A Canonical Discrete-Time Nonlinear Form for Reduced Order

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

A canonical discrete-time nonlinear form for reduced order observers design D. Boutat, ∗ L. Boutat-Baddas ∗∗ and M. Darouach ∗∗ Loire Valley University, ENSI-Bourges, Institut PRISME, 88 Bd. Lahitolle, 18020 Bourges Cedex, France (e-mail: [email protected]). ∗∗ CRAN-CNRS, UHP NancyI, IUT de Longwy 186, rue de Lorraine, 54400 Cosnes-et-Romain (e-mail: latifa.boutat-baddas, [email protected]) ∗

Abstract: This paper deals with a normal form of a class of nonlinear dynamical systems. This form enables us to design a reduced-order observer for a class of nonlinear dynamical system. Necessary and sufficient geometrical conditions for the existence of a coordinate change to transform a given singular dynamical system into such normal form are also given. Keywords: Nonlinear singular systems, Observability normal form, Reduced-order observer. 1. INTRODUCTION The problem of constructing observers for nonlinear systems has attracted significant attention during the last decades. From a practical point of view the observers design for nonlinear systems is often approached by linearizing the system. One technique for constructing nonlinear observers is to linearize the error dynamics by using a change of coordinates and an output injection (see Krener and Isidori [1983], Krener and Respondek [1985], Xiao and Gao [1988], Boutat et al. [2009]). However, it turned out that the conditions for the linearizability of the error dynamics are quite stringent and hard to verify in practice. Recently, a new approach for the design of nonlinear observers which can be applied to a wider class of systems was presented (see Krener and Xiao [2002], Kazantzis and Kravaris [1998]). In (Lee and Nam [1990]); W. Lee and K. Nam, gives a necessary and sufficient condition for a discrete-time nonlinear system to be equivalent to a nonlinear full observer form. On the other hand reduced–order observers design has involving various researchers in the control. The first results on the reduced order observers for linear systems were presented by Luenberger (Luenberger [1964]). In (Sundarapandian [2004] and Sundarapandian [2006]), the reduced order observer design for discrete-time nonlinear systems is presented. This work is a generalization of the construction of reduced order observers for linear systems developed by Luenberger (Luenberger [1964]). However, normal nonlinear forms which support a reduced nonlinear observer, with a linear error, remains unknown. The main contribution of this paper is to gives a class of these normal forms and then to characterize the family of nonlinear dynamical systems which can be transformed by mean of a change of coordinate to these class. The paper is organized as follows. Section 2 contains notations and a definition. Section 3 presents a nonlinear Copyright by the International Federation of Automatic Control (IFAC)

normal form and the associated reduced-order observer. Section 4 deals with geometrical conditions under which a nonlinear discrete-time dynamic system can be transformed into such normal form. Section 5 presents an example to illustrate the previous results, and Section 6 concludes the paper. 2. NOTATIONS AND DEFINITION Let us consider the following discrete-time nonlinear dynamic system: xk+1 = F (xk ) (1a) yk = h(xk ) (1b) where the xk ∈ Rn is the state, yk ∈ Rm is the output. We assume that the map F is a diffeomorphism. We assume, also, that the pair (h, F ) satisfies the rank observability condition. Thus, the following 1-forms: d(hoF i−1 ) 1 ≤ i ≤ n are linearly independent, where F i−1 = F {z } is the | oF...oF times (i−1)

th

(i − 1)

composition of F .

Under this assumption see for e.g. (Sundarapandian [2006]) the dynamical systems can be rewritten in new coordinates as follows: x1k+1 = f1 (xk ) (2a) x2k+1 = f2 (xk ) (2b) yk = x2k (2c) 1 xk ∈ Rn , x1k ∈ Rn−m and x2k ∈ Rm where vector xk = x2k denote the global state, the unmeasured state and the measured state respectively. It is easy to see that the pair (h, F ) satisfies to the observability rank condition if and only if the pair (f1 , f2 ) as well satisfies the observability rank condition.

10002


Now, let us define the so-called reduced order observer. Definition 1. We say that the following discrete dynamical system x ˆ1 = fe1 (ˆ x1 , x2 ) (3) k+1

k

k

is a reduced order observer for the discrete time dynamical system (1) if the error ek = x ˆ1k − x1k −→ 0 as k −→ +∞. In the case where maps f1 and f2 are linear. Thus, if we set



 1 z1,k  ..   .   1  z   r1, k   .  where ..  ∈ Rn−m−p is the unmeasured state,  m   z   1,k   .   ..  zrmm ,k T

f1 (xk ) = A11 x1k + A12 x2k f2 (xk ) = A21 x1k + A22 x2k yk = x2k

and A = diag(A1 , ..., Ap )

its easy to see that the reduced order observer is as follows: x ˆ1k+1 = A11 x ˆ1k + A12 x2k + K(A21 x1k − A21 x ˆ1k ).

(4)

where Ai is an ri × ri is in the following form: 

0 1 . Ai =   .. 0 0

Then the observation error dynamics behaves as follows: ek+1 = (A11 − KA21 )ek . which is stabilizable as a long as the pair (A11 , A21 ) is observable. We will end this section by a remark. Remark 1. The reduced order observer (4) introduced by Leunberger ( Luenberger [1964]) enables us to overcome on the redundancy of measurement. However, the new output A21 x1k can be contains more information that we are needing. For example: x1k+1 = f1 (x1k , x2k ) = x2k+1 = f2 (x1k , x2k ) = ξ yk = x2k = k ηk

"

x1,k+1 = yk x2,k+1 = x1,k x3,k+1 = x2,k

#

ξk+1 = x3,k + ξk , ηk+1 = ax1,k + bx2,k + Cx3,k

Thus A21 =

T

ξk = ( ξ1,k · · · ξm,k ) ∈ Rm , ηk = ( η1,k · · · ηp,k ) ∈ Rp are the measurable outputs, z0,k = Czk = zr11, k zr22 ,k · · · zrmm ,k

0 0 1 a b c

This shows that we need to use only the dynamic ξk+1 = x3,k + ξk to design a reduced order observer. Thereafter, we will take into account this fact in the proposed normal form. 3. NORMAL FORM AND ITS REDUCED ORDER OBSERVERS DESIGN

 0 0 0 0 .  · · · ..  0 0 1 0

We will assume that γ1 (yk ) is an invertible map such that from (5b) we obtain: −1

z0,k = Czk = (γ1 (yk )) (ξk+1 − γ2 (yk )) . Therefore the pair (C, A) is observable.

(6)

Now, let us consider the following system ζk+1 = N ζk + Ψ(yk , yk−1 ) (7a) zˆk = ζk + G(yk , yk−1 ) (7b) Where N is a matrix of appropriate dimension, G and Ψ are nonlinear maps which must be determined such that (7) is an asymptotic observer for system (5). One can see that the form (7) is a generalization of the functional and reduced order observers forms considered in (Darouach [2000]) for example.

G(yk , yk−1 ) = L(yk−1 )γ1−1 (yk−1 )(ξk − γ2 (yk−1 )), (8c) Ψ(yk , yk−1 ) = N G(yk , yk−1 ) + β(yk ). (8d) Proof 1. Let ek = zk − zˆk = zk − ζk − G(yk , yk−1 ) to be the estimation error, then its dynamic is: ek+1 = zk+1 − ζk+1 − G(yk+1 , yk )

In this section we will give a nonlinear normal form which admits a reduced order observer. We consider a multivariable discrete-time nonlinear system described by:

yk = (ξk , ηk )T = (yk1 , yk2 )T

0 0 .. . 1 0

We have the following result. Proposition 1. For the canonical form (5), where the pair (C, A) is observable; there always exists a gain κ such that the observer (7) is asymptotically stable, with: N = A − κC, is a stability matrix, (8a) L(yk ) = α(yk ) + κ (8b)

.

zk+1 = Azk + α (yk ) z0,k + β (yk ) ξk+1 = γ1 (yk ) z0,k + γ2 (yk ) ηk+1 = µ(zk , yk )

··· ··· .. . ··· ···

(5a) (5b) (5c)

= −N ζk − Ψ(yk , yk−1 ) − G(yk+1 , yk ) + Azk + α (yk ) z0,k + β (yk ) By using the fact that −ζk = ek − zk + G(yk , yk−1 ), we obtain

(5d) 10003

ek+1 = N ek − N zk + N G(yk , yk−1 ) − Ψ(yk , yk−1 ) + Azk + α (yk ) z0,k + β (yk ) − G(yk+1 , yk )


and by induction we define the following family of vector fields: τi,j (p) = Adf |F −1 (p) (τi,j−1 (F −1 (p))) for 2 ≤ j ≤ ri

If (8b) is satisfied, we obtain ek+1 = N ek − (N − A + κC)zk + (N G(yk , yk−1 ) − Ψ(yk , yk−1 ) + β (yk )) − G(yk+1 , yk ) + L(yk )γ1−1 (yk )(yk+1 − γ2 (yk )) If (8a), (8c) and (8d) are satisfied we obtain ek+1 = (A − κC) ek Therefore we can choose the gain κ such that the eigenvalues of A − κC lie within a unit disk, since the pair (C, A) is observable . 4. CANONICAL DISCRETE-TIME NONLINEAR FORM CALCULATION In this section we present the main result of the paper, namely a tool for constructing a normal form observer. We consider the following dynamical system: xk+1 = f1 (xk , yk ) ξ¯k+1 = γ1 (yk )(H(xk )) + γ2 (yk ) η¯k+1 = µ(xk , yk ) yk = (ξ¯k , η¯k )

(9a) (9b)

From the rank observability condition of the pair (f1 , H) we can see that the family of vector fields τ = (τi,j ) are independent. Before to go ahead we must check if the vector fields τi,j commute thus: for 1 ≤ i ≤ m, 1 ≤ j ≤ ri and 1 ≤ l ≤ rs [τi,j , τs,l ] = 0. (11) where [, ] is the Lie bracket. As will be shown later, equations (11) are necessary for the existence of change of coordinates. Under these conditions we construct σ1 , ..., σm and ν1 , ..., νp m + p vector fields such that: 1 for i = j and dξ¯i,k (νj ) = 0 (1) dξ¯i,k (σj ) = δij = 0 for i 6= j 1 for i = j j (2) d¯ ηi,k (σj ) = 0 and d¯ ηi,k (νj ) = δi = 0 for i 6= j (3) {τ, σi , νj } is a basis of the hall space

(9c)

(9d) where xk ∈ Rn−m−p is the unmeasured state and ξk ∈ Rm and ηk ∈ Rp are the measurable states. We assume that γ1 (yk ) is invertible and the pair (f1 , H) satisfies the observability rank condition, thus there exist indices r1 ≥ r2 ≥ ... ≥ rm such that the codistribution

(4) [X, Y ] = 0 where X, Y ∈ {τ, σi , νj } Now consider the following matrix:   θ Λ =  dξ¯k  (τ, σi , νj ). d¯ ηk

∆ = span{θi,j ; 1 ≤ i ≤ m and 1 ≤ j ≤ ri } is of dimension n, where the 1−forms θi,j are given by: θi,j = d Hi of1j−1 for 1 ≤ j ≤ ri−1 .

with f1j−1 = f1 of1 o...of1 and f10 = I. {z } | times j−1

In the rest of the paper we will call these observability 1−forms.

Next we need the following important definition introduced by Lee and Nam in (Lee and Nam [1990]). Definition 2. Let σ : U1 → U2 be a diffeomorphism between to be a vector field on U1 and U2 ant let X be a vector field on U1 , we define Adσ X to be a vector field on U2 such that: Adσ X(p) = Dσ|σ−1 (p) X(σ −1 (p)), where Dσ is the Jacobian of σ. Hereafter, we will make the following assumption: Assumption that 1. We will assume within this work f1 ξ¯k+1 is a diffeomorphism. Where f2 = . F = f2 η¯k+1 Now, for i = 1 : m we define the vector fields τi,1 by the following equations: θi,ri (τi,1 ) = 1, for 1 ≤ i ≤ m θi,l (τi,1 ) = 0, for 1 ≤ l ≤ ri − 1 (10) θj,l (τi,1 ) = 0, for j < i and 1 ≤ l ≤ ri θj,l (τi,1 ) = 0, for j > i and 1 ≤ l ≤ rj 10004



θ1,1 (τ1,1 ) ..   .   θ1,r1 (τ1,1 )  ..   .   θm,1 (τm,1 )  ..   . Λ=  θm,,rm (τ2,1 )  dξ¯ (τ )  1,k 1,1  ..  .   dξ¯ (τ )  m,k m,1  d¯  η1,k (τ1,1 )  ..  .

···

··· ···

θ1,1 (τ1,r1 ) .. .

θ1,r1 (τ1,r1 ) .. .

··· · · · θm,1 (τm,,rm ) .. ··· . · · · θm,rm (τm,,rm ) · · · dξ¯1,k (τ1,r1 ) .. ··· . · · · dξ¯m,k (τm,rm ) · · · d¯ η1,k (τ1,r1 ) .. ··· .

d¯ ηp,k (τp,1 ) · · · d¯ ηp,k (τp,rm )  θ1,1 (σ1 ) θ1,1 (ν1 ) .. ..   . .  θ1,r1 (σ1 ) θ1,r1 (ν1 )   .. ..   . .  θm,1 (σm ) θm,1 (νm )   .. ..   . .  θm,rm (σm ) θm,rm (νm )  ¯ ¯ dξ1,k (σ1 ) dξ1,k (ν1 )    .. ..  . .  dξ¯m,k (σm ) dξ¯m,k (νm )   d¯ η1,k (σ1 ) dη1,k (ν1 )    .. ..  . . d¯ ηp,k (σ1 ) d¯ ηp,k (νm )


By the rank condition observability it easy to see that the matrix Λ is invertible. Therefore, we can consider the following multi 1-forms:   θ ω = Λ−1  dξ¯k  . d¯ ηk " # τ Set by definition that : ω σi = In×n and let the multi νj ω1 1-forms be ω = where ω1 is formed by the m first ω2 1-forms of ω. Then we have the following result. Theorem 1. There exists a diffeomorphism  " #  zk φ(xk , ξ¯k , η¯k )  ξk =  ξ¯k ηk η¯k

which transforms the dynamical system (9) into the canonical form (5) if and only if the following conditions are satisfied:

Sufficiency: The evaluation of the differential of ω for any X, Y ∈ {τ, σi , νj } give dω(X, Y ) = LX ω(Y ) − LY ω(X) − ω[X, Y ] = −ω[X, Y ] because ω(X) and ω(Y ) are constant. As ω is an isomorphism, this implies the equivalence between [X, Y ] = 0 and dω = 0 According to theorem of Poincaré (Poincaré [1892]), dω = 0 implies that there exists a local diffeomorphism φ such that ω = Dφ. Therefore we can write: ∂ Dφ(τi,j ) = i ∂zj,k Now let us clarify the effect of the diffeomorphism φ on f1 the global map F = defined in (9a). To do so for f2 1 ≤ i ≤ m and 1 ≤ j ≤ ri − 1, we get

(1) the commutativity conditions (11) are fulfilled, ∂ (φ∗ (F )) = ∂zi,j

(2) Adf2 (τi,j ) ∈ ker ω1 for 1 ≤ j ≤ ri



(3) Adf1 (τi,ri ) = V (y) spanνj , for 1 ≤ i ≤ m, 1 ≤ j ≤ ri

=

(4) θj,1 (τi,l ) = 0, for j > i, 1 ≤ i ≤ m, rj + 1 ≤ l ≤ ri

ω1 (Adf1 (τi,j ) + ω1 (Adf2 (τi,j ) ω2 (Adf1 (τi,j ) + ω2 (Adf2 (τi,j ) ∂



 ∂zi,j+1 ω2 (Adf1 (τi,j )) + ω2 (Adf2 (τi,j ))

where V (yk ) is a smooth map and spanνj modulo a combination of vector fields νj .

From condition (3) we have [τi,j , f2 ] ∈ ker ω1 for 1 ≤ i ≤ m and 1 ≤ j ≤ ri − 1 implies that ω1 (Adf2 )(τi,j ) = 0.

Before giving the proof let us explain these conditions in the following Remark 2.

By integrating we obtain: φof1 oφ−1 = Azk + ̺(yk , z0,k ) Now by the condition (2), we have

(1) Conditions (1) on commutativity of vector fields unsure the existence of a diffeomorphism zk = φ(xk , ξk , ηk ) such that:

(12)

∂(φof1 oφ−1 ) = Dφ(τi,ri+1 ) ∂zri i ,k ri X = Dφ( Vi,j (yk )τi,j )

φ ∗ (f1 ) = Dφ (f1 )

j=1

and

=

φof1 = Aφ + β(y, z0,k ) (2) Condition (2) implies that β is linear in z0,k . l (3) Condition (3) means that zri i ,k doesn’t depend on zj,k for l < i. The proof which we will give here uses the same materials as in (Krener and Respondek [1985], Xiao and Gao [1988], Boutat et al. [2007], Boutat et al. [2009]) for continuous time. We will fellow the method of (Lee and Nam [1990]) for discrete time. Proof 2. Necessity: Indeed, if (9) can be transformed into (5) via the diffeomorphism zk = φ(xk , ξk , ηk )), then τi,j = ∂z∂i .

=

j=1 ri X j=1

Vi,j (yk )Dφ(τi,j ) Vi,j (yk )

∂ i ∂zj,k

which means ̺(yk , z0,k ) in (12) can be decomposed as: ̺(yk , z0,k ) = β(yk )z0,k + γ(yk ) and this implies that (9a) is transformed to form (5a) via φ(xk ). Finally, let us prove that the diffeomorphism φ(xk ) will transform (9b) to (5b). As we know ∂ Hj ◦ φ = dHj (τi,l ) = θj,1 (τi,l ) i ∂zl,k

j,k

And it is easy to check that all conditions of Theorem 1 are satisfied.

ri X

According to the definition of τi,1 in (10), we get

10005


θj,1 (τi,l ) = θj,1 ([τi,l−1 , f ]) = θj,2 (τi,l−1 ) .. .



 −2χ2,k χ3,k 1

= θj,l (τi,1 ) = 0 = θj,l (τi,1 ) = 0

Df1 F −1 (χk ) =  χ4,k − χ2,k − χ22,k −4 χ4,k − χ2,k χ2,k χ23,k  2χ3,k χ4,k − χ2,k

A simple calculation gives the 1-forms of observabilty as follows:

for j < i and 1 ≤ l ≤ ri . Following the same procedure, we have θj,1 (τi,l ) = 0, for j > i and 1 ≤ l ≤ rj . Combined with condition (3) in Theorem 1, we have 1, i = j, l = ri θj,1 (τi,l ) = 0, otherwise

θ1,2 = d(x1,k + ξ¯k x22,k ) = dx1,k + 2ξ¯k x2,k dx2,k + x22 dξ¯k , Moreover, we can determine the following vector fields:

which implies that (9b) is transformed to form (5b) through the diffeomorphism .

τ1,1 =

5. EXAMPLE

The dynamic of this system is given by the following diffeomorphism: 2     x1,k η¯k + ξ¯k x2,k − x2,k x1,k + ξ¯k x22,k  x   x1,k + ξ¯k x22,k . F  ¯2,k  =    ξk x2,k 2 ¯ ¯ η¯k ξk + x1,k + ξk x2,k    χ1,k x1,k χ  x  xk =  ¯2,k  and χk =  2,k  χ3,k ξk η¯k χ4,k

∂ ∂ − 2x2,k ξ¯k , ∂x2,k ∂x1,k

∂ ∂ − x22,k , ¯ ∂x1,k ∂ ξk ∂ , υ1 = ∂ η¯k

σ1 =

This section gives an academic example in order to high light the proposed results. For this, consider the following nonlinear discrete time dynamical system:  2  x1,k+1 = η¯k + ξ¯k x2,k − x2,k x1,k + ξ¯k x22,k     x2,k+1 = x1,k + ξ¯k x22,k (13) ξ¯k+1 = x2,k  2  ¯ ¯  η¯ = ξ + x1,k + ξk x2,k   k+1 ¯ k yk = (ξk , η¯k )



∂ , ∂x1,k

τ1,2 = Df1 F −1 (xk ) (τ1 F −1 (xk ) ) =

Let us consider an example to highlight our propose.

Let

θ1,1 = dx2,k ,

Thus one has  θ1,1 (τ1,1 ) θ1,1 (τ1,2 )  θ1,2 (τ1,1 ) θ1,2 (τ1,2 ) Λ= ¯ dξ1,k (τ1,1 ) dξ¯1,k (τ1,2 ) d¯ η1,k (τ1,1 ) d¯ η1,k (τ1,2 )   0 1 0 0 1 0 0 0 = . 0 0 1 0 0 0 0 1 Then    θ1 0 1 −1  θ2  ω=Λ  ¯ = 0 dξk 0 d¯ ηk

θ1,1 (σ1 ) θ1,2 (σ1 ) dξ¯1,k (σ1 ) d¯ η1,k (σ1 )

1 0 0 0

0 0 1 0

 θ1,1 (ν1 ) θ1,2 (ν1 )  dξ¯1,k (ν1 )  dη1,k (ν1 )

  θ1,1 0 0   θ1,2  . 0   dξ¯k  1 d¯ ηk

Set the following change of coordinates: z1,k = x1,k + ξk x22,k z2,k = x2,k Therefore the canonical form is:  z = ηk + ξk z2,k 1,k+1     z2,k+1 = z1,k ξk+1 = z2,k   η   k+1 = ξk + z1,k yk = (ξk , ηk )

Its inverse map is given by:   χ2,k − χ23,k χ4,k − χ2,k   (14) χ3,k  xk = F −1 (χk ) =    χ4,k − χ2,k χ1,k − χ4,k − χ2,k χ3,k + χ3,k χ22,k ηk and ξk are invariant. x1,k+1 represented The differential of the part f1 = x2,k+1 Now we design an observer for system (14). Using the by the two first dynamics is notations of proposition 1 we have: Df1 (xk ) 0 0 1 0 ξk A= , α (yk ) = = 1 0 0 0 η ! k   ξ¯k − x1,k + ξ¯k x2 2 − 2 2,k ¯  −2x2,k x1,k + ξk x2,k  4ξ¯k x22 x1,k + ξ¯k x22,k   0 1 ξk   C = [ 0 1 ] , β (yk ) = , 0 0 ηk ¯ 1 2ξk x2,k γ1 (yk ) = 1, γ2 (yk ) = 0, Ψ(zk , yk ) = ξk + z1,k

then the differential of f1 at F −1 (χk ) is: 10006


κ=

L(yk ) =

G(yk , yk−1 ) =

Ψ(yk , yk−1 ) =

0 N12 κ1 ,N = 1 N12 κ2

20 z1k zobs1k 15

10

ξk + κ1 κ2

ξk−1 ξk + κ1 ξk κ2 ξk

κ2 N12 ξk + ηk ξk−1 ξk + (κ + κ2 N22 ) ξk

5

0 0

2

4

6

8

10

12

14

16

18

20

State

and

20 z2 zobs2k 15

10

5

Then the reduced-order observer is: ζ1,k+1 = N12 ζ2,k + κ2 N12 ξk + ηk ζ2,k+1 = ζ1,k + N22 ζ2,k ξk−1 ξk + (κ1 + κ2 N22 ) ξk

0 0

2

4

6

8

10 Time

12

14

16

18

20

Fig. 1. zi,k (solid lines) and zî,k (dashed lines)

and zˆ1,k = ζ1,k + ξk−1 ξk + κ1 ξk zˆ2,k = ζ2,k + κ2 ξk

0.6 e1=z1k−zobs1k 0.5 0.4

Simulation results are presented in figures 1 and 2. They show the good performances of our results.

0.3 0.2 0.1 0

6. CONCLUSION

−0.1 −0.2 0

2

4

6

8

10

12

14

16

18

20

Error

In this paper a reduced-order observer design procedure is proposed for nonlinear discrete-time systems. The approach relies on the equivalence to a canonical observer form through coordinates change. Numerical example was given to show the applicability of our approach.

0.6 e2=z2k−zobs2k 0.5 0.4 0.3 0.2 0.1

REFERENCES

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Boutat, D., Benali, A., and Hammouri, H. (2007). Geometrical conditions for observer error linearization with a diffeomorphism on the outputs. In Proceedings of the 7th IFAC symposium on nonlinear control systems. Boutat, D., Benali, A., Hammouri, H., and Busawon, K. (2009). New algorithm for observer error linearization with a diffeomorphism on the outputs. Automatica, 45, 2187–2193. Darouach, M. (2000). Existence and design of functional observers for linear systems. IEEE Trans. Automat. Control, AC-45, 940–943. Kazantzis, N. and Kravaris, C. (1998). Nonlinear observer design using lyapunovs auxiliary theorem. System Control Letters, 34, 241–247. Krener, A. and Isidori, A. (1983). Linearization by output injection and nonlinear observer. Systems and Control Letters, 3, 47–52. Krener, A. and Respondek, W. (1985). Nonlinear observer design with linearizable error dynamics. SIAM Journal on Control and Optimization, 23, 197–216. Krener, A. and Xiao, M.Q. (2002). Nonlinear observer design in the siegel domain. SIAM J. Control Optim, 41, 932–953. Lee, W. and Nam, K. (1990). Observer design for nonlinear discrete-time systems. Proceeding of the 29th CDC. Luenberger, D.G. (1964). Observing the state of a linear system. IEEE Trans. Mil. Electron, 8, 74–80.

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Fig. 2. The error between zi,k and zî,k Poincaré, H. (1892). Solutions périodiques dans le voisinage d’un point d’équilibre-lunes sans quadrature, volume 1. dans Les Méthodes Nouvelles de la Mécanique Céleste, Gauthier-Villar. Sundarapandian, V. (2004). New results on general observers for discrete-time nonlinear systems. Appl. Math. Lett., 17, 1415–1420. Sundarapandian, V. (2006). Reduced order observer design for discrete-time nonlinear systems. Appl. Math. Lett., 19, 1013–1018. Xiao, X. and Gao, W. (1988). Nonlinear observer design by canonical form. Internat. J. Control, 47, 1081–1100.

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