8CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
is interpreted to mean that the signal yis the sum of the response of system P21 to
input signal vand system P22 to input signal u. In general, we will not be specific abo ut
the representation of thesystem P. Ifwe do need to be more specific about P,then
P(s) is the Laplace representation and p(t) is the impulse response.
Note that Figure 2.1 is drawnfrom right to left. We use this form of diagram because it
more closely represents the orderin which the systems are written in the corresponding
mathematical equations. We will latersee that the particular block diagram shown in
Figure 2.1 is used as a generic descriptionof a robust control system.
In the case where we are considering a state-space representation, the following notation
is also used. Given P(s), with state-space representation,
sx(s)=Ax(s)+Bu(s)
y(s)=Cx(s)+Du(s),
we associate this description with the notation,
P(s)=AB
C D.
The motivation for this notation comes from the example presented in Section 2.2.4. We
will also use this notation to for state-space representation of discrete time systems
(where sin the above is replaced by z). The usage will be clear from the contextof the
discussion.
2.1.2 An Introduction to Norms
A norm is simply a measure of the size of a vector, matrix, signal, or system. We will
define and concentrateo n particularnorms for each of these entities. This gives us a
formal wayof assessing whether or not the size of a signal is large or small enough. It
allows us to quantify the performance of a system in terms of the size of the input and
output signals.
Unless stated otherwise, when talking of the size of a vector, we will be using the