NonlinearBook10pt November 20, 2007
814 CHAPTER 13
appropriate modifications, is also valid for discrete-time systems.
13.11 Linearization of Dissipative Dynamical Systems
In this section, we present several key results on linearization of dissipative,
geometrically dissipative, passive, geometrically passive, nonexpansive, and
geometrically nonexpansive systems. For these results, we assume that there
exists a function κ : R
l
→ R
m
such that κ(0) = 0 and r(κ(y), y) < 0,
y 6= 0, and the available storage function (respectively, geometrically storage
function) V
a
(x), x ∈ R
n
, is a three-times continuously differentiable function.
Theorem 13.24. Let Q ∈ S
l
, S ∈ R
l×m
, R ∈ S
m
, and suppose G given
by (13.135) and (13.136) is completely reachable and dissipative with respect
to the quadratic supply rate r(u, y) = y
T
Qy + 2y
T
Su + u
T
Ru. Then, there
exist matrices P ∈ R
n×n
, L ∈ R
p×n
, and W ∈ R
p×m
, with P nonnegative
definite, such that
P = A
T
P A − C
T
QC + L
T
L, (13.179)
0 = A
T
P B − C
T
(QD + S) + L
T
W, (13.180)
0 = R + S
T
D + D
T
S + D
T
QD − B
T
P B − W
T
W, (13.181)
where
A =
∂f
∂x
x=0
, B = G(0), C =
∂h
∂x
x=0
, D = J(0). (13.182)
If, in addition, (A, C) is observable, then P > 0.
Proof. First note that since G is dissipative with respect to a quadratic
supply rate there exists a continuous nonnegative function V
s
: R
n
→ R such
that
V
s
(f(x) + G(x)u) −V
s
(x) ≤ r(u, y), x ∈ R
n
, u ∈ R
m
. (13.183)
Next, it follows from (13.183) that there exists a three-times continuously
differentiable function d : R
n
× R
m
→ R such that d(x, u) ≥ 0, d(0, 0) = 0,
and
0 = V
s
(f(x) + G(x)u) −V
s
(x) − r(u, h(x) + J(x)u) + d(x, u). (13.184)
Now, expanding V
s
(·) and d(·, ·) via a Taylor series expansion about x = 0
and u = 0, and using the fact that V
s
(·) and d(·, ·) are non negative definite
and V
s
(0) = 0, d(0, 0) = 0, it follows that there exist matrices P ∈ R
n×n
,
L ∈ R
p×n
, and W ∈ R
p×m
, with P nonnegative defin ite, such that
V
s
(x) = x
T
P x + V
sr
(x), (13.185)
d(x, u) = (Lx + W u)
T
(Lx + W u) + d
sr
(x, u), (13.186)