E L S E V I E R Systems & Control Letters 26 (1995) 41-53

l ] , l n l l s 6 i lllllll

Bounded smooth state feedback and a global separation principle for non-affine nonlinear systems*

Wei Lin *

Department of Systems Science and Mathematics, Washinoton University, St. Louis, MO 63130, USA

Received 8 September 1994; revised 17 December 1994

Abstract

In this paper we show that a single-input nonlinear system L': ~ = fo(x)+gl (x)u+. . . +g:(x)d is globally asymptotically stabilizable by arbitrarily small smooth state feedback if unforced dynamics of Z" are Lyapunov stable and S satisfies appropriate controllability-like rank conditions characterized by the Lie bracket of vector fields fo(x) and el(x). An explicit smooth state feedback control law is constructed. This, in turn, leads to an analytic (resp. smooth) state feedback control law which solves the problem of global stabilization for analytic (resp. smooth) systems i = f (x ,u) . Based on these results, we then establish a global separation principle for the system 27 with an output. Finally, we briefly discuss how the results can be extended to a multi-input system.

Keywords." Nonlinear systems; Global stabilization; Smooth saturation technique; State feedback; Dynamic output feedback

1. Introduction

In recent years feedback stabilization o f affine nonlinear systems with stable free dynamics has received a great deal o f attention in the nonlinear control literature. Several systematic design methodologies have been developed for local and global stabilization problems of continuous-time or discrete-time nonlinear systems; see for instance [1-11, 13, 14] as well as the references therein. In this paper, we address problems o f global stabilization, via smooth state feedback, o f non-afline nonlinear control systems

X: £c = f o ( x ) + 91(x)u + 92(x)u 2 + " " + O:(x)u: (i.1)

and

.f = f ( x , u ) , (1.2)

where x E R n is the state, u E • is the control input, f 0 : ~n ~ ~n, gi : R~ ~ ~ , i = 1,2 . . . . . : , and f : R ~ x R ---, R ~, are smooth mappings (i.e., o f class o f C ~ ) . Without loss o f generality, we assume that x = 0 is an equilibrium of the vector field f o (x ) .

* Research supported in part by grants from the AFOSR and NSF. * Tel.: 314-935-6092. Fax: 314-935-6121.

0167-6911/95/$9.50 (~ 1995 Elsevier Science B.V. All rights reserved SSDI 0167-6911(94)00109-X

42 w. LinISystems & Control Letters 26 (1995) 41~3

Our interest in non-affine nonlinear systems of the form (1.1) has been inspired by the observation that the nonlinear system (1.1) is the fth order Taylor approximation of a general nonlinear system (1.2). As a matter of fact, since the right-hand side of (1.2) is smooth, it can always be written as

l ~ u 1 O/f / 1 i = f ( x , 0 ) + (x ,O)u+. . .+~.~Tuf(X,O)u + ( f + l ) ~ R ( x , u ) u ~+'.

Let fo (x ) = f (x ,O) and 9i(x) = (1/i!)(Oif/Sui)(x,O) for 1 ~< i ~< { and for arbitrarily positive integer E. It is clear that the non-affine system (1.1) is the Taylor approximation of the system (1.2). Moreover, if the function f : A n x ~ --, ~" is analytic with respect to u in an open neiohborhood o f u = O, the following relation

f ( x , u) = lim o(X) -1- g i (x)u i , te----~ o o

holds uniformly Vx E ~ and Vu E {u :lul < R} for some R > 0. We expect that under certain appropriate conditions, studying the nonlinear system (1.1) might lead to a solution to the problem of global stabilization by smooth state feedback, for general nonlinear systems of the form (1.2).

Throughout the paper we make the following hypothesis:

(HI) There exists a C r (r~> 1) function V : ~ n ---, ~, which is positive definite and proper on R n, such that the unforced dynamic system ~ = fo (x ) is Lyapunov stable, or equivalently, Lfo V(x)<~0 VX E A n.

The objective of this paper is twofold: to present sufficient conditions under which a class of non-affine nonlinear control systems (1.1) or (1.2) satisfying (H1) is globally asymptotically stabilizable by smooth or analytic state feedback, and to explicitly construct a state feedback control law u = u(x) with u(0) = 0, for systems (1.1) and (1.2).

In the previous paper [11], we have derived sufficient conditions for a multi-input nonlinear control system (1.2) with stable unforced dynamics to be asymptotically stabilizable by virtue of smooth state feedback. It was shown in [11] that the system (1.2) can be locally asymptotically stabilizable provided that there exists a positive definite Lyapunov function and a neighborhood U of x = 0 in R n, such that f2u ¢q Su = {0}, where

f2 u ~= {x E U'LkfoV(X) =- O, l<~k<~r} and Su ~ {x E U:LkfoL~V(x ) = O, Vz E D, O<~k<~r- 1}, with

D ~ span{ad~0g° :O<~k<<.n- 1, l<~i<~m} and 9 o ~= (~f/Oui)(x,O), i = 1 . . . . . m. In addition, it was also shown [11] that a local smooth stabilizing state feedback control law can be obtained by solving the function equation

3V ,~V ~xO(X,U)) + U = 0, (1.3)

where 9(x,u) = f~(Of/Occ)(x, ~)l~=0u dO. The results proposed in [1 1] are based on the development of non- affine passive systems theory established in [11], and the technique o f renderin9 system (1.2) with a dummy output passive. In the affine system case (i.e., y(x,u) = (Of/Su)(x,O) = 91(x)), the stabilization results of [11] recover a number of stabilization theorems independently proposed in the literature by various authors [1,2,6-9]. However, the main result described in [11] has essentially a weak point: it is required to solve an implicit function equation in order to get a smooth state feedback control law, and no systematic approach is given on how to solve u = u(x) with u(0) = 0 from Eq. (1.3).

In this paper we address this issue for nonlinear control systems of the form (1.1) or (1.2). For system (1.1), we show how a globally stabilizing smooth state feedback control law can be constructed explicitly without solving the implicit function Eq. (1.3). On the basis of this result, we then derive an analytic (resp. smooth) state feedback control law which globally asymptotically stabilizes the analytic (resp. smooth) system (1.2). The crucial idea behind the construction of such an explicit state feedback control law is to use the technique of input smooth saturation. The idea is inspired by our recent work [10], in which the problem of

14C Lin/Systems & Control Letters 26 (1995) 41-53 43

global stabilization via dynamic output feedback has been solved by using the technique of input saturation, for a class of affine nonlinear systems.

The paper is organized as follows. In Section 2, we introduce some standard notations and preliminary results to be used in the sequel. Section 3 deals with the problem of global stabilization by state feedback, for both single-input and multi-input non-affine nonlinear systems. Two typical examples are presented to illustrate the application of global stabilization theorems developed in the section. In Section 4, we study the problem of global stabilization via smooth dynamic output feedback for a class of input/output non-affine systems. A global separation principle is established based on the results developed in Section 3. Concluding remarks are included in Section 5.

2. Preliminaries

In this section, we present a preliminary result on global stabilization of affine nonlinear systems via arbitrarily small smooth state feedback. Motivated by this result, we will show in the next section how an explicit smooth state feedback control law can be constructed, which solves the problem of global stabilization of non-affine nonlinear systems (I. 1 ) and (1.2).

To begin with, we briefly recall some standard notations. Given smooth vector fields fo (x ) and 91(x), [f0(x), 9J (x)] standards for the Lie bracket. For each integer k = 0, 1,2 . . . . . we can define inductively ad°fogl =

01, adlfoOl = [f0, gl],...,UUfo-'~k+l~Vl = [fo, adkfoOl]. Let V" •" ~ R be a class of C r (r~> 1) functions.

L k+l V(x) = The Lie derivative of V with respect to f o and higher-order Lie derivatives are defined as fo

Lfo(Lkfo V(x)), 0 ~<k ~<r - 1, with L°fo V(x) = V(x). With vector fields f0 and 01 we introduce the distribution

D1 = span {adkfoOt:O~k<~n- 1},

and two sets I2 and S associated with D1, which are defined by

~'~ = (X E ~n:LkfoV(X ) ~- O, 1 <~k~r} ,

S = {x E ~":L~oL~V(x) = O, Vz E 01, O<~k<~r- I} .

Then we can prove the following stabilization result which is a refinement of Theorem A.1 of [10].

Proposition 2.1. Consider an affine system EA : ~ = f o(x) + 91(x)u satisfying hypothesis (HI). Suppose f2 r3 S = {0}. Then EA is globally asymptotically stabilizable at the equilibrium x = 0 by arbitrarily small smooth state feedback, i.e., for any given fl > O, there exists a smooth state feedback control law u = u(x) with u(O) = O, which renders the closed-loop system globally asymptotically stable and satisfies lu(x)l </~. In particular, a candidate o f such a u(x) is given by

u(x) = - f l Lo, V(x) I +(Lo, V(x)) 2 for any fl > 0. (2.1)

The proof of this result can be carried out, as illustrated in [12], by using the theory of passive systems and the technique of feedback equivalence.

Remark 2.2. The key feature of Proposition 2.1 can be described as follows: a class of affine nonlinear systems satisfying (HI) and f2 N S = {0} is always globally asymptotically stabilizable by a smooth state feedback control law whose absolute magnitude can be rendered arbitrarily small. Because of this important property, it is natural to expect that the modified smooth state feedback control law

Lo, V(x) u(x) = -~ (x ) 1 + (Lo, V(x)) 2' ~(x) > 0 (2.2)

44 W. LinlSystems & Control Letters 26 (1995) 41-53

might globally asymptotically stabilize non-affine systems of the form (1.1) or (1.2), provided that ct(x) can be chosen in such a way that lu(x)I <fl<< 1. Intuitively, asymptotic stability property of the equilibrium x = 0 of the closed-loop system (1.1)-(2.4) is determined or dominated by the affine part of the closed-loop system if the absolute value of u(x) is sufficiently small (i.e., lu(x)l<< 1). This observation plays a crucial role in constructing a globally stabilizing state feedback control law for the system (1.1), and is indeed the main motivation of the global stabilization results proposed in the next two sections.

Finally, we introduce a technical lemma which is a direct extension of Theorem B.1 of [10].

Lemma 2.3. Consider a perturbed system

Xp: A = f ( x ) + d ( t ) . (2.3)

Suppose there exists a C 1 function V:ff~ n ---* R, with V(O) = O, such that

(A1) V(x)>~al [Lxll q+l, I[~V/&ll ~ a2llxllq and LfV(x)<~O, for any positive integer q.

Assume that the disturbance d : ~ --~ ~n is piecewise continuous and satisfies

(A2) fo°~]ld(t)[ldt<~a3 < + ~ ,

where ai, 1 <~i <~3, are positive real constants. Then all the trajectories of the system (2.3) are bounded.

The proof of the Lemma follows the same line of reasoning as in the proof of Theorem B.1 in [10] and is therefore omitted here.

3. Global stabilization via bounded smooth state feedback

For the simplicity of exposition, we first restrict our attention to the single-input non-affine system (1.1). Motivated by Proposition 2.1 and Remark 2.2, we propose in this section a solution to the problem of global stabilization via smooth state feedback.

Theorem 3.1. Consider a class of non-affine nonlinear systems of the form (1.1) whose free dynamics satisfy hypothesis (HI). I f t2 f)S = {0}, then the equilibrium x = 0 of (1.1) is globally asymptotically stabilizable by arbitrarily small smooth state feedback. A possible choice is

ul(x) = -o~e(x) Lo, V(x) 1 + (Lo, V(x)) 2 (3.1)

with

o~/(x) = 1 + ~]/~=2(Lo, V(x)) 2 for any 0 < fl < 1. (3.2)

Proof. Choose V(x) satisfying hypothesis (HI) as a Lyapunov function for the closed-loop system (1.1)-(3.1). Along the trajectory of the closed-loop system,

{ (/(x) = Lfo V(x) -F Lo, V(x)u l (x ) + ~ Lo, g ( x ) (u { ( x ) ) i

i=2

(L°' V(x))2 -1 - ~-]Lo, V(x)(-~e(x)) i-l = Lfo V(x) + ~t(x)- 1 + (Lg, V(x)) 2 i=2

(Lo, V(x)) i-2 ] (1 + (Lo, V ( X ) ) 2 ) i - ! "

(3.3)

W. LinlSystems & Control Letters 26 (1995) 41-53 45

Observe that (

- ~ Loi V(X)(--O~(X)) i-I i=2

(Lo, V(x)) i-2 (1 + (Lo, g(x))2) i-I

t" (~E(X)) i -1 ILg ' V(x)l i-2 t ( IZ01V(x)] ~i-2 E ILgiV(x)l (1 Ar(Lo, V(x))2)i-1 ~ ~ ILoiV(x)l(°~f(x))i-I i=2 i=2 1 ~ g l - ~ - ) ) 2 J

~ ~( ,i--' {ILo'V(X)IL 2 '-~ ~ , ILg, V(x)l l ~ 2~_ ~ <.13 i=2 (1 q- y~'~/=2 (o ,Z(x) )2 ) i - t = l+~'f=2(Lo, V(x))22i-z<' /3:

< 1 .

With this estimation in mind, it follows immediately from (3.3) that

(Lo, V(x)) 2 (/(x) <~ Lfo V(x) + ~¢(x) I + (Lg, V(x)) 2 ( - 1 + fl) ~< 0 . (3.4)

This shows that the closed-loop system (1.1)-(3.1) is Lyapunov stable. In order to prove global asymptotic stability, we set V(x) = 0. This, in turn, yields

LfoV(x) = 0 and Lg, V(x) = 0. (3.5)

The remaining part of the proof can be carried out as in Theorem A.1 of [10]. For the sake of completeness, we briefly repeat the argument in the following.

From (3.5), it is clear that ul(x) = 0. Let x(t, xo) be a trajectory of the unforced dynamic system i = fo(x) starting from x(0) = xo. Let r ° denote its ~o-limit set. Since the unforced dynamic system is stable, r ° is nonempty, compact and invariant. By invariance, for any £ C r ° the corresponding trajectory x(t,£) of

i = fo(x) stays in r ° for all t~>0. Since LfoV(X)<~O, V(x(t)) is nonincreasing along every trajectory of the unforced dynamics. Thus limt__+~ V(x(t)) exists and is equal to a nonnegative constant, say c. By continuity of V(.), V(£) = c for every .f = limtr_,~x(tj,xo) in r °. This shows that V(.) is constant over r °. Hence

L~fV(x(t,£)) = 0 Vt>~O and 1 <~k<<.r. (3.6)

On the other hand, since Lo, V(x) vanishes on the set {x : I?(x) = 0}, so does Lo, V(£) V£ E r ° C{x : l?(x) = 0}. Thus Lo, V(x(t,£)) = 0 Vt>~O. This, together with (3.6), implies that for z = [f0,91],

L¢ V (x(t,£)) = LfLo, V (x(t,£)) - Lg, L f V (x(t,£)) = O.

By using the inductive argument, it is easy to show that

LkfLrV(x(t,£)) = 0 Vz E D1 and O<~k<<.r- 1. (3.7)

From (3.6) and (3.7), we conclude that all trajectories of the closed-loop system (1.1)-(3.1) eventually approach the largest invariant subset of {x E ~n :/?(x) = 0} which is contained in S A t2. Since t2 A S = {0}, this implies {x E R n : l?(x) = 0} = {0}, thus completing the proof of asymptotic stability by LaSalle's invariance principle. Note that by hypothesis (HI), V(-) is proper on R n. Then the equilibrium x = 0 of the closed-loop system is globally asymptotically stable.

Finally, it is trivial to show that by construction, the smooth state feedback control law (3.1) is bounded and can be rendered as small as possible by choosing the coefficient/3. In particular,

lug(x)[ < ~ </3 < 1. [] Z

As an immediate consequence of Theorem 3.1, we have the following important global stabilization result for analytic systems of the form (1.2).

Theorem 3.2. Consider a sinole-input analytic system o~

S,o~: i : f ( x , u ) : f o ( x ) + ~ g i ( x ) u i, lul < R , i=1

46 W. Linl Systems & Control Letters 26 (1995) 41-53

with R > 0 and Or(x) = (1/i!)(Oif/Oui)(x,O), i = 1,2 . . . . Suppose ( H I ) is satisfied and O f ' I S = {0}. Then S~, is globally asymptotically stabilizable by analytic state feedback. In particular, a possible choice of the analytic state feedback control law is

uoo(x) = -~oo(x) Lo, Z(x) 1 + (Lo, V(X)) 2 '

where

~ ( x ) = 1 "4- E~2(fli-2Lgi V(x)) 2

for any 0 < /~ < min{R, 1}.

Proof. Clearly, all the arguments used in the proof o f Theorem 3.1 can be applied here if we are able to show that ~¢(x ) is a well-defined analytic function. To see this is the case, we note that by assumption

1 o t f i f(x, ,S) = f(x,O) + ~ ~.--~(x,O)fl = fo(x) + ~_, gi(x)[3 i.

i=1 i=l

Thus

OV ~Oi(x)fli_ 2 = ~Lg , V(x)[3i_ 2 OV f (x , fl) - fo(x) - 91(x)~ An" -Y;x ~:2 i:2 = -gZx I~ 2 Vx

By analyticity of f ( x , u ) on {u: lul < R}, the series Z°°i=2 ]Lo~V(x)~i-2[ converges uniformly Vx E R n because/~<min{R, 1}. Therefore, we conclude immediately that the series o~ i-z Zi=2(/~ Loi V(x)) 2 also converges uniformly for all x in ~n. The conclusion follows from the well-known fact that E °°i=l ]ai] < +oo ==~ ~i=lai°° Z < +C~. Since ~¢ i-2 Ei=z(/~ Lo, V(x)) 2 converges uniformly, ~oo(x) apparently is a well-defined analytic function Vx E R n. []

Inspired by Theorem 3.1, next we will present a global stabilization theorem via state feedback, for a smooth nonlinear system in the general form (1.2). To this end, we observe that as illustrated in [11], any smooth system (1.2) can be represented as

= fo(x) + gl(X)U "-[- R(x,u)u z ,

where R : R n x R --, ~n is a smooth function. Moreover, since R(x, u) is smooth over R n × R, the function

O(x) ~= max IIR(x,u)II lul.<l

is well-defined on R n and at least continuous. With these observations, we are able to prove the following result.

T h e o r e m 3.3. Suppose a smooth nonlinear system (1.2) satisfies hypothesis (H1) and 12 N S = {0}. Then there exists a C k (k >~O) state feedback control law

u(x) = - fl L°' V(x)T for any 0 < fl < 1 , (3.8) 1 + IIOV/OxllZ~2(x) 1 -4- (Lo~ V(x)) 2

with any C k (k >~O) function ~(x) satisfying

~(x) >>. O(x), (3.9)

such that the equilibrium x = 0 of the closed-loop system (1.2)-(3.8) is globally asymptotically stable.

W. Lin/ Systems & Control Letters 26 (1995) 41-53 47

The proof of this theorem is similar to that of Theorem 3.1 and is therefore omitted for reasons of space. The reader is referred to [12] for additional details.

Remark. Since the submission of this paper, we became aware of the work [5] by Coron. Specifically, Coron has also discussed the stabilization issue of general nonlinear systems with stable free dynamics. Some of the results in [5] such as Lemma B.1 and Corollary 1.6 are related to ours (e.g., Theorems 3.1 and 3.3), but have been derived from different points of view. In particular, a difference between the two papers is that we apply passivity-based and smooth saturation techniques developed in the previous papers [10, 11] to design a globally stabilizing state feedback control law expl ic i t l y while the only existence of such a state feedback control law is proven in [5] under slightly different assumptions.

To illustrate the application of Theorem 3.1, we consider the following example. Note that the affine system of the nonlinear system given below has been studied in [1, 15].

Example 1. Consider a single-input non-affine system

)~1 = X 3 ,

Xa: ~2 = x 3 - x l e x3u4,

3f 3 : --X~ d- U -- XIX2U 3 if" ln(x~ + x 4 + 1)u 7 .

Clearly, Za can be expressed in the form (1.1), with

Ii] I°°l f0(x) ----- x 3 1 , g l ( x ) = , g 3 ( x ) = ,

J -x,x

0 . 0 4 ( x ) = - e x3 , g 2 ( x ) = o s ( x ) = o 6 ( x ) = , ge (x )= l n ( x ~ + x ~ + l )

Let V ( x ) = ½(xl +x3) 2 + 1 (x~ +x~). Then it is easy to show that • = N 3. Moreover,

0 = Lo, V ( x ) = xl +x3 +x~ , (3.10)

0 = Ladrog, V ( x ) : LfoLg , V ( x ) - Lg, Lfo V ( x ) = LfoLg , V (x ) = - 3 x ~ x 3 , (3.11 )

0 = Lad~og , V ( x ) = L[fo,adyog,] V ( x ) : LfoLadlog , V ( x ) = - 9 x 2 x ~ + 6x3x 6 ,

0 = LfoLad~og , V ( x ) = - 6 x 9 - 18xvx 8 + 81x52x 4 , (3.12)

0 = L2foL~d~og, V ( x ) = Lfo(LfoL~d~og ~ V ( x ) ) = -18x~ l - 378xSx 3 + 549x4x37 . (3.13)

From (3.10)-(3.13), it follows immediately that xl = x2 = x3 = 0. Hence S = {0}. By Theorem 3.1, the equilibrium x = 0 of Xa is globally asymptotically stabilizable by the smooth state feedback control law

XI "q-X3 -[-X 3 u7(x) = - ~ 7 ( x )

1 +(Xl +x3 + x 3)e '

with ~7(x) : 1(1 + E7=2(Lg, and E7=2(Lg, V(x) )2 = (x, + x 3 + x 3 ) 2 ( x 2 x a +ln2(x2 + x 4 + l ) ) +xlx2e262~3.

In the remainder of this section, we briefly discuss how Theorem 3.1 can be extended to a multi-input non-affine nonlinear system in the form

X: fo(X)-I- ~ gOil(X)Uil -t- ~ ~ gilil(X)UiiUil-t-'''-t- ~ ~ "'" ~ gil...i/(X)Uii "''nil. (3.14) i i=l i1=1 i2=1 i1=1 i2:1 i,,=1

48 W. Linl Systems & Control Letters 26 (1995) 41-53

For simplicity, in what follows we focus our attention on the system (3.14) with d --- 2. In this case, (3.14) can be expressed in the condensed form

2 = fo(x) + Ol(X)U + ~ u i R i ( x ) u , (3.15) i=1

where 9, (x) = [0°(x) . . . . . 9°(x)] E R n x,,, and Ri(x)E R n×m, 1 <~i<~ m, are smooth functions. Then the following global stabilization result via smooth state feedback can be proven.

Theorem 3.4. Suppose the multi-input non-affine system (3.15) satisfies hypothesis (H1) and St; N O v = {0} Vx E U - R", where St: and f2v are defined in Section 1. Then the system (3.15) is globally asymp- totically stabilizable by bounded smooth state feedback. In particular, the smooth state feedback control law

(Lg, V(x)) T (3.16) U2(X) -~2.x 1('t- + iiLo ' V(x)]l z

with

LR, V(x) 1 [3 0 < [3 < l, LRV(x) a= . ,

• 2(x) = 1 -t- IILRV(x)II 2' LLie. V(x)J

renders the equilibrium x = 0 of the closed-loop system (3.15)-(3.16) 91obally asymptotically stable.

(3.17)

Proof. Similar to the proof of Theorem 3.1, we choose V(x) which satisfies (HI) as a Lyapunov function. Along the trajectory of the closed-loop system (3.15)-(3.16),

V(x) = Lfo V(x) + Lo, V(x)u + ~ UiLR, V(x)u i=1

= Lfo V(x) H-Ls, V(x)u2(x) + uT(x)LR V(x)u2(x)

( LieV(x) 1 ) 0~2(X) . . . . . - I + [3 (Lo, V(x)) y. = Lfo V(x) + 1 + IlLs, V(x)ll 2~s' vtx) 1 + [ILRV(x)II z 1 + [ILo, V(x)lt 2

Since

fl 1 Lie V(x) 1 fl + IlZRV(x)ll z 1 + Ilzo, V(x)ll = ~< 2 '

it is clear that

--1 + [3/2 L V(x)(Ls, V(X)) T ~'(X) <.LfoV(x ) + 0~2(x ) 1 + IlLs, V(x)ll 2 s,

(3.18) ( = Z f ° V ( x ) + 1 + IlZslV(X)t[ 2 Iltg'V(x)[[2 -1 + <~0.

So the closed-loop system is Lyapunov stable. The proof of global asymptotic stability remains the same and one can proceed with exactly the same arguments as those used in the proof of Theorem 3.1. []

Remark 3.5. It can be proven that Theorem 3.1 also holds in the multi-input case. However, the proof becomes very complicated if •/> 3 [12]. The reader is referred to [12] for detailed discussions. Note that Theorem 3.4 is a refinement of Theorem 2.3 proposed in [11]. In particular, the assumption of (H2) of Theorem 2.3, namely the invertibility of the matrix Imxm + (LRV(x)) x Vx E •n, has been shown to be dispensable.

We close this section with an example which illustrates how the problem of global stabilization of a multi-input non-affine system can be solved by bounded smooth state feedback.

I~. Lin/Systems & Control Letters 26 (1995) 41-53 49

Example 2. Let us consider a two-input nonlinear system

ffl = --X2X3 + Ul --X23Ul U2 q- sin(x2x3 )U 2 ,

~ b : 3C2 = 2XIX2X3 + U2 -- (X3 COS Xl )Ul 2 ,

ff3 = XlX2 •

Obviously, it is easy to check that Zb can be written in the form (3.15) with

So(x) = j2x,x~x~ , o , ( ~ ) = [o° (x ) o ° (x ) ] = 1 , L xlx2 0

0 0

For V(x) = 1 (x~ +x~ + (x2 -x~)2) , one is able to show that e = N 3 and

Loo ' V(x) = xl = 0, (3.19)

LoT V(x) = x2 - x~ = 0, (3.20)

LadlooO l/(x) = Ltfo,gO, l V ( x ) = LfoLgo Y(x ) = -x2x3. (3.21 )

Thus it follows from (3.19)-(3.21) that S = {0}. By Theorem 3.3, the bounded smooth state feedback control law

I U l ] = ~2(X) II V(x) )T (3.22) u2 1 + IlZg, V(x),, i(zg'

globally asymptotically stabilizes the equilibrium x = 0 of Zb, where

OV L.,v(x) = ~ [d(x) o°(x)} = Ix, x~-x~].

IlZg, V(x)ll 2 = Zg, V(x) (G V ( x) ) T = x~ + (x2 - x~) 2 , (3 .23)

1 • 2(X) =

1 -I-IILRV(x)IF '

with

-x,.,,; 1 LR V(x ) = L LR2 V(x ) J = xl sin(x2x3 ) J

We should point out that in the present example, the system fails to satisfy hypothesis (H2) of Theorem 2.3 proposed in [11]. In fact, the matrix A(x) := I2x2 +(LR V(x)) T is apparently not invertible Vx E R 3. Therefore,

50 W. LmlSystems & Control Letters 26 (1995) 41-53

global asymptotic stability of the closed-loop system does not follow from Theorem 2.3 is also interesting to note that Byrnes and Isidori [1] considered the affine system of Eb,

)fl ~ Vl ,

Xc : 22 = v2, (3.24)

3f 3 • XlX2,

and proposed a globally stabilizing feedback control law. From the arguments above, we see that the following simplified feedback law,

[ x x3x ] ,3. , V ~ V2 = 2XlX2X3 -~- X 2 - - X2 '

also does the job. This can be verified easily by using the same V(x) for the system (3.24)-(3.25).

[11]. In addition, it

4. A global separation principle

As an application of the global stabilization results developed in the last section, we show how a global separation principle can be proven for a class of input-output non-affine systems of the form

Z l / O : .~ = Ax + g l ( x ) u + g2(x)u 2 + "'" + g:(x)u:, (4.1) y = C x .

As we shall see in a moment, the technique of input smooth saturation whose power has been demonstrated in the previous section also plays an important role in the development of a nonlinear enhancement of the global separation principle.

Theorem 4.1. Consider a SISO non-affine system (4.1) satisfying hypothesis (HI) with V(x) = ½xrPx, P > O, and fo(x) = Ax. Suppose the pair (A, C) is observable, f2 N S = {0} and the functions gi(x), 1 <<. i <~ :, are global Lipschitz on R". Then a Luenberger-observer-like based output feedback control law

= A~ + gl(~)u + g2(~)u 2 + ' " + g:(~)u: + K(y - C~),

flgT(~)p~ (4.2) u = [1 + E~=2(g~(¢)P¢) 2] [1 + (g~(~)p~)2] for some ½ >>.fl > 0

renders the equilibrium (x, ~) = (0,0) of the closed-loop system (4.1)-(4.2) globally asymptotically stable, where K is a constant matrix such that A - KC is Hurwitz.

Proof. Let e(t) = ~ ( t ) - x(t) be the error signal. The closed-loop system can be rewritten as

= (A - KC)e + ~,(gi(~) - gi(x))ui(~), (4.3) i~l

: A ~ -~- E gi(~)ui(~) -- KCe. (4.4) i=1

By assumption, (A, C) is observable. Thus there always exists a constant matrix K such that A - K C is Hurwitz. This, in turn, implies that there is a unique positive definite matrix R satisfying

(A - KC)TR + R(A - KC) = -I .

Let Q(e) =- eTRe. Then a routine calculation shows that

(eTRe) = --eTe + 2eTR Y~.(gi(~) -- gi(x))ui(~)~< -- eTe 1 -- 2iIRll c~/~ i i=1

W. Lin/Systems & Control Letters 26 (1995) 41-53 51

because lu(¢)l < /~ and IIg~(~)-g,(x)ll ~<c;llel[ for ci > O, 1 <~i<<.E. Let fl = min{l,(4llRllE[=ici)-I}. Clearly,

( <) l , (eTRe)<~--eTe 1 -2 f l l IR l [~c i < . - ~ e e<. 22 1 R eTRe"

i=l max( ) From the inequality above, it follows that

Ile(t)[I <~Me -at for some M > 0 and a > 0. (4.5)

On the other hand, let d(t) = -KCe(t) in (4.4). Recall that by Theorem 3.1, the system (4.4) with d(t) = 0 is globally asymptotically stable at ~ = 0 for V(¢) = ½~Vp~. Let f (~ ) = A~ + E[=lOi(~)ui(Q. Obviously, Lf(~)V(~)~O. Moreover, it is easy to check that

V(~)>>-½2m~.(e)ll~ll 2, - ~ ~<llellll~ll

and

f0 +°° f0 +°° Ild(t)lldt<~ [[gfllMe-at dt < +oo.

With this in mind, one concludes from Lemma 2.3 that ~(t) is bounded for all t~>0. The remaining part of the proof is similar to Theorem 2 of [10] for affine systems. Here we give a sketch

proof. Let (e(t), ¢(t)) be a trajectory of the system (4.3)-(4.4) with the initial value (~(0), e(0)). Let r ° denote its to-limit set. Obviously, r ° is nonempty, compact and invariant because every trajectory (e(t), ~(t)) of the closed-loop system (4.3)-(4.4) has been proven to be bounded V(e(O), ~(0)) E R n x R n and Vt >>.0. Moreover, the estimation (4.5) leads to limt_-.o~ e(t) = 0. Therefore, any point in r ° must be a pair of the form (0, ~(t)). Let (0, ~) E r ° and (0, ~(t)) be the corresponding trajectory. This trajectory is characterized by the equation

~(t) = A~(t) + ~ Oi(~(t))ui(~(t)), (4.6) i=1

which is globally asymptotically stable by Theorem 3.1. To put it differently, asymptotic stability of the system (4.3)-(4.4) at (e,~) = (0,0) is completely determined by the flow on the invariant manifold governed by the differential equation (4.6). Since the latter is globally asymptotically stable, so is the closed-loop system (4.3)-(4.4). []

Remark 4.2. It is predictable that Theorem 4.1 can be extended to the case of multi-input systems. In fact, the proof of the global separation principle for MIMO systems has been carried out and the details can be found in [12].

Remark 4.3. In the case of f -- 1 (i.e., affine systems), Theorem 4.1 provides an alternative to the problem of global stabilization by dynamic output feedback. The difference between Theorem 4.1 and Theorem 2 of [10] is that the dynamic compensator (4.2) proposed here is smooth while the dynamic output feedback controller given in Theorem 2 of [10] is only continuous.

5. Conclusions

In this paper, we have proved that a class of non-affine nonlinear systems which satisfy hypothesis (H1) and certain controllability-like rank conditions can be globally asymptotically stabilized by arbitrarily small smooth state feedback. A globally stabilizing smooth (resp. analytic) state feedback control law has been constructed explicitly. Based on this result, we have been able to establish a global separation principle for a class of non-affine systems which are "controllable" and "observable". It should be pointed out that the

52 W. Lin/Systems & Control Letters 26 (1995) 41-53

global stabilization results developed in the paper rely on the powerful technique, called the i n p u t s m o o t h

s a t u r a t i o n , which has been used in the previous paper [10] for affine systems. Theorem 3.2 shows that a n y a n a l y t i c s y s t e m (1.2) can always be globally asymptotically stabilizable by

analytic state feedback if appropriate conditions are met. Since system (1.2) can be approximated by (1.1), one might naturally ask if for a large enough integer : > 0, the smooth state feedback law (1.1) also globally asymptotically stabilizes analytic nonlinear systems. This is an interesting and important question raised in this paper. The problem remains unsolved and open at this moment. However, the following simple example gives a positive answer to the question.

Example 3. Consider an analytic nonlinear system

(xu)i Zc: :~ = e xu - 1 : f ( x , u ) : i! (5.1)

i=1

Clearly, f o ( x ) = f ( x , O ) : 0 and 9 1 ( x ) = ( O f / O u ) ( x , O ) : x . Choose V ( x ) = ix2 for the system (5.1). It is trivial to check that [2 n S = {0}. Hence, the {th order approximation system of (5.1),

: (xu)i = E ; i--T- ' ( 5 . 2 )

i=l

is globally asymptotically stabilizable by the smooth state feedback law

1 x2 u : ( x ) = 2 z~ C : ( x ) x 2. (5.3)

( x i + l ~ 2 1 "~-X 4 I +

It is interesting to note that (5.3) also globally asymptotically stabilizes the original system (5.1). In fact, using u:(x) yields the closed-loop system

= e -c:(x)x' - 1, (5.4)

with 0 < c : ( x ) < 1 Vx E R n. This system is globally asymptotically stable at the equilibrium x = 0, simply because~ > 0 i f x < 0 a n d ~ < 0 i f x > 0 .

Acknowledgements

The author would like to thank Chris Bymes and David Elliott for helpful discussions.

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