IEEE
TRANSACTIONS
ON
AUTOMATIC
como~,
VOL.
AC23,
NO.
4,
AUGUST
1978
557
Adaptive Control
of
SingleInput,
SingleOutput Linear Systems
AbsrrucA
procedure
is
presented for
designing
parameteradaptive
control
for
a singleinput, singleoutput process admitting
an
essentiauy
unknown
but
fiied
linear
model,
so
ttmt the
resulting
closedloop syshn
is
globally stable
with
zero
steadystate
tracking
error
been
the
output
of
the
process
and
the
output
of a
prespecified
hear
reference
model.
The
adaptive controller
is
a differentiatorfree
dynamical
system forced
only
by
the
process
input
and
output,
as
well
as
by
a
reference
input.
INTRODKCTIOW
HERE ARE many examples of physical processes
Twhich require feedback control syestems capable of
functioning at a number of different process operating
points. In some instances, the parameters of the linearized
process model upon which closedloop control is based
assume such a wide range of values during process opera
tion that a single, fixedparameter control system proves
inadequate to regulate the process. In such cases paramet
ric changes are often dealt with by using several fixed
parameter controls and an appropriate switching logic;
alternatively, in some cases model parameter values are
precomputed and stored as functions of operating point,
and a single control system with gains functionally depen
dent on stored model parameters is used. If the number of
process model parameters is large, if the parameters
cannot be computed with sufficient accuracy or if “tight”
control is required to meet rigid specifications, neither
type
of
control system may be capable of providing ade
quate regulation.
A
promising alternative, potentially applicable in situa
tions such as these, is
a
parameteradaptive control sys
tem. Roughly speaking, a parameteradaptive system
is
an
adaptive control system with the capability of adjusting its
own parameters to compensate for the slow but significant
changes in process characteristics resulting from process
transfer from one operating point to another. The idea of
a parameteradaptive system is, of course, not new
[
11[3].
Nevertheless, it is fair to say that the basic principles
governing the design and operation of such systems are
only now just beginning to be understood.
March 8, 1978. Paper recommended
by
R.
V.
Monopoli, Past
Chairman
Manuscript received April
25,
1977;
revised September
19,
1977
and
of
the
Adaptive,
Learning
Systems, Pattern Recognition Committee.
This
work
was
supported
by
the United States
Air
Force
Office
of
Scientific Research under Grant
773176.
The authors are with the Department
of
Engineering
and
Applied
Science, Yale University, New Haven,
CT
06520.
Fig.
1.
Parameteradaptive sytstem.
In this paper we consider the problem of designing a
parameteradaptive control system for a singleinput,
sin
gleoutput process admitting an essentially
unknown
but
fixed linear model,
so
that for any reference input
r(t),
the
tracking error
e(t)
between the output
y(t)
of the resulting
controlled system
(Fig.
1) and the output
y,(t)
of
a
pre
specified linear reference model
is
regulated
to
zero
asymptotically. We assume that only the process input
u(t)
and output
y(t)
can be measured (but
not
the process
model state) and we require the controller to be a dif
ferentiatorfree dynamical system realizable with conven
tional analog components.
Although
a
great many parameteradaptive controllers
have been proposed in the literature, only under very
restrictive assumptions have any actually been shown to
result
in
stable closedloop systems. For example, in
[3]
Parks puts forth the idea of Lyapunov redesign to achieve
stable adaptive operation, but the stability analysis given
there is incomplete and the overall approach is limited to
process transfer functions of relative degree one. In
[4],
Astrom and Wittenmark propose a parameteradaptive
system consisting of an online recursive parameter
identifier and
a
minimumvariance control law generator;
the asymptotic properties
of
such “selftuning regulators”
have recently been examined in
[5],
[6],
but global stability
of these systems has not yet been established. In
[7],
Monopoli suggests an alternative configuration (a simpli
fication
of
which is used in this paper) based on the
important observation that under certain conditions it is
not necessary to separately identify process model param
eters and control feedback gains; but the arguments
in
[7]
concerning stability contain errors and do not justify the
paper’s main claims
[8].
In addition to these references
there are numerous others dealing with parameteradap
tive control, but for one reason or another none apparen
00189286/78/08000557$00.75
01978 IEEE
558
IEEE
TRANSA~ONS
ON
AUTOMATIC
CONTROL,
VOL.
AC23,
NO.
4,
AUGUST
1978
tly give an adequate answer to the following fundamental I.
SYSTEM STRUCTURE
question:
Do
parameteradaptive controllers which yield
globally stable closedloop systems actually exist for sin
gleinput, singleoutput linear processes?
The purpose of this paper is to provide an affirmative
answer to this question by presenting what we believe is
the first parameteradaptive control configuration, appli
cable to a reasonably large class of linear process models,
which
is
known
to result in a globally stable closedloop
system in which
all
signals and gains are guaranteed to
remain bounded. The proposed controller requires no
differentiators and can be realized with integrators,
summers, gains, and multipliers. The only assumptions
made about the process are that it admits
a
transfer
function model with lefthalf plane zeros, and that:
1)
an
.
upper bound for the transfer function’s “dimension,”
2)
the “relative degree”
of
the transfer function, and
3)
the
sign of the transfer function’s “gain” are
known.
In Section I the general structure of the controller is
discussed
as
are the crucial process model assumptions
upon which it is based; an interpretation of this structure
in statespace terms reveals that the function of the con
troller is,
in
essence, to adaptively shift one subset of
process model poles to prescribed locations, while adap
tively cancelling process transfer function zeros with the
others. Detailed descriptions of the control parameter
adjustment law and the auxiliary control signal needed to
guarantee system stability are given in Section
11.
Finally
in Section I11 it is shown that application of the controller
to any process satisfying the assumptions
of
Section
I
results in a globally stable closedloop system which
follows a prespecified linear reference model with zero
steadystate output tracking error.
Notation
It
is
perhaps most natural to think of a parameteradap
tive controller
as
a system consisting of two distinct sub
systemsone which dynamically generates asymptotic
estimates
act)
and
2(t)
of process model parameters
p
and
state
x(t)
respectively, the other which generates an esti
mate
At)
of desired feedback gains
f
as a function of
estimated model parameters
d(t).
However, in spite of its
intuitive appeal, this particular configuration is dfficult to
analyze and at present is not well understood. There are
at least three reasons for this.
1)
Known results [9][ll] characterizing the behavior
of dynamic identifiers (e.g., adaptive observers) are not
directly applicable, since such results usually require all
identifier inputs to be bounded; in the present situation,
neither the process input
u
nor output
y
can be assumed
bounded
a priori,
since the identifier is
in
feedback with
the process.
2)
The relationship between process model parameters
p
and desired feedback gains
f
is typically a complicated
nonlinear functionf(p). Roughly speaking, this is because
the type of parameterized model which is amenable to
online identification and state estimation is a linear sys
tem which is observable for all values of its free parame
ters, whereas the type of parameterized model which
might conceivably yield a linear relationship between its
parameters and desired feedback gains is one which must
remain controllable for all values of its free parameters.
3)
f(p)
is usually not welldefined at those points
in
parameter space at which
the
parameterized model used
for identification is not controllable; thus,
if
the control
generator shown in Fig.
2
is a memoryless realization of
this function, then the possibility of an unbounded gain
vectorjcannot be ruled out
[lo].
By adopting a somewhat different concept of a parame
are
two
x
matrices
of
time
functions,
we
write
M=
ble
to
avoid the preceding problems. The key idea is to
(E)
if
each
element
of
the
matrix
M
is
a
linear
combi
generate a feedback control without utilizing
distinct
nation
of
decaying
exponentials.
If
a(s)
is
a
polyno~al, estimates of desired feedback gains
f
and process model
function
a(s)/p(s),
written
(a(s)/P(s))O,
is the integer adaptive system
be
described
ear system with inputf(t) and transfer function
a(s)/p(s)
the
process input
and
Output
y can
be
by
a
is
sometimes
written
as
(a/Plf(t).
A
dynamical
system
of
canonical (i.e., controllable and observable) linear system
the form
f
=
g(x,
r)
with piecewisecontinuous input
r(t)
is
smooth
if
g(
.)
is a continuous function of its arguments;
such a system is
global&
stuble
if for each initial time
At>= c,x,(t>
In the sequel, prime denotes transpose. If
M(t)
and
N(t)
teradaptive system than
that
in
Fig.
2,
it
is
POssi
(a(s))O
denotes its &gee. The
relative degree
of
a rational state
x(t)*
In
the
the
general structure
Of
such
an
(p(s))o

(a(s))o.
me zerostate output response of a fin Our basic assumption is that the relationship
between
ip(t)=ApXp(t)+ b,u(t)
to>
0
and state
x,,,
the state response
x(x,,t,t)
exists for
t
>to
and the Euclidean norm
IIx(xo, to, tll
is bounded by a
with strictly proper transfer function
finite constant not depending on
to.
In the special case of
a linear system,
stable
means asymptotically stable (i.e., all
T,(s)=g,pPo
system eigenvalues are in the open lefthalf plane.)
Simi
larly, a polynomial transfer matrix or square matrix is where
gp
is a constant gain and
$(s)
and
p,(s)
are
stable if its zeros, poles, or eigenvalues, respectively, lie in coprime monic polynomials. For reasons to be made clear
the open lefthalf plane.
in
the sequel, we make the following
4
(s)
r
1
6,.(s)
such that
J
I
I
I
y(t)=gp*(
b(s>
u(t)
u(f)syoy(~))
8
Y
(3)
(s)
Y
(4
(E)(4)
d
T
DY?;>MC
1
IDEX'IIII'IEII
where C?u(s)/y(s) and .li,(s)/y(s) are strictly proper and
I
I'
I
I
I
I
Ll
I
I
proper transfer functions, respectively.
To
understand the
I
I
role of this representation, which has its origin in
[7],
first
e(t)
=y(t)
Y,(f)
(5)
GEXETUTCI
+
CONTROLLER
observe that
(4)
implies that the
output
tracking
error
Fig.
2.
A
parameteradaptive system with separate identification and
control.
can be written as
Process Model Assumptions
1)
o$(s)
is a stable polynomial.
2)
The following data are known:
a) The sign of
g,;
b) An integer
n
>
(P,(s))O;
c) The relative degree
n*=(o$(s)/pp(s))O.
The principal function of the adaptive controller shown
in Fig.
1
is to force the process output
y
to approach and
track the output
y,
of a prespecified linear reference
model. The reference model is a stable, canonical linear
system
with strictly proper transfer function
T,(s)
and bounded,
piecewisecontinuous, reference input
r(
t).
To
determine
what else
is
required of
Z,
for zero tracking error to be
possible, let
us
note that if the adaptive controller shown
in Fig.
1
were replaced by any linear dynamical (i.e.,
differentiatorfree) compensator
Z
with reference input
r,
measured input
y
and output
u,
and
if
Tz
were the
resulting closedloop transfer function from
r
to
y,
then
the relative degree
of
T2
would not be less than
n*.
Indeed, this fundamental constraint on
Tz
can be relaxed
only by incorporating differentiators in
2.
Clearly, any
reference model not respecting
this
constraint must in
volve some form of differentiation. Since we have stipu
lated that our adaptive controller be differentiatorfree,
we must require that
(T,(s))O
>n*.
The process model assumptions imply that the process
can be represented in an especially useful way.
To
de
scribe this representation, let
a(s),
b(~)
and y(s) be any
three polynomials which have been selected with knowl
edge
of
n
and
n*
so
that
a)
a(s),p(s)
and y(s) are monic and stable.
b)
a(s)
and
p(s)
are coprime and
(a(s)//?(~))~
=
n*.
c)
a(s)
divides
y(s)
and
(y(s))"
>n

1.
i
(3)
where T(s)r
(b(s)/a(s))T,(s);
the assumptions on
T,(s)
(i.e., stability and
(T,(s))O
>n*),
together with the con
straints on
a(s)
and
p(s)
dictated by (3a) and (3b)
guarantee that
T(s)
is a stable, proper transfer function.
Next, observe that since S,(s)/y(s) and G,(s)/y(s) are
strictly proper and proper transfer functions, respectively,
it
is
possible to write
and
where the ki are constants,
ny
=
(~(s))",
and
{y,(s); ,y,(s)} is any preselected basis for the vector
space
of
polynomials
of
degree less than
nr
(e.g.,
yi(s)=
si').'
The implication of these expressions is that it is
now possible to rewrite
(6)
as
where
k
[
k,,
k,,
.


,
k,,,
+
,,
1
/g,]'
is a vector of unknown
constants;
t9(t)r[Ol(t),02(t),.
.
,02%+,(t)]'
is
a
vector
of
known sensitivity functions obtained by passing
u,
y,
and
r
through the stable, canonical, linear system
x,:
{
}
(8)
~e(t)=ABxe(t)+b,u(t)+b,y(t)+b,r(t)
B(t)
=
Cexe
(t)
+
dYy(t)
+
drr(t)
with transfer matrix To=block diag
{
T,,T,,,T},
where
The expression for
e(t)
in
(7)
shows that
If
k were
known,
then zerooutput tracking error could be achieved
Tu=[Yl/Y,
.
,Y,/Yl'
and
Ty
=[YI/Y,'.

,Y,,/YI
1Y.
It
be
shown
in
the
that
if
p('>
and Y(')
IIf
(?(s))">n
1,
hen
Gy(s)/y(s)
turns out to be strictly proper [cf.
are
SO
defined, then there must exist polynomials
6Js)
and
Proposition
11;
thus,
in
this case
k%+,
=o.
560
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TRANSACTIONS
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CO~OL,
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AC23,
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4,
AUGUST
1978
(asymptotically) by setting
u(t)
=
O'(t)k.
Since
k
is not to select polynomials
a,
P,
and
y
for which
(4)
is guaran
known, what we shall do instead is to set teed to hold, it is not only sufficient to know
n*
and
n,
but
necessary as well. Thus, even though process model
u(t)=B'(t)L(t)+a(t)
(9)
assumptions 2b) and 2c) may appear somewhat severe,
where
i(t)
is a suitable defined estimate of
k
and
g(t)
is
they
are unavoidable consequences of any approach
an
auxiliary whose sole function is
to
guarantee
based
on
system representation
(4).
Whether or not one
system stability. In Section
11
we explain how to select
can design an adaptive system under weaker hWotheses
i(t)
and
a(
5)
SO
that for any reference input
r(t),
all
(e*& bY only assuming bounds for
n*
are known, rather
system signals are bounded and
e(t)+O
as
tw.
than
n*
itself) is very much an open matter.
The structure of the control law
(9)
is motivated prim
Remark
1:
It is of Some interest to know whether Or
kly
by the expression for
e
in
(7)
which, in turn, is a
not a particular selection of
a,
P,
and
Y
proides a
consequence of (4).
T~
establish the validity
of
(4),
first
''minimal parameterization," i.e., a parameterization
in
note that since
y(s)
is a stable polynomial,
y(t)
and
u(t)
which
ap
and consequently
k
are uniquely determined
will satisfy (4) just in case
y(t)
is a solution to the
by
gp.
$9
and
Pp.
It can be shown that
if
&
and
Y
are
differential equation
chosen to satisfy (14), (15), and the hypotheses of Proposi
tion 1, then for any pair of coprime, monic polynomials
o$
fi(s)y(s)~(t)=gpa(s)((y(s)GU(s))u(t)8,(s)y(t))
(E).
and
Pp
satisfying (12) there exist unique polynomials
6,
and
4
satisfying (13) provided: 1)
a
and
5
are coprime,
(lo) and 2) either
y"=(P,)"
1
or
yo
=(/?')"
and (13b) is
Since process model assumption 1) asserts that
%(s)
is required to hold with inequality. While coprimeness of
a
stable, (1) will imply (10) if and only if and
L$
can be guaranteed by simply choosing
a
=
1, it is
clearly not possible to choose
y
to satisfy
2)
above, unless,

4s) 4S)(Y(S)

aU(s))
(11) of course, if
(j3')"
is
known
exactly.
PPW

Y(S)P(S)+8p(s)ay(s)
.
Justification of
(4)
thus amounts to showing that if
a(s),
P(s)
and
y(s)
are polynomials satisfying (3), then there Before concluding this section we briefly outline
an
must exist polynomials
8Js)
and
6Js)
such that (1 1) holds alternative approach leading to
(9),
based on statespace
where
du(s)/y(s)
and
G,(s)/y(s)
are strictly proper and considerations.
This
approach, which was what originally
proper transfer functions respectively. The following pro led
us
to
(9),
provides further insight by characterizing
(9)
position implies this and more. as an estimate of a desired statefeedback control law. For
simplicity, we outline the approach under the assumption
that the polynomials
a(s),
P(s),
and
y(s)
satisfying
(3)
(y(s))"
=
n;
we further assume that
T,(s)=g,/P(s),
where
g,
is a constant.
The approach is based on two easily proved facts: First,
acteristic polynomial of
A,
then there exist nvectors
hp
and
bp
such that
StateSpace Interpretation
Proposition
I:
Let
a,&
y
be fixed monic polynomials
fixed positive integers with
n
>
n*.
For each nonzero con
stant
gp
and each pair of monic and coprime polynomials
05
and
Pp
satisfving
with
a
and
P
''Prime
and
./P
Proper;
and
let
and
n*
be
have been selected
so
that
a(s)=
1,
(P(s))O
=
n*
and
($lPp)"
=
n*
(12a) if
(clxn,Anxn)
is any observable pair, with
y(s)
the char
and
(P,)"
Qn
(12b)
o$
(4
there exist polynomials
8,
and
8,
satisfving
c(slAh,c)'b,=

Bp(4
*
(16)
(44/Y)O>O
(13a)
Second, for at least
one
pair
(hp,bp)
satisfying (16), there
@,/Y)"
2
0
(13b) exists
a
row vector
f,
such
that
A
+
hpc
+
bpgJp
is
stable
and
(1 l),
if
and oniy
if
a
divides
y,
and
(14)
(a/P)"=n*,
c(sIAh,cbpgJp)'bp=.
P(s)
(17)
1
and
Equation (1
6)
implies that
(y)"
>n
1
(15)
ip(t)=(A
+h&,(t)+b,g,u(t)
If
a,
fi
and
y
have the required properties and
if
the
y(t)=cxp(t)
inequality in
(15)
is strict, then there exist
8,
and
6,
satisfv
ing
(11)
and
(13)
for which the inequality in (136)
is
strict.
is
a process model. sine
ip
cafl
also be written
as
A
proof of this proposition appears in the Appendix.
The proposition clearly implies that in order to be able
ip
(t)
=
(A
+
hpC+ bpg&)Xp (t) bpgp
(u(t)
f,Xp(t>)
FEUER
AND
MORSE:
ADAF’TNE
CONTROL
OF
LINEAR
561
and since
A
+
h,c
+
bpgJp is stable, we can use (17) to
obtain
By assumption,
y,(t)
=(g,/P(s))r(t), so the output track
ing error can now be written as
e(t>
=
BP(U(t)
f,x,(t)

(g,/gp)r(t))
(4
(19)
P(s)
This suggests that desired system behavior might be
achieved by setting u(t)=f(t)f(t)+g(t)r(t) wheref(t),I(t)
and
g(t)
are suitably defined estimates of
f,,
x,(t)
and
g,/g,, respectivelybut as mentioned at the beginning of
this section, this approach leads to difficult problems
which
so
far have not been overcome.
The alternative approach pursued here is to use an
estimate of the product
f,x,(t)
rather than the product of
estimates off, and
x,(t),
respectively. The structure of the
proposed estimate
is
motivated by the fact that the state
xp(
t)
of system (1
8)
can be written as
.,(t)=
Eu(t)bpgp
+
Ey(t)h,
+
eA‘q (20)
where
E,([)
and
E,(t)
are any pair of solutions to the
matrix differential equations
&(t)=AE,(t)+lu(t)
ky(t)=AEy(t)+Zy(t)
(21)
root set of
%(s)P(s).
The more general situation, when
a,
/I,
y,
and
T,(s)
are not constrained by these special
assumptions, admits a similar statespace interpretation,
provided system (18) is replaced with a more general
linear model, representing the process together with a
dynamic compensator.
11.
CONTROL
EQUATIONS
To
complete our description of the proposed adaptive
system, we first select
a($)
and
P(s)
so that
(3)
holds, and
in addition
so
that
P(s)=
P*(s)S(s),
where
P*(s)
is monic
and
a(s)/P*(s)
is a strictly proper, strictly positive real
transfer function. It is easy to check that these constraints
imply that
S(s)
must be a monic, stable polynomial of
degree
n*

1.
It turns out in what follows that nothing essential is lost
if
a(s)/P*(s)
is assumed to be the transfer function of a
onedimensional system, i.e.,
a(s)/P*(s)=
l/(s
+xo>
for
some
b>O.
For the sake
of
simplicity we henceforth
make this assumption; the more general situation
in
which
a(s)/P*(s)
is the positive real transfer function of a higher
dimensional system can be treated along similar lines
using the KalmanYakubovich lemma.
These assumptions together with
(5)
and the control law
defined by (9) imply that the output tracking error
e(t)
can
be
written
as
I
is the n
X
n identity and
q
is
a constant vector depending where
is
the
parameter
error
on the initial values of
x,,
E,,
and
E,
[lo].
Since
A
is a
stable matrix by hypotheiis,
it
followifrom (20) that
Z(t)=Jc(t)k.
f,xp(t)=f,E,(t)b,gp+f,E,(t)h,
(€1
(22) We wish to develop a parameter adjustment law for
k(t)
and an expression for
a(t).
Since these equations are
for the case
n*
=
1,
these cases will be treated separately.
Using
just
(21), (22),
and
the
Of
A,
it
is
not considerably more compficated
for
the
n*
>
1
than
difficult to verify that
f,x,(t)
can also be written
as
will yield an estimate off,x,(t)+(g,/g,)r(t) provided
&(t)
is an estimate of
ki
for iE{1,2,...,2n} and
k(2n+I)(t)
is
an
estimate of l/g,.
The preceding development shows, at least under the
special assumptions on
a,
P,
y,
and
T,(s),
that the term
eyt)L(t)
appearing in (9) is really an estimate off,x,(r)+
(g,/g,)r(t) where
f,x,(t)
is a state feedback law, whch
if
applied to process model (18), would have the effect of
shifting the model’s poles from the root set of
P,(s)
to the
Z,:i(t)=
(sign(g,))@(t)e(t), (26)
where
Q
is any prespecified, positive definite, constant
gain matrix, then for any reference input r(t), bounded for
t
>
0,
and for any initial time
to
>
0
and state
xz(to),
the
state response
xx(t)
=
(x,(
t),
x,(
t),
xs(t),
k(
t))
of the result
ing closedloop adaptive system
Z
described by
(I),
(2),
(5),
(8),
(9),
and (26) exists for
t
>to
and is bounded
uniformly
in
to.
To
understand why this is
so,
first observe
that we can now use (24)(26)
to
write