Filter
Top Products
14 CHAPTER 2. OVERVIEW OF THE UNDERLYING THEORY
robust control model is therefore a set description and we hope that some members of
this set capture some of the uncertain or unmodeled aspects of our physical system.
For example, consider the “uncertain” model illustrated in Figure 2.2. This picture is
equivalent to the input-output relationship,
y =[(I+∆W
m
)P
nom
]u. (2.2)
Figure 2.2: Generic output multiplicative perturbation model
In this figure, ∆, W
m
and P
nom
are dynamic systems. The most general form for the
theory can be stated with these blocks as elements of H
∞
. For the purposes of
calculation we will be dealing with Xmath Dynamic Systems, and in keeping with this
we will tend to restrict the theoretical discussion to RH
∞
, stable, proper real rational
transfer function matrices.
The only thing that we know about the perturbation, ∆, is that ∆
∞
≤ 1. Each ∆,
with ∆
∞
≤ 1 gives a different transfer function between u and y. The set of all
possible transfer functions, generated in this manner, is called P. More formally,
P =
(I +∆W
m
)P
nom
∆
∞
≤ 1
. (2.3)
Now we are looking at a set of possible transfer functions,
y(s)=P(s)u(s),
where P(s) ∈P.
Equation 2.2 represents what is known as a multiplicative output perturbation
structure. This is perhaps one of the easiest to look at initially as W(s) can be viewed