ADAPTIVE AND MINIMAX METHODS OF PREDICTING DYNAMIC SYSTEMS USING THE KALMAN ALGORITHM Текст научной статьи по специальности «Математика»

PHYSICS AND MATHEMATICS

АДАПТИВНЫЕ И МИНИМАКСНЫЕ МЕТОДЫ ПРОГНОЗИРОВАНИЯ ДИНАМИЧЕСКИХ СИСТЕМ С ИСПОЛЬЗОВАНИЕМ АЛГОРИТМА КАЛМАНА

Сидоров И.Г.

к.т.н., генеральный директор ООО "Алгол - М", Москва

ADAPTIVE AND MINIMAX METHODS OF PREDICTING DYNAMIC SYSTEMS USING THE

KALMAN ALGORITHM

Sidorov I. G.

PhD, general director of Limited Liability Company "Algol-M", Moscow

Аннотация

В статье рассматривается проблема линейной экстраполяции стационарных в широком смысле случайных процессов для непрерывного и дискретного времени в отсутствии априорной информации о статистических характеристиках возмущения, когда отсутствуют ошибки измерения в скалярном наблюдении.

Предполагается лишь, что известна верхняя оценка дисперсии возмущения. В этих условиях естественно применять минимаксный подход, имеющий наилучшее гарантированное значение критерия, то есть алгоритм обработки, имеющий самую высокую точность в указанном выше смысле при наихудших условиях работы. Простая реализация оптимального адаптивного минимаксного предиктора и предиктора по фильтру Калмана и их сравнительные характеристики представлены в работе. В этом случае мы имеем дело с комбинированным адаптивно-минимаксным методом прогнозирования. Существование седловой точки в игре экстраполяции в терминах экстремальных свойств допустимых спектральных плотностей линейного фильтра и второго игрока - природы также обсуждаются.

Abstract

In the article we consider the problem of linear extrapolation of zero-mean, wide-sense-stationary random process with discrete and continuous in time under conditions of the absence of a priori information about the statistical characteristics of perturbation when the measuring errors under scalar observation are absent. Only perturbation dispersion limitedness is assumed. The minimax approach, which guarantees the prognosis of high quality at the most unfavourable perturbation spectrum is used. The simple implementation of an optimal adaptive minimax predictors and forecast based on Kalman filter and their comparative characteristics are presented. In this case we dealing with combined adaptive-minimax prediction method. The existence of the saddle-point of extrapolation game in terms of the extreme properties of permissible spectral densities of linear filter and nature are also discussed.

Ключевые слова: минимаксный, фильтрация, линейный, экстраполяция, стационарный, седловая точка, предиктор, дисперсия

Keywords: minimax, filtering, linear, extrapolation, stationaty, saddle-point, dispersion

1. Introduction

The solution to a similar problem for continuous in time processes was obtained for the first time by U. Grenander [1-3]. Unlike the Kalman method, or a more general case of Bayes evaluation, the essence of the minimax method is that the perturbations are not considered probability-described to the full, i.e., as for instance in the Kalman model, the perturbations are defined by a noncorrelated in time process. Such a problem, as well as other more general problems may be solved using the minimax approach which guarantees the prognosis of high quality at the most unfavourable spectrum. The methods of solving similar problems which were not sufficiently or at all reflected in scientific literature, are widely represented in the given article.

2. Statement of a problem of minimax extrapolation in case of scalar observation under the absence of measuring mistakes with continuous

Let us suppose that the actual component of the measured signal y (t) was formed from a certain perturbation u(t) by means of the dynamic system:

i(t) = Ax(t) + bu(t) (1)

Here A - is the constant matrix of ^ X n dimension; - the phase vector of the condition of the system; x(t ) €= R" - the constant vector; u{i ) - the

perturbation representing a scalar stationary occasional process with zero mean value with the only information concerning its correlation function about the constraint on its dispersion

Mu\t) < a.

And, perhaps, he constraint on the concentration area of its spectral density - h (A) at Is A, A -

the given submultiplicity of the frequency axis. The measured signal

i(t) = J q(t — t)u(t)ít

y(i) = CTx(t>

(2)

where c G r" - is the constant vector. Let us make the following assumptions concerning A,b and C matrixes:

1) the meter-object system (1),(2) is the observed system, i.e.:

rang(C, ATC,..., (A"-1 )TC) = n

2) system (1) is "disguised" by the perturbation ,

i.e.:

where q (t) - is a real value mean-square integrable function with frequency characteristics Q(X).

The quality criterion is the dispersion of the evaluation error:

minmaxM [s(t)-s(t))]2 = D(G h)

(4)

The spectral density of the evaluation s(t ) is given by:

s(A) = G(A)Y (A),

(5)

x(t) = j eA(t~T)bu(t)dz

—C»

or the equivalent condition :

rangib, Ab,..., An~lb ) = n It is required to find out the process:

(3)

t

s(t) = J g(t — t)y(z)dz-

i.e., to find the transitional function g(t) of a

physically realized filter , which evaluates linear functional

The frequency characteristics of linear transformation can be expressed as :

w

G(A) = J e~iAtg)t)dt.

—w

Thus, the problem comes to finding the frequency characteristics G(A) according to minimax criterion

(4).

The problem should be interpreted as an antagonistic interplay f(d, n, I), where the gain functional is

D(G, h) with corresponding strategy spaces of the players: nature - n and the investigator - i, n - the multiplicity of permissible spectral densities h(A), I

- the multiplicity of permissible spectral densities of linear filters - G(A):

(6)

N = {h(A) G A(A)| J h(A) Y j (A)dA ^ a j, j = 1,..., m}

a G R™ is given vector ; X) E R™ is given

vector-function satisfying the Paley-Wiener condition, A - the given submultiplicity of the frequency axis of

I = {G(A) e L2 (A) | Im J emG(A)dA = 0} (7)

L (—w, +w) - the multiplicity of functions integrated according to Lebesque. together with the square of the module of functions given on the real axis.

Let (h0, G0) be the saddle point of the extrapolation game f (D,N,I) Then

the spectral density of y (t) signal from (2) may be presented by:

Xu (A) = T (A) + h(A),

positive measure ( the infinite measure is possible), mesA> 0,L(A) - the multiplicity of functions

summed up over a measurable submultiplicity A of an actual straight line according to Lebesque.

where T(A) - is the known component (T (A) > 0). Then, on the one hand, G( A) -is the Wiener filter [2]

G(A) =

Q(A)[T (A) + h(A)] X— (A)

1

X+ (A)

(8)

here the equation symbols for function a+ (A) and A_ (A) were introduced, which correspond to the separation operation of A(A) function in the lower and upper analytic semi-planes, correspondingly; A+ (A) corresponds to the operation of A(A) factorization. And on the other hand, h(A) is the solution to the Markov moments problem:

—œ

u

h (A) = max where

. [Q(A)x;(A)f u; — i (A)

<ä,x¥(X)>

fl, x eA

X (A)

(9)

aeR"

+oo

p = â-jh(À)x¥(À)dÀeR^;

(10)

Xa (A) = ■

0, x £ A

is the characteristic function of A multiplicity and a is the Lagrange factor vector, satisfying a set satisfies the equation: of simultaneous ratios:

J 1 f, r rr , r - \Q(®)K(®]-\2) Xu (A) = exp^ — lnmax[r(ffl);/A(ffl) „

2m i <a,x¥(a)>

<a,p>= 0.

Here h(A) was determined in (9) and X* (A)

da A —a

(11)

Formulas (9) - (11) could be p(A) considerably simplified if T (A) = 0, m = l, A = (—», ») and

G3 (A) = 0(A)^t(A)Xu(A)/[zXM+(A)], (17)

where

C_x C

—2 U....+, z = e"iA

Y(A) =|p(A)|2

(12)

where <p(A) is the result of ^(A) factorization. Conditions (9) - (11) are transformed into

[ X (A)]_--- + . .

z z

and [X(A)]_ does not contain a zero component in the Loran expansion.

h(A) =

|[Q(A) x; (A)]_

3. Statement of a problem of extrapolation with discrete time

Let us consider the dynamic model of a step-by-(13) step vector process:

o*F(A)

» |[Q(A)x; (A)]_ _» oW(A)

Equation (11) is transformed as follows :

I Q*(A)X_(A)]; |2

I = a.

(14)

yn = cTl>

(18)

where A„ - is the column-vector of the model

x; (a)=■

(15)

^p(A)

condition, yn - is the measured quantity; <Z)n, is the

column-vector of the excitation in the model; A - is the

transition matrix of the order T ; C is the bonding col-

The expression for frequency of the characteristics umn"vector ; n " is the time-moment.

Let us pass from vector record (18) to the scalar

one by choosing the corresponding coefficients A, C (16) and p. in case if (18) - is the observed system [4]. Let

of the extrapolation is given by:

[0(A) X (A)].

G3 (A):

XU (A)

The extrapolation error dispersion is presented by

D

l ;»

= — f | G(A) — 0(A) I2 h(A)dA = a^i2 2n L

us confine ourselves to the case of when

m

S Piyn-i = un

i=0

(19)

where u - is the scalar perturbation with limited

dispersion. In z -symbolization the latter representa-where j is the maximum positive eigenvalue tion is given by P(z)y(z) = U(z), where

corresponding to the eigenfunction

X(Ä) in the equation (15).

The expression for frequency characteristics of the filter in discrete time can be expressed as :

P(z)=S pz •

Let us label the desired evaluation Y^+n by ln •

Then

i=0

i =t gy

n ^^OlS n—l

(20) If the spectrum of u (z) excitation is given by :

i=0

In Z -representation expression (20) can be expressed as :

l ( z) = G( z) y( z).

hu P) =2 e~j0mKu„ '

where K = M (uu+ ) - are correlational moments, then the expression of the prediction error dispersion will be as follows :

1 +S h (a)

D (G=—+(7PhöF

[G(e~ja ) — ;"N ]2 dco

The given problem has a saddle point due to the fact that D (G, h ) is linear over hu and the set of all h (®) with restricted integral dispersion

1

D — I h (rn)drn 2n —n

is convex weak compact and D (G, h ) is

square over G(e~ }C° ), i.e. the conditions of convexity-

concavity which are obligatory when the well-known theorems from the game theory are used [5], are met here

minmax dn (G,hu) = maxmin dn (G,hu) (22)

g h.

g

Minimum (21) gives Wiener - Kolmogorov filter

[3]:

[^x]+

G(z) = ■

(21)

(23)

where X — x(Z) - is analytical function (together with the inversed one ) inside an isolated

fx (z)

circle , obtained by factorizing the function :

X (a) = I

hu (a)

| P(e~ja |2

= x(e"a) x(e}a).

Maximum (21) is assured if the spectral density h (p) satisfies the restriction

2n

only in the case

1 +S

— J huU.o)dm = Du

<œ

/2 X (a) =| ejwNx+— [e]0Mx+ ]

jaN ^

(24)

| P(e~p) |2

Equations (25) have a solution relative x with

Lagrange factor Y determines mean-square error . ,. ™

66 ' _ ^ an accuracy to an arbitrary multiplier. Therefore,

of the prognosis & = va & = D Equation for x = i may be added to (25). Then at N = 1

pr t u ' u V u ' N—1

Y and nonzero x+ (z) follows from expression (24):

rt, PiXn-i = XN_n_i1(N-n-1),

(25)

i=0

where

ln = Y (N),

i.e. there is no difference between the minimax prediction for one step and the prediction without taking into consideration the perturbation, and at N > 1:

1(0 =

0, i<0,

ln = Yn ( N ) — t Xn—i—1(ln—I — Yn (n — i)),

(26)

[1, i > 0.

Thus, minimax prediction method, concerning the

algorithm does not depend on the level D , which ' 1

makes its applicability domain wider. i.e. minimax predictions are calculated according

Now let us consider a more compact representa- to recurrent ratios with filter memory depth of N — 1

tion of minimax prediction process. Let us study the steps, as at the evaluation process only N — 1 last val-

prognosis in accordance with (19) in the absence of per- ues of the previous prognoses estimations

turbation. / i are memorized and the same number

n—1 ' ' n—N+1'

of values of prognoses estimations Yi(N — 1),...,Y(1) according to the model, the

z

x

perturbations being neglected. Thus, minimax predictions require corrections which should be more profound at long-term prognoses.

However, it should be noted, that unlike in the Kalman filter, there should be a separate time - set run over recurrent ratios (26) for each N.

The initial outset of algorithm (26) is evident it is assumed that the first N — 1 residuals in (26) / — Y (n — i) equal zero. Formula (26) is the main

in the suggested algorithm of minimax prediction. As will be seen from analytical examples, minimax prognoses may not prove to be effective at low N. As N increases the efficiency of the prognosis grows. Let us consider the examples:

1. y = y + ^ . Equation (25) can be expressed as:

K xn — xn—1) = XN-n-1-

Its solutions at N > 1 is as follows: n(n +1) 1

7 =

2N ; l sin(^N /(2N ; l))

sin(^N /(2N ; l)) l

sin(W(2Af +1)) 2sin(W(2Ar +1)) The filtration equation can be written as :

N—l

ln = yn -Z -N-i-1(ln-i - yn )' (27) i=\

At high N the maximal gain obtained by the minimax filter in comparison with the model prognosis totals n /2. The criterion at the comparison of two filters is a mean-square error of the prognosis under the most favourable perturbation spectrum.

2. yn = 2 yn-1 - yn-2 + un • Equation (25) can

be expressed as:

H- - 2--\ + --2 ) = XN.n.\. Their solutions at N > 2 may be presented by:

~ sina(n + \-N/2) coshß(n + \-N/2) ~

sin(aN /2)

coshß /2)

where cos a = 1 - (1 / 2y); cos JJ = 1 + (1 / 2y) and y may be calculated using the equation :

ylr +1/4 tanh(JN/2)-Jy-1/4 ctg(aN/2) = 1 (28)

At N = 2, y = 42 +1; X0 -1; x = 1.

The maximal gain k = 3/ (JT +1)« 1,21. The filtration equation can be expressed as:

h = (NU l)y — Nyn_l —^Xn—^ — (N;l — i)yn ;(N—i)y—).

(29)

a

At N ^ » asymptotically 7 ^ »,

r aN \ ( ßN

►ß^ 0,

ctg

From equation

v 2 ,

■■th

( aN > / aN y

= tan

v 2 v 2 ,

= l

giving the evaluations of the vector Xn in (18) has the recurrent form:

1

= ^ + ^^ KnC^n - ( Kl

C1 KfC

we obtain

aN 1

— -4 = 0.937552, r — —, 2 a2

The maximal gain by the minimax filter in comparison with the model prognosis totals is k = 24 = 1.87. In the comparison of filters the criterion of the evaluation of mean - square error of prognosis under the most favourable perturbation spectrum can be chosen also here.

4. Adaptive minimax prediction. a. Adaptation based on the Kalman filter.

In the assumption that the A matrix is known and the measurements are correct, the Kalman filter [6,7],

1

Kn = K3n - K3CCTK3n —-

n ^T TS~3

(30)

c K„C

Here the extrapolation of y into

N steps may

be presented by:

y>cTA»A

The adaptation is based on the selection of the elements of the matrix K at every moment n, for ex-

4n

ample , the selection of the minimized sum of residuals squared:

xn =

2

i=l

r-1

) = &„-* - У-r (r - к)]2,

k=0

(31)

where Г = dim X .

The condition of notnegative definiteness should be imposed on K* . Thus the space of time, measured

in

back from the present moment, during which the measurements are used effectively is regulated.

b. Adaptation of model coefficients

Let us resent formula (19) in the following way:

а у =un

(32)

1

1+Ttk^T '

(33)

where U - are perturbations with zero mean and

restricted dispersion and the following columns are introduced

a = [a0,...,aNy,...f ,yn =[yn,...,yn_N^JT.

Here it is assumed that Y are precisely measured

and a coefficients are unknown. In some cases the following representation may be known a priori.

a = Aaaä,

where ä -is the column-vector smaller than a; A - is the familiar matrix.

aa

The adaption filter in ref. [7] which provides a estimations, can be expressed as:

«„ = «Vi + KJ-iO-a^Aly") = СE-KnrrTAaa)ä,

where a - is the estimation of an ; E - is a single matrix ; K n is the covariance error matrix of an coefficients of the order of Ta .The algorithm outset is conducted according to the first Ta measurements , where ra = dim«, in the correlation assumption of ll[ .The minimax approach is used after the adaptation of à

coefficients to the model. In this case we are dealing with a combined adaptive-minimax method.

References

1. Grenander U. " A prediction problem in game theory " // Endless antagonistic games. M. : Fizmatgiz, 1963, pp. 403-413.

2. Kurkin O.M. Minimax linear filtering of a stationary occasional process with restricted perturbation dispersion // Radiotechnique and electronics, 1981, № 8,pp. 1689-1697.

3. Korobochkin Yu.B. Minimax linear estimation of a stationary occasional sequence in the presence of perturbation with restricted dispersion // Radiotechnique and electronics, 1983, № 11,pp. 2189-2190.

4. Atans M., and Falb P. Optimal control . M.: Machine - building, 1968.

5. Fedotov A.M. Incorrect problems with occasional mistakes in the basic data. M.: Nauka, 1990.

6. Sage E., Mels J. The evaluation theory and its application in communication and management. M.: Communication, 1976.

7. Albert A. Regression, pseudoinversion and recurrent evalution. M.: Nauka, 1977.

ВЛИЯНИЕ ФОТОМЕТРИЧЕСКИХ ФАКТОРОВ НА СТРУКТУРУ И ДИНАМИКУ ПОЛЕЙ

СЕРЕБРИСТЫХ ОБЛАКОВ

Солодовник А.А.

Северо-Казахстанский госуниверситет, профессор кафедры «Физика», доцент астрономии

Баймурзин А. С.

Северо-Казахстанский госуниверситет, студент кафедры «Физика»

INFLUENCE OF PHOTOMETRIC FACTORS ON THE STRUCTURE AND DYNAMICS OF FIELDS

OF SILVERY CLOUDS

Solodovnik A.A.,

North Kazakhstan state University, Professor of the Department of Physics,

associate Professor of astronomy Baymurzin A.S.

North Kazakhstan state University, student of the Department of Physics»

Аннотация

Рассматривается вопрос влияния на видимую структуру и движение полей серебристых облаков условий их освещения. Показано, что положение центра яркости облачного поля связано не с положением Солнца, а с концентрацией облачных частиц. Обосновываетсяважностьучётаэффектовзатенениянаструк-туруоблачногополя.