IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 1 1 , NOVEMBER 1991 1241

Systematic Design of Adaptive Controllers for Feedback Linearizable Systems

Ioannis Kanellakopoulos, Student Member, IEEE, Petar V. Kokotovic, Fellow, IEEE, and A. Stephen Morse, Fellow, IEEE

Abstract-A systematic procedure is developed for the design of new adaptive regulation and tracking schemes for a class of feedback linearizable nonlinear systems. The coordinate-free geometric conditions, which characterize this class of systems, do not constrain the growth of the nonlinearities. Instead, they require that the nonlinear system be transformable into the so-called parametric-pure-feedback form. When this form is strict, the proposed scheme guarantees global regulation and tracking properties, and substantially enlarges the class of non- linear systems with unknown parameters for which global stabi- lization can be achieved. The main results of this paper use simple analytical tools, familiar to most control engineers.

I. INTRODUCTION OST of the research activity on adaptive control of M nonlinear systems [1]-[15] is still focused on the

full-state feedback case [ 11 - [ 131, although output-feedback results are beginning to appear [14], [15]. The full-state feedback case continues to be a challenge because of the severe restrictions of the two currently available types of schemes: the uncertainty-constrained schemes [ 11 - [4], [lo], [ 1 11 assume restrictive matching conditions, and the non- linearity-constrained schemes [5] - [9], [ 121 impose restric- tions on the type of nonlinearities.

The systems to which uncertainty-constrained schemes can be applied may contain all types of smooth nonlinearities and are fully characterized by coordinate-free geometric con- ditions [2], [3], [ 1 11, which, unfortunately, are quite restric- tive. On the other hand, the applicability of nonlinearity- constrained schemes is restricted by coordinate-dependent growth conditions on the nonlinearities, which may exclude even certain linear systems [ 131. Less restrictive coordinate-free growth conditions, written in terms of a control Lyapunov function, are used in the schemes of [6]-[8]. Unfortunately, the existence of such a Lyapunov function cannot be ascertained a priori.

Manuscript received September 18, 1990; revised May 14, 1991. Paper recommended by Associate Editor, J. W. Grizzle. The work of I. Kanel- lakopoulos and P. V. Kokotovic was supported in part by the National Science Foundation under Grant ECS-87.15811 and in part by the Air Force Office of Scientific Research under Grant AFOSR 90-0011. The work of A. S . Morse was supported by the National Science Foundation under Grant ECS-88-05611 and EDS-90-12551.

I. Kanellakopoulos is with the Coordinated Science Laboratory, Univer- sity of Illinois, Urbana, IL 61801.

P. V. Kokotovic is with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93 106.

A. S. Morse is with the Department of Electrical Engineering, Yale University, New Haven, CT 06520-1968.

IEEE Log Number 9103022.

The new adaptive control scheme developed in this paper can be classified as the least restrictive uncertainty-con- strained scheme available for feedback linearizable systems. It significantly extends the class of nonlinear systems for which adaptive controllers can be systematically designed.

Among the advantages of the new scheme are its concep- tual clarity and wide applicability. Its stability proof, based on a straightforward Lyapunov argument, is particularly simple. The coordinate-free geometric conditions, character- izing the class of systems to which the new scheme is applicable, do not constrain the growth of the nonlinearities. Instead, they require that the nonlinear system be trans- formable into the so-called parametric-pure- feedback form. Furthermore, in the case of systems transformable into the more restrictive parametric-strict- feedback form, the new adaptive scheme guarantees global regulation and tracking properties.

The presentation is organized as follows: First, we address the regulation problem. In Section I1 we characterize the class of single-input nonlinear systems to which the new scheme is applicable. The design procedure is presented in Section 111, and the simple proof of stability is given in Section IV. In Section V we give the conditions under which the stability of the closed-loop system is global. Then, in Section VI, we use the design procedure to solve the tracking problem for a class of input-output linearizable systems with exponentially stable zero dynamics. In Section VI1 we illus- trate this procedure on some benchmark examples, and discuss its properties in comparison to previous results. Fi- nally, some concluding remarks are given in Section VIII. The reader unfamiliar with differential geometric results for nonlinear systems can follow the presentation starting with Section I11 and then omitting Propositions 5.3 and 6.3.

11. THE CLASS OF NONLINEAR SYSTEMS The adaptive regulation problem will first be solved for

single-input feedback linearizable systems that are linear in the unknown parameters:

4 r 4 1

where {E W is the state, U E W is the input, 8 = [ O I , - - , BPIT is the vector of unknown constant parameters, and f i , g;, 0 5 i 5 p , are smooth vector fields in a neigh-

0018-9286/91$01.00 01991 IEEE

~ -~ ~ _ _ ~-

1242 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 11, NOVEMBER 1991

, O s i r n - 1

(2.11)

where a 1 a x I , . . . , a /ax, are the coordinate vector fields associated with the x-coordinates. Because of (2.1 l), the parametric-pure-feedback condition (2. 5 ) , expressed in the x-coordinates, states that

borhood of the origin c = 0 with f i (0) = 0, 0 I i I a , go(0) f 0.

The design of the adaptive scheme assumes that, using a parameter-independent diffeomorphism x = +( r ) , the sys- tem (2.1) can be transformed into the ~arametric-~ure- feedback form:

x 1 = x2 + OTT,( XI, x*) i 2 = x3 + OTY2(XI , x2, xg)

with 1 I i ~ p . (2.12)

y0(O) = 0,y1(O) = * * . = y,(o) = 0 , PO(O) f 0 . But (2.12) can be equivalently rewritten as

a (2.3) g i = P i ( X) - 3

ax, Necessary and sufficient conditions for the existence of such a diffeomorphism are given in the following proposition.

= 0, transforming (2.1) into (2.2), exists in a neighborhood Proposition 2. I : A diffeomorphism x = +( r ) , with +(O)

B, C U of the origin if and only if the following conditions are satisfied in U . a a

a 8x1 8x2

+ yn-1 , i (xI ,* .* , xn)- + y n , ; ( x l , * - * , XJ-,

a f i = y l , i ( ~ l , x2)- + y2,i(X1, X 2 , X3)- + * . *

ax, i) Feedback Linearization Condition: The distributions

0 I i 5 n - 1

(24)

ax,- I 9 = span { g o , adfo g o , - , ad), go} , 1 4 i I p . (2.13)

Furthermore, since +(O) = 0 and f i (0) = 0, 1 I i I p , we conclude from (2.13) that

yl(o) = - * . = y,(o) = 0. (2.14) are involutive and of constant rank i + 1.

ii) Parametric-Pure-Feedback Condition:

g i E go,

[x, f i ] E % J + I , V X E ~ J , 0 4 j 1 n - 3 , 1 s i ~ p . (2.5)

Suficiency: As proved in [16], condition i) is sufficient for the existence of a diffeomorphism x = +({) with +(O) = 0 which transforms the system

Proof:

3: = fO(T) + g , ( r ) u , f o p ) = 0 , go(0) + 0 (2.6) into the system

x. I = x.

i n = Y ~ ( x ) + P o ( x ) ~ ,

1 I i s n - 1

(2.7)

YO(0) = 0 , Po(0) f 0 . (2.8)

with

Combining (2.9), (2. lo), (2.13), and (2.14), we see that in the x-coordinates the system (2.1) becomes (2.2).

Necessity: If there exists a diffeomorphism x = 4({) that transforms (2.1) into (2.2), one can directly verify that the coordinate-free conditions i) and ii) are satisfied for the

U Remark 2.2: A special case of the parametric-pure-feed-

g i E go, f i e Y1, 1 I i s p (2.15)

introduced in [2], [3] and formulated in [l] as a strong linearizability condition. This is clear from the proof of Proposition 2.1 : if (2.5) is replaced by (2.15), then (2.13) still holds, but with y1 = 0, e , yn-2 = 0. Then, the system (2.1) is expressed in the x-coordinates as

system (2.2), and hence for the system (2.1).

back condition (2.5) is the extended-matching condition

.x1 = x2 x2 = x3

Hence, in the coordinates of (2.7) we have

a a a x , 4 = x,-1 (2.9) xnpl = x, + e T y n p l ( x 1 , - . , x,) +X,--- f - x - + . . . ax1 ax,- 1 ax, 0 - 2

U

KANELLAKOPOULOS et al. : SYSTEMATIC DESIGN OF ADAPTIVE CONTROLLERS 1243

Remark 2.3: The expressions given in (2.4) and (2.5) for the feedback linearization and parametric-pure-feedback con- ditions are convenient for the proof of Proposition 2.1, but they are not minimal. As shown in [17], [18], the equivalent minimal form of (2.4) is

is involutive and 9 n - has constant rank n . gn-2 (2.17)

The minimal form of (2.5) is

[ go > f;] E gj+1 , O s j s n - 3 , l s i r p . (2.18)

The equivalence of (2.5) and (2.18) follows from the involu-

Remark 2.4: The term parametric-pure-feedback indi- cates that the nonlinearities multiplying unknown parameters are allowed to depend only on state variables that are fed back when the system is written in the x-coordinates. This term should not be confused with the term pure-feedback systems used in [19] to denote the class of feedback lin- earizable systems, nor with the pure-feedback systems for which the nonlinearity-constrained scheme of [5] was devel-

tivity of 2 . 0

oped. 0

111. ADAPTIVE SCHEME DESIGN

Since the diffeomorphism x = 4( { ) does not depend on the unknown parameter vector 8 , Proposition 2.1 gives an a priori verifiable characterization of the class of nonlinear systems to which the new adaptive scheme is applicable. Assuming that the transformation of (2.1) into (2.2) has been performed, the new adaptive scheme is designed for the parametric-pure-feedback system

x. I = x. + BTyj(x1;*., x i + , ) , i n = Y o ( X ) +BTyn(X) + [Po(.) + 8TP(X)]U, (3.1)

1 s is n - 1

with

yo(o) = O,y,(O) = * * . = yn(o) = 0 , PO(O) f 0. (3 4

Recall that yo, Po, and the components of P and y;, 1 s i 5 n, are smooth nonlinear functions in B,, a neighborhood of the origin x = 0.

The following step-by-step procedure was inspired by an idea contained in an early linear result [20]. However, the intuition behind our nonlinear development becomes much clearer if the procedure is interpreted as the interlacing of the steps of the chain of integrators method [2 11 - [23] with the design of a new parameter estimator at each step.

Step 0: Define z1 = x,, and denote by c,, c 2 , * a , c, constant coefficients to be chosen later.

Step I : Starting with

il = x2 + BTy1(x,, x2) (3.3) and following the chain of integrators method, we see that, if x2 were the control input and 8 were known, the

control law

x2 = - c l z l - e T Y 1 ( X l , x2) (3.4) would stabilize and regulate the system (3.3). As 8 is un- known, this control law is modified to its certainty- equivalence form

x2 = - c , z , - 6 h ( x , , x2) (3.5) where 29, is an estimate of 8. Then, (3.5) together with the update law

dl = ~ l Y I ( X 1 9 x2) (3 4 would render the closed-loop system stable and would achieve the regulation of 2, . However, x2 is not the control. There- fore, define the new state z2 as the difference between the actual x2 and its desired expression (3.5):

22 ClZ, + x2 + qy1( XI 3 x2). (3.7) To complete Step 1, substitute (3.7) into (3.3)

i , = -CA + z2 + ( 6 - d,)Ty,(Xl, x,) - c , z ~ + z2 + ( e - 6 1 ) T ~ , ( ~ 1 , z2,dI) (3.8)

and rewrite the update law (3.6) for the parameter estimate 6, in the form

9, = Z l W , ( 21, 2 2 9 6 , ) . (3.9) In addition to the error system (3.8) and the update law (3.9), Step 1 has introduced the new state variable z2, which is to be regulated in Step 2.

Step 2: Using the definitions for z , , z 2 , and d,, write i2 as

= ( 1 + 8:- aa::) [x3 + e T y 2 ( ~ , , x 2 , x3)] In this system, we will think of x3 as our control input. As in Step 1, we need an estimate for 8. Since the update law for 19, has already been defined in (3.9), 29, cannot be used again. Therefore, let 62 be a new estimate of 8 and define the new state z3 as

1244 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 1 1 , NOVEMBER 1 9 9 1

Substitute (3.1 1) into (3.10) to obtain be a new estimate of 0 and choose the control U as

Then, let the update law for the new estimate 6, be Substitute (3.19) into (3.18) to obtain

8 2 = z2w,(z1, 2 2 , 2 3 r f 4 , % ) . (3.13)

Step i (2 I i I n. - I): Using the definitions for e , 6;- 1, express the derivative of z; as z I , - * , zi and 6,,

with pi, rC/; appropriately defined smooth functions. Let 6; be a new estimate of 0 and define the new state z ; + ~ as

.[$;+ ( l + @ ) ... ( l+6:- lT)T;] ay;- 1 ax,

A - C J , + z ~ + ~ + ( e - 8;) * w;(z1;-, z ; + l , ~ 1 , * - * , ? g i ) . (3.16)

Then, let the update law for di be

4; = ziwi( zl; * , z ; + ~ , 6,; * , 6;).

express the derivative of z , as

(3.17)

Step n: Using the definitions for zl; a , z , and 8,, ... , 6, -

where (3.19) is used in the definition of w,. Finally, let the update law for the estimate 6, be

The feasibility of this design procedure and the stability of the resulting closed-loop adaptive system are analyzed in the next section.

IV . FEASIBILITY AND STABILITY The above design procedure has introduced a control law

defined by (3.19)-(3.20) and a set of new coordinates z1;--, z , defined by (3.15). In order to ensure that the procedure is feasible, we construct in Proposition 4.1 an estimate BC Rn(l+p) of the feasibility region such that for all (x, 6,, - , 6,) E 9 the denominator in (3.19) is nonzero and the coordinate change (3.15) is one-to-one, onto, contin- uous, and has a continuous inverse.

Proposition 4. I : Suppose the parametric-pure-feedback form (3.1) of the system (2.1) exists in B,, and let B, C R be an open set such that 6 E B, and

V X E B , , V ~ ; E B , , 1s i I n - 1 (4.1)

V X E B , , V ~ , E E , . (4.2) IPo(x) + 6:P(x)I > 0, Then, the set 9= B, x B, is a subset of the region in which the design procedure of Section I11 is feasible.

Proof: Obvious, since (4.1) and (4.2) guarantee that in B, x B, the denominator in (3.19) is nonzero and (3.15) is

Remark 4.2: In general, the feasibility region is not global. However, this is not due to the adaptive scheme because, even when the parameters 0 are known, the feedback lin- earization of the system (3.1) can only be guaranteed for

8 % - 1 i n = ( l + @ ) * * . ( l + 6 : p l - ) ax2 ax,

uniquely solvable for x i . 0 [Po(.) + e@(.)] U + p n ( z , 61, * * * 6,- I ) + eT$,( Z 61 * * ,d,- I ) ,

(3.18)

with p,, $, appropriately defined smooth functions. Let 6,

KANELLAKOF'OULOS et al. : SYSTEMATIC DESIGN OF ADAPTIVE CONTROLLERS 1245

19 E Bo C W p , an open set such that

V X E B , , V ~ E B ~ , ~ ~ ~ ~ ~ - 1 (4.3)

1/3,(x) + eT/3(x)l > 0, V X E B , , V O E B ~ . (4.4) 0

In the feasibility region, the adaptive system resulting from the design procedure can be expressed in the z-coordinates as

in-, = - C n - , Z n - , + 2, + ( e - dn-,)' * W n - , ( 2 , ? , z n , 8 , , - , 8 , - J

2, = -C,Z, + ( e - 8n)TWn(z,61,..*,6n) 6; = z i w i , 1 I i I n. (4.5)

A nice property of this system is that its stability can be established using the quadratic Lyapunov function

1 1 "

2 2 ; = I v ( ~ , ~ , , - - , ~ , , ) = - z T z + - E ( e - 6;)'(e - 6;).

(4 4 The derivative of V( z , I?, , * (4.5) is

, 6,) along the sohtions of

n - 1

c;z; + ( e - 6 ; ) T ( z i W ; + si)] + z;z;+1 i = 1 i = 1

n n - 1

i = 1 i= 1 = - c;z? + z;z;+1. (4.7)

At this point we can choose the coefficients c,, - - , c, to guarantee that V is negative semidefinite. The choice ci 2 2, for all i = 1, * * , n, yields

Ps -112((2. (4.8)

z = 0 , 8, = e,- ,d , = e (4.9) This proves the uniform stability of the equilibrium

of the adaptive system (4.5). To give an estimate Q of the region of attraction of this equilibrium, we note that Q must be a subset of our estimate 8 of the feasibility region. We also note that, by definition, the point x = 0 is contained in B,, 0 E B,, and ~ ~ ( 0 ) = = ~ ~ ( 0 ) = 0. Combining these facts with the definitions of zl,. - e , z , , 6,, - - , 8,, it is straightforward to show that the equilibrium (4.9) coincides with the point x = 0, 8, = 6, - - , 8, = e , and is therefore contained in F. Let Q ( c ) be the invariant set of (4.5) defined by V < c, and let c* be the largest constant c such that Q(c) C 9. Then, an estimate Q of the region of attraction is

* *

Q = Q ( C * ) = { ( z ,81 ,* . . , 79n) : V(z,61,- ,I?n) < c * } , c* = arg sup { c } . (4.10)

O ( C ) C 9

Remark 4.3: It can be expected that the above estimate is not tight because the choice of the unity gains in the update laws was made for simplicity. With some a priori knowl- edge about the shape of 8, different adaptation gains can be

U Next, from the invariance theorem of LaSalle, we con-

clude that for all initial conditions ( z , 6,, * * - , 6,) = , E Q , the adaptive system (4.5) has the following regulation properties:

lim z ( t ) = 0 ,

found so that Q is maximized by a better fit of F.

lim i ( t ) = 0 , lim 8;( t ) = 0 , t + m t+m i-+ 03

1 I i I n. (4.11)

Finally, to establish that the original coordinates { are regu- lated to zero, we note that (4.1) guarantees, first, that the solution x 2 = - = x , = 0 of the system of equations

x i + , + eTyi(o, x 2 ; - - , x i + l ) = 0, 1 5 i I n - 1 (4.12)

is unique in B, x B,, and, second, that x , , * * , x , can be expressed as smooth functions of z , 6,; * , 6, using (3.15). Combining these two facts with (4.11), we obtain

lim x l ( t ) = 0, lim x i ( t ) = 0, 1 I i 5 n. (4.13)

Using an induction argument, it is now shown that x j ( t ) + 0 a s t + m , l I i < n .

t-m t-m

For i = 1, we have x l ( t ) + 0 as t -+ 00. For i = k , 2 5 k I n, we assume that x j ( t ) + 0 as

t -+ CO, 1 I j 5 k - 1. Then, from (4.13) we have

= o (4.14) and from the uniqueness of solutions of (4.12) we conclude that x k ( t ) -+ 0 as t + CO.

Hence, x ( t ) + 0 as t + CO. Finally, since x = 4({) is a diffeomorphism with 4(0) = 0, regulation is achieved in the original coordinates r, namely

lim { ( t ) = 0. (4.15) t + m

The above facts prove the following result. Theorem 4.4: Suppose that the system (2.1) satisfies

Proposition 2.1 and that the design procedure of Section 111 is applied to its parametric-pure-feedback form (3.1). Then, the equilibrium (4.9) of the resulting adaptive system (4.5) is uniformly stable and its region of attraction includes the set Q defined in (4.10). Furthermore, regulation of the state ( ( t ) is achieved for all initial conditions in Q. 0

V. GLOBAL REGULATION

There are strong theoretical and practical reasons for in- vestigating whether the stability properties of an adaptive system can be made global in the space of the states and parameter estimates. Systems with a finite region of attraction may not possess a wide enough robustness margin for distur- bances and unmodeled dynamics. Furthermore, it is usually

JI n

1246 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 11, NOVEMBER 1991

hard to find nonconservative estimates of finite regions of attraction.

Another aspect of the global stability issue is the compari- son of the proposed adaptive controller to its deterministic counterpart, that is, the controller that would be used if the parameter vector 8 were known. Suppose that for all values of 8 there exists a deterministic controller which achieves global stabilization and regulation of the system. If, with 8 unknown, the proposed adaptive controller does not achieve the same global stability, this loss is clearly due to adapta- tion.

The stability result of Theorem 4.4 is not global. How- ever, as pointed out in Remark 4.2, this is not due to adaptation, because for parametric-pure-feedback systems global stability may not be achievable even with 8 known. In Proposition 5.3, we define the class of parametric-strict- feedback systems, for which a globally stabilizing con- troller exists when 8 is known. We then prove that for this class of systems our adaptive scheme guarantees global stabil- ity when 8 is unknown.

In order to characterize the class of parametric-strict- feedback systems, we use the following assumption about the part of the system (2.1) that does not contain unknown parameters.

Assumption 5 . I : There exists a global diffeomorphism x = +({), with 4(0) = 0, transforming the system

s = f O ( H + g o ( s ) u ( 5 4 into the system

x . I = x.

i n = Y ~ ( x ) + P o ( x ) ~ , 1 I is n - 1

(5 .2)

y0(O) = 0, Po(.) # 0 VXEW,. (5.3) with

Remark 5.2: The local existence of such a diffeomor- phism is equivalent to the feedback linearization condition (2.4). At present there are no necessary and sufficient condi- tions verifying the global validity of this assumption. Suffi- cient conditions for Assumption 5 .1 are given in [24], while necessary and sufficient conditions for the case where p0(x)

0 Proposition 5.3: Under assumption 5 . 1 , the system (2.1)

is globally diffeomorphically equivalent through x = $({) to the parametric-strict- feedback system

= const. can be found in [25], [26].

X i = x , + ~ + 8 T ~ ; ( ~ I , * * * , x i ) , l s i s n - 1 i n = YO( X) + oTy,( X ) + Po( X) U (5 .4)

if and only if the following condition holds globally. Parametric-Strict-Feedback Condition:

g ; = 0 , [ x , ~ ; ] E Y ~ , V X E Y J , O s j s n - 2 ,

1 I i s p , ( 5 . 5 )

with Y J , 0 5 j I n - 1 , as defined in (2.4). Proof: The proof is very similar to that of Proposition

2.1. First, because of the assumptions that the diffeomor- phism x = 4( {) is global and Po( x) # 0 Vx E W, the dis- tributions 9 I , 0 I j I n - 1 , are globally defined and can be expressed in the x-coordinates as

, 0 1 j 1 n - 1 .

(5 4 To prove sufficiency, note that if the parametric-pure-feed- back condition (2.5) of Proposition 2.1 is replaced by the parametric-strict-feedback condition (5 3, then (2.12) is re- placed by

g; = 0 ,

2 s j 1 n ,

1s i ~ p . (5 .7 )

Thus, the expression for f i in (2.13) becomes

l i i i p . ( 5 . 8 ) The necessity is again straightforward. 0

The above proposition gives a geometric characterization of the class of systems for which the following global proper- ties can be achieved.

Theorem 5.4: Suppose that the system (2.1) satisfies Proposition 5 . 3 and that the design procedure of Section I11 is applied to its parametric-strict-feedback form (5.4). Then, the equilibrium

z = 0 , 29, = 8 , * . . , 1 9 , = 8

of the resulting adaptive system is globally uniformly stable. Furthermore, regulation of the state { ( t ) is achieved:

lim { ( t ) = 0 (5 .9) tm

for all initial conditions in Q = W n ( l + p ) . Proof: When the adaptive design procedure (3 .3 ) -

(3.22) is applied to the system (5.4), then for all f l i E Wp, 1 I i I n , the change of coordinates (3 .15) is one-to-one, onto, continuous, and has a continuous inverse, and the control (3.19) is well-defined, since

(x) = 0, P(x) = 0, Po(.) # 0, VXEW. a Y; ax;+ 1

(5.10)

Hence, (4.1)-(4.2) are trivially satisfied on 8= B, X B, = W(+p), and from (4.10) we conclude that Q = Wn(l+). 0

Remark 5.5: The results of Sections 11-V can be ex- tended to multiinput systems. We do not present this exten-

0 sion here, but refer the reader to [27].

KANELLAKOPOULOS et al. : SYSTEMATIC DESIGN OF ADAPTIVE CONTROLLERS 1247

VI. GLOBAL TRACKING

Every regulation result in Sections 11-V has its tracking counterpart. For brevity, we restrict our presentation to the tracking version of the global regulation result in Section V. The counterparts of nonglobal regulation results can be ob- tained using the same Lyapunov function argument as in this section to determine an invariant set in which asymptotic tracking is guaranteed.

Consider the nonlinear system P

3: = f o ( r ) + C e i f i ( r ) + go(r)u i = 1

g P - ' is involutive and of constant rank p . (6.9)

The sufficiency of these conditions is a consequence of Proposition 10 in [28]. The necessity can be easily estab- lished by verifying that (6.7)-(6.9) hold in the coordinates of (6.4).

Furthermore (see Theorem 5.5 and Corollary 5.7 of [29]), (6.3) is globally equivalent to (6.4) if and only if the following conditions are satisfied for all {E W":

0 I i I p - 2 (6.10)

LgoLJo- h # 0 (6.11)

LgoLffoh = 0 ,

9 p - is involutive and of constant rank p (6.12) Y = h ( r ) (6.1) where W" is the state, U E R is the input, y E R is the output, 8 = [ e ,, - e , BPIT is the vector of unknown constant parameters, h is a smooth function on W" with h(0) = 0, and the vector fields g o , f i , 0 I i I p , are smooth on W" with g ( r ) # 0 v ~ E W " , f i (0) = 0, 0 5 i I p. We first for- mulate the input-output counterpart of Assumption 5.1.

Assumption 6.1: There exist n - p smooth functions $ i ( r ) , p + 1 I i I n , such that the change of coordinates

XI = h ( r )

x2 = L f o h ( r )

x3 = L2fh ( r )

X, = LTO-'h(r)

xi = $ i ( r ) , p + 1 s i I n (6.2) is a global diffeomorphism x = 9(r) transforming the sys- tem

3: = f o ( r ) + go(r)u Y = h ( r ) (6.3)

into the special normal form

x, = x2

Xp-1 = x,

X, = Y o ( 4 + P O ( X b X' = *o( y , x')

Y = X I , (6.4) with

yo(0) = LJoh(0) = 0 , @o(O,O) = 0 (6.5)

&(x) = LgoLT0-'h(r) # 0 , VXEW". (6.6)

Remark 6.2: In order for (6.3) to be locally equivalent to (6.4), it is necessary and sufficient that the following condi- tions hold in a neighborhood of the origin r = 0:

LgoL>oh = 0 , 0 I i I p - 2 (6.7)

the manifold

M = { (ER": h ( r ) = L f o h ( r ) = =LTo- 'h(r) = O } (6.13)

is connected, and the vector fields go, adfo go, * - , ad yo- 'go are complete where

0 We are now ready to formulate the input-output counterpart of Proposition 5.3.

Proposition 6.3: Under Assumption 6.1, the system (6.1) is globally diffeomorphically equivalent to the parametric- strict-feedback normal form

x. ' I = x. + 8 T ~ i ( ~ 1 , * * * , xi, x'), X p = Y o ( 4 + ~ T Y p ( X ) + Po(.>.

X' = ao( y , x') + 8,ai( y , x')

1 I i I p - 1

P

i = 1

Y = X I (6.15)

if and only if the following condition holds globally. Parametric-Strict-Feedback Condition:

[ x , f i ] E g', V X E gj, 0 I j I p - 2,11 i ~p (6.16)

with Y J , 0 I j I p - 1, as defined in (2.4). Proof: The proof follows closely that of Proposition

5.3. First, because of the assumptions that the diffeomor- phism x = $( C) defined in (6.2) is global and that Po( x) # 0 vx E W", the distributions 59 ', 0 I j 5 p - 1, are globally defined and can be expressed in the x-coordinates as

(6.17)

LgoL;o-lh(0) # 0 (6.8) The sufficiency follows from the fact that, by (6.16) and

I248 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 11, NOVEMBER 1991

Step 1: Starting with

i, = x2 + eTyl( XI, x') - P, (6.22) let 8, be an estimate of 0 and define the new state z2 as

2 I j I p , 1 I i I p . (6.18) z2 clzl + x2 + 8Tyl( x l , X') - L, Thus, the expression for f i in the x-coordinates is Li c,z, + x2 + 6~w,(z,, x', Y , ) - L,, Cl 2 2.

a a (6.23) f; = y , , i ( x , , x')- + Y 2 , i ( X l t x2, xr)- + * . *

ax2 Substitute (6.23) into (6.22) to obtain ax1 a i, = -C,Z~ + z2 + ( e - 81)T~l(~1, XI, U , ) . (6.24) + Tp- l , ; (X1,** . , xp-19 xr)-

axp- 1 Then, let the update law for the parameter estimate 6, be

a ax,

+ T,.i(X,,"*, x,, x')- d, = ZIWI( z1 > x', Y , ) . (6.25)

a + 2 + i , j ( x r ) axi, 1 I i I p , (6.19) Step 2: Using the definitions for zI, z 2 , and d,, Write i2

j = p + l as

The necessity is again straightforward. 0 Remark 6.4: To obtain the input-output counterpart of

Proposition 2.1, one just needs to replace the feedback linearization condition (2.4) with conditions (6.7)-(6.9), and the parametric-pure-feedback condition (2.5) with

i 2 = Cl[ -C,Z1 + 22 + (0 - 29,)TWl(Z1, X', Y , ) ] + x3 + OTy2( XI, x2, x') + ZIW,(ZI, x', Y, )Tyl (X, , x')

g i E go,

[ x, f;] E 9J+I , V X E Y J , 0 1 j 1 p - 2 , 1 I i I p . (6.20)

0 As in most tracking problems, we need an assumption

about the stability of the x'-subsystem of (6.15). Assumption 6.5: The x'-subsystem of (6.15) has the

bounded-input bounded-state (BIBS) property with respect to y as its input.

It was shown in [9, Proposition 2.11 that the following conditions are sufficient for Assumption 6.5 to be satisfied:

i) the zero dynamics of (6.1) are globally exponentially stable, and

ii) the vector field CP = CPo + C~=)=O,CP., in (6.15) is glob- ally Lipschitz in x.

These conditions are more convenient for nonglobal re- sults, where i) can be used to estimate the region of attraction via a converse Lyapunov theorem. However, they are too restrictive for global results. For example, the system x' = - ( x ' ) ~ + y 2 violates both i) and ii), but is easily seen to satisfy Assumption 6.5.

The control objective is to force the output y of the system (6.1) to asymptotically track a known reference signal y,( t ) , while keeping all the closed-loop signals bounded.

Assumption 6.6: The reference signal y,( t ) and its first p derivatives are known and bounded.

To achieve the asymptotic tracking objective, the design procedure presented in Section I11 is modified as follows:

Step 0: Define z, = X I - y r . (6.21)

P

Let d2 be a new estimate of 8 and define the new state z3 as

~ 3 ~ ~ 2 ~ 2 + ~ 3 + ( ~ 2 ( ~ 1 , ~ 2 , ~ ~ , 8 , , ~ l , L r , j i r )

+ 8;w2(z,, z2, x', 81, Y , , L,), c2 2 2. (6.27)

i2 = -c2z2 + 2 3 + (e - 6 2 ) T ~ 2 ( ~ 1 , ~ 2 , x r , 81, Y r , Lr). (6.28)

Substitute (6.27) into (6.26) to obtain

Then, let the update law for the new estimate 8, be

9 2 =z~w,(z~, ~ 2 9 ~ ~ 2 8 1 3 ~ r 9 Y r ) . (6.29)

Step i (2 5 i I p - I): Using the definitions for express the derivative of zi as zl,. . - , zi and d, , * , 4-

z. ., = x. , + I + Cpi(Z1,"', zi, x r , 6 1 , * * * , ~ ; - , 9 Yr 3 * * 9 Y?)

+ eTW;(ZI,"-, zi, x ' , 8 1 , * * - , 6 ; - , , y , , * , y:i- 1)) . (6.30)

Let 8i be a new estimate of 8 and define the new state zi+,

KANELLAKOPOULOS et al. : SYSTEMATIC DESIGN OF ADAPTIVE CONTROLLERS 1249

as

z; , , c,z; + X j + ] + $o;(z1,.**, z ; , x r , 6 1 , * * * , d i - , , Y,,*.*J:) + I 9 ~ w ; ( z 1 , * - , zi, x r ,d , , * - * , t 9 ; j - 1 ,

y,, * * - , J y - l ) ) , c; L 2 . (6.31) Substitute (6.31) into (6.30) to obtain

2 . = - c . z , I + 2. r + l + (0 - di)T * w i ( z 1 ; * - , z i , x, d l , * * - , d;- , , y , . , * * * , $- ) ) . (6.32)

Then, let the update law for f i i be

9; = qw;( 2 , ; * , z i , x r , 6, ; * , d i & , , Y,. * - 7 Y:;- ). (6.33)

Step .p: Using the definitions for z , ; * * , z , and 9, , - -, 6, - , , express the derivative of z , as

z , = p,(x)u + $ o p ( z l , , z,, x r , ~ 1 ? * ? ~ , - 1 ~ Y , 7 . * 7 Y:,) + e T w , ( z l , - . , Z , , X ~ , ~ ~ , - . J J , - ~ ,

y,, - , y y ) ) . (6.34) Let 6, be a new estimate of 0 and choose the control U as

Substitute (6.35) into (6.34) to obtain

i, = -c,z, + ( e - ? 9 p ) T w p ( z 1 , . . - , z,, x, ? 9 1 , ~ ~ ~ , 6 p ~ l , y r , ~ ~ ~ , ~ ~ p ~ ~ ) . (6.36)

Finally, let the update law for the estimate do be

9, = z ,wp(z , , * , z p , X r , ~ l , . . . , 6 p q p - l , Y , . . . 3 J ) ; P - U ) . (6.37)

As was the case in the regulation result of Section V, the assumptions of Proposition 6.3 guarantee that the design procedure (6.21)-(6.37) is globally feasible. The resulting closed-loop adaptive system is given by

i l = - c C , z , + z 2 + ( e - 6 , ) T ~ 1 ( z 1 , X r ~ Y r )

l s i l p

Y = z1 + Y r . (6.38)

The stability and tracking properties of (6.38) will be estab- lished using the quadratic function

1 p

2 = - [ z ; + ( e - fii)(e - 4)]. (6.39)

The derivative of V, along the solutions of (6.38) with ci 2 2 , l I i I p , is

D 0- 1

= - ciz; + E zizi,, i = I i= 1

P

5 - C Z 5 0 . (6.40) i = I

This proves that V, is bounded. Hence zI; * * , z , and 6,; - a , 19, are bounded, and V, is bounded and integrable. The boundedness of zl and y, implies that y is bounded. Combining> this with Assumption 6.5 proves that x r is bounded. Therefore, the state vector of (6.38) is bounded. This fact, combined with (6.31) and Assumption 6.6, implies the boundedness of x , r, and U. Thus, the derivatives i , , * , 2, are bounded, which implies that V, is bounded. Hence V, -+ 0 as t -+ 00, which, combined with (6.40), proves that

lim z i ( t ) = 0 , 1 I i I p . (6.41) t - r m

In particular, this means that asymptotic tracking is achieved:

lim z 1 ( t ) = lim [ y ( t ) - y , ( t ) ] = 0. t-m c- m

(6.42)

These results are summarized as follows. Theorem 6.7: Suppose that the system (6.1) satisfies

Proposition 6.3 and Assumption 6.5, and that the design procedure (6.21)-(6.37) is applied to its parametric-strict- feedback normal form (6.15). Furthermore, suppose that the reference signal yr satisfies Assumption 6.6. Then, all the signals in the resulting closed-loop adaptive system (6.38) are bounded and asymptotic tracking (6.42) is achieved for all

Remark 6.8: Since in (6.15) we allowed y,, i = 1; * , p - 1 to depend on x, we had to restrict a,, i = 0; - a , p not to depend on x 2 ; e . , x,. If +, also depended on x 2 , - , x, and Assumption 6.5 were modified to read with respect to x,, - * e , x, as its inputs, the boundedness of y would not guarantee the boundedness of x, and the argu- ments after (6.40) would be invalid. However, if y r , i = 1 , - - , q - 1 (1 5 q 5 p ) , were restricted not to depend on x r , then +, could be allowed to depend on x2; * a , x, also. In that case, the boundedness of 2,; . , z , and Assumption 6.6 would guarantee the boundedness of xl, - e , x,, and

initial conditions in R n + p p . 0

1250 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 1 1 , NOVEMBER 1991

hence the boundedness of x'. The boundedness of xq+ I , , x, would then follow from arguments similar to those after (6.40). This means that our design procedure can be easily modified to be applicable to systems of the form (6.1) which are globally diffeomorphically equivalent to the following parametric-strict- feedback normal form :

XI = x, + OTTl( XI)

X,-, = X, + eTYq(X1,-, X,-,,Xr) x, = YO(X) + eTr,(x) + PO(X)U

P x r = 'P,(x,,-., x,, x') + ej'Pj(xl,-, x,, x')

i= 1

y = XI. (6.43)

Using the results of [28] and [29], it is straightforward to show that there exists a parameter-independent global dif- feomorphism x = $(r) transforming (6.1) into (6.43) if in addition to (6.10), (6.11), (6.13), and (6.14), the following conditions are satisfied for all {E W": gP-' is involutive and of constant rank p - q + 1 (6.44)

0 sj 5 p - q - 1, 1s i s p (6.45) [ X, f;] E Y J , VXE Y J ,

d ( Lf, Lj0h) E span { dh , * . , d ( Ljoh) ) , 0 I j I q - 2, 1 cc i I p . (6.46)

U

VII. DISCUSSION AND EXAMPLES

With the help of two examples, we now discuss some of the main features of the new adaptive scheme. The first example illustrates the systematic nature of the design proce- dure, while the second one compares the stability properties of the new scheme to those of the nonlinearity-constrained scheme of [9].

Example 7.1 (Regulation): We first consider a "be- nchmark" example of adaptive nonlinear regulation:

xl = x2 + ex: x, = x3

x3 = U , (7.1)

where 0 is an unknown constant parameter. This system violates both the geometric conditions of [ 11-[3] and the growth assumptions of [5], [6], [9], [12]. In fact, the only available global result for this example was obtained in [7].

The system (7.1) is already in the form of (5.4) with Po = 1. Hence, this system satisfies the conditions of Theo- rem 5.4, which guarantees that the point x = 0, 6, = 8, =

6, = 0 is a globally stable equilibrium of the adaptive sys- tem. Moreover, for any initial conditions x(0) E W3, (8,(0), f12(0), 6,(0)) E W3, the regulation of the state x(t) is achieved

l i m x ( t ) = 0 . ( 7 4 ?+a

The design procedure of Section 111, applied to (7.1), is as follows.

Step 0: Define 2 , = x,. Step I: Let 6, be an estimate of 0 and define the new

state zz as

z2 = 22, + x, + 29,z:. (7.3) Substitute (7.3) into (7.1) to obtain

i l = -22, + 2 2 + Z : ( e - 6J. (7.4)

9, = 2: . (7.5) Then, let the update law for 29, be

Step 2: Using (7.3) and (7.5), write i2 as

2, = 2 ( ~ , + ex:) + x3 + 6,22,(x2 + ex:) + 2: . (7.6) ,Let 29, be a new estimate of 8, and define the new state

z3 = 22, + 2(x, + 6,x:)(1 + 8,2J + 2: + x3. (7.7)

2, = -22, + 23 + 22:(1 + 8 , 2 , ) ( 8 - 6 2 ) . (7.8)

9, = 22,2:(1 + 61z1). (7.9)

Substitute (7.7) into (7.6) to obtain

Then, let the update law for 192 be

Step 3: Using (7.3), (7.5), (7.7), and (7.8), write i, as

Let be a new estimate of 8 , and define the control U as

Substitute (7.11) into (7.10) to obtain

i3 = -223 + [ 4 ~ : ( 1 + 61Z1)(1 + 8221) +219,(~, + f12x:)x: + s ~ p ] ( e - f 1 3 ) . (7.12)

KANELLAKOPOULOS et al. : SYSTEMATIC DESIGN OF ADAPTIVE CONTROLLERS

Finally, let the parameter update law for 6, be

d3 = 2, [4~:(1 + g121)(1 + 6221) +2t9,(x2 + 6,x:)x: + 5 z f ] . (7.13)

The resulting adaptive system is

i , = - 2 2 , + 2, + z:(e - sl) i2 = - 2 2 , + 2, + 22:(1 + g , z , ) ( e - 19,)

i3 = - 2 2 , + [ 4 ~ : ( 1 + iJ,Zl)(l + 6221) + 2 6 , ( ~ , + ~ ,x : )x : + 5 ~ f ] ( e - 6,)

9, = 2: 9, = 22,2?(1 + ?9,2,)

Using the Lyapunov function

1 2

v = - [ 2: + 22 + 23 + ( e - 6,), + ( e - g2), + ( e - l j ) ,12] (7 .15)

regulation properties of (7.14). 0 it is straightforward to establish the global stability and

Example 7.2 (Tracking): Consider now the problem in which the output y of the nonlinear system

XI = X, + ex: x2 = U + x, i, = -x3 + y

Y = XI (7.16)

is required to asymptotically track the reference signal y, = 0.1 sin 1.

For the sake of comparison, let us first solve this problem using the scheme of [9]. This scheme employs the control

U = -x3 + k , ( x , -U,) + k,(x , + e^ ,.: - L,) + j j , - 2e^,x,x, - 2e^,x:, (7.17)

where e^, , e^,, the estimates of 8 , 8 tained from the update laws

respectively, are ob-

e141 e2 = e142 (7.18) e , = 1 + 4 : + E ; 1 + 4 : + 4 ; * Using a relative-degree-two stable filter M( s), the variables e , , t,, E, in (7.18) are defined as

e, = y - y , + w - e^,tl - e^,E2 (7.19) 4 1 = M ( s ) [ 2 x , x , + k 2 4 1 (7.20)

4 2 = M ( 4 [ 2 x : ] (7.21)

w = M ( s ) [ ~ , ( ~ x , x , + k 2 4 ) + e^ , (2~: ) ] . (7.22)

1251

Simulations of this system were performed with

1 s2 + 5s + 6 6 = 1 , k l = - 6 , k, = - 5 M ( s ) =

(7.23)

and all the initial conditions zero, except for x,(O), which was varied between 0 and 0.45. The results of these simula- tions, shown in Fig. 1 , indicate that the response of the closed-loop system is bounded for x,(O) sufficiently small, that is, for xl(0) < 0.45. However, for x,(O) 1 0 . 4 5 , the response is unbounded. This behavior is consistent with the proof of Theorem 3.3 in [9], which guarantees boundedness for all initial conditions only under a global Lipschitz as- sumption. In the above system, the presence of the term x: leads to the violation of this assumption, and, as the simula- tions show, to unbounded response. Simulations with other schemes based on linear growth conditions [SI, 1121 show that the behavior illustrated by Fig. 1 is typical.

The unbounded behavior in Fig. 1 is avoided by the new scheme, which results in globally stable tracking. The design procedure of Section VII, applied to the system (7.16), results in the control

and the update laws

9, = qx:, 9, = 22 ,x :p + 6 ,z , ) , (7.25)

2, = 2 ( ~ , - y r ) + X, + 2 9 , ~ : - j , . (7.26) Theorem 6.7 establishes that uniform stability and asymptotic tracking are achieved for all x,(O), x2(0), x,(O), flI(O), 29,(0).

0 This is illustrated by simulations in Fig. 2.

The above example illustrates an obvious advantage of the new scheme when applied to parametric-strict-feedback sys- tems: it guarantees global stability for all types of smooth nonlinearities. For parametric-pure-feedback systems, when the feedback linearization is not global, the new scheme provides an estimate of the region of attraction. An advantage of the schemes in [I], [5] - [9], [ 121 is that they provide local results without assuming the parametric-pure-feedback form. However, estimates of the region of attraction are given only in [l], [6]-[8]. A quantitative comparison of the regions of attraction and robustness properties guaranteed by different schemes is a topic of future research.

VIII. CONCLUSIONS

The results of this paper have advanced in several direc- tions our ability to control nonlinear systems with unknown constant parameters. The most significant progress has been made in solving the global adaptive regulation and tracking problems. The class of nonlinear systems for which these problems can be solved systematically has been substantially enlarged. The parametric-strict-feedback condition precisely

1252 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 1 1 , NOVEMBER 1991

Tracking error y - yr

r 0 1 2 3 4 i

Time t Fig. I .

Time t Fig. 2.

characterizes the class of systems for which the global results hold with any type of smooth nonlinearities. For the broader class of systems satisfying the parametric-pure-feedback con- dition, the regulation and tracking results may not be global, but are guaranteed in regions for which a priori estimates are given. It is crucial that the loss of globality, when it occurs, is not due to adaptation, but is inherited from the deterministic part of the problem. All these results are ob- tained using a step-by-step procedure which, at each step, interlaces a change of coordinates with the construction of an update law. Apart from the geometric conditions, this paper uses simple analytical tools, familiar to most control engi- neers.

ACKNOWLEDGMENT The authors are grateful to R. Marino and L. Praly for

their insightful comments and helpful suggestions.

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[I31 P. V. Kokotovic and I. Kanellakopoulos, Adaptive nonlinear control: a critical appraisal, in Proc. 6th Yale Workshop on Adaptive and Learning Systems, New Haven, CT, 1990, pp. 1-6.

[I41 I. Kanellakopoulos, P. V. Kokotovic, and R. H. Middleton, Ob- server-based adaptive control of nonlinear systems under matching conditions, in Proc. 1990 Amer. Contr. Conf., San Diego, CA, May 1990, pp. 549-555.

[I51 -, Indirect adaptive output-feedback control of a class of nonlinear systems, in Proc. 29th IEEE Conf. Decision Contr., Honolulu, HI, Dec. 1990, pp. 2714-2719.

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[I81 L. R. Hunt, R. Su, and G. Meyer, Design for multi-input nonlinear systems, in Diflerential Geometric Control Theory, R. W. Brock- ett, R. S. Millman, and H. S. Sussmann, Eds. Boston, MA: Birkhauser, 1983, pp. 268-297.

[I91 R. Su and L. R. Hunt, A canonical expansion for nonlinear systems, IEEE Trans. Automat. Contr., vol. AC-31, pp. 670-673, July 1986.

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[21] J. Tsinias, Sufficient Lyapunov-like conditions for stabilization, Math. Contr. Signal. Syst., vol. 2, pp. 343-357, 1989.

[22] C . I. Byrnes and A. Isidori, New results and examples in nonlinear feedback stabilization, Syst. Contr. Lett., vol. 12, pp. 437-442, 1989.

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[29] C. 1. Byrnes and A. Isidori, Asymptotic stabilization of minimum phase nonlinear systems, ZEEE Trans. Automat. Contr., to be published.

Petar V. Kokotovic (SM74-F80), for a photograph and biography, see p. 440 of the April 1991 issue of this TRANSACTIONS.

Ioannis Kanellakopoulos (S86) was born in Athens, Greece, in 1964. He received the Diploma in electrical engineering from the National Techni- cal University of Athens in 1987, and the M.S. degree in electrical engineering from the Univer- sity of Illinois, Urbana, in 1989.

He was a University of Illinois Fellow in 1987-1988 and a Grainger Fellow in 1990-1991. Since 1987 he has been a Research Assistant at the Coordinated Science Laboratory in Urbana. He is currently working towards the Ph.D. degree in

electrical engineering at the University of Illinois, Urbana. His research interests include adaptive and nonlinear control with applications to robotics, electric motors, and automotive vehicle controls.

I

A. Stephen Morse (S62-M67-F84) was born in Mt. Vernon, NY, on June 18, 1939. He re- ceived the B.S.E.E. degree from Cornel1 Univer- sity, Ithaca, NY, in 1962, the M.S. degree from the University of Arizona, Tucson, in 1964, and the Ph.D. degree from Purdue University, West Lafayette, IN, in 1967, all in electrical engineer- ing.

From 1967 to 1970 he was associated with the Office of Control Theory and Application, NASA Electronics Research Center, Cambridge, MA.

Since July 1970 he has been with Yale University, New Haven, CT, where he is a Professor of Electrical Engineering. His main interest is in system theory, and he has done research in network synthesis, optimal control, multivariable control, adaptive control, urban transportation, and nonlinear systems.

Dr. Morse is a Member of Sigma Xi and Eta Kappa Nu. He has served as an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL and as a Director of the American Automatic Control Council representing the Society for Industrial and Applied Mathematics.

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