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Internat. 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 multi-
dimensional frequency domain. Conditions are then determined for which this
series solution will converge. 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. Assoon of vabl, Multidimensional frequency domain,
Nonlinear transfer function.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES: 34-00, 34A45.
i. INTRODUCTION.
This paper considers a nonlinear system described by the equation of the form
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536 S.B. KARMAKAR
Ly + a3y3 + a5Y5 x(t) (i.i)
L is a linear operator defined as
n-iL(s) bnsn + bn-lS + + bo
a5; b are constants, x(t) is the input to the nonlinear system and y(t) is3 n
the response to it. Many physical systems encountered in practice can be reiated
to above kind of equation. 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:
y(t)
I hl(t-l)X(l)dl + I[h2(t-l,t-2)X(Tl)X(2)didT2+. (2.1)
We assume a causal system, i.e., x(t) 0 for t < 0. The limits of integrations
are [0,]. Let
Yn(t) I---I hn(t-wl...t-Wn)X(Wl)X(W2)’’-X(n)dWld2--’dwnn-fold
Then equation (2.1) can be written as
y(t) Z Yn (t) (2.2)
hl(t- I) is the ist order Volterra Kernel and h _(t-l...t-w is the nth ordern n
Volterra Kernel.
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STEADY STATE RESPONSE OF A NONLINEAR SYSTEM 537
3. DETERMINATION OF THE VOLTERRA KERNELS
By applying the technique or reversion of algebraic series [2] from (i.i)
we get
Y L-I -i( 3
{3a3L (L-ix)- a3L L-ix) + L-I 2-i 5a5(L-ix)
5} + (3.1)
It is clearly understood that x and y are both functions of time t.
define
L-lx
Ih(t-Y)x()d
where h(t) is the impulse response of the linear part of the system.
(2.2)
Yl ] h(t-w)x(Y)d[
Y3 -a3L-i (Ih(t-)x(w)d
1-i
-a3L Ill h(t-l)h(t-2)h(t-3 )x( Yl)X( 2)x( 3 didw2dr3
Let us
(3.2)
Hence from
-a3 h(t-u)h(U-Tl)h(u-T2)h(u-T3)X(Tl)X(X2)x(3)dXldT2dx3du
Similarly
)x(1
)’" .x( 5)dl" .d 5dudv
-a5 I’’’I h(t-u)h(U-Tl)’’’h(u-x5)x(T1)’’’x(T5)dTl’’’dT5du"6
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538 S.B. KARMAKAR
Hence by comparing (2.1) and (2.2)
hl(t-T) h(t-T)
h2(t--Tl,t--T2) 0
h3 (t-l’t-2’t-3) =-a3 I 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 IIh(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 (3.1).
4. CONVERGENCY OF THE FUNCTIONAL EXPANSION
Let
maxlx(t) <_ X
0<t<
HX
where
Ih(t)lct H <
Y3 <- 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= la31H4X3
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STEADY STATE RESPONSE OF A NONLINEAR SYSTEM 539
Similarly
Let
Hence
Again by applying the technique of reversion of series 3 it is easy to show
that (4.1) is the solution of (4.2)
Y HX + H(Ia31Y3+Ia51Y5) (4.2)
and
i y3 y5Y la31 la51 (4.3)
dY
XI YI . Hence if 0 < X < XIthenet X and Y corresponding to the value
dX
0 < Y ! YI" Therefore in this above region a bounded input will produce a bounded
output. YI can be determined from
4 2
5HIasIYI + 3HIa31YI i 0
corresponding to
dY
or
-3Hla31 + 9H2a +20Hla51
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540 S.B. KARMAKAR
considering only the positive value of YI’ YI is found from YIFrom (4.3)
Therefore for x(t) < X1, the functional expansion (2.1) will be convergent.
REMARKS:
With the assumed solution of equation (1.1)
y(t) hn(T1 T2 .Tn)X(t--TI) ..x(t--T )d
1..d
n n
n=l
where
Y= cXnnn=l
c I...IIh(T1 T2 Tn) IdTl dn n
And for the time invariant system, C is independent of time [3] which has then
radius of convergence X1, which in turn implies the convergence of the Volterra
series (2.1) for
maIx(t)l < x0<t <
5. REVIEW OF MULTIDIMENSIONAL TRANSFORM.
As an analogy to the linear system, the nth order transform of a nonlinear
system [4] is defined to be
H (Sls ..s ): |...lh(l e . )exp[-SlXl+S2X2+...+s T )]dx...dxn 2" n n n n n(5.1)
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’STEADY STATE RESPONSE OF A NONLINEAR SYSTEM 541
One method of determining Hn (sl’s2"’’sn) is the harmonic input method [5]. When
the system described by (i.i) isexcited
with a set of n unit amplitude exponen-
tials at the noncommensurate frequencies sI s2...s n
x(t) exp(slt) + exp(s2t) + + exp(s t)n
(5.2)
the output will contain exponential components of the form
y(t
ml !m2 nn=l M
exp mIsl+m2 s
2+ +mn s
n(5.3)
while M under the summation indicates that for each n the sum is to be taken over
all distinct values of m where i < m_ < m_ < m such that m_ + m_ + ...m n.n +/- n +/- n
Thus to determine Hn(Sl,S 2...s n) we substitute the values of x(t) and y(t)
from (5.2) and (5.3), 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-
1(5 )Hl(SI) L(Sl
-a3
H3(Sl,S2,S 3) L(sI+s2+s3)L(Sl)L(s2)L(s3)
(5.5)
H5(Sl..-s 5)a3
L(sI+s2+... s5)L(sI+s2+s3)L(s4)L(s 5)
a>L(sl+s2+...s5)L(sl)L(s2)...L(s5)
(5.6)
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542 S.B. KARMAKAR
Let A mean associate variables Sl,S2.../s. Then
Y(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
lira y(t) lira sY(s)
t s+o
Subject to the convergency of (2.1) it is often enough to consider the first two
nonzero terms of (5.7) to represent the system in the frequency domain.
6. EXAMPLE.
We consider a nonlinear system of the form (i.i) whose linear transfer
function is given by
i i
L(s)H(s)
e es +2
s+ o
n n
Let its impulse response be h(t).
It is easy to see for <
0,11h(t)IdtH is unbounded. Hence the Volterra
series expansion (2.1) will be divergent, and the system will not have any
bounded steady state response. If > i it is easy to evaluate Ilh(t)Idt since
2 2 2s +2 + 0 will have real roots. If H is bounded, convergency of the
n n
Volterra series can further be investigated. If 0 < < i the evaluation of H
is a little more involved.
For 0 < < I
h(t) -t
n
sin (n-i 2 )t [7]
Let
n’ 8 nl-2
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STEADY STATE RESPONSE OF A NONLINEAR SYSTEM 543
henc e
-ath(t) esin 8t
We consider the integral
atH’ e- Is in 8tldt
Now
axax (a sin px-p cos px)
e sin pxdx
e
2 2a +p
sin 8t < 0 in the intervals ,m etc.
H2
a +6
o -m 2m+ 3m
Ilie +e I + {e-m+e-2ml + {e- e- +
6
[2+2e-m+2e-2m+ -e
]m=2e o aw
a +6 6
Henc e
2 2 n- a2 -m+6 i + i -e
Now -coth(-u)= coth u
Therefore,
l+e
-2ul-e
hence
H’ 2
132 coth()+6
H - coth
2n C
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544 S.B. KARM
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 + y A cos mt
The above approximation is quite valid if y < i. Here
(6.2)
Hence
i
=--=2
4 n
2 -3x0.64984 x_+ 9x(O 64984)2 1 _2
-+ 2oo. 6498
15
Y1 lOO. 64984
+0.94432 hence YI 0.9717658
XI0.9717658 i 3
0.64984(0"9717658) 5 (0"9717658)5
1.49539- 0.305888- 0.1155438 1.07395
Hence the system will have a steady response for IA cos 0t < 1.07395. Response of
i As2
the linear part lim ----- 2------0. It is not necessary to determine the
s+0 s +s+4 s +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. Hence
the system will be stable for Ix(t)l < 11.07395 cos t I. If x(t) A, then for
A < 1.07309, the approximate value of the steady state response is given by
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STEADY STATE RESPONSE OF A NONLINEAR SYSTEM 545
lira y(t) lira Yl (t) + lira Y3 (t)
t+ t t+
Sl A Alira Yl(t) lira
2 m2 iSl/0 sI+ 2nSI
+n n
In order to determine lira Y3(t) we first obtain Y3(Sl,S2,S3)t+
YB(s,se,s3) HB(s,se,sB)X(s)X(se)X(s3)
From (5.5)
Y3(Sl,S2,S3)a3 A
3
L(sI+s2+s3)L(Sl)L(s2)L(s3 SlS2S 3
Where
2 2L(s) s’ + 2a s +
n n
Now by the technique of association of variables [7,8,9]
Y3(Sl,S2,S3translated in one frequency domain, thus we obtain
is
Y3(s)( 2+2s nS+n2) 2+2o2+oo2)n A1 A8
i 2
AI
A2
A8
iform Hences
I--are functions of (s,,) but do not contain any factor of the
n
sA3a3 ilira Y3(t)=- lira
(2 2 2) 4nS2+_ + 2)+ s+o s +2n+n n s n
+ lim s +
s+o
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546 S.B. KARMAKAR
a3A3im Y3(t) 8-- + 0t+
n
The steady state response of the system is approximately given by
Yss
For
=4, A 1
Yss "i +
3.64 - +76--
If x(t) 6(t). Again by associating the variables [8] we get
Slim
Yss 2 2s+o s +2 s +n n
+
lims+o
l+
B2
i i i iSince
i’ 2’ 3d nt cntain any factr f the frm-’s
ACKNOWLEDGMENT: 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.
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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).
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Mathematical Problems in Engineering
Special Issue on
Modeling Experimental Nonlinear Dynamics andChaotic Scenarios
Call for Papers
Thinking about nonlinearity in engineering areas, up to the70s, was focused on intentionally built nonlinear parts inorder to improve the operational characteristics of a deviceor system. Keying, saturation, hysteretic phenomena, anddead zones were added to existing devices increasing their
behavior diversity and precision. In this context, an intrinsicnonlinearity was treated just as a linear approximation,around equilibrium points.
Inspired on the rediscovering of the richness of nonlinearand chaotic phenomena, engineers started using analyticaltools from “Qualitative Theory of Diff erential Equations,”allowing more precise analysis and synthesis, in order toproduce new vital products and services. Bifurcation theory,dynamical systems and chaos started to be part of themandatory set of tools for design engineers.
This proposed special edition of the Mathematical Prob-lems in Engineering aims to provide a picture of the impor-tance of the bifurcation theory, relating it with nonlinear
and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understandingby the combination of specific tools that associate dynamicalsystem theory and geometric tools in a very clever, sophis-ticated, and at the same time simple and unique analyticalenvironment are the subject of this issue, allowing newmethods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems inEngineering manuscript format described at http://www.hindawi.com/journals/mpe/. Prospective authors shouldsubmit an electronic copy of their complete manuscriptthrough the journal Manuscript Tracking System at http://
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Manuscript Due December 1, 2008
First Round of Reviews March 1, 2009
Publication Date June 1, 2009
GuestEditors
José Roberto CastilhoPiqueira, Telecommunication andControl Engineering Department, Polytechnic School, TheUniversity of São Paulo, 05508-970 São Paulo, Brazil;
[email protected] E. NeherMacau, Laboratório Associado deMatemática Aplicada e Computação (LAC), InstitutoNacional de Pesquisas Espaciais (INPE), São Josè dosCampos, 12227-010 São Paulo, Brazil ; [email protected]
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