N A S A C O N T R A C T O R N A S A C R - 7 5 6 R E P O R T -

SYSTEM IDENTIFICATION AND PATTERN RECOGNITION

by Rob Roy und Jumes Shermun

Prepared by RENSSELAER POLYTECHNIC INSTITUTE Troy, N. Y.

for

N A T I O N A L A E R O N A U T I C S A N D SPACE A D M I N I S T R A T I O N W A S H I N G T O N , D. C. APRIL 1967

https://ntrs.nasa.gov/search.jsp?R=19670014387 2020-03-24T16:51:30+00:00Z

27 NASA CR-756 END

5 SYSTEM IDENTIFICATION AND PATTERN RECOGNITION b B p b Roy and James Sherman

Distribution of this report is provided in the interest of information exchange. Responsibility for the contents resides in the author or organization that prepared it.

Prepared under Grant No. \ NGR-33-018-01 d by RENSSELAER POLYTECHNIC INSTITUTE

Troy, N.Y. 3 \

for

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For sal. by tho CIooringhouro for Fodoral Sciontific and fochnical lnformotion Springfiold, Virginio 22151 - CFSTl prieo $3.00

SYSTEM IDENTIFICATION AND PATTERN RECOGNITION

Rob Roy James Sherman Associate Professor Research Assistant Elec t r ica l Engineering Department and Elec t r ica l Engineering Department Rensselaer Polytechnic Ins t i t u t e Troy, New York

Rensselaer Polytechnic Ins t i t u t e Troy, New York

Introduction

This paper presents a new technique f o r determining an input-output model of L

a system with f i n i t e s e t t l i ng t i m e .

t ha t the problem of system ident i f icat ion is closely related t o the problem of

pattern recognition. ”* given, an explanation of the concept of the technique is presented.

This technique grew out of a real izat ion

Therefore, before the mathematical relationships are

Consider a f i n i t e s e t t l i ng time system. The relevant input past of such a

system can be represented by a set of N uniformly spaced samples, such as those

obtained at the taps of a lossless delay l ine.

forms an orthogonal set of coordinates, such tha t the input past is a vector i n

t h i s coordinate system.

input vector.

scalar output value.

N + 1 dimensional space.

hypersurface is the value of the system output.

surface i s a hyperplane, the system output being a l i nea r combination of the

input sample values.

format ion hypersurf ace.

The set of sample time points

The output of the system is a scalar function of the

The system performs a transformation of the input vector t o the

This transformation can be viewed as a hypersurface i n an

The projection of the t i p of the input vector t o the

For a l i nea r system the hyper-

System identification i s the determination of the trans-

Pat tern recognition i s closely related t o system ident i f icat ion. A pattern

i s characterized by a set of pattern features, such t h a t the pattern can be viewed

as a point i n an N dimensional feature space. Each pattern belongs t o a specific

1

category. The problem is t o sort the patterns in to their proper categories.

i s done by observing a t ra ining set of patterns, where the correct category f o r

each pattern i s known.

must correctly categorize test patterns whose correct category i s unknown.

This

After observing the training set, the pattern recognizer

Pattern recognition is usually performed by determining a s e t of surfaces,

or discriminant functions, which separate the t ra ining set in to t h e l r correct

categories. These surfaces are determined by first assuming a general form for

the surfaces and then i t e r a t ive ly adjusting these surfaces after observing each

member of the t ra ining set. 3 This i s called "nonparametric" training.

The analogy between Pattern Recognition and System Ident i f icat ion i s clear.

Both require the determination of a hypersurface.

is viewed as a sequence of patterns, then the w e l l developed techniques of

pattern recognition can be applied.

transformation and the error-correcting t ra ining algorithms are used t o determine

the specific hypersurface.

this viewpoint naturally leads t o the use of techniques from a seemingly unrelated

f ie ld .

If the input data t o the system

A general form I s asswd f o r the system

Note that it is the viewpoint which i s important, as

4 Volterra Series Representation

Consider the linear system of F i g . (1). The output y( t ) i s given by the

convolution integral

y ( t ) =JR.h(T) x ( t - T ) d.7" - w

C l e a r l y ,

2

then a bounded input t o the system produces a bounded output.

cal led "stable".

Such a system i s

The systems that w i l l be considered here are those systems

which are called "stable".

N e x t , examine the nonlinear system of F i g . (2). Since

z ( t ) = Y 2 ( t >

the output of the system is given by

then

The two dimensional kernel h&T', T2) i s called a "regular homogeneous"

functional of second degree. This kernel is "realizable" i f

f o r either Tl or r2 L 0 h2('1, 72) = 0

* . and "stable" if

(4)

3

Those functionals w i t h realizable kernels are called vol te r ra kernels.

Kernels of t h i s type play an important role in the a n a l y ~ i s ~ ' ~ and synthesis 738

of nonlinear systems.

Next, consider the case where the nonlinear block is an a rb i t ra ry continuous

f'unction.

The f'unction f ( y ) can be approximated* by a f i n i t e sequence of polynomials N

i=1

Consequently %(t) can be expressed a s

+ . . . . . . . . . . .

or N

where

_ _ *

The Weierstrass theorem assures tha t a sequence of polynomials ex i s t which con-

verge everywhere t o f (y) .

the mean. Thus, discontinuous nod inea r i t i e s are excluded.

For bounded m c t i o n s , t h i s implies convergence i n

4

If f (y) i s analytic i n a given region, then f ( y ) can be expanded i n a

p e r series

and w

z ( t ) = bi hi(T1 ,..., Ti) x(t-Tl) ... x ( t - 5 ) dTl ... dTi (16) i=l -OD -00

Since the power ser ies (15) will converge for all

the functional power series will converge for all

Systems which can be represented by a functional power series with a nonzero

radius of convergence are cal led “analytic systems”.9 Although the l i m i t of (13)

and expression (16) are the same i n the region of convergence of the functional

power series,expression (16) i s res t r ic ted i n i t s range of val idi ty .

If the system w a s time varying, then Eq. (13) would be extended t o the more

general expression

The systems which will be investigated are the c lass of nonlinear systems

whose output depends t o an a rb i t r a r i l y smaJJ. extent on the remote past .

words, f i n i t e s e t t l i ng time systems.

with f i n i t e s e t t l i n g t i m e could be characterized by a l i nea r network which charac-

t e r i zed the input past, followedby a zero memory nonlinearity.

I n other

Wiener’’ shared t h a t any nonlinear system

Dr. Wiener used a

5

Laguerre network t o produce an orthQgonal representation of the input past, then

followed this network with a set of Hermite p o l y n d a l s which represented the

zero memory nonlinearity. This cascade of two operations i s essent ia l ly a specif ic

form of the functional approach of volterra.

functionals depend on the values of a real function over a f i n i t e interval.

In t h i s case the values of the

"he

functions are continuous and square integrable over a f i n i t e interval.

approach was also studied by Cameron and Martin.

This

ll

Consider the case where the representation of the input past consists of a

set of sample values. Thus

- X(t) = input vector = col [%(t) x;?(t) ..... xn(tjJ

xi( t ) = x( t .. (i-1)T)

T = sampling interval

nT = se t t l i ng t i m e of system

Furthermore, the input w i l l be assumed t o be piecewise constant

This type of input i s inherent i n a d i g i t a l computer controlled system.

12 Under these assumptions, Eq. (13) becomes

where

6

(ki+l)T f o r k i e m

t f o r ki = m M={

If Eq. (21) i s expanded, taking in to account the symmetry of the kernels,

then the form of the transformation surface is seen.

n

(23) ( t ) xi 0 . . x + % * e c H i . . . q 9

f . . . . . . i=1 9= -

N S U W

Note that If N = 1 ( l inea r system) the t r a n s f o m t i o n surface is a hyper-

plane, f o r N = 2 a quadric surface, o r i n general, an ~ t h order pol~nomial type

surf ace.

$ Learning Machine13

The term " 9 learning machine" refers t o the generic form of a pat tern

recognition device.

The first operation is a transformation of the input vector (pat tern)

vector - F i n 9 space. Vector 2 is a set of l i n e a r u independent functions

fi(2). The coordinates i n

specific examples of 3 functions Bpe: (g has d dimensions)

The general block diagram of t h i s device i s shown i n F i g . (3).

,X i n to a

space a re a s e t of functions which span the space

1. Linea r functions fi(g) = xi i = 1, ... d

m 2. Quadric functions fi(3) has the form sn xi

k,R = l,...,d and n , m = O and 1

7

3. r t h order polynomial functions: f i (X) has the form -

f o r r

1, ..., a and 1, 5, ..., n = 0 and 1 r

If the or iginal vector g was defined i n a d dimensional space, the vector

F(5) = If,(z), f2(s), ..., fM(5)) i s defined i n an M dimensional space where

M = ( ' l r ) - 1 r = order of polynomial (24)

The second operation is a l i nea r summation of the functions fi(_X). The

function

i=1

represents a hyperplane i n space, and an rth order polynomial surface i n the

or iginal 2 space.

I n the transformed space, or h space, the separating surface $(,X) i s

adjustable by an i t e r a t ive e r ror correcting algorithm.

transformation t o a nonlinear space considerably eases the conceptual and com-

Consequently the use of a

putational d i f f i cu l t i e s i n achieving a given separating hypersurface i n the

original. l inear space.

regression,13 where a least squares f i t t o a given surface i s achieved.

The general procedure i s qui te similar t o tha t of multiple

Equivalence of Volterra Series and 5 Machine

The equivalence between a vol te r ra series expansion f o r a nonlinear system

and a 9 learning machine w i l l now be demonstrated. The case that w i l l be con-

sidered i s the time invariant expansion (13).

8

n

x. ... x 9

? + H i . . .q 1

i=l q=

This represents an Nth order polynomial hypersurface i n - X space. On the

other hand, this i s exactly the same type of surface implemented by an Nth order

f machine

d d d d d

Therefore, the techniques used i n determining the separating surfaces f o r

pat tern recognition can be d i rec t ly applied t o the problem of determining the

transformation surface f o r nonlinear systems. This is indeed a useful analogy,

as the techniques f o r pattern recognition are well developed.

Training Procedure

The training procedure uses an error-correcting algorithm t o t r a i n the

$-machine.

portion of the $-machine based upon the no& operating record of the system.14

This algorithm i te ra t ive ly adjusts the weight vector of the l inear

The algorithm uses the following nomenclature:

= output of the system at the ith iterat ion. Y i

X. = input vector t o the system a t the ith iterat ion.

z = $(Xi) = output of 3-machine at the ith iterat ion.

W. = weight vector of 5 -machine a t the ith iterat ion.

-1

i

-1

9

gi =

11 gill = ziT zi (squared Euclidian nom).

= output of 9 -processor at the i t h i t e r a t ion

a = convergence factor of algorithm.

"he sequences of steps used with the algorithm i s as follows:

1.

2. Determine yi and &. 3. Generate zi and zi.

Set the i n i t i a l weight vector. A zero weight vector i s adequate.

4. Calculate new weight vector using the following error-correcting

-or ithm .3 A'+

where 0 C CY 4 2 .

Repeat starting with s tep 2. 5.

This procedure generates a sequences of W ' s whose components w u 1 con- -i

verge t o the kernels of the Volterra representation of the system under cer ta in

conditions.

The system sham i n Fig. (4) w f f l be used t o i l l u s t r a t e t h i s algorithm. The

input-output relationship f o r this system i s

The correct weight vector i s co l (lTr 17). Assume an input sequence such as

10

.

N 1 0 1 2 3 4 5 6 7 8

x[NT]lo 1 1 1 -4 2 -1 0 -1

Table 1 lists the results a t each i terat ion.

This sequence converges t o the correct weight vector.

The sequence of weight vectors

i s plot ted i n Fig. ( 5 ) .

The weight vector e r ror i s the difference between the correct weight vector and

the $-machine weight vector. The sequence of weight vector e r rors i s also

plot ted i n Fig. ( 5 ) .

Wi+l -&, is the projection of the weight vector error, si, on the input vector

Xi. The weight vector e r ror

w i l l remain the same, if two successive input vectors are l inear dependent. The

input vectors f o r i = 2, 3 and f o r i = 5, 6 are examples of this.

The difference between two successive weight vectors,

Therefore the weight vector e r ro r cannot increase.

Convergence

Consider the system of Fig. (6). The output of the system i s corrupted by

additive noise N. Ex i s the weight vector whose components are the kernels of

Eq. (26). Equation (26) can be writ ten as:

For systems that are approximated by Eq. (26) the truncated terms of their

Volerra representations are combined With the additive noise N. Therefore Ni

is the additive noise corresponding t o zi.

The $ -machine learns the mean of the noise along with the &ea@ state

value of the process. Thus, the noise Ni has zero mean but otherwise a rb i t ra ry t

character is t ics .

T Y i = 5

Then,

F + N i -i

11

Now the convergence properties of the system i n Fig. (4) w i l l be studied.

Let Ei be the weight vector error .

-1 E. = % - -1 W. ( 3 3 )

Define convergence of the learning machine as the norm of

as i increases without bound. From Eqs. ( 2 8 ) , ( 2 9 ) , ( 3 2 ) and ( 3 3 )

E. approaching zero -1

a Ni a E -i+l -1 jIEiII -1 - =E.-- F. FIT Ei - - F. II Eiil -= Let

ana i

a N b . =-

II-FiIl Then

E = Ai Ei - bi Ei -i+l

Taking the norm of E gives -i+l

F T Ai E . =zi T Ei - - a iI x i I \ -i -1 F.T F F E

-1 -i -i -i

12

.

Separating the terms containing N gives i

( 3 6 ) L e t

where 0 L- c . 6 1 since O < a < 2. L e t 1

Then

Note that

i

j =1 OGT ( l - c j ) s 1 since 0 6 c i s 1

Consider the case where Ni = 0 for all i.

Eliminate the poss ib i l i ty t ha t

and c equaling one implies immediate convergence, IIE. I ! = 0. Thus, one has

ci

i -1+1

ever equals one because it is quite improbable

only t o consider c less than one. Then a necessary and suff ic ient condition i

f o r convergence of the limit of Eq. (41) t o zero as i increases is tha t the sum

of the ci diverges.

c j = c o

j =1

13

Therefore, the i n p t vector must probe the input vector space so that an

inf in i ty of ci are nonzero. Also, the ci cannot approach zero too quickly.

Thus, the sequence of Fits must probe i t s vector space i n all directions

in f in i t l y often i n the t ra ining sequence.

must probe i t s vector space i n both magnitude and direction.

Therefore, the sequence of' 3 ' s

Consider the case where the noise Ni i s nonzero and assume t h a t condition

(42) i s sat isf ied. If Ni does not approach zero as i increases the e r ror

vector norm cannot approach zero. - L e t 11 Ei ( 1 be the conditional expected value of the e r ror norm given the

sequence of I&'s. only di depends on Ni i n Eq. (39). The conditional ex-

pected value of di i s

This exists assuming Fi # 0 and that N:, the variance of the noise, exists.

Taking the conditional expected value of Eq. (39) gives

- The %-machine is converging if IIEill decreases as i increases otherwise

it i s diverging. Assume that is bounded fo r a l l i and that there are only

a f i n i t e number of ci l e s s than a preset posit ive constant.

- -

fo r N - 1 values of i.

14

I

.- A bound on \\Ei\) i s found by operating on Eq. (45).

Taking the l i m i t as 1 3 - This is a very conservative bound.

of the noise fo r stationary noise characterist ics.

decrease as

However, it is proportional t o the variance

Therefore the e r ror norm w i l l

i increases i f the noise variance is not too large,

Tests - The ident i f icat ion method was studied by s w a t i o n on an IBM 360 Model 50.

All noise signals were generated by pseudo-random number generators.

system shown i n Fig. (7) was used i n t e s t s of the ident i f icat ion procedure.

input, x(t), used w a s correlated guassian noise passed through a sampler and a

zero order hold.

sampling interval.

The cubic

The

The 9 -machine was trained t o the average output over the

The sampling interval used f o r aJJ. t e s t s was 1 sec.

Polynomial terms of greater than th i rd order are not needed f o r the system

shown in Fig. (7).

input.

noise w a s assumed t o have zero mean.

This system also has a n u l l steady state output f o r a nu l l

"he settling tFme of the system was approximated as 10 sec. The additive

Thus, the weight vector has 285 components.

15

The correct weight vector, &, w a s determined by using special input sequences.

These input sequences were chosen so that the corresponding F. 's w a d form a

l inear ly independent set . The corresponding outputs of the system were used t o

form the following equations.

Y i - W T F . -4 -1

-1

i = 1, 285

These equations were solved f o r &. Two types of t e s t s were made. In the f i r s t type the$-machine w a s t ra ined

t o the average output over the sampling interval of the cubic system. In the

second type the $-machine w a s trained t o the output of the system shown i n Fig. (6).

The L used W ~ G the one f o r the cubic system. The Ni was sampled uncorrelated

gaussian noise with zero mean. Only the variance of the input w a s varied i n both

t e s t s . Variances of ten, one, and one-tenth were used. The input noise had zero

mean and exponential correlation, = 0.707.

The following er ror measures were used. e

The normalized weight e r ror w a s used

t o judge the extent of convergence of the $-machine. The normalized weight error

i s given by

The R.M.S. error between system output and $-machine oiltput

the performance of the &-machine as a model for the system.

i s given by

16

was used t o i l l u s t r a t e

This e r ror measure

I

I'

Graphs of the normalized weight e r ror verses the number of i t e ra t ions are

i n Fig. (8) f o r the cases where the input variance w a s ten. The ident i f icat ion

procedure diverged f o r the second type of t e s t when additive uncorrelated noise

with a variance of f i f t y was used.

f i rs t type of t e s t where the additive noise was correlated and had a variance of

176.

i f the noise i s correlated.

However, t h i s procedure convergedfor the

Thus, the ident i f icat ion procedure can withstand greater noise variances,

The results from 1000 i te ra t ions are i n Tables 2-4. i s an approximate - h i exponential convergence rate. An exponential function, e , w a s f i t t e d t o the

curve of normalized weight e r ror verses the number of i t e ra t ions f o r the last

for ty i te ra t ions .

the ident i f icat ion procedure did not obtain any s ignif icant amount of convergence.

The r m s e r ror did not decrease significantly i n these cases. Thus, the input

mst not be t o small. The amount of decrease i n r m s e r ror is well correlated

w i t h the amount of convergence.

a constant weight vector after a large i n i t i a l error.

are all the same order of magnitude for the cases where the convergence was good.

In the cases where the variance of the input w a s one-tenth,

The cases that did not converge seem t o approach

The convergence r a t e s

Conclusions

This paper has presented a new method f o r system identification. Based upon

the relaxation technique used i n pat tern recognition, t h i s method of ident i f icat ion

produces the hypersurface of the input-output transformation.

the describing hypersMace is equivalent t o a vol terra series representation of

the system and that the ident i f icat ion technique produces the kernels of the

vol t erra representat ion.

It is shown that

17

The relaxation (or error-correcting) technique will converge to the correct

Any final error in solution, even when the measurements are corrupted by noise.

the weight vectors (constants of the hypersurface) is shown to be bounded by a

constant which is directly proportional to the variance of the measurement noise.

In addition, experimental results have shown that the range of acceptable noise

variance under which the system will converge can be greatly increased if the

noise is correlated.

support these conclusions.

Acknowledgement

Both theoretical and experimental results are presented to

This study was supported by the National Aeronautics and Space Administration

under Research Grant NGR-33-018-014.

References

1.

2 .

3.

4.

5.

6.

Roy, R. and R. W. Miller, Nonlinear Process Identification Using Decision

Theory, IEEF: Transactions on Automatic Control, Oct. 1964, Vol. AC-9, No. 4,

PP. 538-5400

Roy, R. and C. Schley, Process Identification Using a Mode Learning Machine,

Proceedings of the 7th Joint Automatic Control Conference, Univ. of Washington,

Seattle, Washington, Aut. 17-19, 1966, pp. 639-648.

Nilsson, N. J., Learning Machines, McGraw-Hill Book Co., N.Y., 1965.

Volterra, V., Theory of Functionals and of Integral and Integro-Differential

Equations, Dover Publicat ions, New York ( 1959).

Smets, H. B., Analysis and Synthesis of Nonlinear Systems, IRE Transactions

Professional Group on Circuit Theory, (3-7, No. 4, pp. 459-469, Dec. 1960.

Barrett, J. F., The Use of Functionals in the Analysis of Nonlinear Physical

Systems, Statistical Advisory Unit, Report No. 1/57, Ministry of Supply,

Great Britain, 1957.

18

7. VanTrees, H. L., Synthesis of Optimum Nonlinear Control Systems, The M.I.T.

Press, Cambridge, Mass., 1962.

8. Shen, D. W. C., A r t i f i c i a l Intelligence i n a Hierachy of Nonlinear Systems,

IEEE Special Publication S - l k Art i f ic ia l Intelligence, Jan. 1963, pp. 18-30.

9. Bri l l iant , M. B., Theory of the Analysis of Nonlinear Systems, Technical

Report 345, Research Laboratory of Electronics, KIT, March 1958.

10. Wiener, N., Nonlinear Problems i n Random Theory, The Technology Press of M.I.T.,

Cambridge, Mass., and John Wiley and Sons, New York, 1949.

ll. Cameron, R. H., and Martin, W. T., The Orthogonal Development of Nonlinear

F'unctionals i n Series of Fourier-Hemite Functionals, Annals of Mathematics,

48, No. 2, pp. 385-389, April 1947.

12. Zaborsky, J., and Humphrey, W., Control Without Model or Plant Identification,

Proceedings of the 5th Joint Automatic Control Conference, Stanford University,

Stanford, California, 1964, pp. 361-366.

13. Ralston, A., and W i l f , H., Mathematical Methods f o r Digital Computers,

"Multiple Regression Analysis," by M. A. Wroymson, John Wiley and Sons, Inc.,

1960, pp. 191-203.

14. Nagumo, J., and No&, A., A Learning Method f o r System Identification,

Proceedings of the 1966 National Electronics Conference, Vol. 22, pp. 584-589.

19

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Variance of additive noise

1 IJ&.)oI I

x x 1000

* Divergent case

F i r s t

.176

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22

. *

TABU 4

Input Noise Variance = 0.1

Type of test F i r s t

Variance of additive noise ,000176

e(1, 40) .00036

e(960, 40) ,00034

e( l , 1000) .00050

' ' ~ 1 0 0 0 ~ ~ -9571

Ax 1000 1 9 557

Second Second Second Second

0.0 .001 .005 . 01

.00043 .00173 ,00820 ,01634

.00038 .00161 .00809 .01622

.00057 .00159 .00756 .01512

-9492 -9725 1.276 1.901

.188 - -352 * *

* Divergent case

23

FIGURE I

FIGURE 2

24

X. I

r --- -- --- - -- 1

I I I

@-PROCESSOR LINEAR PROCESSOR

a- MACHINE

FIGURE 3

25

FIGURE 4

26

w2

16

12

8

4

- 0 I I 1 I I

4 8 12 16 20 24 28

, wI

28

WEIGHT VECTOR SEQUENCE

16

12

8

4

-8 -4 0

/ E -5

4 8 12 16

WEIGHT VECTOR ERROR SEQUENCE

FIGURE 5

27

FIGURE 6

CUBIC SYSTEM

FIGURE 7

28

0 0 0

0 0 0

3 3 r)

0 0 pc

0 0 (D

9

0 0 In

0 0 e

0 0 m

0 0 (u

0 0 -

0

NASA-Langley, 1967 - 10 CR-756 29

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