Nonlinear Control Sy.. - Free

9.3. QSR DISSIPATIVITY 227
Definition 9.3 Given constant matrices Q E 1Rp> , S E R' m, and R E Rmxm with Q and
R symmetric, we define the supply rate w(t) = w(u, y) as follows:
It is immediately obvious that
and
w(t) = yTQy+2yTSu+uTRu
00
f w
[YT, UT) L
Q RSi 1
L U I (9.6)
(t) dt = (y, Qy) + 2(y, Su) + (u, Ru) (9.7)
T
w(t) dt = (y, QY)T + 2(y, SU)T + (u, Ru)T (9.8)
J0
moreover, using time invariance, w(t) defined in (9.6) is such that
to
to+T
w(t) dt = JO
Thus, we can now state the following definition
Definition 9.4 The system V) is said to be QSR-dissipative if there exist a storage function
X -- ]R+ such that 11x(0) = xo E X and for all u E U we have that
J
0
T
w(t) dt. (9.9)
T
w(t) dt = (y, Qy)T + 2(y, Su)T + (u, Ru)T ? O(xi) - 1(xo).
(9.10)
Definition 9.4 is clearly a special case of Definition (9.1) with one interesting twist. Notice
that with this supply rate, the state space realization of the system V) is no longer essential
or necessary. Indeed, QSR dissipativity can be interpreted as an input-output property.
Instead of pursuing this idea, we will continue to assume that the input output relationship
is obtained from the state space realization as defined in (9.1). As we will see, doing so
will allow us to study connections between certain input-output properties and stability in
the sense of Lyapunov.
We now single out several special cases of interest, which appear for specific choices
of the parameters QS and R.
1- Passive systems: The system V) is passive if and only if it is dissipative with respect
to Q = 0, R = 0, and S = 21. Equivalently, V) is passive if and only if it is (0, 2, 0)dissipative.
The equivalence is immediate since in this case (9.10) implies that
(y, u)T ? O(xi) - O(xo) ? -O(xo) (9.11)
(since O(x) > 0 Vx, by assumption)