202
zyxw
zyxz
zyxwvutsrqpon
IEEE TRAMACTTONS ON AUTOH&flC COXTROL. VOL. AC-29. NO. 3. M4RCH
1984
Robust Redesign of Adaptive Control
PETROS A. IOANNOU AND PETAR V. KOKOTOVIC. FELLOW.
Absiruct-Effects ofunmodeledhighfrequencgdynamicsonstability
and performance of adaptive control schemes are analped. In the regulation problemglobal stabiity propertiesarenolongerguaranteed,but
a
region of attraction exists for exact adaptive regulation. The dependence
of
the region of attraction on unmodeled parasiticsis examined first. Then the
general case ofmodel reference adaptivecontrol is consideredinwhich
parasitics can destroy stabilie andboundednessproperties.
A modified
adaptive law is proposed guaranteeing the existence of a region of attraction from whichall signals converge to a residual set which contains the
equilibrium for exact tracking.The size of this set dependsondesign
parameten,thefrequencyrange
of parasitics, andthe reference input
signal Characteristics.
IEEE
zy
the frequency content of the plant input. The lack of this mechanism has caused the loss of robustness reported in [12]. [13j.
The two main results of this paper are: first, an estimate of the
region of attraction for adaptive regulation, and second. a modification of the adaptive laws to guarantee boundedness in the case
of tracking. The frequency content and magnitude of the reference input signal. the speed ratio p of slow versus fast phenomena, the adaptive gain, and initial conditions are shown to have
crucial effects on the stability of the adaptive control schemes.
These results are first analytical conditions for robustness of
direct adaptivc control with respect to high frequency dynamics.
They are obtained for a continuous-time SISO adaptive control
scheme 113. The same methodology can be extended to more
complicated continuous and discrete-time adaptive control problems. The paper is organized in two main sections. The first
section containsa simple motivating scalar example which illustrates the salient features of the general methodology developed in the second section.
zyxwvutsrqpo
INTRODUCTION
LOBAL stability of adaptive control systems, an open
problem for almost two decades, was recentlysolved for
both continuous and discrete SISO (single-input single-output)
systems [1]-[5]. However, there still remains a significant gap
between the available theoretical methodologies and the potential
applications of such adaptive schemes. Global stability properties
are guaranteed under the “matching assumption” that the model
order is not lower than the order of the unknown plant. Since this
restrictive assumption is likely to be violated in applications, it is
important to determine the robustness of adaptive schemes with
respect to such modelin, errors.
Several attempts have been made to formulate and anal5ze
reduced-order adaptive systems.Specific results such as error
bounds have been obtained for adaptive observers and identifiers
[6]-[lo]. In [ l l ] local stability has been proved for a reduced-order
indirect adaptive regulator. Efforts on reduced-order direct adaptive control [12], [13] concentrated on simple examples where it
wasshown by “linearization” [12] or demonstrated by simulations [13] that unmodeled parasitics can lead to an unstable
closed-loop system. Analysis [6]. [9] of the effects of high
frequency plantinputson the performance of identifiers and
adaptive observers with parasitics has determined that the inputs
should be restricted to dominantly rich inputs. As a design
concept, the dominant richness requires that in the presence of
parasitics the richness condition be satisfied outside the parasitic
range. It excludes wideband inputs such as noise and square
waves as undesirable. The situation in adaptive control is more
difficult because the plant input is generated by adaptive feedback which incorporates the u r h o w n plant with parasitics. The
schemes proposed thus far do not contain a mechanism to restrict
G
0
zyxw
zy
zyxwvutsr
zyxw
17. 1983. Paper
Manuscript received May 7. 1982; revisedMarch
recommended by G. C. Goodwin. Past Chairman of the Adaptive. Learning Systems. Pattern Recognition Committee. This work was supported in
part by the U.S. Am Force under Grant AFOSR 78-3633.the Joint
ServicesElectronicsProgram
underContract N00014-79-C-0414. Ford
Motor Company. and by the U.S. Department of Energy under Contract
DE-AC01-81RA50568 with Dvnamic Svstems.
Urbana. IL 61801.
-.
P. A. Ioannou~iswith thi Univeriity of’ Southem CalifoAa. Los
Angeles, CA 90089-0781.
P. K. Kokotokic is with the Coordinated Science Laboratory. University of Illinois, Urbana, IL 61801.
THE SCALAR
REDUCED-ORDER ADAPTIVE CONTROL
PROBLEM
We start with a simple example of reduced-order adaptive
control in which the output yp of a second-order plant
-i.P = a P -~P +
2 z - u .
=-Z S u,
(1.1)
(1.2)
with unknown constant parameters u p and p, is required to track
the state rg of a first-order model
jn,= - a,,,yn1+ r
a,, > 0
(1.3)
where u is the control input and r = r ( r ) is a reference input. a
uniformly bounded function of time. This example serves as a
motivation for and an introduction to the general methodology to
be developed in the next section. As in our earlier work [6], the
model-plant mismatch is due to some “parasitic” time constants
which appear as multiples of a singular perturbation parameter p
and introduce the “parasitic” state 7. In (1.1).(1.2) the parasitic
state is defined as 9 = z - u resulting into the following representation:
j p = a,Fp + 2 q
+u
(1.4)
(1.5)
p*=-q-pU
where the “dominant” part (1.4) and “parasitic” part (1.5) of the
plant appear explicitly.
Ifwe apply to the plant with parasitics (1.4), (1.5) the same
adaptive law which we would have applied to the plant without
parasitics: that is, if we use the control
+
(14
y > 0,
(1.7)
u=- K(C)J> r(t)
~~~
and the adaptive law
0018-9286/84/0300-0202%01.00 01984 IEEE
K
= yevp
203
IOANNOU AND KOKOTOVIC: ROBUST REDESIGN OF ADAPTIVE CONTROL
and the objective is to drive yp to zero despite the presence of
parasitics while assuring that all the signals in the closed-loop
system (1.15)-(1.18) remain bounded. It is important to note that
the open-loop system (1.15):(1.16) might not be stabilizable by
constant gain output feedback for a given value of p . If this is the
case, then there is no hope that the adaptive controller (1.17)?
(1.18) will stabilize the equilibrium of (1.15), (1.16). The following
lemma characterizes parasitics for which a linear output stabilizing feedback law exists.
Lemma 2: There exists a p1 > 0 and a constant KO such that
for all p E (0, pl] the system (1.15), (1.16) with the feedback law
we obtain
zyxw
zyxwvutsrqpon
zyxwvutsrq
zyxwvutsr
zyxwvu
zyxwvutsrq
zyxwvutsr
zyxwv
zyxwvuts
where
zyxwvutsr
e=yP-yfll.
K*=a,+a,.
(1.11)
The existing theory of adaptive control [14],[15] guarantees
stability properties for the case without parasitics p = 0, when
(1.8)-(1.10) reduce to
t=-a,,,~-(K(t)-~*)(~+y,,,)
x = y2( 2 + y,,,).
(1.19)
u - KOyp
is an asymptotically stable closed-loop system. Furthermore,
(1.20)
(1.12)
(1.13)
and
1
Lemma I : For any bounded initial conditions i?(O)? X ( 0 ) the
solution F ( t ) , K ( t ) of (1.12), (1.13) is bounded and lim,+=F(t)
= 0, limrzXF( t ) = K, where constant K , is in general a function
of i?(O), K(0). Furthermore, .if r ( t ) is sufficiently rich, then
K ( t ) = K *, independent of F(0).K(0).
lim,
The above example illustrates some of the robustness questions
to be answered, in this paper. Given that the adaptive system
without parasitics, in this case (1.12), (1.13), possesses properties
such as in Lemma 1, how will these properties be altered by the
parasitics, that is, what are the stability properties of (1.8)-(1.10)?
Which modification of the adaptive law would help to preserve
some of the desirable properties? The perturbation parameter p
provides us with a means to answer such questions in a semiquantitative way using the orders of magnitude O(p’), noting that
for p small, the quantity O( p ” ) is small when v > 0 and large when
v 0. The smallness of p implies that the parasitics are fast and
that neglecting them, p = 0, we concentrate on the slow. that is.
the “dominant,” part of the plant.
As we shall see, a first property to be lost due to parasitics is
global stability. In the case of regulation, that is. when y,,, = 0,
r ( t ) = 0, the boundedness of the solutions e( t ) , K( t ) and the
convergence of e ( r ) to zero as t + ce is preserved, but is not
global. It possesses a domain of attraction whose size we describe
by estimating the orders of magnitudes of the axes of an ellipsoid
D(p). In the tracking problem, when r( r ) 0 the adaptive system
with parasitics such as (1.8)-(1.10) may not converge to or may
not evenpossess an equilibrium. A practical goal is then to
guarantee some boundedness properties. Weshow that a redesign, which may sacrifice some propeoies of the ideal system
without parasitics, results in the convergence from any point in
D(p) to a uniformly bounded residual set D,(p). Thedesign
objective is then to make D(p) as large as possible and D,(p) as
small as possible. Let us illustrate this discussion by analqzing the
regulation problem and the tracking problem for the example
(1.1)-(1.3).
+
-
P
- up > K, > u p .
(1.21)
We now establish the stability properties of the adaptive control
system (1.15)-(1.18) for p < p , .
Theorem 1: There exists p* < p , and positive numbers a < 1/2,
cl, c2 such that for each p s ( O , p * ] any solutionyp(r),q(r):K(t)
of (1.15)-(1.18) starting from the set
D(p)
=
{ ~ b , qK:
, lypl+ IKl< c,p-=, 191 < c 2 p - u - 1 / 21
(1.22)
is bounded and y ( t )+ 0: q ( t ) + 0, K ( t ) ---* constant as t + m.
ProoJ Let
> u p be a finite constant and consider the
function
k,
Observe that for each p > 0, c > 0, a > 0 the equality
V(yp,q,K)=cp-*“
(1.24)
defines a closed surface . Y ( p , a , c) in R 3 . The time derivative of
V(y,,q, K ) along the solution of (1.15)-(1.18) is
A . Regulation
In the regulation problem expressions (1.8)-(1.10) become
r ( t ) = O , y,,,,(t)=O, e ( t ) = y p ( t )
(1.14)
,$ = ap-”p+ u + 2 9
(1.15)
pQ=-q-pU
(1.16)
u = - K(t)yp
(1.17)
K = yy;
(1.18)
where /3 = K , - a p > 0.
Inside 9 ( p t a , c), IynI,1K1 can grow up to O ( P - ~ ) . whereas 191
can grow up to O(pLil/Z-a).Therefore, there exist constants
204
zyxwvutsrqponm
zyxwvutsr
IEEETRANSACTIONS Oh' AUTOMATICCONTROL,VOL.
p l , p 2 , p 3 such that insideY(p,a,c)
AC-29, NO.
3,
MARCH
1984
1.5
zyxw
zyx
zyxwvuts
zyxwvutsrq
zyxwvutsrqpo
1.0
- 82(+
-
2p2p1-a
-
(1.26a)
4p)
for some positive constants 6,, 6,.
Choosing a < 1/2 we can find a p* > 0 such that for any
P E (0,P*l
> 6 , p 1 - ~ a+ a2p2(1-la) , 1 > 2p2p1-" +4p.
2
-0.5
1
1
,
,
,
1
,
,
,
,
1
,
,
,
,
1
1
,
,
,
1
- ' - - m ~ - ~
zyxwvu
0
1
2
3
4
Time,t (sed
5
6
(a)
Therefore, ~ O ; p , ~ , K ) ~ O . f o r e a c h ~ ~ ( O , p * ] a n d a l l ~ , , ~ . K
enclosed in 9(
p , a, c), and V = 0 only at the equilibrium J, = 0,
7 = 0, K = constant. Moreover, there exist positive constants cl, c2
20
2'5s$ \
such that the set D(p) given by (1.22) is enclosed by the surface
:!
Y ( p , a, c) and any solution of (1.15)-(1.18) starting from D(p)
v 15; \
remains inside Y ( p , a, c). Furthermore, inside Y ( p , a , c),
- I
V (y,, q , K ) is a nonincreasing function of time which is bounded
1.03
* ii
from below, and hence converges to a finite value V,. Since
V ( q , q , K ) is bounded, V ( y , ( r ) , q ( t ) , K ( t ) ) is uniformly conj i
0.5i
y p , q, K enclosed in Y ( p , a, c) and t > 0. Theretinuous for
1
fore, lim,+zV(y,(t), q ( t ) , K(rN = 0, i.e., q ( t ) 0, q ( t ) -,0.
; \
and K ( t )+ constant as t + w .
0.0 , , ,Ll
1 , ' , , , , , , , I , I , , , I
0
1
2
3
4
5
6
Remark I: It can also be shown that increasing adaptive gain
Tlme,t (sed
y for a fixed p reduces the size of the domain D(p) and the
stability properties of Theorem 1 can no longer be guaranteed if
(bl
Y 3 0(1/F).
Fig.
1.
Adaptive
regulation
for
p
= 0.05. ~ ~ =
( 1,
0 p(0)
)
= 1. K(0) = 3,
Remark 2: As p -,0, domain D(p) becomes the whole space
and y = 5. Response ma. stable.
R3; that is, the adaptive regulation problem (1.15)-(1.18) is well
posed with respect t o parasitics.
Remark 3: Theorem 1 is more than a local result because it
:
shows that given any bounded initial condition y (0).~ ( 0 )K(0).
there always exist p* such that for each p E (0,p*? the solution of
(1.15)-(1.18)isboundedand~~(r)-,O,~(t)+O.K(t)-,constant
I\
+
I
,
,
,
ast-m.
Remark 4: Since Theorem 1 is only a sufficient condition it is
of interest to examine whether the stability properties of Lemma
1 are indeed lost for initial conditions outside the set (1.22). From
Lemma 2 and the fact that K ( r ) is nondecreasing it can be seen
that instability occurs if K(0) > (l/p)- a,.
As an illustration of the stability properties established by
Theorem 1 simulation results for (1.15)-(1.18) with a p = 4 and
different values of p, y and initial conditions are plotted In Figs.
1-4. In addition to y,(t), V ( y p ,17, K ) = V with K , .= 7 is plotted
against time to show whether all the signals in the closed loop
remain bounded. In, Fig. 1 where p = 0.05, y = 5 . ~ ~ (= 1.0.
0 )
q(O)=l.O, K(0) = 3, the objective of the regulator is acheved
since y,
0 and Vis bounded. Increasingy,(O) from 1 to 2.4 and
keeping all the other conditions the same as in Fig. 1, the
regulator fails its objective and y, -,oc as s h o m in Fig. 2. With
the same initial conditions as in Fig. 1, but with p = 0.07 instead
of p = 0.05. y, -, 03 as indicated in Fig. 3. Fig. 4 shows the effect
of increasing the adaptive gain y. With the same initial conditions
as in Fig. 1, but with y = 30 instead of y = 5. regulation fails and
-
zyxwvuts
Y, --*a.
B. Tracking
Returning now to the tracking problem we note that for a
general r ( t ) + 0 the system(1.8)-(1,10)need
not possess an
equilibrium. The best we can expect to achieve in this case is to
guarantee that the solutions startingin D(p) remain bounded
and converge to a residual set Do(p).
Fig. 2. Adaptive regulation for p = 0.05, ~ ~ (= 02.4.) q(0) = 1, K(O) = 3,
and y = 5. Instability due to largerxD(0).
To prove such a result we modify the adaptive law (1.10) as
k=-aK+ye(e+yn2)
(1.27)
where is a positive design parameter. In viewof (1.27) the
equations describing the stability properties of the tracking problem in the presence of parasitics are:
(J
k=-a,e-(~(t)-~*)(e+~,,,)+2q
,++,
= - ?I + [ ye( e + yn2 1'+ 2 ~ +q Kr -
4
k=-aK+ye(e+yn,).
(1.28)
K ( K - a, - .)(e + y m ) .
(1.29)
(1.30)
zyxwvutsrqpo
zyxwvutsrqpo
z
zy
zyxwvutsrqponmlkjihgfedc
zyxwvutsrqpon
zyxwvutsrqpo
zyxwvutsrqpo
zyxwvutsrq
205
IOANNOU AND KOKOTOVIC: ROBUST REDESIGN OF ADAPTI\T CONTROI
=-a,,,e'-aK(K-K*)-q2+p(q+2e)
Y
~[ye(e+ynz)'-K(K-ap-u)(e+y,,,)+2Kq+Kr
YP 50+
1
Equation (1.36) can be rewritten as
o i
a
P ( e , q , K ) = - ~ ~ [4~ - p ( 2 - y e ~ + 4 y y ~ , e + 2 y y ~ - 2 K '
A
I-----
(1.36)
-2an,e-i-2(K-K*)(e+ym)+4q].
1
' ' ' '
0.0
ID
05
2.0
1.5
Tlme,t
!
' ' ' ' I
2.5
30
+2Kap-4am-4K+4K*+2Ko)
(sed
Fig. 3. Adaptive regulation for 1.1 = 0.07, ~ ~ (= I0, ~) ( 0=I.
) K(O) = 3.
and y = 5. Instabilitylarger
due to
1.1.
-p2(4K+8+ye'+2yyme+yy,i
-K2+Kap-2a,,,-2K+2K*+oK
I')
- - [1p . - 2 p e ( 4 K + 8 + y e 2 + 2 y y m e + ~ y y ~
4
-K2+Kap-2a,-2K+2K*+oK)]'
zyxwvutsrqponmlk
1
-
-100
1,
4 [ 71 - 2 ~ ~ ( a p . ~-n~z y
2
+
m -2ym + u ~ m ) ]
- - [41q - 2 p ( 2 K * y m - i ) ] * - q ' ( 8 -1p ( 4 + 2 K ) )
1
- 1 2 5 . ~ y r - - y r ~ ~ - ~ - - r - - - - . ~ I
0.0
0.2
0.4
C16
0.8
LO
12
Tlme,t (sed
Fig. 4. Adaptive regulation for 1.1 = 0.05, y (0) =1, q(0)= 1, K(0) = 3,
and y = 30. Instability due & larger y.
Theorem 2: Let the reference input r ( t ) satisfy
Ir(t)l<r,,
li.(t>l<rz
Vt>O
(1.31)
where r,, r, are given positive constants. Then there exist positive
constants ti, p*, r3,a < 1/2, c1 to c 3 , and c4 >, 1 such that for each
p E (O,p*] every solution of (1.28)-(1.30) starting at r = 0 from
the set
D(p) = { e , K , qlei+
: IKI < clp-",
(1.37)
PI,
1711 < c~"~-'.''}
(1.32)
enters the residual set
h i d e Y ( p , a, c,),
lKl can grow up to O(p-") and 1711 can
grow up to O(p-" -"). Hence, there exists positive constants
11,5,. 5; such that inside Y (p , a,c,)
le1 <
1K1< S Z P - ~ , Iql<
5 3 ~ - ~ ' ~ - ~ (1.38)
.
For e, 7
1
, K inside Y ( p , a,c,), (1.37) can be simplified to
for some positive constants PI, p2, and
we can see thatfor each p > 0, co > 0, a > 0 the equality
V ( eK, )q ,
= coppZa
(1.35)
defiies a closed surfaceY(p, a, cp) in R 3 space. The time derivative of V( e , 7, K ) along the solution of (1.28)-(1.30) is
,, >
,
c3p2(1-a)
/I3.
Choose
(1.40)
where cj = 2yP,, and a < 1/2. Then there exists a p* such that
for each p E (0, p * ]
206
zyxwvutsrqponm
zyxwvut
AC-29.NO. 3,
1984
zyxwvutsr
zyxwvutsrq
zyxwvuts
zyxwvutsr
zyxwvutsr
zyxwvuts
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL.
MARCH
7
(1.41)
Hence, for each p E (0, p * ] and all e, q , K inside Y ( p , a, co)
i
+ p 2 I + - 12K*yn,- i j 2 .
a:,)
(1.42)
Since r, i. are uniformly bounded the set Do@) is uniformly
bounded and is enclosed by Y ( p ,a, co). Outside D,(p) and
inside Y ( p , a , co), k(e , q , K ) < 0, and therefore V (e, q , K ) decreases. Hence, there exists positive constants c,, c2 such that the
set D(p) is enclosed by Y ( p , a , co) and any solution e( t ) . K( t ) .
q ( t ) which starts in D(p) remains inside Y ( p , a , c,). Furthermore, there exist constants c4 > 1 and t1 such that every solution
of(1.28)-(1.30)
starting at t = 0 from D(p)/DO(p) mill enter
Do(p) at r = r, and remain in D,(p) thereafter. Similarly, any
solution starting at t = 0 from Do(p ) will remain in D,(p) for all
t > 0.
Remark 5: The set Do(p) depends on r, and rz, i.e., the
bounds for the magnitude and frequency content of the reference
input signal. For, a given p an increase in Irl or 1i.l can no longer
guarantee that V(e, q , K ) < 0 everywhere in D(p)/D,(p). For
this reason, ow formulation excludes high frequency or high
amplitude reference' input signals such as square or random
waveforms. the traditional favorites of the adaptive control literature.
Remark 6: It can also be shown that increasing the adaptive
gain y for given p , r , and i reduces the size of the domain D(p).
For y > O(l/p) the stability properties of Theorem 2 can no
longer be guaranteed.
Remark 7: The analysis in the proof of Theorem 2 shows that
the useof u is essential in obtaining sufficient conditions for
boundedness in the presence of parasitics. However, in the absence of parasitics ( p = 01, o > o causes an output error of o(&).
This is a tradeoff between boundedness of all signals in the
presence of parasitics and the loss of exact convergence of the
output error to zero in the absence of parasitics. The sizeof u
reflects our ignorance about p. For high frequency parasitics p is
small, and therefore u can be small.
Remark 8: Conditions (1.20),(1.21) of Lemma 2 specify the
relation between the range of parasitics and the open-loop and
desired closed-loop poles. From (1.21) we cansee that for a
desired closed-loop pole a,, the gain K ( t ) has to converge to
K * = a,, + a,, which has to satisfy
"ri
zyxwvuts
zyxwvu
1 - a p> K * > a p .
P
(1.43)
From (1.43) it is clear that a,, should satisfy
1 -Zap > a,, > 0
-
P
(1.44)
indicating that the desired closed-loop system has to beslower
than the fast subsystem. Therefore, for stability the poles of the
reference model have to be chosen away from the parasitic range.
It is of interest to examine whether for initial conditions
outside the set D(p) we can loose boundedness. Simulation
results with a p = 4, a,, = 3, and y = 5 are summarized in Figs.
5-12. Plots of the output error and the function V ( e ,q . K ) = V
versus time are obtained for different initial conditions, p, u, and
reference input characteristics. In Fig. 5 the output error e and
function V are plotted for p = 0.01. e(0)=l. q(0)=1, K(0) = 3.
o = 0.06, and r( t ) = 3sin2t. The output error decreases and
remains close to zero. The function V is strictly decreasing for
LO
I'"'I-1
0.0
0.0
2.5
5.0
7.5
no.15.0
Tlme,t (sed
12.5
(b)
Fig. 5. Tracking for p
= 0.01. e(0)= 1, q(0) = 1. K ( 0 ) = 3.
r( t ) = 3 sin 2 t. Bounded response.
(I =
0.06 and,
V > 0.05, but V changes sign in the region V < 0.05 as shown in
Fig. 5(b). Keeping the same conditions as in Fig. 5, but increasing
p from 0.01 to 0.05 we can still achieve similar results as shown in
Fig. 6. However, in this case the steady-state error and V are
larger and V changes sign for V < 0.4. Increasing the value of p
from 0.05 to 0.08 the output error becomes unbounded for all
u 2 0 as indicated in Fig. 7. The effects of the input characteristics are summarized in Figs. 8-10. In Fig. 8, p = 0.05, e(0)= 1,
q(0) = 1, K(0) = 3, u = 0.06. and r( t ) = 3sin10t results into an
unbounded output error due to the increase of the frequency of
r ( t ) from 2 to 10. The same instability result has been observed
for u = 0.0,0.02. However, for u = 0.08 the output error became
bounded as shown in Fig. 9 illustrating the beneficial effects of u
when parasitics are present. The effect of the amplitude of the
reference input r( t ) is shown in Fig. 10. With p = 0.05, u = 0 or
0.06 and the same initial conditions as before, but with r( r) =
15sin2t, the output error goes unbounded. Fig. 11 shows the
effect of initial conditions on boundedness. By increasing e(0)
from 1 to 2.5 and keeping p = 0.05, q(0)=1, K(0) = 3, and
r( t ) = 3 sin2r the output error becomes unbounded for all u > 0.
In Fig. 12 we show the loss of exact convergence of the output
error to zero in the absence of parasitics ( p = 0) due to the design
parameter u.
ADAPTIVE
CONTROL
OF A SISO PLANTIN THE PRESENCE
OF PARASITICS
.
zyxwv
We now consider the general problem of adaptive control of a
SISO time-invariant plant of order n + m where n is the order of
the dominantpart of the plant and rn is the order of the
parasitics. The plant is assumed to possessslow and fast parts
zyxwvutsrqp
zyxwvutsrqp
zyxwvutsrqponm
zyx
I
zyxwvutsr
zyxwvutsrqponmlkjih
207
IOANNOL! AND KOKOTOWC: ROBUST REDESIGN OF ADAPTIVE C O m R O L
L5 -l
-1
__
__ -
-.
.
__- -
-__ -
-
-- - -
e
- - - -- -
200
___ ---
-- -
-a5-
zyxwvutsr
- l . O ~ y ~ - r ~ - - - ~ - r - -TO
- 2
4
IO
8
6
I
12
-203
O
-
P
r
7
,
Q2
0.1
0.0
Tm,t (sed
(a)
-.
-
0.4
0.3‘
(sed
T1me.t
0.5
Fig. 8. Tracking for p = 0.05, e(0) = 1, q(0) =1, K(0) = 3, (I = 0, or 0.06
and r ( t ) = 3sin10t. Response unbounded due to high reference input
frequency.
e o
zyxwvutsrqpon
zyxwvutsrqpo
zyxwvut
1
m
IO
0.0
0
2
4
6
12
8
Time,t (sed
-1
(b)
Fig. 6 . Tracking for p = 0.05, e(0)= 1, q(0) = 1, K ( 0 ) = 3, u = 0.06, and
r ( t ) = 3sin2t. Bounded response, error and V larger due to larger p.
-2
ao
1
1
1
1
,
1
25
1
1
1
1
1
5.0
I I I I I I
1
Tm,t
2000
:
no
7.5
I
I
,
I
125
,
,
,
1-
(sed
Fig. 9. Tracking for p = 0.05, e(0) =1, q(0)=1, K(0) = 3, u = 0.08, and
r( t ) = 3 sinlot. Input frequency high, but e is bounded due to larger u.
~
1500 -
e IOOO-
500-
e 200
0
-
v
a2
0.0
,
I
OS
a4
Time,t
-?
I
a8
(sed
Fig. 7. Tracking for p = 0.08, e(0) = 1, q(0) = 1, K(0) = 3, u 2 0.0, and
r( t ) = 3 sin2t. Response unbounded due to large p.
OD
a2
~4
and is represented in the explicit singular perturbation form
k = A 1 , x + A 1 2 Z + b,u
pi
y
= Az,x
+ A,?” + b2u,
LO
IZ
(sed
Fig. 10. Tracking for p = 0.05, e(0) =1, q(0) =1, K(0) = 3, u = 0, or
0.06 and r( t ) = 15sin2t. Response unbounded due to large reference
input amplitude.
(2.1)
ReA(A,,) < 0
(2.2)
(2.3)
= c;x
where x, z are n and m vectors, respectively, and u , y are the
scalar input andoutput of the plant, respectively. State z is
formed of a “fast transient” and a “quasi-steady state” defined
as the solution of (2.2) with p i = 0. This motivates the definition
of the fast parasitic state as
q =z
aa
~6
Tirne,t
+ A,’(A2,x + b,u).
(2.4)
The two important restrictions on the unknown plantare
Re A(A2,) 0 and y = c r x . The first restriction, which prohibits
unstable or oscillatory parasitic modes, is natural and cannot be
removed. The second restriction allows the parasitics to be only
“weakly observable,” that is observable through the slow part of
the plant. For plants with strongly observable parasitics y = clx
+ c2z, the static output feedback is nonrobust, that is, it can
destabilize a stable plant [16]. In this case a dynamic compensator must be used, containing a low-pass filter. As shown in [9],
the original plant with a first-order filter can be represented as an
208
zyxwvutsrqponm
zyxwvutsrqponz
IEEE TRAWSACTIONS ON AUTOMATIC CONTROL, VOL. AC-29. NO.
6oo-
I
3,
p+=Ar2q+p(A1x+Azu+A3q+A4k)
MARCH
1984
(2.7)
(2.8)
I'= c;x.
The output y of the system (2.6)-(2.8) is required to track the
output;,'I of an nth order reference model
m
i = AnrXnt + bnrr
(2-9)
zyxwvutsrqponmlkjihgf
zyxwvutsrqp
zyxwvutsr
zyxwvutsr
ynt=
(2.10)
CLXm
where r( t ) is a uniformly bounded reference input signal. Without
loss of generality let us assume that the transfer function W n t ( s )
-2000
8
.
0
0
.
1
, , , I
I ' I I ! I I I I I
L O L ! 5 2 ~ 2
TIW,~ (sed
,
,
5
I
I
I
5
3
D
~
Fig. 11. Tracking,forp = 0.05, e(0) = 2.5. ~ ( 0=)1. K ( 0 ) = 3. w > 0. and
r ( t ) = 3an2t. Response unbounded due to large e ( 0 ) .
is strictly positive real.
The reduced-order plant obtained by setting p = 0 in (2.6)-(2.8)
is assumed to satisfy the following conditions.
i) The triple ( A o , b o co)
, is completely controllable and completely observable.
ii) In the transfer function
W o ( s )= c:(sI- A o ) - ' b 0 = K Np ( sm)
(2.12)
Hurwitz polynomial of degree n - 1 and D ( s ) is
a monic polynomial of degree n. For ease of exposition we
assume that K p = K , = 1.
The controller structure has the same form as that used in [l]
for the parasitic-free plant, that is for p = 0 in (2.6)-(2.8). In this
controller the plant input u and measured output y are used to
generate a (2 n - 2) dimensional auxiliary vector u as
X ( s) is a monk
-1
2.5
0.0
zyxwvutsrqpon
7.5
50
Time)
10.0
12.5
15.0
(set)
8 , = Av,
w1= c'(
B,
v
+ gu
(2.13)
t ) v1
= Av,
+ gy
W, = d o ( t ) y + dT(t)v2
1.0-
(2.14)
where A is an ( n - l ) ~ ( n - l ) stable matrix and ( A , g ) is a
controllable pair. The plant input is given by
0.5-
(2.15)
u=r+era
0.0
1
-
I
I
,
2.5
loa
7.5
Tlme. t
Eo
(sed
(b)
Fig. 12. Tracking for p = 0. e(0)= 1. ~ ( 0=)1, K(0) = 3. w = 0.08. and
r ( r ) = 3sin2r. Loss of u.a.s. in the parasitic freecase. p = 0. due to
w > 0.
wherewT=[~~,v~,~]and8(t)=[cT(t),d~(t),do(t)lTiSa(
1) dimensional adjustable parameter vector. It has been shown in
[l] that a constant vector 8* exists such that for O ( t ) = 8* the
transfer function of the parasitic-free plant (2.12) with controller
(2.13)-(2.15) matches that of the model (2.11).
Ifwe apply to the plant with parasitics (2.6)-(2.8) the controller described by (2.13)-(2.15) we obtain the following set of
equations for the overall feedback system
zyxwvutsrqp
zyxwvutsr
0.05.0
augmented plant in the form (2.1)-(2.3) with weakly observable
parasitics. Defining
[tl)
c2 =
A , =A l l - A12A,'A21,
[ I: :][rl]+
gc,
$r
A
0
u,
[:](era
+
[*,I.
zyxwvutsr
A , = A$A,,AO
A,
bO=bl-A,,A,'b2,
A? = AglAzlbo,
= A G I A Z I A ,I Z ,
= AG'bz
(2.16)
(2.5)
and substituting (2.4) into (Z.l), (2.2) we obtain a representation
of (2.1)-(2.3) with the dominant part (2.6) and the parasitic part
(2.7) appearing explicitly
i= A o x
+ bou + A12q
(2.6)
pQ = A,2q i p ( A l x + A28'o
-k
A,d'a
+ A , r + A,q
+ A,B'cb + A d ? )
(2.18)
y =.x:.
Introducing 8 * , Y'
(2.17)
', a,: t
,
:
] and
= [X
zyxwvutsrqpon
zyxwvutsrqpo
zyxwvutsrq
209
IOANNOU AND KOKOTOVIC: ROBUST REDESIGN OF ADMTIVE CONTROL
where u is a design scalar parameter. The resulting adaptive
control system with parasitics is described by
zyxwvutsrqp
zyxwvutsrq
zyxwvutsrqp
zyxwvutsrq
(2.19)
we rewrite (2.16), (2.17) in a form convenient for our stability
analysis
Y=
A , Y + b,((e
+ r)+&2p
-
(2.32)
(2.20)
p3=~22p+p(&~+~2eTW+~2r+~3p
+ A,dTw + A4B% + A , i )
- ~ , ~ e + x , , , ~ ~ r ~ e + x , , ~ e , + ~ , ~ ~ ~ ~ , e
(2.21)
where 4, = [AT, 0 0IT and Kl = [AT 0 O]? An advantage
of this form is that for e( t ) = e* in the parasitic-free case (2.20)
becomes a nonminimal representation of the reference model
... Ole
d-rel(e+x,,).
e,= [I O
e=-
(2.33)
(2.34)
Theorem 3: Let the reference input r( t) satisfy
zyxw
for some given positive constants r l , r2. Then there exist positive
constants p*, u, (Y < 1/2, c1 to c,, y1 and t , such that for each
p E (0, p * ] every solution of (2.31)-(2.34) starting at ( = 0 from
the set
,
The equations for the error e = Y - x
can be expressed as
t.=A,e+b,(e-e*)'(P+x,,)+&~p
(2.23)
p 3 = ~ ~ ~ p 1 p [ [ ( e + x , , ) + ~ ~ e ~ ~ , + ~ ~ , , )enters the residual set
+ A 2 r + A 3 p + A , P ( P + x,,)
+ A407f(e, e , s ) + A4i]
e,=kTe=[l
O
...
Ole
(2.24)
(2.25)
where
where X, to X, are positive constants, at t
D,(p) for all t > t,. Furthermore,
=t,
and remains in
The -corresponding regulation result follows as a corollary.
Corollary I: Assume r( t ) = 0, x,, = 0. Then there exists a p*
such that for each p E (0, p * ] and u = 0 in (2.34) any solution
e ( ? )= Y ( t ) ,p ( t ) , e( t ) of (2.31)-(2.34) which starts from D(p)
given by (2.36) is bounded, l i m , + x ~ [ Y ( =
r )0,
~ ~lim,+xllp(()]l =
0, lirn,+,llO(t)ll= 0, and l i m , + J p ( r ) - O * ~ ~ r - ~=constant.
Proof of Theorem 3: Choose the function
e'"
=
zyxwv
x - x,,
e("
=
u1 - u l n , , e(2)= u2 - van,. (2.28)
We now need to design an adaptive law for updating the parameter vector e ( t ) . For the parasitic-free case [l]the adaptive law
zyxwvutsr
[
zyxw
(2.39)
P, p - P;'( e'PA,,A;')']
guarantees that the output error goes to zero as t 4c ~ jand the
signals in the closed loop remain bounded for any uniformly where P = P T > 0 satisfies
bounded reference input r ( t ) . As demonstrated in the first secATP+PA,=-qqT-~L
tion for the scalar tracking problem, the best we can expect in the
presence of parasitics is to guarantee that the solutions starting in
Pb, = h
a domain D(p) remain bounded and converge to a residual
bounded set D,(p). To achieve this we modify the adaptive law for some vector q, matrix L = LT> 0, and scalar
(2.29) as
P, = PT> O satisfies
(2.40)
(2.41)
4
0, and
210
zyxwvutsrqpon
zyxwvutsrqp
zyxwvu
zyxw
zyxwvutsr
zyxwvutsrq
zyxwvuts
zyxwvutsr
IEEE TRANSACTIONS ON AUTOhL4TTC CONTROL. VOL. AC-29. NO.
Equations (2.40), (2.41) follow from the fact that hT(sZ - A,)-'b,
is strictly positive real [l] and (2.42) follows from the assumption
that Re h ( A), < 0.
Observe that for each p > 0, p o > 0, and a > 0 the equality
v(e,q.8)=pop-2a
3.
XIARCH
1984
for some positive constants 6, to 6, which depend on r , , r, and
the norms of matrices. For a < 1/2 there exists a p* such that for
each p E (0, p * ]
(2.43)
(2.48)
defines a closed surface S(p,a , p o ) in R S ' 1 + m .- 3The time derivative of V ( e ,q , 8 ) along the solution of (2.31)-(2.34) is
V(.,v,e)
1
= - -eT(qq'+
2
rL)e-
o(e-
e*)'r-le
-
1
?qTQIq
+p[9-pll(e~p~,~,')~]Tp~[~,(e;x,,,)
+ A28T( 2 + x , , ) + A , r + A39
T
-~A48'(e+x,,)-A,(P+x,,)T(e+x,,)e,
+A,0Tf(8,e,q,r)+A4t
-p-
A~,P(A,.~+~=(B-~*)~(~~X,,,)+A,~~)I.
22t7-7~
(2.44)
Let
E
h,=-minh(L),
2
1
h,=minA(P).
A , = ~ ~ ~ ~ A ( Qx 4 J= ,l p
ll.
(2.45)
Then (2.45) is simplified to
Choose u 3 c3p2("-"' where c3 = (4/X,)6,.
Then there exist constants c1 to c2 such that the set D,(p) is enclosed by [[P(p).and
D(p) isenclosed by S ( p , a . p o ) . Furthermore. V ( e ,77.6) < 0
everywhere inside S(p. a , p , ) except possibly in D,(p). The
constants cl, c2 are chosen such that every solution of (2.31)-(2.34)
starting from D ( p ) remains in S ( p , a , p,). Since in D(p)/[[P,(p).
I/( e, q , 6 ) is strictly decreasing there exist constants c4 2 1 and t ,
such that any solution starting at t = 0 from D ( p ) / ~ , ( p ) will
enter Do(p) at t = r , and remain in [[P,,(p) thereafter. Any
solution starting from D,(p) at t = 0 cannot escape and remains
in D o ( p ) for all t > 0.
Pruuf of Curulhy I: The proof of Corollary 1 follows directly from the proof of Theorem 3 by noting that when r(r) = 0,
.xn, = 0, and u = 0, the set D,(p) reduces to the origin e = 0,
q = 0, i.e.? in (2.38) y1 =.O. Therefore, V(e,q,8)< 0 everywhere
inside S ( p , a , p o ) and V( e. 17.8) = 0 at the origin e = 0, 17 = 0.
Hence, any solution e(t) = Y ( t ) ,q ( t ) ,O ( t ) which starts from
D(p) is bounded. Furthermore, V ( e( t ) , q( t ) , 8( t ) ) is uniformly
continuous, and therefore lim, ,V( e( t ) , q( t ) . 8( t ) ) = 0. Hence,
lim,+,ll~(t)ll=0. and Lm,,,ll8(r)ll=O.
limt,,llY(t)ll=O,
Since inside S ( p , a , p , ) , V(e, q, 8) is nonincreasing and bounded
below, it reaches a limit k
'
,which is a finite constant. Therefore,
lirn,,,(6(r)-e*)'r-'(6(r)-6*)
= 27/,. Le., limr-xlle(t)8*llr-1 =constant.
In Theorem 3 and Corollary 1, it is assumed that p* < p1where
p1is defined in the following lemma.
Lemma 3: There exists a p1> 0 such that constant output
feedback u = B&J stabilizes (2.16)-(2.18) for all p E (0.pl].
The proof of Lemma 3 is more complicated than that of
Lemma 2 and can be found in [17] where an explicit expression
has been obtained for p,.
Remark 9: The dependence of constants 6, to 8, and y, on
r l . r, shows that for a given p a reference input signal with high
magnitude or high frequencies can no longer guarantee that
V ( e ,17.8) < 0 everywhere in D(p)/D,(p). Such a reference signal introduces frequencies in the input control signal which are in
the parasitic range. Thus, the control signal is no longer dominantly rich [6] and, hence, it excites the parasitics considerably
and leads to instability. This explains the instability phenomena
observed by other authors in simulations such as [18] where a
square wave was used as a reference input signal.
Remark IO: For ease of exposition we have assumed that r( t )
has bounded derivative everywhere on [O.m). This assumption
can be relaxed byallowing a finite number of jump points or
comer points for r ( t ) on the t-axis d h o u t any significant
changes on the results of Theorems 2 and 3.
Similar comments as in Remark 8 apply here also.
zyx
zyxwvutsr
where a, to a,,PI to & and y1 are positive constants determined
from r,, r, and the norms of the system and the reference model
matrices. Constant y1 depends on rlr r, and y1 = 0 when r, = 0.
r, = 0.For all e, 8, and q enclosed in S(p,a , pol. (2.46) becomes
CONCLUSION
We have analyzed reduced-order adaptive control schemes in
which reference models can match the dominant part of the
plant, while the model-plant mismatch is caused by the neglected
high frequency parasitic modes. In presence of parasitics the
global stability properties of the parasitic-free schemes can be
lost. However. we have shown that in the regulation problem a
region of attraction exists for exact adaptive regulation. This
region is a function of the adaptive gains and the speed ratio p ,
and as p -+ 0, it becomes the wholespace. Thus the adaptive
regulation problem is well posed with respect to parasitics. In the
case of tracking we proposed a modified adaptive law. The
modified scheme guarantees the existence of a region of attrac-
D
zyxwvutsrqpon
zyxwvutsrqp
zyxwvutsr
KOICOTOVlC: ROBUST REDESIGN OF ADAPTIVE CONTROL
IOANNOU
tion from which all signals converge to a residual set which
contains the equilibrium for exact tracking. The dependence of
the size of this set on design parameters indicates that a tradeoff
can be made sacrificing some of the ideal parasitic-free properties, in order to achieve robustness in presence of parasitics. The
crucial effects of the frequency range of parasitics, the adaptive
gains, and the reference input signal characteristics on the stability properties of adaptive control schemes explain the undesirable
phenomena observed in [12], [13]. The results of this paper are
obtainedfor a continuous-time SISO adaptive control scheme
where the transfer function of the dominant part of the plant has
a relative degree of one. The same methodology can be extended
to more complicated continuous and discrete-time adaptive control problems.
REFERENCES
~
[16] H. K. Khalil, “On the robustness of output feedback control
methods to modehng errors,” IEEE Trans. Aurotmt. Contr., 1701.
AC-26, Apr. 1981.
[17] P. A. Ioannou, “Robustness of absolute stability,” Znr. J . Confr.,
vol. 34, no. 5, pp. 1027-1033, 1981.
[18] C. Rohrs, L. Valavani, and M. Athans, “Convergence studm of
adaptive control algorithms, Part I: Analysis,” in Proc. 19th IEEE
Conf. on Decision and Conrr., Albuquerque, NM, Dec. 1980.
Petros .A. Ioannou was born in Cyprus, on
February 3, 1953. He received the B.Sc. degree
with First Class Honors from University College.
London, England. in 1978 and the M.S. and
Ph.D. degrees from the University of Illinois,
Urbana, in 1980 and 1983, respectively.
In the summer of 1979 he worked with the
Ford Motor Company on the control problem of
the car engine. From 1979 to 1982 he was a
Research Assistant at the Coordinated Science
Laboratory, University of Illinois. During his
graduate studies, he also worked on research projects from the Ford
Motor Company, General Electric, and Dynamic Systems. In 1982 he
joined the Department of Electrical Engineering-Systems, University of
Southern California, Los Angeles, as an Assistant Professor. He currently
teaches and conducts research in the areas of adaptive systems. large scale
systems, and singular perturbations.
Dr.Ioannou is a member of the American Society of Mechanical
Engineers, EtaKappaNu,and
Tau Beta Pi. He has held a Commonwealth Scholarship from The Association of Common\vealth Universities, London. England, and was awarded the Goldsmid Prize and the
A.P. Head Prize from University College.
zyxw
zyxwvutsrqp
zyxwvuts
[l] K. S. Narendra. Y. H. Lin, and L. S. Valavani, “Stable adaptive
controller desi Part 11: Proof of stability.” IEEE Trans. Automar.
. Conrr.. vol. A F h , pp. 440-448, June 1980.
[2]A. Feuer and S. Morse, “Adaptive control of single-input singleoutput linear systems,” IEEE Trans. Automar. Contr., vol. AC-23,
pp. 557-570, Aug. 1978.
[3] G. Kreisselmeier, “Adaptive control via adaptive observation and
asymptotic feedback matrix synthesis.” IEEE Trans. Automar. Conrrl., vol. AC-25, Aug. 1980. .
[4] G. C . Goodwin, P. S. Ramadge, and P. E. Caines. “Discrete time
multivariable adaptive control.” IEEE Trans. Automar. Conrr., vol.
AC-25. June 1980.
~[SI I. D. Landau and H. M. Silveira, “A stability theorem with applications to adaptive control,” IEEE Trans. Auromat. Contr.. vol. AC-24.
pp. 305-312. Apr. 1979.
[6] P.A. Ioannou and P. V. Kokotovic. “An asymptotic error analysis
of identifiers and adaptive observers in the presence of parasitiis,”
IEEE Trans. Auromat. Contr., vol. AC-27, Aug. 1982.
[7] B. D. 0. Anderson and C. R.Johnson, Jr.. “On reduced order
adaptive output error identification and adaptive IIR filtering,”
IEEE Trans. Automat. Conrr., vol. A5;27. Aug. 1982.
[SI P. A. Ioannou and C. R. Johnson, Jr., Reduced-order performance
of parallel and series-parallel identifiers of two-time-scale systems.”
in Proc. Workshop on Applications of .Adaptice SJst. Theoy, Yale
Univ., New Haven, CT, May 1981.
[9] P. V. Kokotovk and P.A.
Ioannou. “Robustness redesign of
continuous-time adaptive schemes,” in Proc. 2Orh IEEE Conf. on
Decision and Conrr.. San Diego, CA, 1981, pp. 522-527.
[lo] M. S. Balas and C. R. Johnson. Jr..“Adaptive identification and
control of large scale or distributed parameter systems using reduced order models.” in Proc. Workshop on Applicutioru of Aduprice
S U I . Theor3., Yale University. New Haven, CT, May 1981.
[ll]G. Kreisselmeier. “On adaptive state regulation:” IEEE Trans.
,4utomut. Contr.. vol. AC-27, Feb. 1982.
[12] C. E. Rohrs, L. Valavani, M. Athans. and G. Stein, “Analytical
verification of undesirable properties of direct model reference
adaptive control algorithms,” in Proc. 20th IEEE Conf. on Decision
und Contr.. San Diego. CA. Dec. 1981.
[13] C. R.Johnson. Jr.. and M.J.Balas.
“Reduced-order adaptive
controller studies.” in Proc. Joinr Automar. Conrr. Conf.. June 1980.
[14] I. D. Landau, “A survey of model reference adaptive techniquesTheory and applications,” Automatica, vol. 10. no. 4, pp. 353-379.
[15] B. B. Peterson and K. S. Narendra,“Boundederror
adaptive
control,” part I, Dep. Eng. AppI. Sci.. Yale Univ., New Haven, CT.
Rep. 8005, Dec. 1980.
~1~~~~~
211
zyxwvuts
zyxwvutsrq
zyxwvutsrqponm
Petar V. Kokotovic (SM’74-F’80) received
graduate degrees in 1962 under Prof. Mitrovic in
Belgrade, Yugoslavia, and in 1965 under Prof.
Feldbaum and Prof. Tsypkin in Moscow, USSR.
He has been active in control theory and its
applications for more than 20 years. From 1959
to 1966 he was writh the Pupin Research Institute.
Belgrade, Yugoslavia. Since1966he
has been
with the Department of Electrical Engineering
and theCoordinated Science Laboratoq? University of Illinois, Urbana, where he teaches control theory and conducts research in modeling. sensitivity analysis, optimal and adaptive control, singular perturbations. and large scale systems.
He has held visiting appointments with ETH. Zurich, Switzerland.
INRIA, France, and has lectured at other institutions in the U.S. and
Europe. His consulting activities include General Electric Company and
Ford Motor Company.
Prof. Kokotovic has served on the IDCand AdCom of the IEEE
Control Systems Society and on committees of the International Federation of Automatic Control. He is an Associate Editor of $vrretm and
Control Lerters.