Adaptive nonlinear control without overparametrization

Systems & Control Letters 19 (1992) 177-185 North-Holland

177

Adaptive nonlinear control without overparametrization * M. Krsti6, I. Kanellakopoulos and P.V. Kokotovi6 Department of Electrical and Computer Engineering, University of California, Santa Barbara CA, USA Received 24 October 1991 Revised 3 April 1992

Abstract: A new design procedure for adaptive nonlinear control is proposed in which the number of parameter estimates is minimal, that is, equal to the number of unknown parameters. The adaptive systems designed by this procedure possess stronger stability properties than those using overparametrization.

Keywords: Adaptive control; nonlinear systems; overparametrization; tuning functions; backstepping design.

1. Introduction

In this paper we present a new design procedure for adaptive control of nonlinear systems transformable into the p a r a m e t r i c - s t r i c t - f e e d b a c k form Xi=Xi+l'4-oT~bi(Xx . . . . . Xi) ,

l
(1.1a)

:on = 6 0 ( x ) + 0T6n(x) +/30(X) u ,

(1.1b)

where 0 ~ ~P is the vector of unknown constant parameters, ~b0, /30, and the components of qbi, 1 ~
Xe+l = - - 0 T ~ e := --0T~bi(0, - - 0 ~

.....

--0~e_l),

i = 1 . . . . . n -- 1.

(1.2)

Correspondence to: Prof. P. Kokotovi6, Dept. of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA. * This work 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 91-147. 0167-6911/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

M. Krstid et al. / Adaptive nonlinear control

178

N o t e t h a t in the special case w h e n (DI(0) ..... ~ n _ l ( 0 ) = 0, t h e e q u i l i b r i u m is x ~ = 0 for all values o f 0. H o w e v e r , as we shall see later, a n o n z e r o e q u i l i b r i u m x ~ 4= 0 i m p r o v e s p a r a m e t e r c o n v e r g e n c e a n d stability p r o p e r t i e s .

2. Backstepping design with tuning functions T h e design p r o c e d u r e is recursive. A t its i-th step a n d i - t h - o r d e r subsystem is stabilized with r e s p e c t to a L y a p u n o v f u n c t i o n V~ by t h e design o f a stabilizing function oli a n d a tuning function ~i. T h e u p d a t e law for the p a r a m e t e r e s t i m a t e / ~ ( t ) a n d the f e e d b a c k c o n t r o l u a r e d e s i g n e d at the final step. Step 1: I n t r o d u c i n g z I = x ~ a n d z 2 = x 2 - 0 q , we r e w r i t e .~ = x 2 + oTd~1(Xl)as (2.1)

Z1 = Z 2 + a l + 0 ~ l ( X l )

a n d use a I as a c o n t r o l to stabilize ( 2 . 1 ) w i t h r e s p e c t to t h e L y a p u n o v function V 1 = ~Z 1 1 2 + 2( ~ --

O)TF- 1(~ _ 0). T h e n /21 =Z'(Z2 '+ ~I "]- o T ~ I ) +

(0- o)TF-I(~- Fz,~1).

(2.2)

If x 2 w e r e o u r a c t u a l control, we w o u l d let z 2 ~ 0, t h a t is, x 2 = a 1. T h e n , we w o u l d e l i m i n a t e 0 - 0 from /21 with t h e u p d a t e law 0 = 21, w h e r e 'r,( x I ) = e z l ¢ 1 ( x , ) .

(2.3)

T o m a k e /21 = - c l z2, we w o u l d c h o o s e ffl(Xl, O) : --CIII--OT~)I(XI).

(2.4)

S i n c e x 2 is not o u r control, we have x 2 ~ 0, a n d we do not use 0 = T 1 as an u p d a t e law. H o w e v e r , we r e t a i n z I as o u r first tuning function a n d a 1 ^as o u r first stabilizing function. W e thus p o s t p o n e t h e decision a b o u t 0 a n d t o l e r a t e t h e p r e s e n c e o f 0 - 0 in/21:

(2.5)

/2,: -ClZ + ZlZ2 + (O-O)Tr

T h e s e c o n d t e r m zlz z in/21 will be c a n c e l l e d at t h e next step. T h e c l o s e d - l o o p f o r m of (2.1) with (2.4) is

2l=-clz

l+z 2+(0-0)

^ T

~l(Xl).

(2.6)

Step 2: I n t r o d u c i n g z 3 = x 3 - a 2 , we r e w r i t e xz = x 3 + OT~z(X~, X2) as Oal X 00ll '~ 22 = Z3 + t~2 + 0 T t ~ 2 - ~---~1( 2 + 0Tt~l) -- " ~ - 0 ,

(2.7)

1 2. 2 Then a n d use a 2 as a c o n t r o l to stabilize t h e ( z l , z2)-system (2.6)-(2.7) with r e s p e c t to V2 = V l + ~z

+z:(+:-

~0ll

oET,

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M. Krstidet al. / Adaptive nonlinear control

]If x 3 were o u r actual control, we would let z 3 - 0 and eliminate 0 - 0 from I22 with the u p d a t e law O = z2, w h e r e

•/Xl x 2 ,

°o,~b,J,1] 0--~1

Then, to m a k e 122 = - C l Z ~ equals - C z Z 2, namely

°°l.,

(2.9)

c2z~, we would design a 2 such that the b r a c k e t e d t e r m multiplying z 2

~0/1

~ffl

(

~ffl

Ol2(Xl' X2' O) = --ZI--C2Z2 + OX-----~X2+ 0-'~-'T2--0Tk~2-- ~(~1)

"

(2.10)

It is i m p o r t a n t to note that in this expression r 2 replaces t~. Since x 3 is not our control, we have z 3 ~ 0 and we do not use 0 = rE as an u p d a t e law. However, we retain r 2 as our second tuning function and a 2 as our second stabilizing function. T h e resulting 122 is

[ 00/1 122 = --CIZ2--C2 z2 "{'-Z2Z 3 + [Z2-~-

]

+ (0- 0)TF -1 (r2- 0).

(2.11)

T h e first two t e r m s in I22 are negative definite, the third t e r m will be cancelled at the next step, and the last t e r m is tolerated at this step, as the decision about 0 is again p o s t p o n e d . T h e closed-loop f o r m of (2.7) with (2.10) is

o0/,. I

)-0-w 0- ~2-- ~-xl~lJ +

Z2 -~" --Z1--C2Z2 +Z3 + (

30/, ~ -~-t,2-

~)

(2.12)

Step 3: Introducing z 4 = x 4 - 0 / 3 , we rewrite x3 = x 4 + 0v~b3(x,, x2, x 3 ) a s

30/2 "X 30/2 30/2 x Z3 = Z4 -{- 0/3 -'l- 0T~t)3 -- -~1 ( 2 -'1-0T~I ) -- 3X------2( g 3 "1-0 T~2 ) -- -"~- 0,

(2.13)

and use 0/3 as a control to stabilize the ( z p z z, z3)-system with respect to V3 = V z + ~z3.1 z Then

30/1

123= -c,z~, -c~z~ + z2 3-~(~-~) [

30/2

30/2

30/2 ~ + ~T (

+ ~3 ~ + ~4+ 0/3- 3~-ix~- 3x~X3- 3o

30/2

30/2

~3- ~ , -

~)

+(~_o)Tr_,[~_r(z,4, +z~(4~2_30/, (4,3_,3,~2

0--~-1~b,

~]

30/2 ,,1

0-~z ~b2))]

(2.14)

ff x 3 were our actual control, we would let z 4 - - 0 and eliminate 0 - 0 from 123 with the u p d a t e law 0 = r 3, w h e r e

• ~(x,,x~,x~,O)=r

[~,~,+~2(~ - ~0Ol,( ,)+~ 30/2

=,a + rz3 ~3- G ~ , -

~-~'-~*~l]

30/2

G~2j.

0o2.,1 (2.15)

Noting that

X " [ 3Ol2 30/2 0 -- "r2 = [~ -- "/'3 + "r3 -- "g2 = 0 -- "r3 + Cz3 [ ~3 -- 0---~1*1 -- a--~2~2 J,

(2.16)

180

M. Krstid et al. / Adaptive n o n l i n e a r

control

we rewrite 123 as

123=

0oQ

+

nt-z 3 Z 2 nt-z 4 q-O~3 - - - X 2 - OX1

--X3-~X2

08 0

q- 0 T

- - Z 2 - ~ -/~) ~ 3 - - ~X1 (DI

~---~-2~)2)]

+ (8 - 0)TF-I(0 -- ~'3). Then, to m a k e I23 = - c ~ z ( z 3 equals -c3z3, n a m e l y

(2.17)

c2z ~ - c 3 z2, we w o u l d design a 3 such that the bracketed term multiplying

00l2 00/2 Oa 2 a3(Xl, X2, X3, 8) = --Z 2 - c 3 Z 3-~- -OX - 1 X2 "~- -t)X2 - X3 -[- - O0 - "I"3

A where r 3 replaces 0. Since x 3 is not our control, we have z 4 ~ 0 and we do not use 0 = z 3 as an u p d a t e /, law. W e again p o s t p o n e the decision about 0 and retain -r3 as our third tuning function and ot3 as our third stabilizing function. T h e resulting 123 is

123= --ClZ2--C2Z2--C3Z2 "~ Z3Z4 q- Z2 0-~ "~-Z3 0--~- Jr ( o - 8 ) ' r

"3-

-

T h e closed-loop form o f (2.13) with (2.18) is

^ T[

0012

OOtl [

00l2

0Or2

0012

OOt2. 1

(2.20)

Step i: Introducing zi+ 1 = x i + ~ - ai, we rewrite 2 i = x i + , + OTcki(Xl . . . . . x i) as

1 (Xk+' -I- oTf~k ) Zi=Zi+l nt-Oli'~-oT~i - iE- 1 U' ~ .~.kOOli-

OOlil)o-1 O, :"

(2.21)

k=l

1 2 Then and use Oti as a control to stabilize the ( z 1. . . . . zi)-system with respect to V~ = V,_ 1 + ~zi.

k=l

\k=l

-~Zi Zi-l"~Zi+l"~-Oli -

i-1 i}O~i-1 E ~ X k Xk+l k=l k=l

0ai-1 ~_t_ 8 T( i-1 ) 1-0 a-i E- l ( ~ f 0-'~ ~)i-k= 1 OXk k

Oxk

(2.22)

1]

If xi+ 1 were our actual control, we w o u l d let zi+ 1 : - 0 and eliminate 8 - 0 from ~ with the update law 0 = r i, where Ti(X 1. . . . . Xi, 8):F~zl(~l--l=1

E

k=l

~k

(

~ Ti-- 1 "~-Fzi ~i --

i-100fi_ll~

E

k=l

1

(2.23)

M. Krsti( et aL / Adaptive nonlinear

181

control

Noting that

(

O--Ti-1 =O--'gi-l-'i'i--q'i-l~'~--Tiq-Czi

~)i--

)

(2.24)

~ X k (~k ,

k=l

we rewrite ~ as

i-1 = --

i-2

03ak [

[

i-1 03ai_1

CkZk "}- E Zk+l 0~ t '[T i¢ 1- ~ ) d-Zi[Zi_, +Zi+l '~-ai-- E k~l k=l k=l

0a,

03Xk Xk+ 1

Ezk+, (2.25)

"

Then, to make I~/= - E ~ = lCk - c i z i, namely

2

Z k,

we would design

a i ( x I . . . . . Xi, O) = --Zi_l--eiziq-

+

E Zk+l

such that the bracketed term multiplying z; equals

i~_~l Oai- 1 Oai- 1 - qTi k=l OXk Xk+l O0 x r-o

k=l

a i

T

O0

~XkCPk

~)i- E

k=l

where ~i replaces 0. Since x i + 1 is not our control, we have z i ~ law. However, we retain 1-i as our i-th tuning function and resulting ~ is

~'=--

k=l

CkZk2 + ZiZi+l +

/k=l

Zk+ , ~

0 a i

(2.26)

and we do not use/~ = r i as an update as our i-th stabilizing function. The

+ (0 -- 0)TF-1 (Ti--O) •

(2.27)

The closed-loop form of (2.21) with (2.26) is

i-I Oai_ 1 . ) Zi=--Zi-l--¢iZiq-Zi+l-[- ( 0 - - 0 ) T C~i-- E ~ X k ~k

k=l "[" O0 (Ti -- O) -I'- k=IEZk+l-~Step n:

With

z . = x . - a n _ l,

] ~i-- k=l"~--"~--~-~-k

we rewrite k . = ~bo(x) +

n--1 0301n_1 (Xk+l "q- oT~ok) Zn = 60 "[- OT6n q- •0 u -- E OXk

k] •

(2.28)

OTdp.(x) + [3o(X)U

as

030ln-1 x 03T O.

k=l

(2.29)

We now design our actual update law 0 = r n and feedback control u to stabilize the full z-system with 1 2 respect to V n = V . _ 1 + ~ z . . Our goal is to make I,;'. nonpositive:

n-1

(n-2

~-Zn Zn-l"l'-~0/g"]'-(~0--

[

+(0 - o)TF-10-Ft

03ak ]

E k=l

OXk Xk+l

z, ~b,-

k=l E

"~

@

030

O--~-k 4~k

•

~)n--

k=l

~Xk

f~k (2.30)

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M. Krsti( et aL / Adaptive nonlinear control

To eliminate 0 - 0 from I)', we choose the update law

(

O= ,n( z, O) = F ~ z, ok,- E l=1 k=l

(2.31)

- ckk = ' n - l + Fzn ck, - ~ l oa~-' " k=l OXk (~k)"

Then, noting that

~lO~n_ ~ )

O - - T n - I = T n - - T n - 1 = F Z n ~)n--

(2.32)

OXk (~k ,

k=l

we rewrite I)n as

.-1

[

v.= - E c~z~2 + z° z°_, q"- ~0 u -{- ~/~0 k=

1

~ ~.-I

O~n-l~ Xk+l

k= 1 OXk

+ 0T- EZ~+l~-~ k=l

~0

n-l__OO/n_l

4'~- k=l ~

ax~ ~

)]

(2.33)

"

Finally, we choose the control u such that the bracketed term multiplying z n equals --CnZn:

1 [ U

n~l ~O~n_l

flO --Zn-1

+

CnZn -- 60 +

E~+,

k=l

2., k=l

^r-~

00

~O/n-I

~Xk Xk+l q-

T

~o-E

k=l

O0

'Tn

(2.34)

ax~

We have thus reached our goal:

vo = - E c~z~.

(2.35)

k=l

With (2.34) the closed-loop form of (2.29) becomes

n~l OO~n_ 1 Zn = - - Z n - l - - C n Z n d - ( O - - O ) T ~)n--

k=l

ax k

~2 k=l

- ~k

(2.36) In the more compact notation

i--1 OO/i_ 1 Wi(XI,''',Xi, O):=~i -

E

k=l

~Xk

(2.37)

(~k,

the overall closed-loop system is rewritten as

^T

2~=-ClZ~+Z2+(O-O ) wl,

(2.38a)

^T W 2 22 = --Z 1 -- C2Z 2 +Z 3 + ( 0 - - O)

~ a ~- -l F z k w k ,

(2.38b)

k=3 Z3 = --Z2--C3Z3"I-Z4 "F ( 0 - - 0 ) T w 3

- k=4 ~ aO~2 ~)--o-FZkWk-t- Z 2 ~O/1 -~FW3,

(2.38c)

M. Krstid et al. / Adaptive nonlinear control

Zi = - - Z i - l - - C i Z i - [ - Z i + l - [ -

^ T

n

( O - O) Wi --

E

O0 FZkWk + E Zk+l - - ~ - F w i ,

k=i+l

k=l

n-2

00~i_ 1

i--2

183 Oa k

(2.38d)

Oa k

i~=--Zn_~--%Z.+(O--O)^ T W~+ ~Zk+l --~-Fw~,

(2.38e)

k=l n

= F E ztw,.

(2.38f)

I=1

R e m a r k 1. Even t h o u g h the new p r o c e d u r e is presented for strict-feedback systems (1.1), its first n - 1 steps remain the same for parametric-pure-feedback systems:

fCi=Xi+l"[-oTt~i(Xl,...,Xi+l),

l <~i<~n-1,

(2.39a)

ffn = ~)0( X) -1- O ~ n ( X) "3t- [ / 3 0 ( X ) "b 0 T / 3 ( X ) ] U ,

(2.39b)

w h e r e ~b0,/30, and the c o m p o n e n t s o f / 3 ~ ~ p and ~ i ~ ~ P, 1 < i < n, are s m o o t h nonlinear functions in B x, a n e i g h b o r h o o d of the origin x = 0, /3o(X)4: O, for all x ~ B x. T h e completion of the n-th step requires that feasibility conditions similar to those in [2] be satisfied. A verifiable geometric characterization of this class of systems is given in [2].

3. Stability and convergence T o prove stability of the closed-loop system (2.38), we express ~bi, ai, Ti, Wi in the z-coordinates. T h e n the global stability of the equilibrium z = 0, 0 = 0 follows from the fact that the derivative of Vn = ~zlT Z + ½(0 -- 0 ) T F - I ( 0 -- 0) for (2.38) is given by (2.35). F r o m LaSalle's invariance theorem, it further follows that the state ( z ( t ) , O(t)) converges to the largest invariant set M of (2.38) contained in E = {(z, 0) ~ ~n+P I z = 0}, that is, in the set where l)n = 0. W e now set out t9 determine M. O n this invariant set, we have z = 0 and 2 = 0. Setting z = 0, 2 = 0 in (2.38) we obtain 0 = 0 and ^ T

(0--0)

Wi=0,

i=1 ..... n,V(z,O)

eM.

(3.1)

Using (2.37) and (3.1) for i = 1 we get ( 0 - 0 ) T i l l ( 0 ) = ( 0 - 0 ) ~ = 0 on M. Recall from (2.4) that a 1 = - C l Z 1 - 0T~br Therefore, on M we have a 1 = - 0T~b~ = --0T~b]. Combining this with z 2 = 0, we get x 2 = x ~ on M. Using (2.37) and (3.1) for i = 2, we obtain (0 - 0)T(~b 2 -- (Oal/aXl)~bl) = 0. Since on M we have (0 - 0)T~b~ = 0 and ~b2(x p x 2) = ~b2(0, - 0T~b~) = ~b~, this implies that (0 - 0)T~b~ = 0. Continuing in the same fashion, we prove that x i = x e and (0 - 0)Tth~ = 0 on M, i = 1 , . . . , n. Thus, the largest invariant set M in E is

M=((z,O)~"+P:

={(x,

z = O , ( O - O ) ^ T ~bei = O , i = l

0)~n+P:x=x

e,(0-0)

^ T ~bie =

0, i

.... ,n I =

1 .... , n ) .

(3.2)

T h e s e two equivalent expressions for M and the convergence of ( z ( t ) , O(t)) to M prove that x ( t ) --, x ~ as t ----~OO.

A n o t h e r important p r o p e r t y of M is its dimension p - r, where r = rank[~b~ . . . . . 6ne]. It is well known from the adaptive control literature that as the dimension of M is reduced, the robustness properties of

M. Krstid et al. / Adaptive nonlinear control

184

the adaptive system are improved. W h e n r = p , the dimension of M is zero, that is, M b e c o m e s the equilibrium point x = x e, 0 = 0. This means that the p a r a m e t e r estimates converge to their true values, so that the equilibrium x = x e, 0 = 0 is globally asymptotically stable. The above facts prove the following result:

Theorem 1. Suppose that the design procedure is appfied to the parametric-strict-feedback system (1.1). Then, the equilibrium x = x e, 0 = 0 o f the resulting adaptive system is globally stable. Furthermore, its state ( x( t ), O( t )) converges to the ( p - r )-dimensional manifold M given by (3.2). The equilibrium x = x e, 0 = 0 is globally asymptotically stable if and only if r = p. [] The new p r o c e d u r e can achieve global asymptotic stability with as many as p = n unknown p a r a m e ters, while in earlier procedures that n u m b e r was at most p = 2.

4. A design example Let us illustrate the difference between the new procedure and the p r o c e d u r e of [2,4] on the ' b e n c h m a r k ' example"

21 = x 2 + 0 ( o ( x l ) ,

(4.1a)

22 = x 3 ,

(4.1b)

23=u.

(4.1c)

The controller of [2,4] employs three estimates 01, O 2, 03: 04, = z l~b,

(4.2a)

Oct1 042 = -z20-~-i 6 ,

(4.2b)

3a 2 043 = - z 3 0-~(~b.

(4.2c)

The corresponding stabilizing functions and the feedback control u are a 1 = - z I - Ol~b ,

(4.3a)

30/1 ( 0al 0l 2 : --Z 2 -- Z 1 "1"- OX--'7- X 2 "t"- ' 0 2 6 ) q'- ~ 1 Z 1 6 ,

(4.3b)

0a2 (

u= -z,-z2+

30/2

ox-7 x2+

+ ox2X,+

30/2

30:2

00:1

- z,4- 2z G4.

(4.3c)

In contrast to the three estimates in (4.2), the new design p r o c e d u r e needs only one estimate 0. The tuning functions are % = zlq~, r 2 = % - Z2(0Odl/0Xl)$, "/'3 = "/'2 -- Z 3 ( 0 0 / 2 / 0 X 1 ) ~ ' and the update law for 0 is 0~ 1 a = r 3 = z,&

00:2

-- z 2 ~ - - - - ~ -- z 3 7

OXl

OXl

},

(4.4)

M. Krstid et al. / Adaptive nonlinear control

185

while the stabilizing functions and the feedback control u are a 1 = --Z l -- 0t]),

a2=

(4.5a)

Oal

- z 2 - z l + Ox--~x(x2+

u= -z3-z2+

Oa2 "X

~al 0th) + 0---~-~'2, Oa2X

(4.5b) Oa2

~)al Oa2

-~1 ( 2+0~b)+ 0x----2-3+ O---~-~'3-z2 0---~-itx---~-~b"

(4.5c)

Comparing (4.2)-(4.3) with (4.4)-(4.5), we see that the new design procedure reduces the dynamic order of the adaptive controller.

5. Concluding remarks

The new design procedure eliminates the need for more than the minimum number of parameter estimates. Among the advantages of having exactly p estimates for p parameters are a reduction in the controller's dynamic order, enhanced stability properties, and improved parameter convergence. In particular, if there are at most n unknown parameters and r =p, then the adaptive system is globally asymptotically stable, and, hence, more robust to disturbances.

References [1] Z.P. Jiang and L. Praly, Iterative designs of adaptive controllers for systems with nonlinear integrators, Proc. 30th IEEE Conf. Decision Control (Brighton, UK, 1991) 2482-2487. [2] I. Kanellakopoulos, P.V. Kokotovi6 and A.S. Morse, Systematic design of adaptive controllers for feedback linearizable systems, IEEE Trans. Automat. Control 36 (1991) 1241-1253. [3] I. Kanellakopoulos, P.V. Kokotovi6 and A.S. Morse, Adaptive output-feedback control of a class of nonlinear systems, Proc. 30th IEEE Conf. Decision Control (Brighton, UK, 1991) 1082-1087. [4] P.V. Kokotovi6, I. KaneUakopoulos and A.S. Morse, Adaptive feedback linearization of nonlinear systems, in: P.V. Kokotovi6, Ed., Foundations of Adaptive Control (Springer-Verlag, Berlin, 1991)311-346. [5] R. Marino and P. Tomei, Global adaptive observers and output-feedback stabilization for a class of nonlinear systems, in: P.V. Kokotovid, Ed., Foundations of Adaptive Control (Springer-Verlag, Berlin, 1991) 455-493. [6] S.S. Sastry and A. lsidori, Adaptive control of linearizable systems, IEEE Trans. Automat. Control 34 (1989) 1123-1131.