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Feedback Stabilization of General Nonlinear Control Systems*
Copyright © IFAC Non1inear Control Systems Design, Tahoe City, California, USA, 1995
Feedback Stabilization of General Nonlinear Control Systems * Wei Lin Department of Systems Science and Mathematics Washington University, St. Louis, MO 63130, U.S.A.
*Research supported in part by grants from the AFOSR and NSF.
Abstract. This paper develops sufficient conditions for a general nonlinear control system El : x = f(x ,u) to be locally (resp. globally) asymptotically stabilizable via smooth state feedback . In particular, it is shown that as in the case of affine systems, this is possible if the wUorced dynamic system of El is Lyapunov stable and appropriate controllability-like rank conditions are satisfied. Our results incorporate a series of well-known stabilization theorems proposed in the literature for affine control systems and extend them to non-affine nonlinear control systems.
Key Words. Non-affine control systems, smooth state feedback, asymptotic stabilization, passive systems, zerostate detectability.
1. Introduction
In the pioneering work [1) , lurdjecvic and Quinn presented a sufficient condition under which an affine nonlinear control system (2) whose unforced dynamics are linear (i.e. , f(x) = Ax) , and the state transition matrix is unitary can be globally asymptotically stabilizable by smooth state feedback. Since then , various Jurdjevic-Quinn type sufficient conditions have been developed during the last fifteen years. Under some restrictive assumptions on the form of Lyapunov function associated to the un forced dynamics , Kalouptsidis and Tsinias [3) proposed a sufficient condition for global stabilization of affine nonlinear control systems via smooth state feedback. Their work extends the results of lurdjevic and Quinn [1) to systems with Lyapunov stable unforced dynamics. In [2) , Gauthier and Bonard assumed that there exists a positive definite, readily unbounded smooth function V(x) such that Lf V(x) == O. Then they proved that if
Consider a general nonlinear control system of the form
I: l
:
i
= f(x,u)
(1)
where x E lRn is the state, u E R m is the control input , f : Rn X lRm -+ lRn is a smooth (i.e. of class of COO) mapping, with f(O , O) = O. Throughout this paper we make the following assumption: -+ lR , with V(O) = 0, which is positive definite and defined on some neighborhood U (resp . lR n) of x = 0 such that the unforced
(HI) There exists a function V : lRn
dynamic system of (1) , namely i Lyapunov stable, i.e., LfoV(x) :S 0
f(x , 0) ~ fo(x) is "Ix E U (resp . lR n ).
=
The purpose of this paper is to present sufficient conditions for such a class of nonlinear control systems of the form (1) to be locally (resp. globally) asymptotically stabilizable, by means of a smooth state feedback control law u = u( x) with u(O) == o.
dim jj
~ dim span [ad~gi: 0 :S k :S n - 1, 1 :S i :S "Ix E lRn - {O},
=n
m}
the affine system (2) is globally asymptotically stabilizable by a smooth state feedback control law
For affine control systems of the form
u == - (LgV(x)f·
(3)
m
i;
= f(x) + g(x)u = f(x) + L
gi(X)Ui
Clearly, this is reminiscent of the well-known rank condition for accessibility of the system (2) and can be viewed as a kind of controllability-like rank condition. Lee and Arapostathis [4) studied the single-input system (2) . They observed that the conditions proposed in [2) [3) can be further weakened, and derived a general sufficient condition which includes the results obtained in [1)-[3) . More specifically, it was shown in [4) that the smooth state feedback control law (3) globally asymptotically stabilizes the affine system (2) if there is a Coo V(x) , which is positive definite and proper , satisfying (HI) and
(2)
i=l
with the property (HI) , stabilization via smooth state feedback has been investigated by many researchers. Among several contributions to the subject , we wish to point out those which are closely related to our work to be reported in this paper; see for instances, the following papers as well as references therein: Jurdjevic and Quinn [1], Gauthier and Bornard [2], Kalouptsidis and Tsinias [3), Lee and Arapostathis [4], Byrnes, Isidori and Willems [5) . For more information and additional details on nonlinear feedback stabilization, the reader is referred to a survey due to Sontag
5 ~ {x
[6).
403
E lR n
:
L} LT V (x) == 0 "IT E
D, i = 0, 1, .. . } ==
{O}.
The later is apparently much weaker than those presented in [1]-[3]. By using a synthesis of the techniques and concepts drawn from passive systems theory [7]-[9] and from geometric nonlinear control theory [10] [11] , Byrnes, Isidori and Willems [5] developed a framework for studying stabilizability of minimum-phase nonlinear systems, and illustrated how the stabilization results proposed in [1]-[3] and some generalization thereof [4] , can all be re-derived from a basic stability property of affine passive systems.
From the notations introduced in (4) and the identity
f(x ,U)-fa(X)=(taafl 0' a=9u
la
d(})U~9(X ' U)U
(8)
with g(x,u) : JR.n x JR.m ..... JR.n x JR.m being a smooth map, we conclude that a smooth nonlinear control system (1) can always be represented as m
X = fa(x)
Since passive systems theory has been developed by Willems [7] for more general classes of nonlinear systems which are not necessarily affine in u , it is very natural to expect the idea and method developed in Byrnes, Isidori and Willem [5] can be generalized to non-affine nonlinear control systems. Indeed , this is precisely the point of view to be pursued in this paper. Starting from the next section , we show how the theory of passive systems developed in [5] can be extended to general input/output nonlinear control systems, and how the problem of asymptotic stabilization of the nonlinear system (1) can be solved by means of passive systems theory which we develop here , and the technique of feedback equivalence.
+ g(x , u)u = fa(x ) + L
gi(X , U)Ui .
(9)
i=l
Using (4) and proceeding essentially the same argument , (9) can be further decomposed as m
X = fa(x)
+ go(x)u + L
Ui (Ri(X , u)u)
(10)
i=l
with Ri(X , U) : IRn 1 S; i S; m .
X
JR.m ..... lR nxm, being a smooth map for
Since a smooth nonlinear control system of the form (1) is equivalent to either system (9) or (10) , we henceforth will used them interchangeably when referring to a smooth nonlinear control plant .
This paper is organized as follows: in Section 2, we introduce necessary notations and present the main results of this paper. Section 3 uses three examples to illustrate the stabilization schemes of non-affine systems proposed in Section 2. In Section 4 , we first extend passive systems theory to input-output nonlinear systems with general structure. We then combine the theory thus developed with the technique of feedback equivalence to prove the stabilizability results stated in Section 2. Section 5 briefly discusses how passive systems theory in continuous-time can be generalized to its discrete-time counterpart.
Now we are ready to state our main results in this paper. The first result is given by the following statement whose proof will be presented in Section 4.
Theorem 2.1 Suppose a non linear control system Bl of the form (1) satisfies Assumption (HI). If Snn = {O} , then Bl is locally asymptotically stabilizable at the equilibrium x = 0 by smooth state feedback. In particular , a local smooth state feedback control law (11)
u=O'(x) , O'(O)=O
2. Notations and Main Results
can be solved uniquely from the equation We present in this section sufficient conditions for the existence of smooth state feedback control laws which locally (resp . globally) asymptotically stabilize a general nonlinear control system of the form (1 ), with Lyapunov stable free dynamics.
u+(Lg(z.u)V(x))
g~(x)
= f(x , O) E JR.n
= g.(x , 0) = ~(x, 0) E JR. n,
1 S; i S; m, ga(x) = *(x , O) = [g~(x), .. . ,g~(x)l E JR.n xm
Remark 2.2 In the case of affine nonlinear systems , the equation (12) reduces to u = O'(x) = - (Lgo(z )V(x ){ , and therefore, (12) always allows a global solution on JR. n. In this case, Theorem 2.1 includes the global stabilization results due to Byrnes, Isidori and Willems [5] (see Section 3, especially, Remark 3.8) and Lee-Arapostathis [4] (see Theorem 1) , etc ., as particular cases .
}
(4)
With the vector fields fa, g~ , . . . ,g~ we introduce the distribution
D= span {ad~og?: OS;kS;n-l , lS;iS;m} ,
(5)
The second main theorem is a stabilization result for a nonaffine nonlinear control system (10) in which Ri(X , u) , 1 S; i S; rn , are independent of u, i.e., R i( x , u) == Ri (X). For convenience, we rewrite (10) as
and two sets nand 5 associated with D , which are defined by
n = {x 5 = {x
= O,k = 1, . . . , r} E U ~ JR.n: L:oLTV(X) = O, Ii. E D, E U ~ JR.n : L:oV(x)
k=O , I , . . . , r-l}
(12)
where g(x , u) is determined by the identity (8). If V is proper , n n 5 = {O} lix E JR.n and equation (12) has a solution u = 0'( x) which is well-defined on IRn , then BJ is globally asymptotically stabilized by (11 ).
To begin with, we first introduce some notations. Throughou t this paper , let fa (x) and g? (x) denote the smooth vector fields
fa(x )
T=u+ (av -axg(x , u ))T =0 ,
m
(6 ) B2
:
X = fa(x)
+ ga(x)u + L i=l
(7 )
404
Ui (Ri(X)U)
(13)
By comparison with the affine system (2), the system (13) is indeed a "second affine approximation"-with a minor abuse of terminology-of the general nonlinear system (1).
Let V(x) = ~(xi + x~). A routine calculation shows that Llo V(x) = 0, so n = IR2. In addition
Theorem 2.3 Consider a nonlinear control system 2:2 of the form (13) with the property (HI). Suppose Snn = {a} . Then the equilibrium x = 0 of 2:2 is locally asymptotically stabilizable by the smooth state feedback control law
0= LUO,gol V(x) = LloLgo V(x) = x~ - 2xix2
0= LgoV(x)
LR, (r) V(x) := Imxm
+ LR=(r) V(x)
sume that
= XIX~
0= LloL[fo ,golV(x) = -7XIX~ + 2xi . Then, we conclude immediately from (7) that S = {O} . Hence n n S = {O}. By Theorem 2.1, there exists a smooth state feedback control law such that 2:" is asymptotically stabilizable. In particular, a typical control law is given by (11) and it can be solved from the equation (12) , which in the present case is
r
Moreover , as-
[
X2][0 XIX2f
and
(14)
where 6.(x)
= [Xl
Thus
(H2) 6.(x) is invertible 'Ix E IRn . u=a(x)= If V is proper and S n n = {O} 'Ix E IRn , then the state feedback control law (14) globally asymptotically stabilizes the equilibrium x 0 of the system (13).
This theorem will be proven in Section 4. It turns out that in the single-input case (m = 1), the previous result yields the following interesting consequence.
(15 )
1
Remark. It is interesting to note that the same arguments show that the system
B,, :
(16)
Xl X2
= X2 + x~u3 - X2U 4 = -Xl + XIX2U - XIX2U 3 + XlU 4
is also globally asymptotically stabilized by u = -XIX~ . In fact , the equation (12) in the present case is indeed globally solvable because , 0 = u + ~~ g(x , u) == u + ~~ go(x) for V(x) = ~ (xi + d).
n
If 1 + Lp(r)V(x) f. 0 'Ix E IR , V is proper and S n = {O} 'Ix E IR n , then the system (15) is globally asymptotically stabilized by (16) . n
.
Wltha(O)=O .
La Salle's Invariance Principle [15] .
satisfying Assumption (HI). Suppose S n n = {O} . Then x = 0 of the system (15) can always be locally asymptotically stabilizable by smooth state feedback. A possible choice is
= - 1 + L per )V()X Lgo(r )V(x) .
2
2XIX 2
V = -(XIX~)4 - (XIX~)2 ~ 0 and V = 0 '* 0 = LgO V(x) . This yields n n S = {O} again . Thus, the claim follows from
Corollary 2.4 Consider a single-input nonlinear system
u
+ VI - 4(XIX~)2
Clearly, a(x) is Coo around the origin in IR2 . More interesting, if one replaces the smooth state feedback control law above by u = -LgOV(x) = -XIX~, it is easy to show that this Coo feedback control law globally asymptotically stabilizes the system 2:". In fact , along the trajectory of the closed-loop system ,
=
x = fo(x) + go(x)u + p(x)u 2
-1
Example 3.2 Consider a continuous-time nonlinear control system with two inputs
Remark 2.5 Locally speaking, Assumption (H2) always holds, simply because ~~ (0) = 0 and hence 6.(x) Ir=o= Imxm. In the case of affine nonlinear systems, (H2) is globally satisfied since R;(x) == 0 (resp. p(x) == 0) for 1 ~ i ~ m . Therefore, Theorem 2.3 and Corollary 2.4 recover the stabilization schemes proposed in [4] [5]. This , in turn , implies as illustrated in Remark 3.8 of [5] that all the stabilization results developed by Jurdjevic and Quinn [1] , Gauthier and Bornard [2], Kalouptsidis and Tsinias [3] can be viewed as consequences of Theorem 2.3 or Corollary 2.4.
+ x~) + t'n(xi + I)UI + COS(X2 + X3)UI U2 + xiu~ = Xl - 2x~ + X2U2 + sin Xl UI U2 = X2 + X3U2 + x2ui
Xl = -2X2(X~ 2:b :
X2 X3
We choose V(x)
=~
ward to verify that
LR,( r )V(x)
(xi + (x~ + X~)2).
n n S = {O}.
It is straightfor-
Moreover , note that
= [0 , Xl COS(X 3 + X2) + 2X2(X~ + x~) sin xd ~ [0 , *] and LR2(r)V(x)
3. Examples
2x~ui
= [0,
xn
Then , Assumption (H2) holds because the matrix 6.(x )
Before giving a rigorous proof of the stabilizability results introduced in the last section, we first use them to solve the stabilization problems of non-affine nonlinear control systems.
1 0 4 ] is invertible 'Ix E IR3. By Theorem 2.3, 2:b is [ * 1 + Xl globally asymptotically stabilized by the smooth state feedback control law
Example 3.1 Consider a planar nonlinear control system
405
Let us end this section with a simple example which demonstrates the application of Corollary 2.4.
Definition 4.1 An input/output nonlinear control system I;3 of the form (17) is said to be passive if there exists a CO nonnegative function V : lR n -- lR with V(O) = 0, called the storage function , such that
Example 3.3 Consider a single-input nonlinear system I;c :
where x
We note that this system has some interesting properties similar to Examples 3.1 and 3.2. Firstly, the first approximation of I;c is determined by the pair (A , B) ([
~ ~], [ ~
=
If V is Cr(r
Hxi + x~).
Llo Vex)
= -xi
Definition 4.2 An input/output nonlinear system I;3 of the form (17) is locally zero-state detectable if there is a neighborhood N of x = 0 such that Vxo = x E N
h( lim
$ 0 and Llo Vex)
X2
'-00
= 0 ::} Xl = O.
= O.
A fundamental property of an affine passive system (2) with y = h( x) is characterized by the well-known KYP Lemma [5]. In the case of non-affine systems, the KYP Lemma does not exist . However, necessary conditions for an input/output nonlinear system (17) to be passive can still be obtained. These conditions, especially relation (ii) given below, will be very useful to derive a checkable zero-state detectability criterion for passive systems of the form (17) .
= - 1 + Lp(x)V(x) L gO Vex) = _ l+xiex2' 2 4. The Proofs of Main Theorems
We now present the proofs of stabilizability results stated in Section 2. The underlying philosophy of the proofs is to use the concepts of passivity and feedback equivalence, which have been developed in [7]-[9] and [5]. The reason why these two concepts are instrument in designing a global stabilizing state feedback control laws for a class of affine nonlinear systems has been interpreted in [5]. In this section , we first explore some intrinsic properties such as zero-state detectability and stabilizability of smooth non-affine passive systems of the form
Y
= f(x , u) ,
= h(x,u),
f(O , 0) = 0 } h(O , O) = 0
= o.
If N = lRn, the system I;3 is zero-state detectable. A system I;3 is locally (resp. globally) zero-state observable if there is a neighborhood U of x = 0 such that Vxo = x E U (resp. lRn) , h(
x3
1
3·
2: 1), the passive inequality is equivalent to
loss less.
Thus n n S = {O} . By Corollary 2.4, I;c is globally asymptotically stabilized by the smooth state feedback control law
i;
f(x , u) starting
Moreover, if (18) becomes a strict equality, I;3 is said to be
Obviously
0= LgoV(x) = x~::}
I; .
=
(18)
On the other hand,
u
is a solution of i;
]), which is not controllable. Secondly,
the center manifold theory cannot be applied to solve the stabilization problem of I;c because two eigenvalues of A are located on the imaginary axis. However , it can be shown that I;c is globally asymptotically stabilizable via smooth state feedback. This claim follows from Corollary 2.4, as illustrated in the following . Let Vex) =
=
x(O)=xo.
Proposition 4.3 Let n l ~ {x E lR n : Li oVex) = O}. Necessary conditions for I;3 to be passive with a C 2 storage function V are that
(i) L lo Vex) $ 0 (ii) LgO Vex) = hT(x , 0)
'Ix E
~ [) 2 fi [)V (iii) ~ ou 2 (x , 0)· OXi $
(17)
n1
[)h T
[)h
[)u (x, 0)
+ [)u (x , 0)
'Ix E
n1
i=l
Where h : lRn x lR m __ lR m is a smooth function and y E lR m is the system output. The essential difficulty in such a development is the lack of the Kalman- Yacubovitch-Popov Lemma (KYP Lemma for short) for a non-affine passive system (17) , whereas in the case of affine passive systems the KYP Lemma does exist and ineed plays an important role in the analysis and design [5]. In what follows , we show how some basic properties of passive systems of the form (17) including zero-state detectability and stabilizability via output feedback can be carried out in the absence of the gyp Lemma. The approach given below in studying passive systems (17) is very much in the spirit of [12]-[14].
where f,ex , u) is the i-th component of the vector function
We first review two useful concepts related to passive systems .
In other words , F(x , u) arrives its maximum at u set n 1 .
f(x, u). Proof. Consider an auxiliary function F : lR n x lR m -- lR defined by
F(x, u)
[)V = Eh f(x, u) -
T
h (x , u)u
Since I;3 is passive , it is clear that F(x , u) $ 0 Vu E lRm . Hence (i) follows by setting u = O. For all x in n1 , F( x , 0) LloV(x) == O. Thus
=
F(x,u) $ F(x , O)
406
=0
'Ix E
n1
and Vu E lRm.
= 0 on the
set. By La Salle's book [lS}, r O is nonempty, compact and invariant. Since V(x(t)) is nonnegative and nonincreasing along the trajectory x(t) = 4>(t, XO; 0), lime_oo V(x(t)) exists and lime_oo V(x(t)) = c ~ O. By continuity of V, V(x) = c for every point x = lime_oo 4>(t, xo; 0) E rO.
This, in view of (4), implies that "Ix E rh
o= o
~~ (x, 0) = L go V ( x) - h T ( x , 0)
~ fu?(x,o) = 8(~!fr lu=o - (~:(x, o) + 8:: (x,o)) _""n -
8'/,( X, O)8V 8Xi
L..,i=l a;;:t"
(8h( ) 8h T ( x , o) ) Ft;' x,o + 8u
-
Let x E r O ~ 0 and 4>(t , x; 0) be the corresponding trajectory of i; = fo(x) . By invariance, 4>(t, x; 0) E r O ~ O. From Step I, it follows that V (4)(t, x; 0)) = V(x) = c "It ~ O. Hence
Therefore, (ii) and (iii) follow immediately. As in the case of affine systems, a non-affine passive system (17) also enjoys a nice stabilizability property by output feedback. The following result basically describes such a stability property.
V (4)(t, x; 0)) = LID V
Theorem 4.4 Consider a passive system 2::3 of the form (17) with a Cl storage function V , which is positive definite. Let s : IR m - JRm be any first sector/third sector function, i.e. , yT s(y) > 0 Vy i= 0 and s(o) = O. Then the equilibrium x = 0 of 2::3 is locally asymptotically stabilizable by
u
= -s(y)
x E n.
Step Ill: tectability.
We show that
=0
"It
2: O.
~ n.
0
Thus
=
(19)
0n
S
=
In addition, since 4>(t,x;O) Proposition 4.3 (ii) that
The proof of this theorem, which is similar to the proof of Theorem 3.2 given in [5], follows immediately from passive inequality (18) with u = -s(y), zero-state detectability and LaSalle's Invariance Principle.
= {O}
=> zero-state de-
=
=
x(t) E n, it follows from
LgOV (x(t)) = hT (x(t) , 0) = 0
"It
2:
From (20) and (21), we deduce that for
Now a natural question arises immediately: When is a passive system (17) zero-state detectable? In the following , we
(21)
O. T
= [fa , gal
ED
Using the inductive argument , it is not difficult to show that "It 2: 0
will give a satisfactory answer to this question and show how zero-state detectability of the passive system (17) can be tested by means of the Lie bracket of vector fields fa (x) and g?(x), which characterize the affine part of the inputstate differential equation .
L~oL7.v (x(t))
=0
"IT E D and 0 ::; k ::; r - 1.
(22)
In view of the definition of S (see (7)) , we conclude that x E S. Hence x E On. If 0 n S = {o},- clearly x = O. Thus V(x) = 0, which implies lime_oo 4>(t , Xo; 0) = O. By Definition 4.2, 2::3 is zero-state detectable.
To begin with, set uxoEUC;Rn
This means
(4)(t , x; 0))
Let x(t) = 4>(t , x; 0) be a trajectory of the unforced system i; fo(x) yielding y(t) h(x(t),O) 0 "It 2: o. From Steps I and Il, "Ix E ra, 4>(t , x; 0) E r O ~ 0 ~ nand
provided that 2::3 is locally zero-state detectable. If V is proper and 2::3 is zero-state detectable , 2::3 is globally asymptoticlaly stabilizable by (19).
0=
0 ~ n.
Step II: We show that
(w -limit set of 4>( ·, xo ;o))
where 4>(-. Xo ; 0) is the trajectory of i;
= fo (x)
If 2: is lossless, then V = Lf(x,u)V(x) = hT(x , u)u JRm. This implies that "Ix E IR n
starting from
Vu E
Xo ·
L foV(x) = 0
Theorem 4.5 Consider a passive system 2::3 of the form (17) with a Cr(r 2: 1) storage function V , which is positive definite and proper . Then (i) 2::3 is zero-state detectable if
0n
S
and
LgO V(x) = hT(x, 0)
(23)
Using the exact same arguments and (23), we can prove that (22) holds for every x(t) E r O yielding h(x(t) , O) = O. Therefore, S = {O} implies zero-state observability.
= {o}. As a consequence of Theorem 4.5 , we have (note that
(ii) 2::3 is zero-state observable if 2::3 is lossless and S = {o}. In the local case, (i) and (ii) remain true without the hypothesis that V is proper.
0~
n)
Corollary 4.6
A passive system 2:3 of the form (17) is zero-state detectable if n n S = {O} .
Proof. The proof is based on Proposition 4.3 and proceeded in three steps.
We now turn to the proofs of Theorems 2.1 and 2.3 by using Theorem 4.4 and Corollary 4.6.
Step I: We show that VC) is constant on the w-limit set r O of each trajectory of the unforced dynamic system i; = fo (x).
Proof of Theorem 2.1: Consider a nonlinear system of the form (9) with the property (HI). Choose a dummy output
By the passive inequality (18), the unforced dynamic system of (17) is Lyapunov stable. Let 4>(t , xo ; 0) be the trajectory of i; = fo(x) starting from Xo in IR n and r O denote its w-limit
(24) Then it is easy to show that the input/output system (9)-
407
(24) is passive. As a matter of fact,
11
established in this paper are based on the development of non-affine passive systems theory and the technique of ren= LID V(x)+Lgex ,u) V(X)U = LID V(X)+yT U :::; yT U Vu E ffi. m. de ring a systems passive via a dummy output.
Thus the claim follows. Moreover, since n n S = {O}, it follows from Corollary 4.6 that the passive system (9)-(24) is locally zero-state detectable. By Theorem 4.4, the memoryless output feedback control law
Finally, it is worthwhile to point out that non-affine passive systems theory developed in Section 4 can be extended to discrete-time nonlinear passive systems of the form
u = -s(y) = -y = - (LgeX,u)V(x))T locally asymptotically stabilizes the equilibrium x = 0 of the system (9)-(24). This, in turn, implies that u = O'(x) locally asymptotically stabilizes the equilibrium x = 0 of the nonlinear system (9) or (1), provided that there exists a Coo solution u = O'(x) with 0'(0) = 0, locally defined on a neighborhood U of x = 0, such that the equation (12) is satisfied. But this is indeed the case, as illustrated hereafter.
6. REFERENCES [1]V. Jurdjevic and J.P. Quinn, Controllability and stability, J . Diff. Equations, 28 (1978), 381-389. [2]J.P. Gauthier and G. Bornard, Stabilization des systems nonlineaires, Outilset methods mathematiques pour L'automatique, 1.D. Landau, Ed., C .N.R.S. (1981),307324.
Consider the following function
H(x, u)
8V )T = u + ( ax-g(x, u)
[3]N. Kalouptsidis and J . Tsinias, Stability improvement of nonlinear systems by feedback, IEEE Trans. Automat. Contr. 29 (1984) , 364-367.
Clearly, H(O,O) = 0 and ~~ (0, 0) = Im xm. By the Implicit Function Theorem, there exists open neighborhood N of x = 0 in ffi.n, open neighborhood M of u = 0 in ffi.m and a unique Coo mapping er : N -- M, with 0'(0) = 0, satisfying H(x , O'(x)) = O. Thus the proof of the local result of Theorem 2.1 is completed. If V is proper and the equation (12) is globally solvable, then Theorem 2.1 also holds globally.
[4]K .K. Lee and A. Arapostathis, Remarks on smooth feedback stabilization of nonlinear systems , Syst. Contr. Lett., 10 (1988), 41-44. [5]C .1. Byrnes , A. Isidori and J .C . Willems, Passivity, feedback equivalence, and global stabilization of minimum phase nonlinear systems, IEEE Trans. Automat. Contr . 36 (1991) , 1228-1240.
Proof of Theorem 2.3: Consider a nonlinear control system (13) and design a dummy output
[6]E.D . Sontag, Feedback stabilization of nonlinear systems , in M.A. Kaashoek et al., Eds. , Robust Control of Linear Systems and Nonlinear Control, Birkhauser, Boston (1990), 61-81.
(25)
[7]J.C. Willems , Dissipative dynamic systems, Arch. Rational Mech. & Analysis , 45 (1972) , 321-393. [8]P. Moylan, Implications of passivity in a class of nonlinear systems , IEEE Trans. Automat. Contr. 19 (1974 ), 373381.
A straightforward computation shows that
11 = L(/O V(x) + LgO V([X)~:le~~(x~i(]L)Riex) V(x)u) :::;
LgoV(x)
+ uT
:
(26)
L;d :
u :::; yT U Vu E ffi.m
LRmex)V(x)
[9]D. Hill and P. Moylan, The stability of nonlinear dissipative systems, IEEE Trans. Automat. Contr ., 21 (1976) , 708-711. [10]A. Isidori, Nonlinear Control Systems, 2nd ed. New York , Springer Verlag, 1989.
In other words, the input/output system (13)-(25) is passive . By assumption and Theorem 4.5, the system (13)-(25) is locally zero-state detectable . Therefore , it follows from Theorem 4.4 that the equilibrium x = 0 of (13)-(25) is locally asymptotically stabilizable by u = -y. This, together with (25), results in (14) immediately. Note that t:.(x) , as illustrated in Remark 2.5, is always invertible on a neighborhood of x = O. Thus , the state feedback control law (14) is locally well-defined and smooth with respect to x . If (H2) holds and V is proper, the global stabilization result of Theorem 2.3 is obvious.
[l1]H . Nijmeijer and A.J. Van der Schaft , Nonlinear Dynamic Control Systems, Springer Verlag , 1990. [12]C.1. Byrnes and W . Lin , Losslessness , feedback equivlaence and the global stabilization of discrete-time nonlinear systems , IEEE Trans. Automat. Contr. 39 , No . 1 (1994),83-98 . [13]W. Lin and C.1. Byrnes, Passivity and absolute stabilization of a class of discrete-time nonlinear systems , Automatica,VoI.31, No .3,1994. [14]C .1. Byrnes and W . Lin, Discrete-time lossless systems, passivity and feedback equivalence, Proc. of the 32nd CDC , 1775-1781.
5. Conclusions We have derived sufficient conditions under which a class of non-affine control systems of the form (1), characterized by (HI), can be locally (resp . globally) asymptotically stabilizable by smooth state feedback . All the stabilization results
[15]J .P. LaSalle , The stability of dynamic systems, Regional ConL Series in Appl. Maths ., 1976.
408
Feedback Stabilization of General Nonlinear Control Systems * Wei Lin Department of Systems Science and Mathematics Washington University, St. Louis, MO 63130, U.S.A.
*Research supported in part by grants from the AFOSR and NSF.
Abstract. This paper develops sufficient conditions for a general nonlinear control system El : x = f(x ,u) to be locally (resp. globally) asymptotically stabilizable via smooth state feedback . In particular, it is shown that as in the case of affine systems, this is possible if the wUorced dynamic system of El is Lyapunov stable and appropriate controllability-like rank conditions are satisfied. Our results incorporate a series of well-known stabilization theorems proposed in the literature for affine control systems and extend them to non-affine nonlinear control systems.
Key Words. Non-affine control systems, smooth state feedback, asymptotic stabilization, passive systems, zerostate detectability.
1. Introduction
In the pioneering work [1) , lurdjecvic and Quinn presented a sufficient condition under which an affine nonlinear control system (2) whose unforced dynamics are linear (i.e. , f(x) = Ax) , and the state transition matrix is unitary can be globally asymptotically stabilizable by smooth state feedback. Since then , various Jurdjevic-Quinn type sufficient conditions have been developed during the last fifteen years. Under some restrictive assumptions on the form of Lyapunov function associated to the un forced dynamics , Kalouptsidis and Tsinias [3) proposed a sufficient condition for global stabilization of affine nonlinear control systems via smooth state feedback. Their work extends the results of lurdjevic and Quinn [1) to systems with Lyapunov stable unforced dynamics. In [2) , Gauthier and Bonard assumed that there exists a positive definite, readily unbounded smooth function V(x) such that Lf V(x) == O. Then they proved that if
Consider a general nonlinear control system of the form
I: l
:
i
= f(x,u)
(1)
where x E lRn is the state, u E R m is the control input , f : Rn X lRm -+ lRn is a smooth (i.e. of class of COO) mapping, with f(O , O) = O. Throughout this paper we make the following assumption: -+ lR , with V(O) = 0, which is positive definite and defined on some neighborhood U (resp . lR n) of x = 0 such that the unforced
(HI) There exists a function V : lRn
dynamic system of (1) , namely i Lyapunov stable, i.e., LfoV(x) :S 0
f(x , 0) ~ fo(x) is "Ix E U (resp . lR n ).
=
The purpose of this paper is to present sufficient conditions for such a class of nonlinear control systems of the form (1) to be locally (resp. globally) asymptotically stabilizable, by means of a smooth state feedback control law u = u( x) with u(O) == o.
dim jj
~ dim span [ad~gi: 0 :S k :S n - 1, 1 :S i :S "Ix E lRn - {O},
=n
m}
the affine system (2) is globally asymptotically stabilizable by a smooth state feedback control law
For affine control systems of the form
u == - (LgV(x)f·
(3)
m
i;
= f(x) + g(x)u = f(x) + L
gi(X)Ui
Clearly, this is reminiscent of the well-known rank condition for accessibility of the system (2) and can be viewed as a kind of controllability-like rank condition. Lee and Arapostathis [4) studied the single-input system (2) . They observed that the conditions proposed in [2) [3) can be further weakened, and derived a general sufficient condition which includes the results obtained in [1)-[3) . More specifically, it was shown in [4) that the smooth state feedback control law (3) globally asymptotically stabilizes the affine system (2) if there is a Coo V(x) , which is positive definite and proper , satisfying (HI) and
(2)
i=l
with the property (HI) , stabilization via smooth state feedback has been investigated by many researchers. Among several contributions to the subject , we wish to point out those which are closely related to our work to be reported in this paper; see for instances, the following papers as well as references therein: Jurdjevic and Quinn [1], Gauthier and Bornard [2], Kalouptsidis and Tsinias [3), Lee and Arapostathis [4], Byrnes, Isidori and Willems [5) . For more information and additional details on nonlinear feedback stabilization, the reader is referred to a survey due to Sontag
5 ~ {x
[6).
403
E lR n
:
L} LT V (x) == 0 "IT E
D, i = 0, 1, .. . } ==
{O}.
The later is apparently much weaker than those presented in [1]-[3]. By using a synthesis of the techniques and concepts drawn from passive systems theory [7]-[9] and from geometric nonlinear control theory [10] [11] , Byrnes, Isidori and Willems [5] developed a framework for studying stabilizability of minimum-phase nonlinear systems, and illustrated how the stabilization results proposed in [1]-[3] and some generalization thereof [4] , can all be re-derived from a basic stability property of affine passive systems.
From the notations introduced in (4) and the identity
f(x ,U)-fa(X)=(taafl 0' a=9u
la
d(})U~9(X ' U)U
(8)
with g(x,u) : JR.n x JR.m ..... JR.n x JR.m being a smooth map, we conclude that a smooth nonlinear control system (1) can always be represented as m
X = fa(x)
Since passive systems theory has been developed by Willems [7] for more general classes of nonlinear systems which are not necessarily affine in u , it is very natural to expect the idea and method developed in Byrnes, Isidori and Willem [5] can be generalized to non-affine nonlinear control systems. Indeed , this is precisely the point of view to be pursued in this paper. Starting from the next section , we show how the theory of passive systems developed in [5] can be extended to general input/output nonlinear control systems, and how the problem of asymptotic stabilization of the nonlinear system (1) can be solved by means of passive systems theory which we develop here , and the technique of feedback equivalence.
+ g(x , u)u = fa(x ) + L
gi(X , U)Ui .
(9)
i=l
Using (4) and proceeding essentially the same argument , (9) can be further decomposed as m
X = fa(x)
+ go(x)u + L
Ui (Ri(X , u)u)
(10)
i=l
with Ri(X , U) : IRn 1 S; i S; m .
X
JR.m ..... lR nxm, being a smooth map for
Since a smooth nonlinear control system of the form (1) is equivalent to either system (9) or (10) , we henceforth will used them interchangeably when referring to a smooth nonlinear control plant .
This paper is organized as follows: in Section 2, we introduce necessary notations and present the main results of this paper. Section 3 uses three examples to illustrate the stabilization schemes of non-affine systems proposed in Section 2. In Section 4 , we first extend passive systems theory to input-output nonlinear systems with general structure. We then combine the theory thus developed with the technique of feedback equivalence to prove the stabilizability results stated in Section 2. Section 5 briefly discusses how passive systems theory in continuous-time can be generalized to its discrete-time counterpart.
Now we are ready to state our main results in this paper. The first result is given by the following statement whose proof will be presented in Section 4.
Theorem 2.1 Suppose a non linear control system Bl of the form (1) satisfies Assumption (HI). If Snn = {O} , then Bl is locally asymptotically stabilizable at the equilibrium x = 0 by smooth state feedback. In particular , a local smooth state feedback control law (11)
u=O'(x) , O'(O)=O
2. Notations and Main Results
can be solved uniquely from the equation We present in this section sufficient conditions for the existence of smooth state feedback control laws which locally (resp . globally) asymptotically stabilize a general nonlinear control system of the form (1 ), with Lyapunov stable free dynamics.
u+(Lg(z.u)V(x))
g~(x)
= f(x , O) E JR.n
= g.(x , 0) = ~(x, 0) E JR. n,
1 S; i S; m, ga(x) = *(x , O) = [g~(x), .. . ,g~(x)l E JR.n xm
Remark 2.2 In the case of affine nonlinear systems , the equation (12) reduces to u = O'(x) = - (Lgo(z )V(x ){ , and therefore, (12) always allows a global solution on JR. n. In this case, Theorem 2.1 includes the global stabilization results due to Byrnes, Isidori and Willems [5] (see Section 3, especially, Remark 3.8) and Lee-Arapostathis [4] (see Theorem 1) , etc ., as particular cases .
}
(4)
With the vector fields fa, g~ , . . . ,g~ we introduce the distribution
D= span {ad~og?: OS;kS;n-l , lS;iS;m} ,
(5)
The second main theorem is a stabilization result for a nonaffine nonlinear control system (10) in which Ri(X , u) , 1 S; i S; rn , are independent of u, i.e., R i( x , u) == Ri (X). For convenience, we rewrite (10) as
and two sets nand 5 associated with D , which are defined by
n = {x 5 = {x
= O,k = 1, . . . , r} E U ~ JR.n: L:oLTV(X) = O, Ii. E D, E U ~ JR.n : L:oV(x)
k=O , I , . . . , r-l}
(12)
where g(x , u) is determined by the identity (8). If V is proper , n n 5 = {O} lix E JR.n and equation (12) has a solution u = 0'( x) which is well-defined on IRn , then BJ is globally asymptotically stabilized by (11 ).
To begin with, we first introduce some notations. Throughou t this paper , let fa (x) and g? (x) denote the smooth vector fields
fa(x )
T=u+ (av -axg(x , u ))T =0 ,
m
(6 ) B2
:
X = fa(x)
+ ga(x)u + L i=l
(7 )
404
Ui (Ri(X)U)
(13)
By comparison with the affine system (2), the system (13) is indeed a "second affine approximation"-with a minor abuse of terminology-of the general nonlinear system (1).
Let V(x) = ~(xi + x~). A routine calculation shows that Llo V(x) = 0, so n = IR2. In addition
Theorem 2.3 Consider a nonlinear control system 2:2 of the form (13) with the property (HI). Suppose Snn = {a} . Then the equilibrium x = 0 of 2:2 is locally asymptotically stabilizable by the smooth state feedback control law
0= LUO,gol V(x) = LloLgo V(x) = x~ - 2xix2
0= LgoV(x)
LR, (r) V(x) := Imxm
+ LR=(r) V(x)
sume that
= XIX~
0= LloL[fo ,golV(x) = -7XIX~ + 2xi . Then, we conclude immediately from (7) that S = {O} . Hence n n S = {O}. By Theorem 2.1, there exists a smooth state feedback control law such that 2:" is asymptotically stabilizable. In particular, a typical control law is given by (11) and it can be solved from the equation (12) , which in the present case is
r
Moreover , as-
[
X2][0 XIX2f
and
(14)
where 6.(x)
= [Xl
Thus
(H2) 6.(x) is invertible 'Ix E IRn . u=a(x)= If V is proper and S n n = {O} 'Ix E IRn , then the state feedback control law (14) globally asymptotically stabilizes the equilibrium x 0 of the system (13).
This theorem will be proven in Section 4. It turns out that in the single-input case (m = 1), the previous result yields the following interesting consequence.
(15 )
1
Remark. It is interesting to note that the same arguments show that the system
B,, :
(16)
Xl X2
= X2 + x~u3 - X2U 4 = -Xl + XIX2U - XIX2U 3 + XlU 4
is also globally asymptotically stabilized by u = -XIX~ . In fact , the equation (12) in the present case is indeed globally solvable because , 0 = u + ~~ g(x , u) == u + ~~ go(x) for V(x) = ~ (xi + d).
n
If 1 + Lp(r)V(x) f. 0 'Ix E IR , V is proper and S n = {O} 'Ix E IR n , then the system (15) is globally asymptotically stabilized by (16) . n
.
Wltha(O)=O .
La Salle's Invariance Principle [15] .
satisfying Assumption (HI). Suppose S n n = {O} . Then x = 0 of the system (15) can always be locally asymptotically stabilizable by smooth state feedback. A possible choice is
= - 1 + L per )V()X Lgo(r )V(x) .
2
2XIX 2
V = -(XIX~)4 - (XIX~)2 ~ 0 and V = 0 '* 0 = LgO V(x) . This yields n n S = {O} again . Thus, the claim follows from
Corollary 2.4 Consider a single-input nonlinear system
u
+ VI - 4(XIX~)2
Clearly, a(x) is Coo around the origin in IR2 . More interesting, if one replaces the smooth state feedback control law above by u = -LgOV(x) = -XIX~, it is easy to show that this Coo feedback control law globally asymptotically stabilizes the system 2:". In fact , along the trajectory of the closed-loop system ,
=
x = fo(x) + go(x)u + p(x)u 2
-1
Example 3.2 Consider a continuous-time nonlinear control system with two inputs
Remark 2.5 Locally speaking, Assumption (H2) always holds, simply because ~~ (0) = 0 and hence 6.(x) Ir=o= Imxm. In the case of affine nonlinear systems, (H2) is globally satisfied since R;(x) == 0 (resp. p(x) == 0) for 1 ~ i ~ m . Therefore, Theorem 2.3 and Corollary 2.4 recover the stabilization schemes proposed in [4] [5]. This , in turn , implies as illustrated in Remark 3.8 of [5] that all the stabilization results developed by Jurdjevic and Quinn [1] , Gauthier and Bornard [2], Kalouptsidis and Tsinias [3] can be viewed as consequences of Theorem 2.3 or Corollary 2.4.
+ x~) + t'n(xi + I)UI + COS(X2 + X3)UI U2 + xiu~ = Xl - 2x~ + X2U2 + sin Xl UI U2 = X2 + X3U2 + x2ui
Xl = -2X2(X~ 2:b :
X2 X3
We choose V(x)
=~
ward to verify that
LR,( r )V(x)
(xi + (x~ + X~)2).
n n S = {O}.
It is straightfor-
Moreover , note that
= [0 , Xl COS(X 3 + X2) + 2X2(X~ + x~) sin xd ~ [0 , *] and LR2(r)V(x)
3. Examples
2x~ui
= [0,
xn
Then , Assumption (H2) holds because the matrix 6.(x )
Before giving a rigorous proof of the stabilizability results introduced in the last section, we first use them to solve the stabilization problems of non-affine nonlinear control systems.
1 0 4 ] is invertible 'Ix E IR3. By Theorem 2.3, 2:b is [ * 1 + Xl globally asymptotically stabilized by the smooth state feedback control law
Example 3.1 Consider a planar nonlinear control system
405
Let us end this section with a simple example which demonstrates the application of Corollary 2.4.
Definition 4.1 An input/output nonlinear control system I;3 of the form (17) is said to be passive if there exists a CO nonnegative function V : lR n -- lR with V(O) = 0, called the storage function , such that
Example 3.3 Consider a single-input nonlinear system I;c :
where x
We note that this system has some interesting properties similar to Examples 3.1 and 3.2. Firstly, the first approximation of I;c is determined by the pair (A , B) ([
~ ~], [ ~
=
If V is Cr(r
Hxi + x~).
Llo Vex)
= -xi
Definition 4.2 An input/output nonlinear system I;3 of the form (17) is locally zero-state detectable if there is a neighborhood N of x = 0 such that Vxo = x E N
h( lim
$ 0 and Llo Vex)
X2
'-00
= 0 ::} Xl = O.
= O.
A fundamental property of an affine passive system (2) with y = h( x) is characterized by the well-known KYP Lemma [5]. In the case of non-affine systems, the KYP Lemma does not exist . However, necessary conditions for an input/output nonlinear system (17) to be passive can still be obtained. These conditions, especially relation (ii) given below, will be very useful to derive a checkable zero-state detectability criterion for passive systems of the form (17) .
= - 1 + Lp(x)V(x) L gO Vex) = _ l+xiex2' 2 4. The Proofs of Main Theorems
We now present the proofs of stabilizability results stated in Section 2. The underlying philosophy of the proofs is to use the concepts of passivity and feedback equivalence, which have been developed in [7]-[9] and [5]. The reason why these two concepts are instrument in designing a global stabilizing state feedback control laws for a class of affine nonlinear systems has been interpreted in [5]. In this section , we first explore some intrinsic properties such as zero-state detectability and stabilizability of smooth non-affine passive systems of the form
Y
= f(x , u) ,
= h(x,u),
f(O , 0) = 0 } h(O , O) = 0
= o.
If N = lRn, the system I;3 is zero-state detectable. A system I;3 is locally (resp. globally) zero-state observable if there is a neighborhood U of x = 0 such that Vxo = x E U (resp. lRn) , h(
x3
1
3·
2: 1), the passive inequality is equivalent to
loss less.
Thus n n S = {O} . By Corollary 2.4, I;c is globally asymptotically stabilized by the smooth state feedback control law
i;
f(x , u) starting
Moreover, if (18) becomes a strict equality, I;3 is said to be
Obviously
0= LgoV(x) = x~::}
I; .
=
(18)
On the other hand,
u
is a solution of i;
]), which is not controllable. Secondly,
the center manifold theory cannot be applied to solve the stabilization problem of I;c because two eigenvalues of A are located on the imaginary axis. However , it can be shown that I;c is globally asymptotically stabilizable via smooth state feedback. This claim follows from Corollary 2.4, as illustrated in the following . Let Vex) =
=
x(O)=xo.
Proposition 4.3 Let n l ~ {x E lR n : Li oVex) = O}. Necessary conditions for I;3 to be passive with a C 2 storage function V are that
(i) L lo Vex) $ 0 (ii) LgO Vex) = hT(x , 0)
'Ix E
~ [) 2 fi [)V (iii) ~ ou 2 (x , 0)· OXi $
(17)
n1
[)h T
[)h
[)u (x, 0)
+ [)u (x , 0)
'Ix E
n1
i=l
Where h : lRn x lR m __ lR m is a smooth function and y E lR m is the system output. The essential difficulty in such a development is the lack of the Kalman- Yacubovitch-Popov Lemma (KYP Lemma for short) for a non-affine passive system (17) , whereas in the case of affine passive systems the KYP Lemma does exist and ineed plays an important role in the analysis and design [5]. In what follows , we show how some basic properties of passive systems of the form (17) including zero-state detectability and stabilizability via output feedback can be carried out in the absence of the gyp Lemma. The approach given below in studying passive systems (17) is very much in the spirit of [12]-[14].
where f,ex , u) is the i-th component of the vector function
We first review two useful concepts related to passive systems .
In other words , F(x , u) arrives its maximum at u set n 1 .
f(x, u). Proof. Consider an auxiliary function F : lR n x lR m -- lR defined by
F(x, u)
[)V = Eh f(x, u) -
T
h (x , u)u
Since I;3 is passive , it is clear that F(x , u) $ 0 Vu E lRm . Hence (i) follows by setting u = O. For all x in n1 , F( x , 0) LloV(x) == O. Thus
=
F(x,u) $ F(x , O)
406
=0
'Ix E
n1
and Vu E lRm.
= 0 on the
set. By La Salle's book [lS}, r O is nonempty, compact and invariant. Since V(x(t)) is nonnegative and nonincreasing along the trajectory x(t) = 4>(t, XO; 0), lime_oo V(x(t)) exists and lime_oo V(x(t)) = c ~ O. By continuity of V, V(x) = c for every point x = lime_oo 4>(t, xo; 0) E rO.
This, in view of (4), implies that "Ix E rh
o= o
~~ (x, 0) = L go V ( x) - h T ( x , 0)
~ fu?(x,o) = 8(~!fr lu=o - (~:(x, o) + 8:: (x,o)) _""n -
8'/,( X, O)8V 8Xi
L..,i=l a;;:t"
(8h( ) 8h T ( x , o) ) Ft;' x,o + 8u
-
Let x E r O ~ 0 and 4>(t , x; 0) be the corresponding trajectory of i; = fo(x) . By invariance, 4>(t, x; 0) E r O ~ O. From Step I, it follows that V (4)(t, x; 0)) = V(x) = c "It ~ O. Hence
Therefore, (ii) and (iii) follow immediately. As in the case of affine systems, a non-affine passive system (17) also enjoys a nice stabilizability property by output feedback. The following result basically describes such a stability property.
V (4)(t, x; 0)) = LID V
Theorem 4.4 Consider a passive system 2::3 of the form (17) with a Cl storage function V , which is positive definite. Let s : IR m - JRm be any first sector/third sector function, i.e. , yT s(y) > 0 Vy i= 0 and s(o) = O. Then the equilibrium x = 0 of 2::3 is locally asymptotically stabilizable by
u
= -s(y)
x E n.
Step Ill: tectability.
We show that
=0
"It
2: O.
~ n.
0
Thus
=
(19)
0n
S
=
In addition, since 4>(t,x;O) Proposition 4.3 (ii) that
The proof of this theorem, which is similar to the proof of Theorem 3.2 given in [5], follows immediately from passive inequality (18) with u = -s(y), zero-state detectability and LaSalle's Invariance Principle.
= {O}
=> zero-state de-
=
=
x(t) E n, it follows from
LgOV (x(t)) = hT (x(t) , 0) = 0
"It
2:
From (20) and (21), we deduce that for
Now a natural question arises immediately: When is a passive system (17) zero-state detectable? In the following , we
(21)
O. T
= [fa , gal
ED
Using the inductive argument , it is not difficult to show that "It 2: 0
will give a satisfactory answer to this question and show how zero-state detectability of the passive system (17) can be tested by means of the Lie bracket of vector fields fa (x) and g?(x), which characterize the affine part of the inputstate differential equation .
L~oL7.v (x(t))
=0
"IT E D and 0 ::; k ::; r - 1.
(22)
In view of the definition of S (see (7)) , we conclude that x E S. Hence x E On. If 0 n S = {o},- clearly x = O. Thus V(x) = 0, which implies lime_oo 4>(t , Xo; 0) = O. By Definition 4.2, 2::3 is zero-state detectable.
To begin with, set uxoEUC;Rn
This means
(4)(t , x; 0))
Let x(t) = 4>(t , x; 0) be a trajectory of the unforced system i; fo(x) yielding y(t) h(x(t),O) 0 "It 2: o. From Steps I and Il, "Ix E ra, 4>(t , x; 0) E r O ~ 0 ~ nand
provided that 2::3 is locally zero-state detectable. If V is proper and 2::3 is zero-state detectable , 2::3 is globally asymptoticlaly stabilizable by (19).
0=
0 ~ n.
Step II: We show that
(w -limit set of 4>( ·, xo ;o))
where 4>(-. Xo ; 0) is the trajectory of i;
= fo (x)
If 2: is lossless, then V = Lf(x,u)V(x) = hT(x , u)u JRm. This implies that "Ix E IR n
starting from
Vu E
Xo ·
L foV(x) = 0
Theorem 4.5 Consider a passive system 2::3 of the form (17) with a Cr(r 2: 1) storage function V , which is positive definite and proper . Then (i) 2::3 is zero-state detectable if
0n
S
and
LgO V(x) = hT(x, 0)
(23)
Using the exact same arguments and (23), we can prove that (22) holds for every x(t) E r O yielding h(x(t) , O) = O. Therefore, S = {O} implies zero-state observability.
= {o}. As a consequence of Theorem 4.5 , we have (note that
(ii) 2::3 is zero-state observable if 2::3 is lossless and S = {o}. In the local case, (i) and (ii) remain true without the hypothesis that V is proper.
0~
n)
Corollary 4.6
A passive system 2:3 of the form (17) is zero-state detectable if n n S = {O} .
Proof. The proof is based on Proposition 4.3 and proceeded in three steps.
We now turn to the proofs of Theorems 2.1 and 2.3 by using Theorem 4.4 and Corollary 4.6.
Step I: We show that VC) is constant on the w-limit set r O of each trajectory of the unforced dynamic system i; = fo (x).
Proof of Theorem 2.1: Consider a nonlinear system of the form (9) with the property (HI). Choose a dummy output
By the passive inequality (18), the unforced dynamic system of (17) is Lyapunov stable. Let 4>(t , xo ; 0) be the trajectory of i; = fo(x) starting from Xo in IR n and r O denote its w-limit
(24) Then it is easy to show that the input/output system (9)-
407
(24) is passive. As a matter of fact,
11
established in this paper are based on the development of non-affine passive systems theory and the technique of ren= LID V(x)+Lgex ,u) V(X)U = LID V(X)+yT U :::; yT U Vu E ffi. m. de ring a systems passive via a dummy output.
Thus the claim follows. Moreover, since n n S = {O}, it follows from Corollary 4.6 that the passive system (9)-(24) is locally zero-state detectable. By Theorem 4.4, the memoryless output feedback control law
Finally, it is worthwhile to point out that non-affine passive systems theory developed in Section 4 can be extended to discrete-time nonlinear passive systems of the form
u = -s(y) = -y = - (LgeX,u)V(x))T locally asymptotically stabilizes the equilibrium x = 0 of the system (9)-(24). This, in turn, implies that u = O'(x) locally asymptotically stabilizes the equilibrium x = 0 of the nonlinear system (9) or (1), provided that there exists a Coo solution u = O'(x) with 0'(0) = 0, locally defined on a neighborhood U of x = 0, such that the equation (12) is satisfied. But this is indeed the case, as illustrated hereafter.
6. REFERENCES [1]V. Jurdjevic and J.P. Quinn, Controllability and stability, J . Diff. Equations, 28 (1978), 381-389. [2]J.P. Gauthier and G. Bornard, Stabilization des systems nonlineaires, Outilset methods mathematiques pour L'automatique, 1.D. Landau, Ed., C .N.R.S. (1981),307324.
Consider the following function
H(x, u)
8V )T = u + ( ax-g(x, u)
[3]N. Kalouptsidis and J . Tsinias, Stability improvement of nonlinear systems by feedback, IEEE Trans. Automat. Contr. 29 (1984) , 364-367.
Clearly, H(O,O) = 0 and ~~ (0, 0) = Im xm. By the Implicit Function Theorem, there exists open neighborhood N of x = 0 in ffi.n, open neighborhood M of u = 0 in ffi.m and a unique Coo mapping er : N -- M, with 0'(0) = 0, satisfying H(x , O'(x)) = O. Thus the proof of the local result of Theorem 2.1 is completed. If V is proper and the equation (12) is globally solvable, then Theorem 2.1 also holds globally.
[4]K .K. Lee and A. Arapostathis, Remarks on smooth feedback stabilization of nonlinear systems , Syst. Contr. Lett., 10 (1988), 41-44. [5]C .1. Byrnes , A. Isidori and J .C . Willems, Passivity, feedback equivalence, and global stabilization of minimum phase nonlinear systems, IEEE Trans. Automat. Contr . 36 (1991) , 1228-1240.
Proof of Theorem 2.3: Consider a nonlinear control system (13) and design a dummy output
[6]E.D . Sontag, Feedback stabilization of nonlinear systems , in M.A. Kaashoek et al., Eds. , Robust Control of Linear Systems and Nonlinear Control, Birkhauser, Boston (1990), 61-81.
(25)
[7]J.C. Willems , Dissipative dynamic systems, Arch. Rational Mech. & Analysis , 45 (1972) , 321-393. [8]P. Moylan, Implications of passivity in a class of nonlinear systems , IEEE Trans. Automat. Contr. 19 (1974 ), 373381.
A straightforward computation shows that
11 = L(/O V(x) + LgO V([X)~:le~~(x~i(]L)Riex) V(x)u) :::;
LgoV(x)
+ uT
:
(26)
L;d :
u :::; yT U Vu E ffi.m
LRmex)V(x)
[9]D. Hill and P. Moylan, The stability of nonlinear dissipative systems, IEEE Trans. Automat. Contr ., 21 (1976) , 708-711. [10]A. Isidori, Nonlinear Control Systems, 2nd ed. New York , Springer Verlag, 1989.
In other words, the input/output system (13)-(25) is passive . By assumption and Theorem 4.5, the system (13)-(25) is locally zero-state detectable . Therefore , it follows from Theorem 4.4 that the equilibrium x = 0 of (13)-(25) is locally asymptotically stabilizable by u = -y. This, together with (25), results in (14) immediately. Note that t:.(x) , as illustrated in Remark 2.5, is always invertible on a neighborhood of x = O. Thus , the state feedback control law (14) is locally well-defined and smooth with respect to x . If (H2) holds and V is proper, the global stabilization result of Theorem 2.3 is obvious.
[l1]H . Nijmeijer and A.J. Van der Schaft , Nonlinear Dynamic Control Systems, Springer Verlag , 1990. [12]C.1. Byrnes and W . Lin , Losslessness , feedback equivlaence and the global stabilization of discrete-time nonlinear systems , IEEE Trans. Automat. Contr. 39 , No . 1 (1994),83-98 . [13]W. Lin and C.1. Byrnes, Passivity and absolute stabilization of a class of discrete-time nonlinear systems , Automatica,VoI.31, No .3,1994. [14]C .1. Byrnes and W . Lin, Discrete-time lossless systems, passivity and feedback equivalence, Proc. of the 32nd CDC , 1775-1781.
5. Conclusions We have derived sufficient conditions under which a class of non-affine control systems of the form (1), characterized by (HI), can be locally (resp . globally) asymptotically stabilizable by smooth state feedback . All the stabilization results
[15]J .P. LaSalle , The stability of dynamic systems, Regional ConL Series in Appl. Maths ., 1976.
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