Discrete-time Normal Form for Left Invertibility Problem - Science Direct

European Journal of Control (2009)2:194–204 # 2009 EUCA DOI:10.3166/EJC.15.194–204

Discrete-time Normal Form for Left Invertibility Problem 

M. Djemaı¨ 1, , J.P. Barbot2,3 and I. Belmouhoub2 1

LAMIH, UMR CNRS 8530, Université de Valenciennes et du Hainaut-Cambrésis, Le Mont Houy, 59313 Valenciennes Cedex 9, France; Equipe Commande des Systèmes (ECS), ENSEA, 6 av. du Ponceau, 95014 Cergy, France; 3 Equipe projet ALIEN-INRIA 2

This paper deals with the design of quadratic and higher order normal forms for the left invertibility problem. The linearly observable case and one-dimensional linearly unobservable case are investigated. The interest of such a study in the design of a delayed discrete-time observer is examined. The example of the Burgers map with unknown input is treated and a delayed discrete-time observer is designed. Finally, some simulated results are commented. Keywords: Discrete-time normal forms, Left invertibility problem, Output injection, Homological equations.

1. Introduction Since the last decade the concept of normal form in control theory was introduced by W. Kang and A. Krener in [21] and [22], (see also [11] for an algebraic point of view). On this basis the appearance of bifurcations under lost of controllability was studied [2, 15, 17, 20, 23, 39]. Following this way of thinking and the original concept of normal form introduced by H. Poincare´ in [33], the observability normal form for continuous-time system was introduced in [8]. In the well known paper [31] the authors demonstrated that unidirectional synchronization of chaotic systems is equivalent to an observer design problem. Starting from this result and considering chaotic Correspondence to: M. Djemai, E-mail: mohamed.djemai@ univ-valenciennes.fr

systems with unknown input, the problem of synchronization with recovering of the input can be seen as a left invertibility problem. Left invertibility was studied in several papers (see [12, 35, 36, 37, 38] . . . ). This work deals with left invertibility for discrete-time systems with observability singularity or/and left invertibility singularity. The results are based on the concept of discrete-time observability normal form [3]. The group of transformations associated with the left invertibility normal form is similar to the one associated with the observability normal form; the only difference is that input injection is not allowed in the first case. This slight difference generates extra resonant terms in the normal form. Clearly, when considering systems without input, both forms are identical. The normal form approach is facilitated in discrete-time by the possible use of geometric differential tools as proposed by S. Monaco and D. Normand-Cyrot (see [25, 27, 28]). Even though exact properties can be satisfied such as linearization or matching conditions, it is often enough in practice to consider approximate solutions so enlarging the domain of applicability. This is one of the main motivations of our approach (see the first paper of A. Krener regarding approximate design [22]). In the sequel, left invertibility normal forms are introduced and, depending on the resonant terms, their use is proposed in the design a delayed discrete-time observer. An application to an academic private communication is given as an illustrative example. This paper follows the lines of our previous work [2, 5, 6, 8] and it is organized as follows. In section 2 the Received 17 April 2008; Accepted 18 December 2008 Recommended by D. Normand-Cyrot and S. Monaco

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definitions and a settlement of the problem are stated. Quadratic observability and the left invertibility problem are presented in section 3. Section 4 is dedicated to the linearly unobservable case followed by the left invertibility problem in the one-dimensional unobservable case. An illustrative example ends the paper by showing the efficiency of the proposed approach.

2. Problem Statement and Quadratic Equivalence We are interested in solving the Left Invertibility Problem (LInP) for a nonlinear SISO (Single Input Single Output) discrete-time system of the form:  uÞ z ¼ ðz; þ

y ¼ hðzÞ ¼ Cz

ð1Þ

where the state vector z(k) is denoted z and z þ denotes z(k þ 1), k 2 N. The unknown input is uðkÞ 2 D  R and yðkÞ 2 R is the output. The vector fields  : U  D!M  Rn and the function h : U   Rn !W  R are assumed to be real analytic. With 0Þ ¼ 0: out loss of generality, we assume that ð0; The left invertibilty notion used in this paper is the following one. Given system (1), recover the state z(k) and the input u(k) from the outputs y(k), . . . , y(k þ n), y(k þ n þ 1) . . . .. Such a problem (LInP), is motivated by the fact that usually, in a control scheme, u is on the left side and y is on the right side of the block diagram. If (1) is invertible with respect to the unknown input u, the construction of a delayed observer like in [5] allows us to completely recover the information u. Such a delayed observer was implemented as a deciphering process for a secure communication application. In [32], the so-called observability matching condition is given. This condition ensures the existence of a unique solution for the LInP in a neighborhood of an equilibrium point. In this paper, we deal with discrete-time systems and study how to design an equivalent class to (1) modulo an extended output injection, under the socalled Discrete-Time Observability Matching Condition (DTOMC). Each class is characterized by a discretetime normal form for LInP. This normal form is reduced to the main terms of the original system while preserving its structural properties. These so-called resonant terms are shown to be the key for recovering the unknown input in the observer. Firstly, the case where system (1) is linearly observable is analyzed. This means that the corresponding observer may be designed on the basis of the linear residue. Secondly, systems which do not satisfy

linear observability in one direction are studied. As the linear residue does not give enough information about the state vector, we have to look for more pertinent information, in higher order terms. In both cases, the linear residue represents the linear part of the equivalent normal form. Moreover, the linear resonant terms are the only ones which can not be cancelled by linear transformation and output injection. These terms characterize the observability and detectability properties. The system is rewritten as: 8 þ > ¼ Az þ Bu þ F½2 ðzÞ : y ¼ Cz :¼ hðzÞ 



with A ¼ @@z ð0; 0Þ; B ¼ @@u ð0; 0Þ where: 2 3 2 ½1 3 ½2 F1 ðzÞ g ðzÞ 6 . 7 6 1. 7 6 . 7 ½2 ½1 6 F ðzÞ ¼ 6 . 7and g ðzÞ ¼ 4 .. 7 5 4 5 ½1 ½2 gn ðzÞ Fn ðzÞ ½2

½1

for 1  i  n; Fi ðzÞ and gi ðzÞ are respectively homogeneous polynomials of degree 2 and 1 in z. Roughly speaking the resonant term is that one which can not be cancelled by coordinates change and output injection. The following definition also sets a criterion which ca be used to check the ‘‘Quadratic Discrete-Time Observability Matching Condition’’ QDTOMC. This criterion means that1, y þ, . . . , y(n  1) þ should not depend on the unknown input, contrary to y(n) þ. We will see later how this condition allows to recover u in the normal form for the LInP. Definition 2.1: The QDTOMC holds a neighborhood of the equilibrium point ðze ; ue Þ of system (2) if J½1 :G½2 ¼ ð0; . . . ; 0; ?ÞT þ O3 ðz; uÞ

ð3Þ

where h i ½1 T ½1 J½1 ¼ ðdhÞT ; . . . ; fdðð½2 Þn1  hÞgT  G½2 ðz; uÞ ¼ Bu þ g½1 ðzÞu þ  ½0 u2 ½2 ¼ Az þ F½2 ðzÞ ‘‘o’’ denotes the usual composition operator and ð½2 Þjo denotes the expansion of  up to order 2 in z and composed j times (i.e. ð½2 Þjo ¼ ½2  ð½2 Þoj1 for 2  j  n with ð½2 Þ1o ¼ ½2 ). 8j 2 N ; yðjÞþ denotes yðk þ jÞ and yðjÞ denotes yðk  jÞ.

1

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The symbol  in (3) represents a first or second order non-trivial function of x and u. This function is non-null almost everywhere around the equilibrium point, containing -if it exists- observability singularity. In order to analyze the local invertibility of system (2), an equivalence modulo an extended output injection must be established at each order i (2  i  m). In the sequel, for simplicity of presentation, only quadratic equivalence is investigated.

Let us first define the notion of quadratic equivalence under coordinates change and output injection. Note that, in this paper output injection denotes injection of all available variables, output and known input, and strict output injection denotes usual injection from the output. This last injection will be used to solve the left invertibility problem. Let us first define the so-called quadratic equivalence. Definition 2.2: The system (2) is said to be quadratically equivalent to the system: 8 þ ¼ Ax þ Bu þ F½2 ðxÞ þ g½1 ðxÞu þ  ½0 u2 :

y

¼

Aobs z þ Bobs u þ F½2 ðzÞ þ g½1 ðzÞu

¼

þ ½0 u½2 þ O3 ðz; uÞ Cobs z ¼ z1

ð8Þ

where: 0

Aobs

B B a2 B B ¼ B .. B . B @ an1

0 .. . 0

.. 1 . .. .. . . 0

an 3

0



2 Bobs

a1

1

0

0

1

C 0C C .. C and .C C C 1A 0

 ½2 ðyÞ

ð9Þ

From Theorem 3.1 and its proof, we deduce the following corollary:

Theorem 3.1: The quadratic normal form associated to system (8), modulo an output injection is: 0

Remark 3.2: We recall that resonant terms -according to Poincare´’s works- are those which are invariant under quadratic transformations and output injection (additional transformation due to the study of observability). Obviously for high-order approximation, some high order resonant terms must be considered.

Since in the left invertibility problem the input is unknown, we used the strict output injection (6)

b1

On this basis we can establish the following theorem:

3

1 In the last component of F½2 , the terms xi,xj, (n j i 1) are resonant. 2 In g½1 , terms xi (n i 2) are resonant. 3 There are no resonant terms in the vector field  ½0 .

3.2. Left invertibility normal form

6b 7 6 27 7 ¼6 6 .. 7 4 . 5

12

3 x1 B C 6 xþ 7 B 6x 7 27 6 2 7 B a2 0 1 0C C6 7¼B 7 6 C6 6 .. 7 B . 6 .. 7 . . . C .. A4 . 5 .. .. 4 . 5 @ .. xn xþ an1 0 0 1 n1 2 Pn 3 3 2 b1 i¼2 ki1 xi 6 Pn 7 6 b 7 6 i¼2 ki2 xi 7 6 2 7 6 7 7 6 þ 6 . 7u þ 6 7u .. 6 7 4 .. 5 . 4 5 Pn bn1 k x i¼2 iðn1Þ i xþ 1

Lemma 3.1: Consider system (8), then:

Proof: See the appendix A for the proof of Theorem 3.1. &

bn

2

To prove Theorem 3.1 we use the following lemma:

a1

xþ n ¼ an x1 þ bn u þ

1

n X j>i¼1

0

.. . .. .

hij xi xj þ

0

n X

Corollary 3.1: The quadratic normal form associated to system (8), modulo a strict output injection (9) is: 1 2 þ 3 0 a 1 0 0 2 x1 3 1 x1 C6 7 .. 6 xþ 7 B x2 7 6 2 7 B a2 0 1 . 0C B C6 6 6 . 7 7 C 6 .. 7 ¼ B 6 .C . 7 .. . . . . . 4 . 5 B @ .. . .. A4 . 5 . . xn xþ an1 0 0 1 n1 2 Pn 3 2 3 b1 i¼1 ki1 xi 6 Pn 7 6 b 7 6 i¼1 ki2 xi 7 6 2 7 6 7 6 7 þ 6 . 7u þ 6 7u .. . 6 7 4 . 5 . 4 5 Pn bn1 k x i¼1 iðn1Þ i 3 2 1 6  7 6 2 7 2 7 þ6 6 .. 7u 4 . 5 n1

! kin xi u

xþ n ¼ an x1 þ bn u þ

hij xi xj

j>i¼1

i¼2

Remark 3.1: The normal form defined in the previous theorem is slightly different from the normal form for continuous-time systems: here we are able to cancel any term in x2i ðn i 1Þ in the last row of the dynamics (instead of any term x1xj, n j 2).

n X

þ

n X

!

kin xi u þ n u2

i¼1

Now, in order to recover the state and the unknown input, let us consider the following lemma and proposition.

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Lemma 3.2: The QDTOMC is invariant by a quadratic coordinates change and strict output injection (9). Proof: Using the following conditions  @y ¼ O3 ðx; uÞ @ux1 .. .

 @yðn1Þþ  ¼ O3 ðx; uÞ @u 

ð10Þ T

the particular form of the coordinates change z ¼ Id þ ½2 ðxÞ, and the fact that the output injection (9) do not change the QDTOMC, we obtain directly ¼ O3 ðz; uÞ .. .

@yðn1Þþ ¼ O3 ðz; uÞ @u From theorem 3.1 and lemma 3.2, we obtain: Proposition 3.1: The quadratic normal form, modulo a strict output injection, for system (8) verifying the QDTOMC, is: 2 6 6 6 6 4

3

xþ 1 xþ 2 .. .

xþ n1

0

7 B 7 B 7¼B 7 B 5 B @

a1

1

0

a2 .. . an1

0 .. .

1 ..

.

1 0 2 x1 3 C6 7 .. x2 7 . 0C C6 6 . 7 C 6 .C . 7 .. . .. A4 . 5 xn 0 1

0 n X xþ ¼ a x þ b u þ hij xi xj n 1 n n j>i¼1

þ

n X

T

where: z~ ¼ ½z1 ; z2 ; ; zn1  and z ¼ ½~ z ; zn  :

x

@y @u

linear change of coordinates (z ¼ T) and a Taylor expansion which transform the system (1) into: 8 > z~þ ¼ Aobs z~ þ Bobs u þ F~½2 ðzÞ þ g~½1 ðzÞu > > > > > > þ ~½0 u2 þ O3 ðz; uÞ > < Xn1 þ z ¼ z þ

z þ bn u þ F½2 n n n ðzÞ > i¼1 i i > > > ½0 2 3 > þ g½1 > n ðzÞu þ n u þ O ðz; uÞ > > : y ¼ Cobs z

!

kin xi u þ n u2

i¼1

From the previous form it will be easier to analyze if the system is Left invertible and if so to design an observer. In the next section we will study the case of linearly unobservable systems in one dimension.

4. One-dimensional Linearly Unobservable Case Let us consider system (1) and assume that the pair (A, C) has one unobservable mode. Then there is a

T

Remark 4.1: System (10) is the general linear canonical form of the unobservable system in one direction. (ii) If is a unique eigenvalue for the linear approximation of (1), then there exists a linear transformation ðz ¼ TÞ which transforms (1) in (10), with i ¼ 0 for any i 2 f1; . . . ; n  1g. (iii) The normal form which follows is structurally different from the controllability discrete-time normal form, given in [15, 14], in the last state dynamics xþ n . For the observability analysis the main structural information is not in the xþ n dynamics but in the previous state evolution ðxþ i for n  1

i 1Þ. The terms i xi ; bn u; F ½2 ðxÞ; g½1 ðxÞu are (i)

n

n

only important in the case of detectability analysis when ¼ 1. 4.1. Observability normal form Hereafter, we particularize the definition of quadratic equivalence to those systems with one unobservable mode. Now, following definition 2.2, system (10) is said to be quadratically equivalent to: 8 ~½2 ðxÞ þ g~½1 ðxÞu þ > ¼ Aobs x~ þ Bobs u þ F > > x~ > > > > ~½1 ðyÞu þ ~½0 u2 þ ~½2 ðyÞ þ  > > > > > > þ ~½0 u2 þ O3 ðx; uÞ > < Xn1 xþ ¼ xn þ

x þ bn u þ F½2 n n ðxÞ > i¼1 i i > > > ½0 2 ½2 ½1 > > þ g½1 n ðxÞu þ n u þ n ðyÞ þ n ðyÞu > > > > > þ  ½0 u2 þ O3 ðx; uÞ > > > : y ¼ Cobs x ð11Þ modulo an output injection:  ½2 ~½1 ðyÞu þ ~½0 u2 ~ ðyÞ þ  ½2

½1

½0

n ðyÞ þ n ðyÞu þ n u2

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Discrete-time Normal Form

if there exists a coordinates change of the form:  ~ ½2 ðzÞ x~ ¼ z~   ½2 xn ¼ zn  n ðzÞ which transforms the quadratic part of (10) into the quadratic part of (11), with: ~ ½2 ðzÞ ¼ ½½2 ðzÞ; . . . ; 2 ðzÞT :  n1 1 Now we determine the set of homological equations which will allow us to construct the quadratic normal form associated to system (10). Proposition 4.1: System (10) is quadratically equivalent to (11), modulo an output injection if and only if ½2 ½2 ½1 ½0 ~ ½2 ; ~½2 ;  ~½1 ðyÞ; ~½0 and n ; n ; n ; n Þ there exists ð which satisfy the following sets of homological equations: (i) 8 ~½2 ðxÞ ¼  ~ ½2 ðAxÞ   Aobs  ~ ½2 ðxÞ > F~½2 ðxÞ  F > > > > < þ ~½2 ðx1 Þ > g~½1 ðxÞ   b ~ ½2 ðAx;  BÞ  þ > ~½1 ðx1 Þ g~½1 ðxÞ ¼  > > > : ½0 ½0 ~ ½2 ðBÞ  þ ~½0 ¼  ~  ~

equations (12) imply, by applying then only to the observable part of system (11), the same assumption of proposition (2.1). The set of homological equations (13) is deduced from the unobservability line of the ½2 system, by considering the output injection: n ðyÞ þ ½1 ½0 2 n ðyÞu þ n u and the change of coordinates: & xn ¼ zn  ½2 n ðzÞ: Thanks to the notion of quadratic observability equivalence presented above, we are able to define an equivalent class for every system of the type (2), reduced to a unique system under the quadratic observability normal form, possessing the same structural properties as those of the corresponding equivalent system. This unique form will be described in the following theorem. Theorem 4.1: The normal form with respect to the quadratic equivalence modulo an output injection of system (10) is: 2 6 6 6 6 4

xþ 1 xþ 2 .. .

3 7 7 7 7 5

xþ n2

ð12Þ (ii) 8 ½2 ½2  ½2 F ðxÞ  F½2 > n ðxÞ ¼ n ðAxÞ  n ðxÞ > > n > X > n1 ½2 < 

 ðxÞ þ n½2 ðx1 Þ i¼1 i i > ½1 ^ ½2   > ½1 g½1 > n ðxÞ  g n ðxÞ ¼ n ðAx; BÞ þ n ðx1 Þ > > : ½0 ½0  n  n½0 ¼ ½2 n ðBÞ þ n

xþ n1



A A ¼ Tobs 0n1

   0n1 Bobs  ; B¼ ; bn

1 0 2 x1 3 B C6 B a x2 7 7 0C B 2 C6 6 ¼B C6 . 7 . . . 7 B . .. C @ . A4 . 5 xn1 an2 0 0 1 2 Pn 3 2 3 b1 i¼2 ki1 xi 6 Pn 7 6 b 7 6 i¼2 ki2 xi 7 6 2 7 6 7 7 þ6 7u .. 6 .. 7u þ 6 6 7 4 . 5 . 4 5 Pn bn2 i¼2 kiðn2Þ xi n X ¼ an1 x1 þ bn1 u þ hij xi xj þ hnn x2n a1

1 0 . 0 1 .. .. . . . . . . .

j>i¼1

þ

ð13Þ where,

0

n X

!

ki ðn  1Þxi u

i¼2

Moreover, by setting: R ¼ AT n A  n ; and by considering condition: 9

ði; jÞ 2 I f1; . . . ; ng such that Ri; j ¼ 02 ð14Þ

and 8 b i¼1 hij xi xj which appears terms in the in the ðn  1Þth line. By isolating the Pn1 , as follows: unobservable direction x n j>i¼1 hij xi xj þ  Pn  x h x we can deduce the manifold of local in i n i¼1   n P unobservability: Sn ¼ x; such that hin xi ¼ 0 . i¼1

Then, outside Sn we recover the observability.

iii)- If 2 ½1; 1 then locally, the system (10) is detectable.

Remark 4.3: In this paper, only the case of left invertibility, without ‘‘approximated zero dynamics’’2 is investigated [16]. Since lemma 3.1 independent of the considered case, we can set the following proposition Proposition 4.2: For system DTOMC, the quadratic normal output injection (9) is: 2 þ 3 0 a 1 0 1 x1 6 xþ 7 B 6 2 7 B a 0 1 6 7 B 2 6 .. 7 ¼ B .. . . . B 4 . 5 @ .. . . xþ an2 0 n2 xþ n1

Now, as in section 3 we will study the left invertibility problem.

xn1 3

an1 0 0 1 2 Pn 2 3 b1 i¼1 ki1 xi 6 Pn 7 6 b 7 6 i¼1 ki2 xi 7 6 2 7 6 7 7 þ6 7u .. 6 .. 7u þ 6 6 7 4 . 5 . 4 5 Pn bn2 i¼1 kiðn2Þ xi 2 3 1 6  7 6 2 7 2 7 þ6 6 .. 7u 4 . 5

þ

n X

!

n X

12

C6 0C C6 6 .. C C6 . A4

n X

x1

3

x2 7 7 .. 7 7 . 5 xn1

!

kiðn1Þ xi u þ n1 u2

i¼2

Corollary 4.1: The quadratic normal form associated to (8), modulo a strict output injection (9) is: 1 2 þ 3 0 a 1 0 0 2 x 3 1 x1 1 C6 .. 6 xþ 7 B 7 B C 6 2 7 B a2 0 1 . 0 C6 x2 7 6 6 7¼B .. 7 6 .. 7 B . 7 .. C .. . . . . C6 . 5 4 . 5 @ .. 4 . .A . .

xþ n1 ¼ an1 x1 þ bn1 u þ

0

0 1 n X ¼ an1 x1 þ bn1 u þ hij xi xj þ hnn x2n þ

From Theorem 4.1 and its proof, one deduces the following corollary:

n2

.. . .. .

j>i¼1

4.2. Left Invertibility Normal Form

xþ n2

(8) verifying the form modulo strict

hij xi xj þ hnn x2n

j>i¼1

kiðn1Þ xi u þ n1 u2

i¼2

Similarly to section 3, we will consider, for the left invertibility problem, a strict output injection of the form (9).

To conclude the paper, an algorithm is established in order to compute the mth (m ! 1) order DTNF. The algorithm Initialization: In this phase, we give the order of DTOMC: OrderðDTOMCÞ ¼ k and set p, the desired approximation order, then m ¼ minfk; pg Beginning of the algorithm Question one: is the system linearly observable? If the answer is positive then go to Case one otherwise go to question two. Question two: is the system linearly unobservable in one dimension? If the answer is positive then go to Case two otherwise go to the End of the algorithm. Case one: Compute the output injection ½i ðyÞ and the coordinates change Id þ ½i , in order to obtain the Left Invertible normal form of ith order. If i ¼ m then go to the End of the algorithm. Else, i: ¼ i þ 1. Go to Case one. 2@yjþ

jþ1 ðx; uÞ for j < n and @u ¼ O approximation order.

@ynþ @u

6¼ 0, where j is the considered

201

Discrete-time Normal Form −0.1

Initialization If YES Beginning of the algorithm

One dimensional unobservable ?

If No

−0.2 −0.3

If No

If YES

−0.4 x2

Linear Observable? Case 2 : Compute: I+Φ[i] and α[i] Normal Form for LinP at order i

Case 1 : Compute: I+Φ[i] and α[i] to obtainNormal Form for LInP at order i

−0.5 −0.6 −0.7

If i=m ?

If i=m ? −0.8 0.85

i = i+1

0.9

0.95

1

1.05

End of Algorithm

Fig. 1. Flow chart of the algorithm.

Case two: Compute the output injection ½i ðyÞ and the coordinates change Id þ ½i , in order to obtain the Left Invertible normal form of ith order. If i ¼ m then, go to the End of the algorithm. Else, i :¼ i þ 1. Go to Case Two. End of the algorithm. In Fig. 1, we sum up the previous algorithm.

5. An Illustrative Example: The Burgers Map As known from the work of H. Nijmeijer and I. Mareels [31] synchronization of a chaotic system can be reformulated as an observer design problem. In this section by means of an example we will show how the quadratic normal form can be used to solve the problem of synchronization with unknown input. Let us study the following two-dimensional system (modulo O3(z,u)): ( zþ ¼ Az þ Bu þ f½2 ðzÞ þ g½1 ðzÞu þ  ½0 u2 y ¼ C z ¼ z1 ; where A ¼ diagf1 þ a; 1  bg; B ¼ ½1; 0T .   f½2 ðzÞ ¼ ½z1 z2 ; a2 þ 2a þ b  1 z21 T g½1 ðzÞ ¼ ½0; 2ða þ 1Þz1 T ;  ½0 ¼ ½0; 1T where a and b are real parameters and u is the unknown input. Let us assume that state z1 is measured so y ¼ z1 is the output. Firstly, the observability of (15) will be studied. System (15) is linearly unobservable in the neighborhood of the equilibrium point (0,0). Moreover,

1.1

1.15

1.2

1.25

1.3

x1

i = i+1

Fig. 2. Burgers map phase portrait.

system (15) satisfies the DTOMC everywhere except in z1 ¼ 0. This allows to recover u outside the set S ¼ {z/z1 ¼ 0} of left invertibility an observability singularity. By applying the change of coordinates x ¼ z  ½2 ðzÞ to system (15); ½2 is deduced as follows: ½2

1 ðzÞ ¼ 0;

½2

2 ðzÞ ¼ z21

and

½2 ¼ 0 F1 ðxÞ ¼ 2 ðxÞ þ 1 x21 > > > > ½2 ½2 ½2 > 2 > > F2 ðxÞ ¼ 2 ðAobs xÞ  3 ðxÞ þ 2 x1 > < .. . : . ¼ .. > > > > ½2 ½2 2 > > Fn1 ðxÞ ¼ n1 ðAobs xÞ  ½2 > n ðxÞ þ n1 x1 > > : ½2 2 Fn ðxÞ ¼ ½2 n ðAobs xÞ þ n x1 ð17Þ Now, in any row i (8n i 2) of system (17) let us ½2 substitute i ðAobs xÞ by its expression deduced from the i  1th row; it follows: The 1st equation do not change: ½2

½2

F1 ðxÞ ¼ 2 ðxÞ þ 1 x21

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Then, by induction the ith equation (8n  1 i 2) may be written as: ½2 Fi ðxÞ

½2 ¼ iþ1 ðxÞ  Xi þ di k¼2 1k Xi1 þ di k¼1 kk

þ

Xi

k¼1

i d1k ð8i

Xi1

½2 ik F ðAobs k¼1 k

xÞ

x1 x k ð3Þ

xk þ i ðx1 ; . . . ; xi Þ

k x2ikþ1

where

k 2Þ,  1 k 1Þ may be written as a linear combination of: j al am and/or ð3Þ j al ð8i j 1; i  1 ; m l 1Þ; and i is a polynomial function of ðx1 ; . . . ; xi Þ of degree 3. So, we remark that in each row i (8n  1 i 1) of system (17): " We can determine by identification, the coordi½2 nates change component iþ1 so as to cancel the ½2 quadratic terms Fi . " Consequently, we can isolate i degrees of freedom: 1 ; . . . ; i . Finally the last equation is: Xn1

b ½2 ðAobs x; Bobs Þ þ ½1 ðx1 Þ g½1 ðxÞ ¼ 

We deduce from this equation, that only the quadratic terms in x1 u may be cancelled by the free vector field ½1 u. Thus, all terms in xi uð8n i 2Þ are resonant in the dynamics.

3rd homological equation: (  ½0 wished equal to zero)  ½0 ¼ ½2 ðBobs Þ þ  ½0

i dkk ð8i

F½2 n ðxÞ ¼ 

2nd homological equation: (with g½1 desired equal to zero)

½2

Fk ðAnk obs xÞ Xn1 dn x x þ dn þ k¼2 1k 1 k k¼1 kk Xn þ nð3Þ ðx1 ; . . . ; xn Þ þ  x2 k¼1 k nkþ1 k¼1 Xn

n n where d1k ; dkk are defined by a linear combination of: j al am and=or j al ð8i j 1; i  1 ; m l 1Þ; ð3Þ and n is a polynomial function ðx1 ; . . . ; xn Þ of degree

3. We can see that in the row n of system (17), there degrees are n degrees of freedom: 1 ; . . . ; n ; for nðnþ1Þ 2 ½2 of freedom in Fn . So the coefficients i ð8n i 1Þ ½2 can cancel only n quadratic terms in Fn , such that 2 each i cancel xniþ1 ð8n i 1Þ. We chose to cancel these terms because they are independent from the entries: ai of the matrix Aobs . We conclude the following result about the 1st homological equation:

" The coordinates change ½2 is completely determined in the (n  1) first rows, which cancel all ½2 ½2 quadratic terms in F1 . . . Fn1 ; hence there is no ½2 resonant terms in Fi ð8n  1 i 1Þ.

" In the last row, the free vector field  ½2 allows to cancel only the quadratic terms in x2i ð8n i 1Þ; consequently the terms xi xj ð8n j > i 1Þ are ½2

resonant in the component Fn .

This equation is trivial, since the free vector field  ½0 u2 may cancel all the quadratic terms in u2 . So, we conclude that there is no resonant terms in u2 . Finally, thanks to the coordinates change ½2 we construct the equivalent system of (8) modulo the output injection (6), restricted to the key dynamics in order to study and analyze this last; these dynamics are no others than the resonant terms established in Lemma 3.1. The equivalent system in question is under the quadratic normal form described in Theorem 3.1. &

Appendix B. Proof of Theorem 4.1 Proof:

The quadratic normal form associated to the linearly observable part of the system (10) is deduced from the theorem 3.1’s proof, by considering the first system of homological equations (12) in Proposition (4.1).

For the linearly unobservable part; let us consider the two last homological equations of (13) in Proposition (4.1), it is easy to deduce that: xi uð8n i

½1 2Þ are the resonant terms issued from gn u, and 2 finally there is no resonance in u . So, let us analyze the 1st homological equation in (13) of the Proposition (4.1): Given , ai and i ð8n  1 i 1Þ; ½2

Case 1: If P 9n , ½2verifying 2(14) so, Fn ðxÞ ½2 ½2 Fn ðxÞ ¼  n1 i¼1 i i ðxÞ þ n x1 . We remark that n allows us to cancel only the quadratic terms in x21 . Consequently, we have xi xj and x1 xj ð8n j i 2Þ as ½2 resonant terms due to Fn . Case 2: If 6 9 n 6¼ 0, such that (14) is verified, we obtain ½2 ½2 nðnþ1Þ degrees of freedom in both n and Fn , which 2 ½2 allows us cancelling all the quadratic terms in Fn . Consequently, in this case there is no resonant terms in & xi xj ð8n j > i 1Þ.