Nonlinear Control Sy.. - Free

Chapter 8
Passivity
The objective of this chapter is to introduce the concept of passivity and to present some
of the stability results that can be obtained using this framework. Throughout this chapter
we focus on the classical input-output definition. State space realizations are considered in
Chapter 9 in the context of the theory of dissipative systems. As with the small gain theorem,
we look for open-loop conditions for closed loop stability of feedback interconnections.
8.1 Power and Energy: Passive Systems
Before we introduce the notion of passivity for abstract systems, it is convenient to motivate
this concept with some examples from circuit theory. We begin by recalling from basic
physics that power is the time rate at which energy is absorbed or spent,
where
Then
P(t) = dwd(t)
power
w(.) . energy
t . time
(8.1)
w(t) = J t p(t) dt. (8.2)
to
Now consider a basic circuit element, represented in Figure 8.1 using a black box.
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