Haddad W.M. Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach
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NonlinearBook10pt November 20, 2007
STABILITY OF FEEDBACK SYSTEMS 445
Now, letting t
1
→ t yields
0 = min
u(t)∈U
dJ
∗
(x(t), t)
dt
+ L(x(t), u(t), t)
. (6.87)
Next, noting that
dJ
∗
(x(t), t)
dt
=
∂J
∗
(x(t), t)
∂t
+
∂J
∗
(x(t), t)
∂x
F (x(t), u(t), t),
(6.87) yields
0 = min
u∈U
∂J
∗
(x, t)
∂t
+
∂J
∗
(x, t)
∂x
F (x, u, t) + L(x, u, t)
, x ∈ D,
which is equivalent to (6.85). Finally, (6.86) can be proved in a similar
manner by replacing u(·) with u
∗
(·), where u
∗
(·) is the optimal control.
Next, we provide the converse result to Theorem 6.9.
Theorem 6.10. Suppose there exists a continuously differentiable fun-
ction V : D×R → R and an optimal control u
∗
(·) such that V (x(t
f
), t
f
) = 0,
0 =
∂V (x, t)
∂t
+ H
x, u
∗
(t),
∂V (x, t)
∂x
T
, t
!
, x ∈ D, t ≥ 0, (6.88)
and
H
x, u
∗
(t),
∂V (x, t)
∂x
T
, t
!
≤ H
x, u(t),
∂V (x, t)
∂x
T
, t
!
,
x ∈ D, u(t) ∈ U, t ≥ 0. (6.89)
Then u
∗
(·) solves th e optimal control problem, that is,
J
∗
(x
0
, t
0
) = J(x
0
, u
∗
(·), t
0
) ≤ J(x
0
, u(·), t
0
), u(·) ∈ U, (6.90)
and
J
∗
(x
0
, t
0
) = V (x
0
, t
0
). (6.91)
Proof. Let x(t), t ≥ 0, satisfy (6.83) and, for all t ∈ [t
0
, t
f
], d efi ne
˙
V (x(t), t)
△
=
∂V (x(t), t)
∂t
+
∂V (x(t), t)
∂x
F (x(t), u(t), t). (6.92)
Then, w ith u(·) = u
∗
(·), it follows from (6.88) that
0 =
˙
V (x(t), t) + L(x(t), u
∗
(t), t).
Now, integrating over [t
0
, t
f
] and noting that V (x
f
, t
f
) = 0 yields
V (x
0
, t
0
) =
Z
t
f
t
0
L(x(t), u
∗
(t), t)dt = J(x
0
, u
∗
(·), t
0
) = J
∗
(x
0
, t
0
).
NonlinearBook10pt November 20, 2007
446 CHAPTER 6
Next, for all u(·) ∈ U it follows from (6.88) and (6.92) that
J(x
0
, u(·), t)
=
Z
t
f
t
0
L(x(t), u(t), t)dt
=
Z
t
f
t
0
−
˙
V (x(t), t) + L(x(t), u(t), t) +
∂V (x(t), t)
∂t
+
∂V (x(t), t)
∂x
F (x(t), u(t), t)
dt
=
Z
t
f
t
0
(
−
˙
V (x(t), t) +
∂V (x(t), t)
∂t
+ H
x, u(t),
∂V (x, t)
∂x
T
, t
!)
dt
≥
Z
t
f
t
0
(
−
˙
V (x(t), t) +
∂V (x(t), t)
∂t
+ H
x, u
∗
(t),
∂V (x, t)
∂x
T
, t
!)
dt
=
Z
t
f
t
0
−
˙
V (x(t), t)dt
= V (x
0
, t
0
)
= J
∗
(x
0
, t
0
),
which completes the proof.
Note that (6.88) and (6.89) imply
0 =
∂V (x(t), t)
∂t
+ min
u(t)∈U
H
x, u(t),
∂V (x, t)
∂x
T
, t
!
, (6.93)
which is known as the Hamilton-Jacobi-Bellman equation. It follows from
Theorems 6.9 and 6.10 that the Hamilton-Jacobi-Bellman equation provides
necessary and sufficient conditions for characterizing the optimal control
for time-varying nonlinear dynamical systems over a finite time interval or
the infinite horizon. In the infinite-horizon, time-invariant case, V (·) is
independ ent of t so that the Hamilton-Jacobi-Bellman equation reduces to
the time-invariant partial differential equation
0 = min
u∈U
H(x, u, V
′
T
(x)), x ∈ D. (6.94)
Example 6.6. Consider the controlled nonlinear s calar system
˙x(t) = x
2
(t) + u(t), x(0) = x
0
, t ≥ 0, (6.95)
with performance functional
J(x
0
, u(·)) =
Z
∞
0
[x
2
(t) + u
2
(t)]dt. (6.96)
NonlinearBook10pt November 20, 2007
STABILITY OF FEEDBACK SYSTEMS 447
To obtain the optimal control u
∗
(t), t ≥ 0, that minimizes (6.96) we use
Theorem 6.10. Specifically, for (6.95) and (6.96) the Hamiltonian is given
by H(x, u, V
′
(x)) = x
2
+ u
2
+ V
′
(x)[x
2
+ u]. Now, if follows from (6.90)
that the optimal control is given by
∂H
∂u
= 2u + V
′
(x) = 0 or, equivalently,
u = −
1
2
V
′
(x). Next, using (6.88) with u
∗
= −
1
2
V
′
(x) it follows that
0 = x
2
+ V
′
(x)x
2
−
1
4
[V
′
(x)]
2
. (6.97)
Solving (6.97) as a quadr atic equation gives V
′
(x) = 2x
2
+2x
√
x
2
+ 1, wh ich
implies, with V (0) = 0, V (x) =
2
3
(x
3
+ (x
2
+ 1)
3/2
−1). Hence, the optimal
control is given by u
∗
(t) = −
1
2
V
′
(x(t)) = −x
2
(t) −x(t)
p
x
2
(t) + 1.
6.7 Feedback Linearization, Ze ro Dynamics, and Minimum-
Phase Systems
Recent work involving differential geometric methods [75, 212, 336] has
made the design of controllers for certain classes of nonlinear systems more
methodical. Such fr ameworks includ e the concepts of zero dyn amics and
feedback linearization. Even though the nonlinear stabilization frameworks
presented in this book are based on Lyapunov theory, in certain cases
feedback linearization simplifies th e construction of Lyapunov functions
for nonlinear sy s tems. Here, we present a brief introduction to feedback
linearization needed to develop some of the results in this book. For an
excellent treatment on this subject, the interested reader is referred to [212].
In this section, we consider square (i.e., m = l) nonlinear dynamical
systems G of the form
˙x(t) = f(x(t)) + G(x(t))u(t), x(0) = x
0
, t ≥ 0, (6.98)
y(t) = h(x(t)), (6.99)
where x ∈ R
n
, u, y ∈ R
m
, f : R
n
→ R
n
, G : R
n
→ R
n×m
, and
h : R
n
→ R
m
. We assume that f(·), G(·), and h(·) are smooth, that is,
infinitely d ifferentiable m ap pings, and f(·) has at least one equilibrium so
that, without loss of generality, f(0) = 0 and h(0) = 0. Furthermore, for the
nonlinear dynamical system G we assume that the required properties for
the existence and uniqu en ess of solutions are satisfied, that is, u(·) satisfies
sufficient regularity conditions such that the system (6.98) has a unique
solution forward in time.
The controlled nonlinear system (6.98) is feedback linearizable [210,419]
if there exist a global invertible state transformation T : R
n
→ R
n
and a
nonlinear feedback control law u = α(x) + β(x)v, where α : R
n
→ R
m
and β : R
n
→ R
m×m
satisfies det β(x) 6= 0, x ∈ R
n
, that transf orms
(6.98) into a linear controllable companion form. In this case, standard
NonlinearBook10pt November 20, 2007
448 CHAPTER 6
linear control design techniques can be us ed to synthesize stabilizing linear
feedback controllers for the linearized system. Since the original nonlinear
system and the feedback linearized system are feedback equivalent, that is,
the two systems exhibit the same input-state behavior [17, 69], the linear
control law can be trans formed through the nonlinear feedback and the
state transformation to yield a stabilizing nonlinear controller for the original
nonlinear system.
To elucidate the above discussion consider the nonlinear dynamical
system (6.98) and suppose there exist an invertible state transformation
T : R
n
→ R
n
such that z = T (x) T (0) = 0, and a nonlinear feedback
control law u = α(x) + β(x)v, wh ere α : R
n
→ R
m
and β : R
n
→ R
m×m
satisfies d etβ(x) 6= 0, x ∈ R
n
, that transforms (6.98) into
˙z(t) = Az(t) + Bβ
−1
(x)[u(t) − α(x(t))], z(0) = z
0
, t ≥ 0, (6.100)
where A ∈ R
n×n
, B ∈ R
n×m
, and (A, B) is controllable. Furthermore,
let P ∈ R
n×n
be a positive-definite matrix satisfying the algebraic Riccati
equation
0 = A
T
P + P A + R
1
−P BR
−1
2
B
T
P, (6.101)
where R
1
> 0 and R
2
> 0. Now, the function V (x) = T
T
(x)P T (x) = z
T
P z
satisfies
inf
u∈U
V
′
(x)[f(x) + G(x)u]
= inf
u∈U
z
T
(A
T
P + P A)z + 2z
T
P Bβ
−1
(x)[u − α(x)]
= inf
u∈U
−z
T
R
1
z + z
T
P B[R
−1
2
B
T
P z + 2β
−1
(x)u
−2β
−1
(x)α(x)]
=
−z
T
R
1
z, z
T
P B = 0,
−∞, z
T
P B 6= 0,
(6.102)
and h en ce, V (·) is a control Lyapunov function for (6.100). Hence, it follows
from the results of Section 6.5 that (6.100) is globally stabilizable. In
particular, one such feedback controller is given by (6.81) with x replaced by
z, and hence, a globally stabilizing feedback nonlinear controller for (6.98)
is given by u = φ(T (x)).
The above discussion raises an interesting question, namely, un der
what conditions d oes there exist an invertible state transformation and a
nonlinear control that f eedback linearizes (6.98)? To present necessary and
sufficient conditions for feedback linearization of the nonlinear system (6.98)
the following definitions are n eeded. For the first defi nition, a k-dimensional
vector field f
1
(x), f
2
(x), . . . , f
k
(x), defined on an open subset D, is a mapping
that assigns a k-dimensional vector to each point x of D.
NonlinearBook10pt November 20, 2007
STABILITY OF FEEDBACK SYSTEMS 449
Definition 6.7. A k-dimensional distribution D(·) on D is a mapp ing
that assigns, to each x ∈ D, a k-dimensional subspace D(x) of R
n
such
that there exist smooth vector fields f
1
(x), f
2
(x), . . ., f
k
(x), x ∈ D, with
{f
1
(x), f
2
(x), . . . , f
k
(x)}, x ∈ D, forming a linearly independent set and
D(x) = span{f
1
(x), f
2
(x), . . . , f
k
(x)}, x ∈ D.
Definition 6.8. Let f, G : R
n
→ R
n
be continuously differentiable
functions. The Lie bracket of f and G is defined as
[f(x), G(x)] = ad
f
G(x)
△
=
∂G
∂x
f(x) −
∂f
∂x
G(x).
Furth ermore, the zeroth-order and higher-order Lie brackets are defined as,
respectively,
ad
0
f
G(x)
△
= G(x), ad
k
f
G(x)
△
= [f (x), ad
k−1
f
G(x)],
where k ≥ 1. Finally, define the Lie derivative of a s calar function V (x)
along the vector field of f (x) by
L
f
V (x)
△
=
∂V (x)
∂x
f(x),
and define the zeroth-order and higher-order Lie derivatives, respectively, by
L
0
f
V (x)
△
= V (x), L
k
f
V (x)
△
= L
f
(L
k−1
f
V (x)),
where k ≥ 1.
Definition 6.9. The distribution D(x), x ∈ D, is involutive if [f
i
(x),
f
j
(x)] ∈ D(x), for all f
i
(x), f
j
(x) ∈ D(x), and i 6= j, i, j = 1, . . . , k, wh ere
D(x) = span{f
1
(x), . . . , f
k
(x)}, x ∈ D.
Example 6.7. Let D = R
4
and D(x) = span{f
1
(x), f
2
(x)}, where
f
1
(x) =
x
1
1
0
x
3
, f
2
(x) =
−e
x
2
0
0
0
. (6.103)
To show that D(x) is involutive note that
∂f
2
∂x
f
1
(x) −
∂f
1
∂x
f
2
(x) = 0, which
shows that rank[f
1
(x), f
2
(x), [f
1
(x), f
2
(x)]] = 2, x ∈ D. △
For the statement of the next theorem define the distrib ution
D
i
(x)
△
= span{ad
k
f
G
j
(x), k = 0, 1, . . . , i, j = 1, 2, . . . , m},
for 0 ≤ i ≤ n − 1, where G
j
(x) denotes the jth column of G(x).
Theorem 6.11 ([210]). Suppose rank G(0) = m. Then (6.98) is
feedback linearizable if and only if the following statements hold:
NonlinearBook10pt November 20, 2007
450 CHAPTER 6
i) For each i ∈ {0, . . . , n − 1}, the distribution D
i
(x) has a constant
dimension in a neighborhood of the origin.
ii) For each i ∈ {0, . . . , n − 2}, the distribution D
i
(x) is involutive in a
neighborhood of the origin.
Example 6.8. The dynamics of an undamped single-link manipulator
with flexible joints are given by (6.98) where ([415])
f(x) =
x
2
−α sin(x
1
) −β(x
1
− x
3
)
x
4
γ(x
1
− x
3
)
, G(x) =
0
0
0
δ
, (6.104)
and where α, β, γ, δ > 0. The undisturbed system has an equilibrium at
x = 0. To show that this system is feedback linearizable we compute the
distribution D
i
(x) for each 0 ≤ i ≤ 3 and check conditions i) and ii) of
Theorem 6.11. Specifically, it can be easily shown that at x = 0
ad
f
G(0) =
0
0
−δ
0
, ad
2
f
G(0) =
0
βδ
0
−γδ
, ad
3
f
G(0) =
−βδ
0
γδ
0
.
Now, since det[G(0), ad
f
G(0), ad
2
f
G(0), ad
3
f
G(0)] = β
2
γ
4
for all x ∈ R
4
it follows that D
i
(x), 0 ≤ i ≤ 3, has constant d imension at x = 0.
Furth ermore, since ad
i
f
G(0), 0 ≤ i ≤ 3, are constant, it follows that the
span{G(0), ad
f
G(0), ad
2
f
G(0)} is involutive. Hence, it follows from Theorem
6.11 that the system is feedback linearizable. △
The zero dynamics of the nonlinear system (6.98) and (6.99) are the
dynamics of the system subject to the constraint th at the output y(t), t ≥ 0,
is identically zero. The s ys tem (6.98) and (6.99) is said to be minimum
phase if its zero dynamics are asymptotically stable, while the system is said
to be critically minimum phase if its zero dynamics are Lyapunov stable.
Furth ermore, (6.98) and (6.99) is said to have relative degree {r
1
, r
2
, . . . , r
m
}
at a point x
0
, if there exists a neighborhood D
0
of x
0
such that, for all x ∈ D
0
,
L
G
i
L
k
f
h
j
(x) = 0, 0 ≤ k < r
j
−1, 1 ≤ i, j ≤ m,
and the matrix
L(x)
△
=
L
G
1
L
r
1
−1
f
h
1
(x) ··· L
G
m
L
r
1
−1
f
h
1
(x)
.
.
.
.
.
.
.
.
.
L
G
1
L
r
m
−1
f
h
m
(x) ··· L
G
m
L
r
m
−1
f
h
m
(x)
is nonsingular. In the above notation, L
f
h
j
(x)
△
= h
′
j
(x)f(x), j ∈ {1, . . . , m}
and L
k
f
h
j
(x)
△
= L
f
(L
k−1
f
h
j
(x)), 2 ≤ k < r
j
−1, 1 ≤ j ≤ m, where L
0
f
h
j
(x)
△
=
NonlinearBook10pt November 20, 2007
STABILITY OF FEEDBACK SYSTEMS 451
h
j
(x), G
i
, i = 1, . . . , m, are the sm ooth column vector fields of G, and h
j
,
j = 1, . . . , m, are the smooth components of h. In th e case where the relative
degree {r
1
, r
2
, . . . , r
m
} = {1, 1, . . . , 1},
L(x) = L
G
h(x)
△
=
L
G
1
h
1
(x) ··· L
G
m
h
1
(x)
.
.
.
.
.
.
.
.
.
L
G
1
h
m
(x) ··· L
G
m
h
m
(x)
,
which is nonsingular for all x ∈ D
0
.
Example 6.9. Consider the controlled Van der Pol oscillator
˙x
1
(t) = x
2
(t), x
1
(0) = x
10
, t ≥ 0, (6.105)
˙x
2
(t) = −x
1
(t) + ε(1 −x
2
1
(t))x
2
(t) + u(t), x
2
(0) = x
20
, (6.106)
y(t) = x
2
(t), (6.107)
where ε > 0. Note that (6.105)–(6.107) can be written in the state space
form (6.98) and (6.99) with x = [x
1
, x
2
]
T
, f(x) = [x
2
, −x
1
+ ε(1 −x
2
1
)x
2
]
T
,
G(x) = [0, 1]
T
, and h(x) = x
2
. Now, computing L
G
h(x) =
∂h
∂x
G(x) =
[0, 1][0, 1]
T
= 1, and hence, the system has a relative degree 1 in R
2
. Next,
setting y(t) = x
2
(t) = 0, t ≥ 0, it f ollows that the zero dynamics given by
˙x
1
(t) = 0, t ≥ 0, are not asymptotically stable, and hence, the system is not
minimum phase. However, the system is critically minimum phase. △
For the nonlinear s y s tem ˙x = f(x), x ∈ R
n
, we say that f is complete
if f is an infinitely differentiable function defined on a manifold M ⊂ R
n
and the flow of f is defined on the whole Cartesian product R × M.
Furth ermore, recall (see Definition 2.35) that a mapping T : D ⊆ R
n
→ R
n
is a diffeomorphism on D if T (x) is invertible on D and T (x) and T
−1
(x),
x ∈ D, are continuously differentiable. For nonlinear affine systems of
the form (6.98) and (6.99), conditions for the existence of globally defined
diffeomorphisms transforming (6.98) and (6.99) into several kinds of normal
forms are given in [76]. Here, we only consider relative degree {1, 1, . . . , 1}
systems with complete and involutive vector fields G(L
G
h)
−1
. The f ollowing
result is used later in the book.
Lemma 6.2 ([76]). Assume (6.98) and (6.99) is minimum phase with
relative degree {1, 1, . . . , 1}. If the vector field G(L
G
h)
−1
is complete and
involutive, then there exists a global diffeomorphism T : R
n
→ R
n
, an
infinitely differentiable fun ction f
0
: R
n−m
→ R
n−m
, and an infinitely
differentiable function r : R
n−m
× R
m
→ R
(n−m)×m
such that, in the
co ordinates
y
z
△
= T (x), (6.108)
NonlinearBook10pt November 20, 2007
452 CHAPTER 6
(6.98) is equivalent to
˙y
˙z
=
L
f
h(x)
f
0
(z) + r(z, y)y
+
L
G
h(x)
0
u. (6.109)
As discussed in Section 5.2, dissipativity can be helpful in constructing
Lyapunov functions for nonlinear dynamical systems. Next, usin g the results
in this section, we present necessary and sufficient conditions for rendering a
system passive and strictly passive via feedback. Specifically, suppose (6.98)
and (6.99) is stabilizable. Then, we construct a feedback transformation
u = α(x) + β(x)v, (6.110)
where det β(x) 6= 0, x ∈ R
n
, such that
˙x(t) = f (x(t)) + G(x(t))α(x(t)) + G(x(t))β(x(t))v(t), x(0) = x
0
, t ≥ 0,
(6.111)
y(t) = h(x(t)), (6.112)
is passive. Now, since (6.111) and (6.112) is rendered passive via (6.110)
or, equivalently, (6.98) and (6.99) is feedback passive, v = −φ(y), where
φ : R
m
→ R
m
is such that y
T
φ(x) > 0, y 6= 0, stabilizes (6.111) and (6.112).
First, we show that if (6.98) and (6.99) is passive, then (6.98) and
(6.99) has relative degree {1, 1, . . . , 1} at x = 0.
Lemma 6.3. Assume (6.98) and (6.99) is passive with a two-times
continuously differentiable storage function V
s
: R
n
→ R. Suppose
rank G(0) = rank
∂h
∂x
(0) = m. Th en (6.98) and (6.99) has relative degree
{1, 1, . . . , 1} at x = 0.
Proof. Since (6.98) and (6.99) is passive with a two-times continuously
differentiable storage function V
s
: R
n
→ R it follows from (5.116) of
Corollary 5.2 with J(x) ≡ 0 that
1
2
G
T
(x)V
′T
s
(x) = h(x). (6.113)
Now, differentiating both sides of (6.113) with respect to x yields
1
2
∂
∂x
[G
T
(x)V
′T
s
(x)] =
∂h
∂x
(x). (6.114)
Forming (6.114)G(x), it follows that
1
2
∂
∂x
[G
T
(x)V
′T
s
(x)]G(x) =
∂h
∂x
(x)G(x). (6.115)
Now, noting that at x = 0, V
′
s
(0) = 0 and V
s
′′
(0) is nonnegative definite, it
NonlinearBook10pt November 20, 2007
STABILITY OF FEEDBACK SYSTEMS 453
follows from (6.115) that
1
2
G
T
(0)V
s
′′
(0)G(0) = L
G
h(0). (6.116)
Since V
s
′′
(0) ≥ 0 it follows from the Schur decomposition that there exists
M such that
1
2
V
s
′′
(0) = M
T
M. Hence, L
G
h(0) = G
T
(0)M
T
MG(0). Next,
evaluating (6.114) at x = 0 it follows that
∂h
∂x
(0) = G
T
(0)M
T
M. Now, since
rank G(0) = rank
∂h
∂x
(0) = m it follows that rank M G(0) = m, and hence,
L
G
h(0) is nonsingular. The result is now immediate f rom th e definition of
relative degree.
Next, we show that if (6.98) and (6.99) is passive, then the zero
dynamics of (6.98) and (6.99) are Lyapunov stable. Since, as discussed
in Section 3.5, a L yapunov stable time-invariant system does not ensu re
the existence of a continuously differentiable time-independent Lyapunov
function, we require the following definition.
Definition 6.10. The system (6.98) and (6.99) is weakly minimum
phase if its zero dynamics, evolving on the smooth (n − m)-dimensional
submanifold Z
△
= {x ∈ R
n
: h(x) = 0} described by ˙z = f
0
(z), are Lyapunov
stable and there exists a continuously differentiable positive-definite function
V
0
: R
n−m
→ R such that V
0
(0) = 0 and V
′
0
(z)f
0
(z) ≤ 0, z ∈ Z.
Lemma 6.4. Assume (6.98) and (6.99) is passive (respectively, strictly
passive) with a continuously differentiable positive-definite storage function
V
s
: R
n
→ R. Then (6.98) and (6.99) is weakly minimum p hase (respectively,
minimum phase).
Proof. Since (6.98) and (6.99) is passive with a continuously
differentiable positive-definite storage function V
s
: R
n
→ R it follows from
(5.116) of Corollary 5.2 with J(x) ≡ 0 that
1
2
V
′
s
(x)G(x) = h
T
(x). (6.117)
Next, since by defi nition the zero dynamics of the nonlinear system (6.98)
and (6.99) are the dynamics su bject to the external constraint y(t) =
h(x(t)) ≡ 0, it follows from (6.117) that V
′
s
(x)G(x) = 0, x ∈ R
n
. Now,
using the above two facts it follows that
˙
V
s
(x) = V
′
s
(x)[f(x) + G(x)u] = V
′
s
(x)f(x) ≤ 2u
T
y = 0,
which shows that
˙
V
s
(x) ≤ 0, x ∈ R
n
, along the submanifold h(x) ≡ 0. Thus,
the zero dynamics of (6.98) and (6.99) are Lyapunov stable, and hence,
(6.98) and (6.99) is weakly minimum phase. Finally, if (6.98) and (6.99) is
strictly passive, th en
˙
V
s
(x) < 2u
T
y, and hence, it follows from (6.118) that
˙
V
s
(x) < 0, x ∈ R
n
, x 6= 0, along the submanifold h(x) ≡ 0. Thus, the zero
dynamics of (6.98) and (6.99) are asymptotically stable, and hence, (6.98)
NonlinearBook10pt November 20, 2007
454 CHAPTER 6
and (6.99) is minimum phase.
Next, we present the main result of this section which gives necessary
and sufficient conditions for rendering a nonlinear system f eedback passive.
Theorem 6.12. Consider the nonlinear dyn amical s ystem G given
by (6.98) and (6.99). Suppose the vector field G(L
G
h)
−1
is complete
and involutive. Then the system (6.98) and (6.99) is feedback passive
(respectively, feedback strictly passive) with a two-times continuously
differentiable positive-definite s torage function V
s
: R
n
→ R if and only
if (6.98) and (6.99) has relative degree {1, 1, . . . , 1} at x = 0 and is weakly
minimum phase (respectively, minimum phase).
Proof. For both cases necessity is a direct consequence of Lemmas 6.3
and 6.4. Specifically, if G is feedback passive (respectively, feedback strictly
passive), then there exist functions α : R
n
→ R
m
and β : R
n
→ R
m×m
such
that det β(x) 6= 0, x ∈ R
n
, and with u = α(x) + β(x)v the nonlinear system
(6.111) and (6.112) is passive (respectively, strictly passive). That is,
˙x(t) =
˜
f(x(t)) +
˜
G(x(t))v, x(0) = x
0
, t ≥ 0, (6.118)
y(t) = h(x(t)), (6.119)
is passive (respectively, strictly passive), where
˜
f(x)
△
= f (x) + G(x)α(x) and
˜
G(x)
△
= G(x)β(x).
Next, noting that
˜
G(L
˜
G
h)
−1
= G(L
G
h)
−1
it f ollows that rank
˜
G(0) =
rank
∂h
∂x
(0) = m, and h en ce, by Lemmas 6.3 and 6.4 it follows that the
nonlinear system (6.118) and (6.119) has relative degree {1, 1, . . . , 1} at
x = 0 and is weakly minimum phase (respectively, minimum phase).
Now, it follows from Lemma 6.2 that there exists a global diffeomorphism
T : R
n
→ R
n
, an infi nitely differentiable fu nction f
0
: R
n−m
→ R
n−m
, and
an infinitely differentiable function r : R
n−m
× R
m
→ R
(n−m)×m
such that,
in the coordinates
y
z
△
= T (x), (6.120)
(6.98) is equivalent to
˙y
˙z
=
L
˜
f
h(x)
f
0
(z) + r(z, y)y
+
L
˜
G
h(x)
0
v
=
L
f
h(x)
f
0
(z) + r(z, y)y
+
L
G
h(x)
0
u. (6.121)
The result now follows immediately from th e structure of (6.121) by noting
that relative degree and zero dynamics are invariant under a static s tate
feedback transformation.