11 Proceedinas
- of the 15th IEEE Intemational Symposium [
on Intelligent Control (ISIC 2000)
Rio, Patras, GREECE
I l l
MC4-3
AN ADAPTIVE LEARNING SYSTEM YIELDING UNBIASED
PARAMETER ESTIMATES
DANIEL
w. REPPERGER~
SEDDIK DJOUADI'
'
Air Force Research Laboratory, AFRTfHECF', WPAFB,Ohio 45433, USA, D.ReDDereer@IEEE.ORG
'National Research Council, AFRUHECP, WPAFB, Ohio 45433, USA
Abstract, A learning system involving model reference adaptive control (MRAC)algorithms is studied in
which the Lyapunov function and its associated time derivative are simultaneously quadratic functions of
both the position tracking error and the parameter estimation error. An implementation method is
described which expands results from [5]. Initially it appears that the condition of persistent excitation
(PE) need not be explicitly satisfied, however, this condition is actually implicit in the requirements for a
solution.
Key Words: Adaptive Control, Learning System
A standard solution to this problem is given by Lemma
1 [3]:
1. INTRODUCTION
Lemma 1
This paper will address a specific class of learning
systems or MRAC algorithms which have the
interesting property that both the Lyapunov function
and its associated time derivative along the motion
trajectory are quadratically dependent on both the
tracking error and the parameter estimation error. If
this type of algorithm can be successfully
implemented, then unbiased parameter estimates can be
obtained. The study of model reference adaptive
control has been well-established [ I ] when over 1,500
papers have been reported at that time with numerous
experimental results obtained. The notation used
herein and a brief description of the conventional
MRAC method is first discussed.
Let a state-space description (x E R", Q E R', v E R", 8 E
R") of Figure 1 be of the form:
2. THE DIRECT METHOD OF THE STANDARD
MRAC PROBLM
Using the nomenclature of [2], the scalar tracking error
q ( t ) = yp(t)-y,(t) in Fig. 1 represents the difference
between the plant's output yp and the reference model
ym. The unknown parameter vector is 8 which is mxl
and its adjustment mechanism only allows knowledge
of yp(t), %(t), and possibly their respective derivatives.
0-7803-6491 -O/OO/$l 0.00 02000 IEEE
x
= A x + b [ k €ITv(t)]
(1)
The remaining matrices are of appropriate dimensions
with A being Hurwitz, the scalar k is unknown (except
for sign), the pair (A,b) is completely controllable with
b known, and v(t) is a measured variable to be defined
in the sequel. For the parameter vector, 8 ' represents
A
-
6
the true value, 8 is the estimate, and 8 = 8 - 8' is
the parameter error. With some abuse of notation
(mixing both the Laplace transform variable p with the
time domain variables), the error vector resulting from
equation (2) admits to the form:
The transfer function H(p) is strictly positive real
(SPR) (stable, minimum phase, and of relative degree
no greater than unity). The adaptation law is given by:
- 139-
parameter estimation problem can be mitigated. An
alternative approach to this problem is now provided.
where y >O. Q(t) and e (t) are globally bounded and if
v(t) is bounded, then Q(t) + 0 as t+ W. Associated
with this problem is a Lyapunov function (Y =(x,
4.
-
e )):
V(Y) = V(x,
Ikl g T g
e- ) = X f P J + Y
For this problem, first define a scalar sliding state
variable s(r) as follows:
(5)
and y > 0 controls the rate of parameter adaptation
such that the parameters change more slowly than the
effects they induce on the error vector e&).. It can-beshown that:
=, lim VW)+
(radial unbounded
(i) As IYI
condition).
(ii) V(Y) > 0 if Y # 0 (positive definite property).
(iii) To show that 3 (Y ) < 0 tl Y # 0 (negative
-
definite property of 3 ), if A is Hurwitz, using the
Kalman-Yakubovich lemma (there exists positive
definite mamces P and Q such that ATP+ PA = -Qand
P b = c) and with the fact that H(p) is SPR, then it
follows:
It is noted that only Q(t) + 0 as t
8
+
-
-
=y p-y ,
x = x(r)-x,(t)
(10)
will be used to represent tracking error. The choice is
now made of the following Lyapunov function:
L
L
.&
where h is identical to the variable used in (9) and will
be defined later with the positive constants y3 and y4.
The following assumptions are implicit in what is to
follow:
is guaranteed
-
to the fact that v
The persistent excitation condition can mitigate this
situation [3,4].
3.
where the term:
Where s(r) is the tracking error of equation (9) and the
positive constants yI and yz will be specified later. The
time derivative of V I along its motion trajectory is
required to satisfy:
+ 0 may not occur due
does not depend explicitly on 8 .
and the requirement that
A NEW ADAPTATION ALGORITHM [SI
4.1 Assumptions
THE PERSISTENT EXCITATION
CONDITION
(1) The true parameter
8
is constant.
(2) s ( f ) satisfies the following relationship [ 5 ] :
If the reference model in Figure 1 satisfies:
Where the measured position variable
via:
Where r(t) is the input forcing function to the reference
model, the persistent excitation condition would
normally require, on
= [r, eo]', through the choice
of r(t) in Figure 1:
v
u(t)
=
x,
-2 h
x"
- h2
v(t)
F
is specified
(14)
Hence it is required to measure the variable x, and its
next two derivatives as well as obtain measurements of
2 and its first derivative.
Where 3 al > 0, I is the identity matrix, and T > 0,
for any t > 0. With this condition in place, the biased
-
The algorithm now follows:
140-
4.2 Derivation of the Algorithm:
Then the following linear equation in 2 has to be
solved:
.
The goal is to simultaneously satisfy (1 1) and (12).
Differentiating VIof (1 1) yields:
VI
=
e
~ ( t .c(t)+yI
)
e'
8
. ..
+%
8 e'
1
zy*+-y4z+zy,=
2
(15)
-
. ..
= s [ - h e - s + ev ( r j ] + y t 8
8+%e' e'
(21)
Since all y > 0, i=1,4 then (21) is Hurwitz for properly
selected 1. This leads to the following methodology
for the selection of the y, in equations (1 1,12):
using the relationship in equation (13), the following
results:
VI
0
(16)
7.
which is required to satisfy V, of equation (12). This
will occur if the following relationship holds:
METHODOFSELECTIONOF
AND
IMPLEMENTATION OF THE ALGORITHM
A three step procedure will implement this algorithm:
Step 1: Picky such that equation (21) is strictly
Hurwitz, i.e. the solution of (21) is of the form:
This equation will now be simplified and various
solutions examined. This extends results from [SI to a
larger class of solutions.
5.
Where the real part of a3 and a 4 are both >O. Then
-
Z(t) + 0
lim
t+
THE NONLINEAR EQUATION TO BE
SATISFIED
at)= e
-
and the parameters are unbiased since
e *.
For notational simplicity, it is easier to denote Z(t) =
8 = 8 - 8 and since 8 . is constant then:
-
(23)
I
=8-
6
i (t)
=
e*
Ster, 2: From (20) this also implies that A(t) must
also be of exponential order since:
(18)
A(t) = -
and the adaptation law is then specified independent of
Also for
knowledge of the true parameter 8'.
brevity, the variable A(t) = v ( t ) s ( t ) is known from
measured quantities (cf. equations (9) and (14)).
Equation (17) now simplifies to the form:
2 2 y2 +
Zyl
+
1
2
- Y3 z
= v(tl s ( t )
(24)
A(t) + 0. This
which would now satisfy lim t+ -,
means that both tracking error variables (containing
s(f) in equation (9) and v ( t ) in equation (14)) would
have to converge to zero. Thus both tracking error and
parameter error convergence are established
simultaneously.
1
.
1
- y 4 z 2 = Z [ - A ( f ) - - y? Z] (19)
2
2
Step 3: There still exists a caveat from the procedure
so far. What is unknown is:
The goal is to provide solutions of (19) which are
stable and not trivial.
6. SOME ALTERNATIVE SOLUTIONS OF (19)
If the right hand side of equation (19) could be set to
zero, the left hand side then becomes linear since the
cancels out. If (sufficient condition):
term
if the estimator 8 (0) = 0 is unbiased, which is usually
the case. This implies we know the true parameter 8
if we know Z(0). To circumvent this difficulty, the
procedure is modified to determine z ( t ) rather than
Z(t) via the following sequence of events:
z
(20)
-
141
-
[2] S. Sastry and M. Bodson, Adaptive Control,
Stabilitv. Convereence. and Robustness, Prentice Hall,
1989.
(a) Solve equation (24) for z(t) and substitute the
results into (21). This yields:
I
2
2 Y2 + - Y4 2 =
2 (71 1 Y3 ) A(t)
(26)
[3] J-J E. Slotine and W. Li, Amlied Nonlinear
Control,Prentice-Hall Inc., 1991.
(b) Now define a new variable:
Y(t) = 2 =
&t)
(27)
[4] K. S. Narenda, A. M. Annasswamy,
Adaptive Svstems, Prentice-Hall Inc., 1989.
(28)
[5] D. W. Repperger and J. H. Lilly, “A Study on a
Class of MRAC Algorithms,” hoceedines of the 1999
E
E
d
December, 1999, Phoenix, Arizona..
Then Y(t) is Hurwitz and satisfies:
Y(O)= 0
1
Y ( t )+ -
2
- Y(t) = f(t)
Y4
Y2
m,
(29)
where f(t) is of exponential order since:
f(t) = 2
- A(t)
(30)
Y2Y3
and A(t) is of exponential order from equation (24).
(b) Thus the adaptation algorithm is to calculate Y(t)
via (28-29) and then:
and
&t) = Y(t)
(32)
Numerical simulations of examples are presented in
figures 2, 3 and 4 which will be discussed at the
conference during this paper’s presentation.
8.
CONCLUSIONS AND DISCUSSION
A simple method of providing both parameter error
convergence and tracking error convergence is
demonstrated by taking a special case solution of a
nonlinear equation, which describes potential
adaptation algorithms. It is possible to guarantee the
tracking error to be Hurwitz as well as the parameter
estimation error in a special case solution of this
nonlinear equation.
9. REFERNCES
[I] K. J. Astrom, “Theory and Applications of
Adaptive Control - A Survey,” Automatica, Vol. 19,
NO. 5, pp. 471-486, 1983.
- 142-
Stable
"t
I
c
cu0
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d)
2
d)
k
&
2
.
.
.
,
1
- 143 -
i
i
0
1
I
0.5
1
j
I
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1.s
Time in sacon&
2
3
2.5
Figure 2- Method from [3]
.-...--
1.4
-..
-..---.
A
..... -.---.-- .--.. .i
;
3 1 2 ....... t.....--.--.---.+
4
i
2
*
E .
f
*
1 .-
2
f 0.8
j
.)
*
;
------------.-;
.i
.........
j.
........ -
.c
*
0.6 --..-...-.i--
i....................
i
*
1.pagb;ma)ion
(M'=t 14 true) - Non PE h(r(t)-0)
4
i.......................
i......................
....--t.-....-..-----...
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0.2 ........................
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i
.
.....................
:.-; ................ 4........................
.
......-..--.-
....
0.4 ....
i
___._.___.______
_. i.....
-. i.
New Algorithm
4.......................
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i
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j
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Figure 3-New Method
- 144-
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