared before. We do
not know how to generalize this interpretation to the case of a transfer matrix, although
it is likely that there is a similar interpretation.
To prove the result (5.27) we consider an arbitrary
, and a complex random
h
2
C
variable uniformly distributed on the disk of radius 1 . We then have
=
2
2
1
2
Z
Z
2
=
h
=
h
E
1
1
r
d
dr
i
;
h
0
0
;
r
e
h
2
2 2
=
log(1
)
;
;
jhj
=
jhj
<
1
jhj
(the integration over can be evaluated by residues). By integrating over , the re-
!
sult (5.27) follows.
Comparing Gains
The result (5.33) is from Boyd and Doyle BD87].
Small Gain Theorem
The small gain theorem from section 5.4.2 is a standard mathematical result. Applications
of this result (and extensions) to feedback system analysis are discussed in Desoer and
Vidyasagar DV75] and Vidyasagar Vid78] see also chapter 10.
NOTES AND REFERENCES
125
State-Space Norm Computations
Using the controllability or observability Gramian to compute 2 norms is standard
H
see for example Fra87]. The Lyapunov equations that arise are nowadays solved nu-
merically by special methods see Bartels and Stewart BS72] and Golub, Nash, and
Van Loan GNL79]. Tables of formulas for the 2 norm of a transfer function, in terms
H
of its numerator and denominator coecients, can be found in Appendix E2 of Newton,
Gould, and Kaiser NGK57]. These tables are based on a method that is equivalent to
solving the Lyapunov equations. (Professor T. Higgins points out that there are several
errors in these tables.)
The result on Hankel norm computation can be found in, e.g., Glo84, 2.3] and Fra87].
x
The result of section 5.6.3 is from Boyd, Balakrishnan, and Kabamba BBK89] see also
Robel Rob89] and Boyd and Balakrishnan BB90]. The method for computing the en-
tropy appears in Mustafa and Glover MG90] and Glover and Mustafa GM89].
A discussion of solving the ARE can be found in AM90]. The method of solving the
ARE based on the Schur form is discussed in Laub Lau79] see also the articles AL84,
Doo81]. Numerical issues of these and other state-space computations are discussed in
Laub Lau85].
Computing Some Other Norms
Computing the peak gain or peak-step norm from a state-space description of an LTI
system is more involved than computing the entropy or the 2 or
norm. Perhaps
H
H
1
the simplest method is to numerically integrate (i.e., solve) the state-space equations
to obtain the impulse or step response matrix.
pk gn could then be computed by
kH
k
numerical integration of the integrals in the formula (5.23). Similarly,
pk step could be
kH
k
determined directly from its denition and the computed step response matrix.
For other norms, e.g.,
wc, there is not even a simple formula like (5.23). Nevertheless
kH
k
it can be computed in several ways we briey mention some here. It can be expressed as
Z
1
wc = sup
( ) ( )
ampl _
slew
(5.44)
kH
k
0 h t w t dt kwk
M
kw
k
M
1
1
which is an innite-dimensional convex optimization problem that can be solved using
the methods described in chapters 13{15. Alternatively, the (innite-dimensional) dual
problem can be solved:
Z
t
wc =
min
ampl
1 + slew
+
( ( ) ( ))
kH
k
M
k
k
M
0 h
;
d
:
1
2
R
: +
R
!
R
This dual problem is unconstrained.
The computation of
wc can also be formulated as an optimal control problem. We
kH
k
include as an additional state, so the dynamics are
w
_ =
+
_ =
=
(0) = (0) = 0
x
Ax
B
w
w
u
z
C
x
x
w
:
The peak and slew-rate limits on can be enforced as the control and state constraints
w
( )
slew
( )
ampl
ju
t
j
M
jw
t
j
M
:
126