Nonlinear Control Sy.. - Free

9.5. AVAILABLE STORAGE
9.4.2 Mass-Spring System without Friction
Consider again the mass-spring system of the previous example, but assume that f3 = 0.
In this case, the state space realization reduces to
I it = x2
i2 = -mxl+f
y = X2
Proceeding as in the previous example, we define
f= E = 2kxi + 2rnx2.
Differentiating m along the trajectories of 0, we obtain:
=x2f =yf
since once again,
rt 0 dt = E(t) > 0. f0 We conclude that the mass-spring system with output i = x2 is dissipative with respect to
the supply rate
w(t) = yf
which corresponds to Q = 0, S = 2, and R = 0. This implies that the mass-spring system
is passive.
9.5 Available Storage
Having defined dissipative systems, along with supply rates and storage functions, we now
turn our attention to a perhaps more abstract question. Given a dissipative dynamical
system, we ask: What is the maximum amount of energy that can be extracted from it, at
any given time? Willems [91] termed this quantity the "available storage." This quantity
plays an important conceptual role in the theory of dissipative systems and appears in the
proofs of certain theorems. For completeness, we now introduce this concept. This section
is not essential and can be skipped in a first reading of this chapter.
Definition 9.5 The available storage, cba of a dynamical system,0 with supply rate w is
defined by
f T
0a(x) = sup - J w(u(t), y(t)) dt, x(0) = X. (9.12)
u() o
T>0
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