485
CORRESPONDENCE
1965
Forced Response of Linear TimeVarying Systems
c
HAVING FREQUENCY
RESPONSE Gliwl
Fig. 1 .
I
The utility of error coefficients in studying thebehavior
of constantparameter
linear systems has been well established [l].
This concept can be extended easily to timevaryinglinearsystemsindeterminingthe
H(5, t ) ,
response toanyinput,knowing
where H(s, t j isZadeh’ssystemfunction
[21-[11.
of a
T h e generalinput-outputrelation
time-varying linear system can be written
as
Control system with one zero memory nonlinearity.
U p , t)ez(t)
K(p, t)edt)
=
(1)
where p = d / d t , e l ( t )is theinput ande2(t),the
output. In other words,
L(p, t)W(t,T ) = k(P,
(2)
f)8(T)
where W ( t , T ) is the impulse response and
S ( T ) thedeltafunction.Themethodsuggested here consists in considering the output of the system as a superposition integral,
given by (for signals applied to t=O)
et(t) =
Fig. 2.
Approximate and exact region of stable initial conditions for example problem.
A
T)el(t
- 7)dT
(3)
and in expanding e l ( t - T ) in terms of Taylor’s series so as to get (assuming that the
first n-derivatives of el(f) exist)
ez(tj =
IiTith theassumptionthat
X ( e ) is reThere is one crossing of 1/G+) and --\‘(e)
(which occurs onthenegative
real axis)
placed by a linear element as in the precedwith2
ing paragraph, the response e ( t ) can be calculated from knowledge of the initial conditions of the linear part;e ( t ) nil1 contain a n
l/GCjw) = - ;V(e) = 0.1
oscillatorycomponentcaused
bythetwo
e = 12.7
poles on the j axis. The steady amplitude
of oscillation can be found from the evaluw = 0.1.
(3)
ation of the residues of thetwoneutrally
stable poles.
Now set
The peak of the steady-state response of
a = O.le = - 0.lc
(4)
e is now set equal to the limit cycle ampli-,V(e) locus a tt h e
tudereadalongthe
andsubstitutein
( 1 ) takingtheLaplace
-iV(e), 1 / G ( j w ) crosing.
equation retransform
sults which contains all of the initial conditions of thelinear-partdifferentialequaO.O1s2c(O)
(O.ls2
O.Ols)t(O)
tions, and which expresses a surface in the
e(s) = -(5)
initial condition space. Every point
on the
(s2
0.01) (S 0.1)
surface of initial conditions produces steadystate oscillation in thesystem(subjectto
where C(0) hasbeenassumedzerointhe
the approximation inherent in the describing interest of simplicity. Byevaluatingthe
function method and the degree with which
residue at s = O . l j , the peak steady state e ( t )
the variable e is represented by the linearis found to be
J-a~(t,
+J
~ ( t7 ), [ e l ( t ) - Te:(t)
-r
Since \ye are interested only in the forced
response of the system, letting t--t CO,
+=
et(t) =
J-, ~ ( t , [ e l ( t ) - Te:(t)
7)
3
Let us now define
h
+
+
+
+
c,, = (-l)=
(
+m
F ( t , T)T%!T.
(6a)
J4
Substituting from (6a), (5) may be written
as
e2(t) = coel(t)
+ cle<(b) +- - .
(6b)
which gives the forced response to any input
e l ( t ) . Also, by Zadeh’s definition, the system
function H(s, t ) is given by [2]-[4]
ized system). A new surface of initial conditions is obtained for eachcrossing of the
l / G ( j w ) and - X ( e ) loci.
T h e following exampleillustratesthe
method. Let the nonlinear element be saturation (with saturation occurringat one unit
of e ) , and let the linear part beexpressed by
e
...
=c
+ 0.16 + 0.01~
(1)
c=-a.
Then
-e
s2
G(s) = - (s) =
a
+ 0.1s + 0.01 .
s3
(2)
By setting (6) equal to 12.7 [from ( 3 ) ] ,the
approximationtotheboundarycurve
of
stable initial conditions is obtained.
4 plot of
the 6(0) vs. c(0) stability boundary approxi2. Thestability
mationisshowninFig.
boundary as determined from analog simulation of (1) is shown for comparison.
H. N. SCOFIELD
-4strionics Lab.
Marshall Space Night Center
or in general
Huntsville, Ala.
= c1
Manuscript received June 1 , 1965.
TR.4KSACTIONS
IEEE
486
+-
an
Lt - H ( S ,
H
asn
(-1),J
t)
r”i$’(t,r)dr
ON AUTOMLIATIC CONTROL
1
2t
-m
- c,.
(10)
Thus, given H(s, t ) we can find the response t o a n y forcing function e l ( t ) using
(6b) and(10). The coefficients CO, cl, . . . are
all functions of time,andbycomparison
with their counterparts in the case
of constant parameter linear systems, these may
be called time-varyingerror
coefficients.
However, the use of these coefficients may
not be very pronounced in the synthesis of
linear time-varying systems.
are all independent of
Since CO, t l , . *
el(t), the forced response of any input may
be quickly determined using (6b).
3
e&) = - - -
e&
412
= coj(t)
1) Let us consider a system,described
by the differential equation,
+ 2et P
12.5‘
,
The authors gratefully acknowledge the
kind help of Dr. hl. D. Srinath and 11.A .
L. Thathachar.
I t isclear
that the function c ( t ) is
viewed as a sum of ramp functions. The approximation of a typical nonlinear character1. I n Fig. 1, c ( t ) is
isticisshowninFig.
approximated by
c [ t ) = K j t ) - (t - 0.2j.L.jt - 0.2).
(2)
R. BALASUBRAMANIAS~
B. L. DEEKSHATULU
One can convert such functions to stan-
Dept. of Elec. Engrg.
Indian Institute of Science
Bangalore, India
REFEREKCES
[I] B. C. Kuo. Anlotnalic Control Syslems. Englervood
Cliffs,
S. J.: Prentice-Hall. pp. 151-153, 1962.
121 L.A. Cadeh. ‘Frequency analysis of variable networks, Proc. I R E , vol. 35. pp. 291-299. March
low
131
*Circuit,analysis of linear varying parameter networks. J . A p p l . Phys.. voL 21. pp. 117111 77. Kovember 1950.
[41 W. Kaplan. O p e r a i w m l .tierhods for Linear S33ferns. Reading. Mass.: Addison-\%’esley, 1962, pp.
515-525.
dard polynomial form by having availablea
table of coefficientsfor approximating the
function (t-tr;)G(t-ttl;). Such a table is used
to find the contribution of each ramp function to the Laguerre coefficients.
If A o , A I , A 2 , . . . , A,, represent the
Laguerre spectra of c ( t ) , one may write
+ 12e2‘+ 2
Hence
u
= Lt
H
+
+2 4
27 + 32e‘
t) = 36(1 + 2 4
H(s, I) =
st0
a
-His,
as
3
6(1
6
6
~
3 +4et
e&) = ___
6(1 2e9
+
Ak
0.4
1.00000
0,00000
it can be easily checked t h a t
Hence
co =
L1 H(s, t)
st0
=
=f ( t )
21
0.2
0.81873
-0.98246
0.01637
-0.01528
0.01439
-0.17iO1
0,67032
-0.93845
0.05363
-0.04648
0.04004
2.77646
GENERAL
Insystemsengineeringitissometimes
advantageous
to
represent
functional
a
relationship in terms of polynomials. However, one may encounter a functional relationship c ( t ) defined for 0 < t < a which cannot be conveniently represented by a polynomial. In such cases, it is possible to approximatethefunctionsbythe
piecewise
linear representation 1‘1
1
-
The approximation of a nonlinear
characteristic curve.
0.6
0.8
0.54881
-0.87810
0.09879
-0.07910
0.09393
0.30335
0,44933
-0,80879
0.14379
0.01549
0.14666
-0.09790
1
.o
I.
-1.00000
0.00000
0.00000
0.00000
The results may be checked by direct evaluation. The response to any input f(t) may
be determined in a similar manner.
2) For the system given by
Fig. 1.
TABLE I
+
+
+ (2P + 6t + 2)]e?(t)
& ( t ) is thekth
normalized Laguerre
pol>-nomial and can be expressed as [3]
This correspondence describes a method
of evaluating the Laguerre coefficients of a
continuous (or piecewise continuous) function which can be approximated by a piecewise linear function. The function is viewed
as a sum of ramp functions. Table I can be
used to find the contribution of each ramp
function to the Laguerre coefficients.
27 32e‘
3 4et
e&) = ___ 6(1 2et) 36(1 2 4
+ +
(3)
Tables of Laguerre Coefficients for
Representation of Piecewise Linear
Functions
and e 2 ( t )n-henf(t)=t would be
[t2p2 (3t2 4t)p
c AtLdt).
L O
tI
Thus e2(t)when f(t) is a unit step would be
+
+
C(I) G
1 R. Balasubramanianis
now withtheDept.
of
Elec. Engrg.. University of New Brunswick. Fredericton. N.B.. Canada.
First we shall find the coefficients CO, cl, . . .
of this system and then find the forced response when f(t) is a unit step input, or a
ramp input.
It can be readily checked that H(s,
the t )
of this system is
61
(t-tk).
*
-.._-_.
Examples
Go =
*
.\CKSOWLEDGYEST
.
1
where L’(t-tk) is the unit step function and
Kr; is a coefficient of therampfunction
and for any otherf(t)
+ clj’(t) + e?f”(t) + -
OCTOBER
0.36388
-0.53576
0.18394
0.12263
0.07664
4.37469
The characterizing coefficients
Laguerre spectrum of c ( t ) are [ 3 ]
A k
= Joic(t)L,G(t)e-tdt
ilk
of the
(5)
where e--L is the weighting factor. c ( t ) is a
continuous (or piece-wise continuous) function such as (3):
An alternative expression for (4)is [2]
“
c(t) E
Kr;(t- Ir;) rr(t - t k )
(1)
L O
Thus whenf(t) = t
hlanuscriiot received Aueust 20. 196-4.:
revised
March 22. 1965 and July 22. 1965. This work is a part
of the research work under the Grant-in-Aid :>rugram
supported by the Department of Electrical Engineering, Ohio State Unlx-erslty. Columbus, Ohio.
The first fen- Laguerre polynomials are
given as follows