Nonlinear Control Sy.. - Free

230 CHAPTER 9. DISSIPATWITY
Figure 9.2: Mass-spring system.
where zmx2 represents the kinetic energy of the mass and Zkxi is the energy stored by
the spring. Since the energy is a positive quantity, we propose E as a "possible" storage
function; thus, we define:
¢r=E=2kxi+2mx2.
Since 0 is continuously differentiable with respect to xl and x2i we can compute the time
derivative of 0 along the trajectories of i/i:
Thus,
a0x.
8x
[kxl 7fl2]L
-/3x2+xf
_Qy2 + Y f.
X2
-m x1 - 1x2
t 0 dt = E(t) > 0
J0 and it follows that the mass-spring system with output i = x2 is dissipative with respect
to the supply rate
w(t) = yf - /3y2.
It is immediately evident that this supply rate corresponds to Q = -0, S = 2, and R = 0,
from where we conclude that the mass-spring system 0 is strictly output-passive.