426
1969
lEEE TRANSACTIONS O S AUTOU1TIC COSTROL, AUGVST
type, an application of the resultant. r i m e
optimal inputstothe
syst.em cawesthe
magnitude of the output, to be less than or
equal t.0 ‘ Y d l in the interval ( f ~ , f f * ) , where
to is t,he init.ia1time.
.Slt,hough many techniques can be used to
solve the preceding timeoptimal problem,
Kranc
and
Sarachik’s
approach
[ l ] is
probablybest because it. does not. require
the solut.ion of a two-point boundav
value problem. I n fact, a particularly
simple and rapid determinat,ion of t f * and
the opt.imal control is possible when their
t,echnique is used. To briefly summarize
their results (for single output plants),
consider an
r-input
singleoutput.
nthorder linear t.imevarying plant. characterized by its impulse response matrix h(t,r)
(an rth-order row vector). If t,he system
input vector is constrained as follows:
Taking magnit.udes of both sides of this
equation and assuming that the elements
of h ( t , ~ )are all nondecreasing functions
of T, we find
u
i
=
1,2, . . ‘ , r
(2)
where L isa
given posit.ive number, hi
is t,he ith element of h,
1
1
- + - = I
P
Q
and tf* is the smallest value of t~ satisfying
the equation
=
is absolutely continuous so that. its derivat.ive exists almost everywhere. The transfer
function of the system G(s) = c’ (SI - A - l
b. The system being assumed to be asympt,ot.icallystable for all constant gains k ( t ) =
K E (0, m ), it.follows that. - i~ < arg
G ( i w ) < T for all real w (Nyquist criterion).
In ot.her words, there exists amnltiplier
X ( s ) , not. necessarily causal, such
t.hat,
for all real o,
Iylct)i
Ibdl.
(6)
Even if the elements of h ( l , r ) are not.
nondecreasing funct.ions of T , (6) isstill
valid. This is easily seen by noting that, the
requirement t.hat, (4) be sat,isfied for the
smallest value of f f is ident.ica1 to stating
that, the process terminates t,he first time
y ( t ) = yd. This completes the proof.
We haveshow1that
when thet.irne
optimal inputs of a linear system satisfy a
constraint of the L , type, the plant output
cannot overshoot the final specified value.
This result has been wed to obtain a lower
bound on t,* for problems which involve
inequality constraints on the out,put. [2]
Re X ( i w ) G ( i w ) 2 0
Re .lf(iw)> 0.
PROBLEV
Find sufficient conditions for asymptotic
stability of thetimevarying
system (1)
in mult.iplier form.
I n preparation for stating theorem
a
which employs a nonrausal function in its
mult.iplier, define A t ) to be a real function
of time with a two-sided Fourier transform
Z ( L ) such that
Z(s) =
R. D. KL-XFTER where
Dept,. of Elec. Engrg.
Drexel Inst. of Technology
Philadelphia, Pa.
~y
Ly
+ 0s -k I’l(S) + Yds)
and 6 are real numbels and
y,(t) = y’(1)
REFERENCES
!1)
It is observed that t.he solution of (4)
only requires that a real root of atranscendental
equation
be found, routine
a
calcdat,ion. Weare now ina position to
prove t.he following theorem.
111 G . RI. Iiranc and P. E. Sarachik. “An application of functional
analysis
to
the
optimal
Trans. A S M E . J . Basic
control
problem,”
Enwrg.. 1-01.8 5 , pp. 143-150, June 1963.
121. R. D. Klafter,
“Time-optlmal
control
of
.
systemsvith
controls of partially
specified
City
College
structure.” Ph.D. dissertation,
of New Uork. Kew P o r k . N a y 1968.
O n the Stability of a Linear
Time-Varying System
Abstr~ct-A single-loop linear system
with a timeinvariant stable block G in the
direct path and a time-varging gain in the
feedbackpath is analyzed for asymptotic
stability in the Popov framework by way of
admittingnoncausal“multipliers”
in the
stability criterion. It is shown that an autocorrelation bound, analogous to O’Shea’s
cross-correlation bounds [l], results in a
constraint on d k / d t morerestrictive than
that of Gruber and Willems [ Z ] .
€i’8(t
1
+ Vi‘)
I’
UP([) = y ” ( t )
+ ,=1
Ejd18(t
-
Yj”)
with y’(f) and y ” ( f ) piecewise continuous
functions; y’(t) = 0 for f > 0 and y ” ( t ) = 0,
t < 0; ly’(t)l 5 lemt and I y ” ( t ) l 5 lepmt
where I, ‘m. are positive nnmbers;and 7;’
(>O), y j ” ( > O ) , ~ i ’ ) and e,” are real numbels
for all 2: and j . Let y ( t ) = y,(t)
y?(t).
THEOREM
System (1) is asymptot,ically stable if,
for some 01, 6 > 0 and Z ( s ) given above,
t.he following conditions occur.
a ) R eZ ( L ) G ( L ) 2 0 for all real o
b)
+
Manuscript received December 30, 1968; revised
March 4. 1969.
6
j=l Eju
< -e,
e >o
e) d k l d i is monot.onically nonincreasing
and/or 5 2(a - O)k(t)!6.
Proof is based on the following lemmas
which are easy to establish.
Lemma 1
For all u1 and
constrifned. if p = ~2 the total-quadratiicontent
(i.e., ener&) of u is ’limited;and if p = m t.he
magnitude of all of the elements of u is hounded.
+ i=
+
Theorem
If thetimeoptimalinputsin
(2) are
applied to a linearsystem whose impulse
response is h(t,r), t.hen
Proof: If theinputstotheplantare
given by (2), the output, at any time
can
be found by wing thesuperposition integral.
Thus
c‘x
where A is an n X n st.able matrix; b, E, and
x (state) are n X 1 vectors: and u, t.he output. of t,he system, is a scalar. k ( t ) E (0, )
Therefore,
t.hen the inputswhich satisfy this constraint.’
and drive theoutput y ( t ) from 0 to y d
in the least t,ime are
FORNTLITIOX
ASD SOLCTION
OF THE PROBLEM
The linear feedback system of Fig. 1 is
governed by
(u12
uLand
f unu,)kif) 2
k(t) 2 0,
(U’2
- u2z)k(t);2.
427
CORRESPOXDEXCE
I
I
RELAY
Fig. 1. Lineartime-varyingfeedbacksystem.
Lentmu. 2
If k ( t - 7 ) - k i t ) 2 0 for all
for all T and ang u ( t ) in I>>(- m , rn ),
T,
then
j:=
k(t)u(t)uft - T j dt
Proof of Theorem: Set.
(+(.,lj,
$T(G,
t) =
0
5t 5T
(C)
i
Fig. 1.
t>T
10,
where $ ( u , f ) = k ( t ) u ( t ) and the subscript.
T denotes hruncat.ion (t,runcation guarantees
the exist.enceof Fourier transform). Define
Y ( T ) U T ( t - 7)d T
(2)
+
and consider
p(2’) =
LT
XT!t)$TIU,
t ) dt.
(3)
Steps leading to the crucial ineqnalit,y (4)
are as folloas:
1) obtain a. bound on p ( T ) using conditions b ) and c ) of the Theorem as well as
Lemma 2,
2) replace U T ( f ) by w p ! t )
UT&),
where u ~ ~ (ist t.he
) mponse of the systemt.0
the feedback signal - $ (u, t ) and o T C ( f )
is that due to initial
conditions,
3) invoke
Parseval’s
theorem
for
the
expression derived in 2) and use condition
a) of the theorem,
4)combine the results of 1) and 3).
Finall;, one obtains
+
E,
lT
k ( t ) u T 2 ( f )dt
This correspondence deals wit.h t,he
definition and development. of the dual
input discrete describing function (DIDDF). A high-frequency signal is injected into t,he nonlinearity to quench the
limit cycles occurring in closed-loop autony . 1:. VENKATESH omom relay-type
sampled-data
sq-st,ems
Dept. of Elec. Engrg.
ill.
Indian Inst.. of Science
This method can be considered t.0 be the
Bangalore 12, India
counterpart of the conventional describing
function developed and applied with success
REFERENCES
by Boper [2]. Kuo [3] analyzes the problem
of forced oscillations andthe
stabilizing
R. P. O’Bhea.,.“Xn improvedfrequencytime
domain
stablllty
criterion
ior
autonomous
effect of an external signal whose frequency
continuoussystems.’’ I E E E Trans. B z l t o n ~ ~ t i c
is assumed t.0 bean integral mult,iple of
Conirol, vol. AC-12, pp.725-731,~December
1967.
the samplingfrequency. Furthermore,the
M. GruberandJ.
L. Willems, “On agenexternal signal is t.0 be applied at the input
Proc. 4th
eralization of t h e circlecriterion,”
I.
~..V
.
.. .. A l l p r t o n C o n f . on C i r m i t and Suctent
side of the cont>rol system. Inthiscorre
Theory:-pp. 827;835 1966.
spondence, however, suchrestrictions
are
8 . A i . Popor.Absolutestability
of nonlinear
control,”
Automation
systems of automatic
removed to render the analysis more realand Remote Control (Englishtransl.), vol. 22,
iat.ic.
pp. 857-876, March 1962.
DERIVATIOX
OF DIDDF
Let
A sin w& error signal
B sin W d t dither signal
Wd
Dual Input Discrete Describing
Function
CONCLESIOX
The requirement that d k l d i be uonposit.ive for the asympt,otic stability of (1) is
so trivial as to make the direct. adoption of
O’Shea’s met.hod of correlation bounds
unattrache.
The
essential Popov
approach, however, retains its power if the
>> ‘lCc
B(2)
+,8k(T)u~~(T)/2
where e, X o , and LIT, are positive numbers
independent of T . Boundedness of u ( T )
anditsasymptotic
stabilit.y follow from
(4)as, of course, with minor modifications,
in Popov’s proof [3] for an autonomous
nonlinear syst.em.
INTRODUCTION
noncaudpart of t.his integral of (2) is
converted to a slightly different. form. By
this means, t,he aut.hor has derived asymptotic stabi1it.y conditions more general
t.han those of Gruber andRillems
[2].
These will be reported elserrhere.
Abstract-The
delinition anddevelop
ment of a modified discrete describing
function, referred to a s dual input discrete
describing function, is considered. Highfrequency signal is injected into the nonlinearity to quench the limit cycles in
closed-loop autonomous relay-type sampleddata systems. An analytic procedure is
described, which aims at deriving the dual
inputdiscrete
describing function for a
relay in the 2 domain.
Manuscript received January 29, 1969.
TC
T
Q
describing function
period of oscillat,ionof system
sampling period
phase lead of error signal
and
z
=
. p C T .
Relay: C = 1; e > 0; C = -1; e < 0;
C = 0 ; e = 0. G ( Z ) = GM G @ ) , and for
oscillations -I/~V(.Z) = G(Zj.
Consider the system shown in Fig. l(a).
The nonlinearity with high-frequency dither
can be replaced bythe
altered charact,eristic of t,he relay. Since the zero-order
hold precedes the nonlinearity, the input t.0
t.he nonlinearity
stepped.
is
The
output of the nonlinearityis
also stepped.
The int.roduct.ionof a synchronized sampler
and a zero-order hold on the output side,
as shown, leaves the situat.ion mahered.