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On the stabilization of homogeneous systems in the plane
ON THE STABILIZATION OF HOMOGENEOUS SYSTEMS IN THE...
14th World Congress of IFAC
E-2c-14-1
Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
ON THE STABILIZATION OF HOMOGENEOUS SYSTEMS IN THE PLANE Ph" Adda and H. Zenati
INRIA-Lorraine (COPlGE project) fj [lniversity of l'.1etz lIe du Soulcy 57045 - METZ (FRANCE) e-mail: {adda~zenati}@saulcy.loria.fr
ISTGMP, Bat A
~t\.bstract:
j
In this paper we consider the stabilization problem of two-dimensional
homogeneous systems. Some sufficient conditions for the asyrrlptotic stabilization by homogeneous feedbacks are derived. A new approach inspired by the well-
known Kalman's method (pole assignment) is developed to construct explicitly the stabilizing feedback lav;.'s. Local stabilization result is obtained for a class of nonlinear systems. Copyright © 1999 IFAC
Keywords: nonlinear systems, pole assignment, feedback stabilization
1+ INTRODUCTION
nonlinear systems in the plane+ In this paper, we apply VPA technique to our system (1) and we
We consider a single input homogeneous system,
prove that under some conditions the homoge-
x = j(x) + u g(x)
neous constructed feedback Ia"vs globally aSYlnptotically stabilize the system at the origin.
(1)
2
, U E lR , f and g are homogeneous (ra.tional) vector fields of degree, respectively, 2k 1 + 1 and k 2 with k 1 ~ 1. The problem is to find a feedback function u(x), which is homogeneous~ smooth on IR? - {O} and \vhich
where x E IR
asymptotically stabilizes the control system (1)
+
In the litera.ture, global stabilization results are often obtained by the Lyapunov machinery, see
The paper is organized as follows. Section 2 considers preliminaries where VPA method is developed. Section 3 contains our main result. In section 4 a homogeneous approximation for nonlinear systems in the plane is discussed, a local result of stabilization is given. Section 5 contains some concluding remarks.
for instance (Kawski, 1990; Jurdjevic and Quin~
1978; Rosier, 1992; Sontag, 1989; Tsinias, 1990). Another stabilization result (Chabour et at, 199:3) is from the bilinearization point of vieVl. Whereas, since the systems studied here may satisfy g(O) == o~ ,'le are not in the scope of (Boothby and Marino, 1989; Dayawansa et al.~ 1990). Our ap-
2. PRELIMINARIES In this section we develop the VPA method ap-
plied to (1) in the plane and recall a known stability result related to planar homogeneous systeIlls.
proach to the stabilization problem of (1) is in-
First of all, let us briefly recall the LPA theory.
spired by the well-known Kalman's linear pole assignment (LPA) technique and is not a Lyapunov like lllethod. Then our results are com-
Given a controllable linear system x = A x + ub, x E lRn ,U E JR. For all manic real polynomial p of degree n, there is some line matrix F such
pleting the preceding \yorks. The idea of variable pole assignment (\rpA) was first introduced by (Clavier J 1995) fOT planar bilinear systems and recently discussed in (Adda and Zenati, 1998) for
that the characteristic polynomial of the matrix ~4 + bP is the desired p. 'Then if the spectrunl of A + bF is chosen to have negative real part, the linear feedback u = Fx stabilizes the system~
2409
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ON THE STABILIZATION OF HOMOGENEOUS SYSTEMS IN THE...
So, let us consider the homogeneous system (1). It is known that by Euler's formula
f(x)
1
= 2k + 1 1
(D[)x x
\\~here (Df)x is the jacobian matrix of !, (2k 1 + 1 is the hornogeniety degree of f). Then (1) can be written in the following form
14th World Congress of IFAC
Working on JR.?, we shall make a constant use of the following result. Theorem 1 (Hahn; 1967) Consider the planar system
Xl == fl(X1J X2) { X2 == f2(X1 X2)
(3)
J
x = A(x) where A(x)
=
2kl
1
+1
(2)
x+u g(x)
(Df)x and its coefficients
are homogeneous of degree 2k 1 .
(i) The restriction of th€ system to each of its
The main idea here is to apply the pole assignment for the homogeneous system (2) in the plane. It means that at each point x E ]R2 - {O}, we apply the above pole assignment technique to the pair (A(x), g(x)). 1v[ore precisely~ for each x 0 we construct, exactly in the same way than for linear systems, a functional matrix F(x) such that the matrix A(x) +g(-x)F(x) has a desired preassigned spectrunl.. The method is as follows:
t
Assume that the following rank condition is satisfied
rank(g(x): A(x) g(x))
(RC)
f = (/1, f'J)T is Lipschitz and homogeneous polynornial veclor field of degree k ~ 1~ The system (3) is asymptotically stable if and only if one of the follou1ing is satisfied:
1L1here
= 2,
one dimensional invariant homogeneous rays is asymptotically stable}·
(ii) The system does not have anyone dimentionnal invariant s'Ubspaces and 27f
j o
cos8f1(Cos8,sine)
+ sin 0f2 (cos 0, sin 0)
dB
cos efz(cos (), sin B) - sin () /1 (cos 8 1 sin 8)
where (p,O) denote the polar coordinates p COS
\/x:f O.
(), X 2
==
0
< Xl
P sin a.
3. GLOBAL STABIL1ZATION
2
For each x E 1Ft - {O}, define the set of desired spectrum Bp == {Al(X), A2(X)}, where Al(X) and A2 (x) are homogeneous functions of the same
degree 2k 1 , and let PA(X) 02(X) and Ps(X) = X 2
=
+
X 2 + Ul(X)X + tJ'l(X)X + 132(X)
the characteristic polynolllials respectively asso-
cied to the matrix .A.(x) and to the spectrum Sp. Put Ft(x) ::::: (a2(x) ~ p'2(x),al(x) - J31(X) a line matrix and let the functional invertible matrix P(x) := (A(x )g(x) + (}:1 (x )g(x), g(x)). Then, we construct the functional line-Inatrix F(x) = F' (x)p(x)-l and we obtain the closed-loop matrix lvl(x) == A(x) + g(x)F(x) which has the preassigned spectrum Bp. Finally, putting
u(x)
we get t.VQ different expressions of the closed-loop
x = A(x)x + u(x)g(x), ==
(~4(x)
+ g(x )F(x)) x == 1\1 (x)x
So, let us consider the two-dimensional homogeneous systenl (2) with A ( x ) --
(ab(x) d(x) (x) ) (x)
c
,g
()
x
~
(9192(X) (x) )
where the coefficients of _A(x) are homogeneous rational functions of the same degree 2k 1 and 9i, i == 1, 2, are also hOlnogeneous rational functions of degree k2; x == (x:J ~ X2) E ]R2. From now
"'le
sumptions hold
It should be noted that an analogous construction holds also in IRn , n > 2.
apply; under RC condition,
ists an invariant homogeneous ray, then the choice A1 (x) and ..\2 (x) negative definite ensures that the restriction of the closed-loop system to this invariant ray is asymptotically stable~ Then, according to (i) of theorem 1. proving global asymptotic stability of the closed-loop system amounts to prove the existence of (at least) one invariant ray.
because u(x) = F(x)x E rn.~
can easily verify that, by construction, the functional matrix lW(x) is homogeneous of degree 2k 1 (its coefficients are homogeneous of degree 2k 1 ) and, hence the right-hand side of the closedloop system x == M (x) x is homogeneous of degree 2k 1 + 1 ~
v.~e
VPA to (2) with negative definite spectrum (A1 (x), A2(X)) for M(x). We prove that if there ex-
== F(x) x
system:
i:
In this section
OIl,
,'ve assume that the following as-
Assum.ption (a): (RC) rank(g(x), A(x)g(;c)) 2, "Vx E rn} ~ {D}, i.e.
det(g(x) A{x )g(x}) j
=
#- 0, Vx i= o.
(where det denotes determinant of matrix). Assumption Cb): the desired spectrum is Sp == (allxIl 2k1 , JJllxI1 2k1 ), where a and 13 are arbitrary
2410
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress ofIFAC
ON THE STABILIZATION OF HOMOGENEOUS SYSTEMS IN THE...
negative real constants. norm).
(11.11 denotes the euclidian
'fhen, the constructed matrix F(x) given in section 2 takes the
follo\~ling form
be distinguished: the case when the degree of homogeneity of g, k 2 , is even and the case when k 2 is odd. In both cases, the problem amounts to
prove the existence of (at least) one invariant ray. Therefore, we have the following theorem.
1
F{x)
= det(g(x), A(x)g(x)) (Fl(x), F 2 (x»)
Theorem 3 Given the hom.ogeneotJ,s system (2) with A{x) and 9 as defined above+ [lnder the assumptions (a) and (b), 'we have
where
+ d - a:llxl1 2k1 + .8I1xI1 2k1 )91 + a( alJxll 2k1 + ,811x11 2k1 ) - a~!Ixl12kl )92,
Case 1: the degree of homogeneity k2 is even. For all (a < 0, j3 < 0), the closed-loop system (4J has at least one invariant homogeneo'Us ray.
+ be - d(all x l1 2k1 + ~llxl!2kl) + + c( a. + d - allxl1 2k1 - j9}l xIl 2k1 )92
Case 2: k 2 is odd~ For a suitable choice of (a,.8), system (4) has at least one ifivar~'ant homogene01jS ray if the Jollowzng extra as,fiumption holds:
F 1 (x) = -b(a
(a
2
+ be F 2 (x)
==
_(d 2
a;3l1xl14k1 )91
Easy computation shows that the components of - k 2 and analytic on n 2 - {O}. Then, setting
F(x) are homogeneous of degree 2k 1
Proof: Case 1: The property in theorem. 3
u(x) == F(x)x on obtains the follo\ving homogeneous closed-loop system of degree 2k 1 + 1
x=
(A(x)
+ g(x)F(x))
x
= .l\JI(x)x
(4)
where A;f(x) has, by construction, the preassigned
spectrum (olfxI1 2k1 , f3llxIJ2k
1
ass'U1nption (e): det (g(x), x) has no constant sign - {a}.
on. IR 2
).
D(x)
+ 1 =s k 2 , the constructed homogeneous feedback may fail to be continuous at the origin. But, this does not cause problems, since the right-hand side of the closed-loop system (4) is homogeneous of degree 2k I + 1 ~ 1 and, hence, at least continuous at the origin. Theorelll 2 Consider the homogeneous system (2) 'U;ith A(x) and 9 as defined above. Under the assu-mptions (a) and (b)~ if the closed-loop system (4) admits at least one invariant homogeneous ray, then it has x 0 as a globally asymptotically
P2(X)
the form
D(x) :;;;
Pa(x) P{3(x) q~q~ det(g(x), A(x )g{x))
which is ¥lell defined on (RC) condition) and
rn? - {O} (because of the
wher~
Pa(X) == (b(X)Xl + (d(x) - alfxI!2k1)X2)PlQ2 + (( aj Jxl12k1 ~ a( x) )XI - c(x )X2)P2Ql, ~6(X)
Proof": Let m. + XQ be an invariant homogeneous ray of (4) . Tt. means that the vector neld M (x) x
= (b(X)Xl + (d(x) ((pllxI1 2k1
-
j61l x l1 2k1 )X2)P1Q2 +
a(x))xl - C(X)X2)p2Ql
such that
Now, since 91 and 92 have the same even degree
from where XQ is an eigenvector of the matrix . i.l-1 (xo) ,vhich has the spectrum (allxoI1 2k1 , pllxol1 2kI ) with et < 0 , {3 < O. So, 2k1 Pxo == allxol1 or .Bllxo 11 2k l . it follows that the restriction of (4) to the ray n+ Xo is
k 2 , it is easy to see that PI q2 and P2 ql are homogeneous polynomials with the same even degree k;. Therefore, the numerator of D(x) is a product of tV{O homogeneous polynomials of odd degree + 2k 1 + 1 and, 80, the equation (6) has at least t\VO distinct real roots if a =I=- 13. Thus, the
Nf(xo)xo =
Xo,
E
IR
Pl(X)
== -(-)' Y2(X) == -(-) ql X q2 X with ql (x )q2(X) i= 0, 'Vx E rn? - {O}, D(x) takes
=
i.e. :3 PXo
(6)
When gl and 92 are rationa.l fractions, i.e_
.fitable equilibruim point.
is radial at
= det (kl(x)x, x) == 0
has at least one real root.
gl(~~)
Relnark 1: When 2k 1
]S
satisfied if the equation
Pxo Xo,
x = rllxtl 2k1
x
(5)
where r == Q; (or f3 )4 ~ow ~ it is obvious that (5) has the san1.e trajectories than x = r x which is asymptotically stable (because a < 0: f.3 < 0). So, the theorem follo\vs from (i) of theorem 1.
k;
system (4) has at least two invariant homogeneous rays.
Case 2: A straightforward COluputation yields
D x ~ Qu(x) x Qf3(x) ( ) - det(g(x}, A( x )g(x))
V\Te will nO"\lv discuss about the existence of invariant. homogeneous rays for (4). Two cases ""rill
(7)
Vv+here
2411
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ON THE STABILIZATION OF HOMOGENEOUS SYSTEMS IN THE...
Qa(X)
== (b(X)Xl + (d(x)
«(a\lxH
2k
l
-
allxI1 2k1 )X2)gl(X) +
-
a(x))xl - C(X)X2)g2(X),
Q{3(X) == (b(X)Xl + (d(x) - ,BllxIJ2k )X2)gl(X) + ((fillxI12k1 - a(x))xl ~ C(X)X2)g2(X)
14th World Congress of IFAC
However, this does not cause problems, since (4)
has the same trajectories and the same invariant homogeneous rays that the system
x ==
1
Observe that Qa and Q(3 are rational homogeneous functions of even degree 2k 1 + k z + 1 which can be y..rritten as follo",rs:
== «A(x)
Qo:(X)
Q p ( x)
((A ( x) -
:=
lIx11 ) ,8B 11 x ll2k 1)
- aB
X,
'lIlhich is polynomial and so theorem 1 applies.
Remark 3: When k 1 == 0 the spectruill of M ( x) 1 Sp: beCOllles constant (Sp = ( a, ,B)) and the closed-loop system (4) turns out to be homogeneous of degree one. However, contrary to the linear systems, the fact t.hat 0' and j3 are chosen strictly negative does not imply the asymptotic stability of (4). (see t.h.e counter-exampIe in
x~ g(x ),
2k1
]det(g(x), A(x)g(x))1 M(x)x
9 ( x ))
(Clavier, 1995)).
where
01)
A(x) = (b(X) d(X)) B _ ( -a(x) -c(x) , -1 0
signs of Qa and Qf) do
homogenelty~ the
::-Jaw, by
.
On the other hand, as (4) is homogeneous of degree one, asymptotic stability implies exponential stability see (Hahn, 1967)~
not change along any ray issuing from the origin.
So, it is sufficient to study their signs along the unit circle Si == {x E lR2 / llxll = I}. Namely, we have t.o show that Qa(x) (or Qf3(x)) vanishes at some point of SI. Indeed) observe that \..I
v
~ x E 51 , l'lID (A(x) .
aBllxl1 2k1 ) -allxH 2k
a-t-oco
x
=
BX
4. LOCAL STABILIZATION BY
HOMOGENEOUS APPROXIMATION Consider a single input, nonlinear system defined on a neighborhood II of the origin
l
(8)
T'his in turn implies that for all x E SI . Qa(X) ( ) hm -a 11 x 112k 1 == < BX,9 x a---+-oo
> == det(g(x),x)
'I'hen, it follo\vs that i) :3 y E Sl, 3 fr
i i) :)
Z
ex
==
Putting aD
<
a1
=>
E Si ~ :3
<
01
<0
0!2
IIyll2k
>0
< 0 such
that
a2
=::} -Cl!
and where
Ilzl1 2k1 < O.
min( G'l, Ct2), then for all a < ao we
Finally ~ by continuity one has
Obviollsly}
(A11
>
u} .; U
0, M 2 > 0), for every .x in a neighborhood of the origin.
Now l with system (8) we associate the "homogeneous" approximation
ao I 3xQ E Si : exitQo' (xo) = 0
,"~e
3;30, 'ri(J
prove in the same
<
+ /2 (x )
Qa(z)
Qa(Y) >0 and Qa(Z) <0
<
/1 (x) = (D 2k l +1 /1)0 x 2k1 + 1
1
get:
Va.
l
such that
Qa(Y)
-a
where 11 and 91 are sufficiently regular (say COO) vector fields such that D i /1 (0) == (D i 11)0 == 0 Vi E {O, 1, ... , 2k1 }, k 1 ?:: 0 and DJ 91 (0) == (Dig1)o = 0 , Vj E {a, 1, ... , k 2 - I}, k 2 2' 1. lTsing the Taylor expansion of fl(X) and gl(X) near x == 0, there exist 12 and 92 such that
(30,3 Yo E 8 1
:
~Nay
that
Qj3(YO) = 0
x ~ I(x) + u. g(x) where f(x)
(D k 2
==
k2 . 91 )0 x
(9)
(D 2k l+ 1 fl)ox 2k1 + 1 and g(x) =
'Therefore, ~Ne prove the follo"\ving
theorem.
Thus, the system (4) admits two invariant rays for
a suitable choice of (0:, fJ) and the theorem follows.
Remark 2: The closed-loop system (4) is ho~ 1110gelleous of degree 2k 1 + 1 but not necessary polynomia.l as it has been assumed in theorell1 1.
TlleoreTll 4 If the homogeneous approximati.on (9) is stabilizable by means of a positively homogeneous feedback of dcg1~ec 2k 1 - k 2 + 1, then
the system (8) is locally stabilizabl€' by the same feedback.
2412
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress ofIFAC
ON THE STABILIZATION OF HOMOGENEOUS SYSTEMS IN THE...
Proof: Let u == u(x) be a positively homogeneous stabilizing feedback of degree 2k 1 - k 2 + 1 (u("\x) == A2k l-k2+1 u(x) 1 for all x and any positive nUDlber A) for systenl (9). Set
F(x)
==
l(x)
Hahn, "'V-t/. (1967). Stabilily of motion. Springer \Terlag. New York. Jurdjevic, V. and J. P. Quin (1978)~ Controllability and stability. J. Diff. Equations 28~ 381-
+ u(x)g(x)
and
G(x)
w. P., C. F. ~vla.l'tin and G. Knovv'les (1990). i\symptotic stabilization of a class of smooth t"\vo-dinlentional systems. SIAM J. Control and Optimization 28, 1321-1349.
Daya,~~ansa,
= f2(X) + U(X)g2(X).
Since u is Cl on [l} - {O}, F and G are locally Lipschitz. ~loreover, we have
389. Kawski , M. (1990). Homogeneous stabilizing feedback. Contr-ol- Theory and ~4dvanced Technology 6, 497-516.
So~
2k 1
F is positively homogeneous function of degree + 1 and G satisfies \:Ix E
ut.
where Ai == -i\11 +1\-10 . 1. \.-[2. It follows from lVlassera's theorem (~{assera, 1956) that the solution x = 0 of the closed-loop system
x == F(x) + u(x)
IvIassera, J. L. (1956). Contribution to stability theory. Anals of Mathematics 64, 1182-206. Rosier, L. (1992). Homogeneous lyapunov function for homogeneous continuous vector field. Systems Control Lett. 19, 467-473. Sontag, E. D. (1989). A universal construction of arstein '8 theorem on nonlinear st.abilization. Systems Control Lett. 10, 263-284. Tsinias, J. (1990). Remarks on feedback stabiliza-
tion of homogeneous systems. Control- Theory
G{x)
and Advanced Technology 6) 533-541.
is asymptoticaly stable. Thus, asyu1ptotically stabilizable.
(8)
IS
locally
5. CONCLUSION
Explicit construction of stabilizing feedbacks for a class of t\vo-dimensional homogeneous systems is given _ T'he application field of this result is of course limited by the low dimension and the restrictive rank condition (RC). Nevertheless this work may seem of some interest. Indeed, it turns out that classical methods that include the construction of a Lyapullov function are difficult to apply to systems that we consider. So, our apj
proach ,vhich provides feedbacks that are analytic
except at zero, allows to remove this difficulty.
6. REFERENCES Adda, P. and H. Zenati (1998). yTariable pole assignment for a class of nonlinear syst.ems. In: 2nd Conference on Cornputational Engineer-
ing in Systems Applications (Pierre Borne, E,d.). Vol. I. pp. 167-170. CESA..98. Hammamat, Tunisia. Boothby, W. and R. Marino (1989). F'eedback stabilization of planar nonlinear systems. Systems and Control Lett. 12, 87-92. Chabour, R., G. Sallet and J~C. \ljvalda (1993). Stabilization of nonlinear two dimensional systems: a bilinear approch. A1ath. Control ~"ignal8 Systems 6, 224-246. CJavier, IVI. (1995). Stabil-isation des Systemes Bilineaires et des System,es mecaniques. Thesis. l\1etz, France.
2413
Copyright 1999 IFAC
ISBN: 008 0432484
14th World Congress of IFAC
E-2c-14-1
Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
ON THE STABILIZATION OF HOMOGENEOUS SYSTEMS IN THE PLANE Ph" Adda and H. Zenati
INRIA-Lorraine (COPlGE project) fj [lniversity of l'.1etz lIe du Soulcy 57045 - METZ (FRANCE) e-mail: {adda~zenati}@saulcy.loria.fr
ISTGMP, Bat A
~t\.bstract:
j
In this paper we consider the stabilization problem of two-dimensional
homogeneous systems. Some sufficient conditions for the asyrrlptotic stabilization by homogeneous feedbacks are derived. A new approach inspired by the well-
known Kalman's method (pole assignment) is developed to construct explicitly the stabilizing feedback lav;.'s. Local stabilization result is obtained for a class of nonlinear systems. Copyright © 1999 IFAC
Keywords: nonlinear systems, pole assignment, feedback stabilization
1+ INTRODUCTION
nonlinear systems in the plane+ In this paper, we apply VPA technique to our system (1) and we
We consider a single input homogeneous system,
prove that under some conditions the homoge-
x = j(x) + u g(x)
neous constructed feedback Ia"vs globally aSYlnptotically stabilize the system at the origin.
(1)
2
, U E lR , f and g are homogeneous (ra.tional) vector fields of degree, respectively, 2k 1 + 1 and k 2 with k 1 ~ 1. The problem is to find a feedback function u(x), which is homogeneous~ smooth on IR? - {O} and \vhich
where x E IR
asymptotically stabilizes the control system (1)
+
In the litera.ture, global stabilization results are often obtained by the Lyapunov machinery, see
The paper is organized as follows. Section 2 considers preliminaries where VPA method is developed. Section 3 contains our main result. In section 4 a homogeneous approximation for nonlinear systems in the plane is discussed, a local result of stabilization is given. Section 5 contains some concluding remarks.
for instance (Kawski, 1990; Jurdjevic and Quin~
1978; Rosier, 1992; Sontag, 1989; Tsinias, 1990). Another stabilization result (Chabour et at, 199:3) is from the bilinearization point of vieVl. Whereas, since the systems studied here may satisfy g(O) == o~ ,'le are not in the scope of (Boothby and Marino, 1989; Dayawansa et al.~ 1990). Our ap-
2. PRELIMINARIES In this section we develop the VPA method ap-
plied to (1) in the plane and recall a known stability result related to planar homogeneous systeIlls.
proach to the stabilization problem of (1) is in-
First of all, let us briefly recall the LPA theory.
spired by the well-known Kalman's linear pole assignment (LPA) technique and is not a Lyapunov like lllethod. Then our results are com-
Given a controllable linear system x = A x + ub, x E lRn ,U E JR. For all manic real polynomial p of degree n, there is some line matrix F such
pleting the preceding \yorks. The idea of variable pole assignment (\rpA) was first introduced by (Clavier J 1995) fOT planar bilinear systems and recently discussed in (Adda and Zenati, 1998) for
that the characteristic polynomial of the matrix ~4 + bP is the desired p. 'Then if the spectrunl of A + bF is chosen to have negative real part, the linear feedback u = Fx stabilizes the system~
2409
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ON THE STABILIZATION OF HOMOGENEOUS SYSTEMS IN THE...
So, let us consider the homogeneous system (1). It is known that by Euler's formula
f(x)
1
= 2k + 1 1
(D[)x x
\\~here (Df)x is the jacobian matrix of !, (2k 1 + 1 is the hornogeniety degree of f). Then (1) can be written in the following form
14th World Congress of IFAC
Working on JR.?, we shall make a constant use of the following result. Theorem 1 (Hahn; 1967) Consider the planar system
Xl == fl(X1J X2) { X2 == f2(X1 X2)
(3)
J
x = A(x) where A(x)
=
2kl
1
+1
(2)
x+u g(x)
(Df)x and its coefficients
are homogeneous of degree 2k 1 .
(i) The restriction of th€ system to each of its
The main idea here is to apply the pole assignment for the homogeneous system (2) in the plane. It means that at each point x E ]R2 - {O}, we apply the above pole assignment technique to the pair (A(x), g(x)). 1v[ore precisely~ for each x 0 we construct, exactly in the same way than for linear systems, a functional matrix F(x) such that the matrix A(x) +g(-x)F(x) has a desired preassigned spectrunl.. The method is as follows:
t
Assume that the following rank condition is satisfied
rank(g(x): A(x) g(x))
(RC)
f = (/1, f'J)T is Lipschitz and homogeneous polynornial veclor field of degree k ~ 1~ The system (3) is asymptotically stable if and only if one of the follou1ing is satisfied:
1L1here
= 2,
one dimensional invariant homogeneous rays is asymptotically stable}·
(ii) The system does not have anyone dimentionnal invariant s'Ubspaces and 27f
j o
cos8f1(Cos8,sine)
+ sin 0f2 (cos 0, sin 0)
dB
cos efz(cos (), sin B) - sin () /1 (cos 8 1 sin 8)
where (p,O) denote the polar coordinates p COS
\/x:f O.
(), X 2
==
0
< Xl
P sin a.
3. GLOBAL STABIL1ZATION
2
For each x E 1Ft - {O}, define the set of desired spectrum Bp == {Al(X), A2(X)}, where Al(X) and A2 (x) are homogeneous functions of the same
degree 2k 1 , and let PA(X) 02(X) and Ps(X) = X 2
=
+
X 2 + Ul(X)X + tJ'l(X)X + 132(X)
the characteristic polynolllials respectively asso-
cied to the matrix .A.(x) and to the spectrum Sp. Put Ft(x) ::::: (a2(x) ~ p'2(x),al(x) - J31(X) a line matrix and let the functional invertible matrix P(x) := (A(x )g(x) + (}:1 (x )g(x), g(x)). Then, we construct the functional line-Inatrix F(x) = F' (x)p(x)-l and we obtain the closed-loop matrix lvl(x) == A(x) + g(x)F(x) which has the preassigned spectrum Bp. Finally, putting
u(x)
we get t.VQ different expressions of the closed-loop
x = A(x)x + u(x)g(x), ==
(~4(x)
+ g(x )F(x)) x == 1\1 (x)x
So, let us consider the two-dimensional homogeneous systenl (2) with A ( x ) --
(ab(x) d(x) (x) ) (x)
c
,g
()
x
~
(9192(X) (x) )
where the coefficients of _A(x) are homogeneous rational functions of the same degree 2k 1 and 9i, i == 1, 2, are also hOlnogeneous rational functions of degree k2; x == (x:J ~ X2) E ]R2. From now
"'le
sumptions hold
It should be noted that an analogous construction holds also in IRn , n > 2.
apply; under RC condition,
ists an invariant homogeneous ray, then the choice A1 (x) and ..\2 (x) negative definite ensures that the restriction of the closed-loop system to this invariant ray is asymptotically stable~ Then, according to (i) of theorem 1. proving global asymptotic stability of the closed-loop system amounts to prove the existence of (at least) one invariant ray.
because u(x) = F(x)x E rn.~
can easily verify that, by construction, the functional matrix lW(x) is homogeneous of degree 2k 1 (its coefficients are homogeneous of degree 2k 1 ) and, hence the right-hand side of the closedloop system x == M (x) x is homogeneous of degree 2k 1 + 1 ~
v.~e
VPA to (2) with negative definite spectrum (A1 (x), A2(X)) for M(x). We prove that if there ex-
== F(x) x
system:
i:
In this section
OIl,
,'ve assume that the following as-
Assum.ption (a): (RC) rank(g(x), A(x)g(;c)) 2, "Vx E rn} ~ {D}, i.e.
det(g(x) A{x )g(x}) j
=
#- 0, Vx i= o.
(where det denotes determinant of matrix). Assumption Cb): the desired spectrum is Sp == (allxIl 2k1 , JJllxI1 2k1 ), where a and 13 are arbitrary
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Copyright 1999 IFAC
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14th World Congress ofIFAC
ON THE STABILIZATION OF HOMOGENEOUS SYSTEMS IN THE...
negative real constants. norm).
(11.11 denotes the euclidian
'fhen, the constructed matrix F(x) given in section 2 takes the
follo\~ling form
be distinguished: the case when the degree of homogeneity of g, k 2 , is even and the case when k 2 is odd. In both cases, the problem amounts to
prove the existence of (at least) one invariant ray. Therefore, we have the following theorem.
1
F{x)
= det(g(x), A(x)g(x)) (Fl(x), F 2 (x»)
Theorem 3 Given the hom.ogeneotJ,s system (2) with A{x) and 9 as defined above+ [lnder the assumptions (a) and (b), 'we have
where
+ d - a:llxl1 2k1 + .8I1xI1 2k1 )91 + a( alJxll 2k1 + ,811x11 2k1 ) - a~!Ixl12kl )92,
Case 1: the degree of homogeneity k2 is even. For all (a < 0, j3 < 0), the closed-loop system (4J has at least one invariant homogeneo'Us ray.
+ be - d(all x l1 2k1 + ~llxl!2kl) + + c( a. + d - allxl1 2k1 - j9}l xIl 2k1 )92
Case 2: k 2 is odd~ For a suitable choice of (a,.8), system (4) has at least one ifivar~'ant homogene01jS ray if the Jollowzng extra as,fiumption holds:
F 1 (x) = -b(a
(a
2
+ be F 2 (x)
==
_(d 2
a;3l1xl14k1 )91
Easy computation shows that the components of - k 2 and analytic on n 2 - {O}. Then, setting
F(x) are homogeneous of degree 2k 1
Proof: Case 1: The property in theorem. 3
u(x) == F(x)x on obtains the follo\ving homogeneous closed-loop system of degree 2k 1 + 1
x=
(A(x)
+ g(x)F(x))
x
= .l\JI(x)x
(4)
where A;f(x) has, by construction, the preassigned
spectrum (olfxI1 2k1 , f3llxIJ2k
1
ass'U1nption (e): det (g(x), x) has no constant sign - {a}.
on. IR 2
).
D(x)
+ 1 =s k 2 , the constructed homogeneous feedback may fail to be continuous at the origin. But, this does not cause problems, since the right-hand side of the closed-loop system (4) is homogeneous of degree 2k I + 1 ~ 1 and, hence, at least continuous at the origin. Theorelll 2 Consider the homogeneous system (2) 'U;ith A(x) and 9 as defined above. Under the assu-mptions (a) and (b)~ if the closed-loop system (4) admits at least one invariant homogeneous ray, then it has x 0 as a globally asymptotically
P2(X)
the form
D(x) :;;;
Pa(x) P{3(x) q~q~ det(g(x), A(x )g{x))
which is ¥lell defined on (RC) condition) and
rn? - {O} (because of the
wher~
Pa(X) == (b(X)Xl + (d(x) - alfxI!2k1)X2)PlQ2 + (( aj Jxl12k1 ~ a( x) )XI - c(x )X2)P2Ql, ~6(X)
Proof": Let m. + XQ be an invariant homogeneous ray of (4) . Tt. means that the vector neld M (x) x
= (b(X)Xl + (d(x) ((pllxI1 2k1
-
j61l x l1 2k1 )X2)P1Q2 +
a(x))xl - C(X)X2)p2Ql
such that
Now, since 91 and 92 have the same even degree
from where XQ is an eigenvector of the matrix . i.l-1 (xo) ,vhich has the spectrum (allxoI1 2k1 , pllxol1 2kI ) with et < 0 , {3 < O. So, 2k1 Pxo == allxol1 or .Bllxo 11 2k l . it follows that the restriction of (4) to the ray n+ Xo is
k 2 , it is easy to see that PI q2 and P2 ql are homogeneous polynomials with the same even degree k;. Therefore, the numerator of D(x) is a product of tV{O homogeneous polynomials of odd degree + 2k 1 + 1 and, 80, the equation (6) has at least t\VO distinct real roots if a =I=- 13. Thus, the
Nf(xo)xo =
Xo,
E
IR
Pl(X)
== -(-)' Y2(X) == -(-) ql X q2 X with ql (x )q2(X) i= 0, 'Vx E rn? - {O}, D(x) takes
=
i.e. :3 PXo
(6)
When gl and 92 are rationa.l fractions, i.e_
.fitable equilibruim point.
is radial at
= det (kl(x)x, x) == 0
has at least one real root.
gl(~~)
Relnark 1: When 2k 1
]S
satisfied if the equation
Pxo Xo,
x = rllxtl 2k1
x
(5)
where r == Q; (or f3 )4 ~ow ~ it is obvious that (5) has the san1.e trajectories than x = r x which is asymptotically stable (because a < 0: f.3 < 0). So, the theorem follo\vs from (i) of theorem 1.
k;
system (4) has at least two invariant homogeneous rays.
Case 2: A straightforward COluputation yields
D x ~ Qu(x) x Qf3(x) ( ) - det(g(x}, A( x )g(x))
V\Te will nO"\lv discuss about the existence of invariant. homogeneous rays for (4). Two cases ""rill
(7)
Vv+here
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Copyright 1999 IFAC
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ON THE STABILIZATION OF HOMOGENEOUS SYSTEMS IN THE...
Qa(X)
== (b(X)Xl + (d(x)
«(a\lxH
2k
l
-
allxI1 2k1 )X2)gl(X) +
-
a(x))xl - C(X)X2)g2(X),
Q{3(X) == (b(X)Xl + (d(x) - ,BllxIJ2k )X2)gl(X) + ((fillxI12k1 - a(x))xl ~ C(X)X2)g2(X)
14th World Congress of IFAC
However, this does not cause problems, since (4)
has the same trajectories and the same invariant homogeneous rays that the system
x ==
1
Observe that Qa and Q(3 are rational homogeneous functions of even degree 2k 1 + k z + 1 which can be y..rritten as follo",rs:
== «A(x)
Qo:(X)
Q p ( x)
((A ( x) -
:=
lIx11 ) ,8B 11 x ll2k 1)
- aB
X,
'lIlhich is polynomial and so theorem 1 applies.
Remark 3: When k 1 == 0 the spectruill of M ( x) 1 Sp: beCOllles constant (Sp = ( a, ,B)) and the closed-loop system (4) turns out to be homogeneous of degree one. However, contrary to the linear systems, the fact t.hat 0' and j3 are chosen strictly negative does not imply the asymptotic stability of (4). (see t.h.e counter-exampIe in
x~ g(x ),
2k1
]det(g(x), A(x)g(x))1 M(x)x
9 ( x ))
(Clavier, 1995)).
where
01)
A(x) = (b(X) d(X)) B _ ( -a(x) -c(x) , -1 0
signs of Qa and Qf) do
homogenelty~ the
::-Jaw, by
.
On the other hand, as (4) is homogeneous of degree one, asymptotic stability implies exponential stability see (Hahn, 1967)~
not change along any ray issuing from the origin.
So, it is sufficient to study their signs along the unit circle Si == {x E lR2 / llxll = I}. Namely, we have t.o show that Qa(x) (or Qf3(x)) vanishes at some point of SI. Indeed) observe that \..I
v
~ x E 51 , l'lID (A(x) .
aBllxl1 2k1 ) -allxH 2k
a-t-oco
x
=
BX
4. LOCAL STABILIZATION BY
HOMOGENEOUS APPROXIMATION Consider a single input, nonlinear system defined on a neighborhood II of the origin
l
(8)
T'his in turn implies that for all x E SI . Qa(X) ( ) hm -a 11 x 112k 1 == < BX,9 x a---+-oo
> == det(g(x),x)
'I'hen, it follo\vs that i) :3 y E Sl, 3 fr
i i) :)
Z
ex
==
Putting aD
<
a1
=>
E Si ~ :3
<
01
<0
0!2
IIyll2k
>0
< 0 such
that
a2
=::} -Cl!
and where
Ilzl1 2k1 < O.
min( G'l, Ct2), then for all a < ao we
Finally ~ by continuity one has
Obviollsly}
(A11
>
u} .; U
0, M 2 > 0), for every .x in a neighborhood of the origin.
Now l with system (8) we associate the "homogeneous" approximation
ao I 3xQ E Si : exitQo' (xo) = 0
,"~e
3;30, 'ri(J
prove in the same
<
+ /2 (x )
Qa(z)
Qa(Y) >0 and Qa(Z) <0
<
/1 (x) = (D 2k l +1 /1)0 x 2k1 + 1
1
get:
Va.
l
such that
Qa(Y)
-a
where 11 and 91 are sufficiently regular (say COO) vector fields such that D i /1 (0) == (D i 11)0 == 0 Vi E {O, 1, ... , 2k1 }, k 1 ?:: 0 and DJ 91 (0) == (Dig1)o = 0 , Vj E {a, 1, ... , k 2 - I}, k 2 2' 1. lTsing the Taylor expansion of fl(X) and gl(X) near x == 0, there exist 12 and 92 such that
(30,3 Yo E 8 1
:
~Nay
that
Qj3(YO) = 0
x ~ I(x) + u. g(x) where f(x)
(D k 2
==
k2 . 91 )0 x
(9)
(D 2k l+ 1 fl)ox 2k1 + 1 and g(x) =
'Therefore, ~Ne prove the follo"\ving
theorem.
Thus, the system (4) admits two invariant rays for
a suitable choice of (0:, fJ) and the theorem follows.
Remark 2: The closed-loop system (4) is ho~ 1110gelleous of degree 2k 1 + 1 but not necessary polynomia.l as it has been assumed in theorell1 1.
TlleoreTll 4 If the homogeneous approximati.on (9) is stabilizable by means of a positively homogeneous feedback of dcg1~ec 2k 1 - k 2 + 1, then
the system (8) is locally stabilizabl€' by the same feedback.
2412
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress ofIFAC
ON THE STABILIZATION OF HOMOGENEOUS SYSTEMS IN THE...
Proof: Let u == u(x) be a positively homogeneous stabilizing feedback of degree 2k 1 - k 2 + 1 (u("\x) == A2k l-k2+1 u(x) 1 for all x and any positive nUDlber A) for systenl (9). Set
F(x)
==
l(x)
Hahn, "'V-t/. (1967). Stabilily of motion. Springer \Terlag. New York. Jurdjevic, V. and J. P. Quin (1978)~ Controllability and stability. J. Diff. Equations 28~ 381-
+ u(x)g(x)
and
G(x)
w. P., C. F. ~vla.l'tin and G. Knovv'les (1990). i\symptotic stabilization of a class of smooth t"\vo-dinlentional systems. SIAM J. Control and Optimization 28, 1321-1349.
Daya,~~ansa,
= f2(X) + U(X)g2(X).
Since u is Cl on [l} - {O}, F and G are locally Lipschitz. ~loreover, we have
389. Kawski , M. (1990). Homogeneous stabilizing feedback. Contr-ol- Theory and ~4dvanced Technology 6, 497-516.
So~
2k 1
F is positively homogeneous function of degree + 1 and G satisfies \:Ix E
ut.
where Ai == -i\11 +1\-10 . 1. \.-[2. It follows from lVlassera's theorem (~{assera, 1956) that the solution x = 0 of the closed-loop system
x == F(x) + u(x)
IvIassera, J. L. (1956). Contribution to stability theory. Anals of Mathematics 64, 1182-206. Rosier, L. (1992). Homogeneous lyapunov function for homogeneous continuous vector field. Systems Control Lett. 19, 467-473. Sontag, E. D. (1989). A universal construction of arstein '8 theorem on nonlinear st.abilization. Systems Control Lett. 10, 263-284. Tsinias, J. (1990). Remarks on feedback stabiliza-
tion of homogeneous systems. Control- Theory
G{x)
and Advanced Technology 6) 533-541.
is asymptoticaly stable. Thus, asyu1ptotically stabilizable.
(8)
IS
locally
5. CONCLUSION
Explicit construction of stabilizing feedbacks for a class of t\vo-dimensional homogeneous systems is given _ T'he application field of this result is of course limited by the low dimension and the restrictive rank condition (RC). Nevertheless this work may seem of some interest. Indeed, it turns out that classical methods that include the construction of a Lyapullov function are difficult to apply to systems that we consider. So, our apj
proach ,vhich provides feedbacks that are analytic
except at zero, allows to remove this difficulty.
6. REFERENCES Adda, P. and H. Zenati (1998). yTariable pole assignment for a class of nonlinear syst.ems. In: 2nd Conference on Cornputational Engineer-
ing in Systems Applications (Pierre Borne, E,d.). Vol. I. pp. 167-170. CESA..98. Hammamat, Tunisia. Boothby, W. and R. Marino (1989). F'eedback stabilization of planar nonlinear systems. Systems and Control Lett. 12, 87-92. Chabour, R., G. Sallet and J~C. \ljvalda (1993). Stabilization of nonlinear two dimensional systems: a bilinear approch. A1ath. Control ~"ignal8 Systems 6, 224-246. CJavier, IVI. (1995). Stabil-isation des Systemes Bilineaires et des System,es mecaniques. Thesis. l\1etz, France.
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