Equivalence of Nonlinear Systems to Prime Systems under Generalized Output Transformations.
Within a linear algebraic framework, we present a new characterization of the class of nonlinear systems which are equivalent to a prime system.  We then introduce a class of generalized output transformations that can be thought of as a generalization to the nonlinear setting of a unimodular transformation in the output space. Our main result gives necessary and sufficient conditions for equivalence to a prime system under a certain group of transformations that includes generalized output transformations.
Introduction
. The problem of characterizing the class of linear systems that
are equivalent to prime systems was first posed and solved by Morse [12]. The group of
transformations considered in [12] included, besides state space change of coordinates
and linear state feedback, output space change of coordinates. Marino, Respondek,
and van der Schaft [10] generalized this result to the nonlinear case. They showed that
the class of smooth affine nonlinear systems that are locally equivalent to prime systems
can be characterized by the properties of two families of involutive distributions
defined on the state manifold.
In this paper we consider the problem of equivalence to a prime system under
a group of transformations that consist of state space diffeomorphism, regular static
state feedback, and generalized output transformation (GOT). In the case where we
restrict ourselves to the output space change of coordinates used by Marino, Respon-
dek, and van der Schaft [10], we obtain a new and simpler characterization for the
class of nonlinear systems which are (locally) equivalent to prime systems. We then
introduce the notion of GOT to identify a larger class of systems equivalent to prime
systems. We proceed in two steps. First, we enlarge the output space by considering
a finite number of the time derivatives of the output as coordinates of new output
space. Then we define a new set of outputs on this enlarged space. This transformation
is invertible in the sense that the new outputs can be expressed as functions of
a finite number of the time derivatives of the original output, and vice versa. That
is, we can recover the original output without any integration. By analogy with the
linear case, these transformations could be called unimodular, in the sense that their
inverse belongs to the same class of transformations. This approach finds a natural
application in control problems such as output tracking and output regulation.
The goal of this paper is to develop a framework and sound theory to study this
Received by the editors July 22, 1996; accepted for publication (in revised form) August 5, 1997;
published electronically September 25, 1998.
http://www.siam.org/journals/sicon/37-1/30706.html
y Secci'on de Control Autom'atico, Departamento de Ingenier'ia El'ectrica, CINVESTAV-IPN,
Apartado Postal 14-740, 07000 M'exico, D.F., M'exico (aranda@ctrl.cinvestav.mx). This work was
done while this author was with the Department of Mathematics and Statistics, Queen's University,
Kingston, ON, Canada.
z Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6, Canada
(ron@rmh.mast.queensu.ca).
new group of transformations and then identify those systems which are equivalent
to a prime system under this group of transformations.
Of course, a necessary condition for this equivalence is that the system be linearizable
by static state feedback. It will be shown that the crucial step is the requirement
that certain linear forms on the extended output space can be constructed such that
their pull-back under the output map coincide with some suitable forms on the state
space. In that respect, it is worth mentioning that the pull-back of a form is always a
well-defined object, as opposed to the push-forward of a distribution, which may fail
to be well defined. Therefore, one completely avoids the projectability-type conditions
as stated in [10].
We present our results within the linear algebraic framework introduced by Di
Benedetto, Grizzle, and Moog [6]. However, it will be shown that our results can be
given a meaningful geometric interpretation in terms of jet bundles [15]. Finally, let us
mention that this work was partially motivated by some results previously obtained
for discrete-time systems [1]. One advantage of our formalism is that it allows a
completely parallel treatment of both the continuous- and the discrete-time cases.
The paper is organized as follows. In section 2 we recall some basic definitions
from the so-called linear algebraic approach [2, 6]. Our main results are contained
in section 3. In subsection 3.1 we obtain new necessary and sufficient conditions
for equivalence to prime system under regular static state feedback, state space dif-
feomorphism, and output space diffeomorphism. In subsection 3.2 we introduce the
notion of GOT and study some of its properties. In subsection 3.3 we derive necessary
and sufficient conditions for equivalence to prime system under regular static state
feedback, state space diffeomorphism, and GOT. Finally, some conclusions and final
remarks are offered in section 4.
2. Linear algebraic framework. To begin with, we recall some basic definitions
from [2, 6]. Consider a nonlinear system \Sigma, described by equations of the form
ae -
(1)
where the state x 2 M , an open and connected subset of R n , the control u
and the output y 2 Y , an open and connected subset of R m . Throughout the paper
the following standing assumptions are made:
A1. The vector fields f(x) and g i (x) and the mapping h(x) are real analytic.
A2. For almost all x 2 M , rank
Let K denote the field of meromorphic functions of a finite number of the variables
0g. The time derivative of a function ' 2 K is defined by
dt
@x
(2)
Notice that the sum in (2) involves only finitely many terms. Let E denote the
K-vector space spanned by fdx; du respectively,
for and fdu (j)
g. The elements of E are differential forms of
degree one, or simply one-forms. The operator d
induces a derivation in E
by
a
a
The relative degree r of a one-form defined to be the least integer such that
fdxg. If such an integer does not exist, set r = 1.
Introduce a sequence of subspaces fH k g of E by
This sequence of subspaces was first introduced in [2, 3] to address the dynamic
feedback linearization problem. It is clear that the sequence (3) is decreasing. Denote
by k   the least integer such that
Assume that We shall explain below the significance of this assumption. In
[2] it was proven that there exists a set of one-forms and a list of
integers such that, for
The integer r i associated to the one-form w i coincides with its relative degree. A set
of one-forms satisfying (5) is called a system of linearizing one-forms.
According to (2), define
@x
@x
Therefore, associated to the system \Sigma, we can define two sequences of subspaces fE k g
and fF k g of E defined by
These two sequences of subspaces were first introduced in [6] to unify different notions
of invertibility for nonlinear systems. The number ae  called
the rank of the system \Sigma. It can be shown [6] that ae
This characterization of rank was introduced in [6] and agrees
with Fliess's definition [7]. Finally, for notational convenience, define fdxg.
Remark 2.1. In paper [6] the notation y (k+1) used instead of
We use the latter notation because, in the next section, the y (k)
will be used to denote the canonical system of coordinates of the extended output
space.
Remark 2.2. Throughout the paper we use the notion of pull-back of a differential
form, as well as the differential forms version of Frobenius theorem. For details, the
reader is referred to [4].
3. Main results.
3.1. Equivalence to prime systems. In this section we present new necessary
and sufficient conditions for equivalence to prime system under state diffeomorphism,
regular static state feedback, and output space diffeomorphism. In a sense, this result
is a particular case of the more general notion of equivalence that we introduce below
and provides a new linear algebraic characterization of the class of systems already
identified in [10].
Definition 3.1. A system \Pi is said to be a prime system if it is of the form
z
z i-
~
.
Definition 3.2. The system \Sigma is said to be equivalent to the prime system \Pi if
there exist
(i) A state diffeomorphism
(ii) a regular static state feedback is, fi(x) is a square
nonsingular matrix;
(iii) an output space diffeomorphism
y 7! ~
such that the transformation of \Sigma under (OE; (ff; fi); /) equals \Pi.
In order to state our first result, we need to introduce some notation. First define
g. If f! is a basis of H k , it is easy to check that
g. However, in general, f!
s g is
not a basis of -
may happen that the latter set is not linearly independent.
Also, recall that F fdh(x)g.
Theorem 3.3. Consider the square nonlinear system \Sigma and suppose that it
satisfies A1 and A2. Then \Sigma is equivalent to prime system \Pi if and only if the
following conditions are satisfied:
completely integrable;
(iv) for completely integrable.
Remark 3.4. Theorem 3.3 can be seen as a dual version of Theorem 4 in [10]. In
particular, conditions (i), (ii), and (iv) of Theorem 3.3 are equivalent, respectively,
to conditions (ii), (i), and (iv) of Theorem 4 in [10]. Also, notice that our conditions
require the construction of a single sequence of subspaces or codistributions.
Remark 3.5. In [2, 3] it has been shown that conditions (i) and (ii) are the
necessary and sufficient conditions for the system \Sigma without outputs to be linearizable
122 E. ARANDA-BRICAIRE AND R. M HIRSCHORN
by state diffeomorphism and regular static state feedback. In particular, condition (i)
is a necessary and sufficient condition for strong accessibility.
Remark 3.6. In the event that the conditions of Theorem 3.3 are satisfied, the
decoupling matrix [9, p. 263], [13, p. 254] of the transformed output ~
full rank m. Therefore, the problem of asymptotically tracking a desired output
y d is transformed into the problem of asymptotically tracking the desired output
~
which is a linear problem in the transformed coordinates.
Proof of Theorem 3.3. Necessity . First notice that the subspaces H k are invariant
under state diffeomorphism and under regular static state feedback [2, 3]. Moreover,
they are independent of the output map. Next we show that the subspace F 0 ae X is
invariant under output space diffeomorphism. Suppose that ~
the chain rule, we have
d ~
@y
This shows that span K fd ~ h(x)g ae span K fdh(x)g. Since y 7! /(y) is a diffeomor-
phism, a similar argument shows that span K fdh(x)g ae span K fd ~ h(x)g, and hence
span K fd ~ fdh(x)g. In a similar manner, one can show that F 0 is invariant
under state space diffeomorphism. Finally, it is clear that F 0 remains unchanged
under state feedback. Thus conditions (i)-(iv) are invariant under all the considered
transformations.
An easy computation shows that conditions (i)-(iv) are satisfied for a prime system
\Pi. Therefore, if system \Sigma is equivalent to a prime system \Pi, conditions (i)-(iv)
necessarily hold true.
Sufficiency . We proceed by induction, going down from First
notice that H k
condition (iii) means that H k
Moreover, by condition (iv) we can assume without loss of generality that the forms
are exact, say By (7), the forms ! k   i can also be written as follows:
Even though the forms ! k   i are linear combinations of the differentials dh j (x),
it is not possible to assert a priori that they are the pull-back of some forms on
the output space. This assertion holds true if and only if the coefficients a ij can be
expressed as functions of the scalar outputs y j . The following lemma states that this
is the case indeed.
Lemma 3.7. The coefficients a ij , for in (8) can be
expressed as functions of y
Proof. Since the forms ! k   i are exact, we have that
Now, taking the exterior product of (9) with the
dhm (x), where
means that that factor is omitted, yields
Since rank that the linear forms dh j (x) are independent. Con-
sequently, (10) implies that da ij 2 span K fdh(x)g. The latter means that a ij are
constant on each submanifold h \Gamma1 . Again, since rank
are coordinates (y of M such that becomes the
canonical submersion (y; q) 7! (y). Therefore,
By Lemma 3.7, we can define ae k   forms on the output space Y by j k
. Then it is clear that the pull-back (see [4]) of the
under the map precisely with the form ! k   i ; that is,
Now suppose that through steps constructed sets of
forms (some of them possibly empty) such that
and
where the notation W (j)
i should be understood elementwise. Therefore, by condition
(iii), we can choose a set of forms W
Moreover, condition (iv) means that we can assume, without loss of generality, that
! ki are exact, say ! and that there are forms j ki defined on the output
space such that, for (j ki ).
Repeat the above construction from
g. Notice that, by construction, each ! i belongs
to one and only one set W k , whence the integer r i is well-defined. Therefore,
is a system of linearizing forms whose list of relative degrees is
g. As a matter of fact, it can be shown (see, e.g., [2, 3]) that
and that
n. Recall that, by construction, the forms ! i are exact, say
It follows that the map x 7! OE(x) is a diffeomorphism. In coordinates z ij , system \Sigma
becomes
z
z ir
The fact that the forms ! (j)
are independent implies that the matrix B(z), whose rows
are b i (z), has full rank. Therefore, the static state feedback
well defined and yields
z
z ir
To conclude the proof, we just need to construct a suitable output space diffeomor-
phism. In order to do so, we need the following result.
Lemma 3.8. Let be the collection of forms defined on the output
space which satisfy
Proof. As in the proof of Lemma 3.7, it is possible to choose a coordinates system
becomes the canonical projection
y. Let w
We have already shown that the a ij can be
expressed as functions of y j only. Since necessarily
@yk . Then, the two-form dj i can be written as dj
jk dy j - dy k . Now, recall that d!
so that, in coordinates
(y; q), the two-form d! i becomes d!
jk dy j - dy k . By construction, the
are exact, so that d! i j 0. Therefore, the coefficients - i
jk must be identically
zero.
By virtue of Lemma 3.8, we can assume, without loss of generality, that, for
Finally, define the output space diffeomorphism y 7! /(y).
In coordinates ~
the system (12) is in prime form.
Example 3.9 (see [10]). Consider the system
defined on M easily compute
Therefore, conditions (i) and (ii) of Theorem 3.3 are satisfied. Moreover, if we choose
we see also that condition (iii)
is satisfied. However, span K fW completely integrable. Therefore,
system (13) is not equivalent to prime system.
3.2. Generalized output transformations. Next we introduce the notion of
GOT. As we point out below, the notion of equivalence studied in the previous section
is a particular case of this new class of transformations.
Definition 3.10. Given two finite nonnegative integers d and d 0 , a GOT consists
of two smooth maps (R m (R m ) d 0
Y such that
~
Even though in general we have ~
called
proper if ~ h is a function of x
In the case when d the GOT (14)-(15) reduces to an output space
diffeomorphism. Notice, however, that in general d 6= d 0 .
Example 3.11. Let Y be two
systems of coordinates of R 3 . Consider the maps
(R 3
It is easy to verify that the pair (/; -) is a GOT with 2.
Remark 3.12. Parallel to the algebraic definition of GOT given by Definition 3.10,
a more geometric interpretation of this class of transformations can be given in terms
of jet bundles (see, e.g., [15]). Consider the output y 2 Y as a function of time and
assume that it is of class C 1 . Then, every trajectory y(t) in the output space defines
a section of the d-jet bundle J d (R; Y ) by t 7! j d
Similarly,
every smooth curve z(t) on R m defines a section of the d 0 -jet bundle J d 0
Therefore, the maps / and - can be interpreted as bundle maps such that the following
diagrams commute:
R \Gamma\Gamma\Gamma\Gamma\Gamma!
id R
R
R \Gamma\Gamma\Gamma\Gamma\Gamma!
id R
R
R is the source map and id R is the identity map. Roughly
speaking, this means that to every smooth trajectory y(t) 2 Y corresponds one and
only one trajectory z(t)
Several types of invariants have been associated with the input-output map of the
system \Sigma, e.g., the relative degrees [9, 13], the structure at infinity [11], the essential
orders [8], and the rank of the system [6, 7]. Among them, the most fundamental is,
without doubt, the rank ae   (see section 2). Theorem 3.13 below states that the rank
is invariant under proper GOTs.
Theorem 3.13. Let (/; -) be a proper GOT, and let ae   and ~ ae   denote, respec-
tively, the rank of the system \Sigma with respect to the output and with respect
to the new output ~
ae
Proof. Let fF k g and f ~
denote, respectively, the sequences of subspaces associated
to the system \Sigma with the output and with the new output ~
Next note that, according to Definition 3.10, we have
d
@y (')
@h (')
126 E. ARANDA-BRICAIRE AND R. M HIRSCHORN
Pick an arbitrary form ! 2 ~
it follows that
there are coefficients a ij such that
This shows that, for
every integer k - 0, we have ~
Similarly, it is easily seen that, for k - 0,
g. Then the subspaces fF k g and f ~
~
Now suppose that ~
ae   ! ae   , say ae
ae   for some integer r ? 0. We will show
that this leads to contradiction. First note that, for every k - n, we have
ae
dim ~
ae
Next let \Theta
some sets of linear forms such that \Theta 0 is a
basis of ~
Fn , \Theta 0 [ \Theta 1 is a basis of Fn+s , \Theta 0 [ \Theta 1 [ \Theta 2 is a basis of ~
Fn+2s , etc. Note
that, by construction, ' i - 0 for every integer i - 1.
By definition of the sets \Theta i and by (18), we have that
dim ~
Solving for ' 2k+2 we obtain
By (19), ' 2k+2 becomes negative for k large enough, which is a contradiction. This
concludes the proof.
3.3. Equivalence under generalized transformations.
Definition 3.14. The system \Sigma is said to be equivalent to the prime system \Pi
under proper GOT if there exist
(i) a state diffeomorphism
(ii) a regular static state feedback
(iii) a proper GOT (/; -)
such that the transformation of \Sigma under (OE; (ff; fi); (/; -)) equals \Pi.
Remark 3.15. We restrict ourselves to proper GOTs because we are studying
equivalence to prime systems, for which the output is a function of the state
only.
Theorem 3.16. Consider the square nonlinear system \Sigma and suppose that it
satisfies A1 and A2. Then \Sigma is equivalent to prime system \Pi under proper GOT if
and only if the following conditions are satisfied:
completely integrable;
Remark 3.17. A system which satisfies the hypothesis of Theorem 3.3 (and
hence is equivalent to prime system \Pi) also satisfies conditions (i)-(iv) above. Clearly,
conditions (i)-(ii) hold. Next, note that condition (iii) of Theorem 3.3 implies H k   ae
Therefore condition (iii)
holds. Finally, from Remark 3.6 we can see that condition (iv) holds as well.
In order to prove Theorem 3.16, we need to introduce some notation. For a given
nonnegative integer d, the extended state space, extended output space, and extended
output map are defined, respectively, by M (R m ) d+1 , Y (R m ) d , and
(R m
Also, we will need the following technical result.
Proposition 3.18. The rank ae   of the system \Sigma is equal to m if and only if, for
every integer N ? 0, dim
Proof. First suppose that ae  but that for some integer N we have
necessarily, there is a form dh (N)
i and coefficients c k
ij such
that
dh (N)
c N
The latter implies that for every integer -
N ? N we have
Let us recall [6] that, for N ? n,
(20) implies that dim E -
In particular, if we choose -
have that ae
which is a contradiction. The converse is
obvious.
Proof of Theorem 3.16. Necessity . It is clear that conditions (i)-(iv) are satisfied
for a system in prime form. Moreover, conditions (i)-(ii) are invariant under state
diffeomorphism and regular static state feedback. On the other hand, Theorem 3.13
states that the rank ae   is invariant under proper GOTs. It remains to prove that
condition (iii) is also invariant under proper GOTs. This part of the proof will be
broken down into two lemmas.
Lemma 3.19. Let
0g. Then we have
Proof. Pick an arbitrary linear . Then there are coefficients b ij
and an integer N - 0 such that
Next note that, by Proposition 3.18, the forms fdh (j)
are
linearly independent. Then, necessarily, dh (j)
concludes that N We have shown that
. On the other hand, it is obvious that X "
Lemma 3.20. Let (/; -) be a proper GOT, and let fF k g and f ~
tively, the sequences of subspaces associated with the system \Sigma with the output
and with the new output ~
128 E. ARANDA-BRICAIRE AND R. M HIRSCHORN
Proof. In the proof of Theorem 3.13 we have shown that, for every integer k - 0,
we have ~
Pick an arbitrary form ! 2 ~
implies that
This proves that ~
is a GOT, a symmetric argument
shows that
Sufficiency . Conditions (i)-(ii) imply that the system \Sigma is linearizable by regular
static state feedback. Therefore, we can assume, without loss of generality, that \Sigma is
in the form (12). Next notice that condition (iii) implies that, for
d
a k
Lemma 3.21. The coefficients a k
ij appearing in (21) can be expressed as functions
of fy (j)
Proof. By Proposition 3.18, the forms fdh (j)
are
independent. Therefore, a similar argument to that used in Lemma 3.7 can be used
to conclude the proof.
By Lemma 3.21, we can define m one-forms on the extended output space Y d by
d
a k
Then it is clear that the pull-back of the forms j i under the extended output map
coincides precisely with the forms dz i1 ; that is, dz
e (j i ).
Lemma 3.22. Let be the collection of forms defined on the extended
output space Y d which satisfy dz
e (j i ). Then, for
Proof. By Proposition 3.18, the extended output map Y d has full rank
equal to m(d 1). Therefore, there exists a system of coordinates of the extended
state space M d such that becomes the canonical projection. A similar
construction as in Lemma 3.8 shows then that dj
By virtue of Lemma 3.22, we can assume, without loss of generality, that, for
Finally define a new output function ~
y by
~
this change of output variables, system (12) is in prime
form. It remains to prove that there is an inverse map
y (j \Gamma1)
. Therefore, it follows
that necessarily
for some integer d 0 - 0.
Example 3.23 (Example 3.9, continued). We have shown that system (13) is
not equivalent to a prime system under standard output space transformations, i.e.,
output space diffeomorphism. We shall show that system (13) is equivalent to prime
system under proper GOTs. We have already shown that conditions (i)-(ii) are sat-
isfied. It is easy to check that
Note that X ae F 2 and, since F 2 ae F 5 , we have thus condition
(iii) is also satisfied. Finally, lengthy but straightforward computations show that
ae  Consequently, system (13) is equivalent to prime system.
Since the state equations are already in the form (1), in order to transform system
into prime form, we just need to find a suitable GOT. This can be accomplished
as follows: first note that
e (dy 3
By integrating the right sides of (22), we find the GOT (y; -
In coordinates (~y 1 ; ~
system (13) is in prime form. The inverse output
transformation is obviously given by (~y; - ~
As pointed out before, one immediate application of Theorems 3.3 and 3.16 is the
possibility of (asymptotically or exactly) tracking a desired output trajectory y d (t).
This is best illustrated by the following example.
Example 3.24 (Example 3.23, continued). The decoupling matrix [9, 13] associated
with the original output system (13) is given by
noninteracting control cannot be achieved by regular static state
feedback.
On the other hand, the output functions which bring system (13) to prime form
are given, as functions of x, by
~
~
~
The decoupling matrix ~
associated with the output ~
simply
~
. Therefore the standard noninteracting feedback [9, 13] can be used to
decouple the scalar output components ~ y i , Now suppose that we want
to asymptotically track a smooth output trajectory y d
system (13). Such trajectory is transformed in the new coordinates into ~ y d
(y d1 (t); y d2
y d3 (t); y d3 (t)). Since (13) has been transformed into a prime
system, the asymptotic output tracking problem is solved by linear state feedback in
the transformed coordinates, namely
y d3 );
are Hurwitz polynomials.
4. Conclusion and final remarks. We have introduced the notion of GOT
for nonlinear systems and have shown that the linear algebraic framework introduced
by Di Benedetto, Grizzle, and Moog [6] provides a rather convenient tool to study
their properties. In particular, it has been shown that the rank of a system remains
unchanged under such transformations.
It is worth mentioning that the class of GOT that we have introduced can be seen
as the "dual" transformation of the class of generalized state feedbacks introduced in
[14] and studied from the differential algebraic viewpoint in [5], where they were called
quasi-static state feedbacks.
As an important application of this new class of transformations, a larger class
of systems which are equivalent to prime systems has been identified. In turn, this
result is applicable to control problems where output transformations are naturally
allowed, such as output tracking and output regulation.
Of course, the conditions of Theorems 3.3 and 3.16 imply that the system \Sigma is
invertible [6, 7], and hence noninteracting control can be achieved by dynamic state
feedback. In that respect, Theorems 3.3 and 3.16 avoid the addition of extra dynamics
to the system, as pointed out in [10].
An open issue for further research is the study of the notion of equivalence of
nonlinear systems under GOTs, not necessarily proper.



--R


A linear algebraic framework for dynamic feedback linearization
Infinitesimal Brunovsky form for non-linear systems with applications to dynamic linearization

Paris S'er.

Automatique et corps diff'erentiels
The essential orders and nonlinear decoupling
Nonlinear Control Systems
Equivalence of nonlinear systems to input-output prime forms

Structural invariants of linear multivariable systems
Nonlinear Dynamical Control Systems
Some canonical properties of nonlinear systems

--TR
prime systems;nonlinear systems;differential forms;output transformation