Nonlinear Control Sy.. - Free

Chapter 9
Dissipativity
In Chapter 8 we introduced the concept of a passive system (FIgure 9.1). Specifically, given
an inner product space X, a system H : Xe -4 Xe is said to be passive if (u, HU)T > Q.
This concept was motivated by circuit theory, where u and y are, respectively, the voltage
v(t) and current i(t) across a network or vice versa. Thus, in that case (assuming X = G2)
(u, Hu)T = MO, i(t))T = v(t)i(t) dt
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which represents the energy supplied to the network at time T or, equivalently, the energy
absorbed by the network during the same time interval. For more general classes of dynamical
systems, "passivity" is a somewhat restrictive property. Many systems fail to be
passive simply because (u, y) may not constitute a suitable candidate for an energy function.
In this chapter we pursue these ideas a bit further and postulate the existence of an input
energy function and introduce the concept of dissipative dynamical system in terms of a nonnegativity
condition on this function. We will also depart from the classical input-output
u
H
Figure 9.1: A Passive system.
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T
y