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77

8

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-30, NO. 8, AUGUST 1985 By (lo), the right side of (1 1) contains the term

Since

&

is symmetric

where

Hence, applying the matrix adjoint result we obtain

where

l i k = f & + I * k + u k ( k = n - I ,

’.‘,

1)

A, = 0. (15)

From (8), (1 l), and (14), the derivative

aJ/aej

can be written as

where, for convenience,

the and Ak’s being given by (12) and (15), respectively.

Iv. SUMMARY

A new expression has been obtained for the derivatives of the prediction error cost function for time-varying linear dynamic systems. The main advantage of this new expression is that it requires fewer computations to obtain the gradient than does the straightforward approach via sensitivity equations.

ACKNOWLEDGMENT

The author wishes to thank Dr. I. L. Maryak of the Applied Physics Laboratory for several useful suggestions.

REFERENCES

P. E. Caines, “Prediction error identification methods for stationary stochastic processes,” IEEE Trans. Automat. Contr., rol. AC-21, pp. 500-505, Aug. 1976.

B. D. 0. Anderson, 1. B. Moore, and R. M. Hawkes, “Model approximations via prediction error identification.” Automatica, vol. 14. pp. 615-622, Nov. 1978. L. Ljung and A. J . M. Van Overbeek, “Validation of approximate models obtained from prediction error identification,” in Pror. Int. Fed. Automat. Contr. 7th T r i m World Cong., Helsinki. Finland. London, England: Perga- K. J. Astrom, “Maximum likelihood and prediction error methods,” Auto-

mon, 1978, pp. 1899-1906.

T. Stiderstram and P. Stoica. “On criterion selection and noise model parametriza-

matica, vol. 16, pp. 551-574. 1980.

tion for prediction error identification methods,” Int. J. Conrr., vol. 34, pp. 801- 811, Oct. 1981.

N. R. Sandell, Jr., and K. I. Yared, “Maximum likelihood identification of state space models for linear dynamic systems,” Electron. Syst. Lab., M.I.T.,

G . A. Van Zee, “Gradient computation in prediction error identification of linear Cambridge, MA, Rep. ESL-R-814, 1978.

discrete-time systems,” IEEE Trans. Automat. Contr., vol. AC-27, pp. 738- 739, June 1982.

D. A. Wilson and A. Kumar, “Derivative computations for the log-likelihood function,” IEEE Trans. Automat. Contr., vol. AC-27, pp. 230-232, Feb. 1982.

On

the Identification of Polynomial Input-Output

Differential Systems

A.

E. PEARSON

AND

F. C.

LEE

Abstract-A least-squares parameter identification technique is formu- lated for a class of deterministic nonlinear systems modeled by polyno- mial input-output differential equations. The basis of the technique is Shinbrot’s method of moment functionals using trigonometric modulat- ing functions. Given the input-output data over a single finite time interval for a one-shot estimate, or over a sequence of finite time intervals for sequential least squares, the underlying computations utilize a fast fourier transform algorithm on polynomials of the data without the need for estimating unknown initial or boundary conditions at the start of each finite time interval.

I. INTRODUCTION

The parameter identification of deterministic nonlinear systems modeled by polynomial type differential equations can be undertaken by the Bellman-Kalaba quasi-linearization technique [I], [2], or by finite- dimensional hiU climbing techniques after approximating all signals with

by Past Associate Editor, B. Friedlander. This work was supported in part by the Manuscript received March 15, 1984; revised November 8, 1984. Paper recommended National Science Foundation under Grant ECS-8111219 and in part by the A i r Force o f f i c e of Scientific Research under Grant AFOSR-824230.

02912.

A. E. Pearson is with the Division of Engineering. Brown University. Providence, RI

University, Hsinchu, Taiwan, Republic of China.

F. C. Lee is with the Institute of Control Engineering, National Chiao Tung

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E E E TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-30, NO. 8, AUGUST 1985

779

piecewise constant Walsh-type orthogonal functions [3]. The former entails the iterative solution to a nonlinear two point boundary value problem while the latter approach gives up the differential equation model in the very beginning. In both approaches unknown initial conditions have to be estimated along with the system parameters for time limited data. A different approach is taken here for a restricted class I of polynomial

differential systems which retains the continuous-time format while avoiding the necessity to deal with all unknown boundary (initial) conditions for input-output data collected over a fued finite time interval, or over a sequence of finite time intervals [t,, l ; + J 3 each of duration T. This is accomplished by projecting the data down into a suitable subspace via Shinbrot's "method of moment functionals" [5] using finite sums of commensurable sinusoids as the modulating functions. As developed by the authors in [6] for linear systems, the underlying computations entail calculating a fixed number of Fourier series coefficients for the data on each [0,

TJ

interval. It is known that these coefficients can be determined with a high degree of accuracy using the discrete Fourier transform (DFT)

which, in turn, can be evaluated very efficiently using the fast Fourier transform

(FFT)

algorithm. These computational aspects, together with the results of a simulated example, will be discussed following the formulation of the problem.

II. FOILMULATION

Let [u(t), y(t)] denote an input-output pair for a single input, single output system which is observed free of significant measurement noise over a fixed finite time interval [0,

a:O

5 t 5 T , or over a sequence of time intervals [ti, t, ,

,], i

= 0, 1

. .

, each of duration T . It is assumed that the data are bounded and piecewise continuous on every finite time interval. Letp denote the differential operator d/dt so thatp2 = d 2 / d t 2 , etc. Corresponding to opriori integers

(n,

rn) the class of models relating

u(t) and y ( t ) is defined by the polynomial input-output differential operator equation

p " y ( t ) + C o , ( j , k ) p n ~ ' t u ( i ) l ' l v ( t ) l k = ~

n m m

i = l j = o k = O

O l t S T , U,(O, O)=O, i = l

. - .

n. (1) The

ci(j,

k ) represent parameters-a total of

rnn(rn

+

2) in the general case-which are to be determined by a least-squares technique without the need for estimating unknown initial conditions at the start of each [ t i , ti+

interval. 7

their components

Defining the column n-vector parameters {a,

8.

~ ( j ) , v(k), ~ ( j , k ) } by

w,=a,(O, 1) q ; ( j ) = u i ( j , O), 2 ~ j s r n B;=u,(l, 0) uj(k)=a,(O, k), 2 5 k 5 m

T U , k ) = a , ( j , k), 1 sj, k s m

i = l , 2,

..-,

n , (2) it can be readily verified that a set of state equations equivalent to the model (1) is specified by i =

[

- a : ! x ! ] x - 0 u - i v ( j ) [ u ] l - i u(k)[xllk 1 = 2 4 = 2 r n m

- X

r(j.

W U I J t X l l k (3) J = I k = l

in survey articles such as Haber and Kevinky [4].

'

The various models commonly used in nonlinear systems identification can be found

'

An alternative projection method w,hich could be brought to bear on the same class of models is the "projected integral equation error" technique presented in 171. However, it is believed that the use of the F'FT algorithm makes the approach of this paper computationally superior to the formulation in (71.

are such that

'

( I ) possesses bounded solutions over the time intervals of interest. Le.. no It will be tacitly assumed that any candidate values assigned to the parameters a,O, k) finite escape times.

where y = x1 is the first component of the column n-vector x. Within the context of analytic differential systems of the form

2 = f ( x , u )

Y = &x)

wherefand

g

are polynomial functions of their arguments, the model (3) is seen to represent the special class wherein the output function g is linear in x, Le., &x) = c ' x for some row vector c ' % and the nonlinear t e r n

inf

are polynomials in the scalar pair (c'x, u). It is easy to specialize certain well-known models to the above form such as the Duffing and Van der Pol

equations.

Shinbrot's method of moment functionals is a classical technique for converting a differential equation to an algebraic equation in the parameters by the use of so-called modulating functions. As introduced by Shinbrot 151,

4(t)

is a modulating function of order

n

relative to a fmed time interval 10, T ] if it is sufficiently smooth and satisfies the end point conditions

Q ("(0) = Q q T ) = 0

i = o , 1,

...,

(n-1) (4)

where &(o((t) meansp'Q(t). The significance of this property for the model (1) relates to the fact that if (1) is multiplied by

4(t)

and integrated over

[O,

T ] , the result is the functional equation ( - 1)" s'y(t)d(")(t) dt

+

2

5

9

u,(j,

k)(-

l y '

ST

[ u ( t ) ] ~ L v ( t ) ) ] ~ ~ ' ~ - ~ ) ( t ) dt=O. ( 5 ) < = I j = O k = O

This follows using integration by parts

n

times and noting the end point conditions

(4).

Moreover, if ( & ( t ) } , i = 1, 2,

. .

X,

is a set of linearly independent modulating functions, a vector algebraic equation results which can be used to obtain a least-squares estimate of the parameters.

As pointed out in [6] the above idea has been pursued by several investigators using a variety of modulating functions such as Hermite polynomials and splines. However, the computational burden associated with these functionals on the data will generally be significant unless a "fast algorithm'' is available. Such is the case for modulating functions comprised of linear combinations of commensurable sinusoids since integrals like

J:

[ u ( t ) ] J ~ ( t ) ] k [sin

/wet

or cos ~ , t l dt

""7

I I=O, 1, 2,

..-,

L

2?r

can be efficiently evaluated by an FFT algorithm especially for large L . With this in mind, letflt) denote the (2L

+

1) column vector of sinusoids defined by

f ( t ) = col [I; cos ad, sin wet; cos h o t , sin 2w0f;

..

.,

cos Lwot, sin Loor]

21r

O S t l T , oo=-

T (6)

where the integer L must be chosen at least as large as n / 2 in order to satisfy (4). Roughly speaking, the pair ( L , T) will be selected by the user so that 2L is somewhat larger than the system model order n and T is sufficiently long to assure that w,, = 2 r / T is small enough to resolve the spectral components of the

data.

This will be discussed further in the next section.

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7 80 E E E TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-30. NO. 8, AUGUST 1985 -

n)

X ( 2 L

+

1 ) matrix constructed such that

a(f)

defined by4

@ ( t ) = C f ( t ) , O S t S T (7) satisfies the end point conditions

W ~ ( O ) = @ ( ' ~ ( T ) = o , i = O , 1 ,

...

(n-1). (8) Thus, @ ( t )

is

a (2L

+

1 -

n)

dimensional vector valued modulating function of order

n

in which the derivatives p i @ ( t ) =

a("(f)

have the representation [cf. (6)]

( - l)'p'@(t)= C D l f ( f ) , i=O, I

. . .

(9) where D is the block diagonal matrix defined by

and D o is defined as the identity matrix. Then multiplying the model equation ( 1 ) with @(f), integrating over [0, T ] , and using integration-by- parts

n

times while noting ( 8 ) , there results the vector analog of (5):

where

20'.

k)

and Yare (finite) Fourier coefficient vectors defined by Z ( j , k ) =

IT

[u(t)]JlY(f)]Zf(r) d f , O s j , k s m

Y=Z(O, 1). (12)

Equation (1 1) can be written in the standard form for least squares by defining the column block partitioned parameter vector 0 [cf. (2)]

e=

col

Iff,

0,

7(2)

...

dm), u(2)

.

' . u(m), y(1, 1)

.. .

r(m,

m)]

(13)

and the coefficient Mjk (each of dimension (2L

+

1) x

n)

according to the partitions

M,P=[D"-'Z(J, k), D"-'Z(j, k)

...

Z ( j , k)] (14) such that the product MO results in the vector sum [cf. ( 2 ) and

(3)]

m m m r n

/ = 2 k = 2 , = I P = I

(1 5 )

That is to say, matrix M is partitioned into row blocks

M=

ROW

[ M o l , M I , , Ma

. . .

M,,, A402

.

.

Mom, M I L

. .

M,,,,] (16)

comformably with the partitioning in 0 such that (15) holds. With these definitions, ( 1 1 ) is equivalent to

CD"Y+CMB=O. (17)

- n ) Vandenonde type matrix equations. Further details can be found in [ll]. It can be shown that C has full rank and that determining C involves solving (2L

+

1

The normal equation for (17) is given by

M'C'CMO=M'C'CD"Y (18)

and a unique solution for a one-shot least-squares estimate of 6 is obtained if and only if the Gram matrix M'C'CM has full rank. Although the matrix C has full rank (2L

+

1 -

n),

it is not enough to assert that a unique solution to (18) exists if M h a s

full

rank. Hence, the uniqueness of the one-shot least-squares estimate is predicated on the condition that CM has full rank equal to

mn(m

+

2). In turn, this implies the inequality ( 2 L

+

I

-

n)

2

mn(m

+

2) since C has dimension (2L

+

1

-

n)

x ( 2 L

+

1). Therefore, the one-shot least-squares estimate has the potential for being well posed only if 2L 2

[mn(m

+

2)

+

n - I ] . This supercedes the basic inequality 2L 2

n

which arises from the end point conditions (8).

The above equations pertain to data observed over a single [0, T ] time interval.

In

the case of data observed over sequential time intervals [ti, ti- i = 0, 1

. . .

,

each of duration T, (17) is replaced by

CDnY(i)

+

CM(i)O = 0 (19)

where Y(fi and M(z7 are computed from (12) and (14) for the data collected over each T-interval. Standard sequential least-squares theory for deterministic discrete models can then be applied to (19) in constructing a recursive solution 0(i).

m.

SOME COMPUTATIONAL CONSIDERATIONS A. Choice of (L, T)

Borrowing from frequency domain ideas for linear systems, the choice of the time interval [0,

TJ

can be based on the heuristic notion that since wo

= 29/T is essentially the resolving frequency, T should be large enough to distinguish the characteristic modes of the system. Likewise, the selection of the ratio L I T can be guided by the consideration that Lw0 = L 2 d T be comparable to the system bandwidth. Although "characteristic mode" and "system bandwidth" are not well-defined concepts for nonlinear systems, their intuitive meanings

bear

some relevance to this

discussion. Also, since the modulating functions act as a Nter on the data through the finite Fourier series coefficients, L should not be too large as to give undue emphasis to whatever high-frequency measurement noise might be present in the data.

B. Computing

20,

k) Via FFT Techniques

The major computational burden in setting up the least-squares identification of ( 1 ) will

be

the determination of the Z u , k ) in (12) at each stage. Let z ( f ) denote a typical function from the

m(m

+

1) set of functions { [ u ( t ) ] J L ~ ( t ) ] ~ , 0 5 j , k 5 m } , 0 5 t

s

T. Then (6) and (12) imply determining the following integrals (complex form) for each ~ ( t ) :

~ : z ( f ) e ' f e o ' dr, I = O , 1

. .

.

L . (20) Although the real and imaginary parts of the above integral can be evaluated by passing

z(t)

through a bank of appropriately tuned harmonic oscillators [ 6 ] , greater flexiblity is offered by using well-known digital approximations. Thus, for example. if uniform sampling of z(t) is used to generate N samples

zi

= z(ih), h = TIN,

i

= 0, 1,

.

. ( N - l), the

standard parabolic rule yields

. r

N- I 1

practical problem, such as initialization and roundoff errors, which attend all sequential

See, for example, Mendel [8] for convergence theorem and a discussion of various

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-30, NO. 8, AUGUST 1985

78

1

where W = eJ2r’N and

o(

e) is the order of the error as a function of the

sampling interval h. Assuming N i s a power of 2, the usual FFT algorithm can be used to evaluate the DFT of the quantity in brackets on the right- hand side of the above approximation yielding the Fourier coefficients for

I

= 0, 1

---(A’

- l), i.e.,

h

3

Z = - FFT [(&+Z,v), 421, 222. 423,

2z4

...

4 Z ~ 7 - 1 1 . (22)

The computational savings of this algorithm for large N are well known; specifically, log, N / N . However, a special FFT-type algorithm can be devised in consideration of the fact that only L Fourier coefficients are needed in the computation and the efficiency of such an algorithm is log,

L/L.

As

pointed out by a reviewer, this represents a kind of FFT “pruning” discussed in Markel [9].

C. A Simulated Example

The following bilinear control system model for nuclear fission is discussed in Mohler [lo, Section 4.11

@=(u-P)y+IhC

It

= By - I

xc.

Eliminating the “average precursor population” variable

at),

the preceding equations in the “neutron population“ y ( t ) become

where u(t) is the control reactivity and (A,

0,

I ) are parameters characterizing the nuclear fission process. Using the sinusoidal forcing function

u ( t ) = 2 + sin

(

f + -

+

sin 2t+-

;>

(

1)

several simulation runs were made over a [0,2r] time interval, i.e., w0 = 2rlT = 1 , for a one-shot least-squares estimate of the following parameters:

fJ,=X+-=2.0

P

I

Fig. 1 shows the output data f i t ) for noise-free conditions (a) and a run @) in which white Gaussian noise was added to the data resulting in a noise- to-signal ratio of approximately 5 percent. Notice that the system is apparently unstable for the above chosen parameter values and control reactivity.

Table I lists the estimated parameter values for the different runs using the modulating function frequencies (0, 1, 2, 3, 4}, i.e., L = 4.6 This

means

that 2L

+

1 - n = 7 algebraic equations in (17) were used to form the normal equation (1 8) for each run. No difficulties were encountered in solving the normal equation (18) for any of the runs. The two columns under each parameter estimate in the table give the results for two different discrete approximations in computing the Fourier coefficients (21) for each z(t) E

{u(r),

At),

u(r)y(t)}, i.e., the FFT orders were N = 128 and N = 256 in (22).’ The results show that good accuracy in the parameter estimates is attained under ideal noise free

combinations of the functions in the two sets {sin i f , 1 5 i 5 4) and {cos it, 0 5 i 5 4) The modulating functions themselves w’ere obtained by separately subjecting linear for t E [O, 2 4 to the end point constraints ( 4 ) . As mentioned earlier. this is an off line calculation involving the solution to Vandermonde type matrix equations yielding the matrix C in (7).

T h e IMSL Library was used to provide the integration routine (DVERK) for eenerating the ”continuous” data and as the source for an FFT algorithm to compute the

DFT’s

of the ”sampled” data.

120 , I I I I I : l ’ I I I I

L

..-.

-

80

-

A NOISE FREE O L T U T SIGNAL w 0 3 a 40 100

L

1

0 2 4 6 TPdE AXIS (b)

Fig. I . Response data for the nuclear fission example.

T A B L E I

ESTIMATED PARAMETER VALUES FOR THE NUCLEAR FISSION EXAMPLE

e,

-e,

-e,

llAell

\Order FFT 1%

I

256 128

I

256

I

256 128

I

256 128 18 21 1 . m 3. 4.7% 36.9% 0.939 0.493 0.986 0.982 1.898 2.110 1.924 0.447 0.947 1.088 3.5% 15.9% 0.9% 0.809 0.993 0.936 1.921 1.625 4. 9.4% 37.4% 1.138 0.484 0.942 1.019 2.129 1.233 3.886 6.8% 1.010

*Ram 1-4 include additive noise io the data with an R W N/S ratio 21 5%.

conditions (Run 0) using the relatively coarse DFT order

(N

= 128), but that a similar degree of accuracy could be maintained under the more realistic measurement noise case (Runs 1-4) only by increasing the order of the DFT ( N = 256). Limiting the highest modulating function frequency by choosing the value L =

4

served to filter out the higher frequencies in the white measurement noise, i.e., increasing L will decrease the estimation accuracy in the noisy case. At the same time, reducing L will result in fewer algebraic equations for the least-squares estimate. A rule of thumb is to choose L such that (2L

+

1 -

n)

is approximately double the unknown parameters for a one-shot estimate

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7 8 2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-30, NO. 8, AUGUST 1985 and to choose

T

such that L 2 r / T is comparable to the system bandwidth

(if known).

IV. CONCLUSIONS

Choosing commensurable sinusoids as modulating functions in the Shinbrot method of moment functionals, it has been shown how the least- squares identification of polynomial input-output differential systems can be formulated in a way that utilizes the computationally efficient FFT algorithm at each stage while avoiding the necessity to estimate unknown initial conditions for time limited data. In addition to the order of the system model and the number of parameters to be identified, the choice in modulating functions can be based to some extent on noise rejection and the heuristic notion of “system bandwidth.” An interesting problem for future investigation is determining an optimal set of Fourier-based modulating functions which minimizes some measure of the error in the parameter estimates for noisy measurements with specified spectral characteristics. Another problem is determining conditions on the input -

that guarantee uniqueness of the least-squares estimate. REFERENCEs

R. E. Bellman and R. E. Kalaba. Quasilinearization and Nonlinear Boundary

R. Kalaba and K. Spingarn, Control, Identification and Input Optimization. Value Problems. New York: American Elsevier. 1965.

New York: Plenum, 1982.

G. P. Rao, Piecewise Constant Orthogonal Functions and Their Application

to Systems and Control. New York: Springer-Verlag. 1983.

R. Haber and L. Keviczky, “Identification of nonlinear dynamic systems.“ in

Zdentificotion and System Parameter Estimation, Part I, N. S . Rajbman. M. Shinbrot, “On the analysis of linear and nonlinear systems,” Trans. ASME,

Ed. Amsterdam, The Netherlands: North-Holland, 1978, pp. 79-126. vol. 79, pp. 547-552, Apr. 1957.

A. E. Pearson and F. C. Lee, “Time limited identification of continuous systems using higonomevic modulating functions,” in Proc. of 3rd Yale Workshop on

Applications of Adaptive Syst., New Haven, CT, 1983. pp, 168-173. A. E. Pearson, “Nonlinear system identification with limited time data.”

Automatica, vol. 15, pp. 73-84. Jan. 1979.

J. M. Mendel, Discrete Techniques of Parameter Estimation. New York: Marcel-Dekker, 1973.

J. D. Markel, “FFT pruning,” in ZEEE Trans. Audio. Electroacoust., vol. AU- R. R. Mohler, Bilinear Control Processes. New York: Academic, 1973. 19, pp. 305-311. Dec. 1971.

F. C. Lee, “Time limited identification of continuous systems using the modulating function method,” Ph.D. dissertation, Division Eng., Brown Univ.. Providence, RI, 1984.

Nonparametric Kernel Algorithm for Recovery of

Functions from Noisy Measurements with Applications

ALEXANDER A. GEORGIEV

Abstract-This note presents a kernel algorithm for recovery of a regression function from noisy data. Conditions are provided that assure pointwise convergence in the mean square and almost sure senses. An application to a class of linear system identification problems is discussed.

I. INTRODUCTION

There are many situations in system identification when a nonlinear memoryless system g(x) is estimated from observations

(xI,

Y l ) , (x2, Y2),

. . .

,

(x,,

Y,),

where the design points

xi

are from a real interval [0, 11 a prior submission of Febtuary 28, 1983.

Wroclaw, Wruclaw, Poland.

Manuscript received December 19, 1983; revised July 30, 1984. This paper is based on The author is with the Institute of Engineering Cybernetics, Technical University of

over which observations are taken, satisfying

y i = g ( x J + & . (1)

Assume that the points xi are selected in some arbitrary fashion by the experimenter and are not necessarily equally spaced. The errors Z1,

. . .

,

Z , are mean-zero independent identically distributed random variables with finite variance d . It is interesting for many purposes to estimate the system g, for example, to predict g at a particular point x and to describe and itlustrate the relationship between the design points x, and the observations Yi.

In the finite-dimensional parameter case, when the function g is known except for a set of parameters, the least square or the maximum likelihood methods are commonly used [7], [ 161.

In this note we propose a nonparametric kernel algorithm for recovery of the function g from noisy measurements. The “nonparametric” property refers to the absence of a finitedimensional continuous parametrization of the space of functions containing g.

Assume that the input signals xi satisfy, without loss of generality, the order condition

Let x, = 0 and x, + = 1. The considered kernel algorithm for recovery of g(x) is

where K (so-called kernel) is a bounded nonnegative function on the real line and { h , } is a sequence of positive real numbers. The estimate (3) can

be regarded as the appropriate average of observations in a neighborhood of the point under consideration. For example, one might consider

where s(x) = i : Ix - xi1

<

h,}

.

The procedure

g,

is obtainable from (3)

for the window kernel

and for the xi.s equally spaced, i.e., xi -

x i - ,

= I / n .

The estimate g, was introduced by Priestley and Chao [11] and has been studied by Benedetti [ 11, Gasser and Muller [3], Schuster and Yakowitz [17], as well as Cheng and Lin [2]. Another method, the “nearest neighbor method,” for the recovery of function g has been introduced by Greblicki 191 and an orthogonal series method has been examined by Rutkowski [14].

The main object of this paper is to introduce control engineers to some techniques afforded by nonparametric methodology. In Section II, the convergence theorems are stated and new results are discussed. Applica- tions to linear dynamical system identification are given in Section III. Finally, some remarks about nonparametric identification procedures are made.

n.

THE CONVERGENCE OF THE ALGORITHM

In this section, we establish the sufficient conditions for consistency of the algorithm (3). The proofs of the following results are given in the Appendix.

Lemma (Bias):

Assume that g(x) is a bounded function and K is a continuous probability density function such that K(u) is nonincreasing for u

>

0, and nondecreasing for u

<

0. Let

數據

Fig.  1 shows the output data  f i t )   for noise-free conditions (a) and a run  @)  in which white Gaussian noise was added to the data resulting in a noise-  to-signal  ratio  of  approximately  5  percent

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