Systems & Control Letters 55 (2006) 71 – 80
www.elsevier.com/locate/sysconle
Normal forms and approximated feedback linearization in discrete time
S. Monacoa , D. Normand-Cyrotb,∗
a Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza”, Via Eudossiana 18, 00184 Rome, Italy
b Laboratoire des Signaux et Systèmes, CNRS-Supélec, Plateau de Moulon, 91190 Gif-sur-Yvette, France
Received 6 January 2004; received in revised form 14 April 2005; accepted 22 April 2005
Available online 27 June 2005
Abstract
The paper discusses approximated feedback linearization of nonlinear discrete-time dynamics which are controllable in first approximation
and introduces two types of normal forms. The study is set in the context of differential/difference representations of discrete-time
dynamics proposed in [Monaco, Normand-Cyrot, in: Normand-Cyrot (Ed.), Perspectives in Control, a Tribute to Ioan Doré Landau,
Springer, Londres, 1998, pp. 191–205].
© 2005 Elsevier B.V. All rights reserved.
Keywords: Normal forms; Approximated linearization; Nonlinear discrete-time systems
1. Introduction
The idea of simplifying the nonlinearities of a given
discrete-time dynamics through coordinates change and
feedback, launched in [14] in continuous-time control theory finds its roots in Cartan’s method of equivalence or
Poincaré’s normal forms [24]. It has been more recently further developed and renewed making reference to controlled
dynamics (see [13,7,11,23] and the references therein). On
these bases, stabilizing strategies for dynamics with bifurcations have been proposed in [12]. While the approach
can be similarly developed for both cases of vector fields
(differential dynamical systems) and maps (discrete-time
systems) [24], such a parallelism becomes difficult when
dealing with forced dynamical systems. Even if many
analogies can be set, differentiated studies are necessary. In
discrete time most of the contributions are concerned with
quadratic or cubic normal forms as this is in general enough
to characterize control properties: quadratic approximated
feedback linearization under dynamic feedback is studied
in [1], stabilization of systems with uncontrollable modes
or bifurcations in [8], observer design for systems with
∗ Corresponding author.
E-mail address: [email protected] (D. Normand-Cyrot).
0167-6911/$ - see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.sysconle.2005.04.016
unobservable modes in [3]. In [15], quadratic and cubic normal forms are introduced to propose a systematic classification of discrete-time bifurcations taking place at equilibria
due to loss of linear stabilizability. Following [15], homogeneous normal forms of degree m have been proposed in
[9] for dynamics with controllable linear part.
With respect to these contributions [1,8,15,9], the problem
is presently set and solved for dynamics controllable in first
approximation in the formalism of differential/difference
representations of discrete-time dynamics proposed in [19].
Such a set up, which does not imply any loss of generality
in the present context, makes it possible to give for the first
time a quite complete answer to the problem: two types of
normal forms are proposed; the generic case of degree m is
solved; the invariants are introduced and their role is clarified for achieving approximated feedback linearization. The
advantage of the proposed approach is even more striking
when considering sampled dynamics as illustrated by the
examples worked out throughout the paper.
The study is addressed step-by-step, through homogeneous approximations of increasing degree of the Taylorlike expansions of the dynamics, coordinates changes and
feedbacks. For each degree of approximation, say m, writing down the so-called homological equations which must
be solved for achieving linearization, normal forms of
72
S. Monaco, D. Normand-Cyrot / Systems & Control Letters 55 (2006) 71 – 80
degree m containing all the nonremovable nonlinear terms
are characterized. As in continuous time [11,23], two kinds
of normal forms are introduced depending whether one privileges cancellation of the nonlinear terms in the drift (dual
normal form) or in the control vector fields (Kang’s normal form). Provided the linear part and lower degree terms
are fixed, homogeneous normal forms at a fixed degree are
unique modulo homogeneous transformations of the same
degree and a set of polynomials which are invariants under
homogeneous transformations, the so called homogeneous
invariants, are defined. The nullity of these invariants characterizes homogeneous feedback linearization at a fixed degree. It must be stressed that the normal forms here developed are different from those introduced in previous work
for discrete-time dynamics in the form of maps. Preliminary results were given in [22] and in [21] with reference to
quadratic approximations.
The paper is organized as follows. Section 2 is devoted
to define the context and set the problem. Homogeneous
transformation, feedback and feedback linear equivalences
are formulated in the proposed differential/difference set up.
Sections 3 and 4 contain the results. Homogeneous feedback
and feedback linear equivalences at degree m are characterized either through the solvability of the homological equations of degree m or the nullity of the invariants. Merging the
results, necessary and sufficient conditions ensuring approximated feedback equivalence are given. On these bases, two
different types of homogeneous normal forms and extended
normal forms are described in Section 4. Two examples are
discussed. Notations are introduced in the sequel.
Notations: The state variables and/or x belong to X,
an open set of R n and the control variables v and/or u belong to U, a neighborhood of zero in R. All the involved
objects, maps, vector fields, control systems are analytic on
their domains of definition, infinitely differentiable admitting convergent Taylor series expansions. A vector field on
X, analytically parameterized by u, G(x, u) ∈ Tx X defines
a u-dependent differential equation of the form dx + (u)/du=
G(x + (u), u) where the notation x + (u) indicates that the
state evolution is a curve in R n , parameterized by u. A R n valued mapping F (., u) : x → F (x, u), denotes a forced
discrete-time dynamics while F : x → F (x) and/or F (., 0)
denotes unforced evolutions. Given a generic map on X, its
evaluation at a point x is denoted indifferently by “(x)” or
“|x ”. Jx F |x=0 = (dF (x)/dx)|x=0 indicates the Jacobian of
the function evaluated at x = 0. Given a vector field G on
X and assuming that F is a diffeomorphism on X, F∗ G denotes the transport of G along F , defined as the vector field
on X verifying F∗ G|F =(Jx F )G; analogously indicating by
p
F p =F ◦. . .◦F , the p-times composition of F , F∗ G denotes
p
the transport of G along F p verifying F∗ G|F p = (Jx F p )G.
The upperscript (.)[m] stands for the homogeneous term of
degree m of the Taylor series expansion of the function or
vector field into the parentheses. Analogously, R [m] (.) (resp.
R m (.)) stands for the space of vector fields or functions
whose components are polynomials (resp. formal power series) of degree m (resp. of degree m) in the variables into
the parentheses. The results are local in nature and convergence problems are not addressed so that the solutions proposed will be referred to as formal ones.
2. Context and problem statement
We consider throughout the paper a single-input discretetime dynamics, → F (, v), which is controllable in first
approximation around the equilibrium pair (0, 0). Without
loss of generality as justified in the sequel, we make use of
the differential/difference representation (DDR) introduced
in [19] to describe such a dynamics; i.e. consider
+ = F (),
d+ (v)
= G(+ (v), v);
dv
(1)
+ (0) = + ,
(2)
where G(., v) admits the
expansion around
Taylor-type
i / i!)G
(v
0; G(., v) := G1 +
i+1 with G1 :=
i 1
i
i
G(., 0); Gi+1 = (j G(., v)/jv )|v=0 for i 1; F (0) = 0
and G1 (0) = 0.
To get more familiar with the representation (1–2), let the
following comments.
• Provided completeness of the vector field G(., v), the associated flow is defined for any v, a nonlinear difference
equation → F (, v) can be recovered integrating (2) between 0 and v(k) with initialization at (1), + (0) = + =
F ((k)); we get
(k + 1) = + (v(k)) = F ((k), v(k))
v(k)
G(+ (w), w) dw.
= F ((k)) +
0
An explicit exponential representation of F (., v) in terms
of the Gi is given in [20].
• Reversing the arguments and starting from a difference
equation → F (, v), the existence of (1–2) follows
from the existence of G(., v) verifying G(F (., v), v)) =
jF (., v)/jv. The invertibility of F (., 0) is sufficient to
prove that G(., v) can be locally uniquely defined as
G(., v) := (jF (., v)/jv)|F −1 (.,v) .
• The proposed formalism provides a new paradigm for
modeling discrete-time as well as hybrid phenomena coupling continuous-time and discrete dynamics with jumps,
switches and resets. It makes possible the complementary
use of geometric and algebraic techniques so providing
equivalent formalism and tools between continuous time
and discrete time; a parallelism which is lost in the usual
context of discrete-time dynamics in the form of maps as
soon as nonlinear dynamics are concerned. Finally, let us
note that the study of sampled dynamics can always be
performed in such a context due to the invertibility of the
drift under sampling.
S. Monaco, D. Normand-Cyrot / Systems & Control Letters 55 (2006) 71 – 80
It is now clear that, due to the controllability assumption,
there is no loss of generality to consider, eventually after
feedback, a representation of the form (1–2). In the sequel
[∞] indicates the DDR
+ = A +
F [m] (); + (0) = + ,
(3)
m2
m
v i−1
d (v)
=B +
G[m−i] (+ (v))
dv
(i − 1)! i
+
(4)
m 2 i=1
modified under preliminary linear feedback to put its linear
approximation
73
’s are, respectively, R n and Rwhere [m] and the [m−i]
i
valued mappings. We immediately note that [m] does not
modify the linear part (A, B) of [m] .
[m]
˜
be another dynamics of the form (8–9). The
Let following definitions are given.
Definition 2.1 (Homogeneous feedback equivalence at de˜ [m] if there exists
gree m). [m] is feedback equivalent to an homogeneous feedback transformation [m] which brings
˜ [m] modulo terms in R m+1 (, v).
[m] into (6)
Definition 2.2 (Homogeneous feedback linear equivalence
at degree m). [m] is locally feedback linear equivalent
if there exists a homogeneous feedback transformation
[m] which brings [m] into (A, B) modulo terms in
R m+1 (, v).
(7)
Definition 2.3 (Feedback linear equivalence). [∞] is locally feedback linear equivalent if there exists a feedback
transformation [∞] which brings [∞] into (A, B). If the
equivalence holds modulo terms in R M+1 (, v), the approximated feedback linear equivalence up to degree M is
obtained.
The pair (A, B) corresponds to the first-order Euler approximation of the sampled Brunovsky form [4] with normalized
sampling period equal to 1.
Given [∞] let, for any degree of approximation m 2,
[m]
be the homogeneous approximation of degree m around
(A, B) of [∞] ; i.e.
The next theorem recalled from [18,5] (see also different
approaches proposed in [6,17,16], expresses in the present
setup necessary and sufficient geometric conditions ensuring
feedback linearization.
+ = J F |0 = A; + (0) = + ,
d+ (v)
= G1 (0) = B
dv
into the controllable
⎛1 1
0
⎜0 ... ...
⎜
⎜.
..
A = ⎜ ..
.
⎜
⎝
0
...
+
form
. . . 0⎞
.. ⎟
.⎟
⎟
..
. 0⎟ ,
⎟
..
. 1⎠
... 0 1
[m]
(5)
+
⎛ ⎞
0
⎜ ... ⎟
⎟
B =⎜
⎝0⎠ .
1
+
= A + F (); (0) = ,
m
d+ (v)
v i−1
=B +
G[m−i] (+ (v)).
dv
(i − 1)! i
(8)
(9)
i=1
Remarks. (i) (8) is the approximation of degree m of (3)
around the linear evolution A, while (9) is the approximation of degree m − 1 of (4) around B.
m
i−1 /(i − 1)!)G[m−i] (+ (v)) in
(ii) The term
i=2 (v
i
(9) models nonlinearities with respect to the control
= 0 for i 2, (9) reduces to
variable. Setting G[m−i]
i
[m−1] +
( (v)) and the results further stated are strongly
B +G1
comparable with those obtained in the continuous-time case
for input-affine dynamics. This will be clarified later on.
As we do not want to modify the linear part in [∞] , a
feedback transformation [∞] is defined in the present context as the successive application of homogeneous feedback
transformations of degree m 2, [m] . Each [m] is composed of a coordinates change and a static-state feedback of
the form
x=+
[m]
(),
(10)
m
ui [m−i]
v = [m] (, u) = u + [m]
()
+
(),
0
i! i
i=1
(11)
Theorem 2.1 (Monaco and Normand-Cyrot [18]). The
discrete-time dynamics (1–2) is locally feedback linear
equivalent if and only if
(i) span(G2 , G3 , . . .) ⊂ span(G1 ),
(ii) the distribution (G1 , . . . , F∗n−2 G1 ) is involutive
around 0,
(iii) rank(G1 (0), . . . , F∗n−1 G1 (0)) = n.
It is now possible to formulate the general question asked
in this paper: up to what extent is it possible to simplify
the nonlinearities of [∞] and thus to achieve linearization
through coordinates change and invertible feedback [∞] .
The problem is solved step by step. For each degree of approximation m2, we look for [m] under which [m] is
simplified at most as possible while leaving unchanged the
linear part and parts of degree < m.
3. Approximated feedback linear equivalence
Given [m] , homogeneous feedback linear equivalence at
degree m corresponds to complete cancellation under [m]
of the terms of degree m in (8) and of degree m − 1 in
(9); when this is not achievable, the remaining terms, the
so called resonance terms of degree m describe the normal
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S. Monaco, D. Normand-Cyrot / Systems & Control Letters 55 (2006) 71 – 80
forms of degree m. Thanks to the introduced formalism, approximated feedback linear equivalence can be reported either to the solvability of a family of equations, the homological equations of degree m or the nullity of the invariants of
degree m, a set of polynomials of degree m − 2 which are
invariant under transformations of degree m. These aspects
are discussed first with reference to a generic degree m and
then generalized to approximated feedback linearization.
3.1. The homological equations
d[m] ()
B,
d
(x) = G[m−1]
() +
Ḡ[m−1]
1
1
(x) = G[m−i−1]
();
Ḡ[m−i]
i
i
i = (2, . . . , m).
(12)
(13)
(14)
Then the feedback action further transforms (12) to (14) into
F̃ [m] (x) = F [m] () + [m] (A)
− A[m] () + [m]
0 ()B,
m
i=1
i=1
× (A
0
Proposition 3.1. The homogeneous feedback equivalence
problem is solvable at degree m if and only if there exist
([m] , [m−i]
; i = (0, . . . , m)) satisfying
i
[m−i]
(A−1 − vA−1 B)B
i
(v) − vA
and up to an error in
+ (v)|v=[m]
The following results are immediate consequences of the
equalities (15–16).
−1
(18)
ui−1
())
(G̃[m−i] (x) − G[m−i]
i
(i − 1)! i
m
d[m] () ui−1 [m−i]
+
dn
(i − 1)! i
× (A
−1
i=1
−1
− uA
B)B.
(19)
Proposition 3.2. The homogeneous feedback linear equivalence problem is solvable at degree m if and only if there
; i = (0, . . . , m)) satisfying
exist ([m] , [m−i]
i
−F [m] () = [m] (A) − A[m] () + [m]
0 ()B,
ui−1 [m−i]
(i − 1)! i
i=1
−1 +
(17)
and G[m−i]
while for i 2, G̃[m−i]
differ from their last
i
i
in
component only. To write down the expression of G̃[m−i]
i
requires
first
to
express
the
expansion
with
terms of G[m−i]
i
(A−1 − uA−1 B).
respect to u of [m−i]
i
=
= A−1 + (v) − vA−1 B;
dv = du +
d[m] ()
dn
+ [m−1]
(A−1 )B,
1
i=1
because, up to an error in R m ()
m
(16)
Remark. Setting u = 0 in (16), we get
i=1
i−1
u
(i − 1)!
ui−1
(i − 1)!
i=1
[m−i]
−1 +
× i
(A x (u) − uA−1 B)B.
m
d[m] () ui−1
+
G[m−i] ()
dn
(i − 1)! i
+
m
+ [m]
0 ()B,
ui−1
G̃[m−i] (x)
(i − 1)! i
m
× G[m−i]
(x + (u)) +
i
F̃ [m] (x) − F [m] () = [m] (A) − A[m] ()
m
=
i=1
G̃[m−1]
(x) = G[m−1]
() +
1
1
Let us work out the action of [m] over [m] . First, the
) into
coordinates change [m] transforms (F [m] , G[m−i]
i
)
below
(F̄ [m] , Ḡ[m−i]
i
F̄ [m] (x) = F [m] () + [m] (A) − A[m] (),
m
dx + (u) d[m] (x + (u)) ui−1
=
+
du
dxn
(i − 1)!
−
B) du
m
i=1
R m+1 (, v)
+
ui−1
d[m] ()
() =
G[m−i]
i
(i − 1)!
dn
m
i=1
= + (0) + [m]
0 B;
(20)
ui−1 [m−i] −1
(A − uA−1 B)B.
(i − 1)! i
(21)
Eqs. (20)–(21) are referred to as the homological equations of degree m.
x + (v) = + (v) + [m] (+ (v)).
[m]
In conclusion, [m] brings the system [m] into ˜
described by
3.2. The invariants of degree m
x + = Ax + F [m] (x) + [m] (Ax)
Homogeneous feedback linear equivalence at degree m
can be formulated rewriting Theorem 2.1 in terms of homogeneous approximations.
− A[m] (x) + [m]
0 (x)B,
(15)
S. Monaco, D. Normand-Cyrot / Systems & Control Letters 55 (2006) 71 – 80
75
Corollary 3.1. The homogeneous feedback linear equivalence problem is solvable at degree m if and only if
Proof. Generalizing the equality (17) to i = (0, . . . , n − 3),
we can show that, up to an error in [m − 1],
, . . . , G[0]
(i) span(G[m−2]
m ) ⊂ span(B),
2
(ii) the distribution (B + G[m−1]
, . . . , An−2 B + (F∗n−2
1
G1 )[m−1] ) is involutive around 0 modulo terms in
R m−1 (),
(iii) rank(B+G[m−1]
(0), . . . , An−1 B+(F∗n−1 G1 )[m−1] (0))
1
= n, modulo terms in R m−1 ().
(F̃∗i G̃1 )[m−1] = (F∗i G1 )[m−1] +
(F∗i G1 )[m−1] indicates the homogeneous part of degree
m − 1 of F∗i G1 computed from F () = A + F [m] () and
() describing [m] . More precisely,
G1 () = B + G[m−1]
1
i
i
F∗ G1 = A B + m 2 (F∗i G1 )[m−1] with
(F∗i G1 )[m−1] = A(F∗i−1 G1 )[m−1] |A−1 +
dF [m] ()
|A−1 Ai−1 B.
d
(22)
It can be shown that when the conditions set in Corollary 3.1
are not satisfied, a certain set of polynomials of degree m−2
cannot be equal to zero. These polynomials are invariant
under homogeneous feedback transformation so that they are
called the homogeneous invariants of degree m. Generalizing
the notion of characteristic numbers proposed in [1] for
quadratic approximations, we introduce these polynomials
below. Let I the identity matrix on R n , C = (1, 0, . . . , 0)T
and let us denote by i the projection on Wi = (x ∈ R n :
xi+1 = · · · = xn = 0); i.e. i (x) = (x1 , . . . , xi , 0, . . . , 0). The
study is performed in the (x, u) variables.
Definition 3.1. Given [m] , two sets of invariants of de[m]j,i+2
[m]j
gree m denoted respectively by a1
(x) and ap (x) for
(p = 2, . . . , m − 1) are defined as follows.
• For 1 j n − 2 and 0 i n − j − 2, the polynomials
[m]j,i+2
a1
(x) are defined as the homogeneous parts of
degree m − 2 of the polynomials
C(A − I )j −1 [(F∗ − I )i G1 , (F∗ − I )i+1 G1 ]
× n−i (x).
(23)
[m]j
• For 1j n − 1, the ap (x) are defined as the polynomials of degree m − p below
[m−p]
C(A − I )j −1 Gp
(x).
(24)
The formal operator (F∗ − I )i works out as follows:
(F∗ − I )G1 = F∗ G1 − G1 ; (F∗ − I )2 G1 = F∗2 G1 − 2F∗ G1 +
G1 ; ...; (F∗ − I )i+1 G1 = F∗ (F∗ − I )i G1 − (F∗ − I )i G1 .
Theorem 3.1. Two homogeneous feedback equivalent sys˜ [m] have the same invariants of degree m.
tems [m] and The homogeneous feedback linear equivalence problem is
solvable at degree m if and only if the invariants of degree
m are equal to zero.
+ span(Ai B)
d[m] i
AB
dx
so that, after easy computations, it can be verified that
for i 0, the bracket of vector fields [(F∗ − I )i G1 , (F∗ −
I )i+1 G1 ][m−2] and [(F̃∗ − I )i G̃1 , (F̃∗ − I )i+1 G̃1 ][m−2] differ from their components n − i − 1 up to n. It results that
[m]
are equal.
the invariants (23) associated with [m] and ˜
From (19) and the choice for B, we immediately notice
[m−p]
that for p = (2, . . . , m − 1), the vector fields Gp
and
[m−p]
G̃p
differ for their last component only. The equality of
the invariants defined in (24) follows.
Being (iii) in Corollary 3.1 obviously verified due to the
controllability of the linear canonical form, it is quite immediate to verify that conditions (i) and (ii) are in fact equivalent to require the nullity of the invariants of degree m.
More precisely, due to the form of C, the parallelism con[m]j
dition (i) is equivalent to the nullity of the ap defined by
(24) while the involutivity condition (ii) is equivalent to the
[m]j,i+2
nullity of the a1
defined by (23). 3.3. Approximated feedback linearization
Merging the results previously stated at any degree m,
Theorem 3.2 below gives a precise answer to the more general problem of approximated feedback linear equivalence.
Given [∞] , let us denote by I[m] (.), the set of invariants of
degree m associated with the homogeneous part of degree m
of the system into the parentheses and indicate by I[m] = 0
the nullity of these invariants. Moreover, let [∞] ([2] ◦[3] ◦
· · · ◦ [m] ) denote the system [∞] ([2] ◦ [3] ◦ · · · ◦ [m−1] )
further modified by the transformation [m] . The following
properties characterize the transformation which linearizes,
up to degree M, a given system [∞] .
Theorem 3.2. The iterated application of the transformations [2] ◦ [3] ◦ · · · ◦ [M] to [∞] provides a linearizing
transformation up to degree M if and only if I[2] ([∞] ) = 0
and for k = (2, . . . , M − 1), I[k+1] ([∞] ([2] ◦ [3] ◦ · · · ◦
[k] )) = 0.
We note that each [m] belongs to the group of transformations cancelling the homogeneous part of degree m in
[∞] ([2] ◦ [3] ◦ · · · ◦ [m−1] ), [23]; its existence being ensured by the nullity of the corresponding set of invariants:
I[m] ([∞] ([2] ◦ · · · ◦ [m−1] )).
4. The normal forms
Merging the results of Proposition 3.2, Corollary 3.1 and
the second item of Theorem 3.1, we immediately conclude
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S. Monaco, D. Normand-Cyrot / Systems & Control Letters 55 (2006) 71 – 80
that to solve the homogeneous homological equations described by (20)–(21) is equivalent to find the resonance
terms in the normal forms as well as it is equivalent to the
nullity of the homogeneous invariants. The homogeneous
normal forms of degree m are thus exactly the dynamics
which contain the resonance terms which cannot be simplified according to (20)–(21) under an appropriate choice
; i = (0, . . . , m)). It results that invariants
of ([m] , [m−i]
i
and resonance terms have the same cardinality and are
linked by linear combinations as illustrated in Section 4.2
below.
4.1. Homogeneous normal forms of degree m
The following theorem shows that an homogeneous transformation of degree m is sufficient to cancel the nonlinear
terms of the same degree and transform the given dynamics
into one of its normal forms of the same degree.
Theorem 4.1. For any degree m 2 and neglecting higher
degree terms, any homogeneous discrete-time dynamics [m]
can be tranformed under an homogeneous transformation
[m] into
x + = Ax + F̄ [m] (x),
dx + (u)
= B + Ḡ[m−1] (x + (u), u),
du
= B), [m]
Second type of normal form (G[m−1]
1
N FB
x1+ = x1 + x2 +
n
[m−2]
xi2 F1;i
(x1 , . . . , xi ),
i=3
···
[m−2]
+
xn−2
= xn−2 + xn−1 + xn2 Fn−2;n
(x1 , . . . , xn ),
+
xn−1
= xn−1 + xn ,
xn+ = xn ,
dxp+ (u)
du
=
m
ui−1
Gi;p[m−i] (x + (u)); p=(1, . . . , n−1),
(i − 1)!
i=2
dxn+ (u)
= 1.
du
The proof is given in the Appendix.
It must be stressed that homogeneous normal forms are
unique modulo homogeneous approximations of the same
degree. The two types of normal forms described above are
different from those proposed in [1,8,15,9] for discrete-time
dynamics given in the form of maps. It is worthy to note that
the same number of resonance terms characterizes all these
equivalent homogeneous normal forms modulo approximations in R m+1 (x, u).
where the pair (Ax + F̄ [m] (x), B + Ḡ[m−1] (x + (u), u)) is
[m]
in one of the homogeneous normal forms [m]
NFA or NFB
below.
First type of normal form, linearity of the drift (F [m] (x)=
Ax), [m]
NFA
Example 1. Consider as in [23] the variable length pendulum equations
x + = Ax,
where x1 denotes the length of the pendulum with mass
normalized to 1, x2 its velocity, x3 the angle with respect to
the horizontal and x4 the angular velocity. Compute its Eulersampled equivalent with normalized sampling period at 1 as
in [9]; its linear part takes the form (7). As homogeneous
normal forms are concerned, it is reasonable to replace the
function sin x3 by its approximation of degree 3, sin x3 =
x3 − (x33 /6) + O(x3 ) 5 so that, writing down the equivalent
DDR we get the discrete-time dynamics of degree 3 below
m
dx1+ (u) ui−1
=
G[m−i] (x + (u)),
du
(i − 1)! i;1
i=2
dx2+ (u)
+
+
= xn+ (u)Q[m−2]
2;n (x1 (u), . . . , xn (u))
du
m
ui−1
+
G[m−i] (x + (u)),
(i − 1)! i;2
i=2
···
+
(u)
dxn−1
du
=
n
+
+
xi+ (u)Q[m−2]
n−1,i (x1 (u), . . . , xi (u))
i=3
+
m
ui−1
G[m−i] (x + (u)),
(i − 1)! i;n−1
i=2
dxn+ (u)
= 1.
du
ẋ1 = x2 ;
ẋ2 = −g sin x3 + x1 x42 ;
ẋ3 = x3 + x4 ;
ẋ4 = u,
x1+ = x1 + x2 ;
x2+ = x2 − gx 3 +
x3+ = x3 + x4 ;
g 3
x + x1 x42 ;
6 3
x4+ = x4 ,
dx1+ (u)
= 0;
du
dx2+ (u)
= 0;
du
dx3+ (u)
= 0;
du
dx4+ (u)
= 1.
du
S. Monaco, D. Normand-Cyrot / Systems & Control Letters 55 (2006) 71 – 80
Corollary 4.1. The nonlinear discrete-time dynamics [∞]
can be tranformed under a suitable transformation [M] into
The coordinates change and feedback
z1 = x 1 ;
z2 = x2 ;
g 3
x ;
6 3
g
z4 = −gx 4 + (x43 + 3x32 x4 + 3x3 x42 ),
6
v
u = − + [3] (x, v),
g
z3 = −gx 3 +
with
v
[3] (x, v) = x43 + x3 x42 −
g
+
x + = Ax +
k=1
+ Ḡ[M−1] (x + (u), u) + O(x)>M−1 ,
v2
v3
(2x4 + x3 ) − 3 ,
2
2g
6g
transform, modulo terms in R 4 (z, v), the discrete-time
pendulum dynamics into its cubic normal form with one
resonance term z1 z42 /g 2
z3+ = z3 + z4 ;
z4+ = z4 ,
F [k] (x) + F̄ [M] (x) + O(x)>M ,
M−2
dx + (u)
G[k] (x + (u), u)
=B +
du
x2
2x42 + 3 + 2x3 x4
2
z2+ = z2 + z3 +
M−1
k=2
z1+ = z1 + z2 ;
z1 z42
;
g2
where the pair (Ax + F̄ [M] (x), B + Ḡ[M−1] (x + (u), u)) is
[M]
in one of the normal forms [M]
NFA or NFB .
Applying iteratively the results of Theorem 4.1 to each
successive homogeneous part of degree m, starting at m = 2
and increasing the degree, Theorem 4.2 below describes the
normal forms of a nonlinear discrete-time dynamics.
Theorem 4.2. The nonlinear discrete-time dynamics [∞]
can be tranformed under transformation [∞] into a dynamics exhibiting one of the two normal forms below;
First type of normal form (dual normal form), linearity of
the drift, [∞]
NFA
dz1+ (v)
= 0;
dv
dz2+ (v)
= 0;
dv
x + = Ax
dz3+ (v)
= 0;
dv
dz4+ (v)
= 1.
dv
dx1+ (u) ui−1
=
Gi;1 (x + (u)),
du
(i − 1)!
∞
i=2
4.2. Invariants and resonance terms
By construction, the number of mth degree invariants is
equal to the number of resonance terms in the mth degree
normal form. More precisely, for any m 2, it can be shown
making reference to the second type of normal forms that,
for 1j n − 2 and 0 i n − j − 2
[m]j,i+2
a1
(x)
j
=
jxn−i
2 F [m−2] (x , . . . , x
jxn−i
1
n−i ) j,n−i
jxn−i
and for p = (2, . . . , m − 1),
[m]j
[m−p]
ap (x) = Gp;j (x).
77
dx2+ (u)
= xn+ (u)Q2;n (x1+ (u), . . . , xn+ (u))
du
∞
ui−1
+
Gi;2 (x + (u)),
(i − 1)!
i=2
···
+
dxn−1
(u)
du
=
A−1 x
As an homogeneous transformation of a given degree does
not modify lower degree terms, Corollary 4.1 below is an
immediate consequence of Theorem 4.1. It puts in light that
neglecting terms of degree greater than M, [∞] can be
transformed into a normal form up to the degree M by a
transformation containing terms of at most degree M.
xi+ (u)Qn−1,i (x1+ (u), . . . , xi+ (u))
i=3
+
1j n − 1;
4.3. The normal forms
n
∞
ui−1
Gi;n−1 (x + (u)),
(i − 1)!
i=2
dxn+ (u)
= 1,
du
where Qi;j (x1+ (u), . . . , xj+ (u)) is a formal series defined by
the formal summation
Qi;j (x1 , . . . , xj ) =
∞
m=0
Q[m]
i;j (x1 , . . . , xj ).
78
S. Monaco, D. Normand-Cyrot / Systems & Control Letters 55 (2006) 71 – 80
Second type of normal form, G1 = B, [∞]
NFB
x1+ = x1 + x2 +
n
xi2 F1;i (x1 , . . . , xi ),
i=3
···
+
xn−2
= xn−2 + xn−1 + xn2 Fn−2;n (x1 , . . . , xn ),
+
xn−1
xn+
u=−
v
+ 3 (x, v),
g
with 3 (x, v) solution of v = −g sin(x3 + 2x4 − (v/g) +
3 (x, v)) + 2g sin(x3 + x4 ) − g sin x3 , brings the discretetime pendulum dynamics into its extended controller normal
form
z1+ = z1 + z2 ;
= xn−1 + xn ,
= xn ,
dxp+ (u)
du
∞
ui−1
=
Gi;p (x + (u)); (p = 1, . . . , n − 1),
(i − 1)!
i=2
dxn+ (u)
= 1,
du
where Fi;j (x1 , . . . , xj ) is a formal series defined by the formal summation
Fi;j (x1 , . . . , xj ) =
∞
[m]
Fi;j
(x1 , . . . , xj ).
z2+ = z2 + z3 +
z1 z42
H (z1 , z2 , z3 , z4 ),
g2
z3+ = z3 + z4 ;
z4+ = z4 ,
dz1+ (v)
= 0;
dv
dz2+ (v)
= 0;
dv
dz3+ (v)
= 0;
dv
dz4+ (v)
= 1,
dv
with H (0) = 1. A similar conclusion as in [9], when representing the pendulum dynamics in the form of a map, is so
obtained.
m=0
Remarks. (i) As in the continuous-time case [14] and making reference to Theorem 3.2, we note that if M represents
the smallest integer such that one of the two conditions (i)
or (ii) in Corollary 3.1 fails, then Theorem 4.2 holds true
with formal series expansions starting at m = M − 2.
(ii) We note that while homogeneous normal forms are
uniquely defined, the normal forms are not because homogeneous transformations which preserve their homogeneous
parts induce higher degree terms which may imply different
homogeneous normal forms of higher degree.
(iii) Making reference to the notion of m-jets of a vector
field which correspond to the first mth degree terms in its
Taylor expansion [10], Theorem 4.2 characterizes the controller normal forms of m-jets for nonlinear discrete-time
controlled dynamics.
(iv) Discrete-time and continuous-time normal forms exhibit strongly comparable structures when the Gi ’s are equal
to zero for i 2; [∞]
NFB is the discrete-time equivalent of the
Kang’s normal form [13,11] while [∞]
NFA is the discrete-time
equivalent of the dual normal form [11,23].
Example 1 (continued). The following computations put in
light that the same conclusion as in [9] can be obtained
in our formalism regarding the normal form of this example. For, it is enough to extend the previous arguments to
higher degree approximations of sin x3 ; i.e. sin x3 = x3 +
i 2i+1
/(2i + 1)!. It results that the transformai 1 (−1) x3
tion given by
z 1 = x1 ;
z2 = x2 ;
z3 = −g sin x3 ;
z4 = −g sin(x3 + x4 ) + g sin x3 ,
Example 2. To illustrate the nonuniqueness of the extended
normal forms, consider the following dynamics in normal
form of degree 3, inspired from a similar example treated in
the continuous-time case [11,23].
2
2
2
+
1 = 1 + 2 + 3 − 21 3 − 22 3 −
+
2 = 2 + 3 ;
+
3 = 3 ,
d+
1 (u)
= 0;
du
d+
2 (u)
= 0;
du
2 3
3 3 ;
d+
3 (u)
= 1.
du
The change of coordinates and feedback
x1 = 1 − 21 −
4 3
3 2 ;
x2 = 2 − 22 − 21 2 ,
x3 = 3 − 23 − 222 − 42 3 − 21 3 − 22 23 − 233 ,
u = v + 62 3 + 623 + 2v(1 + 32 + 33 ) + v 2
+ 43 (31 2 + 31 3 + 922 + 182 3 + 1023 )
+ 8v(112 3 + 31 2 + 31 3 )
+ 4v(21 + 922 + 1523 ) + 2v 2
× (31 + 102 + 133 ) + 4v 3
transform, modulo terms in R 4 (x, v), the (, u) dynamics
into
x1+ = x1 + x2 + x32 + 0(x 4 );
x2+ = x2 + x3 ;
dx1+ (v)
= 0;
dv
x3+ = x3 ,
dx2+ (v)
= 0;
dv
dx3+ (v)
= 1,
dv
S. Monaco, D. Normand-Cyrot / Systems & Control Letters 55 (2006) 71 – 80
which is still in normal form of degree 3 but does not contain
cubic terms. This follows from the fact that the quadratic
part of the feedback leaves invariant the quadratic terms of
the dynamics while it modifies the cubic ones.
5. Conclusion
Linearization under coordinates change and feedback has
been studied through approximations of increasing degree
putting in light normal forms and invariants of degree m.
The study, developed for nonlinear single-input discretetime dynamics with controllable linear part represented as
differential/difference equations, can be generalized to the
multi-input case. The same techniques can be used to investigate continuous-time nonlinear dynamics of the form
ẋ(t) = f (x(t), u(t)).
Appendix. Proof of Theorem 4.1
Even quite tedious when approximations of order m > 3
are performed, the algorithmic procedure involved to obtain
the normal forms is simple. First, let us write down the expansion of (21) according to u to get m independent equations and then test the solvability of (20–21) component by
component. Looking at the first set of homological equations
(20) rewritten component by component, we have
[m]
[m]
−F1[m] () = [m]
1 (A) − 1 () − 2 (),
(25)
···
[m]
[m]
[m]
−Fn−1
() = [m]
n−1 (A) − n−1 () − n (),
(26)
[m]
[m]
−Fn[m] () = [m]
n (A) − n () + 0 (),
(27)
where Fi[m] : R n → R (resp. [m]
: R n → R) indicates the
i
[m]
[m]
ith component of F
(resp. ), that is an homogeneous
polynomial of order m in the variables (1 , . . . , n ). From
(26), we immediately deduce that [m]
n can be used to cancel all the terms of order m into the (n − 1)th component
[m]
of the drift so getting the solution [m]
n () = n−1 (A) −
[m]
[m]
n−1 () + Fn−1 (). From (27), we immediately deduce that
[m]
0 can be used to cancel all the terms of order m into the
last component of the drift so getting the solution [m]
0 () =
[m]
[m]
[m]
−Fn () − n (A) + n ().
Looking now at the second set of homological equations
, the j th component of G[m−i]
(21), indicating by G[m−i]
i
i;j
we easily verify that for i = (1, . . . , m), [m−i]
can be used
i
while
its
remaining
to cancel the last component of G[m−i]
i
.
n−1 components cannot be modified except that of G[m−1]
1
In fact, from (17), G[m−1]
has
to
satisfy
for
i
=(1,
.
.
.
,
n−1)
1
the equality
() =
−G[m−1]
1;i
79
d[m]
i ()
.
dn
(28)
[m]
= 0, G[m−i]
= 0)
It results that in both forms (Fn[m] = 0, Fn−1
i;n
m
[m−i]
i−1
and that the term
/(i − 1)!)Gi;j
cannot be
i=2 (u
cancelled in general for j = (1, . . . , n − 1); it is a resonance
term.
The proof now differs depending if one chooses to cancel
the terms in the drift F [m] solving (25) to (26) to obtain
the dual normal form [m]
NFA or to cancel the terms in the
solving Eq. (28) to obtain
controlled vector field G[m−1]
1
.
[m]
NFB
How to find [m]
NFA : It is immediate to note from equalities
can be used
(25) to (26) that for i = (2, . . . , n), each [m]
i
[m]
thus obtaining
to cancel all the terms of degree m in Fi−1
[m]
[m]
[m]
the solutions [m]
i () = i−1 (A) − i−1 () + Fi−1 (). It
[m]
follows that all the coefficients of 1 are kept free; i.e. all
the coefficients [m]
1;i1 ...im in the expansion
[m]
1 () :=
1 i1 i2 ··· im n
[m]
1;i1 ...im i1 . . . im
can be used to cancel terms in G[m−1]
; i = (1, . . . , n − 1)
1;i
solving partially (28). To do so, we first notice that the relations (25) to (26) being satisfied, the induced relation between coefficients of the [m]
makes possible to use the coi
[m]
efficients (1;i1 ...im−1 n−j +1 ; 1 i1 · · · im−1 n − j + 1)
in place of the coefficients [m]
j ;i1 ...im−1 n already used and
thus to solve partially (28). More precisely, the equa[m]
tion −G[m−1]
1;1 () = d1 ()/dn , is solved completely
by adequately choosing the coefficients [m]
1;i1 ...im−1 n while
[m]
the equation −G[m−1]
1;2 () = d2 ()/dn , can be solved
only partially by adequately choosing the coefficients
([m]
1;i1 ...im−1 n−1 ; 1 i1 i2 · · · im n − 1) in place of the
[m−2]
[m]
2;i1 ...im−1 n . It follows that terms in n Q2;n (1 , . . . , n )
cannot be cancelled and define the remaining terms in
the second component of G1 . Iterating the reasoning we
[m]
show that the equation −G[m−1]
1;3 () = d3 ()/dn can
be solved only partially by adequately choosing the coefficients ([m]
1;i1 ...im−1 n−2 ; 1 i1 i2 · · · im n − 2) in
[m]
place of the [m]
2;i1 ...im−1 n−1 and thus of the 3;i1 ...im−1 n .
It follows that terms in n−1 Q[m−2]
3;n−1 (1 , . . . , n−1 ) and
in n Q[m−2]
3;n (1 , . . . , n ) cannot be cancelled and define the remaining terms in the third component of
G1 . At the last iteration, we find that the coefficients
([m]
1;i1 ...im−1 2 ; 1 i1 i2 · · · im 2) can be used in
place of the [m]
n−1;i1 ...im−1 n to solve partially the equation
[m]
−G[m−1]
1;n−1 () = dn−1 ()/dn , so that the remaining terms
n
are of the form i=3 i Q[m−2]
n−1;i (1 , . . . , i ).
80
S. Monaco, D. Normand-Cyrot / Systems & Control Letters 55 (2006) 71 – 80
How to find [m]
NFB : In that case, (28) being satisfied, it follows that the coefficients kept free are
all those defining the (i ; i = (1, . . . , n − 1)) except the [m]
i;i1 ...im−1 n . As previously, the coefficients
([m]
n−2;i1 ...im−1 n−1 ; 1 i1 i2 · · · im n − 1) can be used
in place of the ([m]
n−1;i1 ...im−1 n ) to solve partially the equality
[m]
[m]
[m]
() = [m]
−Fn−2
n−2 (A) − n−2 () − n−1 () so that the re[m−2]
(1 , . . . , n ). Analogously, the
maining terms are 2n Fn−2;n
coefficients ([m]
n−3;i1 ...im−1 n−2 ; 1 i1 i2 · · · im n − 2)
can be used in place of the ([m]
n−2;i1 ...im−1 n−1 ) and the coef-
ficients ([m]
n−3;i1 ...im−1 n−1 ; 1 i1 i2 · · · im n − 1) can
be used in place of the ([m]
n−2;i1 ...im−1 n ) to solve partially the
[m]
[m]
[m]
equality −Fn−3
() = [m]
n−3 (A) − n−3 () − n−2 () so
[m−2]
that the remaining terms are 2n−1 Fn−3;n−1 (1 , . . . , n−1 )
[m−2]
(1 , . . . , n ). Iterating the reasoning, we find
and 2n Fn−3;n
out that the coefficients ([m]
1;i1 ...im−1 n−j +1 ; 1 i1 i2 · · ·
im n−j +1) can be used in place of the ([m]
2;i1 ...im−1 n−j +2 )
for j = (2, . . . , n − 1) to solve partially equality (25) so
n 2 [m−2]
that the remaining terms are of the form
i=3 i F1;i
(1 , . . . , i ).
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