THE USE OF MODULATING FUNCTIONS FOR

IDENTIFICATION OF CONTINUOUS SYSTEMS WITH

TIME-VARYING PARAMETERS

Witold Byrski, Jedrzej Byrski

Institute of Automatics, AGH University of Science and Technology, Av.Mickiewicza 30, 30059 Krakow, Poland

Keywords: Parameter identification, modulating functions, linear continuous systems.

Abstract: In the paper the use of modulating functions for the optimal identification of the structure and parameters of

continuous linear systems is presented. The modulating functions with compact support [0, h] are used in

convolution filter for transformation of input/output signal derivatives. Based on pre-filtered functions

continuous moving window [t-T, t] is used for on-line identification of piecewise constant parameters Θ

changes of linear system. Optimal quadratic method for identification is presented – with the use of

quadratic constraints on parameters Θ. The numerical results of some examples are shown.

1 INTRODUCTION

One of the continuous model identification method

is based on the use of modulating functions. In 1957

an application of integral transformation with the use

of compact support functions for parameter

identification of differential equation based on input

and output continuous measurement signals given on

[0,T] interval was proposed (Shinbrot, 1957). Such

FIR type filter may be used directly for

identification of continuous systems because of

special properties of convolution.

During primary stage of identification each term

in differential equation is convoluted (multiplying

and integrating) with known modulating functions

on assumed interval [0,T]. Modulating functions and

their derivatives have compact support of length h,

where h<<T, i.e. vanish at the ends of the interval h,

hence initial condition terms of the model also

vanish and finally one can have the new algebraic

(non differential) model with the same unknown

parameters. In the second stage of identification

algorithm, optimal parameter identification problem

is formulated as a problem of minimization of norm

of the equation error in function space L

2

(Byrski

W., 1995, 1996). For nontrivial optimal parameters

solution in optimisation task the constraints for

parameters should be assumed. Assumption of

quadratic constrains of parameters, leads to Gram

matrix G and calculation of their eigenvectors.

Above described methodology one can use for

identification of changes in time-varying parameters,

which are piecewise constant. To this aim the on-

line version of the above method with convolution

filter was prepared and tested. For precise

identification of parameters changes the short

interval T in on-line moving observation window

should be assumed.

Linear time-varying systems were investigated in

many monographs (D’Angelo H., 1970),

Niedzwiecki M., 2000). Publications and references

on identification methods for time-varying

parameters one can find e.g. in survey (Kleiman E.,

2000). However the use of modulating function was

not proposed and tested.

2 MODULATING FUNCTIONS

The idea of using modulating functions and integral

transformation follows from the fact that

convolution of unknown signal derivative y’ and

some known function ϕ is equal to convolution of

original measured signal y and known derivative ϕ’.

Integration by parts shows that the proper choice of

modulating function ϕ with special properties

enables omitting the initial conditions problem

(which are also unknown). Moreover the use of

integral transformation to signals reduces the level

of noise. For identification procedure different

201

Byrski W. and Byrski J. (2006).

THE USE OF MODULATING FUNCTIONS FOR IDENTIFICATION OF CONTINUOUS SYSTEMS WITH TIME-VARYING PARAMETERS.

In Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics, pages 201-204

DOI: 10.5220/0001217802010204

Copyright

c

SciTePress

modulating function can be chosen. Convolution

filter with different modulation functions has

different filtration properties. The required

properties of modulating functions follow from

below presented identification problem.

Given LTI model of the continuous SISO system

which has to be identified

a

i

y

(i)

(t)

i0

n

b

i

u

(i)

i0

m

(t)=

=

∑

=

∑

(1)

where y

(i)

(t), u

(i)

(t) are the i-th derivatives of the

output and input, respectively, m≤n and the n+m+2

unknown parameters a

i

, b

i

are constant. It can be

assumed that measurements of y and u on the

interval [t

0

,T] are given, where T can be also

considered as the current time.

In order to avoid the difficulties caused by the

presence of derivatives y

(i)

, u

(i)

in the model (1) this

model can be transformed into a more convenient

form by means of convolution. Choosing some

special filtering function ϕ with known derivatives

ϕ

(i)

one can calculate the convolution of the both

sides of model (1) and the function ϕ. This function

ϕ is supposed to be nonzero in interval [0,h] and

zero outside this interval (function with compact

support). The convolution represents continuous

shifting window h along time axis.

[]

[][]

==== (t)*y(t)*y(t)*y(t)y

(i)(i)

(i)

df

i

ϕϕϕ

=ττ−ϕτ=

∫

−

)d(t)y(

(i)

t

ht

dτ)(ττ)y(t

(i)

h

0

ϕ

∫

−

(2)

This operation generates the new functions y

i

(t),

u

i

(t), for i=0,..,n(m) for t∈[t

o

+h, T]. Hence the

differential model (1) becomes an algebraic one (3).

Different modulating functions were proposed in

literature (Loeb J., 1965), (Maletinsky V., 1979),

(Preisig H., 1993). In our approach for numerical

tests we will use Loeb-Cahen functions

h][0,t,t)(ht(t)

MN

∈−=

ϕ

,

with min(M,N)≥n, N≠M, and n is order of system.

ay(t) bu(t)

ii

i0

n

ii

i0

m

=

==

∑∑

(3)

3 OPTIMAL IDENTIFICATION

For t

0

=0 continuous measurements of the input u and

output y on interval [0,T] are given. We assume that

u, y ∈ L

2

(0,T). After the convolution of both sides of

(1) with an arbitrary assumed function ϕ and with

their derivatives ϕ

(i)

one can obtain new functions

y

i

∈L

2

(h,T), u

i

∈L

2

(h,T) according to (2). The term

ε∈ L

2

(h,T) added to algebraic equation (3) denotes

the combined effects of immeasurable noise or

general equation error EE

a y (t) b u (t) (t)

ii

i0

n

ii

i0

m

=+

==

∑∑

ε

The norm of difference ε(t) of both sides of model

(3) can represent the performance index of

identification.

Denoting by c(t), the vector of convolutions and by

Θ the vector of parameters a

i

, b

i

one can have

[]

θε

)()(),....,(),(),....,()(

00

tc

b

a

tututytyt

T

mn

=

⎥

⎦

⎤

⎢

⎣

⎡

−−=

The statement of the minimization problem is:

[]

2

L

T

Θ

2

Θ

2

Θ

2

Th,

2

L

(t)Θcminε(t)minJmin ==

Jctct c(tct

TT

L

TT T2

2

===() , () ), ()θθθ θθθG

(4)

The squared norm (4) has a form of inner product in

space L

2

. The real symmetric matrix G is the Gram

matrix of inner products of functions which are

elements of vector c(t)

G =

−−

−−

−−

−−

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

y

0

,y

0

,.., y

0

,y

n

y

0

,u

0

,.., y

0

,u

m

,.., ,.., ,.., ,.., ,.., ,..,

y

n

,y

0

,.., y

n

,y

n

y

n

,u

0

,.., y

n

,u

m

u

0

,y

0

,.., u

0

,y

n

u

0

,u

0

,.., u

0

,u

m

,.., ,.., ,.., ,.., ,.., ,..,

u

m

,y

0

,.., u

m

,y

n

u

m

,u

0

,.., u

m

,u

m

⎥

⎥

⎥

⎥

⎥

,

where inner products are given by formulae

dτds)sτu((s)ds)sτy((s)

τd)(τu)(τyu,y

h

0

(j)

T

h

h

0

(i)

T

h

jiji

⎥

⎦

⎤

⎢

⎣

⎡

−

⎥

⎦

⎤

⎢

⎣

⎡

−

==

∫∫∫

∫

ϕϕ

(5)

ICINCO 2006 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL

202

For avoiding trivial solution Θ = 0 one must chose

the constraints for parameters. The common

constraint used by all researchers is for instance a

n

=1

or a

0

=1 what is interpreted as normalization.

However quadratic constraints are more general.

3.1 Optimal Solution Under

Quadratic Constraints

One can formulate the optimization problem (4) with

unit ball Θ∈B(0,1) constraint for the parameters

}1:{

T22mn

=ΘΘ=Θ∈Θ=

++

RB

(6)

The Lagrangian functional L for the above problem

has a form

)1(GL

TT

ΘΘ−λ+ΘΘ=

From the necessary condition of minimum it follows

directly that

oo

022

L

Θλ=Θ⇒=Θλ−Θ=

Θ∂

∂

GG "

(7)

Hence the optimal element

o

Θ

is an eigenvector of

the Gram matrix G and the Lagrange multiplier λ is

its eigenvalue. From the definition of spectral norm

it follows that the minimum of J

2

on unit ball B(0,1)

is equal to the minimum eigenvalue

[

]

min

T

B

2

B

minJmin λ=ΘΘ= G

(8)

The optimal eigenvector

Θ

o

=w

min

should thus be

chosen, as that which correspond to the minimum

eigenvalue λ

min

of G. Gram matrix G is real and

non-negatively definite. From the formulae (8) one

can find the optimal parameters of (1) and (3) based

on the calculations on interval [0,T]. Such a

parameters represent average value for overall

interval [0,T], (Byrski W., 1999, 2000).

For the current time t>T (in on-line applications)

one should repeated calculations in moving window

[t-T, t]. Then elements of Gram matrix start to be

functions of t and also w(t) and λ

min

(t). Hence

calculations of the minimal λ

min

(t) and eigenvectors

of G(t) should be repeated for every t in interval

[t-T ,t]. The value of Θ(t) represent average value of

parameters for whole interval [t-T ,t]. If the interval

T is short enough it is possible to detect the changes

in system parameters.

4 NUMERICAL TESTS

We present application results of above described

methodology for third order system with time-

varying parameters a

1

, a

2

(piecewise constant

parameters as in Fig.2 and Fig.3 - solid line), b

0

=2.

)()()()()()()(

012

tubtytytatytaty =+

+

+

The input and output signals were measured as in

Figure1 and Figure 4. For given input/output vectors

(10000 samples each) program written in C++

automatically searches for the best continuous model

within presumed possibilities.

For starting the program one should prepare the

data file with samples. In this file also, many

possible orders m

i

of input derivative (numerator

degrees of Transfer Function) and many possible

orders n

i

of output derivative (denominator degrees

of TF) should be placed. Also many possible values

of supports h

i

for convolution filter and many

different M, N in ϕ can be proposed. Parameters of

each proper model structure (m

i

≤n

i

) are identified by

the use of different filters based on (2) and (8). Next

each identified model is simulated (by Runge-Kutta

method) and the Output Errors are calculated. The

best model is automatically chosen. Sometimes

different structures give similar small Output Errors

– then special procedure search for the structure,

which is less sensitive to changes of support h

i

in

convoluting filter.

Figure 1: The control signal.

THE USE OF MODULATING FUNCTIONS FOR IDENTIFICATION OF CONTINUOUS SYSTEMS WITH

TIME-VARYING PARAMETERS

203

Figure 2: Time-varying parameter a

2

.

Figure 3: Time-varying parameter a

1

.

In Figure 2 and Figure 3 high quality of detection

of rapid parameter changes is observed. Exemplary

result was obtained for T=5, h=1, M=5, N=4. The

differences between measured output and simulated

output presented in Figure 4 are almost invisible.

Figure 4: The outputs of system and model.

5 CONCLUSIONS

In the paper the optimal parameter estimator for

linear continuous systems with time-varying

parameters was presented. The identification method

is based on the convolution of the input/output

measurements with some chosen modulating

function. Preprocessed data are used in Moving

Window Identifier which operates on finite time

interval [t-T ,t], for T<t<T

0

. Hence the solution

gives optimal parameters which are window average

values.

T)(t,

o

Θ

is function of time t and T.

REFERENCES

D’Angelo H., 1970. Linear Time-Varying Systems,

Allyn&Bacon,Boston, USA.

Byrski W., S.Fuksa, 1995. Optimal identification of

continuous systems in L

2

space by the use of compact

support filter. In International Journal of Modelling &

Simulation. IASTED. vol.15 .No.4 .1995. 125-131

Byrski W., S.Fuksa, 1996. Linear adaptive controller for

continuous system with convolution filter. In IFAC,

13th Triennial World Congress, San Francisco USA.

Byrski W., S.Fuksa, 1999. Time variable Gram matrix

eigenproblem and its application to optimal

identification of continuous systems. In Conference

ECC’99, Karlsruhe, Germany.

Byrski W., S.Fuksa, 2000. Optimal identification of

continuous systems and a new fast algorithm for

on-line mode In Proc. of IFAC SYSID2000, Santa

Barbara, USA.

Kleiman E.G., 2000, Identification of Time-varying

Systems-Survey, Proc. of IFAC SYSID2000, Santa

Barbara, USA.

Loeb J, G.Cahen, 1965. More about process identification,

In IEEE Trans.Automat.Control, AC-10.

Maletinsky V.,1979. Identification of continuous

dynamical systems with spline-type modulating

functions method. In Proc. of V IFAC Symp. on

Identification and SPE, Darmstadt, Vol.1.,

Niedźwiecki M., 2000. Identification of Time-varying

Processes, John Wiley, NY.

Preisig H.,A., D.W.T.Rippin, 1993. Theory and

application of the modulating function method.

Review and theory of the method and theory of spline

type modulating functions. In Computers Chem.. Eng.,

Vol.17, No 1,1-40, Pergamon Press.

Shinbrot M., 1957. On the analysis of linear and Nonlinear

Systems. In ASME Trans.. 547-552.

ICINCO 2006 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL

204