Syslems & Control
North-Holland
Letters
4 (1984)
347-358
September
Design of adaptive pole-placement
for plants of unknown order
1984
controllers
N. MINAMIDE
Deportment
of Electrical
Engineering.
P.N. NIKIFORUK
College
of
Engineering.
Faculty
and M.M.
University
of Engineering
Nagoya
University,
Furo - cho, Chikwa
- ku. Nagoya.
464,
Japan
GUPTA
of Saskarchewan,
Saskatoon.
Saskatchewan,
S7N OWO, Canada
Received 17 March 1984
Revised 11 June 1984
By an indirect control approach,
an adaptive pole-placement
control problem
assuming the knowledge
of an upper bound of the plant order. A class of models
to the plant is first constructed
based on the parrrneter
estimate generated
minimization
problem
is then derived that enables one to determine
the model’s
that the desired closed-loop
poles are assigned in a limit.
Keywords:
Adaptive
control,
Pole-placement,
Deterministic
discrete
systems,
is considered
for a scalar discrete-time
linear plant
that can be regarded IO be input-output
equivalent
by a least-squares-type
identification
scheme. A
coefficients
and the pole-placing
feedback gains so
Least-squares
methods.
Recursive
optimization
methods.
1. Introduction
During the past few years several approaches to the control of unknown plants have been suggested.
Among these, the indirect adaptive control approach seems to be most promising in developing globally
stable adaptive control algorithms applicable to a general class of plants. Many successful applications of
such schemes have been reported in the literature [l-8]. It has been demonstrated in [3] that a large class of
identification methods can be associated with a large class of control methods to perform the unconditional
stabilization
of deterministic
linear systems. However,
there still remains a theoretical difficulty in
establishing global convergence of such schemes. Since the stabilizing feedback compensator is designed
based on the model’s dynamics, the model system is required to be controllable, or at least stabilizable
asymptotically.
What has been established in [l-6] relies on the hypothesis that such singular situations do
not occur. This hypothesis, however, imposes certain restrictions on the plant parameter and the class of
the input applied. In particular, such restrictions become crucial when when only the upper bound of the
plant order is assumed to be known, which is the case of interest here.
The present paper proposes a method for handling the problem of the stabilizability of the identified
model when using an adaptive pole-placement control algorithm for a single-input, single-output discretetime linear plant with the knowledge of the upper bound of the plant order. It is shown that a solution to
this problem can be reduced to solving a minimization problem after each iteration of the identification
algorithm. For minimization, the Quasi-Newton
algorithm is proposed. Recently, Larminat [7] considered a
similar problem of the stabilizability of the identified model in indirect adaptive control and showed that
the usual least-squares adaptation scheme could be effectively modified in order to ensure the above
stabilizability
condition. The algorithm proposed in [7], however, is based on a type of a random search
and seems not to be very efficient for real-time calculation.
In Section 3, some characterization
of a class of parameter vectors satisfying the input-output
relation is
given. In Section 4, a least-squares-type
adaptation scheme of exponential convergence is proposed and a
0167-6911/84/$3.00
0 1984, Elsevier
Science
Publishers
B.V. (North-Holland)
341
class of parallel models is characterized in terms of adjusting vectors. Selection of an appropriate
stabilizable model and determination of an appropriate pole-placing functional observer are considered in
Section 5, and global convergence of the algorithm is established. In Section 6, some simulation results are
given.
2. Problem statement
Consider a linear time-invariant
of the form
plant having an autoregressive
moving average (ARMA)
representation
(2.1)
where u, and y, are scalar plant input and output,
B*( q- ’ ) are scalar polynomials given by
A*(q-‘)=1+a;q-‘+
0.. +un*q-“,
respectively,
B*(q-‘)=b;q-‘+
q -’ is a unit delay operator and A*(q-‘),
..f
+b,:q-“’
(2.2)
where the coefficients {UT}, i = 1, 2,. . . ,n, and {b:), i = 1, 2,. . . , m, are unknown. In the sequel, the
following assumptions on the plant will be used.
A,: The upper bound r of the plant order r* = max(n, m) is known.
A,: A*(q-‘)
and B*(q-‘)
are relatively prime.
A,: The scalar polynomial D( q-‘) = 1 + d,q-’ + . . . + d,q-’ whose roots define the desired closed-loop
poles is stable and has at least one real root.
Now, from assumption A, the input { aI} and the output {y,) satisfying (2.1) may be considered to
obey the following ARMA model of order r:
4P)Y,
(2.3)
= ~Wb,
where
A(q-‘)=l+a,q-‘+
-.-+a,q-‘Bl+u’[(r),
B(q-‘)=b,q-‘+
s-f +b,q-‘PU[(r).
Here, the prime denotes the transpose, t(r) = [q- ’ qm2 . . . qmr]’ and coefficient vectors a = [a, a, . . . a,]‘,
b=[b, b,.. * 6,]’ are unknown. It is to be noted here that the pair (a, 6) satisfying (2.3) may not be
determined uniquely and there may be infinite such pairs in general.
Define a 2r X 1 vector $B,-, by
+,-I=
[Y,-,
y,-2 “‘Y,-,
u,-1 U,-2”’
%,I’.
(2.4)
Then, (2.3) can be described in a (nonminimal) state space representation:
+,-1
+clh%,-l
+c2u,7
Y, = c;+, = h’+,-’ 9
(2.6)
where c, = [l 0 . . . 01’ E R2r, c2 = [0 . . . 0 1 0 . . . 01’ E R2’ (unity appearing at the (r + 1)st component),
h = [-a’ ,‘I’ E R2’ is an unknown parameter vector, and D is an r X r shift matrix defined by
0
1
0
1
(2.7)
Note here that the state of the system (2.5) may not be controllable but it is stabilizable for an appropriate
choice of the parameter vector satisfying (2.3).
348
An adaptive pole-placement control problem is to determine the parameter vector h and the coefficients
fof the state feedback control law u, = -f’+,,-,
+ u, which cause {u,} and { y,) to remain bounded for all
time and such that the resulting closed-loop characteristic polynomial of (2.5) approaches D(q-‘)
for the
uniformly bounded input u,.
3. Some characterization
of a class of parameter vectors in R*’
To clarify the structure of the parameter space R2’ under consideration, let us consider the problem of
determining the set of parameter vectors h that satisfy
y&=h’qQ-,
(k=l,2
(..., t)
(3.1)
where ~u~~~--~+~.-~+~.....~-~ and (Y~)~--~+~.-~+z,....~
Define a 2r X 2r matrix S, by
are data satisfyingt2.1).
(3.2)
and consider the orthogonal decomposition of R2’ associated with the transformation S,:
R2’=9(S,)W’-(S,)
(3.3)
where .%‘(S,) and N(S,) denote the range and null spacesof S,, respectively.
Let Ar = r - r* be the difference between the plant and model order, and form a 2r
defined by
10
-by
.. .
-br%
0
F=
-6;
Lemma.3.1. Let 9(F)
x
1
Ar matrix F
’
(3.4)
denote the range spaceof F. Then,
(t=1,2
~?(F)c.N-(S,)
,... ),
(3.5)
where equality holds if the condition
r+l
U-,+2
...
ut--r--r’
=r+r*
(3.6)
. ..
is satisfied. Thus, dim 9(S,), the dimension of L%‘(S,), or equivalently, the number of linearly independent
vectors in {#I~-, }k-,.2 .,.,., is, at most, r + r*.
Proof. We first show the implication 5Q F) c M(S,). Since for Ar = 0, the result follows trivially by
definition, we suppose that Ar 2 1. Let h = [hi hi]’ E a(F) (hi E R’, i = 1, 2). Then, h = Fp for some
p E R”. Observe that
h’t(2r)
=h,(q-‘)+q-‘h2(q-I),
(Fp)‘&(Zr)
=p( q-‘)A*(q-‘)
- q-rp(q-‘)B*(q-‘)I
(3-T)
349
where hi(q-I)=
gives
h,(P)
hit(r)
(i = 1, 2), and p(q-‘)=p’[(Ar).
h,(q-‘)=
=P(q-‘P*(q-‘)~
Thus, comparison
of the powers
in q-’
-p(q-‘)B*(q-1).
in (3.7)
(3.8)
Therefore,
h,(q-‘)yk=p(q-‘)A*(q-‘)yk=p(q-‘)B*(q-’)Uk=
-h,(q-‘)u,
(k=l,2,...,t)
(3.9)
whichimpIiesh’cp,-,=Ofork=1,2
,..., r.Hence,hEM(&).
Conversely, let h = [hi h;]’ E JV( S,) (hi E R’, i = 1, 2). Then,
h,(q-‘)y,+h,&‘)u,=O
Multiplying
(3.10)
(k=l,2,...,f).
(3.10) by A*(q-‘)
and using (2.1) yields
{h,(q-‘)B*(q-‘)+h2(q-‘).4*(q-‘)}uk=0
It then follows
from the condition
(k=r*+l,...,t).
(3.11)
(3.6) that
h,(q-‘)l?*(q-‘)+hz(q-‘)A*(q-‘)=O.
(3.12)
Since A*(q-‘)
and B*(q-‘)
are relatively prime, there exists a polynomial p(q-‘)
=p’t(Ar)
(p E RAr)
suchthat(3.8)isvalid.Here,forAr=O,p=Oandtherefore,h,(q-‘)=h,(q-’)=O.Thus,h=O~~(F).
For Ar 2 1, (3.7) is valid and h = Fp E 9( F). Now, since 9(F) c JV( S,) implies JV( F’) 3.4?(S,), we have
dim9(S,)<dim.X(F’)=2r-dim9(F)=2r-Ar=r+r*.
Remark. The input { u~}~- -r+l .. .. ..-. is said to be sufficiently
Corollary.
h,i,,=
Under the condition (3.6), the solution h,i,
exciting if it satisfies the condition
(3.6).
of minimum Euclidean norm satisfying (3.1) is given by
{I-F(F’F)-‘F’}h*
(3.13)
where
h* E [ma: . . . - a;a 0 . . . 0 bf . . . b$ 0 . . . 01’ E R2’
(3.14)
is a particular solution of (3.1). The general solution h of (3.1) is given by
h=h,i,+Fp
(PER”).
4. Adaptation schemeof exponential convergence
Consider the following least-squares-type adaptation scheme:
- (P:-,4-J, () = o
e,=e,-,+Kr-A,(Y,
x
0
r-1
K,=
I-
+
r--l
+4-,K,-,+,-,
1,-,,K,=I,
L&*6,
h
9
i
i2,=K,iz,-l+8,,
(4.la)
4-,K,-I%-,
jZo=ho,
K
(4.lb)
(4.lc)
where X,-, is a positive number satisfying 0 <A,-, I .ro for some positive constant Ed. Note that
K, = [I f S,]- ’ is non-negative, non-increasing and hence converges.
350
4.1. The adaptation
scheme (4.1) has the properties:
(1) Let t, (it)
be the lower boundfor
which%‘(S,,)=%‘(S,)
Lemma
Ilh,-P,h*-(I-P,)h&
holds. Then,
fi(l+/~~)-‘llh,,--hill
(4.2)
i-1,
where P, is orthogonal
projection
to %‘( S,) and p, is a minimum eigenvalue of S, restricted to the range space
9?( S,) which is positive and nondecreasing.
(2) For any uniformly bounded vector sequence (v, } (v, E R”, Ilv,ll< M,,, t = 1, 2,. . .), let
K;=K,+
Then,
K,v,,
jy=R:‘+,,_,.
there exists a non-negative
IY, -Xl
5 +i+
sequence
ll~,-,ll)y
(3) For any vector h, satisfying
(t=1,2
h;=h,+K,v,=h,
,liza,
(a, } such that
= 0.
(4.3)
(3.1) there exists a uniformly
bounded
vector sequence
{ v, } such that
,... ).
Proof. (1): Note first that 0, is a unique solution minimizing the functional
INIl + i
IYk - ~,‘+,-,12/L,
k-l
and is given by 8, = K,S,h*
= (I - K,)h*.
Hence,
A,-h*=K,(h,-,-h*)LK,h,-,=K,K,-,
..+K,,h
,,-,.
(4.4)
Now, since P, and Sj (j < t) commute, so do P, and K, = (I + S,)-’ (j I t). Moreover, (I - P,)S, = 0
implies (I - P,) K, = Z - P,. Therefore, operating on (4.4) with P, and (I - P,), respectively, yields
IPAll
s Ii (1 + P,)-‘llhr,-ill
i-1,
(I-P,)h,=(Z-P,)h,-,=
(4.5)
5 Ii (1 + P,)-‘llkJlll
i-r,
*.*
=(z-P,)h,.
(4.6)
Combining (4.5) and (4.6) now gives
lli,-P,h*
-(Z-P,)h~ll=Il’,-(Z-P,)~,ll=llP,~,ll
which is (4.2).
(2): By +,,-, Ed?,
s fi (1 +Pj)-‘ll’oll
j=r,
(4.lb) and taking the trace of (4.lb), we have
IY,-KI~I~:-,~,I+I~:~~,~,I~~~~:~,~,~,I+~~
L,M;-,K,-,v,I
+dK_ ~I 1
I 1 I 1 I
1
where { IIP,h,ll} and {tr( K,-, - K,)} both converge to zero.
(3): Let JV( Z - K,) denote the null space of Z - K,. Then, since X(1
can also be expressed as 8, = (I - K,)h,. Therefore,
A, = K,jr,-,
- K,) =M(S,),
0, = (Z - K,)h*
+(I-K,)h,=h,+K,(&-,-h,).
351
It therefore suffices to set v1= h, - A,- i (t = 1, 2,. . .) where
Ib,II = llhp - LII
is uniformly
= IIW,
- LNI
5 . . . 5 IV, - holl
bounded.
Corollary. If the input 1 uk )k= --r+ 1.-~+2 ..__.t.., satisfies
F( F’F)-‘F’h,
exponentially for t 2 t,,,.
the condition
(3.6), then h, converges to h,i, +
In the scheme(4.1) applying (4.1~) repeatedly, i.e.,
Remark.
qk=K,va-,+8,
(k=1,2
,..., N,),
710= A,- 1,
(4.lc’)
k=%J,,
can improve the convergence, where N, is a possible maximum recursion number.
5. Adaptive
pole-placement
control
We now turn attention to the derivation of the adaptive pole-placement algorithm and its convergence
analysis.
Let h, and K, be the parameter estimate and the time-varying gain matrix generated by the adaptation
schemedescribed in the previous section. Define a modified parameter estimate A, by
h;=h,+K,v,+ri;‘b,!“]’
(a;,i)rY~R~)
(5.1)
where v, E R2’ is an adjusting vector. The nonminimal state space representation (2.5) can then be rewritten
as
+ +cle,
+,=.
D1k1+c,$
[D0
c24
where e, =y, -j,” and J,” = A:‘$,,- ,. When e,, the error between the plant and model output, is regarded as
an external disturbance in (5.2), then (5.2) can be viewed as the state space representation of the parallel
model having the adjusting vector v,. Thus, if the input u, is generated by the causal feedback control law
u,= -14
P,‘]$L,+v,
b,J,-‘),
(5.3)
the characteristic polynomial of the model system (5.2) becomes
{1+~,‘~(r)}{1+Ci~‘~(r)}+(~~~(r)i)IY’~(r)~~(q~’)A”(q~‘)+(~(q~’)B”(q~‘).
5.1. At each sampling time t, there exist uniformly bounded vectors v,, a, and /i, satisfying
Lemma
P(q-‘)A’(q-‘)+a(q-‘)B’(q-‘)=D(q-’).
(5.4)
Theseare characterized as any solution z = [ f3’, a’, v’]’ to the following minimization problem:
,,$“M lldz; A,, K,)ll
= 0
(5.5)
where
dz;
A,,
K,)
=S(ci;,
w[f]-[j+[y
and MO is a constant independentoft. Here, d is a coefficient vector of D(q-‘),
352
(5.6)
i.e., D(q-‘)
= 1 + d’t(r),
and
S(a, 6) E R 2rx2r is defined by
1
a
0;
1
. ..
.S(a,b)=
0
;b
1
...
1;
-0
al
0
0
‘..
..
‘..
0
0
b-
Proof. Since (5.4) is satisfied if and only if the vectors (v, OL,P} solve the minimization
suffices to show the existence of the solution to (5.5).
From assumption A,, the polynomial D( q-‘) can be factored as
0(q-‘)=d,(q-‘)p(q-I)~-d,(q-‘){l
+p’Wr))
where d,(q-‘)
and p(q-‘)
are real manic polynomials
by h, = h* + Fp. By Lemma 2.1, h, satisfies
Yk
=
(k=
+#‘k-,
of degree r* and Ar, respectively.
Define h, E R2’
1,2,...).
Thus, by Lemma 4.1, there exists a uniformly
equation holds:
ir; = A, + K,v:
problem (5.5) it
bounded vector VT (]]v:]] I I/h, - h,,ll) such that the following
= h,.
For this choice of v,*, both sides of (5.4) can be divided by p( q-l)
to obtain
,8(q-1)A*(q-‘)+ol(q-‘)B*(q-‘)=d,(q-’).
(5.7)
Since A*(q-‘)
and B*( q-l) are coprime, a pair of coefficient vectors (a*, p*) satisfying (5.7) exists and is
independent of t. Hence the minimization problem (5.5) has the solution z* = [p*‘, a*‘, v,*‘]’ in a bounded
region of radius Ma (2 ]I[p*‘, (r*‘, hb - hb]‘ll) which is independent of 1.
To solve the minimization
problem (5.5) the following
Quasi-Newton
Quasi-Newton
algorithm
(1) Let z, = zi,, and k = 1.
(2) If llg(z&; h,, K,>ll < E, then set [b,‘, a:, v,‘]’ = zk. Otherwise,
(3) Compute zk + , by
‘&+I
=
Zk
[
is proposed.
go to step (3).
A,,
K,)
1dz&;
adz &; h,,K,)
-
algorithm
+
aZ
(4) Increase k by one and go to step (2).
Here At is the (Moore-Penrose)
pseudoinverse
of A, E is a small positive number,
and zin, is an initial estimate of the solution of (5.5) which may be set to zinc = [/?,‘- i, a:-,, v;-r]’ except at
an early stage of adaptation. A recursive algorithm for computing the vector Atb called a best approximate
solution is available (see [9]).
To study convergence of the Quasi-Newton
algorithm, it is convenient to introduce the shorthand
notations
g(z) =g(z; h, K,),
G(Z)=
adz; L K,)
aZ
353
where g(z) is a vector function
expansions [lo], we have
having a second derivative.
By the Taylor
formula
for a second-order
g(z+h)=g(z)+G(z)h+R(z,h)h
(5.9)
where
(5.10)
R(z, h) =/,l(l -1) a2g’;z: rh) (h) dt.
In the present case, the 2r
a$(4
,z,(Y)=
X
4r matrix a2g(z)(y>/az2
is given by
(5.11)
[~(-~,y,,K~y~)iS(y~,~~~~)Diag(-Z,,,,z,,,)K,l.
Here, y’= [y;,, y;*, y;] E R4’ (yli E R’ (i = 1, 2)) and K,‘= [K,‘jK;]
( Ki E Rrxlr (i = 1, 2)). Since, by
(5.11), a2g(z)(y)/az2
is independent of z, the matrix norm llR(z, h)ll induced by the Euclidean norm l]hll is
bounded by
(5.12)
llR(z, h)ll s m,lVll
where m, is a positive constant.
Proposition 5.1. Let ( zk } be a sequence generated by the Quasi-Newton algorithm which converges to a point
z,, in U(M,,) = {I I llzll< M,}. Suppose further that G(z) has a maximum rank at z = zO. Then, g(zO) = 0.
Furthermore, g( zk) + 0 and zk + z0 quadratically, respectively.
Proof. By assumption,
where { G( z,))
there is a number m,, such that rank{ G( z,)} = 2r for k > m,,. Therefore,
is uniformly
bounded. To show g( zk) + 0 quadratically,
observe that by (5.9)
dz,+,)=dz,c)
+G(zk)(zk+,
- zk)+R(z,v
zk+l-zk)(z,c+,
- z,c)
=g(zk)-G(dG+(zkhdzd
-+,v -G+(zk)g(zk))G+(zk)g(zt).
Hence, by (5.12),
lldzk+l)ll s ~~ll~t~~~)l1211s(~~)l12
I mlm~lls(zk)l12 A Wdzk)l12
(5.13)
where m2 is a positive constant such that I]Gt(zk)l] 5 m2 for k = 1, 2,. . . .
TO show zk --, z0 quadratically,
let m be a number such that
lldz,)ll
< a/M
with 0 < 6 < 1. Then, for n > m,
II-m-1
II% - znl+1II s
c
II-m-1
Ilz,n+,c+~
-z,+,cll~
c
Il~+k,+dll
Ildzm+~)ll
k=O
5 C*mp/Cl- ~~)llAJll.
Letting
n
+ cc in (5.14) thus establishes the quadratic
(5.14)
convergence
of zk to zo.
The following proposition establishes the criteria under which the Quasi-Newton
algorithm generates a
sequence ( zk ) which converges to a solution of g(z) = 0. Let U( zo; E) denote an e-neighborhood of zo, i.e.,
u&J; E) = {z ( 112- zoll I E}.
354
Proposition 5.2. Let zo be a solution of g(z) = 0 at which G(z,) has a maximum rank. Then, there are
neighborhoodsU(z,; E,) and U(z,; Ed) of z0 with E, < Edsuch that if the initial estimateI, is in U(z,; E,), then
the sequence{ zk ) generated by the Quasi-Newton algorithm remains in U( z,,; Ed) and convergesto a solution
of g(z)=0 in U(z,; .eZ).
Proof. Note first that by assumption and boundedness of G+(z) near z,,, there exists an &,-neighborhood
w e; E*) of ze such that
rank{G(z)}
=2r
and I(Gt(z)l(lm2
for z E U( zO; e2).
With M and 6 as defined in Proposition 5.1, let .z3be a positive number such that
Ildz)II~~n
i
s
2’
(1-6)E2
2m
2
1
for I(z - tall 5 &3
and let E, = min{ .z3, ie, ). Select a point z, in U(z a; E,) and let { zk ) be the sequencegenerated by the
Quasi-Newton algorithm. Observe that if {I,) (j = 1, 2,. . . ,n - 1) are in U(z,; e2), then by letting m = 1 in
(5.14)
Ilz, - GJII5 Ilz, - z,ll+ lb, - ZOII5
m211dz1)ll + II
Zl
1-s
-zzollIfe2+~E2sE2.
Thus, z, is in U(z,; Ed). By induction, the whole sequence { zk ) is in U( z,,; Ed). Since IJzk+, - zkl( I
m 211
g( zk) I( and since { g( zk)) converges to zero quadratically, { zk ) is a Cauchy sequencewhich converges
to a solution of g(z) = 0 in U(z,; Ed).
Under the assumption that the Quasi-Newton algorithm works properly, we have the following theorem.
Theorem 5.1. Consider the adaptation scheme(4.1) and the state feedback control law (5.3). Supposethat at
each sampling time t, the Quasi-Newton algorithm obtains a suboptimal solution of (5.5) satisfying llz,ll I Me
and llg( z,; h,, K,)ll < E where M,, and Eare positive numberschosenappropriately. Then, the state +, of (5.2) is
uniformly bounded, and the coefficient vector of the resulting closed-loopcharacteristic polynomial approaches
that of D(q-‘) with norm error at most E, where Ecan be specified arbitrarily small.
Proof. Substituting the state feedback control law (5.3) into (5.2) gives the closed-loop system described by
$5= @4+,-1+ v,
+ c2u,
(5.15)
where
@,=[f ;]+c,h;~-c2[;]’
(5.16)
Note that since by Lemma 4.1 { &, ) and { K, ) converge and since { z, ) is determined based on these values
{it,, K,), {z,) also converges. Now, since by Lemma 4.1 e, satisfies the estimate (4.3) c2u, is uniformly
bounded, and {a,) converges to a stable matrix by the way {z,) is determined, it follows from (5.15) and
(5.16) that II+,,11
is uniformly bounded.
6. Simulation studies
To test the convergence behavior of the proposed algorithms, several simulation studies were performed.
A typical set of results is now discussed.
355
Consider
the following
A*(q-1)
second order system:
= 1 - osq-’
B*( q-1)
- o.f34q-2,
+ 1.oq-2.
= osq-’
The system has poles at -0.7 and 1.2, and one unstable zero at -2.0. The following conditions were used
for simulation:
D(q-‘)
= 1.0,
A, = .EO
= 10-6,
h, = 0,
E= 10-3,
M, = 102,
o, = sin(0.29) + 1.5 sin(0.9) + 2.5 sin t.
The initial conditions used were
u-r+,=
. . . =u()= 0,
y-,+,
= . . . =y-,
=o,
y,=3.
For minimization of the functional (5.5), the Quasi-Newton algorithm was employed.
-10
0
5
10
15
20
15
20
TIME
Fig. 1. Comparison
of output
trajectories
(r*
= r = 2).
-10
0
5
10
TIME
Fig. 2. Comparison
356
of output
trajectories
(r*
= 2. r = 3).
Various values of the upper bound r of the system order r * = 2 were considered. These were 2, 3, 5 and
7. Figures 1-4 show the corresponding
simulation results in which the output y, is compared with the
output $’ that will result from an appropriate feedback control law using known parameters.
For each selection of the upper bound r of the system order r*, the parameter estimate ?I, converged to
h min within 2r steps. With the parameter estimate A, = h min, the transfer functions of the identified models
of order 3, 5 and 7 have common factors located at (0.131}, (0.279, -0.0426 + j 0.566) and (0.387,
-0.321 f j 0.526, 0.232 + j 0.518), respectively. Although these common factors are all stable and
therefore, the identified models are stabilizable without modification in this example, this is not the case in
general. The convergence of the Quasi-Newton
algorithm was considerably rapid by the choice of the initial
value tin, = [O.. .O 0.. .O 1 0.. .O]‘. A few iterations were almost sufficient to guarantee the convergence
throughout the simulation.
-10
I
10
I
5
0
TIME
Fig. 3.
-101
Comparison of output trajectories (r* = 2,
0
I
5
r = 5).
I
10
I
I
15
20
TIME
Fig. 4. Comparison of
output
trajectories
(r*
= 2, r = 7).
357
7. Conclusion
Using the indirect control approach, a discrete-time adaptive pole-placement control problem has been
investigated, and the global convergence of the algorithm has been shown. Although the algorithm requires
the minimization of the functional, the stability analysis does not assume that the plant is of minimum
phase, that the input is sufficiently exciting or that the plant degree is known. The essential assumption
made is that an upper bound on the plant order is known.
Acknowledgements
The assistance of research assistant
Miss E. Nikiforuk
is acknowledged.
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