On the factorization of trajectory lifting maps
Paulo Tabuada
Dept. of Electrical Engineering
University of Notre Dame
Notre Dame, IN 46556
E-mail: ptabuada@nd.edu
Abstract— Trajectory preserving and lifting maps have been
implicitly used in many recursive or hierarchical control design
techniques. Well known systems theoretic concepts such as
differential flatness or more recent ones such as bisimulations
can be also understood through the trajectory lifting maps
they define. In this paper we initiate a study of trajectory
preserving and lifting maps between affine control systems. Our
main result shows that any trajectory lifting map between two
single-input control affine systems can be locally factored as the
composition of two special trajectory lifting maps: a projection
onto a quotient system followed by a differentially flat output
with respect to another control system.
I. INTRODUCTION
This paper initiates the study of a special class of maps
between control systems having the property of preserving
and lifting (or reflecting) trajectories. The importance of
this class of maps can be recognized by realizing that
several hierarchical or recursive control design techniques
are implicitly based on the existence of such maps. The most
popular example is probably backstepping [SJK97] where the
existence of a stabilizing controller for a control system of
the form:
˙y = f (y)+g(y)v (I.1)
with y ∈ R
n
being the state and v ∈ R being the input can
be extended to a stabilizing controller for the larger system:
˙y = f(y)+g(y)v
˙v = f
(y, v)+g
(y, v)u (I.2)
where (y, v) ∈ R
n+1
is now the state, u ∈ R the input and
g
is assumed to be non-zero in the region of interest. What
is interesting in this design technique, from the perspective
of this paper, is that we can define the map φ(y, v)=y
from the state space of (I.2) to the state space (I.1) with the
following two remarkable properties:
1) For any state trajectory x(t)=(y(t), v(t)) of (I.2),
φ(x(t)) = y(t) is a trajectory of (I.1);
2) For any trajectory y(t) of (I.1) there exists a trajectory
x(t) of (I.2) such that φ(x(t)) = y(t).
Indeed, if x(t)=(y(t), v(t)) is a trajectory of (I.2) then
y(t)=φ(y(t), v(t)) is the trajectory of (I.1) corresponding
This research was partially supported by the National Science Foundation
CAREER award 0446716.
to input v(t). Conversely, if y(t) is a trajectory of (I.1) then
(y(t), v(t)) is the trajectory of (I.2) corresponding to input:
˙v(t) − f
(y(t), v(t))
g
(y(t), v(t))
and satisfying φ(y(t), v(t)) = y(t).
A different scenario where trajectory preserving and lifting
maps also appear is in the study of abstractions of control
systems initiated by Pappas and co-workers [PLS00]. Here,
one starts with a control system Σ
F
defined on some
manifold M and a map φ : M → N to some lower
dimensional manifold and one seeks to construct a control
system Σ
G
with state space N such that φ has property (1).
The motivation behind the construction of Σ
G
is that the
lower dimensionality of Σ
G
renders its analysis simpler and
hopefully properties studied in Σ
G
will lift to Σ
F
under the
right technical assumptions. An instance of this approach
is described in [TP05a] where the problem of designing
trajectories for Σ
F
joining point a to point b is converted
into the problem of designing trajectories for Σ
G
joining
point φ(a) to point φ(b) followed by a constructive procedure
lifting designed trajectories from Σ
G
to Σ
F
.
Differential flatness can also be understood under the
light of trajectory preserving and lifting maps. Given a
differentially flat system Σ
F
equipped with a flat output
φ : R
m
→ R
n
we can always construct the trivial control
system Σ
G
on R
n
defined by ˙y = v where y ∈ R
n
is
the state and v ∈ R
n
the input. Since any curve in R
n
is a trajectory of Σ
G
we immediately have that φ satisfies
property (1). Furthermore, being φ a flat output we also
know that for every trajectory y(t) there exists a trajectory
x(t) of Σ
F
satisfying φ(x(t)) = y(t) which shows that
(2) is also satisfied. However, more is true in this case.
Not only trajectories of Σ
G
can be lifted to trajectories of
Σ
F
as this lifting operation is unique, that is, for every
trajectory y(t) of Σ
G
there is one and only one trajectory
of Σ
F
mapping to y(t) under φ. On the other extreme we
have bisimilar control systems. If Σ
F
is bisimilar to control
system Σ
G
through a relation defined by the graph of a map
φ : M → N, then by definition
1
of bisimulation, (1) is
satisfied and every trajectory of Σ
G
can be lifted not to one
but to a family of trajectories. In more detail we have that
1
See for example [vdS04], [TP04], [Pap03] for a discussion of bisimula-
tion in a systems theoretic context.
Proceedings of the
44th IEEE Conference on Decision and Control, and
the European Control Conference 2005
Seville, Spain, December 12-15, 2005
WeA02.5
0-7803-9568-9/05/$20.00 ©2005 IEEE
4225
for every trajectory y(t) of Σ
G
and for every point x ∈ M
satisfying φ(x)=y(0) there exists a lifting trajectory x
x
(t)
of Σ
F
satisfying φ(x
x
(t)) = y(t) and x
x
(0) = x. The
situations just described correspond to two extreme cases
since in general a trajectory preserving and lifting map does
not admit unique liftings neither admits lifting for every
possible initial condition. However, as we prove in this paper,
every trajectory preserving and lifting map between single-
input control affine systems can be locally factored as the
composition of two trajectory preserving and lifting maps of
the kinds just described.
A related line of inquiry is the study of maps satisfying
property (2) but not necessarily property (1) as was done
in [Gra05] for the extreme case where trajectories can be
lifted for all possible initial conditions. We believe that the
results presented in this paper also offer some insight into
this ”one-sided” aspect of the question of which kinematic
reductions [BLL02] can be seen as particular examples.
The results presented in this paper rely on the so called
geometric approach to nonlinear control [Jur97], [NvdS95]
and are presented in the setting of category theory [Lan71].
Even though category theory only plays a moderate role
in the proof of our results, it the provides a convenient
conceptual setting to study many problems in systems and
control theory. Such approach has already been proved useful
in the study of quotients [TP05b], bisimulations for dynami-
cal, control and hybrid systems [HTP03], mechanical control
systems [Lew00] as well as other problems in systems and
control theory [Elk98]. Due to space limitations we were
forced to eliminate the proofs of the most elementary results.
The interested reader can consult such proofs in [Tab05].
II. N
OTATIONAL PRELIMINARIES
We follow standard terminology and notation in differ-
ential geometry [AMR88]. We will assume all objects to
be smooth unless stated otherwise and by smooth we mean
infinitely differentiable. We will denote by TM the tangent
bundle of a manifold M and by T
x
M the tangent space of M
at x ∈ M spanned by {
∂
∂x
1
,...,
∂
∂x
m
} where (x
1
,...,x
m
)
are the coordinates of x. Similarly we denote by Tf the
tangent map of a map f : M → N while T
x
f denotes
the tangent map of f evaluated at x ∈ M . Recall that T
x
f
maps tangent vectors in T
x
M to tangent vectors T
x
f · X =
Y ∈ T
f(x)
N. For each x ∈ M , T
x
f ∈ L(R
m
, R
n
) where
L(R
m
, R
n
) denotes the space of linear maps from R
m
to
R
n
and m =dim(M ), n =dim(N ). When the dimension
of the kernel of T
x
f does not change with x we say that f
has constant rank. By an affine distribution we will mean a
function assigning to each x ∈ M a an affine space of T
x
M.
Recall that a subset S of a vector space is said to be an affine
space when for any s, s
∈ S we have λs + λ
s
∈ S for any
λ+λ
=1and λ, λ
∈ R. Similarly, a function f(x, y) is said
to be affine in y when f(x, λy+λ
y
)=λf(x, y)+λ
f(x, y
)
in which case it can be written as f (x, y)=α(x)+β(x)u.
The exterior derivative of a real valued map f will be denoted
by df while the Lie derivative of f along vector field X will
be denoted by L
X
f. Iterated Lie derivatives are defined by
the recursion L
0
X
f = f and L
i+1
X
f = L
X
(L
i
X
f).
III. T
HE CATEGORY OF AFFINE CONTROL SYSTEMS
Informally speaking, a category is a collection of objects
and morphisms between the objects and relating the structure
of the objects. If one is interested in understanding vector
spaces, it is natural to consider vector spaces as objects and
linear maps as morphisms since they preserve the vector
space structure. This choice for objects and morphisms
defines Vect, the category of vector spaces. Choosing man-
ifolds for objects leads to the the natural choice of smooth
maps for morphisms and defines Man, the category of
smooth manifolds. In this section we introduce the category
of affine control systems which we regard as the natural
framework to study trajectory lifting morphisms. Besides
providing an elegant language to describe the constructions
to be presented, category theory also offers a conceptual
methodology for the study of objects, affine control systems,
in this case. Since our results are of local nature we define
affine control systems directly on open subsets of Euclidean
space.
Definition 3.1: A local affine control system Σ=
(M,R
o
,F) is defined by the following elements:
1) The state space M , an open subset of R
m
;
2) The input space R
o
;
3) The system map F : M × R
o
→ TM defined by:
F (x, u)=X(x)+
o
i=1
Z
i
(x)u
i
where x ∈ M , u =(u
1
,...,u
o
) ∈ R
o
, X is a vector
field on M and Z
1
,...,Z
o
are linearly independent
vector fields on M.
A local affine control system is said to be single-input when
o =1.
Since we are working locally there is no loss in generality
in assuming that vector fields X, Z
1
,...,Z
o
are globally
defined in M. Furthermore, as we are interested in local
results we will not distinguish between a control system
Σ
F
and its restriction to an open subset M
⊂ M . The
linear independence assumption also results in no loss of
generality when the distribution spanned by Z
1
,...,Z
o
has
constant rank. In this case if, for example, vector field
Z
o
is linearly dependent on the remaining vector fields
Z
1
,...,Z
o−1
we have Z
o
(x)=
o−1
i=1
c
i
(x)Z
i
(x) and the
feedback u
i
= −c
i
(x)+u
i
can be used to cancel Z
o
. The
resulting control system F
(x, u
)=X(x)+
o−1
i=1
Z
i
(x)u
i
can now be identified with a control system with input space
R
o−1
where the linear independence assumption is valid.
Definition 3.2: Let Σ
F
=(M,R
o
,F) and Σ
G
=
(N,R
p
,G) be affine control systems. A map f =(f
1
,f
2
):
M ×R
o
→ N ×R
p
with f
1
: M → N and f
2
: M ×R
o
→ N
is a morphism from Σ
F
to Σ
G
if the following equality holds:
T
x
f
1
(x) · F (x, u)=G(f
1
(x),f
2
(x, u)) (III.1)
4226
To illustrate the notion of morphism consider affine control
system Σ
F
defined by:
˙x
1
= x
2
1
+ x
1
x
2
(III.2)
˙x
2
= x
1
x
2
2
+ x
1
u (III.3)
˙x
3
= x
3
1
x
2
x
2
3
+ x
1
x
3
u (III.4)
and affine control system Σ
G
defined by ˙y = v. To show
that:
f
1
(x
1
,x
2
,x
3
)=x
1
(III.5)
f
2
(x
1
,x
2
,x
3
,u)=x
2
1
+ x
1
x
2
(III.6)
defines a morphism from Σ
F
to Σ
G
we need to show
that (III.1) holds. We first note that T
x
f
1
(x) · F (x, u)=
x
2
1
+ x
1
x
2
. Since G(f
1
(x),f
2
(x, u)) = f
2
(x)=x
2
1
+ x
1
x
2
we conclude that equality (III.1) is satisfied and that f
1
and
f
2
define a morphism from Σ
F
to Σ
G
.
The notion of morphism generalizes the notion of feedback
equivalence so many times used in systems and control
theory. Recall that control systems Σ
F
and Σ
G
, defined by
F (x, u)=X(x)+
o
i=1
Z
i
(x)u
i
and G(y, v)=Y (y)+
o
i=1
W
i
(y)v
i
, respectively, are said to be feedback equiv-
alent when there exists a diffeomorphism in the state space
g(x)=y and an invertible feedback h(x, v)=u = h
x
(v)
such that the feedback transformed system:
F
(y, v)=T
g
−1
(y)
g · X ◦ g
−1
(y)
+
o
i=1
T
g
−1
(y)
g · Z
i
◦ g
−1
(y)h(g
−1
(y),v)
is equal to G(y, v). Note that by using x = g
−1
(y) and
v = h
−1
x
(u) the equality between F
(y, v) and G(y, v) can
be written as:
T
x
g·X(x)+
o
i=1
T
x
g·Z
i
(x)u = Y ◦g(x)+
o
i=1
W ◦g(x)h
−1
x
(u)
which is no more than (III.1) with f
1
(x)=g(x) and
f
2
(x, u)=h
−1
x
(u).
Local affine control systems introduced in Definition 3.1
and morphisms between local affine control systems intro-
duced in Definition 3.2 define the category of local affine
control systems denoted by ACon
l
. It follows from the
affine nature of the considered control systems that mor-
phisms are also affine in the following sense:
Proposition 3.3: Let Σ
F
f
-
Σ
G
be a morphism in
ACon
l
. Then, f
2
(x, u)=α(x)+β(x) · v where α : M →
R
p
and for each x ∈ M , β(x) ∈ L(R
o
, R
p
).
Properties of affine control systems are sometimes easily
studied with the help of a naturally induced affine distribu-
tion.
Definition 3.4: With each local affine control system Σ
we associate an affine distribution A defined by:
A(x)=X(x)+span
R
{Z
1
(x),...,Z
o
(x)}
One can show that studying local affine control systems is
in many ways equivalent to studying affine distributions and
their morphisms [Elk98]. The essence of this correspondence
is the following result that we will use later in the paper.
Proposition 3.5: Let Σ
F
f
-
Σ
G
be a morphism in
ACon
l
, then:
T
x
f
1
(x)(A
F
(x)) ⊆A
G
◦ f
1
(x) (III.7)
Conversely, for any map f
1
: M → N satisfying (III.7)
there exists a unique map f
2
: M × R
o
→ R
p
such that
f =(f
1
,f
2
) is a morphism from Σ
F
to Σ
G
.
This correspondence between morphisms in ACon
l
and
affine distribution preserving maps critically relies on the
affine structure of the control systems. For non-affine control
systems additional assumptions are necessary to conclude
regularity of f
2
as discussed in [Gra03].
IV. T
RAJECTORIES OF AFFINE CONTROL SYSTEMS
A. The path subcategory
Even tough we have already introduced the objects of
study, affine control systems, and presented some of its
properties we have not yet defined the fundamental notion
of trajectory. Once again we will follow a categorical ap-
proach based on Joyal’s and co-workers work on bisimula-
tion [JNW96]. There are two main reasons for following this
approach. One, is that this approach has alredeady proved
useful in studying notions of bisimulation for dynamical,
control and hybrid systems [HTP03]. The other reason, is
that by altering the notion of path objects, defined below,
we can use similar techniques to study different properties
lifted by morphisms.
Definition 4.1: An object Σ
T
of ACon
l
is a path object
if the following hold:
1) M is a connected subset of R containing the origin;
2) The input space is R
0
= {0};
3) The system map T is given by T (t)=(t, 1).
A path or trajectory in a local affine control system Σ
F
is a
morphism Σ
T
p
-
Σ
F
.
Morphism p =(p
1
,p
2
):Σ
T
→ Σ
F
captures the usual
notion of trajectory since equality (III.1) reduces to:
d
dt
p
1
(t)=T
t
p
1
(t) · 1=F (p
1
(t),p
2
(t))
where we have identified the function p
2
defined on M ×{0}
with a function p
2
defined on M. The above definition
is no more than an elegant way of expressing trajectories
through the use of morphisms. At this point it is important
to show that morphisms of control systems have property
(1) mentioned in the Introduction. This immediately follows
from our definition since given a path Σ
T
p
-
Σ
F
in Σ
F
and a morphism Σ
F
f
-
Σ
G
from Σ
F
to Σ
G
it follows
immediately that f ◦ p is a morphism from Σ
T
to Σ
G
,
therefore a path in Σ
G
.
B. Path lifting morphisms
Although morphisms in ACon
l
preserve trajectories by
construction not every morphism reflects or lifts trajectories.
4227
Definition 4.2: Let Σ
F
f
-
Σ
G
be a morphism in
ACon
l
. Morphism is said to be path lifting if for any path
object Σ
T
and any morphism Σ
T
p
-
Σ
G
there exists
a morphism Σ
T
p
-
Σ
F
making the following diagram
commutative:
Σ
F
Σ
T
p
-
p
-
Σ
G
f
?
(IV.1)
A path lifting morphism f is said to be:
• Singular when p
is unique;
• Total when for every x ∈ f
−1
1
(p
1
(0)) there exists
a morphism Σ
T
p
x
-
Σ
F
making diagram (IV.1)
commutative and satisfying p
x1
(0) = x.
It follows immediately from diagram (IV.1) that a necessary
condition for f to be a path lifting morphism is surjectivity of
f
1
. In addition to surjectivity other conditions must hold for
a morphism to be path lifting. The study of such conditions
requires the use of extensions of affine control systems
introduced in the next section.
V. E
XTENSIONS
The operation of extension allows to increase the state
space dimension of a control system while retaining many
of its properties. Extensions will play an important role in
the factorization of path lifting morphisms.
Definition 5.1: Let Σ=(M,R
o
,F) be a local affine
control system. The extension of Σ, denoted by Σ
e
, is defined
by Σ
e
=(M
e
, R
o
,F
e
) where:
1) M
e
= M × R
o
;
2) F
e
((x, u),v)=X(x)+
o
i=1
Z
i
(x)u
i
+
o
i=1
v
i
∂
∂v
i
.
The extension of a control system models the addition of
a pre-integrator to the original dynamics. If we start with a
system of the form ˙x = X(x)+Z
1
(x)u
1
+ ...+ Z
o
(x)u
o
its extension is described by:
˙x = X(x)+Z
1
(x)u
1
+ ...+ Z
o
(x)u
o
˙u
1
= v
1
.
.
.
˙u
o
= v
o
where u
1
,...,u
o
are now regarded as states and v
1
,...,v
o
are new inputs.
Note that the extension Σ
e
of a local affine control system
comes equipped with a morphism Σ
e
π
-
Σ defined by
π
1
(x, u)=x and π
2
((x, u),v)=u. Furthermore, morphism
π is a singular path lifting morphism since any trajectory
p(t)=(p
1
(t),p
2
(t)) in Σ defines a unique trajectory p
e
(t)=
(p
1
(t),p
2
(t)),
d
dt
p
2
(t)
in Σ
e
satisfying π ◦ p
e
= p.
Proposition 5.2: Let Σ
F
f
-
Σ
G
be a morphism in
ACon
l
for Σ
F
=(M, R
o
,F) and Σ
G
=(N, R
p
,G), and
assume that Tf
1
· Z
i
=0for i =1,...,o. Then, f
2
can be
identified with a map f
2
: M → R
p
and there exists a unique
morphism f
e
making the following diagram commutative:
Σ
e
G
Σ
F
f
-
f
e
-
Σ
G
π
?
Furthermore, if f is path lifting and f
1
has constant rank
there exists a vector field K defined on a neighborhood of
every x ∈ M satisfying Tf
1
· K =0and Tf
e
1
· K =0.
Proof: Since f is a morphism we have T
x
f
1
(x)·F (x, u)=
G(f
1
(x),f
2
(x, u)) and assumption Tf
1
· Z
i
=0implies
that T
x
f
1
(x) · F (x, u)=T
x
f
1
(x) · X(x). Therefore, for
any u, u
∈ R
o
it follows that G(f
1
(x),f
2
(x, u)) =
G(f
1
(x),f
2
(x, u
)). From injectivity of G(y, v) in v we
conclude that f
2
(x, u)=f
2
(x, u
) so that we can identify
f
2
with a function on M. Let now f
e
1
=(f
1
,f
2
):M →
N × R
p
.IfV ∈A
F
(x), then:
T
x
f
e
1
· V =(T
x
f
1
· V, T
x
f
2
· V )
=
G(f
1
(x),f
2
(x)),T
x
f
2
· V
∈A
G
◦ f
1
(x) × T
f
2
(x)
R
p
= A
e
G
◦ f
e
1
(x)
It now follows from Proposition 3.5 applied to f
e
1
the
existence of a unique map f
e
2
making (f
e
1
,f
e
2
) a morphism
from Σ
F
to Σ
e
G
. To conclude uniqueness of f
e
assume that g
is another morphism satisfying π◦g = f . Since π◦g = g
1
we
conclude that g
1
= f = f
e
1
and as f
2
e
is uniquely determined
by f
e
1
= g
1
it follows that f
e
= g.
We now turn to the second part of the result and start
by showing that if f is path lifting then for every x ∈ M ,
f
2
|
L
is surjective where L is the submanifold
2
L = f
−1
1
◦
f
1
(x) of M . For any trajectory p in Σ
G
starting at f
1
(x),
there exists a trajectory p
of Σ
F
satisfying f ◦ p
= p,by
assumption. Differentiating f
1
◦ p
1
= p
1
at t =0we get
T
x
f
1
(x) · ˙p
1
(0) = ˙p
1
(0). Since ˙p
1
(0) can be any vector in
A
G
◦ f
1
(x), there must exist a x
∈ M such that T
x
f
1
·
F (x
,u)=G(f
1
(x
),f
2
(x
)) = ˙p
1
(0) and f
1
(x
)=f
1
(x),
that is x
∈ L. We thus conclude that f
2
|
L
must be surjective
in order for A
G
◦ f
1
(x) to be contained in the image of
G(f
1
(x
),f
2
(x
)) with x
∈ L since G(y, v) is injective on
v. Having proved surjectivity of f
2
|
L
we now assume, for the
sake of contradiction, that no vector field K satisfies Tf
1
·
K =0. But this implies that L is a manifold of dimension 0
since the tangent space of L is described by the vector fields
V satisfying Tf
1
·V =0. We thus reach a contradiction since
level set L has at most a countable number of connected
components which prevents f
2
|
L
from being surjective (on
the codomain R). Therefore we conclude the existence of
vector fields K satisfying Tf
1
· K =0and to finalize the
proof we assume, again for the sake of contradiction, that
every vector field satisfying Tf
1
· K =0also satisfies Tf
2
·
K =0. However, this assumption implies that f
2
is constant
2
Recall that since f
1
has constant rank L = f
−1
1
◦f
1
(x) is a submanifold
of M .
4228
on every connected component of L since the tangent space
of L consists of all vector fields V satisfying Tf
1
· V =
0. Furthermore, since L has at most a countable number
of connected components we contradict again surjectivity of
f
2
|
L
thus finishing the proof. 2
VI. M
AIN RESULT
In this section we present and prove our main result.
Its statement requires a variation on the notion of rela-
tive degree usually found in the geometric control theory
literature [Isi96], [NvdS95]. The slightly different notion
presented here will simplify the statement of the main results.
Definition 6.1: Let Σ
F
f
-
Σ
G
be a morphism in
ACon
l
where Σ
F
and Σ
G
are single-input systems. The
relative degree of Σ
F
with respect to f is the natural number
k satisfying:
1) k =0if Tf
1
· Z =0otherwise:
2) k =1if L
Z
f
2
=0;
3) k = i +1 if L
Z
L
j
X
f
2
=0for j =0,...,i− 1 and
L
Z
L
i
X
f
2
=0.
Note that the relative degree is not necessarily well defined at
every point in the state space. However, since our results are
local in nature, we will assume that the state space has been
reduced in order to contain only points where the relative
degree is well defined.
Theorem 6.2: Let Σ
F
f
-
Σ
G
be a path lifting mor-
phism in ACon
l
where Σ
F
and Σ
G
are single input systems.
If Σ
F
has relative degree k with respect to f and f
1
has
constant rank, then there exists a unique total path lifting
morphism Σ
F
h
-
Σ
e
k
G
and a unique singular path lifting
morphism Σ
e
k
G
g
-
Σ
G
making the following diagram
commutative:
Σ
F
f
-
Σ
G
Σ
e
k
G
g
-
h
-
Furthermore, g =(g
1
,g
2
) is given by the natural projections
on the first factor g
1
: N × R
kp
→ N and g
2
:(N × R
p
) ×
R
kp
→ N × R
p
.
Proof: We start by considering the case where Tf
1
· Z =
0,thatisk =0. Let F (x, u)=X(x)+Z(x)u and
G(y, v)=Y (y)+W (y)v and recall that by Proposition 3.3,
f
2
(x, u)=α(x)+β(x)u. Evaluating T
x
f
1
· F (x, u)=
G(f
1
(x),f
2
(x, u)) at u =0provides:
T
x
f
1
· X(x)=Y ◦ f
1
(x)+W ◦ f
1
(x)α(x) (VI.1)
Evaluating now T
x
f
1
· F (x, u)=G(f
1
(x),f
2
(x, u)) for an
arbitrary u ∈ R and using (VI.1) we obtain:
T
x
f
1
· Z(x)=W ◦ f
1
(x)β(x)
Since the left hand side is, by assumption, nonzero it follows
that β(x) must also be nonzero. We can therefore consider
the feedback equivalent system Σ
F
defined by F
(x, u
)=
F
x,
u
−α(x)
β(x)
= X
(x)+Z
(x)u
. Note that f is also a
morphism from Σ
F
to Σ
G
and equality T
x
f
1
· F
(x, u
)=
G(f
1
(x),f
2
(x, u
)) now reduces to:
T
x
f
1
· X
(x)+T
x
f
1
· Z
(x)u
= Y ◦ f
1
(x)+W ◦ f
1
(x)α(x)
+W ◦ f
1
(x)β(x)
u
− α(x)
β(x)
= Y ◦ f
1
(x)+W ◦ f
1
(x)u
(VI.2)
Let now p(t)=(p
1
(t),p
2
(t)) be any trajectory in Σ
G
starting at any y ∈ N, that is, p
1
(0) = y. Consider also
the trajectory p
(t) in Σ
F
satisfying p
2
= p
2
and starting
at any x ∈ M such that f
1
(x)=y, that is, p
1
(0) = x.
Differentiating equality f
1
◦ p
1
(t)=p
1
(t) with respect to
time and using (VI.2) we obtain:
d
dt
f ◦ p
1
(t)=T
p
1
(t)
f
1
· X
◦ p
1
(t)+Z
◦ p
1
(t)p
2
(t)
= Y ◦ f
1
(p
1
(t)) + W ◦ f
1
(p
1
(t))p
2
(t)
= Y ◦ f
1
(p
1
(t)) + W ◦ f
1
(p
1
(t))p
2
(t)
thus showing that f ◦p
1
(t) is the trajectory of Σ
G
correspond-
ing to input p
2
(t). Since trajectories are necessarily unique
it follows that we must have f
1
◦ p
1
(t)=p
1
(t) from which
we conclude that for every trajectory p(t) in Σ
G
starting at
any y ∈ N and for any x ∈ M satisfying f
1
(x)=y there
exists a trajectory p
(t) in Σ
F
starting at x and satisfying
f
1
◦ p
= p. Morphism f is therefore a total path lifting
morphism from Σ
F
to Σ
G
and therefore also a total path
lifting morphism from Σ
F
to Σ
G
as Σ
F
is isomorphic to
Σ
F
.
We now consider the case where Tf
1
· Z =0.By
assumption f
1
has constant rank so that we can apply Propo-
sition 5.2 to factor Σ
F
f
-
Σ
G
as Σ
F
f
e
-
Σ
e
G
π
-
Σ
G
.
Recall that f
e
1
=(f
1
,f
2
) and since by Proposition 5.2
there exists a vector field K such that Tf
1
· K =0and
Tf
e
1
· K =0we conclude that Tf
2
· K =0. This shows
that df
2
is linearly independent of dh
1
,...,dh
n
for any
coordinate description f
1
=[h
1
... h
n
]
T
of f
1
. Therefore,
dim ker(Tf
e
1
)=dimker(Tf
1
) − 1 and f
e
1
=(f
1
,f
2
) has
constant rank since f
1
has constant rank. Note also that
if the relative degree of Σ
F
with respect to f is greater
than one we have L
Z
f
2
=0which combined with Tf
1
·
Z =0implies Tf
e
1
· Z =0. We can therefore apply
Proposition 5.2 again to factor Σ
F
f
e
-
Σ
e
G
π
-
Σ
G
as Σ
F
f
e
2
-
Σ
e
2
G
π
2
-
Σ
e
G
π
-
Σ
G
.Wenowhave
f
e
2
1
= f
e
=(f, Tf
2
· F ) since by Proposition 5.2 f
e
is the
unique morphism determined by f and f
e
is a morphism as
can be seen from:
Tf
e
· F =(Tf
1
· F, Tf
2
· F )=(G ◦ f, Tf
2
· F )=G
e
◦ f
e
Provided that the relative degree of Σ
F
is greater than 2, it
follows that L
Z
L
X
f
2
=0=L
Z
(Tf
2
· X)=L
Z
(f
e
2
)=
Tf
e
2
· Z =0leading to Tf
e
2
1
· Z =0. Also since there exists
a vector field K satisfying Tf
e
1
· K =0and Tf
e
2
1
· K =0
we conclude that dim ker(Tf
e
2
1
)=dimker(Tf
e
1
) − 1=
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