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Recent Advances in The Stabilization Problem for Low Dimensional Systems
Copyright © IFAC Nonlinear Control Systems Design. Bordeaux. France. 1992
PLENARY LECfURES
RECENT ADVANCES IN THE STABILIZATION PROBLEM FOR LOW DIMENSIONAL SYSTEMS W.P. Dayawansa* DepartmenJ of Electrical EngiMering and Systems Research CenJer. University of Maryland. College Park. MD 20742, USA
Abstract We survey recent advances on the stabilization problem for two and three dimensional, single input, affine nonlinear systems. Among the new results given here is a theorem which states that a generic, single input, three dimensional, homogeneous polynomial system of a fixed odd degree p can be asymptotically stabilized by using homogeneous feedback of degree p.
Key Words:
Asymptotic Stabilization, Nonlinear Systems,
Homogeneous Systems, Two Dimensional Systems, Quadratic Systems, Three Dimensional Systems.
1
Introduction
2
There has been a tremendous interest in the nonlinear stabilization problem in the recent past, as evidenced by numerous research articles, and the recent book by Bacciotti [Ba]. One of the main contributing factors has been the realization that modern robotic systems and advanced aircraft etc. cannot be analyzed by using linear techniques alone, and more advanced theories are necessary in order to meet these design challenges. This has lead to the generalization of well known linear theories such as stabilization of passive systems (see [BIW], [HM]. (KS]. [Ma), (MO] etc.) and generalization of the notion of minimum phase systems to the nonlinear setting (see [B12] etc.). On the other hand, it has been pointed out that there are interesting classes of highly nonlinear, nongeneric systems which arise as low dimensional subsystems after various types of dimension reduction techniques such as center manifold reduction (see [Ay, [Ba]]), modification of zero dynamics by redefining the output function ( see [Ba]), and partial feedback linearization (see [Mar1],(Mar2]. [Ba] etc.). Analysis of the stabilization problem for these latter classes of systems require innovative techniques which have no counterparts in the theory of linear systems. Our focus in this paper will be on this latter class. Due to the complexity of the problem we will only consider the two and three dimensional cases. We will consider a single input, affine, nonlinear system given by,
=
f(x)
Interested reader is referred to [Ba] for a recent account on the stabilization problem for bilinear systems. Already, there is an excellent review article [S02]. written about the methods based on control Lyapunov functions, and we will not elaborate on this aspect here. This paper is organized as follows. In section two we will give the basic definitions applicable to the problems discussed here. In section three, we will discuss Brockett's, Zabcyk's and Coron's necessary conditions and give some examples. In sections four and five we will review the work on the asymptotic stabilization problem for two and three dimensional systems respectively. In section six, we will conclude the paper by stating a few challenging unsolved problems in the context of low dimensional stabilization.
+
g(x)u
(1.1)
where,xE~n,(n =20r3). uE~,andf,gareek (k>O) vector fields. We will assume that the origin is an equilibrium point ofthe unforcedsystem, i.e. f(O) = 0, and that g(O) # O.
Our aim here is to review recent developments on the stabilization problem for (1.1), when it describes a highly nonlinear system, and give a fiavor of some of the techniques that have been developed in order to study this problem. Even though there has been much interesting work done on bilinear systems in low dimensional cases, we will not discuss this class here. ·Supported in Part by NSF Grant #ECS 9096121 and the Engineering Research Center Program Grant # CD 8803012
Basic Definitions and N otation
Throughout the paper 11.11 denotes the Euclidean norm, and B. denotes the open ball of radius f in an Euclidean space. An overline over an already defined set denotes the closure of the set and 8 before a set denotes its boundary. Here, we will consider a nonlinear control system having the structure,
F(x,u),
(2.1)
e
where, x E ~n, U E ~m, F is a family of vector fields. We will assume that the origin of ~n is an equilibrium point of the unforced system. i.e. F(O.O) = O. Sometimes. we will assume that (2.1) has the affine structure,
=
f(x)
+
g(x)u,
k
(2.2)
where, x E ~n, f and 9 are smooth and u E lR m , and we will assume that f(O) = 0, and that f and 9 are e k . We will use the terms, 6mall time locally controllable, locally a6umptotically controllable to the origin etc., in the way they have been defined in [Is]. Terms such as locally continuoU61y 6tabi/izable, locally continuou61y 6tabilizing feedback control law, locally alm06t continuous 6tabi/izable, locally almost er stabi/izable, locally er stabi/izable, etc. will be used in accordance with the definitions given in [Ba]. In the sequel, we will refer to the notions of degree and index. The notion of oriented degree of a map cP : M _ N be a Cl map. where M and N are oriented manifolds of the same dimension and M is compact, (denoted by deg(cP,M,N) ),is used in the sense it is defined in [GP]. The notion of topological degree, denoted by d(cP,V,y) ), where, U be a relatively compact open subset of M and let yEN be such that y It 8U, where, cP, M and N be as before except that M may not be compact, is defined as in [Sp].[De] etc. Notion of index is a local version of the degree, defined in the following way. Let cP, M, N be as in the previous paragraph. Let yEN, and let K C cP - I {y} be a closed subset such that there exists an open neighborhood U of K in M with the property that V n cP- I {y} = K. Then. index of cP with respect to (K, y) (written as ind(cP, K. y) ) is defined as d(cP, V, y)
). IT K is a singleton {x}, then we write ind(.p, x, y) instead of ind(.p,{x},y) (see [Sp], [De] etc.). Let us now consider a system of differential equations, :i:
One can "almost" project (2.5) onto the orbit space lRn\{O}/G. This follows from the observation that x(t,xo)
sn-l
~
8.(x)
(2.4)
=
Definition 2.1 Inde:r: of (2.3) lit the origin is deg(8., sn-l, sn-l) for amllll enough f
>
(2.6)
for all s E !R where, x(t,xo) denotes the solution of (2.5) at time t starting at Xo at zero time. But , this just means that, if two solutions start at G - equivalent points, then it is possible to rescale the time in one of the solutions in such a way that the two solutions will remain G - equivalent at all times. In other words, the phase portrait of a "projected system" is well defined on the orbit space. In particular, we can obtain a representation of the "projected system" on any sphere as long as the ~r - homogeneous Euler field is transversal to it. Here we will consider the projected system on the unit Euclidean sphere sn-l , and call it projected dynamic6 of X, and denote it by 1I"(X).
We will assume that'" is continuous, and that 0 E !Rn is an isolated equilibrium point. Now, for small f > 0 we may define a map,
8. : Sn-l
= exp(sR) x(exp(-sp)t,exp(-sR)xo),
(2.3)
=
O.
We will now define several notions associated to homogeneous and weighted homogeneous systems, which will be used in the proceeding sections.
3
Definition 2.2 System (2.2) is called a homogeneoU6 ay6tem (respectively, a poaititle/y homogeneoU6 ayatem) of degree p if g(x) is a nonzero constant n X m matrix, and f(x) is a vector field which is homogeneous of degree p, i.e., f(>.x) = (>.)P f(x) for all >. E !R (respectively, positively homogeneous of degree p, i.e., f(>.x) = (>.)P f(x) for all >. > 0).
In this section, we will consider a nonlinear control system having the structure, :i:
Stabilization problem for homogeneous systems arise naturally as that of stabilizing the leading set of tenns in a system with null linear part. It is well known that (see [Ha] and the discussion in the proceeding sections) that addition of higher order terms will not affect the stability of a homogeneous system. A very important recent observation made by Kawski (see [Ka2]' (Ka3]) and Hermes (see [He1]) is that for certain highly nonlinear systems, one can select coordinates of the state space in such a way that the leading terms form a weighted homogeneous system. This is done by considering a certain nilpotent approximation of the system (see [Ka2], [He1],[He2]' [He3] etc. for details). We will define this notion below ( for the most part we will follow [Ka3] here). Let r = {rl,··· ,r n }, where rj are positive integers. Let (Xl,··· ,xn) be a fixed set of coordinates on !Rn. A one p"r"meter fllmi/y of di/lltion6 pllrllmetrized by f > 0 is ~r :!R+ X !Rn ~ !Rn, defined by, ~.r(x) = (fr,xl, ... , fr nXn ). A function .p : !Rn ~ !R is ~r - homogeneou6 of order p if .p 0 ~r fm.p. A vector field X (x) X J (x) 8~ is ~r -
= 2::
=
~r _ homogeneou6 Eu/er field is defined as
f(x)
J
Solu-
J
tion curves of the ~ r - homogeneous Euler field will be called ~r _ homogeneoua r1lY6. Anosov and Arnold [AA] use the term, quasihomogeneous, instead of weighted homogeneous, or ~ r - homogeneous. Also, according to the terminology of Hermes in [Hel] etc., the above definition refers to a ~r - homogeneous of order -p object (instead of order p).
+
g(x)u,
(3.2)
Theorem 3.1 [KZ] Con6ider the 6y6tem of differential equation6 , :i: = X(x) ; x E lR n , where X i6 continuou6. Supp06e that the 6y6tem i6 locally a6ymptoticallll 6table at the origin, and that it ha6 unique solution6 in forward time for 1111 initial conditiona in an open neighborhood of the origin. Then , the inde:r: of the all6tem at the origin i6 equal to (_l)n.
=
Definition 2.3 Let r {rl,···, r n }, where rj are positive integers. The system (2.2) is a ~r - homogeneoua ay6tem of order p if f is a ~r - homogeneous vector field of order p, and g(x) is ~r - homogeneous system of order O.
More recently Coron (see [Cor]) has strengthened (B3). Coron's theorems still depend on Zabreiko's theorem. Let M be a topological space, and let F(M) denotes either the singular homology functor with real or integral coefficients, or the homotopy functor, and for a continuous map 8 : M ~ N, let F(8) : F(M) ~ F(N) denotes the induced map.
In this paper we will only consider single input systems and in the case of ~r - homogeneous systems, we will assume that g(x) = [0, ... ,O,l]T. Let r = {rl,··· ,rn}, and let us denote R ._ diag{rl,··· ,rn}. Let G denote the one parameter group generated by the ~ r - homogeneous Euler field, i.e. G = {exp(aR) I a E !R}. By definition, G acts on !Rn and the G orbits are just ~ r - homogeneous rays. We say that two points in !Rn are G equivalent, if they lie on the same G orbit. Let us consider a ~r - homogeneous vector field in !Rn, :i: = X(x).
(3.1)
where, x E lR n , f and 9 are smooth, and u E lR m , and we will assume that f(O) = 0, and that f and 9 are C k vector fields. We will assume that 8F~~.O) i- 0 in (3.1) (respectively, g(O) i- 0 in (3.2)). It is clear that local smooth stabilizability of (3.1) implies that the linear approximation (8F~~.O) , 8F~~.O») is a stabilizable pair, and that (3.1) is locally asymptotically controllable to the origin. Brockett showed in [Br] that there is a more interesting topological obstruction to asymptotic stabilizability. This is commonly reffered to as (B3) . Two previously stated necessary conditions are reffered to as (Bl) and (B2) respectively. An interesting application of (B3) can be found in [BIl]. Zabcyk [Za] observed that an intermediate step in Brockett's proof is already contained as a theorem in [KZ] . This theorem is commonly referred to as Zabreiko's theorem in the control theory literature.
J
2:: rjxj 8~ "
F(x,u),
where, x E lR n , u E lR m , F is a family of C k vector fields. We will assume that the origin of lR n is an equilibrium point of the unCorced system, i.e. F(O,O) = O. Sometimes, we will assume that (3.1) has the affine structure,
homogeneoua of order p if Xj(x) is ~r - homogeneous of order p + r j. Corresponding to the usual Euler field x j 8~ . , the
2::
Necessary Conditions for the Asymptotic Stabilization of N onlinear Systems
Theorem 3.2 [Cor] For 6mall f > o define, I:. .{(x, u)1 II(x, u)11 < f and F(x, u) i- O}. SupP06e thllt (3.1) i6 continuou6 feedback 6tllbi/izllble. Then the induced map F (F) : F(I:.) ~ F(lRn\{O}) i6 onto . Coron also derived (see [Cor]) a corollary of this theorem, which is applicable for (3.2) in the single input case. Let, us assume
(2 .5)
2
the same degree of homogenei ty pas]. It has been known for a fairly long time (see [Ha]) that a Cl homogeneous asymptotically stable system:i: = X (x) admits a C Ie , positively homogeneous Lyapunov function. Recently, it has been shown by Rosier (see [Ros]) that this assertion remains valid even if we replace Cl by continuOU8 in the statement. We will sketch a slightly modified version of Rosier's argument below. First we observe that it follows from Kurzweil's theorem ([9] ) that the system admits a local, Coo Lyapunov function W (x) defined on a sm&ll neighborhood of the origin. Let f > 0 be small enough such that the level set of W -1 {c5} is a (homotopy) sphere for all c5 < 3f. Let, a : [0,00) -+ [0,00) be a nondecreasing smooth function such that al[o ,.) 0 and al[2"00) 1. Now, define a smooth, homogeneous, degree p function, V : lR n -+ lR by,
without any loss of generality that the system has been transformed into the form,
lex)
= =
(3.3)
u
by using a change of coordinates, and a feedback transformation. For sm&ll ~ > 0 let t. be defined as {x E sn- 1 1](x) #: O}. Let H. denotes the singular homology functor. Coron established the following: Corollary 3.1 F(/) : H n - 2 (t.) onto.
-+
H n _2(lR n - 1 \{0}) i6
=
Now we describe another necessary condition, which we believe is equivalent to the condition in the corollary. It is trivial to see the equivalence for homogeneous and weighted homogeneous systems, but it is less clear in the general case. This is an expansion of a necessary condition given for homogeneous systems in [DM1] (Theorem 3.7) .
Vex)
(i) Ci C (j)-1{0}. (iii)
= (/18B«0»)-1{0} , where Ki
ind(]l8B«0) , Kl, 0)
3.1
=
Ci noB.(O) .
1.
where, x E lR , y E lR, u E lR, and f is positively homogeneous of some degree p, i.e. f(>'x, >.y) = (>.)P f(x, y) for all
>.
..46 = Let A6 =
{(x, y) E lRm+llf(x, y)
..46 n
= c5x} .
(3.6)
sm . Let, u6>oA6,
A+ A_
u6
A+o AiR
Coron gave the following example in [Cor] to illustrate that small time local controllability, and (83) (see theorem 3.1) are not sufficient for continuos local stabilizability. We will apply theorem 3.4 to this example.
=
U6. EiR A 6
(3.7)
Theorem 3.5 [DMl) SUpp OH that there exi8t$ a co ntinuou8 curve I-' : [0, 1] -+ sm 8u ch that , north pole and 1-'(1) 60uth pole, ( i) 1-'(0) (ii) I-' C A+o . Then the 8118tem doe6 n ot admit a continuo6 pO$itiveill homogeneou6 a611mpt oti call1l 6tabiliz ing feedback function.
=
+ iX2),
=
Observe that this obstruction does not depend on whether or not one can find a feedback function such that the index of the closed loop vector field is equal to (-1) n . At this stage we don ' t know whether the restriction on the class of feedback functions can be removed . The discussion in subsection 3.1.1 generalizes without exception to the stabilization problem for t. r - Homogeneous Systems by using t. r - Homogeneous Feedback. Rosier's theorem is valid in this context also (see [Ros]) (simply replace the integrand in (3.4) by {l/(>.)Ie+I )a{W(>.r I XI ,· ··, >.rnXn))}) . It now follows that theorem 3 .5 is valid for t. r - homogeneous systems as well, provided that one is seeking for a t. r - homogeneous, continuous feedback function. Th~ only modification needed is to replace the definition of A6 in (3.6) by, ..46 {xlf(x) c5 [rJXI, · ··,rnxn]T} . The following example due to Kawski (see [Ka2]' [Ka3]) established the fact that, small time locally controllable and (83) or more generally Coron 's necessary condition (theorem 3.2) does not imply asymptotic stabilizabili ty by using t. r - homogeneous feedback .
= i(Z-X3 )3
= u.
It is easy to verify by computing Lie brackets that this system is small time locally controllable. It clearly satisfies (83). However, there are two points in the preimage of 0 by (z , X3) ...... i(z - X3)3 on S2 and each has multiplicity equal to three. Therefore, this system violates the necessary condition given in theorem 3.4, and therefore it is not locally continuously stabilizable.
3.1.1
> o.
Let us refer to the points (0,1) and (0, -1) of sm as the north pole and the south pole respectively. For arbitrary c5 E lR let us define
Theorem 3.4 Let {Vi}iEI denote the connected componen16 of (/1 8B l(0»)-1 {O} . Then, there eri6t6 a continuou8 feedback function or defined on a neighborhood of the origin 6Uch that th e inder oJ[]T(x),or(x)JT i8 equal to (_l)n fi and only if there eri8b a partition I = huI2 6Uch that Kj := UiElj Vi ,j = 1,2 are c108ed 8ubut6 of oB 1 (0) and ind(/18B 1 (0),K) ,0) = 1.
X'3
(3.5)
m
Here we consider (3.3) under the hypothesis that , ] is homogeneous of degree p (p > 0). In this case, the necessary condition for continuous local stabilizability in theorem 3.3 reduces to the following:
Example 3.1 Let z denote the compler number, (XI and let X3 be a 6calar variable. Con6ider,
(3.4)
f(x , y),
u,
Necessary Conditions for the Stabilizability of Single Input Homogeneous Systems
i
= 100 a{W(x/p.l/ P))}d>.,
where, a{W(x/( ,V /P))} is taken to be equal to 1 when x/>.I/p is not in the domain of W . It is easy to verify that V is CP, homogeneous of degree p, and VV(x)X(x) is negative definite (interested readers are referred to [Ros] for further details) . Rosier's theorem has very important consequences with regard to the stabilization problem for positively homogeneous systems. We will fir!;t define certain important subsets related to a system and then describe a necessary condition for stabilizability. It is convenient for us to write our single input positively homogeneous control system in the form,
Theorem 3.3 [DMSt) There eri6t6 a continu06 function or defined on a neighhorhood of the origin in lR n 6Uch that the index of [JT(x),or(x)]T i6 equal to (_l)n if and only if for 8mall f > 0 there end c106ed 6ub8et8 Cl and C2 of B.(O)\ {O} 6Uch that the following hold:
(ii) Kl UK2
=
Necessary Conditions for the Stabilizability of Single Input Homogeneous Systems by Using Homogeneous Feedback
=
In this section, we will discuss some necessary conditions for the (global) continuous feedback stabilizability of (3.3) by using a continuos, positively homogeneous, feedback function of
3
=
Solve,
Example 3.2
-Loa
= =
=
(2iwoI - L»b
u,
Then, (32
(3.8)
We note here that theorem 3 .5 also establishes that this ex· ample is not asymptotic stabilizable by using A r • homogeneous feedback. Clearly, [(.$2 n {Xl = X2!,s}) n (S_ u S+)] u {equator} C A+ . Thus, the hypotheses of theorem 3 .5 are satisfied, and therefore we conclude that the system is not stabilizable by any A {9 ,3,l}. homogeneous , continuous, feedback.
=
Sufficient Conditions for Asymptotic Stabilizability of Low Dimensional Systems
Current interest on the low dimensional stabilization problem seems to have started from the work of Ayels (see [Ay]), in which he showed that it is possible to extract out from a given nonlinear control system, a highly nonlinear part which essentially contains all difficulties associated wi th the stabilization problem. This, alongside with Brockett's observation that even small time local controllability does not imply stabilizability sparked a vigorous study on the stabilization problem for smooth two dimensional systems of the form,
=
F(x,u)
where,:r: E
~n, ~
= XI'(x),
8X;?).
+
IQ(r,b)
+
=
(4.4)
= 0, 9(0)
:;:. O.
Necessary and sufficient conditions for the asymptotic stabi· lizability of (4.5) (hence (4.4) were given by Dayawansa, Martin and Knowles in [DMK]. Without any loss of generality we may assume that (4.4) has the following form.
(4 .1)
Xl where,
h
E C'" and
= =
h (0, 0) =
h(Xl,X2) u
(4.5)
O.
Theorem 4.2 Con&ider the &y&tem (4.5). The following con· dition~ are equivalent. (i) The &y&tem (hence (4 .4) i& locally almo&t COO a&ymptot· ically .tabilizable. (ii) Brockett condition (B2) i& &ati6fied. (iii) For all f > 0 there exi&t PfB«O)n~~ and qfB«O)()~:' .uch that h(p) < 0 and h(q) > O. (Here ~~ = {(Xl,X2)lxl > O) and~:' = {(Xl,X2)1Xl < O). Further, a Holder continuou& &tabi/izing feedback control law can be found.
(4.2)
We will assume that (4 .2) satisfy
(i) (n - 2) • eigenvalues of Ao have negative real parts, and the remaining eigenvalues are ±iwo, where wo is a nonzero real number. (ii) Two of the eigenvalues of AI' are of the form , where, ao = 0 and (d/dlJ.)(al'(O» > o.
rn.
Theorem 4.1 [Kal] SUppOH that (4 ..0 i~ &mall time locally controllable. Then there exi&/s a Holder continuou& locally &ta· bi/izing feedback function.
E (-5,5) is a parameter, and X,..(O) = 0
for alllJ.. Let AI' := the following:
(1/2)Q(r,r) for a and b.
2 Re{2IQ(r,a)
:i; = f(x) + g(x)u , where, x E R2 t I,g are GIN - vector fields, J(O)
where, X E ~2, u E ~ ,and F(O,O) = O. One of the first such studies was carried out by Abed and Fu (see [AF1]). An important aspect of their work is that their stabilization procedure can be carried out without first performing a reduction step (such as Lyapunov, Schmidt reduction) on the original system. The key tool used in their work is the Hopf Bifurcation Theorem. Let us consider a smooth one parameter family of ordinary differential equations, :i;
(1/2)Q(r,r)
(3/4)IC(r, r, The main contribution in [AF1] is an analysis of the problem of modifying f32 by using smooth feedback control. To apply their procedure it is not necessary to assume that the state space of (4 .1) is two dimensional. They key hy· pothesis is that all except two of the eigenvalues of 8Fb~'O) have negative real parts, and the two remaining eigenvalues are nonzero, and lie on the imaginary axis. They consid· ered a parametrized family of feedback functions of the form u a(x) eT x + xTQ"x + C,,(x, x,x) and obtained an expression for {3f of the closed loop system as a sum of (3~ of the open loop system, and another term which depends on c, Q" and C,,' and the order three approximation of (4.1). The work by Abed and Fu which began in [AF1] in 1986 has been greatly expanded in [AF2J, [AF3] , [AF4] etc. to include the cases of Stationary Bifurcation, degenerate Hopf Bifurcation, Robustness Analysis etc. Perhaps, the most important step in the low dimensional stabilization problem was taken by Kawski in [Kal] . Let us consider,
where, (Xt.X2,Y) E~, and u E~ . Thi~ i~ a A{9,3 ,l}. homo· geneo '' '' .y.tem.
4
= = =
Cl and Coo feedback stabilizability are much more subtle even in the two dimensional case. Dayawansa, Martin and Knowles derived some sufficient conditions in [DMK]. We first define two indices. Since multiplication of h by a strictly positive function and coordinate changes do not affect stabilizability of (4.5), we may assume without any loss of generality that h is a Weierstrass polynomial, xi" + al (X2)X~-1 + ... + am(X2) and ai(O) = 0 , 1 ~ i ~ m . It is well known that the zero set of a Weierstrass polynomial can be written locally as the finite union of graphs of convergent rational power series X2 =
a,.. ± iw,.. ,
Then, Hopf Bifurcation Theorem (see [GS], [MC] etc.) asserts that there exists a smooth map f ...... ~(f) a2kf2k + O(f2k) : (-fO,fO) _ (-5,5) for some strictly positive integer k and a2k =F 0, such that (4 .2) has a family of periodic orbits x,..«)(t) having period near 211'w- l . Exactly one of the characteristic exponents is near zero, and it is given by a smooth even function, (4 .3) where, f32 q =F 0 and depends only on Fo . Hence, the periodic orbit is stable if f32 q < 0 and unstable if {32q > o. Abed and Fu gave the following algorithm (due to Howard [Ho]) for computing (32 Let us write Xo(x) Lx + Q(x,x) + C(x,x,x) + h.o.t, where, Lx is linear, and Q(x,x) and C(x , x, x) are homogeneous symmetric quadratic and cubic functions respectively. Let r and I be the right and left eigenvectors of L associated with the eigenvalue iwo . Normalize r and I such that the first component of r is equal to one and Ir 1. Algorithm:
D+
=
bEQ+lh(Xl ,
>0
and for some convergent rational power series
=
=
bEQ+lh(-Xl ,
>0
and for some convergent rational power series
=
4
Definition 4.1 The ability of f is max{ inf
..,eD+
index in!
b},
..,eD-
It follows from Rosier's theorem that a sufficient condition for the continuous stabilizability of (4.6) is the continuous stabilizabilityof (4.8) by using a positively homogeneous feedback function. Recently, Kawslci (see [Ka2].[Ka3)) and Hermes (see [He1)) observed that for a class of three dimensional, affine, small time local controllable systems, it is possible to find a llr homogeneous "dominant part» by constructing a Nilpotent approximation of the system. Once again by Rosier's theorem, it follows that continuous stabilizability of this llr - homogeneous approximation by using II r - homogeneous feedback of the same order implies the continuous stabilizability of the original system. In view of these discussions, we will only consider the stabilization problem for homogeneous and II r - homogeneous systems here. There are two fundamental theorems on the asymptotic stability of two and three dimensional homogeneous systems. Let us consider the system of ordinary differential equations,
of .tabilizThe fundamental
b}}.
.tabilizabilit, degree of h is the order of the zero of am(X2) at X2 O. The .econda,., .tabilizability degree of h is the order of the zero of am-t{X2) at X2 = O.
=
Notation:
I
Index of stabilizability of
h
FUndamentalstabilizability degree of Secondary stabilizability degree of
h
h.
Theorem 4.3 The .,.tem (4.5) and hence (4.4)) i. CI.tabilizable if 31 > 21 - 1 If 31 :5 1 + 232 and 31 i. odd, then (4.5) i. CUI .tabilizable. If 61 < 1 + 2.2, then (4.5) i. not COO .tabilizable. Recently, Coron and Praly have given a more elegant proof of theorem 4.2, (see [CP]) by using control Lyapunov functions. Another interesting approach to the local asymptotic stabilization problem for two dimensional systems was taken by Jakubczykand Respondek (see (JR)). They considered the case in which g(O) and ad~g(O) are linearly independent for some k :5 3 in (4.4). They employed techniques from Singularity Theory (see [GS], (Ma] , [TL] etc.) to show that it is possible to find coordinates in lR2 such that, up to a multiplication by a strictly positive real analytic function, h takes one of the following forms. X2;
(x~ (x~
+
a(xt}x 2
+ +
lXn; lXn;
x=
o=
k
= 1 = 2 k = 3,
f27r cos BXdcos B, sinB) + sin BX2 (cos B, sin B) dB cosBX 2 (cosB,sinB) - sinBXdcosB,sinB)
where X
+
[b
+
H(x,u)]u,
(4.6)
o.
Let us
(4.7)
=
where, b E ~, and H(O,O) O. Here again we assume that this system is highly nonlinear in the sense that the linear approximation of (4.6) is not stabilizable. The principal interest here is in the cases in which (4.7) contains a "dominant part» of a particularly simple form, such that the stabilization problem for this part yields a locally stabilizing feedback function forJ..4.6).The simplest such case is the following. Let F(x,o) tPi(X), where, tPi is homogeneous of degree i. Let k be the sm;illest positive integer such that tPk 1= O. Now let f := tPk and consider the stabilization problem for,
= L..::
x=
f(x)
+
bu.
= (X I ,X2).
Note that the w - limit set C of the projected dynamics in theorem 4.5 consists of a finite union of periodic orbits, equilibrium points, and some homoclinic and heteroclinic orbits between them. Determination of stability on rays corresponding to the equilibria is trivial and on the cones generated by an equilibrium point can be carried out by using (b) in theorem 4.4. If the restriction of the system to the cones generated by equilibria and periodic orbits is asymptotically stable, then the same holds true for the cones generated by the homo clinic and heteroclinic orbits. Theorem 4.5 is due to Coleman (see [Col)). Kawslci observed in [Ka3] that it remains valid for all integers n. Coleman attributes theorem 4.4 to Forster (see[For)), and it has been discovered independently by Haimo [Hai). These two theorems follow rather easily from the G - equivalence of the solutions of a homogeneous system, i.e. (2.6). Complete proofs can be found in [Ha), [Col) etc. Kawslci and Hermes have shown that these two theorems remain valid (with obvious modifications) in the case of II r - homogeneous systems (see [Ka2),[Ka3) and [He1) for details). In particular Kawslci noted in [Ka2) and [Ka3) that in the case n = 3, if the w - limit set of (4.10) consists of a Jordan curve, if the set of equilibrium points is nonempty and that the system is asymptotically stable on each ray corresponding to the equilibria, then (4.9) is asymptotically stable. Asymptotic stabilization problem for homogeneous systems was first considered by Samardzija [Sam). There was a resurgence of interest on this class after Andreini, Bacciotti and Stefani proved the following theorem in [ABS).
Let us first consider the local asymptotic stabilization problem for the system,
= F(x,u),
0
<
Theorem 4.5 Let n = 3. Let C denote. the w - limit .et of (4.10), and let cn denote. the cone generated by C. Then, (4.9) i. a.ymptotically .table if and only if the re.triction of (4.10) to cn i. a.ymptotically .table.
Unlike in the two dimensional case, the picture is much less clear in the three dimensional stabilization problem. Here we attempt to summarize the key work on this problem.
= F(x,O)
(4.10)
(b) Re.triction of the .y.tem to each of it. one dimen.ional invariant .ub.pace. i. a.ymptotically .table.
Asymptotic Stabilization Problem for Three Dimensional Systems
x
1r(X)(B), BE S"-I
io
=
where, x E ~, u E lR, F is Coo , and F(O,O) = write down the Taylor polynomial in the form,
(4.9)
(a) The .y.tem doe. not have anyone dimen.ional invariant .ub.pacu and
k
=
x
= 2 or 3,
Theorem 4.4 Let n = 2. Then (4 .9) i. a.ymptotically .table if and only if one of the following hold:
where, l ±1, a(xt) is a CUI function and a(O) O. They carried out a detailed analysis of these three cases. Even though stabilizability can be determined in each of these cases by using theorem 4.3, the work reported in (JR] is more constructive. We also note here the interesting work of Boothby and Marino (see [BM1], [BM2], [BM3] etc.) and Crouch and Igheneiwa [Cl]. Boothby and Marino focussed on finding obstructions to Cl stabilizability. In [Cl] the main aim is to use Newton Polygons in order to find suitable coordinate changes which can simplify the stabilization problem, and hence to give some sufficient conditions for stabilizability.
4.1
X(x), x E lR", n
where, X(x) is almost Cl, positively homogeneous vector field of degree p (our standing hypothesis is that p is an integer not less than 1). Let us denote the projected dynamics (see the last paragraph of section 2 for the definition) by,
(4.8)
5
5
Let us consider a system,
= =
j(x,y) u
Concluding Remarks
We have summarized some of the work that has been done on the asymptotic stabilization problem for two and three dimensionalsystems in the recent past. Much of the discussion reflects our personal biases, and scant attention was given to important aspects of the problem such as the work based on Control Lyapunov Functions. The interested readers are referred to [S02) for a detailed review of this aspect. In our opinion, we, the nonlinear community, has acquired a reasonable understanding of the complexity of the.stabilization problem for homogeneous and weighted homogeneous systems. Theorem 4.8 is very encouraging in the sense that it shows that there cannot be "too many other unknown obstructions" to stabilizability. However, finding them ought to be an important task ahead. There is at least one other theoretical question raised by theorem 4.8. Does the genericity statement hold in higher dimensions? . Theorem 3.5 gives a necessary condition for the stabilizability of homogeneous systems by using homogeneoua feedback . Does this remain as an obstruction, if we remove the restriction on homogeneity of the feedback function? We believe that much interesting work waits ahead in answering these questions. This is bound to be a highly fertile area for trying out everyone's favorite tools from algebraic geometry, algebraic topology, optimal control theory etc . We hope that these attempts will be successful .
(4.11)
where, x E Rn-I, 11 E R, u E R, and j is homogeneous of degree p for some p ~ 1. Theorem •• 6 {A BS], {DM4] Suppo.e that i: = j(x,O) i. /uymptotically lid/e. Then the feedback function u = -yP globally a,ymptotically ,tdi/ize, (4.11). This theorem follows from the observation that the closed loop system has the structure of a "block upper triangular" system with stable diagonal blocks. Therefore, it follows that the cloeed loop flystem is locally asymptotically stable (see [Vi), [S03) etc.). Now, the G - equivarianceproperty (2.6) establishes global asymptotic stability. A I1 r - homogeneous version of this theorem was given by Kawski in (Ka3). In the three dimensional case, theorem 4.6 can be strengthened in the following way. Definition •• 2 Let r c 52 be an embedded circle. A point fJ E r is on an upper arc, (respectively on a lower arc) if a particular meridian meets r for the first time (respectively, for the last time) at fJ. Theorem •. 7 {DMSf],{DMS3] Con,ider (4 .11) in the ca8e when n 3. Suppo,e that there e:ri,t. r c 52, an embedded circle, which meet. A_ at a point on a lower arc or an upper arc, and that r doe. not meet A+ o , and the pole8. Then (4.11) i, almo,t Coo .tabi/izd/e by u.ing p08itiveiy homogeneou, feedb4ck.
6
=
Acknowledgements
We wish to thank Professor Clyde Martin, Dr. Gareth Knowles, Dr. Sandy Samelson, Dr. D . Chen for the highly fruitful collaborations with us on this subject in the recent past. We also wish to thank Professors David Gilliam and Christopher Byrnes for helpful discussions related to algebraic curves, which eventually lead us to theorem 4.8 .
Theorem 4.7 can be used to prove that stabilizability is generic in the space of three dimensional homogeneous polunomialsystems of a given odd degree.
References
Theorem •. 8 Let p be 4n odd integer. Then, aingle input homogeneou, polynomial Iy,tem, of degree p in three at4tea 4re generically ,t4bi/iz4ble. Moreover, the feedb4ck functiona C4n be found to be almod Coo and homogeneoU8 of degree p .
[AA) D. V . Anosov, V. I. Arnold eds. Dynamical SY8tem8 I , Encyclopedia of Mathematical Sciences , Vol.l ; Springer Verlag, Berlin, 1988.
=
[AFl) E. H . Abed and J .H . Fu, "Local feedback stabilization and bifurcation control ,I , Hopf bifurcation" System8 and Control Letter8 7, (1986) 11-17.
In the case when p 2, theorem 4.7 has been used in [DMS2) to give algebraic necessary and sufficient conditions for the asymptotic stabilizability of a generic system. A comprehensive analysis of the quadratic case, including the nongeneric cases, will be given in [DMS2). The same set of arguments has been used in [DMS3) to give necessary and sufficient conditions for the stabilizability of a generic homogeneous three dimensional systems of a given even degree. Of course, the arguments given here, and the conclusions drawn remain valid for weighted homogeneous systems as well. Here the odd degree case corresponds to 11 r - homogeneous systems of order p, when q, r2, r3 are odd integers and p is an even integer, or when rlo r2, r3 are even integers and p is an odd integer, and the even degree case corresponds to the case when either ri, i 1,2,3 and p are all odd or all even. The exposition in this section has followed along lines which best describes our personal tastes, and biases. An entirely different approach to the stabilization problem for three dimensional systems has been taken by Hermes. His objective has been to extend the quadratic regulator theory to the case of weighted homogeneous systems, by considering a weighted homogeneous cost function. He has been successful in giving an alternate proof of Kawski's theorem (theorem 4.1) by using this approach. He has also worked out some three dimensional examples. However, as far as we know , there hasn't been any characterization of a reasonable set of intrinsic conditions under which this approach leads to a solution in the three dimensional case. Interested readers are referred to [He3). [He4) etc. for further details.
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[MM] J. E. Marsden and M . McCracken, Th e Hopf Bifurcation Theorem and ill Application8, Springer Verlag, New York, 1976. [Mo] P. J. Moylan, "Implications of passivity in a class of nonlinear systems," IEEE Tran84c. A.tomat. Contr. , AC19, (1974),373-381.
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8
PLENARY LECfURES
RECENT ADVANCES IN THE STABILIZATION PROBLEM FOR LOW DIMENSIONAL SYSTEMS W.P. Dayawansa* DepartmenJ of Electrical EngiMering and Systems Research CenJer. University of Maryland. College Park. MD 20742, USA
Abstract We survey recent advances on the stabilization problem for two and three dimensional, single input, affine nonlinear systems. Among the new results given here is a theorem which states that a generic, single input, three dimensional, homogeneous polynomial system of a fixed odd degree p can be asymptotically stabilized by using homogeneous feedback of degree p.
Key Words:
Asymptotic Stabilization, Nonlinear Systems,
Homogeneous Systems, Two Dimensional Systems, Quadratic Systems, Three Dimensional Systems.
1
Introduction
2
There has been a tremendous interest in the nonlinear stabilization problem in the recent past, as evidenced by numerous research articles, and the recent book by Bacciotti [Ba]. One of the main contributing factors has been the realization that modern robotic systems and advanced aircraft etc. cannot be analyzed by using linear techniques alone, and more advanced theories are necessary in order to meet these design challenges. This has lead to the generalization of well known linear theories such as stabilization of passive systems (see [BIW], [HM]. (KS]. [Ma), (MO] etc.) and generalization of the notion of minimum phase systems to the nonlinear setting (see [B12] etc.). On the other hand, it has been pointed out that there are interesting classes of highly nonlinear, nongeneric systems which arise as low dimensional subsystems after various types of dimension reduction techniques such as center manifold reduction (see [Ay, [Ba]]), modification of zero dynamics by redefining the output function ( see [Ba]), and partial feedback linearization (see [Mar1],(Mar2]. [Ba] etc.). Analysis of the stabilization problem for these latter classes of systems require innovative techniques which have no counterparts in the theory of linear systems. Our focus in this paper will be on this latter class. Due to the complexity of the problem we will only consider the two and three dimensional cases. We will consider a single input, affine, nonlinear system given by,
=
f(x)
Interested reader is referred to [Ba] for a recent account on the stabilization problem for bilinear systems. Already, there is an excellent review article [S02]. written about the methods based on control Lyapunov functions, and we will not elaborate on this aspect here. This paper is organized as follows. In section two we will give the basic definitions applicable to the problems discussed here. In section three, we will discuss Brockett's, Zabcyk's and Coron's necessary conditions and give some examples. In sections four and five we will review the work on the asymptotic stabilization problem for two and three dimensional systems respectively. In section six, we will conclude the paper by stating a few challenging unsolved problems in the context of low dimensional stabilization.
+
g(x)u
(1.1)
where,xE~n,(n =20r3). uE~,andf,gareek (k>O) vector fields. We will assume that the origin is an equilibrium point ofthe unforcedsystem, i.e. f(O) = 0, and that g(O) # O.
Our aim here is to review recent developments on the stabilization problem for (1.1), when it describes a highly nonlinear system, and give a fiavor of some of the techniques that have been developed in order to study this problem. Even though there has been much interesting work done on bilinear systems in low dimensional cases, we will not discuss this class here. ·Supported in Part by NSF Grant #ECS 9096121 and the Engineering Research Center Program Grant # CD 8803012
Basic Definitions and N otation
Throughout the paper 11.11 denotes the Euclidean norm, and B. denotes the open ball of radius f in an Euclidean space. An overline over an already defined set denotes the closure of the set and 8 before a set denotes its boundary. Here, we will consider a nonlinear control system having the structure,
F(x,u),
(2.1)
e
where, x E ~n, U E ~m, F is a family of vector fields. We will assume that the origin of ~n is an equilibrium point of the unforced system. i.e. F(O.O) = O. Sometimes. we will assume that (2.1) has the affine structure,
=
f(x)
+
g(x)u,
k
(2.2)
where, x E ~n, f and 9 are smooth and u E lR m , and we will assume that f(O) = 0, and that f and 9 are e k . We will use the terms, 6mall time locally controllable, locally a6umptotically controllable to the origin etc., in the way they have been defined in [Is]. Terms such as locally continuoU61y 6tabi/izable, locally continuou61y 6tabilizing feedback control law, locally alm06t continuous 6tabi/izable, locally almost er stabi/izable, locally er stabi/izable, etc. will be used in accordance with the definitions given in [Ba]. In the sequel, we will refer to the notions of degree and index. The notion of oriented degree of a map cP : M _ N be a Cl map. where M and N are oriented manifolds of the same dimension and M is compact, (denoted by deg(cP,M,N) ),is used in the sense it is defined in [GP]. The notion of topological degree, denoted by d(cP,V,y) ), where, U be a relatively compact open subset of M and let yEN be such that y It 8U, where, cP, M and N be as before except that M may not be compact, is defined as in [Sp].[De] etc. Notion of index is a local version of the degree, defined in the following way. Let cP, M, N be as in the previous paragraph. Let yEN, and let K C cP - I {y} be a closed subset such that there exists an open neighborhood U of K in M with the property that V n cP- I {y} = K. Then. index of cP with respect to (K, y) (written as ind(cP, K. y) ) is defined as d(cP, V, y)
). IT K is a singleton {x}, then we write ind(.p, x, y) instead of ind(.p,{x},y) (see [Sp], [De] etc.). Let us now consider a system of differential equations, :i:
One can "almost" project (2.5) onto the orbit space lRn\{O}/G. This follows from the observation that x(t,xo)
sn-l
~
8.(x)
(2.4)
=
Definition 2.1 Inde:r: of (2.3) lit the origin is deg(8., sn-l, sn-l) for amllll enough f
>
(2.6)
for all s E !R where, x(t,xo) denotes the solution of (2.5) at time t starting at Xo at zero time. But , this just means that, if two solutions start at G - equivalent points, then it is possible to rescale the time in one of the solutions in such a way that the two solutions will remain G - equivalent at all times. In other words, the phase portrait of a "projected system" is well defined on the orbit space. In particular, we can obtain a representation of the "projected system" on any sphere as long as the ~r - homogeneous Euler field is transversal to it. Here we will consider the projected system on the unit Euclidean sphere sn-l , and call it projected dynamic6 of X, and denote it by 1I"(X).
We will assume that'" is continuous, and that 0 E !Rn is an isolated equilibrium point. Now, for small f > 0 we may define a map,
8. : Sn-l
= exp(sR) x(exp(-sp)t,exp(-sR)xo),
(2.3)
=
O.
We will now define several notions associated to homogeneous and weighted homogeneous systems, which will be used in the proceeding sections.
3
Definition 2.2 System (2.2) is called a homogeneoU6 ay6tem (respectively, a poaititle/y homogeneoU6 ayatem) of degree p if g(x) is a nonzero constant n X m matrix, and f(x) is a vector field which is homogeneous of degree p, i.e., f(>.x) = (>.)P f(x) for all >. E !R (respectively, positively homogeneous of degree p, i.e., f(>.x) = (>.)P f(x) for all >. > 0).
In this section, we will consider a nonlinear control system having the structure, :i:
Stabilization problem for homogeneous systems arise naturally as that of stabilizing the leading set of tenns in a system with null linear part. It is well known that (see [Ha] and the discussion in the proceeding sections) that addition of higher order terms will not affect the stability of a homogeneous system. A very important recent observation made by Kawski (see [Ka2]' (Ka3]) and Hermes (see [He1]) is that for certain highly nonlinear systems, one can select coordinates of the state space in such a way that the leading terms form a weighted homogeneous system. This is done by considering a certain nilpotent approximation of the system (see [Ka2], [He1],[He2]' [He3] etc. for details). We will define this notion below ( for the most part we will follow [Ka3] here). Let r = {rl,··· ,r n }, where rj are positive integers. Let (Xl,··· ,xn) be a fixed set of coordinates on !Rn. A one p"r"meter fllmi/y of di/lltion6 pllrllmetrized by f > 0 is ~r :!R+ X !Rn ~ !Rn, defined by, ~.r(x) = (fr,xl, ... , fr nXn ). A function .p : !Rn ~ !R is ~r - homogeneou6 of order p if .p 0 ~r fm.p. A vector field X (x) X J (x) 8~ is ~r -
= 2::
=
~r _ homogeneou6 Eu/er field is defined as
f(x)
J
Solu-
J
tion curves of the ~ r - homogeneous Euler field will be called ~r _ homogeneoua r1lY6. Anosov and Arnold [AA] use the term, quasihomogeneous, instead of weighted homogeneous, or ~ r - homogeneous. Also, according to the terminology of Hermes in [Hel] etc., the above definition refers to a ~r - homogeneous of order -p object (instead of order p).
+
g(x)u,
(3.2)
Theorem 3.1 [KZ] Con6ider the 6y6tem of differential equation6 , :i: = X(x) ; x E lR n , where X i6 continuou6. Supp06e that the 6y6tem i6 locally a6ymptoticallll 6table at the origin, and that it ha6 unique solution6 in forward time for 1111 initial conditiona in an open neighborhood of the origin. Then , the inde:r: of the all6tem at the origin i6 equal to (_l)n.
=
Definition 2.3 Let r {rl,···, r n }, where rj are positive integers. The system (2.2) is a ~r - homogeneoua ay6tem of order p if f is a ~r - homogeneous vector field of order p, and g(x) is ~r - homogeneous system of order O.
More recently Coron (see [Cor]) has strengthened (B3). Coron's theorems still depend on Zabreiko's theorem. Let M be a topological space, and let F(M) denotes either the singular homology functor with real or integral coefficients, or the homotopy functor, and for a continuous map 8 : M ~ N, let F(8) : F(M) ~ F(N) denotes the induced map.
In this paper we will only consider single input systems and in the case of ~r - homogeneous systems, we will assume that g(x) = [0, ... ,O,l]T. Let r = {rl,··· ,rn}, and let us denote R ._ diag{rl,··· ,rn}. Let G denote the one parameter group generated by the ~ r - homogeneous Euler field, i.e. G = {exp(aR) I a E !R}. By definition, G acts on !Rn and the G orbits are just ~ r - homogeneous rays. We say that two points in !Rn are G equivalent, if they lie on the same G orbit. Let us consider a ~r - homogeneous vector field in !Rn, :i: = X(x).
(3.1)
where, x E lR n , f and 9 are smooth, and u E lR m , and we will assume that f(O) = 0, and that f and 9 are C k vector fields. We will assume that 8F~~.O) i- 0 in (3.1) (respectively, g(O) i- 0 in (3.2)). It is clear that local smooth stabilizability of (3.1) implies that the linear approximation (8F~~.O) , 8F~~.O») is a stabilizable pair, and that (3.1) is locally asymptotically controllable to the origin. Brockett showed in [Br] that there is a more interesting topological obstruction to asymptotic stabilizability. This is commonly reffered to as (B3) . Two previously stated necessary conditions are reffered to as (Bl) and (B2) respectively. An interesting application of (B3) can be found in [BIl]. Zabcyk [Za] observed that an intermediate step in Brockett's proof is already contained as a theorem in [KZ] . This theorem is commonly referred to as Zabreiko's theorem in the control theory literature.
J
2:: rjxj 8~ "
F(x,u),
where, x E lR n , u E lR m , F is a family of C k vector fields. We will assume that the origin of lR n is an equilibrium point of the unCorced system, i.e. F(O,O) = O. Sometimes, we will assume that (3.1) has the affine structure,
homogeneoua of order p if Xj(x) is ~r - homogeneous of order p + r j. Corresponding to the usual Euler field x j 8~ . , the
2::
Necessary Conditions for the Asymptotic Stabilization of N onlinear Systems
Theorem 3.2 [Cor] For 6mall f > o define, I:. .{(x, u)1 II(x, u)11 < f and F(x, u) i- O}. SupP06e thllt (3.1) i6 continuou6 feedback 6tllbi/izllble. Then the induced map F (F) : F(I:.) ~ F(lRn\{O}) i6 onto . Coron also derived (see [Cor]) a corollary of this theorem, which is applicable for (3.2) in the single input case. Let, us assume
(2 .5)
2
the same degree of homogenei ty pas]. It has been known for a fairly long time (see [Ha]) that a Cl homogeneous asymptotically stable system:i: = X (x) admits a C Ie , positively homogeneous Lyapunov function. Recently, it has been shown by Rosier (see [Ros]) that this assertion remains valid even if we replace Cl by continuOU8 in the statement. We will sketch a slightly modified version of Rosier's argument below. First we observe that it follows from Kurzweil's theorem ([9] ) that the system admits a local, Coo Lyapunov function W (x) defined on a sm&ll neighborhood of the origin. Let f > 0 be small enough such that the level set of W -1 {c5} is a (homotopy) sphere for all c5 < 3f. Let, a : [0,00) -+ [0,00) be a nondecreasing smooth function such that al[o ,.) 0 and al[2"00) 1. Now, define a smooth, homogeneous, degree p function, V : lR n -+ lR by,
without any loss of generality that the system has been transformed into the form,
lex)
= =
(3.3)
u
by using a change of coordinates, and a feedback transformation. For sm&ll ~ > 0 let t. be defined as {x E sn- 1 1](x) #: O}. Let H. denotes the singular homology functor. Coron established the following: Corollary 3.1 F(/) : H n - 2 (t.) onto.
-+
H n _2(lR n - 1 \{0}) i6
=
Now we describe another necessary condition, which we believe is equivalent to the condition in the corollary. It is trivial to see the equivalence for homogeneous and weighted homogeneous systems, but it is less clear in the general case. This is an expansion of a necessary condition given for homogeneous systems in [DM1] (Theorem 3.7) .
Vex)
(i) Ci C (j)-1{0}. (iii)
= (/18B«0»)-1{0} , where Ki
ind(]l8B«0) , Kl, 0)
3.1
=
Ci noB.(O) .
1.
where, x E lR , y E lR, u E lR, and f is positively homogeneous of some degree p, i.e. f(>'x, >.y) = (>.)P f(x, y) for all
>.
..46 = Let A6 =
{(x, y) E lRm+llf(x, y)
..46 n
= c5x} .
(3.6)
sm . Let, u6>oA6,
A+ A_
u6
A+o AiR
Coron gave the following example in [Cor] to illustrate that small time local controllability, and (83) (see theorem 3.1) are not sufficient for continuos local stabilizability. We will apply theorem 3.4 to this example.
=
U6. EiR A 6
(3.7)
Theorem 3.5 [DMl) SUpp OH that there exi8t$ a co ntinuou8 curve I-' : [0, 1] -+ sm 8u ch that , north pole and 1-'(1) 60uth pole, ( i) 1-'(0) (ii) I-' C A+o . Then the 8118tem doe6 n ot admit a continuo6 pO$itiveill homogeneou6 a611mpt oti call1l 6tabiliz ing feedback function.
=
+ iX2),
=
Observe that this obstruction does not depend on whether or not one can find a feedback function such that the index of the closed loop vector field is equal to (-1) n . At this stage we don ' t know whether the restriction on the class of feedback functions can be removed . The discussion in subsection 3.1.1 generalizes without exception to the stabilization problem for t. r - Homogeneous Systems by using t. r - Homogeneous Feedback. Rosier's theorem is valid in this context also (see [Ros]) (simply replace the integrand in (3.4) by {l/(>.)Ie+I )a{W(>.r I XI ,· ··, >.rnXn))}) . It now follows that theorem 3 .5 is valid for t. r - homogeneous systems as well, provided that one is seeking for a t. r - homogeneous, continuous feedback function. Th~ only modification needed is to replace the definition of A6 in (3.6) by, ..46 {xlf(x) c5 [rJXI, · ··,rnxn]T} . The following example due to Kawski (see [Ka2]' [Ka3]) established the fact that, small time locally controllable and (83) or more generally Coron 's necessary condition (theorem 3.2) does not imply asymptotic stabilizabili ty by using t. r - homogeneous feedback .
= i(Z-X3 )3
= u.
It is easy to verify by computing Lie brackets that this system is small time locally controllable. It clearly satisfies (83). However, there are two points in the preimage of 0 by (z , X3) ...... i(z - X3)3 on S2 and each has multiplicity equal to three. Therefore, this system violates the necessary condition given in theorem 3.4, and therefore it is not locally continuously stabilizable.
3.1.1
> o.
Let us refer to the points (0,1) and (0, -1) of sm as the north pole and the south pole respectively. For arbitrary c5 E lR let us define
Theorem 3.4 Let {Vi}iEI denote the connected componen16 of (/1 8B l(0»)-1 {O} . Then, there eri6t6 a continuou8 feedback function or defined on a neighborhood of the origin 6Uch that th e inder oJ[]T(x),or(x)JT i8 equal to (_l)n fi and only if there eri8b a partition I = huI2 6Uch that Kj := UiElj Vi ,j = 1,2 are c108ed 8ubut6 of oB 1 (0) and ind(/18B 1 (0),K) ,0) = 1.
X'3
(3.5)
m
Here we consider (3.3) under the hypothesis that , ] is homogeneous of degree p (p > 0). In this case, the necessary condition for continuous local stabilizability in theorem 3.3 reduces to the following:
Example 3.1 Let z denote the compler number, (XI and let X3 be a 6calar variable. Con6ider,
(3.4)
f(x , y),
u,
Necessary Conditions for the Stabilizability of Single Input Homogeneous Systems
i
= 100 a{W(x/p.l/ P))}d>.,
where, a{W(x/( ,V /P))} is taken to be equal to 1 when x/>.I/p is not in the domain of W . It is easy to verify that V is CP, homogeneous of degree p, and VV(x)X(x) is negative definite (interested readers are referred to [Ros] for further details) . Rosier's theorem has very important consequences with regard to the stabilization problem for positively homogeneous systems. We will fir!;t define certain important subsets related to a system and then describe a necessary condition for stabilizability. It is convenient for us to write our single input positively homogeneous control system in the form,
Theorem 3.3 [DMSt) There eri6t6 a continu06 function or defined on a neighhorhood of the origin in lR n 6Uch that the index of [JT(x),or(x)]T i6 equal to (_l)n if and only if for 8mall f > 0 there end c106ed 6ub8et8 Cl and C2 of B.(O)\ {O} 6Uch that the following hold:
(ii) Kl UK2
=
Necessary Conditions for the Stabilizability of Single Input Homogeneous Systems by Using Homogeneous Feedback
=
In this section, we will discuss some necessary conditions for the (global) continuous feedback stabilizability of (3.3) by using a continuos, positively homogeneous, feedback function of
3
=
Solve,
Example 3.2
-Loa
= =
=
(2iwoI - L»b
u,
Then, (32
(3.8)
We note here that theorem 3 .5 also establishes that this ex· ample is not asymptotic stabilizable by using A r • homogeneous feedback. Clearly, [(.$2 n {Xl = X2!,s}) n (S_ u S+)] u {equator} C A+ . Thus, the hypotheses of theorem 3 .5 are satisfied, and therefore we conclude that the system is not stabilizable by any A {9 ,3,l}. homogeneous , continuous, feedback.
=
Sufficient Conditions for Asymptotic Stabilizability of Low Dimensional Systems
Current interest on the low dimensional stabilization problem seems to have started from the work of Ayels (see [Ay]), in which he showed that it is possible to extract out from a given nonlinear control system, a highly nonlinear part which essentially contains all difficulties associated wi th the stabilization problem. This, alongside with Brockett's observation that even small time local controllability does not imply stabilizability sparked a vigorous study on the stabilization problem for smooth two dimensional systems of the form,
=
F(x,u)
where,:r: E
~n, ~
= XI'(x),
8X;?).
+
IQ(r,b)
+
=
(4.4)
= 0, 9(0)
:;:. O.
Necessary and sufficient conditions for the asymptotic stabi· lizability of (4.5) (hence (4.4) were given by Dayawansa, Martin and Knowles in [DMK]. Without any loss of generality we may assume that (4.4) has the following form.
(4 .1)
Xl where,
h
E C'" and
= =
h (0, 0) =
h(Xl,X2) u
(4.5)
O.
Theorem 4.2 Con&ider the &y&tem (4.5). The following con· dition~ are equivalent. (i) The &y&tem (hence (4 .4) i& locally almo&t COO a&ymptot· ically .tabilizable. (ii) Brockett condition (B2) i& &ati6fied. (iii) For all f > 0 there exi&t PfB«O)n~~ and qfB«O)()~:' .uch that h(p) < 0 and h(q) > O. (Here ~~ = {(Xl,X2)lxl > O) and~:' = {(Xl,X2)1Xl < O). Further, a Holder continuou& &tabi/izing feedback control law can be found.
(4.2)
We will assume that (4 .2) satisfy
(i) (n - 2) • eigenvalues of Ao have negative real parts, and the remaining eigenvalues are ±iwo, where wo is a nonzero real number. (ii) Two of the eigenvalues of AI' are of the form , where, ao = 0 and (d/dlJ.)(al'(O» > o.
rn.
Theorem 4.1 [Kal] SUppOH that (4 ..0 i~ &mall time locally controllable. Then there exi&/s a Holder continuou& locally &ta· bi/izing feedback function.
E (-5,5) is a parameter, and X,..(O) = 0
for alllJ.. Let AI' := the following:
(1/2)Q(r,r) for a and b.
2 Re{2IQ(r,a)
:i; = f(x) + g(x)u , where, x E R2 t I,g are GIN - vector fields, J(O)
where, X E ~2, u E ~ ,and F(O,O) = O. One of the first such studies was carried out by Abed and Fu (see [AF1]). An important aspect of their work is that their stabilization procedure can be carried out without first performing a reduction step (such as Lyapunov, Schmidt reduction) on the original system. The key tool used in their work is the Hopf Bifurcation Theorem. Let us consider a smooth one parameter family of ordinary differential equations, :i;
(1/2)Q(r,r)
(3/4)IC(r, r, The main contribution in [AF1] is an analysis of the problem of modifying f32 by using smooth feedback control. To apply their procedure it is not necessary to assume that the state space of (4 .1) is two dimensional. They key hy· pothesis is that all except two of the eigenvalues of 8Fb~'O) have negative real parts, and the two remaining eigenvalues are nonzero, and lie on the imaginary axis. They consid· ered a parametrized family of feedback functions of the form u a(x) eT x + xTQ"x + C,,(x, x,x) and obtained an expression for {3f of the closed loop system as a sum of (3~ of the open loop system, and another term which depends on c, Q" and C,,' and the order three approximation of (4.1). The work by Abed and Fu which began in [AF1] in 1986 has been greatly expanded in [AF2J, [AF3] , [AF4] etc. to include the cases of Stationary Bifurcation, degenerate Hopf Bifurcation, Robustness Analysis etc. Perhaps, the most important step in the low dimensional stabilization problem was taken by Kawski in [Kal] . Let us consider,
where, (Xt.X2,Y) E~, and u E~ . Thi~ i~ a A{9,3 ,l}. homo· geneo '' '' .y.tem.
4
= = =
Cl and Coo feedback stabilizability are much more subtle even in the two dimensional case. Dayawansa, Martin and Knowles derived some sufficient conditions in [DMK]. We first define two indices. Since multiplication of h by a strictly positive function and coordinate changes do not affect stabilizability of (4.5), we may assume without any loss of generality that h is a Weierstrass polynomial, xi" + al (X2)X~-1 + ... + am(X2) and ai(O) = 0 , 1 ~ i ~ m . It is well known that the zero set of a Weierstrass polynomial can be written locally as the finite union of graphs of convergent rational power series X2 =
a,.. ± iw,.. ,
Then, Hopf Bifurcation Theorem (see [GS], [MC] etc.) asserts that there exists a smooth map f ...... ~(f) a2kf2k + O(f2k) : (-fO,fO) _ (-5,5) for some strictly positive integer k and a2k =F 0, such that (4 .2) has a family of periodic orbits x,..«)(t) having period near 211'w- l . Exactly one of the characteristic exponents is near zero, and it is given by a smooth even function, (4 .3) where, f32 q =F 0 and depends only on Fo . Hence, the periodic orbit is stable if f32 q < 0 and unstable if {32q > o. Abed and Fu gave the following algorithm (due to Howard [Ho]) for computing (32 Let us write Xo(x) Lx + Q(x,x) + C(x,x,x) + h.o.t, where, Lx is linear, and Q(x,x) and C(x , x, x) are homogeneous symmetric quadratic and cubic functions respectively. Let r and I be the right and left eigenvectors of L associated with the eigenvalue iwo . Normalize r and I such that the first component of r is equal to one and Ir 1. Algorithm:
D+
=
bEQ+lh(Xl ,
>0
and for some convergent rational power series
=
=
bEQ+lh(-Xl ,
>0
and for some convergent rational power series
=
4
Definition 4.1 The ability of f is max{ inf
..,eD+
index in!
b},
..,eD-
It follows from Rosier's theorem that a sufficient condition for the continuous stabilizability of (4.6) is the continuous stabilizabilityof (4.8) by using a positively homogeneous feedback function. Recently, Kawslci (see [Ka2].[Ka3)) and Hermes (see [He1)) observed that for a class of three dimensional, affine, small time local controllable systems, it is possible to find a llr homogeneous "dominant part» by constructing a Nilpotent approximation of the system. Once again by Rosier's theorem, it follows that continuous stabilizability of this llr - homogeneous approximation by using II r - homogeneous feedback of the same order implies the continuous stabilizability of the original system. In view of these discussions, we will only consider the stabilization problem for homogeneous and II r - homogeneous systems here. There are two fundamental theorems on the asymptotic stability of two and three dimensional homogeneous systems. Let us consider the system of ordinary differential equations,
of .tabilizThe fundamental
b}}.
.tabilizabilit, degree of h is the order of the zero of am(X2) at X2 O. The .econda,., .tabilizability degree of h is the order of the zero of am-t{X2) at X2 = O.
=
Notation:
I
Index of stabilizability of
h
FUndamentalstabilizability degree of Secondary stabilizability degree of
h
h.
Theorem 4.3 The .,.tem (4.5) and hence (4.4)) i. CI.tabilizable if 31 > 21 - 1 If 31 :5 1 + 232 and 31 i. odd, then (4.5) i. CUI .tabilizable. If 61 < 1 + 2.2, then (4.5) i. not COO .tabilizable. Recently, Coron and Praly have given a more elegant proof of theorem 4.2, (see [CP]) by using control Lyapunov functions. Another interesting approach to the local asymptotic stabilization problem for two dimensional systems was taken by Jakubczykand Respondek (see (JR)). They considered the case in which g(O) and ad~g(O) are linearly independent for some k :5 3 in (4.4). They employed techniques from Singularity Theory (see [GS], (Ma] , [TL] etc.) to show that it is possible to find coordinates in lR2 such that, up to a multiplication by a strictly positive real analytic function, h takes one of the following forms. X2;
(x~ (x~
+
a(xt}x 2
+ +
lXn; lXn;
x=
o=
k
= 1 = 2 k = 3,
f27r cos BXdcos B, sinB) + sin BX2 (cos B, sin B) dB cosBX 2 (cosB,sinB) - sinBXdcosB,sinB)
where X
+
[b
+
H(x,u)]u,
(4.6)
o.
Let us
(4.7)
=
where, b E ~, and H(O,O) O. Here again we assume that this system is highly nonlinear in the sense that the linear approximation of (4.6) is not stabilizable. The principal interest here is in the cases in which (4.7) contains a "dominant part» of a particularly simple form, such that the stabilization problem for this part yields a locally stabilizing feedback function forJ..4.6).The simplest such case is the following. Let F(x,o) tPi(X), where, tPi is homogeneous of degree i. Let k be the sm;illest positive integer such that tPk 1= O. Now let f := tPk and consider the stabilization problem for,
= L..::
x=
f(x)
+
bu.
= (X I ,X2).
Note that the w - limit set C of the projected dynamics in theorem 4.5 consists of a finite union of periodic orbits, equilibrium points, and some homoclinic and heteroclinic orbits between them. Determination of stability on rays corresponding to the equilibria is trivial and on the cones generated by an equilibrium point can be carried out by using (b) in theorem 4.4. If the restriction of the system to the cones generated by equilibria and periodic orbits is asymptotically stable, then the same holds true for the cones generated by the homo clinic and heteroclinic orbits. Theorem 4.5 is due to Coleman (see [Col)). Kawslci observed in [Ka3] that it remains valid for all integers n. Coleman attributes theorem 4.4 to Forster (see[For)), and it has been discovered independently by Haimo [Hai). These two theorems follow rather easily from the G - equivalence of the solutions of a homogeneous system, i.e. (2.6). Complete proofs can be found in [Ha), [Col) etc. Kawslci and Hermes have shown that these two theorems remain valid (with obvious modifications) in the case of II r - homogeneous systems (see [Ka2),[Ka3) and [He1) for details). In particular Kawslci noted in [Ka2) and [Ka3) that in the case n = 3, if the w - limit set of (4.10) consists of a Jordan curve, if the set of equilibrium points is nonempty and that the system is asymptotically stable on each ray corresponding to the equilibria, then (4.9) is asymptotically stable. Asymptotic stabilization problem for homogeneous systems was first considered by Samardzija [Sam). There was a resurgence of interest on this class after Andreini, Bacciotti and Stefani proved the following theorem in [ABS).
Let us first consider the local asymptotic stabilization problem for the system,
= F(x,u),
0
<
Theorem 4.5 Let n = 3. Let C denote. the w - limit .et of (4.10), and let cn denote. the cone generated by C. Then, (4.9) i. a.ymptotically .table if and only if the re.triction of (4.10) to cn i. a.ymptotically .table.
Unlike in the two dimensional case, the picture is much less clear in the three dimensional stabilization problem. Here we attempt to summarize the key work on this problem.
= F(x,O)
(4.10)
(b) Re.triction of the .y.tem to each of it. one dimen.ional invariant .ub.pace. i. a.ymptotically .table.
Asymptotic Stabilization Problem for Three Dimensional Systems
x
1r(X)(B), BE S"-I
io
=
where, x E ~, u E lR, F is Coo , and F(O,O) = write down the Taylor polynomial in the form,
(4.9)
(a) The .y.tem doe. not have anyone dimen.ional invariant .ub.pacu and
k
=
x
= 2 or 3,
Theorem 4.4 Let n = 2. Then (4 .9) i. a.ymptotically .table if and only if one of the following hold:
where, l ±1, a(xt) is a CUI function and a(O) O. They carried out a detailed analysis of these three cases. Even though stabilizability can be determined in each of these cases by using theorem 4.3, the work reported in (JR] is more constructive. We also note here the interesting work of Boothby and Marino (see [BM1], [BM2], [BM3] etc.) and Crouch and Igheneiwa [Cl]. Boothby and Marino focussed on finding obstructions to Cl stabilizability. In [Cl] the main aim is to use Newton Polygons in order to find suitable coordinate changes which can simplify the stabilization problem, and hence to give some sufficient conditions for stabilizability.
4.1
X(x), x E lR", n
where, X(x) is almost Cl, positively homogeneous vector field of degree p (our standing hypothesis is that p is an integer not less than 1). Let us denote the projected dynamics (see the last paragraph of section 2 for the definition) by,
(4.8)
5
5
Let us consider a system,
= =
j(x,y) u
Concluding Remarks
We have summarized some of the work that has been done on the asymptotic stabilization problem for two and three dimensionalsystems in the recent past. Much of the discussion reflects our personal biases, and scant attention was given to important aspects of the problem such as the work based on Control Lyapunov Functions. The interested readers are referred to [S02) for a detailed review of this aspect. In our opinion, we, the nonlinear community, has acquired a reasonable understanding of the complexity of the.stabilization problem for homogeneous and weighted homogeneous systems. Theorem 4.8 is very encouraging in the sense that it shows that there cannot be "too many other unknown obstructions" to stabilizability. However, finding them ought to be an important task ahead. There is at least one other theoretical question raised by theorem 4.8. Does the genericity statement hold in higher dimensions? . Theorem 3.5 gives a necessary condition for the stabilizability of homogeneous systems by using homogeneoua feedback . Does this remain as an obstruction, if we remove the restriction on homogeneity of the feedback function? We believe that much interesting work waits ahead in answering these questions. This is bound to be a highly fertile area for trying out everyone's favorite tools from algebraic geometry, algebraic topology, optimal control theory etc . We hope that these attempts will be successful .
(4.11)
where, x E Rn-I, 11 E R, u E R, and j is homogeneous of degree p for some p ~ 1. Theorem •• 6 {A BS], {DM4] Suppo.e that i: = j(x,O) i. /uymptotically lid/e. Then the feedback function u = -yP globally a,ymptotically ,tdi/ize, (4.11). This theorem follows from the observation that the closed loop system has the structure of a "block upper triangular" system with stable diagonal blocks. Therefore, it follows that the cloeed loop flystem is locally asymptotically stable (see [Vi), [S03) etc.). Now, the G - equivarianceproperty (2.6) establishes global asymptotic stability. A I1 r - homogeneous version of this theorem was given by Kawski in (Ka3). In the three dimensional case, theorem 4.6 can be strengthened in the following way. Definition •• 2 Let r c 52 be an embedded circle. A point fJ E r is on an upper arc, (respectively on a lower arc) if a particular meridian meets r for the first time (respectively, for the last time) at fJ. Theorem •. 7 {DMSf],{DMS3] Con,ider (4 .11) in the ca8e when n 3. Suppo,e that there e:ri,t. r c 52, an embedded circle, which meet. A_ at a point on a lower arc or an upper arc, and that r doe. not meet A+ o , and the pole8. Then (4.11) i, almo,t Coo .tabi/izd/e by u.ing p08itiveiy homogeneou, feedb4ck.
6
=
Acknowledgements
We wish to thank Professor Clyde Martin, Dr. Gareth Knowles, Dr. Sandy Samelson, Dr. D . Chen for the highly fruitful collaborations with us on this subject in the recent past. We also wish to thank Professors David Gilliam and Christopher Byrnes for helpful discussions related to algebraic curves, which eventually lead us to theorem 4.8 .
Theorem 4.7 can be used to prove that stabilizability is generic in the space of three dimensional homogeneous polunomialsystems of a given odd degree.
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Theorem •. 8 Let p be 4n odd integer. Then, aingle input homogeneou, polynomial Iy,tem, of degree p in three at4tea 4re generically ,t4bi/iz4ble. Moreover, the feedb4ck functiona C4n be found to be almod Coo and homogeneoU8 of degree p .
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In the case when p 2, theorem 4.7 has been used in [DMS2) to give algebraic necessary and sufficient conditions for the asymptotic stabilizability of a generic system. A comprehensive analysis of the quadratic case, including the nongeneric cases, will be given in [DMS2). The same set of arguments has been used in [DMS3) to give necessary and sufficient conditions for the stabilizability of a generic homogeneous three dimensional systems of a given even degree. Of course, the arguments given here, and the conclusions drawn remain valid for weighted homogeneous systems as well. Here the odd degree case corresponds to 11 r - homogeneous systems of order p, when q, r2, r3 are odd integers and p is an even integer, or when rlo r2, r3 are even integers and p is an odd integer, and the even degree case corresponds to the case when either ri, i 1,2,3 and p are all odd or all even. The exposition in this section has followed along lines which best describes our personal tastes, and biases. An entirely different approach to the stabilization problem for three dimensional systems has been taken by Hermes. His objective has been to extend the quadratic regulator theory to the case of weighted homogeneous systems, by considering a weighted homogeneous cost function. He has been successful in giving an alternate proof of Kawski's theorem (theorem 4.1) by using this approach. He has also worked out some three dimensional examples. However, as far as we know , there hasn't been any characterization of a reasonable set of intrinsic conditions under which this approach leads to a solution in the three dimensional case. Interested readers are referred to [He3). [He4) etc. for further details.
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