Nonlinear Control Sy.. - Free

224 CHAPTER 9. DISSIPATIVITY
theory of systems and consider state space realizations. The use of an internal description
will bring more freedom in dealing with initial conditions on differential equations and will
also allow us to study connections between input-output stability, and stability in the sense
of Lyapunov.
9.1 Dissipative Systems
Throughout most of this chapter we will assume that the dynamical systems to be studied
are given by a state space realization of the form
where
(i=f(x,u), uEU, XEX
Sl y = h(x, u), y c Y
U is the input space of functions, u E U : SZ C R --> 1R'. In other words, the functions
in U map a subset of the real numbers into 1R1.
Y the output space, which consists of functions, y E U : SZ C R -> IItp.
X The set X C 1R" represents the state space.
Associated with this system we have defined a function w(t) = w(u(t), y(t)) : U x y -+ ]R,
called the supply rate, that satisfies
J
tl
Iw(t)I dt < oo,
to
that is, is a locally integrable function of the input and output u and y of the system
Definition 9.1 A dynamical system t' 1is said to be dissipative with respect to the supply
rate w(t) if there exists a function ,, : X -+ 1R+, called the storage function, such that for
all xo E X and for all inputs u E U we have
t,
O(x1) < q5(xo) + f
to
w(t) dt. (9.2)
Inequality (9.2) is called the dissipation inequality, and the several terms in (9.2) represent
the following: