OBSERVABLE SYSTEMS TRANSFORMABLE INTO IMPLICIT AFFINE FORMS
Guillaume, D. Rouchon, P.
Centre Automatique et Systemes, Ecole des Mines de Paris, 60,boulevard Saint-Michel, 75272 Paris Cedex, FRANCE E-mail:guillaume@cas.ensmp.fr
idem.
E-mail:rouchon@cas.ensmp.fr
Abstract: A smooth observable system, _x = f(x), y = h(x), that can be put, after a change of state coordinates x 7! X, into _X = A(y)X +b(y) with an implicit observation equation C(y)X = d(y), is said to be transformable into an implicit ane form. Characterization of such systems is an open problem. For single output systems, a necessary constructive condition on the structure of the dierential equation satised by the output is given. When dim(x) = 2, this necessary condition,
y =f0(y) +f1(y) _y+f2(y) _y2+f3(y) _y3, is shown to be sucient. Copyright c1998 IFAC
Keywords: observers, nonlinear control, ane systems, input injection 1. INTRODUCTION
Observers for nonlinear systems were much stud- ied in the last decade, and real advances were done during this period. One way of building observers is to nd new coordinates in which the system is in a special form. Two important classes of systems were successively studied with such techniques, systems linearizable by output injection (Krener and Respondek, 1985), and sys- tems which are ane regarding to the unmeasured states (Besancon and Bornard, 1997; Hammouri and Gauthier, 1992). In both classes, a part of the states can be considered as measured, e.g, the observation equation admits an explicit structure.
In this paper, a larger class of systems is consid- ered: systems transformable into an implicit ane form. This new class of systems, for which ane asymptotic observers can be designed, are systems
_
x = f(x), y = h(x), that can be put, after a change of state coordinates x 7! X, into _X =
A(y)X +b(y) with an implicit observation equa- tion of the formC(y)X =d(y). For single output systems, a necessary condition for arbitrary state dimension is derived here. This necessary condi- tion (see lemma 4) is shown to be sucient when dim(x) = 2 (see theorem 1). This indicates, even for single output planar systems, that exist sys- tems transformable into implicit ane forms that are not ane regarding to the unmeasured states in the sense of (Krener and Respondek, 1985; Be- sancon and Bornard, 1997; Hammouri and Gau- thier, 1992): taking implicit observation equations enlarges the class of systems for which asymptotic observers derived from linear and least squares techniques can be designed.
This study of single output systems is local and around "generic" states x: open subsets where
x can be expressed as a smooth function of (h;Lfh;:::;Ln?1f h) = (y ;y;_ :::;y(n?1)) (n = dim(x)), are only considered here (i.e. open sub- sets where the tangent manifold is observable).
More precisely, we consider here single output observable systems of the form
_
x=f(x); y =h(x) (1) where f and h are smooth and where x lies an open subset of IRn such that exists a local dieomorphism between x and (y ;y;_ :::;y(n?1)).
Such systems are called observable, in the sequel.
They are said to be transformable into an implicit ane form, if, and only if, exists a local state dif- feomorphismx=(X) such that the observation equation y = h((X)) denes locally an ane subspace in theX-space and the restriction of the dynamics in theX-space, _X =F(X), to the ane subspace y=h((X)) is ane (the restriction of
F to y =h((X)) is ane). In other words, this means that in theX coordinates (1) reads
_
X=A(y)X +b(y); C(y)X =d(y) (2) where the matrices A, b, C and d are smooth functions ofy.
Such systems admit an ane observer of the form:
_^
X=A(y) ^X +b(y) (3)
?P
?1
C
0(y)(C(y) ^X ?d(y)) _
P=? P ?At(y)P?PA(y)
+Ct(y)C(y) (4) which converges exponentially when y is per- sistent (in the sense of (Guillaume and Rou- chon, 1997), i.e.P >Idfor some>0).
The paper is organized as follows. In section 2, single output planar observable systems trans- formable into implicit ane forms are character- ized. Section 3 deals with the necessary condition for single output systems with an arbitrary state dimension.
2. SINGLE INPUT PLANAR SYSTEMS 2.1 Necessary condition
Assume that the observable system (1) is trans- formable into an implicit ane form. Set
A(y) =
a
11(y) a12(y)
a
21(y) a22(y)
b(y) =?b1(y) b2(y)t
C(y) =?c1(y) c2(y)
After derivation, (0 stands for derivation with respect toy):
CX=d (C0y_+CA)X+Cb=d0y_
d(CX)
dt
=PX+Q=R
withP = (C00y_2+C0y+ 2C0yA_ +CA0y_+CA2),
Q = 2C0by_ +CAb+Cb0y_ and R = d00y_2+d0y. Since
C
C
0y_+CA
is assumed invertible,X can be eliminated from the last equation. This yields a second order dierential equation for y. Examine the structure of this equation. Up to renumbering the component ofx, assume thatc1(y)6= 0. Thus
X
1 = d(y)=c1(y)?c2(y)=c1(y)X2. Without lost of generality, assume also that the rst column of
A is zero (substitute X1 into the state equation) andc11. This leads to the following simplied structure:
A(y) =
0a1(y) 0a2(y)
b(y) = ?b1(y) b2(y)t
C(y) = ?1c(y):
(5) Denote
1=a1+ca2; 1=b1+cb2: (6) Elimination ofX leads to
g(y)y =g0(y) +g1(y) _y+g2(y) _y2+g3(y) _y3 with
g= (c01+d01)
g
0= (?11a2+21b2)
g
1= (?1(01+c0a2) +d01a2+ 2c01b2+101)
g
2= (?d001?c001+ (01+c0a2)d0+ (10+c0b2)c0)
g
3= (d0c00?c0d00):
Observability of (1), means here that exists a local dieomorphism between X and (y ;y_). This implies that, locally, g cannot vanish. Thus the following lemma is proved:
Lemma 1.(Necessary condition). If the observ- able planar system (1) is transformable into an implicit ane system (3) then, its output y sat- ises a second order dierential equation of the form
y=f0(y) +f1(y) _y+f2(y) _y2+f3(y) _y3 withf0,f1,f2,f3 smooth functions.
2.2 Sucient condition
Assume that the output y of planar observable system (1) satises a second order dierential equation of the form
y=f0(y) +f1(y) _y+f2(y) _y2+f3(y) _y3
with the f0,f1, f2 and f3 smooth function of y. A dieomorphismx=(X) that put the system into an implicit ane form 3 can explicitely be built.
From previous computation,it appears that the problem to solve is equivalent to nd smooth y- functionc,d,1,1,b2 anda2such that
f
0(c01+d01) = (?11a2+21b2)
f
1(c01+d01) = (?1(01+c0a2) +d01a2 +2c01b2+110)
f
2(c01+d01) = (?d001?c001
+(01+c0a2)d0+ (10+c0b2)c0)
f
3(c01+d01) = (d0c00?c0d00):
(7)
Basic algebraic manipulations of this system yield to the following three equations:
1
1
0
1=?(c01+d01)(2c0f0+1f1)
?c 0
1
1 a
2+d021a2+2110?2c011a2
2
1
1(c01+d01)c00=c0F+311f3(c01+d01)
2
1
1(c01+d01)d00=d0F?2112f3(c01+d01(8)) where
F=1(21(c0a2d0+10c0)?c0211a2) +1d0(1(?1c0a2+d01a2+110)
?(c01+d01)(1d0(2c0f0+1f1) +1(c02f0+21f2))?2c011a2): Denote by x the state around which the trans- formationx=(X) is looked for and denote by (y;y_) the corresponding values of (y ;y_).
Set 1(y) 1 and a2(y) 0. By the Cauchy- Lipschitz theorem, there exists locally around y, a unique solution y 7! (1;c;d) of the dieren- tial system (8) with the following initial values:
1(y) = (_y)2 + 1, c(y) = d(y) = d0(y) = 0,
c
0(y) = 1: Such a solution yields to b2(y) via
2
1 b
2=f0(c01+d01) +11a2.
For A, b and C derived from the above solution
y7!(1;1;a2;b2;c;d) of (7) via the formulae (5) and (6), the relations
C
C
0y_+CA
X=
d
d 0y_?Cb
(9) dene a local dieomorphism between (y ;y_) and
X, that is betweenxandX. This results from the following arguments:
since c0(y)_y+1(y) = y2+ y+ 1 > 0, the
matrix
C
C
0y_+CA
is locally invertible; thus X is a smooth function of (y ;y_).
denote by X the solution of (9) for (y;y_);
since
(1(y) +c0(y)_y) X2=d0(y)_y?1(y);
X
2
6= 0 andc0(y) X2?d0(y)6= 0; the implicit function theorem for X1 +c(y)X2 = d(y) ensures thatycan be expressed as a smooth function ofX;
similarly, _y is also a smooth function ofX. Thus, the following lemma is proved:
Lemma 2.(Sucient condition). Assume that the scalar output of the planar observable systems (1) satises a smooth second order dierential equation of the form
y=f0(y) +f1(y) _y+f2(y) _y2+f3(y) _y3: Then, locally, there exists a change of state coor- dinates putting the system into an implicit ane form (3).
Lemmas 1 et 2 lead to the following
Theorem 1. A single output observable planar system (1) is transformable into an implicit ane form (3), if, and only if, the second-order deriva- tive of the output is a polynomial of third degree in the rst-order derivative of the output with coecients smooth function of the output.
Remark that:
planar observable systems transformable into implicit ane forms admit the following nor- mal forms
_
X
1 =a(y)X2+b1(y) _
X
2 =b2(y)
X
1+c(y)X2 =d(y):
This results from the proof of the sucient condition.
The class of planar observable systems, ane regarding to the unmeasured states in the sense of (Krener and Respondek, 1985; Be- sancon and Bornard, 1997; Hammouri and Gauthier, 1992), admit the following output dierential equation:
y =k0(y) +k1(y) _y+k2(y) _y2: This former class is thus smaller than the class considered here.
3. NECESSARY CONDITION FOR GENERAL SINGLE OUTPUT SYSTEMS
Denote by n the state dimension and take the implicit ane form. After n derivations of the observation equation, the following system:
CX=d
d(CX)
dt
=d0y_
d 2(CX)
dt
2 =d"_y2+d0y (10)
:::=:::
d n(CX)
dt
n =d0y(n)+:::+d(n)y_n
is a linear overdetermined system with respect to
X. Eliminating of X yields to a polynomial im- plicit scalar equation with respect to (_y;:::;y(n)).
A careful inspection of the dierent partial degrees of this polynomial equation yields to a necessary condition. Introduce the following denitions bor- rowed from (Krener and Respondek, 1985).
Denition 1. NoteP the ring of polynomials with indeterminates the successive derivative y(k ) of
y and with coecients C1 functions of y. By denition the degree ofy(k ) iskand the degree of
y (j
1 )
:::y (j
r
)isj1+:::+jr. NotePk the subring of polynomials, the degree of which is less than or equal tokandPik the subring of polynomials, the degree of which is less than or equal tokspanned byPi, with ik.
Lemma 3. The system (10) can be written:
P
0
X+Q0(y) =R0
:::=:::
P
n?1
X+Qn?1=Rn?1 (11)
P
n
X+Qn=Rn
withPi andRi elements ofPi fori= 1:::nand
Q
i element dePi?1 fori= 2:::n.
Proof: Immediate fori= 1;2;3:::The derivative of a monomial of the formf(y)y(k ) is a monomial of degree k+ 1 because d(f(y )ydt(k)) =f0(y) _yy(k )+
f(y)y(k +1).
The structure of the (vector) polynomials can be more precisely described:
P= (pij) =
0
B
B
B
@ P
0
P
...1 P
n?1 1
C
C
C
A
with
P
0=C
P
1=C0y_+CA ...
P
n?1=C(n?1)y_n?1
+(n?1)C(n?1)yy_n?3+:::+CAn?2
Q =
0
B
B
B
@ Q
0
Q
...1 Q
n 1
C
C
C
A
, R=
0
B
B
B
@ R
0
R
...1 R
n 1
C
C
C
A
, = detP, ij the cofactors ofP andM = (ij), the comatrix.
The elimination ofX from the previous equations leads to:
P
n P
?1(R?Q) +Qn=Rn that is:
P
n M
t(R?Q) + Qn= Rn (12) The coecientDofy(n)in (12) is:D= (C0X?
d
0) =C0Mt(R?Q)?d0 withRandQelements of Pn?1. M is an element of Pn?1n(n?1)=2?1and an element of Pn?1n(n?1)=2. We have the following degrees for the lines M(i), Q(i) and R(i) of the corresponding matrices
d
M(1) =n(n?1)=2
d
M(i) =n(n?1)=2?(i?1);i= 2:::n
d
Q(1) = 0
d
Q(i) =i?2;i= 2:::n
d
R(i) =i?1;i= 1:::n
Then,Dis an element ofPn?1n(n?1)=2. Isolating the terms withd0inX and using:
D=Xn
k =1 c
0(k)(?1)n+knkd0y(n?1)?d0 +::: with = Pnk =1(?1)n+kpnknk and pnk =
c
0(k)y(n?1)+:::, the term in y(n?1) disappears.
Dis an element of a subring spanned byPn?2. From the equation (C0X?d0)_y=?(CAX+Cb) and after multiplication by , it appears that the degree of the right term is less than equal to
n(n?1)=2 (degree de X and of ) and the one of the left term less than or equaln(n?1)=2+ 1.
This is possible only if the degree of (C0X?d0) is less than or equal to n(n?1)=2?1. There is no term of degree n(n?1)=2 inD. Finally,D is an element ofP(n(n?1)=2?1)
n?2 .
Consider the degree of the remaining termN. N is got from the equation (12), in which the term withy(n) is suppressed:
N = (Pn(y ;:::;y(n?1))X
+Qn(y ;:::;y(n?2))?Rn(y ;:::;y(n?1)))
P
nis an element ofPn?1n (polynomial of degreen without derivatives of order n) and X element ofPn?1(n(n?1)=2)so N is an element ofPn?1n(n+1)=2. Sum up these results in the following proposition:
Lemma 4. If the single output observable system (1) is transformable into an ane system (3) then the output y satises the following dierential equation:
y
(n)=N=D (13)
with N element of Pn?1n(n+1)=2 and D element of
P
(n(n?1)=2?1)
n?2 (see denition 1).
For n = 3, the following output equation exists necessarily:
y
(3) = N(y ;y;_ y)
d
0(y) +d1(y) _y+d2(y) _y2 with
N=n1(y) (y)3+n2(y) (y)2y_2+n3(y) yy_4+n4(y) _y6 +n5(y) (y)2y_+n6(y) yy_3+n7(y) _y5
+n8(y) (y)2+n9(y) yy_2+n10(y) _y4 +n11(y) yy_+n12(y) _y3
+n13(y) y+n14(y) _y2 +n15(y) _y
+n16(y)
The fact that the denominator depends only on _y is not obvious.
Remark: the gap between the dimension 2 and 3 appears here. For the 3 dimensional case, there are 19 equations to satisfy for 12 functions (overdeter- mined), when for the 2 dimensional case, there are 5 equations (4 ifgis set to 1) for 6 functions (un- derdetermined). So we see that there are degrees of freedoms for the 2 dimensional and relations to satisfy in the 3 dimensional case.
4. CONCLUSION
Here is establishesd a necessary condition so that an nonlinear system is equivalent to an ane system (3) with an observation equation, ane regarding to the state and implicit regarding to the the measured output. A such class of systems contains linearizable systems by output injection, and also ane systems described in (Besancon and Bornard, 1997). An observer for the system described here can be found in (Guillaume and Rouchon, 1997). Our present work is on the in- trinsic characterization of such systems and also on the way to compute the coordinates change when it exists.
5. REFERENCES
Besancon, G. and G. Bornard (1997). On charac- terizing classes of observers forms for nonlin-
ear systems.Proceedings of the 4st European Control Conference ECC97.
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