Nonlinear Control Sy.. - Free

9.2. DIFFERENTIABLE STORAGE FUNCTIONS 225
the storage function; O(x(t`)) represents the "energy" stored by the system V) at
time t'.
fto' w(t) dt: represents the energy externally supplied to the system zG during the
interval [to, t1].
Thus, according to (9.2), the stored energy l4(xi) at time t1 > to is, at most, equal to
the sum of the energy O(xo) initially stored at time to, plus the total energy externally
supplied during the interval [to, t1]. In this way there is no internal "creation" of energy. It
is important to notice that if a motion is such that it takes the system V) from a particular
state to the same terminal state along a certain trajectory in the state space, then we have
(since x1 = xo)
0(Xo) < 0(Xo) + i w(t) dt
= J w(t)dt > 0 (9.3)
where f indicates a closed trajectory with identical initial and final states. Inequality (9.3)
states that in order to complete a closed trajectory, a dissipative system requires external
energy.
9.2 Differentiable Storage Functions
In general, the storage function 0 of a dissipative system, defined in Definition 9.1 need
not be differentiable. Throughout the rest of this chapter, however, we will see that many
important results can be obtained by strengthening the conditions imposed on 0. First we
notice that if 0 is continuously differentiable, then dividing (9.2) by (t1 - to), and denoting
O(xi) the value of O(x) when t = ti, we can write
but
and thus (9.4) is satisfied if and only if
_(xl) - 0(Xo) < 1
tl - to tl - to
w(t) dt
lim O(x1) - 0(Xo) = dO(x) = 00(x) f(x, u)
tl-*to t1 - to dt 8x
0a(x) f (x, u) < w(t) = w(u, y) = w(u, h(x, u)) Vx, it. (9.5)