Nonlinear Control Sy.. - Free

8.2. DEFINITIONS 205
Example 8.1 Let X be Rn. Then the usual "dot product" in IRn, defined by
x.y=xTy=xlyl+x2y2+...±xnyn
defines an inner product in IRn. It is straightforward to verify that defining properties (i)-(v)
are satisfied.
For the most part, our attention will be centered on continuous-time systems. Virtually all
of the literature dealing with this type of system makes use of the following inner product
(x, y) = f 0 00 x(t)
y(t) dt (8.7)
where x y indicates the usual dot product in Htn. This inner product is usually referred to
as the natural inner product in G2. Indeed, with this inner product, we have that X = G2,
and moreover
r 00
IIxIIc, = (x, x) = f IIx(t)II2 dt. (8.8)
We have chosen, however, to state our definitions in more general terms, given that our
discussion will not be restricted to continuous-time systems. Notice also that, even for
continuous time systems, there is no a priori reason to assume that this is the only interesting
inner product that one can find.
In the sequel, we will need the extension of the space X (defined as usual as the space
of all functions whose truncation belongs to X), and assume that the inner product satisfies
the following:
(XT, y) = (x, yT) = (xT, yT) (x, y)T
Example 8.2 : Let X = £2, the space of finite energy functions:
X = {x : IR -+ ]R, and satisfy}
f00
IIxIIc2 = (x, x) = x2(t) dt < oo
0
Thus, X,, = Lee is the space of all functions whose truncation XT belongs to £2, regardless
of whether x(t) itself belongs to £2. For instance, the function x(t) = et belongs to Gee even
though it is not in G2. El
Definition 8.2 : (Passivity) A system H : Xe 4 Xe is said to be passive if
(u, Hu)T > 0 Vu E Xe, VT E R+. (8.10)