LIDS-P-132 2
AUGUST 1983
ROBUSTNESS STUDIES IN ADAPTIVE CONTROL
by
James Krause, Michael Athans, Shankar S. Sastry and Lena Valavani
Laboratory for Information and Decision Systems
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
ABSTRACT
This paper examines robustness issues in .Model Reference Adaptive Control systems in the presence of unmodeled dynamics and output disturbances. We present
an approximate technique, trend analysis, by which we
can study the evolution of the parameter error trajectory under periodic excitation. This analysis provides
new insights upon the size and spectral content of the
excitation sufficient to guarantee local stability.
I.
INTRODUCTION
Several similar Model Reference Adaptive Controllers (,MRAC's) have been shown to be globally stable
under certain restrictive assumptions, including the
assumption that the order and relative degree of the
plant are exactly known, and that no disturbances are
Under certain "sufficiently rich"
present ((1]-(31).
excitation conditions, the origin is globally asymptotically stable for the adaptive controller parameter
error ([5], 6]).
When the restrictive assumptions are violated, as
they always are in practice, no proof of stability
Furthermore, instability can
exists [4],[9], [10].
occur under excitations which are "sufficiently rich"
At the present state of our studies, the new richness
conditions provide substantial insight into the type
and size of excitation required for stability. With
reasonable designer-known information, one can determine a desirable frequency range for the command input
energy, and the most desirable subset of this range.
One can also estimate the amplitude of command input
required to prevent a dangerous slow drift of the parameter error due to an output disturbance of any frequency.
We present the problem and establish notation in
Section II. The approximate analysis technique is
presented in Section III, and theorems which justify its
use are stated. Sufficient excitation conditions for
stability are given and discussed. The primary insights
and results derived from the analysis are briefly stated
and discussed.
II.
THE ADAPTIVE CONTROL SYSTEM
The designer assumes that the low frecuency portion
of the single-input-single-output plant can be described
by
y(s) =
u(s)
(1)
P
oles and zeroes of
where G (s) is of order n.
The
P
G (s) are unknown but are assumed to lie within some
in the sense mentioned above. A stronger definition
of sufficient richness is required to guarantee stability of the adaptive controller in the presence of
urnmodeled dynamics and disturbances.
The key factor to the stability of the adaptive
known bounds.
controller is the time-evolution of the parameter er-ut/output
relationship is
The actual
plant
input/output
relationship
is
ror vector.
This time-evolution is described by a
complicated set of time-varying nonlinear differenv(s) = G (s)u(s) + d (s)
(2)
o
p
equations, putting a closed-form analytic solutial
tion of reach.
where
This paper presents an approximate analysis techG (s) = G (s)
(1+E (s))
(3)
p
p
p
nique for studying the long term trends of the parameter error vector trajectory for an adaptive controlThe uantity d s) is an ouut disturbance, and
S
To
ler under periodic excitation. Certain measures of
is a modeling error due to neglected high-frequency
the error of this
technique have been proven to applant d-namics.
An upper bound E (X) on the magnitude
proach zero as the adaptive gain (which controls the
The relevant
rate of adaptation) approaches zero.
of E (jw) is generally known in practice [7].
p
[8].
Thus the
theorems are stated here and proven in
A model output is constructed:
analysis technique can be termed a trend analysis for
slowly adapting controllers.
y (s) = G (s)r(s)
(4)
The trend analysis provides a vector field, defined
M
M)
where r(s) is the command input.
as a function of the command input, which approximates
The plant control input for adaptive control is
(in a long term sense) the time-derivative of the parameter error vector.
From this
vector field
one can
(5)
w (t)k(t)
u(t) =
determine potential limit sets of the parameter error
vector, and associated regions of attraction and apwhere- k(t) is a 2n-vector of adjustable parameters, and
vector, and associated regions of attraction and apNew excitation conproximate rates of convergence.
ditions are defined which are sufficient for stability
Of the adaptive controller in tIe presence of unmodeled
(6)
u(t)
,
W(t) = H(t)
dynamics.
Direction of future research are indicated
which are required to enable analytic evaluation of
some of the sufficient condition for a controller with
a given adaptive cain.
' Research supported by ONR/N00014-82-K-0582 (NR 606-003), NSF/ECS-8210960 and NASA/NGL-22-009-124.
To appear, Proc. 22nd IEEE Conf. on Decision and Control, San Antonio, TX, Dec. 1983.
where,* denotes convolution and H(t) is the impulse
response of a known causal linear time-invariant svstem. The vector W(t) is constructed by the adaptive
controller.
If k(t) is assigned a constant value k , a time-o
invariant closed-loop plant results.
Classical robustness techniques can be used to detera
mine which values of k result in guaranteed stability
-o
of the closed-loon plant. Define -K s :o be the set of
all such stabilizing k .
a-o
B.
ered a sum of a constant desired value and an error:
k* is such that k(t)=0 and F (s)=O0 results in'y(s)/r(s)
-=y(s)/r(s). With an appropriately chosen model, such
an(t)
is
proximate analysis echniques. A vector
defined which has a much simpler trajectory and which
represents the long term trends of the vector k(t). The
accurracy of the trend analysis is shown to improve as
the rate of adaptation is reduced.
The trend vector trajectory is a function of the
model matching can result in a desirable performance
system excitation.
improvement over a nonadaptive alternative design [8].
An error signal is defined:
ic
The adjustable parameter vector k(t) can be consid-
k(t) = k* + k(t)
(7)
Presentation and Validation of the Trend Analvsis
The trajectory of the adjustable parameter vector
k(t) is critical to the stability of the system.
If
k(t) approaches a limit inside K , the system is stable.
--
--
-s
If
k (t) strays outside of K , a rapid onset of instabily may occur and has been observed.
The differential equations describing the time evolution of k(t) are highly nonlinear and rime-varying,
and cannot be solved analytically, hence we employ ap-
The command input studied is period-
and of the form
Z
r(t) = r
(11)
r. sin(wit+rni);Z<
+
e(t)= y(t) - yC(t)
(t)*(wT
T(t)k(t) )+T (t) *r(t) +T
(t) *d (t)
(8)
3
0
2
1
where T1 (t), T2 (t), and T3 (t) are impulse responses of
-
causal linear systems.
The lapiace
transform Tl(s) of
To show the relationship between the trajectory of
the trend vector k' (t) and the actual trajectory of
and to aid the definition of the trend, we first
k(t),
present the limiting case in which the rate of adaotation is zero:
Tl(t) is of the form
Pe K
k(t) = k
TL(s) = T
(s)(l+E (s))
(12)
vt
(9)
where T (s) is a strictly positive real known linear
o
E (s) is unknown but shares the characterissystem.
By choosing a grid on the space
tic shape or E (s).
The usual parameter update law (10) may be modified to
function as an observer:
T
(13)
k(t) = -yw (t)e(t)
of possible plant parameters, one can numerically man
the known upper bound on 1E (jW) to an upper bound on
With r(t) as in
tity
i(j')jT
2(jw),
E
E(j)
and
IT3(u)
.
in the aoplication of the trend analysis.
The parameter update law is
k(t)
=
k~t)
-yw~t~e~t)
k(t)
_(t) == -yw(t)e(t)
--
(10)
(10)
0-ac
bounded. With nonzero unmodeled dynamics, no stability
proof has been given, and potential instability has
been demonstrated [4],[9],[10].
A.
PARPTME`R ERROR TRENDS WITH PERIODIC EXCITATION
Introduction
It is shown in this section that instability is a
potential but not necessary consequence of unmodeledal
high frequency dynamics. Whether or not instability
occurs is largely a function of the command input.
'Low frequency components of the command aid in correct
parameter adjustment, while high frequency components
contribute -o misadjustment. .n output disturancenever
of
any frequency contributes to a hazardous "drift"
the parameter error. An excitation condition is given
which is bothr necessary and sufficient for local (with
respect to the parameter error) stability of the adaptive controller. Rate of adaptation is also a factor;
an excitation condition which is sufficient for the
stability of a slowly adapting system may not be sufficient for the stability of a rapidly adapting systaem
(14)
yield
k=~y(A
where the adaptive gain y is any strictly positive
scalar constant.
Equations (8) and (10) characterize the parameter
error behavior as a function of w(t). When E (s)-O
_?
and d (s)=O, the adaptive control system is stable in(t
o
the sense -that e(t)e L2 and all signals are uniformly
III.
(8), (13) and the iden-
sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
Wnen Z (w)=O,
These upper bounds are important
= T2 (jw)=0.
(11), equations
where
=
c
A
+A
ac
(t))k
-
+ yt
-dc
-ac
(t) are 2nx2n and b
,b
(t) are 2nxl.
c
-dc -ac
The elements of A
and b
are constants, and the ele-dc
-do
ments of A
(t) and b
(t) are sums of sinusoids of
nonzero frequencV.
A
readily-computed
-a are
-do and bc
functions of accessible signals (w(t) which is a funcand as such can
tion of r(t) and y(t)) and as such can be considered
Aiding computation and analysis is the
user-known.
property that each of the frequency components
present
s present
contributes independently
in r(t)
-de
-cc
In the absence of an output measurement, A.c and
b
are a function of r(t) and k , and are defined for
-O
-dc
all r(t) of the form of equation (11) and all k B Ks
Exact computation of
c and b
is no longer so easy.
can be computed a priori
values of A
for each possible k
B K .
The actual values of A
-do
-o
s
b
will be a perturbed version of these nominal values.
The
A bound on the perturbation can be calculated.
compute
sign of the real part of the eige-nvalues of the computed
be changed by inclusion of any stable un-do
modeled dynamics, but the presence of unnodeled dynamics
in Tl(s) can cause such a change.
To avoid confusion between the various sources of
subsequently ignore the issue of a rior
we will
eror
b
without an
an outnut
outpu measuremeasurecalculation of
of AA
and bd
calculation
and
without
-do
-dc
ment, and shall concentrate only on the errors relevant
Nominal values of A
and b
/
to the stability of the system.
city, the dependence of A
and
A',
For notational simpli, and later A and
on either r(t), y(t) cr r(t), k (t),
-o
shall usually
not be shown explicitly.
ntefine the trend
= YA.
k'(t)
k
A.ll
o
-
theorems
Thecrem 1:
+ yb.
If the seauence
(15)
o
{k(t +mT)},m={1,2,3...j
<
_D
for some constant E.
to follow are proven in
[83.
we have thus provided analytic justification for
the use of a trend analysis in the study of an adaptive
controller.
The trend analysis can be made arbitrarily
There exists a T>O such that
k' (t +mT)=k(t +mT) for all positive integers m.
o
- o
Theorem 2:
Corollary:
ot+T]
converges to a limit k , then k(t) converces to a ball
defined by
tk
k'(t ) = k(t )
-
tt
i- (t)-(t)
accurrate by choice of a suitably small adaptive gain y,
which is the designer's prerogative.
If A. k + b. f0, then
-c-o
-cc
(ki(t0+t I)-k (to t1)
C.
1
1m
Sufficient Conditions
for Stability
we now proceed to discuss the qualities of the trend
itself and to present sufficient conditions for stabil-
= O
l l (t +t11
)IIity
of the adaptive system.
Given r(t) only, A' is a continuous function of
is
1nd
)-(t
jlk(to+t
t)|
uniformly bounded.
hne adaptive control problem is fundamentally different than the parameter observer problem above. TAS
while Theorems 1 and 2 and eqn. (15) aid in the understazding of the trend analysis approach, they do not
provide justification for use of the trend analysis in
.ie control problem. We now provide such justification.
(t ). Consider the trend vector field F(R ,r,y)
2
defined on K c R n by associating the corresponding
s
A'(k
in Ks
-0 ,r,y) with each point I
As Y approaches zero, the direction of A' approaches
a limit (see equation (17)):
A'
Let the parameter update law be given by eqn. (10).
'et A,
and b
be computed as before, assuming cons---c
dc
tant parameters.
k' = yAd
-dc-
c yb
+
-dc
0
0
(17)
S
onsider any T satisfying theorem 1.
Defne
A = k(t +T)
--
o
k(t
0o
Condition 1:
-0o
such that for all
Y<~Y
0
°
Corcllary:
x = F (x,r)
(20)
has a locally stable equilibrium point.
For the case of a specific choice of a nonzero y, we
must take into account the error of the approximate
trend analysis.
Because this can be done a variety of
ways, a variety of sufficient conditions can be gener-
<6
p
ated.
Given any 6 there exists a y1 such that for
all Y<y
Two will
be presented here.
In the process we will define constant upper bounds
on various tvlpes of errors.
A discussion of the computation of such bounds shall be given later.
Given any scalar constant p, and vector constant x,
1
KL/(P'·e!L~~~ >I
1-
r(t) is sufficient for local stability in the limit as
y approaches zero if the system
(18)
Ila-AIl
I
2
F (R ,r) be defined by associating the corresponding
vector A' (: ,r) with each point k
in K .
s
We now state, without proof, the first sufficient
richness condition of this paper:
Given any 6>0 and choice of a p-norm there
exists a y >0
(19)
-c
O
-
'= k' (t +T) - k(t )
-heorem 3:
t
i 2
¥°
`L a
k'(t ) = k(t )e K
=
' =I
By the corollary to Theorem 3, the direction of A approaches this same limit as y approaches zero. Let
Define the trend vector
f'
A k(to)+bc
=
-
1
1
2I'A|2
define
<6
~2~~2
~z
where <-,-> denotes the dot product.
Theorem 3 is somewhat analogous to Theorem 1.
.at=-er than exact matching of the trend and the actual
trajectory at a selected point in time, we now have
.=
1
=
il |
{!:!l|-
defines
2
}
2
(21)
)||
_Inx- l'(K
)-H(K
K
1
-_ -°
-=
--
2 < P-
1
near matching with a fractional error which can be made
arbitrarily small. The corollary states that the angle
bert-een the vector directions A and A' can be made ar.it-arily small by sufficiently slow adaptation.
Theorem 3 treats the error at a specific point in
t-re.
For the intermediate points, we have the
following:
(23)
(24)
-o
t
1
eft t +T]
1
o o
2
V
{v: vER n
lvil < 1}
2K' = {K :i +z v~K,
vev}
-S
Theorem 4:
(22)
2
-
-a
2-
2'
(25)
(26)
-
Given a choice of a p-norm, there exists a
constant C such that
The bounds z1
small y,
and z2 are known to exist for sufficiently
and in fact approach zero as y approaches zero.
Condition 2:
The proof is simple and is left to the reader.
r(t) is sufficient for local stability of tne adaptive
controller if there exists a S >0 and an x such that
31
1s
words must now be said about the practical application
of these conditions, lest the reader be mislead as to
the present state of our research.
Testing of condition 1 for stability in the limit
_
,
+
Partial oroof:
f231
- (~7:-o) 6,
(27)
-
(27)
in the
The use of K' rather than K
S
-
) =
o
(28)
-o
k(t
+T) =
k = k +6
(to+T)
= - = k +
k
=t
=
(29)
(29)
+ A'
(30)
if condition 2 is satisfied, then
'
~'
tI i -
fi-
P
z
(31)
2
Thus
i- <1 I-iz -R{-xI
FL
Equilibri= points R_ are
-
(37)
=
S
condition is a subtlety that shall not be discussed
here.
enot
k(t
is currently feasible.
defined by
A few
11!3
-{
Since FL(
,r) is continuous, one can check for local
stability of an equilibrium point through linearization.
Testing of conditions 2 and 3 is not feasible without further research. Computation of the bounds z
2' and z3 is not currently possible.
The theorems
contained herein guarantee that any desired value for
any of these bounds can be achieved through the use of
some sufficiently small y, but the value of y required
is not clear due to the nature of the proofs.
The stabilization technique employed in [8] involves
selection of a reference input which is sufficient for
stability in the limit as y approaches zero, followed
by an arbitrary selection of a "small" value for y.
Simulations indicate that
1) thi-s process is often
2
< z1 + p -z1 = p
Therefore
e3
m=0,1,2,...
;
-n- m
and 3,
is a locally stable region for 2(t).
2(32) very overconservative, and 2) the meaning of "small"
changes substantially with different plants, models,
and excitations. Tight bounds on zl, z,
z3 as a
2
1
function of these different elements shows promise of
allowing the use of much a larger y (hence faster adap(33)
tation) with confidence.
Completicn
D. Trend Analysis Results
D. Trend Analysis Results
that equathat equaThe trend analysis technique constitutes an analytother sysic quantification of the relationship between parameter
of the proof would involve a demonstration
thewould
proofinvolve a demonstration
tion (33) and
6Ks implies boundedness of
tem signals.
emn signals. 1ondition 2 requires that
n effect, condition 2 requires that F
'
......
inward arou.nd the boundary of a region, and
nward ound
the boundary
xco
of a region, and
~of
~
be oriented
be oriented
that this
tgat this
inward orientation exceed the error bound using a
?ar_-icular standard of measurement. Condition 3 to
follow has a similar interpretation but employs a diflraent
star.dard of measurement.
For any positive p <P , define
For Positive
P3 P,
define
1
'3 {i|2x
= fI
<
}
(34)
3
~3
2---3342
(35)
i,
ij~c-xI
I <+pCzr:
2
2+3
z
=max
3=saxI
-0o
1
-- m-o
arccos
-t~'(~ ,r),A~k ,r)>
)
----
*k ,r)
fco
1
-o0°
i
2
Ii(k
,r)
-oo
_-
|
2
(36)
-hat is
'' .
z
r'3
is an upper bound on the angle between L'
--
and A for all :~ in
exists;
arccos
Determination of such an excitation is simplified by
an empirically observed property of Lhe matrix
(_,r).
A
and b
vary relatively slowly
neighborhood of an equilibrium point k
by considering the FL resulting from
3:
(k ,r)k' +
k' =
(t o
(x--o ),A'(-O
arcc _o (x-i7-s
t
it
)
dc
~
2
-I
°
2
where -93 denotes the complement of
3
3
=
requirement that all
the eigenvalues of A
(_ ,r)
-dc '
in the open left half plane
in
the
oen
left
half
con- -1
lie
ane.
We can place further requirements on A
which enters
(38)
Hence we have, as a minimal condition for asymptotic
iu
i
the
the
stability in the limit of an ecuilibrium point
90- z3
-3
voreovser
any trajectory of i(t)
verges to 54'
are revealed
the re-defined
trend
3 K' and
_15X s
for all k oS1 n
1
'3'
Given r,
with i, such that the properties of F (X,r, ) in some
r(t) is sufficient for local stability of the adaptive
controller if there exists P
P3 and
a, x such that
0<P < I
The excitation should be chosen to provide stability
in the limit (condition 1), and allow wide margins for
the errors with the unknown bounds z1 , z2 , and z3 .
For a sufficiently small Y, z3
1.
as y approaches zero, z3 approaches zero.
Condition
convergence and the spectral components of the excitai
iton.
As such it has provided new nfo.mat:on
and
insights into the stability of model reference adaptive
shall e stated
control systems. Some of these results shall be stated
here witchout derivation.
Though the stability problems of adaptive controllers has been well publicized recently, it is worth
emphasizing that the trend analysis provides analytic
verification of the potential instability of MRAC's.
It also provides verification of a potential stability
of the controllers and a means of obtaining stability
without a modification of the basic algorithm. One
can determine a periodic excitation which is capable
of stabilizing the system, and add this excitation to
the command input.
(and
hence indirectly on r(t)) to allow a margin for errors
of the type appearing in conditions 2 and 3. For
example, a large margin for z3 of equation (36) would
be provided by an A.
which gives a small value
ain te
(large
provides a tool for designing a remedy to the stability
negative
to z , where
direction)
4
problem through the filtering of the command input and
the addition of a persistent excitation to the command.
x A. x
(39)
T
xx
For the case of two adjustable parameters, the smallest
values of Z4 are obtained by choice of a command with
z - max
4
max
IV.
For several promising adaptive algorit-hms, in the
absence of u.nodeled dynamics and output disturbances,
weak richness conditions on the excitation guarantee
convergence of th-e adaptive controller parameter error
conveencef
e adative cntoller aameer eror
to zero. Under more realistic assumptions, convergence
to zero
frequency content concentrated near the model bandwi
Spectral components of the command above a threshold
frequency
to ncrease
serve
z
.
Acting alone these
to zero parameter error must be given up.
T
4
high frequency components yield an A.
with positive
eigenvalues.
The threshold frequency wT is the frequency at which
Tl(ju) has a phase of 900 (see equation (8)).
This has
an interesting correspondence to the strictly positive
real requirement placed on T (s) in nominal-system
stability proofs [1].
Under the mild assumption that the designer knows
1) an upper bound on the magnitude of the open loop
unmodeled dynamics as a function of frequency, and
2) bounds on the low frequency plant parameters (modeled uncertainty as opposed to unmodeled dynamics), -he
designer can determine a lower bound on _T.
Thus, the
designer can determine a priori a low frequency band in
which command energy is guaranteed to aid in parameter
identification and stability of the adaptive controller.
This is useful in determining the desired frequency
content of a dither signal added to the command to
maintain stability, and also suggests the desirability
of prefiltering the command input to remove high frequency content.
The amplitude of low frequency command required to
maintain stability is largely a function of the spectr-=n of the output disturbance. Through algebraic
manipulation, the effect of the output disturbance can
be entirely contained in b
.
Thus an output distur-
REFERENCES
[1]
[2]
[3]
[4]
[5]
after leaving K .
s
The frequency content of the disturbance is not a
critical issue except for the fact that disturbance rejection varies with frequency. If unopposed by proper
command-generated excitation, output disturbance at
any frecuencyv contributes to a drift in the parameter
error such that the loop gain increases without bound.
For a given output disturbance, a sufficiently
large low frequency reference input will give the systern described by equation (19) a stable equilibrium
point in K .
That is,
local stability
in the limit as
y approaches zero will be achieved. Given a sinusoidal
reference of a specified low frequency, the designer
can conservatively determine an amplitude sufficient to
guarantee stability
of the system using the following
-information: 1) bounds on the low frequency plant parameters, 2) the classical
(gain margin) robustness constraint, and 3) the model choice.
In summary, one can state
that the trend analysis
reveals the need for a different perspective on "sufficient richness of excitation" than that present in
[51 and [6].
When unmodeled dynamics are considered,
the frequency of excitation becomes an issue.
The
presence of output disturbances at any frequency creates
amplitude constraints on the excitation.
More than
just highlighting these issues, the trend analysis
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ua
s Arising in Adaptive Identification"
IEEE Trans. on Auto. Control, Vol. AC-22, Feb.
and beyond, where tine trend is not
defined. In this situation we typically see the parameter error vector follow its predicted trend while in
K , and then rapidly progress toward infinity shortly
To achieve
the weaker goal of stability and bounded parameter error without a modification of the algorithm, one must
impose stronger conditions on the excitation.
One can
develop such conditions through the approximate analysis technique sketched in this paper.
Valuable
insights have resulted as to the type and size of excitation required.
Further research is required to
enhance the usefulness of the new richness conditions.
bar.ce shifts the equilibrium point of the parameter
error vector without
seriously
affecting
the local
error vector
seriously
wiout
afecting te6]
stability or instability of the equilibrium point. The
danger is that the equilibrium point may be shifted to
the edge of K
CONCLUSIONS
1977, pp. 83-88.
[
7]
£8]
[9]
[10]
J. Dovle and G. Stein, "Multivariable Feedback
Design: Concepts for a Classical/Modern Synthesis",
IEEE Tran.s. on Auto. Control, Vol. AC-26, Feb.
1981, Dp. 4-16.
J.M. Krause, "Adaptive Control in the Presence of
High Frequency Modelling Errors," S.M. Thesis,
MIT, May 1983.
C.E. Rohrs, L. Valavani, M. Athans, and G. Stein,
"Robustness of Adaptive Control Algorithms in the
Presence of Unmodeled Dynamics," Proc. 21st I-=~
Conf. on Decision and Control, Orlando, Fla.
Dec. 1982, pp. 3-11.
C.E. Rohrs, L. Valavani, M. Athans and G. Stein,
"Robustness of Continuous-Time Adaptive Control
Algorilthms in the Presence of Unmodeled Dynamic,"
Paper LIDS-P-1302, MIT, Cambridge, Mass. April
1983 (submitted to IEEE Trans. on Auto. Control).