I nternat. J. Math. & Math. Sci.
Vol. 3 No. 3 (1980) 535-547
535
STEADY STATE RESPONSE OF A NONLINEAR SYSTEM
SUDHANGSHU B. KARMAKAR
Western Electric Company
Whippany, New Jersey 07981 U.S.A.
(Received November i, 1979)
ABSTRACT.
This paper presents a method of the determination of the steady state
response for a class of nonlinear systems.
The response of a nonlinear system
to a given input is first obtained in the form of a series solution in the multidimensional frequency domain.
series solution will converge.
Conditions are then determined for which this
The conversion from multidimensions to a single
dimension is then made by the method of association of variables, and thus an
equivalent linear model of the nonlinear system is obtained.
The steady state
response is then found by any technique employed with linear system.
KEYWORDS AND PHRASES.
Nonlinear
Assoon of vabl, Multidimensional frequency domain,
transfer function.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES:
i.
34-00, 34A45.
INTRODUCTION.
This paper considers a nonlinear system described by the equation of the form
S.B. KARMAKAR
536
a3y3 + a5Y5
Ly +
x(t)
(i.i)
L is a linear operator defined as
L(s)
a
3
a5;
b
n
are constants,
the response to it.
bnsn + bn-lS
x(t)
n-i
+
+
bo
is the input to the nonlinear system and
Many physical systems encountered in practice can be
to above kind of equation.
y(t)
is
reiated
We shall investigate the conditions under which a
bounded input to the system will produce a bounded output.
This will lead to the
stability conditions of the above type of nonlinear system.
2.
ANALYSIS OF A NONLINEAR SYSTEM
Equation
(I.i) can be viewed as follows:
Volterra
[i]
has shown that the
response y(t) can be represented as some functional of the input x(t).
The
following functional expansion was suggested:
I
y(t)
hl(t-l)X(l)dl
+
I[h2(t-l,t-2)X(Tl)X(2)didT2+.
We assume a causal system, i.e., x(t)
are
[0,].
0 for t < 0.
(2.1)
The limits of integrations
Let
Yn(t)
I- -I
hn(t-wl...t-Wn)X(Wl)X(W2)’’-X(n)dWld2--’dw n
n-fold
Then equation
(2.1)
can be written as
y(t)
hl(t- I)
Z Yn
(t)
is the ist order Volterra Kernel and h
Volterra Kernel.
n
_(t-l...t-w n
(2.2)
is the nth order
537
STEADY STATE RESPONSE OF A NONLINEAR SYSTEM
3.
DETERMINATION OF THE VOLTERRA KERNELS
By applying the technique or reversion of algebraic series [2] from (i.i)
we get
Y
L-Ix- a3L-i( L-ix) 3 + L-I {3a32-i
L (L-ix) 5 a5(L-ix) 5 +
}
(3.1)
It is clearly understood that x and y are both functions of time t.
Let us
define
L-lx
where
h(t)
I
h(t-Y)x()d
(3.2)
is the impulse response of the linear part of the system.
Hence from
(2.2)
Yl
]
Y3
-a3L-i
h(t-w)x(Y)d[
-a3L
-a
-i
(I
1
h(t-)x(w)d
Il
h(t-l)h(t-2)h(t-3 )x( Yl)X( 2)x( 3 didw2dr3
h(t-u)h(U-Tl)h(u-T2)h(u-T3)X(Tl)X(X2)x(3)dXldT2dx3du
3
Similarly
)x( 1 )’" .x(
-a5
I’ ’I
6
5)dl"
.d
5dudv
h(t-u)h(U-Tl)’’’h(u-x5)x(T1)’’’x(T5)dTl’’’dT5du"
S.B. KARMAKAR
538
Hence by comparing (2.1) and (2.2)
hl(t-T)
h(t-T)
h2(t--Tl,t--T2)
0
I
h3 (t-l’t-2’t-3) =-a3 h(t-u)h(u-l)h(u-2)h(u-3)du
h4(t-l...t-T 4)
0
h5(t-T1--.t-Ts)
3a
I
h(t-u)h(v-u)h(U-l)...h(u-w5)dudv
a5
h(t-u)h(u-l)’’’h(u-5)du"
Higher order kernels can thus be determined by expanding further the
equation
4.
(3.1).
CONVERGENCY OF THE FUNCTIONAL EXPANSION
Let
maxlx(t) <_ X
0<t<
HX
where
Ih(t)lct
Y3 <-
H <
la31i]h(t-l)l Ix(m)l IIh(t-)llx(2)2 IIh(t- 3) Ix()1:3 IIh(t-u)Iu.
or
4
Y3
<
la31(Ilh(t)Idt), X3= la31H4X
3
539
STEADY STATE RESPONSE OF A NONLINEAR SYSTEM
Similarly
Let
Hence
Again by applying the technique of reversion of series
that
(4.1)
is the solution of
Y
3
it is easy to show
(4.2)
HX +
H(Ia31Y3+Ia51Y 5)
(4.2)
la31 y3 la51 y5
(4.3)
and
X
i
Y
.
dY
XI and Y YI corresponding to the value dX
0 < Y !
YI" Therefore in this above region a bounded
output.
YI can be determined from
Let X
4
2
5HIasIYI + 3HIa31YI
i
0
dY
or
+
I
then
input will produce a bounded
corresponding to
-3Hla31
Hence if 0 < X < X
9H2a +20Hla51
S.B. KARMAKAR
540
considering only the positive value of
YI’ YI
YI
is found from
From (4.3)
Therefore for
x(t)
<
X1,
the functional expansion
(2.1)
will be convergent.
REMARKS:
(1.1)
With the assumed solution of equation
hn(T 1
y(t)
T
2
.Tn)X(t--TI)
..x(t--T n )d 1
..d
n
n=l
Y= cXnn
n=l
where
I...IIh(T1
cn
And for the time invariant system, C
radius of convergence
series
(2.1)
X1,
n
Tn) IdTl
T2
d
n
is independent of time
[3]
which has the
which in turn implies the convergence of the Volterra
for
maIx(t)l
<
x
0<t <
REVIEW OF MULTIDIMENSIONAL TRANSFORM.
5.
As an analogy to the linear system, the nth order transform of a nonlinear
system [4] is defined to be
H
n
(Sls 2" ..s n ): |...lh(l e
. )exp[-SlXl+S2X2+...+s
n
T
n n
)]dx...dx
n
(5.1)
541
’STEADY STATE RESPONSE OF A NONLINEAR SYSTEM
Hn (sl’s2"’’sn)
One method of determining
is the harmonic input method
[5].
When
the system described by (i.i) is excited with a set of n unit amplitude exponentials at the noncommensurate frequencies s
exp(slt)
x(t)
+
s2...s n
I
exp(s2t)
(5.2)
+ exp(s t)
+
n
the output will contain exponential components of the form
y(t
exp m s
ml !m2
n=l M
I
n
l+m2 s 2 +
+mn n
(5.3)
s
while M under the summation indicates that for each n the sum is to be taken over
all distinct values of m
Thus to determine
from
(5.2)
and
(5.3),
n
< m
where i < m_ < m_
+/-
Hn(Sl,S 2...s n)
n
such that m_ + m_ + ...m
+/-
we substitute the values of
x(t)
and
n
n.
y(t)
respectively, in the system equation (I.i) and equate the
coefficients of
n!
exp(sl+s2 +. .+Sn)t
By substituting the values of y(t) and x(t) from (5.3) and (5.2), respectively,
in
(I.i)
the first three nonzero transfer functions are found to be-
Hl(S I)
1
(5 )
L(Sl
-a
3
H3(Sl,S2,S 3) L(sI+s2+s3)L(Sl)L(s2)L(s3)
a
H5(Sl..-s 5)
(5.5)
3
L(sI+s2+... s5)L(sI+s2+s 3)L(s 4)L(s 5)
+
a>
L(sl+s2+...s5)L(sl)L(s2)...L(s5)
(5.6)
S.B. KARMAKAR
542
Let A mean associate variables
Y(s)
Then
Sl,S 2.../s.
HI(S)X(s) +A{H3(Sl,S2,s3)X(Sl)X(s2)X(s3)} +A{H5(Sl...s5)X(Sl)...X(s5)} +
Now by applying the Final Value Theorem we get
y(t)
lira
nonzero terms of
6.
s+o
(2.1)
Subject to the convergency of
sY(s)
lira
t
(5.7) to represent
it is often enough to consider the first two
the system in the frequency domain.
EXAMPLE.
We consider a nonlinear system of the form (i.i) whose linear transfer
function is given by
i
i
H(s)
L(s)
s
e
s
e
n
n
+2 +o
Let its impulse response be h(t).
It is easy to see for
series expansion
(2.1)
<
s
2
+2
n
2
+n
H is unbounded.
> i it is easy to evaluate
If
0 will have real roots.
is a little more involved.
< I
h(t)
-
n
If 0 <
(n-i2 )t
Let
n’
8
since
< i the evaluation of H
t
sin
Ilh(t)Idt
If H is bounded, convergency of the
Volterra series can further be investigated.
For 0 <
Hence the Volterra
will be divergent, and the system will not have any
bounded steady state response.
2
0,11h(t)Idt
nl-
2
[7]
543
STEADY STATE RESPONSE OF A NONLINEAR SYSTEM
henc e
e
h(t)
-at
sin 8t
We consider the integral
e- at Is in 8tldt
H’
Now
e
ax
sin
e
pxdx
ax
(a
sin px-p cos
2
2
a
a
2
+6
2
etc.
lie I {e-m+e-2ml {e- 2m+ e- I
o
H
+p
,m
sin 8t < 0 in the intervals
px)
+e
-m
3m
+
+
]
m
e +2e-m+2e-2m+ -e
=2
2
a +6
[2
6
2
Now
-coth(-u)=
+6
coth u
2
o
n- a2 +
i
l+e
l-e
-2u
hence
13
H’
Therefore,
H
-m
i -e
2
n
+6
2
coth()
coth
C
2
aw
6
Henc e
+
S.B. KARM
544
As a numerical example we consider the following system"
+ 3y + tanh y
+
A cos mt
By expanding tanh y and by retaining the first three terms we get
+ 4y
+
y
+
Here
The above approximation is quite valid if y < i.
i
=--4
(6.2)
A cos mt
y
n =2
Hence
2
-3x0.64984
Y1
9x(O
lOO.
+0.94432
XI
x_+
hence
0.9717658
0.64984
i
---
64984)2
64984
0.9717658
YI
(0"9717658) 3
Hence the system will have a steady response for
lim
s+0 s
i
As
_2
15
+ 2oo. 6498
5 (0"9717658)5
1.49539- 0.305888- 0.1155438
the linear part
1
2
1.07395
IA
cos 0t
< 1.07395.
Response of
0. It is not necessary to determine the
+s+4 s 2-----+co
response of the nonlinear parts because being convergent, the other terms of the
Volterra expansion can not exceed in absolute value of the linear term.
the system will be stable for
Ix(t)l
<
11.07395
cos t
I.
If
x(t)
Hence
A, then for
A < 1.07309, the approximate value of the steady state response is given by
545
STEADY STATE RESPONSE OF A NONLINEAR SYSTEM
lira
y(t)
Yl(t)
t
Sl/0
Y3(t)
+ lira
t+
Y3 (t)
Sl
lira
lira
In order to determine lira
t+
Yl (t)
lira
t
t+
s
2
I
2nS I + m2n
+
A
A
i
n
Y3(Sl,S2,S3)
we first obtain
YB(s,se,s3) HB(s,se,sB)X(s)X(se)X(s3)
From (5.5)
3
A3
L(sI+s2+s3)L(Sl)L(s2)L(s3
SlS2S 3
a
Y3(Sl,S2,S3)
Where
2
n
2
s’ + 2a s +
n
L(s)
Now by the technique of association of variables
[7,8,9]
Y3(Sl,S2,S3
is
translated in one frequency domain, thus we obtain
Y3(s)
i
A
I
form
2
A2
i
s
+n
( 2+2s nS+n2)
I
A
--are functions of
8
(s,,)
n
2+2o2+oo2)
n
A1
but do not contain any factor of the
Hence
lira
t+
Y3(t)=-
lira
s+o
(
+ lim s
s+o
sA3a3
s
2
+2n+n2)
2
+
A8
i
4nS2 n
+_
s+
n2)
546
S.B. KARMAKAR
Jim
t+
a3A38--
Y3(t)
+ 0
n
The steady state response of the system is approximately given by
Yss
For
=4,
Yss
If
x(t)
6(t).
i
i
3.64
-
1
+
76--
Again by associating the variables
lim
Yss
Since
"
i +
A
i
i’ 2’ 3
s+o s
S
2
2
+2 n s +n
+
lims+o
l
+
we
get
B2
d nt cntain any factr f the
ACKNOWLEDGMENT:
[8]
i
frm-’s
The author gratefully acknowledges the assistance offered
to him by Prof. J. F. Barrett, Department of Mathematics, Eindhoven University of
Technology, The Netherlands.
REFERENCES
i.
Volterra, V., Theory of Functionals and of Integral and Integro Differential
Equations, Dover Publications, New York, 1959.
2.
Hodgman, C. D., Standard Mathematical Tables, Chemical Rubber Publishing
Company, Cleveland, Ohio, 1959.
3.
Barrett, J. F., The series-reversion method for solving forced nonlinear
differential equations, Math. Balkanica, Vol.
4.9, 1974, pp. 43-60.
STEADY STATE RESPONSE OF A NONLINEAR SYSTEM
547
4.
Barrett, J. F., The Use of Functionals in the Analysis of Nonlinear Physical
Systems, J. Electronics and Control 1__5, 567-615, 1963.
5.
Kuo, Y. L., Frequency-Domain Analysis of Weakly Nonlinear Networks, Circuits
and Systems, (IEEE) Vol. ii, No. 4, 2-8, August 1977.
6.
Bedrosian, E., Rice, S. 0., Output Properties of Volterra Systems (Nonlinear
Systems with Memory) Driven by Harmonic and Gaussian Inputs, Proc. IEEE 5_9
1688-1707, December 1971.
7.
Dorf, R. C., Modern Control Systems, Addison-Wesley Publishing Company,
Reading, Massachusetts, 197.
8.
George, D. A., Continuous Nonlinear Systems MIT Research Laboratory of
Electronics, Technical Report 355, July 24, 1959.
9.
Lubbock, J. K., Bansal, V. S., Multidimensional Laplace transforms for
solution of nonlinear equation, Proc. IEEE, Vol. 116 (12) 2075-2082,
December 1969.
I0.
Koh, E. L., Association of variables in n-dimensional Laplace transform,
Int. J. Systems Sci., Vol. 6 (2).