optimal discrete filtering for time-delayed systems with respect to mean-square continuous-time...

¹his paper was recommended for publication by editor M. J. Grimble.

*Correspondence to: B. P. Lampe, University of Rostock, Department of Electrical Engineering, D-18051 Rostock,Germany. E-mail: [email protected]

Received 4 March 1996CCC 0890—6327/98/050389—18$17.50 Revised 30 October 1997( 1998 John Wiley & Sons, Ltd. Accepted 1 December 1997

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING

Int. J. Adapt. Control Signal Process. 12, 389—406 (1998)

OPTIMAL DISCRETE FILTERING FOR TIME-DELAYEDSYSTEMS WITH RESPECT TO MEAN-SQUARE

CONTINUOUS-TIME ERROR CRITERION

YEPHIM N. ROSENWASSER1, KONSTANTIN YU. POLYAKOV1 AND BERNHARD P. LAMPE2,*

1 Marine Technical University Saint Petersburg, Department of Ship-Automation, Lotzmanskaja ul. D3,190008 St. Petersburg, Russia

2 University of Rostock, Department of Electrical Engineering, D-18051 Rostock, Germany

SUMMARY

An optimal linear digital filtering problem for continuous-time dynamic processes is considered. Analgorithm for optimal filter design taking into account pure delays in continuous-time networks is presented.The performance of systems, being optimal with respect to discrete-time and continuous-time mean-squareerrors, is compared. An optimal stochastic model-matching problem is proposed and its solution ispresented. ( 1998 John Wiley & Sons, Ltd.

Key words: sampled-data; optimal filtering; parametric transfer function; model matching

1. INTRODUCTION

Many problems in filtering systems design and signal processing call for an analysis of theproperties of a continuous-time process on the basis of sampled data. This is always the casewhen, due to technological reasons, measurements of the continuous-time process are onlyavailable at discrete time points, for instance, by means of radar or sonar systems. The presence ofdigital sensors or computers on-line also causes a discretization of continuous-time signals.

If the reference signal has a bounded spectrum then, by the Khotelnikov—Shannon theorem, weare able to reconstruct the signal completely, provided that the measurements are sufficientlyfrequent and the time for reconstruction is large enough. Nevertheless, application signals withbounded spectrums are, strictly speaking, an idealization. In this connection it is often required toobtain (in some sense) approximations of the reference continuous-time process on the basis of itssampling.

Figure 1. Hybrid discrete-time and continuous-time system

The block-diagram of a system with discrete-time filtering of a continuous-time reference signalis shown in Figure 1. Here, r (t) is the reference signal and n (t) is the noise. The system includesa computer and stable continuous networks with known transfer functions F

0(s) and F (s). Besides

that, H(s) is the desired transfer function of a stable linear transformation of the reference signal,e~qs is the pure delay operator, u(t) is the computer control output, y (t) is the filter output, andy0(t) is the desired output. The signal y

0(t) is approximately generated by the continuous—discrete

filter that consists of the continuous networks with transfer functions F0(s), F(s) and e~qs, as well

as the computer, whose program has to be chosen so that the filtering error e(t)"y (t)!y0(t) is

minimal in a certain sense. Hereafter, we assume the signals r (t) and n (t) to be independentcentred stationary random processes.

Since the filter includes both, continuous-time and discrete-time networks, the system underconsideration is a hybrid continuous—discrete system. Therefore, the output signal y (t) is non-stationary and its variance depends on time, even if the stochastic processes r (t) and n (t) arestationary. If the sampling is carried out with a constant time interval ¹, the system shown inFigure 1 is periodically non-stationary and the error variance l

ein the steady-state mode is

periodic with period ¹, i.e.

le(t)"l

e(t#¹ )

Then to appraise the quality of the filter we will use the cost functional

lNe"

1¹ P

T

0

le(t) dt (1)

which will be called the mean error variance. This paper deals with the design of an optimal digitalfilter with a minimal cost functional (1).

Recently, the investigation of continuous processes in continuous—discrete systems has attrac-ted much attention.1~6 Because of the time-variant character of those systems common transferfunction concepts fail. For an input—output description of discrete-continuous systems, theparametric transfer function method appeared to be very helpful.7~13 A method of optimal digitalfilter design for reconstructing continuous-time processes without pure time delay was describedindependently by Chen and Francis5 and Rosenwasser.11 In this paper, the last method, based onthe parametric transfer function concept, is extended to the case where pure delays in continuousnetworks have to be taken into account. The present method is applicable also for handlingcomputational delay if it is assumed to be known a priori.

In applications instead of the cost functional (1) the variance of the discrete process at thesampling instants is often taken as a performance index. In this case the cost functional is

le(k¹)"l

e(0), k—integer (2)

390 Y. N. ROSENWASSER, K. YU. POLYAKOV AND B. P. LAMPE

Int. J. Adapt. Control Signal Process. 12, 389—406 (1998)( 1998 John Wiley & Sons, Ltd.

Figure 2. Two equivalent models of the computer

which will be called the discrete variance. The latter approach leads, in certain cases, to a discreteWiener—Kolmogorov problem. Algorithms minimising the discrete variance (2) will be calledminimum variance filtering (M» filtering), while programs minimising the average variance (1) willbe referred to as continuous-minimum-variance filtering (CM» filtering). In this paper the perfor-mances of optimal systems with respect to the criteria (1) and (2) are compared.

The examples considered below show that MV filtering sometimes leads to a considerableincrease in the average variance. Hence the continuous-time performance of the systemdeteriorates.

The accuracy of the optimal system depending on the magnitude of the time delay is alsoconsidered. It is shown that MV filtering for time-delayed systems sometimes cause an increase inthe average variance.

A seemingly novel problem in the stochastic model matching is proposed. It is shown that theproblem can be solved by means of our method as a particular case of the optimal filteringproblem.

2. STATEMENT OF THE PROBLEM

The block diagram of the computer is shown in Figure 2(a). The analogue-to-digital converter(ADC) implements a sampling of the input x (t) with period ¹, so that

mk"x (k¹#0), k an integer

The computer converts the input sequence MmkN into the control sequence Mt

kN according to the

difference equation

a0tk#a

1tk~1

#2#antk~n

"b0mk#bm

k~1#2#b

mmk~m

that corresponds to the discrete filter transfer function C (z). The sequence MtkN acts on the input

of the digital-to-analogue converter (DAC), generating the continuous output u(t). The DAC canbe treated as an extrapolator with the transfer function G

h(s).

With respect to input—output analysis of continuous—discrete systems, it has been reported inReference 9 that the block-diagram shown in Figure 2(a) is equivalent to a serial connection of

DISCRETE FILTERING FOR TIME DELAY SYSTEMS 391


a stationary network with the transfer function

C(s)"C(z) Dz/%sT

"

b0#b

1e~sT#2#b

me~msT

a0#a

1e~sT#2#a

ne~nsT

(3)

together with the sampling unit (ADC) and the extrapolator Gh(s) as shown in Figure 2(b). Note

that a0O0 in (3) because of the causality of the algorithm. An operator C(s) of the form (3) will be

called the program operator. Obviously, C(s#kjus)"C(s) follows from (3), where k is any

integer, and us"2n/¹ is the sampling frequency. The program operator can be formally placed

either before or after the sampling unit (see Figure 2(b)).After these preparations the problem is formulated as follows:

Problem. For the structure of Figure 1 let the transfer functions F0(s), F (s) and H(s) of the

continuous-time networks and the transfer function Gh(s) for the hold networks be given. Giving

the spectral densities of the stationary and independent stochastic processes r(t) and n (t) andvalues of ¹ and q find the program of the stable digital filter (3) that minimizes the average errorvariance (1) in the steady-state mode.

3. PARAMETRIC TRANSFER FUNCTION METHOD

The earliest works applying the notion of parametric transfer function (PTF) are due to Blanch14

and Shtokalo15. The notion PTF for linear non-stationary systems was introduced in controltheory by Zadeh16,17. The idea of applying this approach to the investigation of pulse systemswas discussed in Ragazzini and Zadeh18 and was applied by Kosyakin19 to a particular problemin continuous—discrete filtering. A systematic theoretical method of parametric transfer functionswith application to linear non-stationary systems of various classes, including digital controlsystems, has been developed in the monographs Rosenwasser7,8,20, and specifically for continu-ous—discrete systems in Reference 13. Principal ideas of the PTF theory with applications tocontinuous—discrete systems were presented in English in References 9—12, 21. Since the PTFmethod is not widely known outside Russia, some of its results are presented in this section. Letfunctions x (t) and y (t) be related by a linear operator L as

y(t)"L[x (t)]

From this operator with its input and output spaces it is assumed that the expression

¼(s, t)"L[est]e~st

makes sense where s"c#ju is a complex variable and t is the time. The so specified function¼(s, t) is called the parametric transfer function (PTF) of the operator L. It depends on twoarguments s, the frequency, and t, the time displacement. Starting from the definition of the PTFfor operators, the PTF for a system is defined as the PTF of a certain motion of this system.13

The PTF of a linear stationary network is equal to its common transfer function, while fora periodic non-stationary system with period ¹ the PTF of the network is periodic with the sameperiod, so that

¼(s, t)"¼ (s, t#¹)

The parametric transfer function of the parallel connection of linear operators with PTFs ¼1(s, t)

and ¼2(s, t) appears as

¼(s, t)"¼1(s, t)#¼

2(s, t) (4)



The PTF of a serial connection of a linear operator with the PTF ¼2(s, t) and a stationary

network with the transfer function ¼1(s) is

¼(s, t)"¼1(s)¼

2(s, t) (5)

For any transfer function G(s) of a stationary network the following discrete transformations areintroduced

uG(¹, s, t)"

1

¹

=+

i/~=

G(s#i jus)eijust (6)

DG(¹, s, t)"

1

¹

=+

i/~=

G(s#ijus)e(s`ijus) t (7)

where us"2n/¹ is the sampling frequency. The sum of the series (6) will be called the displaced

pulse-frequency response characteristic and the sum (7) the discrete ¸aplace transform of thetransfer function G(s) with period ¹. Comparing (6) with (7) yields

uG(¹, s, t)"D

G(¹, s, t) e~st

In the following, for G(s)"G1(s)#G

2(s) the notations

uG(¹, s, t)"u

G1`G2(¹, s, t) and D

G(¹, s, t)"D

G1`G2(¹, s, t)

are used. Similarly, for the product G(s)"G1(s)G

2(s) we define

uG(¹, s, t)"u

G1G2(¹, s, t) and D

G(¹, s, t)"D

G1G2(¹, s, t)

Furthermore,

G* (s)"G(!s)

For real rational F(s) the sums uF(¹, s, t), D

F(¹, s, t) can be given as analytic expressions. Let

F (s)"n+i/1

ai

s!si

. (8)

Then, with the notation di"esiT, the result for 0)t(¹ is

DF(¹, s, t)"

n+i/1

aie~st

1!die~sT

(9)

Moreover, for 0)t(¹ and any hold device with transfer function Gh(s)

DFGh

(¹, s, t)"h (t)#n+i/1

aidiG

h(si)esit

esT!di

(10)

where h (t) is the Dirac impulse response of the process with hold and it appears, for the zero-orderhold, as

h (t)"n+i/1

ai

si

(esi t!1)

Formulas for h(t) in case of transfer functions F(s) with multiple poles and arbitrary hold aregiven by Rosenwasser and Lampe.13

If the continuous network F1(s) contains a pure delay q, i.e.

F1(s)"F (s)e~sq (11)



then, with reference to (7)

DF1Gh

(¹, s, t)"DFGh

(¹, s, t!q)

is valid. Let t!q"d¹#0 , where d is an integer and 0(0(¹. Then,

DF1Gh

(¹, s, t)"e~dsTDFGh

(¹, s, 0 ) . (12)

owing to the periodicity of the parametric transfer function with respect to t.The PTF of an elementary open-loop system consisting of serial connected pulse element,

discrete filter C (s) of the form (3), hold element Gh(s) and stationary network F(s) takes the form

¼(s, t)"C(s)uFGh

(¹, s, t) (13)

Let the input of a system with PTF ¼(s, t) be excited by a stationary random signal x (t) withspectral density S

x(s). Then, the output process y (t) is non-stationary with a time-dependent

variance

ly(t)"

1

2nj Pj=

~j=

¼(s, t)¼ (!s, t)Sx(s) ds (14)

Because ¼ (s, t)"¼ (s, t#¹ ), the variance ly(t) is also ¹-periodic. The continuous-time perfor-

mance of the system can be naturally measured according to the variance averaged over theperiod ¹, so that

l6y"

1

¹ PT

0

ly(t) dt

It is convenient, for practical computation, to reduce equation (14) to the integral with finiteintegration limits as

ly(t)"

1

2nj Pjus@2

~jus@2

=+

i/~=

¼ (s#ijus, t)¼(!s!iju

s, t)S

x(s#iju

s) ds (15)

An effective method for computing integrals of the form (15) on the basis of the residue theorem ispresented by Rosenwasser and Lampe.13

4. CALCULATION OF ERROR VARIANCE

Let us assume that r (t) and n (t) are signals of independent processes. Then the parametric transferfunction from r to e due to (6), (7) and (13), can be written as

¼r(s, t)"F

0(s)C (s)u

F1Gh(¹, s, t)!H (s) (16)

where F1(s)"F (s)e~sq. Similarly, the PTF from n to e results in

¼n(s, t)"F (s)C(s)u

F1Gh(¹, s, t) (17)

Owing to the independence of exogenous inputs, by virtue of (14) the error variance is

le(t)"

1

2nj Pj=

~j=

[¼r(¹, s, t)¼

r(¹,!s, t)S

r(s)#¼

n(¹, s, t)¼

n(¹,!s, t)S

n(s)] ds

where Sr(s) and S

n(s) are the spectral densities of r(t) and n (t), respectively. Since the PTFs ¼

r(s, t)

and ¼n(s, t) are periodic with period ¹, the same is valid for the variances. Substituting (6) and (7)



in the last equation, one gets

le(t)"a#

1

2nj Pj=

~j=

[C(s)C* (s)uF1Gh

(¹, s, t)uF1Gh

(¹,!s, t)Sa(s)

!C(s)uF1Gh

(¹, s, t)º (s)!C* (s)uF1Gh

(¹,!s, t)º* (s)] ds (18)

where

a"1

2nj Pj=

~j=

H (s)H*(s)Sr(s) ds, Sa(s)"F

0(s)F

0(!s)[S

r(s)#S

n(s)]

º(s)"F0(s)H* (s)S

r(s)

Notice that a can be evaluated by a well known technique if H (s) and Sr(s) are real rational

functions. Let us denote the integral in (18) by I (t) and produced to finite integration limits.Taking into account

uF1Gh

(¹, s, t)uF1Gh

(¹,!s, t)"DF1Gh

(¹, s, t)DF1Gh

(¹,!s, t)

uF1Gh

(¹, s, t)º(s)"DF1Gh

(¹, s, t)º(s)e~st

and, for any integer k

C(s#kjus)"C(s)

DF1Gh

(¹, s#kjus, t)"D

F1Gh(¹, s, t)

one obtains the integral with finite bounds

I (t)"¹

2nj Pjus@2

~jus@2

[C(s)C* (s)A(s, t)!C(s)B(s, t)!C*(s)B*(s, t)] ds (19)

where

A(s, t)i DF1Gh

(¹, s, t)DF1Gh

(¹,!s, t)DSa

(¹, s, 0)

B (s, t)iDF1Gh

(¹, s, t)DF*0Gh

(¹,!s, t)

Now let us evaluate the mean error variance (1). It follows immediately from (18),

l6e#a#IM

with

IM"1

¹ PT

0

I(t) dt

For certain G1(s), G

2(s)

PT

0

DG1

(¹, s, t)DG2

(¹,!s, t) dt"DG1G*

2(¹, s, 0)

is valid. Hence

IM"¹

2nj Pjus@2

~jus@2

[C(s)C*(s)A(s)!C (s)B (s)!C* (s)B* (s)] ds (20)



where

A(s)i1

¹

DF1F*

1GhG*h(¹, s, 0)D

Sa(¹, s, 0)

B (s)i1

¹

DF1GhF0H*Sr

(¹, s, 0)

Rosenwasser and Lampe13 have shown how the integrals (19) and (20) can be computed by theapplication of the residue theorem. Hereafter, we shall use the variable f"e~sT, that is moreconvenient. Thus (19) and (20) appear as

I(t)"1

2nj Q![C(f)C(f~1)A(f, t)!C(f)B(f, t)!C(f~1)B(f~1, t)]

dff

(21)

IM"1

2nj Q![C(f)C(f~1)A(f)!C (f)B(f)!C(f~1)B(f~1)]

dff

(22)

respectively, where the integrals are taken along the unit circle counter clockwise.

5. OPTICAL FILTER DESIGN

Since the form of the integral (21) with a fixed t coincides with (22), both functionals can beminimized by means of the same procedure. Thus the algorithm for minimization of thefunctional (22) over the class of stable causal algorithms C (f) given by Chang22 is applied. Underthe given assumptions the following factorization holds:

A(f)"»(f)»(f~1) (23)

where all poles and zeros of » (f) lie outside the unit circle in the f-plane. Introducing the function

R(f)"B(f~1)/» (f~1) (24)

that is real rational with respect to f and can be separated as

R(f)"R`

(f)#R~

(f) (25)

where R`

(f) has all its poles outside the unit circle in the f-plane, while all poles of R~

(f) areinside the unit circle, and

limDfD?=

R~

(f)"0 (26)

This separation will be referred to as a proper separation of the function R(f). Then, integral (22)has a minimal value for

C(f)"R`

(f)/»(f) (27)

with this (optimal) choice of the algorithm the minimal value of (22) is

IMo"!

1

2n$N Q!R

`(f)R

~(f~1)

dff

(28)

Thus, the mean error variance of the optimal system is

l6e"a#IM

o



It is noteworthy that the given algorithm provides a stable and causal digital filter. If all thestationary continuous-time networks have real rational transfer functions, then general expres-sions can be derived for the transfer function of the optimal algorithm. Thus the structure andperformance of MV and CMV-optimal filter systems could be compared. Let the continuousstable network have transfer functions with simple poles, so that

F (s)"n+i/1

ai

s!si

, F0(s)"f

0#

m+j/1

bj

s!rj

where ReMsiN(0 and ReMr

jN(0 for all s

iand r

j. The ideal operator may contain an exponential

multiplier by an integer multiple of the period ¹ for smoothing or prediction. Thus

H (s)"H0(s)ecTs

where c is an integer and

H0(s)"h

0#

l

+i/1

ci

s!qi

, ReMqiN(0, i"1, 2,2 , l

The spectral densities of the exogenous signals are assumed to be real rational with simple poles,so that

Sr(s)"

p+i/1

oi

s!/i

!

p+i/1

oi

s#/i

, Sn(s)"

k+i/1

gi

s!ti

!

k+i/1

gi

s#ti

with ReM/iN(0 and ReMt

iN(0 for all /

iand t

i. In principle, the problem can also be solved

when n (t) is white noise with spectral density n0

and F0(s) is strictly proper. In that case it has to

be assumed that Sn(s)"n

0. If F

0(s)"const, the disturbance can be described as a discrete white

noise with variance nJ0, so that D

Sn(¹, s, 0)"nJ

0.13

Let all the numbers

esiT, erjT, eqkT, e/pT, et

hT

be distinct. A zero-order hold is used as an extrapolator. Thus,

Gh(s)"

1!e~sT

s(29)

Let us introduce the following notation for any real rational transfer function F (s) of theform (8):

PF(f)i

n<i/1

(1!esiTf) , QF(f)i

n<i/1

(f!esiT )

Let G1(s) and G

2(s) be real rational transfer functions. For brevity

PG1G2

(f)iPG1

(f)PG2

(f) , QG1G2

(f)iQG1

(f)QG2

(f)

is written. Similar notations will also be applied for the spectral densities, though in this case theproduct is only extended to all stable poles, so that

PSr(f)"

p<i/1

(1!e/iTf) , Q

Sr(f)"

p<i/1

(f!e/iT )



Consider the minimization of the mean variance. We wish to write closed expressions for A(f) andB(f). It was shown by Rosenwasser11 that

1

¹

DF1F*

1GhG*h(¹, s, 0)"

A1(f)A

1(f~1)

PF(f)P

F(f~1)

DSa

(¹, s, 0)"A

2(f)A

2(f~1)

PF0SrSn

(f)PF0SrSn

(f~1)

is possible with polynomials A1(f) and A

2(f). Let us assume that A

1(f), A

2(f) can be chosen so

that all their zeros are outside the unit circle in the f-plane. The degree of the polynomial A1(f) is

calculated to be

degA1(f)"n

If F0(s)"cost. and the disturbance n(t) is described as a discrete white noise, then

degA2(f)"n#m#p#k

while for F0(s)"const. or/and the disturbance n (t) being a coloured noise one obtains

degA2(f)"n#m#p#k#1.

Factorizing A(f) with respect to the unit circle gives

K(f)"A

1(f)A

2(f)

PFF0SrSn

(f)(30)

K(f~1)"AI (f)

QFF0SrSn

(f)(31)

where

AI (f)"fn`m`p`kA1(f~1)A

2(f~1)

Applying transformational (10) in the calculation of B (f), taking into account the time delay q,yields

B (f~1)"f~(c`1)B

1(f)

PFF0Sr

(f)PSrH0

(f)(32)

where degB1(f)"n#m#2p#k. For q"0 the absolute term of B

1(f) is zero. Substituting (31)

and (32) in (24) and implementing the proper separation (25), one gets

R`

(f)"R1(f)/P

SrH0(f) (33)

The degree of R1(f) depends on the exponential multiplier in the ideal operator transfer functions,

i.e. on the value of c. It can be readily seen that for the case of properness of the separation (26), thedegree conditions degR

1(f)"l#p!c!1 for c(0 (optimal smoothing), and

degR1(f)"l#p!1 for c*0 hold. Substituting (30) and (33) into (27), the transfer function of

the CMV-optimal filter algorithm is

C(f)"R

1(f)P

FF0Sn(f)

A1(f)A

2(f)P

H0(f)

(34)



Equation (34) allows the following conclusions.

1. The algorithm always leads to a stable and realizable algorithm of the form (34) since theabsolute terms of the polynomials A

1(f) and A

2(f) are non-zero by construction.

2. The set of poles of the CMV-optimal digital filter transfer function consists of the poles ofthe rational function H

0(s) and the zeros of K(f).

3. The set of the zeros of the CMV-optimal digital filter transfer function consists of the polesof the plant and the prefilter transfer functions, F (s) and F

0(s) respectively, and the poles of

the stable generating filter, corresponding to the disturbance spectral density Sn(s).

4. The degree of the numerator of the transfer function (34) with respect to f can exceed thedenominator degree. That is the case, for example, for the solution of the optimal smoothingproblem.

Now let us consider the design of the MV-filter. The functional I(0) takes the form of (22), whereA(s, 0) and B (s, 0) must be instead of A(s) and B (s), respectively. It can be shown that for0(q(¹

DF1Gh

(¹, f, 0)"fDFGh

(¹, f,0 )"fA`

1(f)A~

1(f)

PF(f)

(35)

where 0"¹!q. The polynomial A`1

(f) is free of zeros inside the unit circle and A~1

(f) is free ofzeros outside the unit circle. It follows from equation (10) for qO0

degA`1

(f)#degA~1

(f)"n.

and for q"0 this sum is equal to n!1. Factorizing A (f, 0), one obtains

K (f)"A`

1(f)AI ~

1(f)A

2(f)

PFF0Sr Sn

(f)

where the polynomial AI ~1

(f) is free of zeros inside the unit circle, and is given by

AI ~1

(f)"f$%'A~1 (f)A~

1(f)

Applying the method revealed above yields the transfer function of the MV-optimal program

C (f)"RI

1(f)P

FF0Sn(f)

A`1

(f)AI ~1

(f)A2(f)P

H0(f)

(36)

where RI1(f) is a polynomial, whose degree depends, as before, on c.

Such being the case, the set of the poles of the MV-optimal filter algorithm consists of the stablezeros of the discrete transfer function (35) and the ‘reflected at the unit circle’ unstable zeros, i.e.the roots of the polynomial AI ~

1(f). As it has been reported by As strom1, for cases where the pole

excess of F(s) is an even number, the discrete transfer function (35) has, for q"0 and ¹P0, a zerothat lies near the oscillation stability boundary m"!1. This zero becomes a pole of theMV-optimal program (36), causing large intersample oscillations, hence the average varianceincreases drastically. In that case, obviously, the criteria of minimum average variance andminimum discrete variance contradict each other. Similar results have been obtained by statespace methods.5,23 It is noteworthy that this effect increases if the sampling period tends to zero,since the ‘dangerous’ zero reaches the stability boundary f"!1 as ¹P0.1 for q'0 the‘dangerous’ zero can appear in (35) for odd pole excess of the transfer function F (s), also if the timedelay is taken into account in the synthesis of the MV-optimal program.24



Figure 3. Delayed digital filter system

6. STOCHASTIC MODEL MATCHING PROBLEM

In many applications it is necessary to match, as close as possible, the performance of a continu-ous—discrete system with those of an ideal linear reference model given by its transfer function. Asan optimal criterion designers usually employ the difference between the transients of the actualand the reference models or an H

=-type functional.3,15 Nevertheless, it is possible to state the

model matching problem for the case of exogenous signals being stochastic stationary processeswith known spectral densities. The optimal stochastic model matching problem for the systemshown in Figure 1 can be formulated as follows:

Let us have a reference continuous-time system with transfer function H(s) and the continuous-digital system, which consists of a sampling unit with period ¹, a digital filter, a hold andcontinuous-time networks with known transfer functions. It is required to design the discretetransfer function of the digital filter so that the average error variance between the outputs of theactual and the ideal system are minimal in the steady-state mode.

It can easily be shown that the above method is applicable with the special assumption that thedisturbance is absent (n(t)"0). Then,

Sa(s)"F0(s)F*

0(s)S

r(s)

and the above optimization procedure still holds.

7. NUMERICAL EXAMPLES

The examples considered below illustrate an application of the proposed method to optimalfiltering with H (s)"1 and F

0(s)"1. A block diagram of the system is given in Figure 3. The

reference signal r(t) is assumed to have the spectral density

Sr(s)"

1

s#2!

1

s!2

The zero-order hold with transfer function (29) is used as the extrapolator.

Example 1

Find the MV and CMV-optimal filter for the system shown in Figure 3 with

F (s)"1

s#1



¹"0)1 and q"0. The disturbance n (t) is described as a discrete white noise with unit variance,i.e.

DSn

(¹, f, 0)"1

Then

DSa

(¹, f, 0)"DSr(¹, f, 0)#D

Sn(¹, f, 0)"

1)574(1!0)520f)(1!0)520f~1)

(1!0)819f)(1!0)819f~1)

1

¹

DFF*GhG*

h(¹, f, 0)"

0)00563(1#0)268f)(1#0)268f~1)

(1!0)905f)(1!0)905f~1)

Factorization gives

K (f)"0)0942(1!0)520f) (1#0)268f)

(1!0)819f)(1!0)905f), K(f~1)"

0)0942(f!0)520)(f#0)268)

(f!0)819)(f!0)905f)

The expression B(f~1) can be evaluated by (10)

B (f~1)"1

¹

DFSrGh

(¹,!s, 0) K%~sT/f

"

!0)00652f2!0)0255f!0)00620

(f!0)819)(f!1)221)(f!0)905)

thus

R (f)"!0)0692f2!0)271f!0)0658

(f!0)520)(f#0)268)(f!1)221)

The proper separation (24), satisfying (25), gives

R`

(f)"0)392

1!0)819f

The CMV-optimal digital filter can be derived from (27) as

CCMV

(f)"4)158!3)762f

1!0)252f!0)139f2

Now let us calculate the continuous-time variance in the optimal system. By the residue theoremone has

a"1

2nj Pj=

~j=

Sr(s) ds"1

The value IMo

is computed by means of (28) and the residue theorem, so that

IMo"!0)465

The mean variance in the CMV-optimal system gets

l6e"a#IM

o"0)535

while the variance at the sampling points is le(0)"0)590.

Similarly the MV-optimal program can be obtained as

CMV

(f)"3)138!2)839f

1!0)520f



Figure 4. Mean variance as function of time delay for Example 2

In the considered case q"0, MV- and CMV-optimal filters have denominators of differentdegrees. The average and discrete variance in the MV-optimal system are l6

e"0)544 and

le(0)"0)574, respectively. Here the advantage of the CMV filter is not significant, but it will be

shown shortly that, in general, it is not reasonable to use the variance at the sampling points asa performance criterion.

Example 2

Now the influence of the computational delay on the optimal system performance is investi-gated. Let the continuous part transfer function be specified by

F(s)"b/ (s!a) (37)

with real a(0. The disturbance is a discrete white noise with variance 0)2. Figure 4 shows themean variance as a function of the time delay q for the optimal systems using MV andCMV-filtering for b"1, a"!1 and ¹"0)1. As it follows from Figure 4, MV filtering leads, forsome q, to an intolerable mean error variance. Thus, in fact, the MV-optimal system becomesinoperable. This effect is caused by a zero of the discrete transfer function (35) that lies closely tof"!1. Let us demonstrate this for the system having the continuous part transfer function (37).

Let 0(q(¹ and 0"¹!q. Then, as was shown above,

DF1Gh

(¹, f, 0)"fDFGh

(¹, f,0 )

Let da"eaT. Referring to (10), after some algebraic transformations

DF1Gh

(¹, f, 0)"b

a(ea0

!1)f(1#cf)1!d

af

is obtained where c"(da!ea0 )/(ea0!1). Thus, one of the zeros of the discrete transfer function is

f0"

1!ea0

da!ea0



Figure 5. Discrete variance over displace-parameter t for Example 2

The zero f0

can be ‘stable’ (outside the unit circle) as well as ‘unstable’. In the case of unstable f0,

one of the poles of the MV filter is f*"1/f0, so that for f

0P!1 one has f*P!1. Therefore

the increase of the mean variance in the MV-optimal system also appears. Thus f0

tends to !1for

0P00"

1

aln

da#1

2

Moreover, under the given conditions if 00(¹. It can be readily shown that

limT?0

00

¹

"

1

2

i.e. if the sampling period tends to zero, the ‘dangerous’ zero of (35) appears in the MV-optimalsystem when the computational delay is about half of the sampling period, provided the delay istaken into account in the synthesis procedure; see also the work of Francis and Doyle.24 Besides,there exists the limit

limT?=

00"

ln 0)5

a"!

0)693

a

i.e. 00

tends to a finite value as ¹ increases. In order to illustrate the differences between the twofilter laws let us plot the curves l

e(t) according to equation (19) for 0(t(¹. These curves are

shown in Figure 5 for the case of (37) with b"1, a"!1, ¹"0)1 and q"0)051. It is easilyverified that the delay q is near to the critical value and yields a zero near the stability boundaryf0"!1. MV-filtering provides the least attainable l

e(0), on the other hand a considerable

variance peak is observed inside the sampling period, so that the mean variance increases as well.

Example 3

Let

F (s)"1/(s2#s#1) (38)



Figure 6. Mean error variance of optimal systems for Example 3

and q"0. Consider the influence of the sampling period on the properties of MV and CMV-optimal filter systems.

Since the system under consideration is discrete-continuous, the mean variance of the error isthe natural stochastic performance criterion. Figure 6(a) shows the mean error variance in theoptimal systems of the two types without disturbance, while Figure 6(b) represents similar plotsfor the system under the disturbance n (t), as described by the model of discrete white noise withunit variance.

The graphs show that for small sampling periods MV-filtering gives considerably larger meanvariance as compared with CMV-filtering. Moreover, this difference increases if the samplingperiod decreases. This effect is due to the discrete transfer function of the continuous network (38)having a zero (stable or unstable) that tends to m"!1 as ¹P0.1 The zero (or, if it is unstable,the corresponding stable zero) becomes a pole of the MV-optimal filter. Under disturbance thedifference between the mean variance of the two systems decreases (Figure 6(b)).



Figure 7. Mean error variance in CMV-optimal system over sampling period

Example 4

Consider an example of the optimal stochastic model matching problem. Let the ideal referencecontinuous system have the same transfer function as the continuous part of the digital system, sothat

F (s)"H (s)"1

(s#1)

The CMV-filter for F0(s)"1, n (t)"0, ¹"0)1 and q"0 is determined as

CCMV

(f)"1)518!0)261f

1!0)268f

Figure 7 shows the mean error variance in the CMV-optimal system as a function of the samplingperiod ¹.

8. CONCLUSIONS

The paper presents how an optimal algorithm for a discrete-time filter as part of continuous-discrete systems can be calculated that reconstructs a reference signal (or its transformationspecified by a linear stationary operator with a known transfer function) when only that signalwith additive noise is measured. The parametric transfer function method is applied, and thismakes it possible to describe continuous-discrete systems by input—output relations, which is notpossible when working with other methods. The presented algorithm takes into account puredelays of the continuous part of the system.

The performance of two types of optimal systems—optimal with respect to the variance at thesampling instants and the mean variance, respectively—are compared. It is shown that MV-filtering sometimes causes a considerable increase in the mean variance if a zero near !1 iscompensated. Thus, in general, it is inexpedient to use the variance at the sampling instants asa performance criterion.

The effects of the pure delay on the stochastic accuracy is investigated. It is shown that for somedelays MV-filtering causes an increase in the mean variance, especially if the sampling period is



relatively small. That is the case for a system with arbitrary (even or odd) pole excess in thetransfer function of the continuous part.

A seemingly new stochastic model matching problem has been presented for matching thestochastic performance of actual continuous-discrete and ideal continuous-time systems in thesteady-state mode. It is shown that the technique developed in this paper allows to solve theproblems of optimal filtering, smoothing, prediction and stochastic model matching.

REFERENCES

1. As strom, K. J., P. Hagender and J. Sternby, ‘Zeros of sampled-data systems’, Automatica, 20(4), 31—38 (1984).2. Chen, T. and B. A. Francis, ‘H

2-optimal sampled-data control’, IEEE ¹rans. Automat. Control. AC-36, 387—397

(1991).3. de Souza, C. E. and G. C. Goodwin, ‘Intersample variance in discrete minimum variance control’, IEEE ¹rans.

Automat. Control AC-29, 759—761 (1984).4. Khargonekar, P. T. and N. Sivarshankar, ‘H

2-optimal control for sampled-data systems’, Systems Control ¸ett., 18,

627—631 (1992).5. Lennartson, B., ‘Sampled-data control for time-delayed plants’, Int. J. Control, 49, 1601—1614 (1989).6. Lennartson, B. and T. Soderstrom, ‘Investigation of the intersample variance in sampled-data control’, Int. J. Control,

50, 1587—1602 (1990).7. Rosenwasser, Y. N., Periodically Nonstationary Control Systems. Nauka, Moscow, 1973 (in Russian).8. Rosenwasser, Y. N., ¸yapunov Indices in ¸inear Control ¹heory, Nauka, Moscow, 1977 (in Russian).9. Rosenwasser, Y. N., ‘Application of parametric transfer functions to analysis of sampled-data systems in continuous-

time. Part I. Analysis of processes with deterministic disturbances’, Automat. Remote Control, 50(11, Part 2),1501—1512 (1990).

10. Rosenwasser, Y. N., ‘Application of parametric transfer functions to analysis of sampled-data systems in continuous-time. Part II. Analysis of processes with random disturbances’, Automat. Remote Control, 51(2, Part 1), 164—175(1991).

11. Rosenwasser, Y. N., ‘Application of parametric transfer functions for synthesis of discrete-time control laws—I.Optimal discrete, filtering of continuous-time processes by the average variance criterian’, Automat. Remote Control,53(6, Part 2), 872—883 (1992).

12. Rosenwasser, Y. N., ‘Method of parametric transfer functions in problems of closed-loop digital control systems’,Automat. Remote Control, 54(1, Part 1), 65—77 (1993).

13. Rosenwasser, Y. N. and B. P. Lampe, Digitale Regelung in kontinuierlicher Zeit—Analyse und Entwurf im Frequen-bereich, Teubner, Stuttgart, 1997.

14. Blanch, Ch., ‘Sur les equation differentielles lineares a coefficients lentement variable’, Bull. ¹echnique de la Suisseromande, 74, 82—189 (1948).

15. Shtokalo, I. Z., ‘Generalisation of symbolic method principal formula onto linear differential equations with variablecoefficients’, Dokl. Akad. Nauk SSR, 42, 9—10 (1945) (in Russian).

16. Zadeh, L. A., ‘Circuit analysis of linear varying-parameter networks’, J. Appl. Phys., 21, 1171—1177 (1950).17. Zadeh, L. A., ‘On stability of linear varying-parameter systems’, J. Appl. Phys., 22, 202—204 (1951).18. Ragazzini, J. R. and L. A. Zadeh, ‘The analysis of sampled-data systems’, AIEE ¹rans., 71, 225—234 (1952).19. Kosyakin, A. A., ‘On investigation of linear pulse systems under stationary random disturbances’, Automatika

telemechanika, 24, 331—340 (1963) (in Russian).20. Rosenwasser, Y. N., »ibrations of Nonlinear Systems, the Method of Integral Equations, Techn. Information Service,

Springfield, VA, 1970.21. Lampe, B. P. and Y. N. Rosenwasser, ‘Design of hybrid analog-digital systems by parametric transfer functions’, In

Proc. 32nd CDC, San Antonio, TX, 1993, pp. 3897—3898.22. Chang, S. S. L., Synthesis of Optimum Control Systems. McGraw-Hill, New York, 1961.23. Lennartson, B., T. Soderstrom and Sun Zeng-Qi, ‘Intersample behavior as measured by continuous-time quadratic

criteria’, Int. J. Control, 49, 2077—2083 (1989).24. Francis, B. A. and J. C. Doyle, ‘Linear control theory with an H

=-optimality criterion’, SIAM J. Control, 25, 815—844

(1987).25. Bamieh, B. A. and J. B. Pearson, ‘The H

2problem for sampled-data systems’, System Control ¸ett., 19(1), 1—12 (1992)

26. Chen, T. and B. A. Francis, Optimal Sampled-data Control Systems. Springer, Berlin, 1995.27. Hara, S., R. Kondo and H. Katori, ‘Properties of zeros in digital control systems with computational time-delay’, Int.

J. Control. 49, 493—511 (1989).28. Shieh, L. S., B. B. Decrocq and J. L. Zhang, ‘Optimal digital redesign of cascaded analogue controllers’, In Optimal

Control Appl. Methods, Vol. 12, 1991, pp. 205—219.29. Toivonen, H. T., ‘Discretization of analogue filters via H

=model-matching theory’, Int. J. Adapt. Control Signal

Process, 6, 499—514 (1992).



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