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On existence of extended observers for nonlinear discrete-time systems H.J.C. Huijberts Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.

1 Introduction In this paper, we consider the design of observers for discrete-time nonlinear systems by means of so called (extended) observer forms. Loosely speaking, a system in observer form is a linear observable (continuous-time or discrete-time) system that is interconnected with an output-dependent nonlinearity. Observers for this kind of systems may be built by building a classical linear Luenberger observer for the linear system, and adding the output-dependent nonlinearity to this observer. Thus, observer design for systems in observer form is relatively easy. By the same token, also observer design for systems that may be transformed into a system in observer form by means of a coordinate transformation and an output transformation is relatively easy. Observer design for systems in observer form was rst studied, in the continuous-time setting, in [10],[11] (see also [15]). In these papers, conditions were given under which a nonlinear continuous-time system may be transformed into a system in observer form by means of a coordinate transformation and an output transformation. Basically, these conditions were given in terms of the integrability of certain codistributions. Later on, the observer design for discrete-time systems in observer form was studied (see [1],[12],[13] and the references therein), and conditions were given under which a nonlinear discrete-time system may be transformed into a system in observer form by means of a coordinate transformation and an output transformation. These conditions came down to the question whether certain functions could be factorized in a certain way. For single-output systems, conditions under which this factorization is indeed possible were given when only output transformations are allowed. (In fact, [13] also claims to give conditions for the multi-output case. However, these conditions seem to be incorrect.) One of the purposes of this paper is to

4

2. On existence of extended observers for nonlinear discrete-time systems

generalize the conditions given in [13] for single-output systems to the case where, besides a coordinate transformation, also an output transformation is allowed. The conditions for the existence of an observer form for continuous-time systems and discrete-time systems given in [10],[11],[1],[12],[13] are quite restrictive. Therefore, generalizations have been considered, both in the continuous-time case and the discrete-time case. In the continuous-time case, so-called generalized observer forms were considered. These generalized observer forms consist of an observable linear system interconnected with a nonlinearity that depends on the output of the system and a nite number of its derivatives. In [6], dierential geometric conditions were given under which a continuous-time system may be transformed into a generalized observer form by means of so called generalized coordinate transformations (i.e., transformations that, besides the state of the system, also depend on a nite number of time-derivatives of the outputs) and an output transformation. In the discrete-time context, the design of so called extended observers by using extended observer forms was studied in [7],[8]. Here, an extended observer is an observer that, besides the output of the system, also depends on a nite number of its past values, while a system in extended observer form consists of an observable linear system interconnected with a nonlinearity that depends on the output of the system and a nite number of past output values. In [7],[8], conditions were given under which a given single-output discrete-time system may be transformed into a system in extended observer form by means of an extended coordinate transformation (i.e., a coordinate transformation that depends on the state of the system and a nite number of past output values) and an output transformation. As in [1],[12], these conditions again boilt down to the question whether a given function may be factorized in a certain way. A corollary of the results obtained in [7],[8] is that when the number of past output values equals n ? 1 (where n is the dimension of the state space of the system under consideration), an extended observer form always exists when the system under consideration is strongly observable (for the exact de nition of strong observability, we refer to Section 3). In this paper, differential geometric conditions for the existence of such a factorization for the cases that the number of past output values is smaller than n ? 1 will be given. It is to be noted that in principle the problem of observer design is a global problem. Therefore one would also like to obtain global conditions for existence of extended observer forms. However, if one knows that the system under consideration evolves on an invariant set, also existence conditions on this invariant set would suce. In this paper, all results obtained will be valid on open invariant sets on which some regularity assumptions hold. Of course, this also includes the case where one really would like to have global conditions. This paper is organized as follows. In the following section, an overview

2. On existence of extended observers for nonlinear discrete-time systems

5

of results from the theory of dierential forms that will be used in this paper is given. In Section 3, we consider the existence of observer forms for single-output discrete-time systems. The results in this section reformulate and generalize the results in [13]. In Section 4, the existence of extended observer forms for single-output discrete-time systems is studied. Section 5 contains some conclusions.

2 Dierential forms In this section we give an overview of results from the theory of differential forms that will be used in this paper. For details, we refer to [2],[3],[4],[5],[14]. Let V be an r-dimensional vector space over IR. A k-form ! on V is a k-linear completely antisymmetric mapping ! : V| 
{z

 V} ! IR, i.e., k times

(8; 2 IR)(8v1 ; v10 ; v2;


; vr 2 V )(!(v1 + v10 ; v2;


; vk ) =

!(v1 ;


; vk ) + !(v10 ; v2;


; vk )) (8v1 ;


; vk 2 V )(!(v1 ;


; vk ) =

(2.1)

(2.2)

?!(v1 ;


; vi?1; vi+1 ; vi; vi+2;


; vk )) Note that a one-form on V is just an element of V  , the dual of V . The space of all k-forms on V is denoted by k (V  ). It is easily checked that the k-linearity and anti-symmetry of a k-form on V implies that all k-forms are zero for k > r. We de ne (V  ) := 0(V  )  1(V  ) 


 r (V  ) (2.3) where 0(V  ) := IR We call (V  ) the exterior algebra over V  . An element ! 2 (V  ) is called a form on V and may be written in a unique way as

! = !0 + !1 +


+ !r

(2.4)

where !i 2 i (V  ) (i = 0;


; r). We next de ne a product on (V  ), the so called wedge product (or exterior product). This product will be denoted by "^". First, let  2 p (V  ),

6

2. On existence of extended observers for nonlinear discrete-time systems

! 2 q (V  ). Then  ^ ! 2 p+q (V  ) is de ned by ( ^ !)(v1 ;


; vp+q ) =

P(sign())(v ;


; v )!(v (1) (p) (p+1) ;


; v(p+q) ) 

(2.5)

where the summation is over all possible permutations of 1;


; p + q, and v1 ;


; vp+q 2 V . For  = 0 +


+ r , ! = !0 +


+ !r , with i ; !i 2 i(V  ) (i = 0;


; r), we de ne

^! =

r X

i ^ !j

i;j =0

(2.6)

Note that the wedge-product is associative and distributive, but not commutative. Instead, it satis es  ^ ! = (?1)pq ! ^ ;  2 p (V  ); ! 2 q (V  ) (2.7) Let v 2 V be given. Then the interior product v : (V  ) ! (V  ) is de ned in the following way. First consider ! 2 k (V  ). Then v ! 2 k?1(V  ) is given by (v !)(v1 ;


; vp?1) = !(v; v1 ;


; vp?1 ) (2.8) where v1 ;


; vp?1 2 V . If ! = !0 +


+ !r , with !i 2 i (V  ) (i = 0;


; r), then (v !) =

r X i=0

(v !i )

(2.9)

Next, consider an n-dimensional manifold M . Let Tx M denote the cotangent space at x 2 M and let T  M denote the cotangent bundle of M . Since Tx M is an n-dimensional vector space over IR, we may de ne k (Tx M ) for all x 2 M , k = 0;


; n, as

X (T  M ) := k (T  M ) n

x

k=0

x

We then de ne the bundles k (T  M ), (T  M ) over M by k (T  M ) :=

[

x2M

k (Tx M ); (k = 0;


; n)

(T  M ) :=

n M k=0

k (T  M )

(2.10) (2.11)

2. On existence of extended observers for nonlinear discrete-time systems

7

A dierential form on M is now de ned to be a smooth section of the bundle (T  M ), while a dierential k-form on M is de ned to be a smooth section of the bundle k (T  M ). So, roughly speaking, a dierential (k?)form on M is a "prescription" that assigns a (k?)form !x on Tx M to every x 2 M in a smooth way. Note that by this de nition a dierential 0-form on M is just a smooth function on M . When no confusion arises, we will simply call a dierential (k?)form on M a (k?)form on M in the sequel. The wedge product of the forms ; ! on M is de ned to be the form ( ^ !) satisfying ( ^ !)x = x ^ !x ; (8x 2 M ) (2.12) Let  be a smooth vector eld on M , and let ! be a form on M . Then the interior product ( !) is de ned to be the form satisfying ( !)x = x !x ; (8x 2 M ) (2.13) The exterior dierential operator d maps a k-form ! into a (k + 1)-form d!, called the exterior derivative of !. The operator d is uniquely de ned by the following properties: 1. d is linear: (8; 2 IR)(d( + !) = d + d!) 2. If  is a k-form, then d( ^ !) = d ^ ! + (?1)k  ^ d! 3. d2 = 0. 4. If f is a 0-form, then df is the ordinary dierential df of f . 5. d is local: if  and ! coincide on an open set U , then d = d! on U . A k-form ! is called closed if d! = 0; it is called exact if there exists a (k ? 1)-form  such that ! = d. Note that, since d2 = 0, an exact k-form is closed. The converse does not need to hold globally. However, it does hold locally, as is re ected by the following theorem. Theorem 1 (Poincare Lemma) If M is smoothly contractible to a point x0 2 M , then every closed form ! on M is exact. Let  be a smooth vector eld on M . The Lie-derivative L maps a k-form ! into a k-form L !. L is uniquely de ned by the following properties: 1. If f is a 0-form on M , then L f =  df

8

2. On existence of extended observers for nonlinear discrete-time systems

2. L is a derivation:

L ( ^ !) =  ^ L ! + L  ^ !

3. L commutes with d:

L (d!) = d(L !)

From the de nitions of interior product, exterior derivative, and Liederivative one may derive the following identities that will be frequently used in the sequel (here ;  denote smooth vector elds on M , and ! denotes a k-form on M ). (2.14) L ! = d( !) +  d! [;  ] ! = L ( !) ?  L ! (Leibniz ? formula)

(2.15)

L[; ] ! = L L ! ? L L ! (2.16) Denote by 1(M ) the set of all one-forms on M . Then 1 (M ) has the

structure of a nitely generated module over the ring of smooth functions on M . A nitely generated submodule of 1(M ) is called a codistribution on M . The minimal number of generators of a codistribution is called its dimension. A codistribution on M is called integrable if it has a set of generators consisting of exact one-forms. The codistribution generated by a set of one-forms !1;


; !d is denoted by spanf!1;


; !dg. For a one-form ! and a d-dimensional codistribution , we will say that d!  0 mod

if and only if d! ^ !1 ^


^ !d = 0 (8!1;


; !d 2 ) Theorem 2 (Frobenius Theorem) Let M be smoothly contractible to a point x0 2 M , and let be a codistribution on M . Then the following statements are equivalent. (i) is integrable. (ii) For every ! 2 we have that d!  0 mod . (iii) Let f!1;


; !dg be a set of generators of and let !d+1 ;


; !n be such that f!1;


; !ng generate 1 (M ). Then is integrable if and only if there exist smooth functions ?kij (i; k = 1;


; d; j = i + 1;


; n) on M such that

d!k =

d X n X

i=1 j =i+1

?kij !i ^ !j (k = 1;


; d)

(2.17)

2. On existence of extended observers for nonlinear discrete-time systems

9

In what follows, we will also need the following result. Theorem 3 (Cartan's Lemma) If !1 ;


; !d are independent one-forms, and 1;


; d are one-forms such that 1 ^ !1 +


+ d ^ !d = 0 then

i 2 spanf!1 ;


; !d g

3 Observer design using observer forms We start our investigation of observer design for nonlinear discrete-time systems by considering a nonlinear discrete-time system ~ of the form  1) = Az (k) + (~y (k)) (2.18) ~ z (k + y~(k) = Cz (k) with state z 2 IRn , output y~ 2 IR, and where A; C are matrices of appropriate dimensions, the mapping  : IR ! IRn is smooth, and the pair (C; A) is in Brunovsky form. Analogously to [10],[11], a system of this form is called a system in observer form. For systems in observer form, observer design is particularly simple. Namely, the fact that the pair (C; A) is in Brunovsky form, and thus in particular observable, implies that there exists a matrix K such that all eigenvalues of the matrix A ? KC are in the open unit disc. It is then straightforwardly checked that the following system is an observer for ~ :

 zb(k + 1) = Azb(k) + K (~y (k) ? by~(k)) + (~y(k)) by~(k) = C zb(k)

(2.19)

We next consider a nonlinear discrete-time system  of the form  1) = f (x(k)) (2.20)  x(k + y(k) = h(x(k)) where x 2 IRn , y 2 IR, and the mappings f : IRn ! IRn and h : IRn ! IR are smooth. We will say that  can be put in observer form if there exist a dieomorphism P : IRn ! IRn of the state space and a dieomorphism p : IR ! IR of the output space such that in the new coordinates z = P (x) and with the new output y~ = p(y) the system  takes the form (2.18), where the pair (C; A) is in Brunovsky form. If  does admit an observer form, an observer for  may then be obtained by rst building an observer (2.19) for the observer form (2.18) and then letting xb(k) := P ?1(zb(k)) be the estimate of x(k).

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2. On existence of extended observers for nonlinear discrete-time systems

We thus see that observer design for a discrete-time system  of the form (2.20) is relatively easy when  can be put in observer form. This raises the question under what conditions  can be put in observer form. To derive these conditions, we rst introduce the so called observable form of . De ne the observability map : IRn ! IRn by

0 h(x) B h  f (x) (x) := B B@ .. .

h  f n?1 (x)

1 CC CA

(2.21)

where f 1 := f , f k := f  f k?1 . Assume that the origin is an equilibrium point of  (i.e., f (0) = 0) and that h(0) = 0. We then call  strongly observable on an open subset U  IRn containing the origin if is a diffeomorphism on U . It follows that if  is strongly observable on U , then s := (x) forms a new set of local coordinates for  on U . In these new coordinates,  takes the form

8 s (k + 1) > 1 > > < > sn?1(k + 1) > : sn(k +y(k1))

= .. . = = =

s2 (k) sn (k) fs (s(k)) s1 (k)

(2.22)

where fs := h  f n?1  ?1 . The form (2.22) is referred to as the observable form of . We now have the following result.

Theorem 4 Consider a nonlinear discrete-time system  of the form (2.20), and assume that f (0) = 0, h(0) = 0. Then  can be put in observer form on an open invariant subset U of (2.20) containing the origin if and only if (i)  is strongly observable on U . (ii) There exist functions p; 1;


; n : IR ! IR, where p is a dieomorphism on h(U ), such that on U the function fs in (2.22) satis es

p  fs (s) = 1 (sn ) + 2(sn?1 ) +


+ n(s1 )

(2.23)

Proof. The proof for the case where only coordinate transformations are considered (or, in other words, the case where p = idIR ) may be found in [13],[8],[9]. The case where also output transformations are considered is an (almost) immediate consequence of the case where p = idIR .

2. On existence of extended observers for nonlinear discrete-time systems

11

If functions p; 1;


; n : IR ! IR exist such that fs satis es (2.23), we de ne the following coordinate change:

zi := p(si ) ?

i?1 X j =1

j (si?j ) (i = 1;


; n)

(2.24)

In these new coordinates, we obtain the observer form for : 8 z (k + 1) = z (k) + ~ (~y(k)) > 1 2 1 > .. > < . (2.25) z ( k ) = zn (k) + ~n?1(~y (k)) n?1 > > > : zny~((kk)) == z~1n((~ky)(k)) where ~i := i  p?1. From Theorem 4, it follows that the question whether or not  can be put in observer form may be reduced to the question whether or not the function fs in (2.22) may be written in the special form (2.23). We will now derive conditions on fs under which this is possible. To this end, we de ne one-forms !1;


; !n by

!i :=

i  X @fs  j =1

@sj dsj (i = 1;


; n)

(2.26)

For the case that p = idIR we then have the following result. Theorem 5 Consider a discrete-time system  of the form (2.20) that is strongly observable on an open invariant subset U  IRn containing the origin. Assume further that U is smoothly contractible to the origin, and that the one-forms !1 ;


; !n in (2.26) generate a codistribution on U . Then  can be put in observer form with p = idIR on U if and only if the one-forms !1 ;


; !n in (2.26) satisfy d!i = 0 (i = 1;


; n) (2.27) Proof. (necessity) Follows by direct veri cation. (suciency) Assume that the one-forms !i in (2.26) satisfy (2.27). It then follows from (2.26,2.27) that

 n  2 i X X @ f s 0 = d!i =


= @sk @sj dsk ^ dsj (i = 1;


; n) (2.28) j =1 k=i+1

which is equivalent to  @2f  s @sk @sj = 0 (j; k = 1;


; n; j 6= k)

(2.29)

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2. On existence of extended observers for nonlinear discrete-time systems

This implies that there exist functions 1;


; n : IR ! IR such that fs satis es (2.23) with p = idIR. We next give conditions under which fs may be written in the form (2.23), where p 6= idIR . To do this, we let 1 ;


; n be vector elds that are dual to the one-forms !1;


; !n in (2.26), i.e.,

i !j = ij (i; j = 1;


; n) (2.30) where ij is the Kronecker delta. Theorem 6 Consider a discrete-time system  of the form (2.20) that is strongly observable on an open invariant subset U  IRn containing the origin. Assume further that U is smoothly contractible to the origin, and that the one-forms !1 ;


; !n in (2.26) generate a codistribution on U . Then the following statements are equivalent: (i)  can be put in observer form on U . (ii) There exists a function S : U ! IR such that

d!i = dS ^ !i (i = 1;


; n)

(2.31)

(iii) The one-forms !1;


; !n and the vector elds 1 ;


; n satisfy

d!i ^ !j + d!j ^ !i = 0 (i; j = 1;


; n)

(2.32)

L ([i; j ] !i) = L ([j ; i] !j )

(2.33)

and j

i

Proof. (i))(ii) Assume that  can be put in observer form on U . De ne

one-forms !~ i by

Xi  @(p  fs) 

!~i :=

@sj

j =1

dsj (i = 1;


; n)

It then follows from Theorem 5 that we have that d~!i = 0 (i = 1;


; n) Together with the fact that !~i = (p0  fs )!i , this gives that 0 = d~!i = d



(p0  fs )d

1 p fs





0

1 p fs 0

^ !~ i



^ !i = d(? log jp0  fs j) ^ !i

(2.34) (2.35)

(2.36)

2. On existence of extended observers for nonlinear discrete-time systems

13

which establishes our claim for S := ? log jp0  fs j. (ii))(i) Assume that there exists a function S : U ! IR such that (2.31) holds. Since !n = dfs , (2.31) for i = n gives that dS ^ dfs = 0. By Cartan's Lemma, this gives that dS 2 spanfdfs g. De ne T := exp(?S ). We then also have that dT 2 spanfdfsg, which implies that there exists aRfunction p~ : IR ! IR such that T = p~  fs . De ne p : IR ! IR by p := p( )d , and de ne one-forms !~1 ;


; !~ n as in (2.34). Note that we then have that !~i = T!i (i = 1;


; n), which implies that d~!i = T d!i + dT ^ !i = T (d!i ? dS ^ !i ) = 0 (i = 1;


; n) (2.37) Together with Theorem 5 this establishes our claim. (ii),(iii) We rst show that (2.32) is equivalent to the existence of a unique one-form  such that d!i =  ^ !i (i = 1;


; n) (2.38) Note that if there exists a one-form  such that (2.38) holds, then (2.32) follows immediately. Conversely, assume that (2.32) holds. For i = j , (2.32) gives in particular that d!i ^ !i = 0 (i = 1;


; n). It then follows from the Frobenius Theorem that there exist one-forms 1 ;


; n such that d!i = i ^ !i (i = 1;


; n) (2.39) P n Writing i = k=1 ik !k (i = 1;


; n), we then obtain from (2.32) that 0 = d!i ^ !j + d!j ^ !i =


=

n X

(ik ? jk )!k ^ !i ^ !j

(2.40)

k=1

k6=i;j

which gives that

ik = jk (i; j; k = 1;


; n; k 6= i; j ) (2.41) This immediately implies that indeed there exists a unique one-form  such that (2.38) holds. What remains to be done, is to show that there locally existsPan function S such that  = dS if and only if (2.33) holds. Writing  = j =1 j !j , we have that d =

XX

n X

n?1 n

j =1

j =1 i=j +1

(dj ^ !j + j d!j ) =


=

(L j ? L i )!i ^ !j i

j

(2.42) By the Poincare Lemma, this gives that there exists a function S : U ! IR such that  = dS if and only if (2.43) L j = L i (i; j = 1;


; n) Using (2.15),(2.14), it is straightforwardly checked that j = [i ; j ] !i . Together with (2.43), this yields (2.33), which establishes our claim. i

j

14

2. On existence of extended observers for nonlinear discrete-time systems

4 Observer design using extended observer forms In the previous section, we have seen that observer design for a nonlinear discrete-time system (2.20) is relatively easy when the system can be put in observer form (2.18). Unfortunately, however, the conditions for existence of an observer form given in Theorem 4 are quite restrictive. In this section, we will relax these conditions by considering so called extended observers and extended observer forms. We will rst explain what is meant by an extended observer. Consider a system  of the form (2.20), and let N 2 IN be given. We now assume that at every time instant k  N we do not only know the output y(k) at time k, but also the past outputs y(k ? 1);


; y(k ? N ). An extended observer with buer N for  then is a dynamical system b of the form b : xb(k + 1) = fb(xb(k); y(k);


; y(k ? N )); k  N where xb 2 IRn and the mapping fb : IRn+N +1 ! IRn is smooth, with the property that x(k) ? xb(k) ! 0 (k ! 1), for all x(0); bx(N ) 2 IRn . To study the design of extended observers for a discrete-time system  of the form (2.20), we rst consider a system ~ e of the form ~ e

 z(k + 1) = Az(k) + (~y(k);


; y~(k ? N ))

(2.44) y~(k) = Cz (k) where the state z 2 IRn, the output y~ 2 IR, A; C are matrices of appropriate dimensions, the mapping  : IRN +1 ! IRn is smooth, and the pair (C; A) is in Brunovsky form. Note that for N = 0 the system ~ e is identical to the system ~ in (2.18). Therefore, a system ~ e of the form (2.44) will be referred to as a system in extended observer form with buer N . As for a system in observer form, the design of an extended observer for a system in extended observer form is relatively easy. Namely, it is straightforwardly checked that the system  zb(k + 1) = Azb(k) + K (~y(k) ? by~(k)) + (~y(k);


; y~(k ? N )) by~(k) = C zb(k) (2.45) where the matrix K is such that all eigenvalues of A ? KC are in the open unit disc, is an extended observer for ~ e . As in the previous section, we now consider the question under which conditions a given discrete-time system can be put in extended observer form for some N 2 IN . The transformations we are going to use here, are more general than the ones in the previous section, in the sense that we also allow them to depend on the past output measurements y(k ? 1);


; y(k ? N ). More speci cally, we will be looking at parametrized transformations z = P (x; n;


; N ), where z 2 IRn , with the property that there exists a

2. On existence of extended observers for nonlinear discrete-time systems

15

mapping P ?1(
; 1;


; N ) : IRn ! IRn parametrized by (1;


; N ) such that for all (1 ;


; N ) we have that P (P ?1(z; 1;


; N ); 1;


; N ) = z A mapping having this property will be referred to as an extended coordinate change. We will then say that the system (2.20) can be put in extended observer form with buer N if there exists an extended coordinate change P (
; 1;


; N ) : IRn ! IRn parametrized by (1 ;


; N ) and a dieomorphism p : IR ! IR of the output space such that the variable z (k) := P (x(k); y(k ? 1);


; y(k ? N )) (2.46) satis es (2.44), where y~ := p(y), and the pair (C; A) is in Brunovsky form. As pointed out above, one may then build an extended observer (2.45) for z (k) in (2.46). From this extended observer, one then obtains an estimate xb(k) for x(k) by inverting the extended transformation P : xb(k) := P ?1(zb(k); y(k ? 1);


; y(k ? N )); k  N (2.47) The following generalization of Theorem 4 gives conditions under which a discrete-time system (2.20) can be put in extended observer form. Theorem 7 Consider a nonlinear discrete-time system  of the form (2.20), and assume that f (0) = 0, h(0) = 0. Let N 2 f0;


; n ? 1g be given. Then  can be put in observer form with buer N on an open invariant subset U  IRn containing the origin if and only if (i)  is strongly observable on U . (ii) There exist functions 1;


; n?N : IRN +1 ! IR and a dieomorphism p on h(U ), such that on U the function fs in (2.22) satis es

p  fs (s) =

X

n?N i=1

n+1?N ?i (si+N ;


; si)

(2.48)

Proof. The proof for the case where only coordinate transformations are considered (or, in other words, the case where p = idIR ) may be found in [8]. The case where also output transformations are considered is an (almost) immediate consequence of the case where p = idIR . If there exist functions p; 1;


; n?N such that fs satis es (2.48), one de nes the following variables: zi (k) := si (k)?

 P j (si?j (k);


; s1 (k); y(k ? 1);


; y(k ? N ? 1 + i ? j )) (2.49) j =1 i

(i = 1;


; n)

16

2. On existence of extended observers for nonlinear discrete-time systems

where i := min(i ? 1; n ? N ). Further, de ne ~1 ;


; ~n?N : IRN +1 ! IRN +1 by ~i (1 ;


; N +1 ) := i(p?1 (1 );


; p?1(N +1 )) (i = 1;


; n ? N ) It may then be shown that in these variables we obtain the following extended observer form: 8 z1 (k + 1) = z2 (k) + ~1(~y (k);


; y~(k ? N )) > > .. > . > > > z ( k + 1) = (k) + ~n?N (~y (k);


; y~(k ? N )) < zn?nN?+1N (k + 1) = zznn??NN +1 +2 (k) (2.50) . > .. > > > zn?1(k + 1) = zn (k) > > zn (k + 1) = 0 : y~(k) = z1 (k) Note that Theorem 7 generalizes Theorem 4. Further, from Theorem 7, we obtain the following result for N = n ? 1 (see also [7],[8]). Corollary 8 Consider a nonlinear discrete-time system  of the form (2.20), and assume that f (0) = 0, h(0) = 0. Then  can be put in observer form with buer N on an open invariant subset U  IRn containing the origin if and only if  is strongly observable on U . Thus, we see that every strongly observable system can be put in extended observer with buer N = n ? 1. From a practical point of view, e.g. when n is large, it may be desirable to reduce the size of the buer. Therefore, we next investigate under which conditions an extended observer with buer N 2 f1;


; n ? 2g exists. As in the previous section, these conditions are again given in terms of the one-forms !i in (2.26) and the vector elds i in (2.30). Using the one-forms !i in (2.26), we de ne the following codistributions:

i := spanf!k ? !k?1 j k = i + 1;


; min(n; i + N )g (i = 1;


; n) (2.51)

~ i := spanf!i ;


; !i+N g (i = 1;


; n ? N ? 1) (2.52) We rst consider the case without output transformations, i.e., the case where in (2.48) we have that p = idIR . This result generalizes Theorem 5. Theorem 9 Consider a discrete-time system  of the form (2.20) that is strongly observable on an open invariant subset U  IRn containing the

2. On existence of extended observers for nonlinear discrete-time systems

17

origin. Assume further that U is smoothly contractible to the origin, and that the one-forms !1 ;


; !n in (2.26) generate a codistribution on U . Let N 2 f1;


; n ? 2g be given. Then for p = idIR,  can be put in extended observer form with buer N on U if and only if the one-forms !i in (2.26) satisfy

d!i  0 mod i (i = 1;


; n) (2.53) Proof. (necessity) Follows by direct veri cation. (suciency) Assume that the one-forms !i in (2.26) satisfy (2.53). Note that by the de nition of the !i we have that

i = spanfdsk j k = i + 1;


; min(n; i + N )g (i = 1;


; n) De ning i := min(n; i + N ) (i = 1;


; n), (2.53) then gives 0 = d!i ^ dsi+1 ^


^ ds =


=

Pi Pn 

j =1 k=i +1

@ 2 fs @sk @sj



i

dsk ^ dsj ^ dsi+1 ^


^ ds

i

(2.54)

(i = 1;


; n) which is equivalent to  @2f  (2.55) @sk sj = 0 (j; k = 1;


; n; jj ? kj > n) It is easily checked that this condition is equivalent to the existence of functions 1 ;


; n?N such that fs satis es (2.48). For the case that in (2.48) we have that p 6= idIR , the following result holds. Theorem 10 Consider a discrete-time system  of the form (2.20) that is strongly observable on an open invariant subset U  IRn containing the origin. Assume further that U is smoothly contractible to the origin, and that the one-forms !1 ;


; !n in (2.26) generate a codistribution on U . Let N 2 f1;


; n ? 2g be given. Then the following statements are equivalent: (i)  can be put in extended observer form with buer N on U . (ii) There exists a function U : IRn ! IR such that d!i ? dS ^ !i  0 mod i (i = 1;


; n) (2.56) (iii) The one-forms !1;


; !n and the vector elds 1 ;


; n satisfy

d!i  0 mod i + spanf!n g (i = 1;


; n ? N ? 1)

(2.57)

18

2. On existence of extended observers for nonlinear discrete-time systems

d!i  0 mod ~ i (i = 1;


; n ? N ? 1)

(2.58)

d!i  0 mod i (i = n ? N;


; n ? 1)

(2.59)

[i; n] !n = [j ; n] !n (i; j = 1;


; n ? N ? 1)

(2.60)

L ([i; n] !i ) = 0 (i = 1;


; n ? N ? 1; j = 1;


; n ? 1) j

(2.61)

where the vector elds 1;


; n?N ?1 are de ned by

i := i +


+ i+N (i = 1;


; n ? N ? 1)

(2.62)

Proof. (i))(ii) Assume that there exist functions p; 1;


; n?N , such that p is a dieomorphism on h(U ) and fs satis es (2.48). De ne one-forms !~1 ;


; !~ n by Xi  @(p  fs)  ds (i = 1;


; n) (2.63) !~ := i

j =1

j

@sj

Then it follows from Theorem 9 that d~!i  0 mod spanf!~ k ? !~ k?1 j k = i + 1;


; min(n; i + N )g

(2.64) (i = 1;


; n) Note that we have that !~ i = (p0  fs )!i (i = 1;


; n). De ning the function S := ? log jp0  fs j, this gives d!i ? dS ^ !i = d 1 !i p0 fs d~



1 ~i p fs ! 0

  +

(i = 1;


; n)

1 0 p fs d(p  fs ) 0



^ p 1f !~ i =


= 0

s

(2.65)

Further, it follows that spanf!~ k ? !~ k?1 j k = i + 1;


; min(n; i + N )g = i (i = 1;


; n) (2.66) Our claim is then established by combining (2.64),(2.65),(2.66). (ii))(i) Assume that there exists a function S : U ! IR such that (2.56) holds. Note that from (2.26) we have that !n = dfs . Thus, (2.56) for i = n gives that 0 = d!n ? dS ^ !n = ?dS ^ !n (2.67)

2. On existence of extended observers for nonlinear discrete-time systems

19

By Cartan's Lemma, this gives that dS 2 spanfdfs g. De ne T := exp(?S ). Then we also have that dT 2 spanfdfsg, and thusR there exists a function p~ : IR ! IR such that T = p~  fs . De ne p := p~( )d , and one-forms !~1 ;


; !~ n as in (2.63). We then have that !~ i = T!i (i = 1;


; n), and thus d~!i = dT ^ !i + T d!i = T (d!i ? dS ^ !i )  0 mod i (i = 1;


; n) (2.68) Together with Theorem 9, this establishes our claim. (ii),(iii) From the fact that !n = dfs, it follows that the function S that needs to exist has to satisfy dS ^ !n . By Cartan's Lemma, this implies that there should exist a function  such that dS = !n and

d ^ !n = 0 (2.69) Thus, the existence of a function S such that (2.56) holds is equivalent to the existence of a function  satisfying (2.69) and d!i ? !n ^ !i  0 mod i (i = 1;


; n ? 1) (2.70) We now have that a two-form !2 satis es !2  0 mod for some codistribution if and only if X Y !2 for all X; Y 2 ? . It is easily checked that we have

?i = spanf1;


; i?1; i+N +1 ;


; n; ig (2.71) (i = 1;


; n ? N ? 1) and

?i = spanf1;


; i?1; ig (i = n ? N;


; n ? 1) (2.72) where, analogously to (2.62), we have

i := i +


+ n (i = n ? N;


; n ? 1)

(2.73)

Now let i 2 f1;


; n ? N ? 1g be given. We then have 0 = k ` (d!i ? !n ^ !i) = k ` d!i = [k ; ` ] !i (k; ` = 1;


; i ? 1; i + N + 1;


; n) 0 = k i (d!i ? !n ^ !i ) = k i d!i = [k ; i] !i (k = 1;


; i ? 1; i + N + 1;


; n ? 1)

(2.74)

(2.75)

20

2. On existence of extended observers for nonlinear discrete-time systems

and 0 = i n (d!i ? !n ^ !i) = i n d!i ?  = [i; n] !i ?  (2.76) Where in (2.74),(2.75),(2.76) the last equality follows by applying (2.15) and (2.14). Combining (2.74) and (2.75), we obtain (2.57) and (2.58). Further, (2.76) gives (2.60), while combining (2.69) and (2.76) we obtain (2.61). Next, let i 2 fn ? N;


; n ? 1g be given. We then have 0 = k ` (d!i ? !n ^ !i) = k ` d!i = [k ; ` ] !i (2.77) (k; ` = 1;


; i ? 1) and 0 = k i (d!i ? !n ^ !i ) = k i d!i = [k ; i] !i (2.78) (k = 1;


; i ? 1) Combining (2.77) and (2.78), we then obtain (2.59).

Remark 11 Theorem 10 generalizes Theorem 6. From the rst two items

of both theorems this is seen immediately. If however, one considers the third item of both theorems the generalization is far from obvious at rst sight. This is due to the fact that the equivalence (ii),(iii) in Theorem 6 holds for general independent one-forms !1;


; !n, while this equivalence in Theorem 10 only holds for one-forms !1 ;


; !n of the form (2.26).

5 Conclusions In this paper, we have given conditions for the existence of extended observer forms and extended observers for single-output nonlinear discretetime systems. All conditions are valid on an open invariant subset of the state space that is smoothly contractible to an equilibrium point of the system and on which some regularity assumptions are satis ed. This raises the question what can be said for the case that (some of) the regularity assumptions are not satis ed. This remains a topic for future research. A further topic for future research would be the question when extended observer forms and extended observers for multi-output discrete-time systems exist. As also mentioned in the Introduction, it seems that the conditions given in [13] for the existence of an observer form when only coordinate transformations are allowed, seem to be incorrect. Preliminary investigations suggest that in fact the problem of coming up with correct conditions may be quite intractable. On the other hand however, it may

2. On existence of extended observers for nonlinear discrete-time systems

21

be shown by using the same techniques as in [7],[8] that a strongly observable multi-output system may always be put in observer form with buer N =  ? 1, where  equals the maximal so called observability index of the system.

Acknowledgment Part of this research was performed while the author was visiting the Laboratoire d'Automatique de Nantes, Nantes, France, supported by a grant from the Region Pays de la Loire. 6 References

[1] Brodmann, M., Beobachterentwurf fur nichtlineare zeitdiskrete Systeme, VDI Verlag, Dusseldorf, 1994. [2] Bryant, R.L., S.S. Chern, R.B. Gardner, H.L. Goldschmidt and P.A. Griths, Exterior dierential systems, Springer, New York, 1991. [3] Cartan, H., Formes dierentielles, Hermann, Paris, 1967. [4] Choquet-Bruhat, Y., and C. DeWitt-Morette (with M. Dillard-Bleick), Anaysis, manifolds and physics, Part I: Basics, North-Holland, Amsterdam, 1991. [5] Flanders, H., Dierential forms with applications to the physical sciences, Dover, New York, 1989. [6] Glumineau, A., C.H. Moog and F. Plestan, New algebro-geometric conditions for the linearization by input-output injection, IEEE Trans. Automat. Control, 41, pp. 598-603, 1996. [7] Huijberts, H.J.C., T. Lilge and H. Nijmeijer, A control perspective

on synchronization and the Takens-Aeyels-Sauer Reconstruction Theorem, to appear in Phys. Rev. E, 1999. [8] Huijberts, H.J.C., T. Lilge and H. Nijmeijer, Synchronization and observers for nonlinear discrete time systems, submitted to European

Control Conference 1999. [9] Huijberts, H.J.C., H. Nijmeijer, and A.Yu. Pogromsky, Discrete-time observers and synchronization, in G. Chen (Ed.), Controlling chaos and bifurcations in engineering systems, CRC Press, Boca Raton, Florida, 1999. [10] Krener, A.J., and A. Isidori, Linearization by output injection and nonlinear observers, Syst. Control Lett., 3, pp. 47{52, 1983.

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2. On existence of extended observers for nonlinear discrete-time systems

[11] Krener, A.J., and W. Respondek, Nonlinear observers with linearizable error dynamics, SIAM J. Control Optimiz., 23, pp. 197{216, 1985. [12] Lilge, T., On observer design for nonlinear discrete-time systems, Eur. J. Control, 4, pp. 306-319, 1998. [13] Lin, W., and C.I. Byrnes, Remarks on linearization of discrete{time autonomous systems and nonlinear observer design, Syst. Control Lett., 25, pp. 31{40, 1995. [14] Spivak, M., A comprehensive introduction to dierential geometry, Volume I, Publish or Perish, Houston, 1979. [15] Xia, X., and W. Gao, Nonlinear observer design by observer canonical forms, Int. J. Control, 47, pp. 1081-1100, 1988.