A constructive algorithm for system immersion into nonlinear observer form

Copyright © IFAC Nonlinear Control Systems, Stuttgart, Gennany, 2004

ELSEVIER

IFAC PUBLICATIONS www.elsevier.comllocalelifac

A CONSTRUCTIVE ALGORlTHM FOR SYSTEM IMMERSION INTO NONLINEAR OBSERVER FORM Juhoon Back and Jin H. Seo

School 01 Electrical Engineering and Computer Science, Seoul National University, Kwanak P.O. Box 3,(., Seoul, 151-7,(.,(., Korea, backhoon@hanmail. net Abstract: This paper investigates when n dimensional nonlinear systems can be immersed into n + m dimensional observer form using output diffeomorphism. We provide a necessary and sufficient condition which involves a differential equation with n+m+ 1 unknowns; n+m unknowns for the state transformation, one for the output diffeomorphism. The algebraic structure of the characteristic equation is investigated, and an algorithm is presented to check the irnmersibility iteratively. Copyright © 2004 IFAC

Keywords: observer form, system immersion, output diffeomorphism.

1. INTRODUCTION

The observer design method by observer error linearization has been studied by many researchers and lots of results can be found in the literature (Krener and Isidori, 1983; Krener and Respondek, 1985; Xia and Gao, 1989; Keller, 1987; Hou and Pugh, 1999). Most of the results solve the problem by transforming the given system into nonlinear observer form via a diffeomorphism. However, this approach has some drawbacks. Firstly, the condition is restrictive in the sense that the exact integrability condition is required. Secondly, one should solve a system of first order partial differential equations to obtain the transformation. There have been many researches to overcome these obstacles. In order to relax the exact integrability condition, immersions rather than diffeomorphisms are used in (Levine and Marino, 1986) and the possibility of using smooth maps with continuous inverse is investigated in (Xia and Zeitz, 1997). On the other hand, constructive algorithms not only to check the possibility to transform the original system into observer form, but also to obtain the transformation via a straightforward procedure have been developed (Glumineau et al., 1996; Banaszuk and Sluis, 1997; Guay, 2001). In this paper, we provide a necessary and sufficient condition for the immersibility of the n dimensional observable nonlinear system into higher n+m dimensional observer form. It should be mentioned that the two

results did not concern the problem of immersion with output diffeomorphism. It is known from the conventional observer error linearization problem that the class of nonlinear systems diffeomorphic to observer form can be broadened if the output diffeomorphism is involved. Thus, it is expected that the class of systems is much enlarged if the output diffeomorphism as well as the system immersion is employed. In general, the output diffeomorphism induces an additional unknown to the problem which makes the problem more difficult. In this work, the problem is solved in a different way compared to the previous works and an algorithm is presented with which the irnmersibility can be checked via a iterative procedure. 2. IMMERSillILITY CONDITION We consider nonlinear systems of the form:

x = I(x),

This work

wall

supported by the Brain Korea 21 Project in 2004.

477

(1)

where I(x), hex) are smooth functions of x E D with D an open connected subset of an. It is assumed that the system (1) satisfies dim span{dh,dL,h," .,dL,-lh}= n '
(2)

We also consider the N(N = n+m) dimensional nonlinear observer form

z=

1

y = hex)

Az + £iCY), Y = t/J(y) = Cz

z E RN

(3)

where the matrices A, C, and a(y) are defined by 0 1

A=

Let ~k := L}-l h, k ~ 1. It should be noted that {6, ... , ~n} forms a coordinate system by the observability condition and that ~n+l, ... , ~N + 1 are functions of {~l, ... , ~n}. Let ~ = (~t,··· ,~N) and P(~) be the ring of polynomials in 6, ... , ~N with coefficients that are Coo functions of ~t. Let

00] 00

. ..... . ,

[.

o ...

1 0

C=[O···Ol]

h = sup{6

and 1jJ(-) is a diffeomorphism.

h = inf

Proposition 1. Consider the observable nonlinear system (1) and let N = n + m. Then, the smooth functions aI, ... , aN and the diffeomorphism aN+t satisfy (5) if and only if they satisfy the n equations shown below:

Remark 1. The basic idea of the immersion is that when two initial conditions Xo and Zo are equivalent in some sense, the outputs of the systems are equivalent. Variants of this problem can be found in the literature, for example, immersion into observable linear system (Levine and Marino, 1986), immersion into nonlinear observer form (Back and Sea, 2004; Jouan, 2003), etc.

a~+t(~d = -

L k=.+1

LJ LJ i=O

k=Q

G)L}-;a~+t(~d (7)

n

Lk-i , (t: )a~n+i-l . , an+k+t .. l at:

+J

.. HI

where i = 0,··· , n - 1 and a'(~d := da(~d/~I. The equations (7) for i = n - 1, n - 2, ... are of the form a~ a~_l

(4)

= Fn - 1 (

= a~_2 =

For simplicity, (4) is rewritten as

F n- 2(

Fn- 3 (

a~+t, ... ,a~+m+t) a~, a~+I'··· , a~+m+l) a~_I' a~, a~+l'··· , a~+m+l)

(8)

= 0, (5) where F;'s are some operators. One can realize that as i decreases by one, the number of unknowns increases only by one. Thus, if the general solutions of an+l, ... , an+m+l are obtained, the unknowns left can be obtained one by one.

where aN+t(h) := -1jJ(h). In general, the differential equation (5) is hard to solve, because it contains N + 1 unknwons and their high order derivatives. It is worth to note that when N = n and 1jJ(-) is the identity map, the problem reduces to that of the observer error linearization with output injection, which can be solved via a step by step procedure (Banaszuk and Sluis, 1997). Unfortunately, when N > n, such a process is not available. Instead of that, we found a condition (shown below) which is useful to construct an algorithm to solve the unknowns. The main objective of this paper is to develope an algorithm with which the immersibility can be checked via a iterative procedure. In other words, it is to provide an algorithm with which one can solve the unknowns of (4) one by one.

3. A NECESSARY CONDITION: A SYSTEM OF LINEAR DIFFERENTIAL EQUATIONS In this section, we derive a necessary condition for the unknwons a n+l, ... , aN+l. In advance, some notations are introduced. Given two integers w and e such that 1 ::; w ::; N and 1 ::; e ::; w, we define a set called set of exponents w.r.t. (w,e). N I[w,e] ={(el'·· ·,eN)

N

I w= I>e;,e= Le;,e; ;=1

(9)

ENt},

;=1

where No+ is the set of nonnegative integers. Let J.L[w,e] = card I[w,ej and rearrange I[w,ej by removing the duplicated elements of I[w,ej. Then, I[w,e] has a property:

We shall consider the observable canonical form of (1):

~l =6

6=6 (6)

~n=fn(6,···,~n),

N

_~ ~ (n + k)

Theorem 1. (Jouan, 2003) The system (1) is immersible into (3) in D if and only if there exist N smooth functions aI, ... , aN and a diffeomorphism 1jJ such that

al(h)+L,a2(h)+· ..+L7-laN(h)+LfaN+t(h)

JR 1(6,0,··· ,0) E <))d(D)},

We provide a variant of Theorem 1.

Zo with T(xo) = Zo, then 1jJ(h(Xt (xo))) = CZt(zo) Vt such that 0 ::; t ::; r:w Here, r zo = sup{ t ~ 0 I Xt(xo) E D}.

+.. .+L7-laN(h).

{~l E

Using hand f!, it is possible to define a set Uh by Uh := {h E JR If! < h < h} and 15 by 15 = {x E D Ih(x) E Uh}.

Definition 1. The system (1) is said to be immersible into (3) in D if there exists a smooth function T : D -+ T(D), D C JRn, T(D) C JRN such that if for every XQ and every

Lf1jJ(h)=at(h)+L,a2(h)

E JR 1(6,0,··· ,0) E <))d(D)}

Proposition 2. For 1 ::; w ::; Nand 1 ::; e ::; w, I[w,e]

i:

0.

By Proposition 2, we obtain a set of N-tuple of integers {v[w, e, 1], .. ', v[w, e, J.L[w,ej]} of which the s-th (1 ::; s ::; J.L[w,ej) element is associated to the s-th element of I[w,ej· For 1::; s::; J.L[w,ej, v[w,e,s] can be denoted by

Y=6

which can be obtained by the diffeomorphism

[6,··· ,~nf = <))d(X):= [h(x) L,h(x) ... L,-lh(x)]T whose existence is guaranteed by (2).

v[w,e,s] = (v[w,e,s,l]' ... ,v[w,e,s,N]).

478

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polynomial of 6,.· ',~w+l whose coefficients are smooth functions of h. Lemma 1 proves this fact (The result is adopted from (Phelps, 1991)[Theorem 2.2] for our case.)

Using the notations introduced above, we define:

C(v[w,e,8))=w!j {~ [ Q \<"V W,e,8 D=

(fi

V[W,e'8,i]!(i!)tl[W.e .••i1) ,

Lemma 1. Let a(~d be a smooth function and 0 :::; w :::; N. Then,

N

IT <'i+l

~tl[w,e .• ,il

.

w PI.... !

i=l We call C(v[w, e, 8]) the leading coefficient function w.r.t. v[w,e, 8] and call Q(~,v[w,e,8)) the monomial function w.r.t. v[w, e, 8].

Lia(~d =

Thanks to the relation (16), (4) can be rewritten as

In order to simplify the notation, a differential operator is introduced as follows:

~(dl""

N

= a~gla~:~ ... a~~n-l' ~ EN:.

Given a differential operator degree W,o. is defined by

W,o.

~(dl .... .d.. -d'

= d 1 + 2d2 + ... + (n -

(11)

n:::; W,o. :::; N}.

w"

(12)

N

w=n e=l

if Wd = WI and if d1 and write ~(dl.··· ,d..-d ~ ~(h,'" ,In-i) if ~(dl"" .dn-d > ~(h,'" ,/n-d or ~(dl"" ,dn-d = ~(h,'" ,In-d' We define an ordered set of ~[n~Nl by ~tn~Nl = {~i E ~[n~NJ I ~i ~ ~i if i > j, 1 :::; i, j :::; J.L~~N]}' By abuse of notation,

we call K* an odered set of ~ if ~ is composed of the differential operators ~(dl •... ,dn - 1 )' It is easy to see the following identity (14)

because the sets in the right hand side of the equation are mutually disjoint. It follows from (14) and from the mutually disjoint property that

I

With the functions C(v[w, e, 8]) and Q(~, v[w, e, 8]) in hand, it is possible to show that the Lia(6) (a(6) is a smooth function) is in P(~). More specifically, it is a

pr..,.]

LL L

Thus, we can obtain a system of J.L~~N] linear differential equations which contain the unknowns {an+l' "', aN+d only. This necessary condition is a good starting point to solve the characteristic equation (5). Unfortunately, it is not easy to solve a system of linear differential equations, except for some special cases. However, we can solve the unknowns one by one under some assumptions. The rest of this paper is devoted to find the conditions under which the system of differential equations can be solved iteratively. 4. SOLVING THE CHARACTERISTIC EQUATION: CONSTRUCTIVE ALGORITHM

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Given the set ~ *[nlI'Vn2 I' we denote the k-th element of the set by ~~*"l"'n2 1k' Similarly, the k-th element of ~*[n1 J is denoted y ~tnd,k'

w

C(v[w, e, 8)) w=n e=1 .=1 (19) . ~(dl.··· ,dn _l) Q(~, V[W, e, S])I~F"=~n=O . a~~l (~1) =0.

= ~(h,'" .1.. -1) + ... + dn-l = h + ... + In-l

J1.~~Nl = J.L~I + J.L~+l1 + ... + J.L~'

(18)

Lemma 2 says that V6 E Uh, the unknowns an+l, "', aN+l necessarily satisfies

~(dl.···,d.. -d

k! ~tn+i-ll )*

(C(v[w, e, s))

8=1

. ~(dl"" ,dn-dQ(~, V[W, e, 8]) . a~~1 (~d) =0.

N

rn+l

w Pi....]

LL L

In addition, we will write

(

o.

Lemma 2. For the system (1), let N = n + m (n ~ 2) and suppose that the characteristic equation (4) admits a solution {aI, "', aN+d· Then, for any ~(dl"" .dn-ll E ~[n~NJ' the functions an+l, "', aN+l satisfy

(13)

+ ... + dn - 1 > h + ... + In-I.

~tn~N] =

(17)

Now, let us provide a necessary condition of (17) (as well as of (4)) which involves only the unknowns an+l, aN+l'

Then, ~[n~N] is a collection of differential operators with weighted degree between n and N. For simplicity, ~[k~kl is abbriviated as ~[kl' We denote the cardinalty of ~[n~N] by J1.~~NI' It is useful to order the elements of ~[n~NI in some sense. Let ~(dl"" .d.. -d' ~(h,'" .I.. -d be two elements of ~[n~Nl and Wd, w I be the weighted degrees of them, respectively. We will write ~(dl"" .d..-d > ~ (h ,'" .I.. -1) if one of the following conditions are satisfied.

(1) Wd > (2) Wd = wI and d1

[C(v[w,e,8])

.=1

. Q(~, v[w, e, 8)) . a~~l (6)] =

Let N ~ n, NE N+ and consider a set defined by ~[n~NI = {~(dl.··· .d.. -lll

pl....]

L L

w=l e=l

the weighted

l)dn-l.

w

al(~d + L

8 d1 +·+ d .. - 1

.d.. -d

L L C(v[w,e, 8])Q(~, v[w, e, 8]) ·a(e) (6). (16) e=l.=l

In this section, we investigate the solvability of the system of linear differential equations (18) which contains the unknowns an+l, "', aN+l. In advance, an algebraic equation will be constructed regarding the unknowns and their derivatives as independent unknowns. Then, we will show that the unknwons can be solved one by one under some assumptions.

479

X(C<" 1)T:

4.1 Algebraic Equation

[(I)

a4

(2)

a4

(3)

(I)

a 4 as

(2)

as

(3)

as

(4)]

as

* 3 0 * * 0 0 : * 0 31 * * 0 0 : * 0 0 * 6 0 0 : * 0 0 * * 12 0 : * 0 0 * * 0 4! : * 0 0 * * 0 0 : * 0 0 * * 0 0 : * 0 0 * * 0 0 where ~[il,k'S are the operators corrsponding to the rows of A(6) and * is for some function of ~I' Note that the matrix has constant rank at least 4. a[3],1 :

We would like to describe the differential equations (19) in a matrix form:

~[31,2 ~[41,1 ~[4J,2 ~(4J,3 ~(51,1 ~[5],2 ~[5J,3

(20)

where Y(~d, A(6), X(6) are J.L~~N] x 1, J.L~~N] X m(n-l+n+m) m(n-l+n+m) X 1 matrices respectively 2 '2 " whose components are functions of output and to be determined. In order to obtain the matrix form (20), we first recall the identity (14). Let i be an integer such that 1 ~ i ~ m+ 1. Let ~[n+i-l],k (1 ~ k ~ J.L~+i-l]) be the kthe element of the ordered set ~[n+i-1J and let Y[il and A[il Ll. m(n-l+n+m) d' . al b e J.LlLl.n+i-l] x 1 an d J.L[n+i-l] x 2 ImenslOn matrices, respectively. After substituting ~[n+i-1J,k into (19), we have

We are in a position to generalize the observations in Example 1 and point out some special structures of A(6) in (20).

Lemma 3. The J.L~+i-IJ x (n+ j -1) matrix A[i,jl(~d can be decomposed into

(21)

where Y[iJ,k (k-th component of y[iJ), the row vector A[iJ,k A[i,jMd = [A~,il(~I) ~jj], (22) (k-th row vector of A[i]) and the column vector X are where A~,j](6) and A~,j] are J.L~+i-l1 x j, J.L~+i-ll x defined as follows: (n - 1) matrices, respectively. Moreover, if we write the N PIN,_] differential operator ~[n+i-I],k as • Y[il,k(~d = C(v[N,e,s]) a d",l+d",2+"+d",ft-' .=1 .=1 (23) ~[n+i-Il,k = a~g""a~:",2 .. . a~~",ft-l' . ~[n+i-1J,kQ(~, v[N, e, sDb=o"" ,{ft=oa~~1(6), then the elements of A["',',1'J are .A[iJ,k(~d = [A[i,lJ,k(6) A[i,2],k(6) ... A[i,m],k(6)], A['"I,3,k,1 'I where A[i,il,k (1 ~ j ~ m) is (n+m- j)-row vector whose 1)' n+t-. .., - . n-I I-th element, denoted by A[i,iJ,k,l, is 2')d (~ 1)~d _ ,lft=; andl+J=Ldk,q ( • ....2 • . . n- . k.n 1

L L

1

P[ft+;-l,l) A[i,il,k,l = C( v[n + j - 1, I, sD .=1 . a[n+i-l],kQ(~, v[n + j - 1, I, sDI{2=O, .. , ,{ft=O·

o

L

(n+,i-l) (C )]T <,,1

r

A[I,li (6) .....

A[m+'1](~d ~[m+:'ll(~d"

•

A[I,~I (6) ] ,

: A[m+I,'m] (6)

yields the desired equation (20).

Example 1. Suppose n = 3 and m = 2. W6

~[3~S] :

W6 W6

X(~d

and

=3 =4

-+~[31

= {~(I,I)'~(3,O)}' = {~(O,2)' ~(2,1)' ~(4,O)}'

-+~[41 = 5 -+~[Sl = {~(1,2)' ~(3,1)' ~(s,o)},

A(~d

of (20) become

6

q=1

-

Let us describe the structure of A(6) using Lemma 3 as follows: A(6)

<

A[I,I#d

[2,ll(~d

Collecting all of the y[il's and A[i]'S as follows:

A(~d =[ A[IJ:(~d ] =

,otherwise

where 1 ~ k ~ J.L[n+i-l] and 1 ~ 1 ~ n - 1.

• X(~d = [X~l X~l X~]]T,where XliI (1 j ~ m) is (n + j - I)-COlumn vector S.t. a n +1

(.

[4.,t] Ar~,46) 0··· Ar~,ml(~d 0 0 Ar1,2J(~d Art2r" Ar~,ml(~d 0

[3,11(6)

0

~,11(6) +1,11(6)

0 0

Ar~,21(~d

0 .. ·

Ar~,2](6) 0 Ar~+1,21(6) 0

Ar~,ml(~d

0

... Ar~,mJ(6) Ar~,m .. A~+1,mMd 0 (24)

where the symbol '0' is for zero matrix with appropriate dimension. From (24), we can deduce Lemma 4:

Lemma 4. The matrix A(~d has at least (n -1) . m rows that are linearly independent for all ~I E Uh . 4.12 IMALGO-OD: IMmersion ALGOrithm with Output DiiJeomorphism Based on the observations in Section 4.1, an algorithm (called IMALGO-OD) to solve the characteristic equation will be derived. In advance, for i < j ~ m, let us define

480

X[i~iMd ~ [X[i) ({d T ... Xli) ({dT]T.

which is, in general, an N-th order linear differential equation for aN+l ('d. Solve the equation and proceed to Step for a n +. with s = m. Step for a n +. (s = m,m -1"" ,1): Suppose the unknowns an+.+l(6), "', aN+l(6) have been obtained. We will treat these functions as known functions even if they have undetermined constants. In order to derive a differential equation for a n +.(6), we decompose the matrix A(6) as follows:

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The algorithm is composed of three parts. The first one is for the differential equation for aN+l({d and the second one is for the steps in which the differential equations for an+ m(6),-", an+l(6) is derived. The last one collects all of the solutions an+l(6)"" ,aN+l({l) and determine whether the algorithm is successful or not. Let us state each steps of the algorithm. Initial Step: In this step, a differential equation for aN+l({d is derived. We first rewrite Y({d as Y(~d ~ wm+l(6)X[m+l)(~d,

A(6)

w.(6) = [A[';,.](6) ... A(",+l,.) (6)r '

(28)

I6=0"" ,en . (29)

Cases 1: A(6) has full column rank on Uh (rank of A(~d is m{2n~m-l): In this case it is possible to construct the equation = A(~dX(~d,

(30)

where Wm+l(6), A(6) are submatrices of wm+l(6), A(6), respectively, and A(~d is nonsingular on Uh. Multiplying [A({l )]-1 to both sides of (29) yields [A(6)t1Wm+l(6)X[m+l](6) = X(~d.

(31)

Now we recall that X(6) is a stack of Xli) (6) (1 ~ j ~ m) and the elements of XU) satisfy 'r/q,1<_q<_n+,·-2.

Choosing one of these relations and decomposing the ~atrix f!(6)t l into m blocks such that each block [A(~d][jJ has n + j - 1 rows, one can obtain

:h [([A(~d]-l)Uj,q Wm+l(6)Xm+1(~d] = ([A(~dtl)[i),q+l ~m+1(6)Xm+l(6),

(32)

where 1 ~ j ~ m, 1 ~ q ~ n + j - 2 and ([A(6)]-1 )[il,q is q-th row of [A(~dliJ{' Solve (32) to obtain aN+l(6) and substitute this to (31). In this way, X (6) can be obtained. In this case, one skips Step for a n +. shown below and jump to Step for a. to solve the unknowns left. Case 2: There exists a nonzero row vector VN+l (6) such that VN+1(6)A(6) = 0 and VN+1(6)wm+l(~d f= O. Then, by multiplying VN+l(6) to both sides of (29), we have vN+l(~dWm+l(6)' X[m+l) (6) = 0,

.

(35)

A[l,m] ('d ] .

[

A.(6)X[1~{.-1)l(6) + W.(6)X[.)(6).

=

Using. (26), the matrix equation (20) becomes

Wm + l (6)X[m+1Md

A[l"+l] (6) .

A[m+l,.-lj(6)

Y(6 )-4».(6)X[{.+l)~ml(6)

=0

I{Im+l(6)X[m+ll(6) = A(~dX(6)·

A[m+:,11(6)

,

A[m+l,~+lJ(~d .. : A[m+l:m](~d Then, we can rewrite the matrix equation (20) as

C(v[N, e, q])

. ~[n+i-1J,kQ(~, v[N, e, q])

dX[ij,q(6)_X. dh UI,q+l (t:) <,1 ,

[

4».(6) =

I'(N,I)

q=l

A[1,.-11(~d ]

A[1,lj(6)

=.

A.(6)

wm+l (6) = [I{I{m+l)[l) (6)T ... w{m+l)[m+lj(6)']T , (27) of which the k-th row and l-th column element is

L

(34)

where

(26)

where X[m+lI(6) = [a~~1(6) ... aWl1(~d]T and wm+l(~d is a J.L~~N) x N dimensional matrix defined by

W{m+l)[ij,k,/(6) = -

= [A.('l) w.(~d
(33)

481

(36)

If there exists a nonzero row vector Vn+.(~l) such that v n+.(6)A.(6) = 0 and v n+.(6)w.(6) f= 0, it is possible to eliminate X[1~{.-1)J(6) from (36) and obtain the (n + s - l)-th order linear differential equation for an+.(~d:

v n+. ('1) (Y (6 )-
a~(6) = - ~ ~: 1)L~-n+la~+l(6) _ ~~ ~n + k)Lk-i , (, ) a'n+;+l ~~ n + j I an+k+l 1 a, . k=Oi=O

(38)

n

We would like to point out that the right hand side of the equation contains an+l(6), "', an+ m+l(6) only and that the left hand side should be a function of h only. From these observations, we can check if the right hand side of (38) can be made a function of h only by proper choice of the constants of an+l (6), "', an+m+l (6). If this turns out to be impossible, the system is not immersible into N dimensional observer form. However, if it is possible, we can proceed to Step for an-I. Suppose the unknowns a.+l, "', an+m+l have been obtained. Check if the right hand side of the equation

a~(~l) =

Le: 1)L~-'+la~+l(6) N

-

10=.

_ ~~ ~n + k)L k-i , (t: ) a~n+i+l ~~ n+' I an+k+l <,1 at: . k=Oi=O ' <,.

(39)

can be made a function of h only by choosing the constants of as+! , ... , an+m+l' When it turns out to be true, proceed to Step for a.-I, Otherwise, the algorithm fails. IMALGO-OD provides the following result:

Theorem 2. The n dimensional observable system (1) is immersible into N(N = n+m) dimensional observer form (3) in 15 if there exist smooth functions at, .. " a n +m and a diffeomorphism a n +m +! to IMALGO-OD.

into the form is characterized by a differential equation (called characteristic equation) in the sense that the solvability of the equation is equivalent to the immersibility. In general, the characteristic equation is hard to solve since several unknowns should be found simultaneously. This paper not only characterizes the class of system immersible into observer form, but also provides a new algorithm to solve the characteristic equation. Several examples are presented to illustrate the ideas and show the efficiency of the algorithm.

Corollary 1. Suppose the row vectors Vn+l (~I), VN+! (6) exist in IMALGO-OD or A(~l) is of full column rank. Then the n dimensional observable system (1) is immersible into N(N = n+m) dimensional observerform (3) in 15 if and only if there exist smooth functions aI, ... , an + m and a diffeomorphism an + m +! (6) to IMALGO-OD.

Example 2. 1

= X2, X2 = (1 + xI)3 VI + X2, Y = hex) = Xt,

Xl

(40)

where the the vector field of the system is smooth over V = {x E ]R2 Illxll < I}. The system is not diffeomorphic to observer form even if the output diffeomorphism is used. Let us try to immerse this system into 3 dimensional observer form using IMALGO-OD. For Initial Step, from n = 2, we have ~[2~3] = {~F),~(3)}, 3/(1 + xI) -3/(1 + xd 0] W2(~I) = [-9/(4(1 + xI)4) 9/(4(1 + xI)3) -6 '

A(~ )

= [-1/(4(1 + Xl)3) 21

3/(8(I+xI)3) O~' l 3 X[2](6) = [ai )(6) ai2)(~I) ai (~djT, and X (~I) = [a~l) (~I) a~2) (~I)jT. As A(~I) is nonsingular on V, it follows that A = A(~I) and W2(6) = W2(~I). We choose j = 1 and q = 1 and consider (32) for a4(6) to obtain 1

ai4 )

3

+[3/(1 + xI) - 1/(2(1 + xd 3 )]ai ) 2 +[3/(8(1 + xd 4 ) - 3/(64(1 + xd 6 )]ai ) l +[3/(64(1 + xI)7) - 3/(8(1 + xd)]ai ) =

(41)

o.

As (41) is a third order linear differential equation, it is required to find 3 linearly independent solutions. Assuming ail) (xI) = (1 + xI)P, one gets p = 1. Fortunately, we don't need to find the other solutions. In fact, let ail)(xl) = c4(1 + xI) (C4 is a constant) and proceed to the next step. Then, we have a~l) (xI) = O. At Step for a2, we substitute a4(xl) and a3(xl) into the equation for a~l) (xI) and get a~l) (xI) = O. At Step for aI, it follows 5C4 Th erelore, r th e system . a (l)() IS .Immersl'ble l Xl = 2(1+z,)6' into 3-dimensional observer form with output diffeomorphism y = 1f;(y) = - C4(1~1I)2 + d4 (d4 is a constant). 5. CONCLUSIONS In this paper, we employed the system immersion as well as the output diffeomorphism to enlarge the class of nonlinear systems that can be transformed into higher dimensional nonlinear observer form. The systems immersible

482

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