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Zubov Theorem and Domain of Attraction for Controlled Dynamic Systems
Zubov Theorem and Domain of Attraction for Controlled Dynamic Systems Wei Kang Mathematics Department Naval Postgraduate School Monterey, CA 93943 (408) 656-3337 email: [email protected]
Abstract. To study feedback stabilization and the domain of attraction for nonlinear control systems, the relation between Zubov's theory and Hamilton-Jacobi equation is derived. Based on this, sufficient condition for a control system to have a largest domain of attraction is addressed. It is a lso proved that if a homogeneous control system is stabilizab le locally by state feedback , then it is stabi lizable in any bounded open set. Key Words. Non lin ear Stability, Zubov Theorem , Homogeneous Systems
y(t) , lR+ -t lR n , is said to be in L2 if
1. Introduction
100 The paper addresses problems related to the do[Ilain of attractioll for controlled nonlinear syst.erns. Zubov's t.heorern 011 the boundary of doIlIain of attraction is applied to some control problems. In S 2, the relation between Zubov equation and t.he Hamilton-.lacobi equation for control syst.eIns is addressed. Then the relation is used to find conditions under which a control system has a largest domain of attraction. In § :~ , we focus on the problem of global st.abilization of homogeneo us systems .
= J(:I:) + g(x)u,
J(O)
= O.
:r E Rn
U
E JRm
= u(x)
u(O)
= O.
00 .
x=J(x) , xElR J(O) = O.
n
Suppose x = 0 is an asymptotically stable equilibrium . The domain of attraction D of the system is defined as
D
= {xo E JRnlx(t , xo) -t 0
as t -t +oo}
where x(t , xo) is the trajectory of the ODE with initial condition x(O, xo) = Xo·
(1)
2. Domain of attraction Consider the family of dynamic systems defined by nonlinear ordinary differential equations. The property of global asymptotic stability is a very strong condition , most nonlinear systems do not satisfy. Local stability is a compromise. In control theory, a system can be locally stabilized by feedback using optimal control, H control or some other stabilization techniques. However , how to obtain a larger or "reliable" domain of attraction is an underexplored problem . In fact , finding the domain of attraction for a nonlinear system is, in general , a difficult problem.
A state feedback is a continuous function u
y(t) 112 dt <
Given an ordinary differential equation
In this paper , the control systems are defined by :i:
11
(2)
The notation :I:(t, Xo . u) represents the trajectory of the dosed loop system under the feedback U(l:) with initial condition x(O , Xo, u) = Xo. The norm. 11 Y 11 . of a vector y in JRk is defined by jYT + y~ + ... + y{ Given an open set D in thE' st.at.e space JRn cont,aining :1: = O. a function Oil) defined in D is said to be positive definite if 0(0) = 0 and, given any '"Y > 0 , there is et> 0 such that 6(:1:) > et whenever 11 :1: 11> '"Y and:l: E D. The boundary of a domain D is denoted by aD. The closure of a set. D is denoted by D. A function
QC
In Zubov [1], it is proved that the solution of certain partial differential equation (Zubov equation)
143
gi ves t.he boundary of the domain of attraction for nonlinear ordinary differential equations. The following theorem is a modified version of Zubov's theorem in [1).
(a)
Theorem 2.1 The domain of attraction containing = 0 for the system
(b)
(6)
X
i:
= f(x)
:2: E!R
n
f(O)
=0 W(x) --+
is [) if and only if there exist a continuous posit.ive definite function q;(:2:) defined in an open set cont.aining D and a Cl positive definite function V (;r) defined in D such that \!~(;r)f(x)
= -q,(x)(1
x-+oA
v (;1:)
I.
is positive definite in D and it has a continuous extension in an open set containing D.
Remark. The equation (6) is in the form of the Hamilton-.1acobi equation in optimal control theory. If it has a solution satisfying (7), then from the condition (c) of part (2) we know that any stabilization feedback with domain of attraction D is in the following form,
J:
EA.
In this t.heorern, the choice for d.>(x) is quite arbit.rary. I t. is proved in [l) that if q) (x) satisfies
1N
d;(x(t))dt <
(7)
(8)
=1 f 07'
oD
- V(x))
(4)
<
as x --+
(c) The function
and V (x) satisfies
lim V(:r)
+00
+00
1t(x) =
(5)
-'21 9T (x)W,!(x) + Ul(X)
xE D
such that ljJ(x)- 11 udx) 112 is positive definite in D and it has a continuous extension in an open set containing D.
for trajectories x(t) in a small neighborhood of J: = 0, then the equation (:3) has a solution V(:.!:) satisfying the condition (4). For instance, if the systern is exponentially asymptotically stable, it is sufficient that. d.>(x) is the quadratic function !I J' 11"
Remark. If 1b( x) is continuously defined in an open set containing D, then the conditions (a) and (b) in the theorem are sufficient conditions for D to be the domain of attraction because (8) is automatically satisfied by
For a control system , the domain of attraction depends on the feedback. If a feedback is fixed , then Zubov 's theorem can be applied to the closed loop system to find out the domain of attraction, provided that an explicit or numerical solution of the Zubov equation (:3) can be found. For the open loop system: what kind of domain can be the domain of at.t.raction for the system uncler certain feedback? The following theorem is another version of Zuhov equation for control syst.ems , which gi vps sufficient coudi t.ion" for a clornain t.o be t.he dornain of attract.ion of a system under feedback.
Proof of Theorem 2.2: "(1) :::} (2)". The closed loop system is ;i:
= f(x) + g(x)u(x)
(9)
Let. (]) (:2:) be a posi ti ve defini te function defined in an open set containing D satisfying the condition (5) for trajectories of (9) in a neighborhood of x O. Then, the function
=
Theorem 2.2 Consider a control system in the form of (I) and acontinuousfeedback u(x) defined in !Rn. The following statements are equivalent.
cP(x)+ 11 u(x) 112 satisfies the condition (5) also. By Theorem 2.1, there is a Cl positive definite function V(x) defined in D satisfying the condition (4) such that
( I) The feed back u = u (x) stabilizes the system with the domain of attraction D and 11 1t(;1:(t , ;2: 0, 1t)) 11 is in L2 for Xo near x = O. (:2) There exist. a continuous positive definite funct.iol1l!' (J:) and a Cl positive definite function ~n ;r) . hoth are defined in D: such that
Vx(x) (f(x) +g(x)ulx))
= -(cP(x)+ 1I u(x) 112)(1 144
V(x))
(10)
Define
W(J:)
a sufficient condition is found for systems having largest domain of attraction under feedbacks which have "finite cost" .
= -In( I -
V(J:)) Corollary 2 ..'3 Suppose the positive definite functions W(x) and 1/;(x) satisfy the equation (6) and (7) in a domain D. Suppose 1/;(x) is bounded in every bounded subset of D. Then, given any continuous feedback u(x) defined in lR n satisfying that the cost function
Then W(J:) is positive definite in D and W(x)--+ +x as J: --+ aD. By the equation (10) , we have
Wx(x)(f(x)
+ g(x)u(x)) = -(q)(x)+
11 u(x)
W).
l.e ,
11 u(;r:)
+ ~gT (x) W; (x)
~
+Wx( ;.c:) f(:r:) -
Wu(xo) =
112 +qi(x)
(11)
100 (1/;(x(t, Xo, u))+ 11 u(x(t , Xo, u)) 112) dt
is finite for Xo in a neighborhood of x = 0, the domain of attraction of the dosed loop system under the feedback u(x) must be a subset of D.
11 Wx(:r:)g(x) 112= 0
Define
=
Proof: To prove this result, notice that u -"219T (x)WT (x) is the optimal control for the
Then , 1/)(;r:) is positive definite in D and "Ib(x)- 11 1 1L + "2 9T (x)W;(x) 112 has a continuous extension
problem
in an open set containing D, which is qi(x). Substitut.e lb(x) int.o (11) , we get the equation (6).
x = fIx) + g(x)u
"(:2) => (I)" . Suppose there are functions W(x) and 'l!!(x) and a feedback u = u(;r:) sat.isfying t.he ('ondition (:2) in t.he theorem. Then, (6) imply
+~
11 gT(x)W;(x) 112 +Wx(x:)g(x)u(x)
I.e
Wx(;r:)(f(:z:)
+ g(x)u(x)) I
= -"Ib(x)-
+ 11 ut;!:) + "2 W; (x)gT (x)
+ 11
= 1 - e-W (x) = lp(X)+ 11 u(x)
u(;r:)
+
( 12)
112
W(x(t, xo, u))
( "" 11 u(J:(t,
::; Wu (x(t, xo, iL))
=
r OO
Remark.
u)) 112 dt ::; W(xu)
This means that. u(x(t , xo , u)) is in L 2 .
it
('IjI(x(s, Xo , iL))+ 11 iL(x(s , Xo , iL)) 112)ds.
where x(t) = x(t , xo, iL). By the assumption that Wu (x) is finite near x = 0 and the assumption that "Ib (x) is bounded , we know that W(x(t, Xo , iL)) is bounded as t --+ O. This contradicts the fact that W(x) --+ 00 as x --+ oD. Therefore. the points in oD are not in the domain of attraction of iL. So , the domain of attraction of u is a subset ciD. •
112
TT') "2 Wx (x)g (x) 111
:./: 1) .
=
100 1/;(x(s, x(t), u))+ 11 u(x(s , x(t), u)) 112 ds
From (1:2), assumption (c) and the definition of IV(;!:) and V(;!:), we can show that V(x) and d>(;r:) sat.isfies (:3) and (4) in Theorem :2.1. Therefore. D i", t.he domain of at.traction. Integral both sides of the equat.ion (1:2). we obtain
./0
(13 )
11 u(x) 112
To prove that D is the domain of attraction. we liSP Theorem :2.1. Define V(l:)
(00 (1/)(x)+ 11 u(;!:) 112) dt ./0
because (6) is the Hamilr,on-Jacobi equation. The optimal cost function is W(x) , which equals Wu(x). Given any continuous feedback u(x), to prove the corollary, it is sufficient to show that any point in oD is not a point in the attraction basin of U. Suppose Xo is in oD such that x(t, xo, u) --+ 0 as t --+ 00, then
+ g(x)u(;r:)) =
\!\Ix(J:)(f(x:) -I/i (:!:)
min u(x)
The inequality
yt,'u(xo)
=
100 1/;(x(t,xo,u))+ 11 u(x(t,xo,u)) 112 dt
In general, the shape and size of the attraction basin are different under different feed backs . However , certain control systems have largest domain of attraction. In the following corollary,
(14)
< +00 , in some sense, measures the speed of a trajectory approaching the equilibrium point x = O. There-
145
fore, another interpretation of Corollary 2.3 is that if 1/)(:1;) is bounded in every bounded subset of D, to obtain a larger domain of attraction, we have to reduce the speed of trajectories approaching
where s
;1:=0.
Then ,
EUl1npll:' .
= mf
'!jJ(x)
- 2m g
+
1. Define
= ,\ 2m,-1 2mg '!jJ('\x)
1 x E \D A
Consider thE' syst.em
= y - x 2y - x(l - x 2 i; = -xv + (2 - 2y2)u
:i.;
y2) - 2xyu
(15 )
1
1
= ~'\Wx('\x)f(x) - 4 11
1 ,\s '\Wx('\x)g(x)
112
Define 1
,\2m,-2m g
and
W(1:) =
-~ln( I
_ x2
_
y2 )
then , it. is easy to check that W(x) and 1/)(x) satisfie::; thp condit.ion of Corollary 2.:3 in the unit disc D
= {(:I: ,y) 111 (x;, y) 11<
This means that W (x) is a Lyapunov function for t.he closed loop system under the feedback it
I}.
-T = -"2IgT (x)W x (x).
+00 as
So , th E' largest domain of attraction under feedbacks satisfying the finite cost condi tion (14) is the unit disc.
x -+ 8(
[IJY. I. Zubov , Methods of A. M. Lyapunov and their application , P. Noordhoff LTD , Groningen, The Netherlands , 1964. [2JW. P. Dayawansa, C. F . Martin and G. Knowles , "Asymptotic stabilization of a class of smooth two-dimensional systems ," SIAM .J. Control and Optimization , Yo!. 28 , No. 6, pp. 1:321-1:349, 1990. [:3JM. Kawski , "Homogeneous feedback stabilization ," Proc . New Trends in Systems Theory, C;enova, Haly, July 1990. [4J1\1 . Kawski , "Homogeneous feedback laws in dimension three ," Proc. 28th IEEE Conf. Decision and Control, pp. 1:370-1:375 , 1989. [5JH . G. Hermes, "Homogeneous coordinates and continuous asymptotically stabilizing feedback controls," Proc. Conf. Differential Equations Applications to Stability and Control , Colorado Springs, 1989 .
for any positivE' ,\ < 1. Proof: By Theorem :2.2 , there exist positive definit.e functions W(:z: ) and 1/'( :1:) defined in D such that
Define I
±D for W(x) and 1j;(x).
4 . REFERENCES
= {x l'\x E D}
,\s
To achieve asymptotic stability in ±D
positive definite in
Th eo rem 3. 1 Suppose that the vector fields f(x) and 51 (J:) in ( I ) are homogeneous of degree m f and my. respecti vely Suppose that u( x) stabilizes t.h e system with domain of att.racti on D. Suppose u(:z:(t , Xo, u )) is in L2 for Xo near x = O. Then , the system can be stabilized in the domain
W(:z:) = -W('\x)
±
the feedback is not necessarily unique. It is sufficient to chose a feedback u(x) such that (8) lS
In this section , we apply Theorem 2.2 to homogeneous control systems. A vector field f(x) is said to be homogeneous of degree k if f(tx) = t k f(1:). Now . WE' focus on cont.rol systE'ms (I) in which the VE'ctor fields f(:I:) and 9(X) are both homogeneous. This kind of systems have been studied by many aut.hors in the literature (for instance , see [2J , [:3], [4J and [5]). We will prove that , if a homogeneous control system can be locally stabilized by a continuolls feedback which is in L2 along every trajectory near x = 0, then any bounded open subset of !Rn containing :z: = 0 can be included in the domain of att.raction under a suitable feedback.
-
±
•
Remark.
1
-
From the fact that W(x) -+
D) we know that D is an invariant subset and it is contained in the domain of attraction of U.
:3. Homogeneous systems
>.D
1/;('\) X
1 x E -D ,\
146
Abstract. To study feedback stabilization and the domain of attraction for nonlinear control systems, the relation between Zubov's theory and Hamilton-Jacobi equation is derived. Based on this, sufficient condition for a control system to have a largest domain of attraction is addressed. It is a lso proved that if a homogeneous control system is stabilizab le locally by state feedback , then it is stabi lizable in any bounded open set. Key Words. Non lin ear Stability, Zubov Theorem , Homogeneous Systems
y(t) , lR+ -t lR n , is said to be in L2 if
1. Introduction
100 The paper addresses problems related to the do[Ilain of attractioll for controlled nonlinear syst.erns. Zubov's t.heorern 011 the boundary of doIlIain of attraction is applied to some control problems. In S 2, the relation between Zubov equation and t.he Hamilton-.lacobi equation for control syst.eIns is addressed. Then the relation is used to find conditions under which a control system has a largest domain of attraction. In § :~ , we focus on the problem of global st.abilization of homogeneo us systems .
= J(:I:) + g(x)u,
J(O)
= O.
:r E Rn
U
E JRm
= u(x)
u(O)
= O.
00 .
x=J(x) , xElR J(O) = O.
n
Suppose x = 0 is an asymptotically stable equilibrium . The domain of attraction D of the system is defined as
D
= {xo E JRnlx(t , xo) -t 0
as t -t +oo}
where x(t , xo) is the trajectory of the ODE with initial condition x(O, xo) = Xo·
(1)
2. Domain of attraction Consider the family of dynamic systems defined by nonlinear ordinary differential equations. The property of global asymptotic stability is a very strong condition , most nonlinear systems do not satisfy. Local stability is a compromise. In control theory, a system can be locally stabilized by feedback using optimal control, H control or some other stabilization techniques. However , how to obtain a larger or "reliable" domain of attraction is an underexplored problem . In fact , finding the domain of attraction for a nonlinear system is, in general , a difficult problem.
A state feedback is a continuous function u
y(t) 112 dt <
Given an ordinary differential equation
In this paper , the control systems are defined by :i:
11
(2)
The notation :I:(t, Xo . u) represents the trajectory of the dosed loop system under the feedback U(l:) with initial condition x(O , Xo, u) = Xo. The norm. 11 Y 11 . of a vector y in JRk is defined by jYT + y~ + ... + y{ Given an open set D in thE' st.at.e space JRn cont,aining :1: = O. a function Oil) defined in D is said to be positive definite if 0(0) = 0 and, given any '"Y > 0 , there is et> 0 such that 6(:1:) > et whenever 11 :1: 11> '"Y and:l: E D. The boundary of a domain D is denoted by aD. The closure of a set. D is denoted by D. A function
QC
In Zubov [1], it is proved that the solution of certain partial differential equation (Zubov equation)
143
gi ves t.he boundary of the domain of attraction for nonlinear ordinary differential equations. The following theorem is a modified version of Zubov's theorem in [1).
(a)
Theorem 2.1 The domain of attraction containing = 0 for the system
(b)
(6)
X
i:
= f(x)
:2: E!R
n
f(O)
=0 W(x) --+
is [) if and only if there exist a continuous posit.ive definite function q;(:2:) defined in an open set cont.aining D and a Cl positive definite function V (;r) defined in D such that \!~(;r)f(x)
= -q,(x)(1
x-+oA
v (;1:)
I.
is positive definite in D and it has a continuous extension in an open set containing D.
Remark. The equation (6) is in the form of the Hamilton-.1acobi equation in optimal control theory. If it has a solution satisfying (7), then from the condition (c) of part (2) we know that any stabilization feedback with domain of attraction D is in the following form,
J:
EA.
In this t.heorern, the choice for d.>(x) is quite arbit.rary. I t. is proved in [l) that if q) (x) satisfies
1N
d;(x(t))dt <
(7)
(8)
=1 f 07'
oD
- V(x))
(4)
<
as x --+
(c) The function
and V (x) satisfies
lim V(:r)
+00
+00
1t(x) =
(5)
-'21 9T (x)W,!(x) + Ul(X)
xE D
such that ljJ(x)- 11 udx) 112 is positive definite in D and it has a continuous extension in an open set containing D.
for trajectories x(t) in a small neighborhood of J: = 0, then the equation (:3) has a solution V(:.!:) satisfying the condition (4). For instance, if the systern is exponentially asymptotically stable, it is sufficient that. d.>(x) is the quadratic function !I J' 11"
Remark. If 1b( x) is continuously defined in an open set containing D, then the conditions (a) and (b) in the theorem are sufficient conditions for D to be the domain of attraction because (8) is automatically satisfied by
For a control system , the domain of attraction depends on the feedback. If a feedback is fixed , then Zubov 's theorem can be applied to the closed loop system to find out the domain of attraction, provided that an explicit or numerical solution of the Zubov equation (:3) can be found. For the open loop system: what kind of domain can be the domain of at.t.raction for the system uncler certain feedback? The following theorem is another version of Zuhov equation for control syst.ems , which gi vps sufficient coudi t.ion" for a clornain t.o be t.he dornain of attract.ion of a system under feedback.
Proof of Theorem 2.2: "(1) :::} (2)". The closed loop system is ;i:
= f(x) + g(x)u(x)
(9)
Let. (]) (:2:) be a posi ti ve defini te function defined in an open set containing D satisfying the condition (5) for trajectories of (9) in a neighborhood of x O. Then, the function
=
Theorem 2.2 Consider a control system in the form of (I) and acontinuousfeedback u(x) defined in !Rn. The following statements are equivalent.
cP(x)+ 11 u(x) 112 satisfies the condition (5) also. By Theorem 2.1, there is a Cl positive definite function V(x) defined in D satisfying the condition (4) such that
( I) The feed back u = u (x) stabilizes the system with the domain of attraction D and 11 1t(;1:(t , ;2: 0, 1t)) 11 is in L2 for Xo near x = O. (:2) There exist. a continuous positive definite funct.iol1l!' (J:) and a Cl positive definite function ~n ;r) . hoth are defined in D: such that
Vx(x) (f(x) +g(x)ulx))
= -(cP(x)+ 1I u(x) 112)(1 144
V(x))
(10)
Define
W(J:)
a sufficient condition is found for systems having largest domain of attraction under feedbacks which have "finite cost" .
= -In( I -
V(J:)) Corollary 2 ..'3 Suppose the positive definite functions W(x) and 1/;(x) satisfy the equation (6) and (7) in a domain D. Suppose 1/;(x) is bounded in every bounded subset of D. Then, given any continuous feedback u(x) defined in lR n satisfying that the cost function
Then W(J:) is positive definite in D and W(x)--+ +x as J: --+ aD. By the equation (10) , we have
Wx(x)(f(x)
+ g(x)u(x)) = -(q)(x)+
11 u(x)
W).
l.e ,
11 u(;r:)
+ ~gT (x) W; (x)
~
+Wx( ;.c:) f(:r:) -
Wu(xo) =
112 +qi(x)
(11)
100 (1/;(x(t, Xo, u))+ 11 u(x(t , Xo, u)) 112) dt
is finite for Xo in a neighborhood of x = 0, the domain of attraction of the dosed loop system under the feedback u(x) must be a subset of D.
11 Wx(:r:)g(x) 112= 0
Define
=
Proof: To prove this result, notice that u -"219T (x)WT (x) is the optimal control for the
Then , 1/)(;r:) is positive definite in D and "Ib(x)- 11 1 1L + "2 9T (x)W;(x) 112 has a continuous extension
problem
in an open set containing D, which is qi(x). Substitut.e lb(x) int.o (11) , we get the equation (6).
x = fIx) + g(x)u
"(:2) => (I)" . Suppose there are functions W(x) and 'l!!(x) and a feedback u = u(;r:) sat.isfying t.he ('ondition (:2) in t.he theorem. Then, (6) imply
+~
11 gT(x)W;(x) 112 +Wx(x:)g(x)u(x)
I.e
Wx(;r:)(f(:z:)
+ g(x)u(x)) I
= -"Ib(x)-
+ 11 ut;!:) + "2 W; (x)gT (x)
+ 11
= 1 - e-W (x) = lp(X)+ 11 u(x)
u(;r:)
+
( 12)
112
W(x(t, xo, u))
( "" 11 u(J:(t,
::; Wu (x(t, xo, iL))
=
r OO
Remark.
u)) 112 dt ::; W(xu)
This means that. u(x(t , xo , u)) is in L 2 .
it
('IjI(x(s, Xo , iL))+ 11 iL(x(s , Xo , iL)) 112)ds.
where x(t) = x(t , xo, iL). By the assumption that Wu (x) is finite near x = 0 and the assumption that "Ib (x) is bounded , we know that W(x(t, Xo , iL)) is bounded as t --+ O. This contradicts the fact that W(x) --+ 00 as x --+ oD. Therefore. the points in oD are not in the domain of attraction of iL. So , the domain of attraction of u is a subset ciD. •
112
TT') "2 Wx (x)g (x) 111
:./: 1) .
=
100 1/;(x(s, x(t), u))+ 11 u(x(s , x(t), u)) 112 ds
From (1:2), assumption (c) and the definition of IV(;!:) and V(;!:), we can show that V(x) and d>(;r:) sat.isfies (:3) and (4) in Theorem :2.1. Therefore. D i", t.he domain of at.traction. Integral both sides of the equat.ion (1:2). we obtain
./0
(13 )
11 u(x) 112
To prove that D is the domain of attraction. we liSP Theorem :2.1. Define V(l:)
(00 (1/)(x)+ 11 u(;!:) 112) dt ./0
because (6) is the Hamilr,on-Jacobi equation. The optimal cost function is W(x) , which equals Wu(x). Given any continuous feedback u(x), to prove the corollary, it is sufficient to show that any point in oD is not a point in the attraction basin of U. Suppose Xo is in oD such that x(t, xo, u) --+ 0 as t --+ 00, then
+ g(x)u(;r:)) =
\!\Ix(J:)(f(x:) -I/i (:!:)
min u(x)
The inequality
yt,'u(xo)
=
100 1/;(x(t,xo,u))+ 11 u(x(t,xo,u)) 112 dt
In general, the shape and size of the attraction basin are different under different feed backs . However , certain control systems have largest domain of attraction. In the following corollary,
(14)
< +00 , in some sense, measures the speed of a trajectory approaching the equilibrium point x = O. There-
145
fore, another interpretation of Corollary 2.3 is that if 1/)(:1;) is bounded in every bounded subset of D, to obtain a larger domain of attraction, we have to reduce the speed of trajectories approaching
where s
;1:=0.
Then ,
EUl1npll:' .
= mf
'!jJ(x)
- 2m g
+
1. Define
= ,\ 2m,-1 2mg '!jJ('\x)
1 x E \D A
Consider thE' syst.em
= y - x 2y - x(l - x 2 i; = -xv + (2 - 2y2)u
:i.;
y2) - 2xyu
(15 )
1
1
= ~'\Wx('\x)f(x) - 4 11
1 ,\s '\Wx('\x)g(x)
112
Define 1
,\2m,-2m g
and
W(1:) =
-~ln( I
_ x2
_
y2 )
then , it. is easy to check that W(x) and 1/)(x) satisfie::; thp condit.ion of Corollary 2.:3 in the unit disc D
= {(:I: ,y) 111 (x;, y) 11<
This means that W (x) is a Lyapunov function for t.he closed loop system under the feedback it
I}.
-T = -"2IgT (x)W x (x).
+00 as
So , th E' largest domain of attraction under feedbacks satisfying the finite cost condi tion (14) is the unit disc.
x -+ 8(
[IJY. I. Zubov , Methods of A. M. Lyapunov and their application , P. Noordhoff LTD , Groningen, The Netherlands , 1964. [2JW. P. Dayawansa, C. F . Martin and G. Knowles , "Asymptotic stabilization of a class of smooth two-dimensional systems ," SIAM .J. Control and Optimization , Yo!. 28 , No. 6, pp. 1:321-1:349, 1990. [:3JM. Kawski , "Homogeneous feedback stabilization ," Proc . New Trends in Systems Theory, C;enova, Haly, July 1990. [4J1\1 . Kawski , "Homogeneous feedback laws in dimension three ," Proc. 28th IEEE Conf. Decision and Control, pp. 1:370-1:375 , 1989. [5JH . G. Hermes, "Homogeneous coordinates and continuous asymptotically stabilizing feedback controls," Proc. Conf. Differential Equations Applications to Stability and Control , Colorado Springs, 1989 .
for any positivE' ,\ < 1. Proof: By Theorem :2.2 , there exist positive definit.e functions W(:z: ) and 1/'( :1:) defined in D such that
Define I
±D for W(x) and 1j;(x).
4 . REFERENCES
= {x l'\x E D}
,\s
To achieve asymptotic stability in ±D
positive definite in
Th eo rem 3. 1 Suppose that the vector fields f(x) and 51 (J:) in ( I ) are homogeneous of degree m f and my. respecti vely Suppose that u( x) stabilizes t.h e system with domain of att.racti on D. Suppose u(:z:(t , Xo, u )) is in L2 for Xo near x = O. Then , the system can be stabilized in the domain
W(:z:) = -W('\x)
±
the feedback is not necessarily unique. It is sufficient to chose a feedback u(x) such that (8) lS
In this section , we apply Theorem 2.2 to homogeneous control systems. A vector field f(x) is said to be homogeneous of degree k if f(tx) = t k f(1:). Now . WE' focus on cont.rol systE'ms (I) in which the VE'ctor fields f(:I:) and 9(X) are both homogeneous. This kind of systems have been studied by many aut.hors in the literature (for instance , see [2J , [:3], [4J and [5]). We will prove that , if a homogeneous control system can be locally stabilized by a continuolls feedback which is in L2 along every trajectory near x = 0, then any bounded open subset of !Rn containing :z: = 0 can be included in the domain of att.raction under a suitable feedback.
-
±
•
Remark.
1
-
From the fact that W(x) -+
D) we know that D is an invariant subset and it is contained in the domain of attraction of U.
:3. Homogeneous systems
>.D
1/;('\) X
1 x E -D ,\
146