Identification of voltera kernels of a class of nonlinear systems by walsh function techniques

Identification of Voltera Kernels of a Class of Nonlinear Systems by Walsh Function Techniques by

MOHAMMAD

Electrical 79968,

MAQUSI*

Engineering

Department,

Universiry

of Texas

at El Paso, El Paso, TX

U.S.A.

This paper discusses the identification problem for a class of nonlinear systems. A member of this class may be represented by a single-valued power-law type nonlinearity preceded and succeeded by linear dyadic invariant systems. Such an arrangement allows for a Voltera functional series representation. The identification problem is then concerned with the specification of the associated Voltera kernels. Two approaches are presented for dealing with this problem. Both approaches are, however, based on Walsh function techniques. The first approach relies on direct output measurements when fhe input is a Walsh function. This approach is suitable for a deterministic case. The second approach assumes ergodic processes for the input. Based on measurements drawn from an input-output dyadic correlation function, determination of the Voltera kernels is made. ABSTRACT:

I. Introduction Several authors (l-5), have discussed the problem of the identification of certain classes of nonlinear systems. A number of these works consider the identification problem in the case where the nonlinear system is representable by a Voltera functional series expansion. Indeed, such a representation provides an adequate description for a large class of nonlinear systems. However, the computations associated with identification schemes based on this representation are often burdensome. In this paper we study the identification of a class of nonlinear systems by Walsh function techniques. As shown in Fig. 1, the nonlinear system under study consists of a linear dyadic invariant (LDI) system followed by a singlevalued nonlinearity and another LDI system. In general, analysis of nonlinear systems with Walsh-type signals yields useful and computationally convenient expressions (6,7).

ZZ. General Analysis We consider the space of real, positive-coordinate x(r) is defined for tr0. In general, an input-output *At Jordan,

0

present on leave Amman, Jordan.

The Franklin htitute

from

the Electrical

00164032/80/07014065302.00/0

Engineering

signals. Thus the input relation for a casual LDI Department,

University

of

65

Mohammad

Maqusi LDI

X&l

q(t)

System,

I

.

h, (t)

system

(8,9)

is given

>

System,

--3o

hz f-t)

1. A class of nonlinear

FIG.

z(t)

LDI

y(t)

N OllllllMr Element

l

systems.

by

q(t) =

h(+(t

CB T) d7 =

%

I0

(1)

-h(t&)x(7)d7,

where x(t), q(t) and h(t) denote the system input, output, and impulse response, respectively. The operator @ denotes modulo-two addition (withoutcarry). The two forms indicated in (1) are made possible by the use of a dyadic translation invariant integral property (10, ll),which states that for a Lebesgue-integrable function f(t) on [0, cc), we have O f(t@T)dt= I-

I0

7 E [O, a).

-f(t)dt,

(2)

Furthermore, relations (1) represent dyadic convolution integrals for the functions x(t) and h(t). Identification of linear dyadic invariant systems has been studied elsewhere, and follows after the case of linear time invariant systems to a large extent (12,13). Referring to Fig. 1, we have cc x(t @ T)h(T) dT. x(T)h(t @ T) dT = (3) q(t)= 50 10 We consider characteristic

the no-memory as Y(O

Employing

X x(t

by a transfer

(4)

final system

h(Th(T2)

@

TI)

dTl+

@

@

72)

dT1dT1 +. . .

IW 0

T1)X(f

. . . kd. d

output

7’2

. . hl(T,,)x(f

@

71)

. . . X(@T,)

dT1 . . . dT,,.

(5)

is given by z(t) =

66

to be represented

r1q(O+ y*qw+...+mq”(t).

kT1)X(f

+‘yn

Substitution

element

(3) in (4) gives

y(t) = Yl

The

=

nonlinear

of (5) in (6), and

m h,(cu)y(t CBa) da.

(6)

50

utilization

of the dyadic

translation

invariant

Journal of The Franklin Institute Pergamon Press Ltd.

of Voltera Kernels of a Class of Nonlinear Systems

Identification integral

property

(2) give the measured ~(T~CB

system

output

a)h,(a)x(t

63 TJ dri da

x x(t CT3TJx(t G3~2) dr1dr2 da +. m +‘yn

. . . I0

form,

h1(71

CB a)

. . . hl(T,

G3 a)h,(a)

x(t G37,) dri . . . dr, da.

xx(t@T1)...

compact

. .

m

I0

In a more

as

we may rewrite 71,

(7)

(7) as

. . . >7,) fi X(t @ up) dTp

(8)

p=l

where

(9) Voltera kernel or the The function gi(rl, _. . , To) is referred to as the ith-order ith-order impulse response of the nonlinear system, and the integral indicated in (8) denotes a multiple (i times) integral ranging from 0 to 0~. The series expansion in (8) is called a Voltera series expansion, and according to the more dictated form in (7) it affords an output expression which is similar in form to that used in the case of linear dyadic invariant systems. Furthermore, the Voltera kernels may be specified in the sequency domain by the use of Walsh transform. Indeed, applying a multidimensional Walsh transform to (9), gives

Ti)+(Ui, Ti) . . . @(ai, 7,) dri . . . drip where the transform kernel +(a, T) denotes function (11). The nonnegative real numbers spatial (e.g., time) (14) variables, respectively. of a suitable (e.g., square-integrable) function

F(u) = and the inverse

Walsh

I0

transform

(11)

by

a, t) da =

Vol. 310, No. 1.July 1980 Printdin Northern Ireland

a generalized Walsh-Fine type (+ and T denote sequency and In general, the Walsh transform f(t) is given by

mfW44~,t) dt = Wf(t)], is given

(10)

T1[F(c)].

(12) 67

Mohammad

Maqusi

For illustration,

we compute

G, (o) using

m G,(o)

= ~1

(9) and (10). Hence,

m

I0 [5 0

h,(o)h,(T,

@ a) do

1

+(u, 71) dr1.

(13)

Interchanging orders of integration and using the (dyadic time-domain) property of Walsh transform; i.e. for a positive real number b, ‘If’Mt @

b)l= F(a)rCr(~,b),

shift

(14)

we obtain

In a similar

manner,

and in general

we obtain

we get

G(al,.

- .v

ai)

By a multidimensional

=

inverse

. . . H,(Ci)H*(Uj

YiyiHICal)

Walsh

transform

x @++(@I, Tl>. . . I/J(u,, ZIZ.Determination

@.

. . @

Ci).

we have

TV)da, . . . dci.

(18)

of the Voltera Kernels

In this section we discuss two methods for determining the Voltera kernels, defined earlier in Section II. Two approaches are used. One approach is based on direct output measurements, while the other approach depends on computations of an input-output dyadic crosscorrelation function. However, both methods utilize Walsh signals for the input. A. Output measurements In this case we assume

approach an input

x(t) given

x(t) = A+(r,

q(t) = IffiAh,(T)$(r, 0

r is real,

t),

where A and r represent the amplitude respectively. Under assumption (19), we obtain

and

sequency

t G37) dT = AH,(r)$(r,

In (20) we have used a multiplicative identity +(r, t)$(r, T), and the definition of the Walsh

68

by (19) values

7).

of the input,

(20)

for Walsh functions: +(r, t @ 7) = transform of h,(t). We note from Jowml

of The Franklin Instihlte Permmon Press Ltd.

of Voltera Kernels of a Class of Nonlinear

Identification

(20) that q(t) is in essence a Walsh function, same sequency as that of the input x(t).

save for a scaling

factor,

Systems with the

Now [$(r, t)li =

(21) where i

@r=r@...@r,itimes.

,=l

Thus,

the output

of the nonlinear

element

is given

by

Y(t) = i nq’(t) = i nA’H;(r)[llr(r, t)T i=l

The quantity H,(r) represents invariant system at the sequency specified by z(t) =

i=l

the steady state gain of the linear r. The measured output of the system

dyadic is now

mh,(a)y(t @ cr)da = i ‘yiAiHl(r) i=l

,cl@

Applying the same procedure (20), we then obtain z(t) = i

to the integral

yiAiH:(r)Hz(

i j=l

i=l

Since Cizl @ r gives either like sequency and rewrite

r, tea)

indicated

CBr)$(

(23)

da.

i

in (23) as was done

CT3r, t).

in

(24)

j=l

r or 0, for all i = 1,2, _. . , n, we may group terms of (24) as

[(n+1)/21

z(t) =

1 ,=I

{y2iA2iH2(0)H:i(r)+

-y2i_1A2i-‘H2(r)H~i-‘(r)

+(r, t)}.

(25)

According to (24), or equivalently (25), and (8) we observe that if the linear dyadic subsystems h,(t) and h2(t) are isolated and identified, then identification of the Voltera kernels is made complete by determining the y-set, If we now take measurements of the output at times t,,,, {rr, Y2,. . ., y,}. Vol. 310. No. 1, July 1980 Rinted in Northern Ireland

69

Mohammad m = 1,2,. unknowns.

Maqusi . . , n, we Hence

obtain,

from

(25),

equations

for

the

+ A3~2(W~(rh3+. . -1d4r, ~1,

m = 1,2, . . . , n,

(26)

where +(r, t,,,)=+l or -1, for all m= 1,2 ,..., n. Without loss of generality, we assume that n is even. By (26), the terms interest in determining the Voltera kernels are defined by Pi = -yiK(r)I%(r)

= G(r)

P2 = Y2#(r)H2(0)

= G2(r, r)

pn-l = y,_,H;-‘(r)H,(r) P,, = x,W(r)~2(0) In terms

y-

. .]

z(f,,,) = [A2H2(0)H~(r)y2+A4H2(O)H;‘(r)y4+.

+[Aff2(rV&(h

determining

= G,_,(r,

of

. . . , r)

= G,(r, . . . , 4.

(27)

of (27), (26) becomes z(fm)=[b2+P4+.

m-l,2

. ~+~nl+[~1+~3+~.

,...,

n.

.+&11$l(r7

k,h

(28)

As mentioned previously, if the two linear dyadic subsystems are identified, then (28) specifies the P-set which in turn specifies the Voltera kernels, and this completes the identification problems. A method for decoupling the identification of the linear subsystems from the identification of the nonlinear element is presented in (1)for the case of linear time invariant systems. This method may be applied to Fig. 1 with minor modifications to suit the dyadic invariant systems. Furthermore, we point out that in general we may evaluate the terms p,, . . . , /3,, for each test sequency (i.e., vary r) which renders a set of magnitude response characteristics from which the sequency transfer function expressions for the two linear dyadic invariant systems may be derived by a least-squares fitting technique, for instance. B. Dyadic Correlation Function Approach Now let

z(t)= f w,(t),

(29)

m=l

where

I

w,(t) = %n

m+l

k(Td. . .h,(TnJh,(a)

xx(t@ri@~)...x(t@r,@o)dri...dr,,,d~. 70

(30)

Identification of Voltera Kernels of a Class of Nonlinear Systems Before proceeding with concepts. For a process

this approach we digress to state a few pertinent x(t), we define a finite dyadic correlation function

03915) by

T

D,(T; T) =_: and a dyadic

correlation

function D,(T)=

A dyadic crosscorrelation

i0

x(t)x(t @ T) dt,

(31)

T x(t)x(t @ T) dt.

(32)

by

lim L T-m T0 j

for two processes x(t) and y(t) is defined by 7 = lim -! x(t)y(t @ 7) dt. (33) T-m TCIj

function D,,(T)

Similarly, we may define a finite dyadic crosscorrelation function without the limit operation. For random processes, these functions are random functions of the parameter T. The description of random processes may be made more appropriately by defining a stayadic correlation function (16), for such a process g(t), by cp(7) = %(t)g(t A corresponding

sequency

power

r,(a)

= w&(T)]

spectral

@ T)].

(34)

density

is given

= jO-c&)$(u,

7) d7.

by (35)

For ergodic processes, dyadic and stayadic correlation functions become interchangeable in the limit as the time interval under consideration becomes infinite. This result is modeled after Birkoff’s theorem for the interchange of time and ensemble averages for such processes, and is discussed in Section IV. We thus assume that the random processes we consider are of the ergodic type. According to (29), an input-output finite dyadic crosscorrelation function is given by D,,(T; T)=$jTx(f)z(l@T)di 0

= i m=l

_: jTx(i)W,(@T)dt 0

(36) Employing

(30), we obtain

X(t)X(t~T~T1~(Y)...X(f~T~T,~(Y)dt I

(37) Vol. 310, No. 1. July 1980 Printed in Northern Ireland

71

Mohammad Denoting A$(r,

Maqusi

the bracketed

integral

in (37) by D:(T;

T), we obtain

for x(t) =

0,

m+ll-,(f @r,a)4( f @r,T)

D,~(T; T) = A

i=l

X 446

~~1

.

.

i=l

.445T,),

(38)

where

For integral to

r and T, we apply orthogonality

I= However,

1,

oddm

0,

even

of Walsh functions

to reduce

(40)

m ’

for real r and T, we have (39) which gives an upper bound

D,,, D,,,

(7; T) according (7; T) = ?,A

to (37) and (38), we have

‘“+‘IH?(r)H,(

i

G3r)+(

dyadic

f

CBr, 7).

(42)

i=l

i=l

In view of (42), the input-output reduces to

on I, as (41)

l~l~-:bl/lL~~~~*~r,r)idr=l.

Evaluating

(39)

crosscorrelation,

as given by (36), now

D,,(T; J-1= D,,(7) ““IHY(r)H,(

f i=l

C3 r)+(

f

CBr, 7).

(43)

i=l

The application of this result is similar to the result obtained earlier for identification by direct output measurements. If the two linear dyadic subsystems are identified, then measurements of D,,(7) at times TV, k = 1,2,. . . , n determine the -y-set by application of (43). On the other hand, by applying different values for the input sequency r, and drawing measurements from interest: terms of different values for the DX,(r), we get v,A*IH,(r)H,(r)$(r, T), yzA31H:(r)H,(0), . . . ,_ From these latter values, the Voltera kernels may be determined as indicated in part (A) of this section. IV. Discussion

of Results

In the previous sections we considered a certain composed essentially of nonlinear elements and discusses a Voltera series expansion representation

class of nonlinear systems LDI systems. Section II for such systems. The

Identification of Voltera Kernels of a Class of Nonlinear Systems representation hinges primarily on the specification of associated Voltera kernels. Consequently, Section III is concerned with the identification of these kernels by the use of Walsh type signals for the excitations. A method of identification based on system output measurements is viewed appropriate for a deterministic case. For random cases, if we assume ergodic signals at the input, then identification of the kernels may still be made using Walsh techniques. But computations are now based on input-output dyadic crosscorrelation functions. However, either approach will ultimately benefit from the various computational advantages rendered to Walsh functions. The assumption of ergodicity in the random case facilitates the interchange of statistical and dyadic correlation functions. In an earlier work, and after Lee (17), Weiser (18) produced such a result by heuristic arguments. Due to the rather difficult accessibility of Weiser’s work, this result is reproduced in this paper. Consider an ensemble of sample functions of a random process, x(t), placed parallel in time as shown in Fig. 2. Stayadic correlation for such a process is then defined by C,(T) = E[x(Qx(t and dyadic

correlation

is defined

@ T)l,

(44)

by

D,(T) = lim $ T+-

I0

Tx(t)x(I @ 7) dt.

Now suppose that each single function is divided into increments of infinitesimal duration At as shown in Fig. 2(b). Since the ensemble members (sample functions) are generated by sources of indentical nature (i.e. ergodic), we make the assumption that all members have the same member aggregate of amplitude values, although the order of these values may be different. Next, we

FIG. 2. (a) An ensemble

Vol. 310. No. 1 July 1980 Printed in Northern hla,,d

of sample functions placed parallel in time. (b) A single sample function divided into increments At.

73

Mohammad

Maqusi

examine the nature of the dyadic time shift. Since the dyadic transformation is measure-preserving (lo), then the aggregate of amplitude values consists of the same members as the aggregate of amplitudes formed by a single sample function. By the ergodic hypothesis, this aggregate is the same as the ensemble aggregate for a fixed time, the elements merely being taken in a different order. Note also that the ordering of the elements in the x(t CI37) aggregate will differ from the ordering of the elements in the x(t) aggregate. Finally, consider the aggregate of pairs formed by first selecting the amplitude value x(t) and then the amplitude value x(t G3 T) over the single ensemble. Since it is assumed that aggregates contain the same elements whether formed from an ensemble or from a single sample function over all time, the aggregate of pairs should also contain the same members whether formulated by scanning the ensemble of sample functions or a single sample function. By the same previous reasoning, it is then possible to say that the mean of the product of the two elements, in the limit as T goes to infinity, is the same no matter how it is expressed whether as a mean with respect to the member aggregate or as a mean with respect to the ensemble aggregate. Thus,

C,(T) = E[x(t)x(t G3T)] = lim E[x,(t)x,(t@

T)]

T-m T =

lim & c x(t)x(t

~-cc

=

This latter correlation

1 Jo

CI37) dt

D,(T).

(46)

result indicates the possible interchange of stayadic functions associated with an ergodic process.

and

dyadic

V. Conclusions This paper presents two methods for the identification of Voltera kernels for a certain class of nonlinear systems. Both methods utilize Walsh signals as inputs. According to the first method, identification is accomplished through output measurements at different input sequencies. The second method relies on measurements taken for input-output dyadic crosscorrelation functions. A primary advantage of employing Walsh signals is in general attributed to their computational advantages.

References “Identification of a class of nonlinear systems and S. Y. Fakhouri, using correlation analysis”, Proc. IEEE, Vol. 125, pp. 691-697, 1978. “Identification of nonlinear systems using the (2) S. A. Billings and S. Y. Fakhouri, Wiener model”, Electron. Letters, Vol. 13, pp. 502-504, 1977.

(1)S. A. Billings

Jo,m,al

74

of The Franklin Institute Pergamon Press Ltd.

Identification of Voltera Kernels of a Class of Nonlinear

Systems

“Identification of simple nonlinear systems”, Proc. 1975. of the Voltera kernels of a process containing (4) R. V. Webb, “Identification single-valued nonlinearities”, Electron. Letters, Vol. 10, pp. 344-346, 1974. (5) E. Economakos, “Identification of a group of internal signals of zero-memory nonlinear systems”, Electron. Letters, Vol. 7, pp. 99-100, 1971. (6) M. Maqusi, “Walsh analysis of power-law systems”, IEEE Trans. Information Theory, Vol. IT-23, pp. 144-146, 1977. (7) M. Maqusi, “On the Walsh analysis of nonlinear systems”, IEEE Trans. elecIromag. Cornpat., Vol. EMC-20, pp. 519-523, 1978. (8) F. Pichler, “Walsh functions and optimal linear systems”, Proc. Symp. Walsh Function Applications, pp. 17-22, Washington, D.C., April 1970. (9) M. Maqusi, “Walsh analysis of linear dyadic invariant and instantaneous nonlinear systems”, Sc.D. Dis., New Mexico State Univ., 1973. (10) N. J. Fine, “On the Walsh functions”, Trans. Am. math. SOL, Vol. 65, pp. 373-414, 1949. (11)N. J. Fine, “On generalized Walsh functions”, Trans. Am. math. Sot., Vol. 69, pp. 66-77, 1950. (12) S. Cohn-Sfetcu and S. T. Nichols, “On the identification of dyadic invariant systems”, IEEE Trans. electramag. Compat. Vol. EMC-17, pp. 111-117, 1975. (13) G. P. Rao and L. Sivakumar, “Identification of time-lag systems via Walsh IEEE Trans. autom. Control, Vol. AC-24, pp. 806-808, 1979. functions”, (14) N. Ahmed, H. H. Schreider and P. V. Lopresti, “On notation and definition of terms related to a class of complete orthogonal functions”, IEEE Trans. electromag. Cornpat., Vol. EMC-15, pp. 76-80, 1973. (15) F. Pichler, “Walsh functions and linear system theory”, Proc. Symp. Walsh Function Applications, pp. 175-182, Washington, D.C., April 1970. (16) M. Maqusi, “A sampling theorem for dyadic stationary processes”, IEEE Trans. Acoust., Speech, Signal Processing, Vol. ASSP-26, pp. 265-267, 1978. (17) Y. W. Lee, “Statistical theory of communication”, John Wiley, New York, 1960. (18) F. W. Weiser, “Walsh function analysis of instantaneous nonlinear stochastic problems”, Ph.D. Dis., Polytechnic Institute of New York, 1964. (3) B. Cooper

and A. H. Falkner,

IEEE, Vol. 122, pp. 753-755,

Vol. 310, No. 1. July 1980 Printed in Northern Ireland

75