Transformation the Nonlinear System into the Observer Form ...

European Journal of Control (2009)2:177–183 # 2009 EUCA DOI:10.3166/EJC.15.177–183

Transformation the Nonlinear System into the Observer Form: Simplification and Extension Tanel Mullari, U¨lle Kotta Institute of Cybernetics at Tallinn Technical University, Akadeemia tee 21, 12618 Tallinn, Estonia

The paper addresses the problem of transforming the discrete-time single-input single-output nonlinear control system into the observer form using the state and output transformations. The necessary and sufficient solvability conditions are formulated in terms of differential one-forms, associated with the input-output equation of the control system. These conditions simplify the existing conditions [10] and extend them for the systems with inputs. The procedures to find the state and output transformations are given. Keywords: Nonlinear control systems, discrete-time systems, observer form, state and output equivalence, differential forms.

1. Introduction Conditions for the existence of the observer form for nonlinear control system using the state coordinate transformation are known to be very restrictive (see [2], [13] for continuous-time systems and [5], [11], [15] for discrete-time systems), motivating various extensions to enlarge the class of systems for which observers with linear error dynamics can be designed. Either the class of transformation was enlarged as in [10], [14] were in addition to state transformation also output transformation is allowed or different generalized observer forms were introduced as in [9], [16], [17], [19], [20] or both as in [1], [4], [7], [10]. Moreover, Correspondence to: T. Mullari, E-mail: [email protected] E-mail: [email protected]

system immersion into a higher dimensional system [12], [18] or output-dependent time scale transformations [8] were also applied to reach the desired goal. This paper addresses the problem of transforming the single-input single-output discrete-time nonlinear control system into the nonlinear observer form using both the state and output transformations. The necessary and sufficient solvability conditions are formulated in terms of exterior derivatives and exterior products of differential one-forms, associated to the input-output equation of the control system. Our results extend and simplify the results obtained by Huijberts in [10] who studied only the systems without input. As for simplification, we will prove that by a different choice of differential 1-forms, associated with the system, the second set of conditions in [10], formulated both in terms of 1-forms and the dual vector fields, is redundant. Moreover, since the new 1forms contain less terms, also the first set of conditions is easier to check. Finally, although the paper [10] suggests the solvability conditions, no procedure was given to compute the output coordinate transformation. We will fill this gap, at least partially. The new 1-forms are also applicable in the case of the generalized observer form with N past measurements of the outputs (and inputs), that is they allow to simplify the conditions of Theorem 10 in [10]. However, this problem is not revisited in this paper and is left for future studies. Note that though the differential 1-forms, introduced in this paper are similar to those of [7], our Received 31 March 2008; Accepted 18 December 2008 Recommended by D. Normand-Cyrot and S. Monaco

T. Mullari and U¨. Kotta

178

results do not remind those in the continuous-time case where only necessary condition has been formulated in terms of the differential 1-forms. The necessary condition yields a partial differential equation, the solution of which provides a candidate output transformation function. However, the necessary condition is very mild and far from being sufficient. To see, if the problem is solvable, one has to apply the output transformation and check if in the new output coordinates the system is transformable into the observer form by state transformation.

2. Necessary and Sufficient Conditions 2.1. Problem Statement Consider the discrete-time system described by the state equations xtþ1 ¼ Fðxt ; ut Þ; yt ¼ hðxt Þ;

ð1Þ

where x 2 X Rn is the state, ut 2 U Rm is the input, yt 2 Y R is the output, F and h are assumed to be meromorphic functions. In this paper we are interested neither in local nor global, but in the generic system properties, i.e., in the properties that hold on open and dense subsets of suitable domains of definition provided that they hold at some point of such domains. This motivates our choice of meromorphic functions since the generic property does not make sense, in general, for systems defined by smooth functions [6]. Find, if possible, the state transformation : X ! X z ¼ ðxÞ

ð2Þ

and the output transformation p : Y ! Y y~ ¼ pðyÞ

ð3Þ

such that in the new coordinates the system (1) is in the observer form yt Þ; ut ; z1; tþ1 ¼ z2; t þ n p1 ð~ yt Þ; ut ; z2; tþ1 ¼ z3; t þ n1 p1 ð~ .. ð4Þ . yt Þ; ut ; zn; tþ1 ¼ 2 p1 ð~ y~ ¼ z1; t : Note that the state equations (1) can be transformed into the form (4) with the original output function

yt ¼ z1; t if and only if the input-output equation, corresponding to (1) ytþn ¼ fðyt ; ytþ1 ; :::; ytþn1 ; ut ; utþ1 ; :::; utþn1 Þ ð5Þ has the special additive form ytþn ¼

1 ðyt ; ut Þ

þ :::

þ

2 ðytþ1 ; utþ1 Þ

n ðytþn1 ; utþn1 Þ:

ð6Þ

This result may be proved in much the same way as Theorem 7 in [10] and is therefore omitted. So, in order to solve the problem introduced, one has to find the output transformation (3) such that its composition with f in (5) is a sum of n two-variable functions pf¼

n1 X

iþ1 ðytþi ; utþi Þ:

ð7Þ

i¼0

Then we can define the new state coordinates as follows z1;t ¼ y~t ;

p1 ð~ yt Þ; ut ; ¼ y~tþ2 n p1 ð~ ytþ1 Þ; utþ1 n1 p1 ð~ yt Þ; ut ; .. . ¼ y~tþn1 n p1 ð~ ytþn2 Þ; utþn2 2 p1 ð~ yt Þ; ut

z2;t ¼ y~tþ1 z3;t

zn;t

n

ð8Þ and one can easily prove by direct computations that this choice leads to the equation (4). To conclude, given the input-output equation (5), corresponding to (1), we search for a single-variable function p whose composition with the function f in (5) yields the form (7). 2.2. The Main Result Define for i ¼ 0; :::; n 1 the 1-forms !tþi ¼

@f @f dytþi þ dutþi : @ytþi @utþi

ð9Þ

Theorem 1: Equation (5) can be transformed by the output transformation (3) into the form (7) if and only if for all 0 i; j n 1 d!tþi ^ !tþj þ d!tþj ^ !tþi ¼ 0:

ð10Þ

179

Transformation into the Observer Form

Proof: Sufficiency. The proof will be performed in two steps. We first show that there exists an 1-form as a total differential of a certain single-variable function S such that d!tþi ¼ dS ^ !tþi holds for all i ¼ 0; :::; n 1, and then we show, that from the existence of function S follows the existence of the output transformation p such that its composition with f yields (7). Note that taking in (10) i ¼ j yields d!tþi ^ !tþi ¼ 0; meaning that if (10) holds, all 1-forms !tþi are integrable. Therefore, there must exist the 1-forms iþ1 such that d!tþi ¼ iþ1 ^ !tþi :

ð11Þ

On the other hand, the integrability of the 1-form !tþi means that there exists the integrating factor iþ1 ðyt ; :::; ytþn1 ; ut ; :::; utþn1 Þ such that !tþi ¼ iþ1 d’iþ1 ðytþi ; utþi Þ

Therefore, the function f can be written as a composite function, though not yet in a form (7) fðyt ; :::; ytþn1 ; ut ; :::; utþn1 Þ ¼ ð’1 ðyt ; ut Þ; :::; ’n ðytþn1 ; utþn1 ÞÞ; and the integrating factors iþ1 can be expressed as composite functions as well iþ1 ð’1 ðyt ; ut Þ; :::; ’n ðytþn1 ; utþn1 ÞÞ ¼

Due to (12, (14) and (17) iþ1 ¼ ¼

ð12Þ ¼

for some function ’iþ1 . Taking the exterior differential of (12) one obtains d!tþi ¼ diþ1 ^ d’iþ1 ¼ d ln jiþ1 j ^ !tþi :

ð13Þ

In general, iþ1 cannot be taken equal to d ln jiþ1 j. However, comparing (13) with (11), and taking into account that !tþi ^ !tþi 0, one may search iþ1 in the form iþ1 ¼ d lnjiþ1 j þ Aiþ1 !tþi ;

diþ1 þ Aiþ1 !tþi iþ1 n1 1 X @iþ1 iþ1 n1 X

ð15Þ

for all i; j ¼ 0; :::; n 1. Since the number of coordinates ðdyt ; :::; dytþn1 ; dut ; :::; dutþn1 Þ in the inputoutput space is 2n and the number of 1-forms !tþi ; i ¼ 0; :::; n 1, is n, they do not form the basis and an arbitrary 1-form can not be written as the linear combination of !t ; :::; !tþn1 . However, as we will show in the sequel, if d!tþi ^ !tþi ¼ 0 holds, the 1-forms iþ1 can be expressed as the linear combinations of !t ; :::; !tþn1 . By d!tþi ^ !tþi ¼ 0, or alternatively, by (12), the total differential of function f reads as df ¼

n1 X i¼0

!tþi ¼

n1 X i¼0

iþi d’iþ1 :

ð16Þ

@’kþ1

d’kþ1

þ Aiþ1 !tþi ¼

iþ1;tþk !tþk ; ð18Þ

where the component of the 1-form iþ1 in the direction of !tþk is iþ1;tþk : ¼

1 @iþ1 1 þ ik Akþ1 : iþ1 @’kþ1 kþ1

By substituting iþ1 from (18) into (11) we get d!tþi ¼

iþ1 ¼ jþ1 ¼

k¼0

k¼0

ð14Þ

where Aiþ1 are certain unknown functions of variables ðyt ; :::; ytþn1 ; ut ; :::; utþn1 Þ. We prove next that under (10),

@f : @’iþ1 ð17Þ

n1 X

iþ1;tþk !tþk ^ !tþi

ð19Þ

k¼0

and substituting the last result into (10) we get for all i; j ¼ 0; :::; n 1 n1 X

iþ1;tþk jþ1;tþk !tþk ^ !tþi ^ !tþj ¼ 0

k¼0

proving (15). To show, that the 1-form is a total differential, note that from (14) jþ1 ; Aiþ1 !tþi Ajþ1 !tþj ¼ d ln iþ1 from which immediately follows that Aiþ1 !tþi for all i ¼ 0; :::; n 1, are total differentials. Therefore, is also total differential, and can be written as ¼ dS: Therefore, by (11) d!tþi ¼ dS ^ !tþi :

ð20Þ


180

Now we will prove, that from the existence of function S satisfying (20) follows the existence of an output transformation p, such that (7) holds. Since df is a total differential, its exterior differential d2 f ¼

n1 X

d!tþi ¼ dS ^

i¼0

n1 X

since the expression in the parentheses is always zero due to the anticommutativity of the wedge product.&

2.3. Comparison with the Earlier Results for Input-free Case

!tþi ¼ dS ^ df

i¼0

equals zero and by Cartan’s Lemma dS 2 spanfdf g. Therefore the function S can be represented as a composite function of f and some other function. We will show below that the choice S ¼ ln jp0 j f ¼ lnjp0 f j guarantees, that the composite function p f has the form (7). First we prove, that p0 f is the common integrating factor for all 1-forms !tþi , that is ðp0 f Þ!tþi ¼ d iþ1 ðytþi ; utþi Þ for some functions iþ1 . Is easy to show by direct calculation, that d½ðp0 f Þ!tþi equals zero, d½ðp0 f Þ!tþi ¼ ðp00 f Þdf ^ !tþi þ ðp0 f Þd!tþi ¼ ðp00 f Þdf ^ !tþi þ ðp0 f ÞdS ^ !tþi ¼ ¼ ðp00 f Þdf ^ !tþi þ dðlnjp0 f jÞ ðp0 f Þ ^ !tþi ¼ 0; ð21Þ meaning the functions iþ1 ðytþi ; utþi Þ really exist. Finally, multiplication of df by p0 f in (16) gives

The necessary and sufficient conditions, alternative to those in Theorem 1, have been given earlier in [10] for the systems without inputs. The 1-forms in [10] were defined in a different manner ~i ¼ !

i1 X @f dytþj ; @y tþj j¼0

¼ ðp f Þ

n1 X

!tþi ¼

i¼0

n1 X

d

iþ1 ðytþi ; utþi Þ;

i¼0

ð22Þ yielding (7). Necessity. Suppose that (7) holds. Taking the total differential of both sides gives ðp0 f Þdf ¼ ðp0 f Þ

n1 X i¼0

!tþi ¼

n1 X

d

~i ¼ !

i1 X

ð23Þ Then ðp0 f Þ!tþi ¼ d

iþ1 :

Its exterior derivative reads

dðp0 f Þ ^ !tþi þ ðp0 f Þd!tþi ¼ 0: Consequently we get d!tþi ¼ dln jp0 f j ^ !tþi for all i ¼ 0; :::; n 1, and substituting it into (10), we get for i; j ¼ 0; :::; n 1 d!tþi ^ !tþj þ d!tþj ^ !tþi ¼ d ln jp0 f j ^ !tþi ^ !tþj þ !tþj ^ !tþi ;

!tþj

ð25Þ

j¼0

and unlike our 1-forms (9), which in the input-free case are !tþi ¼

@f dytþi ; @ytþi

ð26Þ

having the integrating factors ð@f=@ytþ1 Þ1 , the 1forms (25) are not necessarily integrable. In [10] the following result has been proved. Theorem 2: Equation ytþn ¼ fðyt ; ytþ1 ; :::; ytþn Þ can be transformed by the output transformation (9) into the form

iþ1 ðytþi ; utþi Þ:

i¼0

ð24Þ

Because in the input-free case one has to operate only in the output space with the coordinates fdyt ; dytþ1 ; :::; dytþn1 g, having dimension n, the independent 1-forms (24) constitute a complete cobasis. The dual basis ~j satisfies the corresponding ~i ; ~j ¼ ij . Note that the 1-forms !~i are conditions ! related to our 1-forms !tþi (specified for the input-free case) by the formula

dðp f Þ ¼ ðp0 f Þdf 0

i ¼ 0; :::; n 1:

pf¼

n1 X

iþ1 ðytþi Þ

i¼0

if and only if for all i; j ¼ 0; :::; n 1 the following conditions are satisfied ~i ^ ! ~ j þ d! ~j ^ ! ~i ¼ 0; d!

ð27Þ

~i ; ~i ; ~j L~j ! ~j ; ~j ; ~i ¼ 0: L~i !

ð28Þ

Note that the necessary and sufficient conditions (27) and (28) are more complicated compared to

181


conditions (10) of Theorem 1. First, they consist of two sets of conditions – (27) – in terms of the 1-forms, and (28) – in terms of the vector fields and 1-forms. Second, due to the simpler form of 1-forms !tþi (compare (24) and (26)), it is also easier to calculate for !tþi the exterior differentials and wedge products in order to check conditions (10) compared with checking conditions (27). ~i Based on the definitions of !tþi , (see (26)), and ! (see (24)), one can easily show that if (10) holds, the validity of (27) follows, but the converse is not necessarily true. Therefore, although the conditions (10) in terms of 1-forms (26) are necessary and sufficient, the analogous conditions (27) in terms of the different 1-forms are only necessary, and one has to give additional set of conditions in terms of vector fields ~j . Since in [10] the different 1-forms have been used, and one needs additional set of conditions, the proof of Theorem 1 is not the generalization of Theorem 6 in [10].

Observe, that under (10) also (12) holds, allowing to compute the functions ’iþ1 by integrating the 1-forms !tþi , see formula (12). Though a complete solution for finding the function p remains a subject for future research, we suggest here the following procedure, that in most cases yields the desired result. Note that from (29), dðp f Þ ¼

n1 X

ðp0 ’iþ1 Þ d’iþ1 :

ð31Þ

i¼0

Alternatively, multiplying (16) by ðp0 fÞ we obtain dðp f Þ ¼ ðp0 f Þ

n1 X

iþ1 d’iþ1 :

i¼0

Consequently, for all i ¼ 0; :::; n 1 one has p0 ’iþ1 ¼ p0 f iþ1 yielding the relations for finding the function p0

2.4. Coordinate Transformations In this subsection we describe the procedure to find the output and state transformations, allowing to transform the state equations (1) into the observer form (4). We calculate the new state coordinates zt according to formula (8) that requires the knowledge of the inverse of the output transformation p1 ðy~Þ, and the functions tþi ðyt ; ut Þ, defined by the form (7). First, in order to find the functions tþi , we write the total differential of f in (5) according to formulae (16) and (17): df ¼

n1 X @f d’iþ1 ; @’ iþ1 i¼1

where the functions ’iþ1 ðytþi ; utþi Þ are the integrals of 1-forms !tþi correspondingly, see (12). Then the total differential of the composite function p f reads dðp f Þ ¼ ðp0 f Þdf ¼ ðp0 f Þ ¼

n1 X

n1 X @f d’iþ1 @’ iþ1 i¼0

dðp ’iþ1 Þ:

ð29Þ

i¼0

Comparing the last result with formula (23) one can see, that the functions for iþ1 ðytþi ; utþi Þ, i ¼ 0; :::; n 1, may be defined as the composite functions of p and ’iþ1 ðytþi ; utþi Þ . . . iþ1 ðytþi ; utþi Þ

¼ p ’iþ1 ðytþi ; utþi Þ:

ð30Þ

p0 ’iþ1 iþ1 ¼ : p0 ’jþ1 jþ1

ð32Þ

for all i; j ¼ 0; :::; n 1, such that i 6¼ j. If the necessary and sufficient conditions (9) hold, all possible pairs of i and j give the same result, of course.

3. Examples We give two examples, both of them transformable into the observer form. The first example is a simple academic one, whereas the second example describes a model of liquid level system of interconnected tanks obtained by identification [3]. In both examples the state equations will be found by (8), (12) and (30). In the first example the output transformation will be found by formula (32), whereas in the second example the output transformation is unnecessary. Example 1: Consider the system described by the state equations x1;tþ1 ¼ x2;t but þ cx1;t þ aut x1;t ; x2;tþ1 ¼ x3;t ut ; x3;t ¼ x1;t ; yt ¼ x1;t : The input-output equation, associated to the state equations, reads as ytþ3 ¼ utþ1 yt ðbutþ2 þ cytþ2 þ aytþ2 utþ2 Þ:

ð33Þ


182

Compute the 1-forms according to (9):

0:135yðt þ 1Þuðt þ 1Þ

!t ¼ utþ1 ðbutþ2 þ cytþ2 þ aytþ2 utþ2 Þdyt ; !tþ1 ¼ yt ðbutþ2 þ cytþ2 þ aytþ2 utþ2 Þdutþ1 ;

0:027y3 ðt þ 1Þ

!tþ2 ¼ utþ1 yt ½ðb þ aytþ2 Þdutþ2 þ ðc þ autþ2 Þdytþ2 : ð34Þ To find the state and the output transformations, we use the method from Subsection 2.4. From (34) and (12), ’1 ðyt ; ut Þ ¼ yt , ’2 ðytþ1 ; utþ1 Þ ¼ utþ1 , ’3 ðytþ2 ; utþ2 Þ ¼ butþ2 þ cytþ2 þ aytþ2 utþ2 , and 1 ¼ ’2 ’3 , 2 ¼ ’1 ’3 , 3 ¼ ’1 ’2 . By (32),

0:108y2 ðt þ 1Þuðt þ 1Þ 0:099u3 ðt þ 1Þ Since this i/o equation already is in the form (7), the output transformation is unnecessary and one can write directly 1 ð ut ; y t Þ ¼

’1 ðyt ; ut Þ ¼ 0; 149yt 0; 071ut ; 2 ðutþ1 ; ytþ1 Þ ¼ ’2 ðytþ1 ; utþ2 Þ ¼ 0; 681ytþ1 þ0; 014utþ1 0; 03y2tþ1

p0 ’i ’j ¼ ; p0 ’j ’i

0; 135ytþ1 utþ1

yielding p0 ðyÞ ¼ 1=y. Integrating p0 ðyÞ, we get y~ ¼ pðyÞ ¼ ln jyj. Then, according to (30), the functions i are ¼ ln j’1 ðyt ; ut Þj ¼ ln jyt j; 2 ðyt ; ut Þ ¼ ln j’2 ðyt ; ut Þj ¼ ln jut j;

0; 351ytþ2 utþ2 :

¼ ln j’3 ðyt ; ut Þj ¼ ln jbut þ cyt þ ayt ut j; ð35Þ

and by (8) one can define the new state coordinates z1;t ¼ y~t ¼ pðyt Þ ¼ ln jyt j; z2;t ¼ y~tþ1 3 p1 ðy~t Þ; ut ¼ yt Þ þ aut expð~ yt Þj; ¼ y~tþ1 ln jbut þ c expð~ 1 z3;t ¼ y~tþ2 3 p ðy~tþ1 Þ; utþ1 2 p1 ðy~t Þ; ut ¼ ¼ y~tþ2 ln jbutþ1 þ c expð~ ytþ1 Þ ytþ1 Þj ln jut j; þ autþ1 expð~ yielding the state equations in the observer form z1;tþ1 ¼ z2;t þ ln but þ c expðz1;t Þ þ aut expðz1;t Þ; z2;tþ1 ¼ z3;t þ ln jut j; z3;tþ1 ¼ z1;t ; y~ ¼ z1;t :

0; 099u3tþ1 ; 3 ðutþ2 ; ytþ2 Þ ¼ ’3 ðytþ2 ; utþ2 Þ ¼ 0; 43ytþ2 þ 0; 396utþ2

1 ðyt ; ut Þ

3 ð yt ; ut Þ

0; 027y3tþ1 0; 108y2tþ1 utþ1

Now, according to (8), the state coordinates are z1;t ¼ yt ; z2;t ¼ ytþ1 0; 43yt 0; 396ut þ 0; 351yt ut ; z3;t ¼ ytþ2 0; 43ytþ1 0; 396utþ1 þ0; 351ytþ1 utþ1 0; 681yt 0; 014ut þ 0; 03y2t þ 0; 135yt ut þ 0; 027y3t þ 0; 108ytt ut þ 0; 099u3t : This choice of the state coordinates gives the state equations in the observer form z1;tþ1 ¼ z2;t þ 0; 43z1;t þ 0; 396ut 0; 351z1;t ut ; z2;tþ3 ¼ z3;t 0; 681z1;t 0; 014ut þ 0; 03z21;t þ0; 135z1;t ut þ þ0; 027z31;t þ 0; 108z21;t ut

ð36Þ

þ0; 099u3t ; z3;tþ3 ¼ 0; 071ut 0; 149z1;t :

Example 2: Consider the system from [3] yðt þ 3Þ ¼ 0:43yðt þ 2Þ þ 0:681yðt þ 1Þ 0:149yðtÞ þ 0:396uðt þ 2Þ þ0:014uðt þ 1Þ 0:071uðtÞ 0:351yðt þ 2Þuðt þ 2Þ 0:03y2 ðt þ 1Þ

Acknowledgement This work was supported by the Estonian Science Foundation Grant nr. 6922.

183


References 1. Besanc¸on G. On output transformations for state linearization up to output injection. IEEE Trans Autom Control 1999; 44(10): 1975–1981. 2. Bestle D, Zeitz M. Canonical form observer for nonlinear time-variable systems. Int J Control 1983; 38(3): 419–431. 3. Billings SA, Fadzil MB. The practical identification of systems with nonlinearities In Proceedings of 7th IFAC/ IFORS Symposium on Identification and System Parameter Estimation, York, UK, 1985, pp 155–160. 4. Califano C, Monaco S, Normand-Cyrot D. On the observer design in discrete-time. Syst Control Lett 2003; 49(4): 255–265. 5. Chung ST, Grizzle JW. Sampled-data observer error linearization. Automatica 1990; 26: 997–1007. 6. Conte G, Moog CH, Perdon AM. Algebraic methods for nonlinear control systems. Theory and Applications, 2nd edn. Springer: London, 2007. 7. Glumineau A, Moog CH, Plestan F. New algebrogeometric conditions for the linearization by inputoutput injection. IEEE Trans Autom Control 1996; 41(4): 598–603. 8. Guay M. Observer linearization by output diffeomorphism and output-dependent time-scale transformations. In NOLCOS01, Saint Petersburg, Russia, 2001, pp 1443–1446. 9. Hammouri M, Kinnaert M. A new procedure for time-varying linearization up to output injection. Syst Control Lett 1996; 28(3): 151–157. 10. Huijberts HJC. On existence of extended observers for nonlinear discrete time systems. In: Nijmeijer H, Fossen

11.

12. 13. 14. 15. 16. 17. 18. 19.

20.

TH (eds), New Directions in Nonlinear Observer Design, Vol. 224. Lecture Notes in Control and Information Sciences: Springer Verlag, 1999. pp 73–92. Ingenbleek R. Transformation of nonlinear discretetime system into observer canonical form. In Preparation of 12th IFAC World Congress, Vol. 4, Sydney, Australia, 1993, pp 347–350. Jouan P. Immersion of nonlinear systems into linear systems modulo output injection. SIAM J Control Optim 2002; 41(6): 1756–1778. Krener AJ, Isidori A. Linearization by output injection and nonlinear observers. Syst Control Lett 1983; 3: 47–52. Krener AJ, Respondek W. Nonlinear observers with linearizable error dynamics. SIAM J Control Optim 1985; 23(2): 197–216. Lee W, Nam K. Observer design for autonomous discrete-time nonlinear systems. Syst Control Lett 1983; 17(1): 49–58. Lilge T. On observer design for nonlinear discrete-time systems. Eur J Control 1998; 4: 306–319. Lin W, Brynes CI. Remarks on linearization of discretetime autonomous systems and nonlinear observer design. Syst Control Lett 1995; 25: 31–40. Noh D, Jo NH, Seo JH. Nonlinear observer design by dynamic observer error linearization. IEEE Trans Autom Control 2004; 49(10): 1746–1753. Xiao MQ. A direct method for the construction of nonlinear discrete-time observer with linearizable error dynamics. IEEE Trans Autom Control 2006; 51(1): 128–135. Zheng G, Boutat D, Barbot J-P. Single output-dependent observability normal form. SIAM J Control Optim 2007; 46(6): 2242–2255.