Control of Unknown Plants in Reduced State Space
AZI.S~TQC~--A method is proposed in this paper for the synthesis of
an adaptive controller for a class of model reference systems in
which the plant is not known exactly, but which is of the following
type: single variable, time varying, either linear or nonlinear, of
nth order, and capable of mth order input differentiation. The model
is linear, stable, and of n'th order, where (n - nz) 5 n' I n. The only
knowledge of the plant that is required in this synthesis procedure is
the form. of the plant equation and the bounds of b,(t), the coefficient
of the mth order plant input derivative. The synthesis procedure
makes use of an unique function, called the characteristic variable,
and Lyapunov type synthesis. The introduction of the characteristic
variable reduces the synthesis problem to one that involves a known,
linear time-invariant lower order plant. The control signal is gen-
erated by measuring the plant and model outputs, and their first
(n - m) derivative signals. This ensures that the norm of the (n -
nz)-dimensional error vector is ultimately bounded by E, an arbitrarily
small positive number provided t(t), the characteristic variable, is
bounded. Two nontrivial simulation examples are included. R
I. INTRODUCTION
ECEKTLY, a number of papers [2]-[4] have dis-
cussed the cont.rol of plants !in a class of model
reference systems using a Lppunov type synthesis. These
t,echniques use a reference model t,hat is of the same order
as the plant,, or one that is of lower order [3]. In order t,hat,
a control signal ma.y be generated using these techniques,
all of the statme variables must be measured and the form. of
the plant eqwtions, the bounds within which the plant
parameters may vary, and the form. of t.he plant non-
linexities must be known. These techniques suffer from
t.he following disadvantages: 1) for an nt.h order plant,
deriva,tive signals up t.o the (n - 1)th order must be mea-
sured which may not be practically possible for higher
order plants; 2) t,he bounds of the plant paramet.ers and
the form. of the plant nonlinea,rities may not be known; and
3) it is usually more practical t.0 use a reference model that,
is of lower order t,han the plant. At.t.empt8 at bypassing
some of these disadvanhges have already been report,ed
[5]-[7]. In these studies methods of synthesizing a con-
trol signal from the plant output. a.nd its lower order de-
rivatives have been considered for a rest,rictive class of
linear plants. From a. practical viewpoint, it is desirable to
cont.inue t,he development, of such synthesis t,echniques,
but to make them applica.ble to a more general class of
plant,s. One such technique has been developed by the
authors and is described in this paper.
In this technique a uniquely defined fundon called the
characterist,ic variable is introduced. It is related im-
sented at the 1969 Joint 4utomatic Control Conference,
Boulder,
Manuscript.
received November 18, 1968.
This paper was pre-
Colo. This work was supported by the National Research Council of
Canada under Grant A-5625 and t,he Defence Research Board of
Canada under Grant 4003-02.
The auuthors are with the Division of Control Engineering, Uni-
versity of Saskatchewan, Saskatoon, Sask., Canada.
plicitly to all of t,he plant parameters and nonlinearities
through the available state variables. By inhroducing t.his
characteristic variable an unknown, nonlinear! and time-
varying pht, of high order is replaced in the procedure by
a known, linear, and time-invariant pla,nt, tha,t. is of lower
order. A Lyapunov type synthesis technique is used to ob-
tain a control signal which is espressed in t,erms of the
cha.ract,eristic variable. It, is t,hen shown that for an nth
order time-varying plant, which may be eit,her linear or
nonlinear, a. control signal can be synthesized Kith the aid
of a suitably defined reference model which may be of
lower order than the plant. This t,ec,hnique uses t.he plant
and model out,puts and t.heir first (n - m) derivative signals,
where m is t.he order of the phnt input different,iation.
The only knowledge of t,he pht that is required is the
form. of t,he plant, equation and t.he bounds of b,(t),
t.he coefficient of the,mth-order plant input derivative, if the
boundedness of t,he characteristic variable is known a
priori. Otherwise, the boundedness of this variable must
be established by simulat.ion.
Throughout this paper the term unknown plant. is used
according to t.his definition: this is a plant whose param-
eters and nonlinearities are not, known exa.ct.lq, but. about,
which sufficient informat,ion is ava.ilable to permit some
meaningful simulations to be made.