12 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
where σmax denotes the maximum singular value. Not all matrix norms are induced
from vector norms. The Froebenius norm (square root of the sum of the squar eso f all
matrix elements) is one such example.
Now consider the case where P(s) is a dynamic system and we define an induced norm
from L2to L2as follows. In this case, y(s) is the output of P(s)u(s)and
kP(s)k=max
u(s)∈L2
ky(s)k2
ku(s)k2
.
Again, for a linear system, this is equivalent to,
kP(s)k=max
u(s)∈BL
2
ky(s)k
2
.
This norm is called the ∞-norm, usually denoted by kP(s)k∞. In the single-input,
single-output case, this is equivalent to,
kP(s)k∞= ess sup
ω|P(ω)|.
This formal definition uses the term ess sup, meaning essential supremum. The
“essential” part means that we drop all isolated points from consideration. We will
always be considering continuous systems so this technical point makes no difference to
us here. The “supremum” is conceptually the same as a maximum. The difference is
that the supremum also includes the case where we need to use a limiting series to
approach the valueof interest. Thesame is true of the terms “infimum” (abbreviated to
“inf”) and “minimum.” For practical purposes, the reader can think instea d in terms of
maximum and minimum.
Actually we could restrict u(s)∈H
2in the above and the answer wouldbe the sa me. In
other words, we can look over all stable input signals u(s) and measure the 2-norm of
the output signal, y(s). The subscript, ∞, comes from the fact that we are looking for
the supremum of the function on the ω axis. Mathematicians sometimes refer to this
norm as the “induced 2-norm.” Beware of the possible confusion when reading so me of
the mathematical literature on this topic.
If we were using the power norm above (Equation 2.1) for the input and output norms,
the induced norm is still kP(s)k∞.