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.
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.
11. PROBLEM FORMULATION
For purposes of discussion let I represent the set of
nonnegat.ive real numbers, I = { tlf >_ 0, t E Rf , and let to
be an arbit,rary but fixed element of I. In addition, let,
Ck(T) be the class of functions t,hat is IC times differentiable
on T, where T = {tlt 2 to E I].
Consider a class of single-input single-out.put dynanlic
systems, each of which is comprised of an unkno~n plant
and a known stable reference model. The plant. is defined
by t,he following time-va.rying nonlinear nth order dif-
fererhal equation
f, rn
a&)z'Q(t) + f(f,z,?i,. . .,x-11 ) = bJ,(f)u!'"'(t) (1)
k=O k=O
where n and 'rn are known positive integers such that. tt >
m; w(t) and z(t) are respect,ively the scalar input and
scalar output of the plant,; a,(t) = 1, ai(t), i = 0,1,. . -,
(n - I), and bj(t), j = 0,1,. . . , nz, are unknown time-vary-
ing parameters; and f is an unknown time-varying non-
linea function of z, i, . . , x("-1). Furthermore, for this
plant the following assumptions apply.
Assum.ptitm 1: For a given w(t) E Cn(T), (1) is defined
for all t E T and enjoys the usual smoothness condit,ions so
tha.t, no quedons arise as to the existence and uniqueness
of a solution on T for a given set of initial conditions.
490 IEEE TRANSACTIONS OK AUTOMATIC CONTROL, OCTOBER 1989
PLANT INPUT PLANT PLANT OUTPUT
[ n th-ORDER , NONLINEAR
coefficient. b,(t), are known; 2) the only restrict.ive condi-
tions which are imposed on the plant are those implied by
TIME - VARYING Assumptions 1 and 2.
111. COhTROLLER SYNTHESIS
CONTROL
SIGNAL
DIFFEREN-
ERROR
I I
r(t)
REFERENCE MODEL
(n'lh-ORDER , LINEAR
y(t)
REFERENCE INPUT
DESIRED OUTPUT TIME - INVARIANT )
Fig. 1. Model reference control system.
Assumption 2: The time-varying parameter b,(t) is
bounded with known bounds; that. is, 0 m'
and n 2 n' 2 (n - .m).
Let A, be a n' X n' constant stable matrix, formed by
the coefficients of the model (2) and given by
...
- a,.-1 - 01
A block diagram of such a system, together with a con-
troller which is yet to be selected, is shown in Fig. 1. From
this figure it is seen that the plant input is
w(t) = u(t) + r(t) (4)
and the error between the plant output and t.he model out-
put is
z(t) = z(t) - y(t). (5)
The problem to be considered in this st,udy is that of syn-
thesizing a cont.rol signal u(t) E Cm(T) using only z(t),
z(t),. - .,~(~-~)(t) so as to ensure that z(t) follows y(t) as
closely as possible.
Before tackling this problem it is mort.hn;hile t.0 empha-
size t.he following points which have already been stated:
1) as far as the plant is concerned, only its general struc-
tures (1) and t.he two constants Bo and 01, the bounds of the
Let, a constant B and a t.ime function O(t) be defined as
follows
B = (Po + Pl)P (6)
e(t) = bm(t) -p, t E T. (7)
From Assumption 2, le(t)l ained, (1) is obt,ained. Lemma 1 is thus proved.
To summarize, it is seen that (l), which describes t,he
t,ime-vaxying nonlinear nt,h order unknown plant., can be
replaced by (9), a (n - p)th order linear equation with
known constant parameters. This is done by introducing
a term ,$,,,(t) which contains all of the time-varying and
nonlinear dynamic characteristics of the plant. It, is termed
the characteristic variable of t,he plant. and is associated
with p and a.
In view of Lemma. 1, the subst,itution of (n - p) = .n',
the order of the model, and (Y = uo = (09, .lo . . . the
model paramet.ers into (9) yields
Using (14) and (2), the system error signal in (5) is found to
sa.tisfy
R'
akoz"i'(t) = pu(m-n+n') 0) + t(n-d),2(Q
k=O
- baOr'"(t). (15)
m'
E=O
Obviously, Lemma 1 is also applica.ble to (15). In par-
t.icular, with p = m - n + n' = s and a = (a: a,+1° . . .
a,.,) (15) is found to be equivalent to
n-m
a.,,o Z'*'(t) = pu(t) + E(t)
k=O
where the subscripts of ((t), the system characteristic
varia.ble, are dropped. In this equation, note that ((t) E
Cm( T) .
Let A,* be the (n - m) X (n - m) principal minor that
is located in the lower right-hand corner of A,; that is,
1 0
0
...
1 "1 0
Equat,ion (16) can now be equivalently written in the vec-
tor differential equation form.
i(t) = Ao*z(t) + (0 0 * . @(t) + ,$(t))' (18)
where z = (zlzz- -z,,)~, zl = z and xi+' = if, for i = 1,
As fa.r as Bo* is concerned, the follosving points should
be noted. For (n - wz) 5 3, it can be shown using the
Hurwitz crit,erion [9] tha.t a stable A, gives a stable Ao*.
If (n - m) > 3, some analysis may ha.ve to be carried out
in order t.0 est,ablish such an 40. However, it is always pos-
sible to select n' = (71 - ,m), from which it follows that
A,* = A,. Thus, in the discussions to follosv it, is assumed
that, a stable A, implies a st.able A,*.
Lemma 2
Let the dimension of z in (18) be 9, A,* be a stable
matrix, /3 be a positive number, and ,$(t) be continuous on
T. If I,$(t)l 5 31 (Bu(t) + t(0) (23)
where
0
r(z> = c PkOZk (24)
k=l
and pi? is a.n element of mat.rix P. The substitution of (19)
into (23) yields
$'(t,z) = -z'Qz + 2(l - t.an-'
.(Kd(t)) tan-' (&y(z)))[(t)y(z). (25)
It is thus seen tha.t v(. , . ) is a continuous mapping from
T X R, into R.
Kow let the behavior. of f7 on T X R, be examined. For
this purpose let Q = I, a g X g identity matrix. Then
zTQz = //z;i2. Also let S = T X R,. Define Bo, S-, and S+
492 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, OCTOBER 1969
- ARCTAN
Fig. 2. Cont.roller.
It follom from the above discussion that, IjzII is ulti-
mately bounded by E as is given in (20).
Comments: In order for Lemma 2 to be of any practical
significance two points remain to be clarified: 1) the
boundedness of the system characterist,ic variable t(t)
in (16), and 2) the arbitrary smallness of E in (20). These
t.wo problems are closely related and an ana.lytica1 treat-
ment, of them is highly complicated and not very obvious.
However, t,he computer simulat.ion studies for various sys-
tems have shown the boundedness of E(t). Also, it has been
observed that a selection of a large Kl and K2 does not af-
fect AI7 the bounds of E(t). Hence, E can be made ar-
bitrarily small by choosing K1 and K? large enough. If the
dependence of Ji = M(K1,K2) on Kl and K2 is not obvious,
then by making K2 dependent on system variables [5] as
K? = K21 + K& where K21 and K,? are positive numbers,
it can be shown with slight modification in t,he proof of
Lemma 2 that E can be made arbitrarily small by choosing
K1, Kg', and K?? large enough. In the sirnulat.ion studies,
some of which are shown in Section V, it has been ob-
served tha.t t,he controller with Kn = 0 is sat,isfactory for
making E suf€icient.ly small in all cases t,hat f is bounded.
Lemmas 1 and 2, and the preceding comments suggest a
procedure, as given in the following sect.ion, for synthesiz-
ing an adaptive controller in relation to the formulation of
the problem st,ated in Section 11.
IV. UKIFIED PROCEDURE FOR CONTROLLER SYNTHESIS
Consider a model reference syst.enl comprised of a plant
and a model characterized respect,ively by (1) and (2).
Lemma. 1 and Lemma 2 with g = n - m suggest a control
signal u(t) of the form. of (19) rewritten below to achieve
the control objective stat,ed previously
u(t) = -(1/8)t(t) tm-' (KlE(t)) tan-' (Kzr(z)) (33)
where 6 = (80 + P1)/2, K1 and Kg a.re positive numbers,
and ~(z) is defined as
n-m
r(z) = Pkh-mJZk (34)
k=O
where pij is an element of P, the solution matrix of (21)
with dimension (n - nz). Recalling (16), the characteristic
X'IKIFORUK et al.: CONTROL OF UWKNOWN PLANTS IS REDUCED STATE SPACE 493
variable ((t) ca.n be synthesized as
[(t) = U~+~:Z(') - pu(t), s = n' - (n - m). (35)
Equat,ions (33)-(35) then generate the required control
signal if ((t) is bounded, for it requires the measurement of
only (z,i,- . -,z:"-~)), u(t) E Cm(T) since l(t) E C"(T) by
Lemma 1, and from Lemma 2 and its comments llzll = 3,
a&) = 1,
boM = 3K(t),
bl(t) =
bp(t) = K(t) 3 + 0.4t,
LJ = ~/2, z@)(O) = 0, for k = 0, 1, 2, 3.
For the model = al0 = boo = 2, = 1, and Ao* = A,.
The input r(t) is a.gain a step function. For the synthesis
j30 = 3, 81 = 7.4 (up to t = ll), j3 = 5.2;
Q= [: ooJ
p = r12 016],
and K1 = Kz = 20. The simulation results are shown in
Fig. 4(a) and 4(b). K0t.e that u(t) and [(t) in Fig. 4(b) ap-
proach zero as t becomes large.
In Fig. 5, M = lt(t)lmm is plotted versus K1 = Kz up to
unrealistically large va.lues of Kl and Kz for each of the
above examples. It is seen that 114 approaches a constant
496 IEEE TRANSACTIONS ON AUTOXITIC CONTROL, OCTOBER 1969
VI. CONCLUSIONS Peter N. Nikiforuk was born in
A synthesis technique has been described in this pa.per
St. Paul, Alberta, Canada. He re-
ceived the BSc. degree in engineer-
for the design of an adapt.ive controller. This technique is ing physics from Queen’s Uni-
based upon the exist,ence and boundedness of the char-
velsity, Kingston, Ont., Canada,
acteristic variable and Lyapunov type synthesis. It has
in 1952 and the Ph.11. degree in
electrical engineering in control
been shown that, this technique may be applied t,o a wide systems from Manchester Univer-
variety of plant,s in a class of model reference control
sit.y, Manchester, England in 1955.
systems. In the generation of the control signal no kno~vl-
Ele has been a Defence Gcient,ific
Service Officer, for the Defence
edge of the pla.nt parameters and it.s nonlinearities is re- Research Board, Quebec, Que.,
quired. Only t.he form. of the plant equation and the
Canada, and a Systenw Engineer,
bounds of b,(t), t.he coefficient. of the .?nth order plant. in-
for Canadair Ltd., ?\Iontreal, Qne.,
Canada. In 1961 he became an ...
put derivat,ive? are needed. Also, this t.echnique requires
the measurements of only t,he output and it8 first (n - m)
derivatives. Simulation results have been given which
show the boundedness of charact,erist,ic variable ((t).
It. appears t.hat, the proposed technique may be appli-
cable to a wider class of control problems than has pre-
viously been possible. A proof of the boundedness of the
cha.ract.eristic variable is required. This has yet to be done,
but in the meanwhile reliance has been placed upon the
results of simulat,ion studies. The effects of measurement
noise and an imperfect controller on adapt.at,ion have not,
been discussed. Further work in this direction is in prog-
ress.