Steady State Response of a Nonlinear System

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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



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.



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



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




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



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)



’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)



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




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



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




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



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.




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).



Mathematical Problems in Engineering

Special Issue on

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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

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Steady State Response of a Nonlinear System