April 1991
UILU-EN G-91-2224
DC-131
Decision and Control Laboratory
ADAPTIVE
OUTPUT-FEEDBACK
CONTROL OF A CLASS
OF NONLINEAR SYSTEMS
I. Kanellakopoulos
P. V. Kokotovic
A. S. Morse
Coordinated Science Laboratory
College o f Engineering
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
Approved for Public Release. Distribution Unlimited.
unclassified
K fljfllT V g m S I E I^ r if lW b F th is M
i
form Approved
O M § No. 07 0 4 4 1 8 *
REPORT DOCUMENTATION PAGE
1b. r e s t r ic t iv e m a rk in g s
ia . r e p o r t s e c u r it y c la s s if ic a t io n
None
Unclassified
3 . DISTRIBUTION/AVAILABILITY OF REPORT
2a. SECURITY CLASSIFICATION AUTHORITY
Approved for public release;
distribution unlimited
2b. DECLASSIFICATION / DOWNGRADING SCHEDULE
S. MONITORING ORGANIZATION REPORT NUMBER(S)
4. PERFORMING ORGANIZATION REPORT NUMBER(S)
UILU-ENG-91- 2224
DC-131
(«. NAME OF PERFORMING ORGANIZATION
Coordinated Science Lab.
University of Illinois
6b. o f f ic e s y m b o l
(If applkabla)
7b. ADDRESS {City, Sttto, and ZIP Cod#)
Washi ngton,
1101 W. Springfield Ave.
Urbana, IL 61801
8b. OFFICE SYMBOL
(If applkabla)
National Science Foundation
DC
20050
9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER
NSF ECS-87-15811
10. SOURCE OF FUNOING NUMBERS
Be ADORESS (City, Sttto, and ZIP Coda)
DC
National Science Foundation
N /A _____________
6 c ADDRESS [City, St* to, and ZIP Cod*)
Ba. NAME OF FUNOING/ SPONSORING
ORGANIZATION
7a. NAME OF MONITORING ORGANIZATION
PROGRAM
ELEMENT NO.
20050
PROJECT
NO.
WORK UNIT
ACCESSION NO.
TASK
NO.
11. TITLE (Induda Security dasafkation)
ADAPTIVE OUTPUT-FEEDBACK CONTROL OF A CLASS OF NONLINEAR SYSTEMS
12. PERSONAL AUTHOR(S)
I. KANELLAKOPOULOS. P. V. KOKOTOVIC AND A. S. MORSE
13a. TYPE OF REPORT
14. DATE OF REPORT (Yaar, Month, Day)
13b. TIME COVERED
FROM ______________ TO
Technical
1991 April
IS . PAGE COUNT
12
16. SUPPLEMENTARY NOTATION
17.
COSATI COOES
FIELD
GROUP
IB. SUBJECT TERMS (Continua on ravarsa ff nacanary and idarrdfy by block nitmbar)
SUB-GROUP
Single-input, single-output, closed-loop
19. ABSTRACT (Continua on ravana If na<anary and Idandfy by block numbar)
For a class of single-input single-output nonlinear systems with unknown constant
parameters, we present a new adaptive design procedure which requires only output,
rather than full-state, measurement.
Even though the nonlinearities are not required
to satisfy any growth conditions, the stability and tracking (or regulation) properties
of the resulting closed-loop adaptive system are global.
21. ABSTRACT SECURITY CLASSIFICATION
20. DISTRIBUTION /AVAILABILITY OF ABSTRACT
□ UNCLASSIFIED/UNLIMITED
□
SAME AS RPT.
22a. NAME OF RESPONSIBLE INDIVIDUAL
0 0 Form 1473, JUN 86
□
OTIC USERS
Unclassified
22b. TELEPHONE (Induda Araa Coda)
Pravious adltlons ara obsolato.
22c. OFFICE SYMBOL
SECURITY CLASSIFICATION OF THIS PAGE
Adaptive Output-Feedback Control of a
Class of Nonlinear Systems*
I. Kanellakopoulos
P. V. Kokotovic
Coordinated Science Laboratory
Dept, of Electrical and Computer Engineering
University of Illinois
University of California
Urbana, IL 61801
Santa Barbara, CA 93106
A. S. Morse
Dept, of Electrical Engineering
Yale University
New Haven, CT 06520-1968
Technical Report DC-131
Coordinated Science Laboratory
University of Illinois at Urbana-Champaign
April 1991
A b stra ct
For a class of single-input single-output nonlinear systems with unknown con­
stant parameters, we present a new adaptive design procedure which requires only
output, rather than full-state, measurement. Even though the nonlinearities are
not required to satisfy any growth conditions, the stability and tracking (or reg­
ulation) properties of the resulting closed-loop adaptive system are global.
*The work of the first two authors 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 the third author was supported by the National Science Foundation under Grants ECS-88-05611
and ECS-90-12551.
1
Introduction
The problem of designing adaptive output-feedback controllers for nonlinear systems with
unknown constant parameters was recently addressed in [1,2]. These papers considered the
class of n-dimensional nonlinear systems which have an input-output description expressed
globally by the n-th order scalar differential equation
n—1
¥>oi(y) + J2dj(Pji(y)
t=0
( 1 .1 )
j= 1
where D denotes the differentiation operator, and
• the coefficients 60, . . . , bm(m < n — 1) of the polynomial B (D ) = bmD m H-------\-b\D + bo
are unknown, but B (D ) is known to be Hurwitz and the sign of bm is known,
• 0 i , . . . , $p are unknown constant parameters, and
• a (y ), <Pji(y), 0 < j < n —1 , 0 < i < p, are smooth nonlinearities with cr(y) ^ 0 Wy € IR,
y?Jt( 0 ) = 0 , 0 < j < n — 1 , 0 < i < p.
In the case of known parameters, systems in this class are linearizable by output (and input)
injection, and input-output linearizable (but not necessarily full-state linearizable) by fullstate feedback.
In [1], Kanellakopoulos, Kokotovic and Morse extended the direct model-reference adap­
tive design developed for linear systems by Feuer and Morse [3] to nonlinear systems of the
form ( 1 . 1 ), under the additional restriction that for * = m + 1 , . . . , n —1 , the functions <Pji(y),
0
j 5: Pi are linear, i.e., that the nonlinearities do not enter the system before the control
does.
This restriction was not present in the design developed by Marino and Tomei in [2],
which combined their “filtered transformations” , introduced in [4], with the adaptive design
procedure introduced by Kanellakopoulos, Kokotovic and Morse [5] in the full-state-feedback
case.
In this paper we consider the same class of nonlinear systems (1.1), and develop a direct
extension of the adaptive design procedure of [5] to the output-feedback case. This new
procedure allows us to remove the additional restriction of [1] without using the filtered
transformations of [2].
As in [1] and [2], the obtained stability and tracking results are
global, despite the fact that the nonlinearities are not restricted by any growth constraints.
2
The Systematic Design Procedure
Consider the class of n-dimensional nonlinear systems with an input-output description
given by the differential equation (1:1).
An equivalent minimal state representation for
such systems is
C =
AC + b<j(y)u + <p0(y)
y
cTc=Ci,
=
(2.1)
where
0
I
A =
b—
:
' 1 *
0
0
bm
I
0
, c=
vn(y)
0 <i<p.
, ^t(y) =
:
. <Pin{y)
. O'.
( 2 .2 )
.
i bo
Suppose now that the control objective is to track a given reference signal yT(t) with the
output y of the system ( 2 .2 ), and assume that the first p derivatives of yT are also given,
where p = n — m. Then, our step-by-step design procedure is as follows:
Step 0: Choose
K 0
such that Aq = A — K
qct
is a Hurwitz matrix, and define the filters
£o =
A)£o + K qV + ^o(y)
(i
=
Aob + <Pi(u) >
Vj
=
A 0Vj
1
+ en_jcr(y)u ,
(2.3)
<P
0< j < m,
where e,- is the i-th coordinate vector in IRn. It is now easy to see that
£
dt
C—( £o+ 53
i= 1
+ 53 bjvj
= A0
jV j
t=l
j =0
3
j- 0
(2.4)
which implies, in particular, that the derivative of the output y is given by
y
=
C2 + <A>i(y) + X ) 0»Y3ti(y)
t'=l
p
=
£02 + <poi (y) + 5 3
( f »2 + ^ i( y ) ) + 53
j=o
t=i
(2.5)
+ e»
where e is an exponentially decaying term. Next, define
x i
=Ci ~Vt = y - y r =
(2.6)
,
and denote by ci, C2, . . . , cp positive coefficients and by T i , . . . , Tp positive definite symmetric
matrices to be chosen later.
For convenience of notation, we also define for i = 0 , . . . , p
C{V
=
[£o,l j • • • >£o,*+l»£l,l» • • • »£l,i+l» * * * >£p»l 1 * * * >£p,*+l]
(^*^)
=
[^0,1 j * * * 5 ^ 0 ,*'+l> • • • 1 Vm—1 ,1 1 •• •>
(^'^)
l,*+lj ^m,l? • • • 5 ^m,t] i
with the understanding that where Ct<f or C{V appear as arguments of a function, that
function may depend on any of their elements.
Step 1 : Using (2.5), write x\ as
m—1
p
Xl
=
{02 + ¥>0l(y)
-
Z
Vr +
5 3
^ (£ ‘2
+
^l(y))
+
x=i
5 3
j=0
bi V &
+
P
=
=
C
m —1 ^
m—l
[£o2 + ^ 01( 2/) “ 2/r] + 5 3 7“ * t^*2
t=i Om
[
b™ V m 2 +
- (c i + i ) Xi + bm {t >m2 +
+ 53 ~L~VJ2
j=0
yT, yr, C ^ , Cit>)} + e,
Vm2 } ^ 6
J
(2.9)
where
T
1
/C,
=
UN
=
9\
9P bo
( 2 . 10 )
bm—i
b l:b
^Ci +
••Jfp2 +
#1 + £02 + ^ 01(2/) — 2/r, 6 2 + V^ll
i;02> •••J v m - 1,2] •
(y)i
(2.11)
Let «i be an estimate of
and define the new state x 2 as
k,
xi
=
Vm2 + KxWx(y,yT,y T,C ii,C iv )
=
um2 + a 2( 2/,y r,^ ,C ,i f,C 'it ;,« i) .
(2.12)
Substitute (2.12) into (2.9) to obtain
= - ^Ci + 5 ) X1 +
(2 .1 3 )
+ €,
where « ! = « ! — «!• Then, let the update law for
be
« i = sgn(6m)riu;ia ;1 = o>i(y, yr, yr, C if, C i u ) .
(2 .1 4)
The time derivative of the nonnegative function
Vl=z\ i x^ I t
(2 .1 5 )
e2(r^ r) + l^ml^r^/ci
along the solutions of (2.13)—(2.14) is
Vi
=
-c ix \ + &mZiz2 - ^x\ + x xe - ^e 2
=
-C ix l - i ( x i - e) 2 +
•
(2 .1 6)
Step 2: Using (2.3), (2.5) and the definitions of a?i, x2, « i, write i 2 as
=
um3 — Ko2vml +
dy*
=
d ct2
dyr *T
Um3 + A ( y , yr, 2/r,
dy
1 02 + ^ O l ( y ) + 5 3 ^ i l ( y ) + D
t=l
ji=0
^ UJ2 +
e
doc2 si j.
da2
,
doc-2 /
'
c ^ \
Cxv + -zr-<jJi(y,yr, yr, Gif, Giv)
Cif +
d«i
3 (0 ,0
3(Cit;)
C ^ f, C 2U, kX) + KT W2(y ^yr, 2/r, G i f , C\V, Vm2, « i ) +
,(2 .1 7 )
dy
where
kt = [01, . . . , 9 p,bo, . . . , b m]
(2 .1 8)
and ft , w2 are defined appropriately, using the fact that the partial derivatives of a 2 with
respect to its arguments are known smooth functions of measured variables.
5
Let k 2 be an estimate of
k
and define the new state x 3 as
1 ( da2\
x 2 + 02 4- K2 w2 + brn2X\
2 \dy)
=
(2.19)
Vm3 + a 3(y , 2/r, 2/r, j/r, C 2( , C 2V, KU « 2 ) .
Substitute (2.19) into (2.17) to obtain
x2 = -
C2+nwJ
1 ( da2\ 2
j
doc2
x 2 + x 3 + k2 w 2 - bmxi + - 3 — e ,
oy
( 2.20)
where k 2 = k — k 2, and
w 2 = w 2 + [ 0, .. . , 0 , x i ] .
( 2 .21)
k2 = r 2t«2a:2 = w2(i/,yr,yI.,C,iC,Ciu,t)m2 ,Ki).
( 2 .22 )
Let the update law for « 2 be
The time derivative of the nonnegative function
V2 = Vi + ^ ( x l + J
e2( T ) d r S
j + k 2 Y 2 xk2
(2.23)
is then given by
t>
2
1/
,2
V2 = - c 1x 1 - - ( z i - e) -
2
c 2x 2
-
1 (d a 2
-
\2
I - g — x 2 - e I + *2X3
Step i (2 < t < p): Using (2.3), (2.5) and the definitions of x t , . . . , Xi, « i , . . . ,
(2.24)
i, write i,-
as
=
C i£ ,C iV ,
Vm,.*+1 +
+/cTu;t(y,yr,
da;
Ci-iU, um,„ ku . . . , /c;_i) + -~ -e
x),
(2.25)
Let /ct-be a new estimate of k and define the new state xt+1 as
.
i f e '
3y,
2
I
X,' + f t + K^tO; + X j _ l
(2.26)
Substitute (2.25) into (2.25) to obtain
—
where /ct=
k
Xi
— /c,-. Let the update law for
= r {W{X{ — U>i(yi Vti •••>2/r
4* z . + i 4-
'r
doix
«,• W i -
a :» - i +
(2.27)
-3 — e ,
dy
be
^5
if ?
(2.28)
Vrn,ii ^ l j •••» ^*—1)
The time derivative of the nonnegative function
K- = K -i + ~
4 - Jt
(2.29)
£2(t)c?t^ + /c^r,"1*,-
is then given by
o
*
1 /5a,-
\ 2~
(2.30)
-f- •^t^'t+i •
= - £
j= l
Step p: Using (2.3), (2.5) and the definitions of x i , . . . , x p, « i , . . . , £p_i, write
xp =
as
cr(y)u + um,p+i + /9p(y,?/r, . . . , ^ p), Cp£ ,C pv, « i, — , « p- i )
4-«Tiyp ( y, yn . . . , C ' p - i f , C p_i u, um,p,
«
p_i ) +
(2.31)
Let /Cp be a new estimate of k and define the control u as
u=
J
\ V77l,P+l 4*
°{y)
Cp
1 ( da? \ 2
x,, + f)p + k jw p + x p- i > .
2 I
(2.32)
Substitute (2.32) into (2.31) to obtain
xn= -
c
+ - ( ^
f + 2\dy
~T
3otp
x p + Kpwp — Xp_i + -^—e ,
(2.33)
where kp = k — kp. Let the update law for kp be
/Cp =
r p W p X p — UJp | y , 2
/r, . . . 5 y i
\ C p—i f 5 O p —i t ? , V m )p ,
, . . . , /Cp_i^ .
(2.34)
Then, the time derivative of the nonnegative function
V„
=
V„_1 + \ { x l + 1
<?(r)drj + k j T ;1^
= 5 iE
i j + it"*ls? r rlsi + i=2
E « J r71''j + f i
=i
e2(T)dr
(2.35)
is rendered nonpositive (since Cj > 0, j = 1,...,/>):
2
1 (d a j
V
n = - £ CjX>+ 2vsT' - e)
j=1
3
< 0.
(2.36)
Stability and Tracking
We are now ready to state and prove our main result:
T h e o re m 3.1. Assume that yT, yT, ..'.,
wise continuous.
are uniformly bounded, and that y ^ is piece-
Then, if the design procedure o f Section 2 is applied to the nonlinear
system (1-1), all the signals in the resulting closed-loop adaptive system are well-defined and
uniformly bounded on [0 , oo), and, in addition,
lim [ y ( t ) - y T(t)] = 0.
t "■^oo
P ro o f.
(3.1)
Due to the piecewise continuity of y ^ and the smoothness of the nonlinearities,
the solution of the closed-loop system has a maximum interval of definition [0, tf). On this
interval, the time derivative of the nonnegative function Vp defined in (2.35) is nonpositive,
as shown in (2.36). We conclude that x i , . . . , x p and k i , . . . , k p are bounded on [0,£f) by
constants depending only on initial conditions. In particular, since x j and yT are bounded,
we have that y is bounded, which, by (2.3), implies that f 0, f i , . . . , f p are bounded and
cr(y) is bounded away from zero.
Furthermore, from the differential equation (1.1), the
boundedness of y , together with the fact that B (D ) is Hurwitz, imply that Hp(D)cr(y)u is
bounded, where H i(s) denotes any asymptotically stable transfer function of relative degree
greater than or equal to i. This in turn implies that FjVm_ j , 0 < j < m, are bounded,
where F { V = [ t ^ i , . . . ,
In particular, it implies that C\V is bounded. By (2.12),
the boundedness of ?/, yT, yr, &i, Cif,
C\V
and x 2 implies that vm2 is bounded.
Hence,
Hp_i(D )cr(y)u is bounded, which means that Fj+iVm^j, 0 < j < m, are bounded. This
again implies that C2v is bounded, which, together with the boundedness of x 3, implies by
(2.19) that vm3 is bounded. Continuing in the same fashion, we can prove that Hi(D)cr(y)u
is bounded, which implies that v is bounded. Since cr(y) is bounded away from zero, we
8
conclude now from (2.32) that u is bounded. From (1.1), this implies that y, y i , . . . , y^n-m^
are bounded. Since the m-dimensional zero dynamics of (1.1) are linear and exponentially
stable, a standard argument proves that the state of any minimal realization of ( 1 .1 ) is
bounded, and, hence, ( is bounded.
We have thus shown that the state of the closed-loop adaptive system is bounded on
[0,£f). Hence, U — oo. To prove the convergence of the tracking error to zero, we first note
that (2.35) and (2.36) imply that Vp is bounded and integrable on [0,oo).
Furthermore,
the boundedness of all the closed-loop signals implies that Vp is bounded. Hence, Vp —►0
as t —> oo, which proves that x \ , . . . , x p —►0 as t —►oo. This, in particular, implies that
y — yT —* 0 as t —►oo.
4
D
The Class of Nonlinear Systems
Most models of nonlinear systems are expressed in specific state coordinates. From that
state-space form it may not always be obvious whether or not the nonlinear system at
hand has an input-output description of the form (1.1). Therefore, we now give coordinatefree geometric conditions which are necessary and sufficient for a single-input single-output
nonlinear system of the form
z =
V =
/(z ;i? ) + g(z;d)u
Kz;0)
.
l
.
;
to have an input-output description of the form ( 1 . 1 ), which is repeated here for convenience:
n
—1
D ny = B ( D ) c r { y ) u + ' £ D '
t= 0
^o»(y) + 5 3 eW ji(y)
j- 1
(4.2)
In (4.1), z G IRn is the state, u E IR is the input, y € IR is the output, i? is a vector of unknown
constant parameters, and / , g , h are smooth vector fields with /(0;i9) = 0 and g(z\’d) ^ 0
Vz E lRn. Accordingly, in (4.2), a(y) and
0 < i < n — 1 , Q < j < p are smooth
nonlinearities with </?j,-(0) = 0 , cr(y) ^ 0 Vy E IR. The coefficients 60? •••, bm(m < n — 1 ) of
the polynomial B( D) , as well as 0 i , . . . , 9P, are unknown parameters resulting from a possible
reparametrization in which functions of the original unknown parameters d are treated as
new parameters.
The following proposition is a corollary of [1, Proposition 5.2].
P ro p o sitio n 4.1. The system (4.1) has an input-output description o f the form (4.2) if
and only if the following conditions are satisfied for all z 6 IRn and for the true value of the
parameter vector
(i) the one-forms dh, d Lf h , . . . , d L ^ h are linearly independent,
(ii) [ad\g, ad^g\ = 0 , i , j — 0 , . . . , n — 1 , where g is uniquely defined by
L} L'jh = |
y
(Hi) adn
fg = £
1
v ’oi (y) + t l ^ v 'd y )
»=o
4 = ° - • • • ." -2
[ 1 ,
2 =
(- lr - 'a d %
j=i
fVv 'n ^ d s , 0 < i < n — 1 ,0 < j <
with
( 4 .3 )
72 — 1 ,
= /
Jo
p,
m
(iv) g = a(y) 5 3 6 t( - l) * a d } ^ and
i=0
(v) the vector fields f and g are complete.
□
E xam ple 4.2 (Sin gle-link flex ib le-join t m a n ip u la tor).
In order to demonstrate why a
reparametrization may be required to write a system of the form (4.1) into the input-output
form (4 .2 ), we consider a single-link robotic manipulator whose rotary motion is controlled
by means of an elastically coupled actuator. If the effect of elastic coupling is modeled as a
linear torsional spring, then the dynamic equations of the system are (cf. [6 , p. 231]):
Jiqi + Fxqx + K (qi -
J m gdcos qx =
L
K (
J ih + F2q2 ~
v
q2\
TVJ
0
( 4 .4 )
=
U’
where q, and q2 are the angular positions of the link and the actuator, and u is the torque
produced at the actuator axis. The inertias J\,J2, the viscous friction constants F\,F2, the
10
elasticity constant K , the link mass M , the position of the link’s center of gravity d, the
transmission gear ratio N and the acceleration of gravity g can all be unknown.
In order to find the input-output description of the system (4.4), where u is the input
and y
=
q\
is the measured output, we use the following minimal state representation of
(4.4), where x x = qu x 2 = qu x3 = q2, x4 = q2:
=
X \
x
2
mgd
x2
J
X 3
=
x4
—
l
Fi
K (
x3
11 - T X2 ~ T, l X l '
n
x4
(4.5)
K
X
i
F »
(
j 2n
=
y
-cos
-L
1
~ rx4 + - r u
V
J
t/ 2
2
.
we obtain x 2 mgd
D
2
=
y
J
i
K
Fx
cos y - — Dy - —
Ji
Ji
y- —
(4.6)
which implies that
x3 =
J\N ( n2 , mgd
Fx
K
— r- ^ y + - j - cosy + — D y + j - y
x4 =
JiN ( 3 .. , mgd
D x 3 = —77 - ( D 3 y
+ ” ‘a“ £
K
\
*
) cos y +
J\
(4.7)
(4.8)
—
J\
J\
Differentiating (4.8) and substituting £3 and x 4 from (4.7) and (4.8), we arrive at the inputoutput description of (4.4):
D 4y
=
K
K
t K
t Fl F2N\
, mgd
+ -T-T7T + “ T-7T- ) V + “ T“ cos V
j 2n 2
j 2k
J
FXK
J xJ2N 2
F2N\
+ — —
)
j2 j
y+
*
mgdF2N
—
j 2k
—
cos
y
mgdK
J\J2N 2
cos y
i
(4.9)
which is in the form (4.2), if we define
K
h- ——
n a -
1 a = * +
if
( Fl F2N
H------- T
TS'
JoI<
J,J2n
' 1
j2
7! ’ 2
Ji
j 2yv2
FXK
( F2iV ^
mgdF2N n
mgdK
u4 = ~ T- A;-0- i -----;— , c/5 = ---- 7-77— , #6 =
JlJ2iV2
•
J 1 J 2W 2
0
n
J
^3
mgd
=
J
i
(4.10)
□
11
5
Concluding Remarks
For the class of nonlinear systems considered in [2], we have developed a new systematic
design procedure for adaptive output-feedback control. The adaptive controller resulting
from this new procedure has dimension n(m -f p + 2) + p(m + p + 1 ). Comparing this to the
controller of [2 ], which has dimension (n — 1 ) \n -f p + p + pp + 2p + n + 1 , we see that,
depending on the values of n, m ,p (recall that p = n — m), either our new procedure or the
procedure of [2] may yield the controller of lower dimension. Finally, we should note that in
both cases the controller dimensions can be reduced if, instead of using the design procedure
of [5], one employs the improved version of that procedure developed by Jiang and Praly [7].
References
1] I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, “ Adaptive output-feedback con­
trol of systems with output nonlinearities,” to appear in Foundations o f Adaptive Con­
trol,, P. V. Kokotovic, ed., Springer-Verlag, Berlin, 1991.
2] R. Marino and P. Tomei, “ Global adaptive observers and output-feedback stabilization
for a class of nonlinear systems,” to appear in Foundations of Adaptive Control, P. V.
Kokotovic, ed., Springer-Verlag, Berlin, 1991.
3] A. Feuer and A. S. Morse, “Adaptive control of single-input single-output linear sys­
tems,” IEEE Transactions on Automatic Control, vol. AC-23, pp. 557-569, Aug. 1978.
4] R. Marino and P. Tomei, “ Global adaptive observers for nonlinear systems via filtered
transformations,” Technical Report R-90.06, Universita di Roma “Tor Vergata” , Sept.
1990.
5] I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, “Systematic design of adap­
tive controllers for feedback linearizable systems,” to appear in IEEE Transactions on
Automatic Control, 1991.
6 ] A. Isidori, Nonlinear Control Systems, 2nd ed., Springer-Verlag, Berlin, 1989.
7] Z. P. Jiang and L. Praly, “Iterative designs of adaptive controllers for systems with
nonlinear integrators,” submitted to 30th IEEE Conf. Dec. Control, March 1991.
12