Identification of Fast Time-Varying Systems Applied to a Turbo-Generator Set

Copyright © IFAC 12th Triennial World Congress Sydney. Australia. 1993 •

IDENTIFICATION OF FAST TIME-VARYING SYSTEMS APPLIED TO A TURBO-GENERATOR SET V. Rachev* and H. Unbehauen** *Departmt!nt of Automation, lligher Institute of Chemical Technology, I J56 Sofia. Bulgaria **Departmt!nI of Electrical Engineering, Ruhr University, 4630 Bochum. Germany

Abstract. An identitication method for systems WiUI fast time-varying parameters is applied to a turbo-generator set. The utilized method approximates the time-varying parameters of a discrete linear model by their Taylor polynomials with respect to an auxiliary variable, which is correlated with the parameter changes. The approximation scheme is applied in such way that the parameter identification problem is turned to a problem of observing the states of a known state space model. TIle results of the identification ofa turbo-generator set are given. Key Words Fast time-varying systems; auxiliary variable; parameter identification; turbogenerator.

engineering are nonlinear block-oriented models, as e.g. the Wiener, the Wiener-Hammerstein cascade and the Hammerstein ones. In spite of their simplicity and possibility to obtain an exact description of some special nonlinearities, they are not always applicable and require a suitable structure of the model.

1. INTRODUCTION

For many technical processes the parameters of the corresponding models are not constant. They alter because of internal or external influences depending on time. Generally, such systems with arbitrary parameter variations can be described by the standard predictor model

When the dynamics of the process depend on external influences, the system can be represented as multivariable by the nonlinear multi-input single-output difference equation given for the first time by Kolmogorov (1942). The number of the parameters of the general model is very high and the model can easily become overparameterized. To avoid the complexity of the model and the possible numerical ill-conditioning, a proper structure selection algorithm should be applied (Kortmann and Unbehauen, 1988).

y(k+l+d»=al(k)y(k) + a2(k)y(k-l) +... + +ana
(1.1)

where y(k) and u(k) are the directly measurable plant output and input respectively, d is the time ~elay, and the paramete~s ai a~d bj (i=1,2, ... , na ; J=I,2, ... , nb) are functIOns of time. When the parameters change frequently and rapidly, only the information of the plant input and output is not sufficient for accurate predictIOn and adaptation. That is why most estimatIOn procedures for fast time-varying systems are based on additional knowledge about the parameter variation.

In both cases, when internal or external influences or perturbations occur, the resultant models are nonlinear, which brings difficulties in the control system synthesis. Often, for practical applications, a simplification under proper assumptions could be done. For practical applications the nonlinear characteristics are usually approximated by a linear model the parameters of which vary according to the changes in the operating conditions. This approach enables afterwards the application of linear theory for process control.

In the cases when the parameters of the system vary because of internal reasons, e.g. the system trajectory or the operating point, the traditional description of the processes is based on functional series, such as the Volterra (1930) or Wiener (1958) series. The usage of these methods in their original form is limited because of the large number of unknown parameters and the difficulty of interpreting the results. Well-known in control

If the operating conditions are well defined and their number is not too large, the process can be described by one linear model with several sets of parameters, corresponding to different operating conditions (Diekmann and Unbehauen, 1985). 1033

Adaptive identification algorithms, which enable the preliminary specified set of models to be widened on-line were proposed by ledner and Unbehauen (1988) and by Uosaki and Yotsuya (1991). Another comprehensible approach is to describe directly the dependence of the linear model parameters upon available auxiliary signals. Habcr and Kewiczky (1985) and Vuchkov et at. (1985, 1986) suggested the usage of linear difference models with parameters depending on the variation of some measurable external signals which specify the working conditions. The changing parameters are approximated by a linear combmation of functions of measurable signals, which gives a generalized quasi linear model with constant parameters. A similar approach was proposed by Xie and Evans (1984) who approximate time variations of the plant parameters by their differentials and developed an adaptive control algorithm for quite rapidly time-va rying systems. The result was generalized by Li (1987), who approximated the time-varying parameters by their Taylor polynomials with respect to timc. Later, Li and Unbehauen (199Ia) extended this idca and utilized the measurable auxiliary variable of gain shcduling for tracking time-varying plant parameters. They applied a polynomial approximation scheme so, that the parameter idcntification prob\cm is turned to a prob\cm of observation of the states of a known state space model. In thc same paper global convergence has been proved for stable invertable systems, the parameters of which are polynomials in an auxiliary variable. The present paper deals with the first practical application of the above mentioned identification method by Li and Unbehauen (1991 a, b). Section 2 gives briefly the theoretical background of the estimation procedure. Section 3 contains a description of thc turbo-generator to which the method is applied and the results of its identification. Some conclusions are given in Section 4.

Fi~'lJrc 2.1. Notation for approximation ) vU)

aj and bj (i= 1,2, ..., na ; j= I ,2, ... , nb) are functions of the directly measurable auxiliary variable v(k) . If PI ( for 1= 1, 2, .. . , n; n = na + nb) is suffiCiently smooth with respect to v, PI (k) can be approximated with the first NI terms of the Taylor polynomial around PI (r), where r < k (Figure 2.1). Neglecting the approximation error, we can write [ v(k) - vCr) ]' II . NI (2.4) (Nil [v(k) - vCr) ] NI! + ... + PI (r)

PI(k) =

PI(r)+ p\I )(r)

and

piNI)(k) =

xr(r) =

[PI(r)

cl'

b 2 Iv(k»)

...

[1 0 ... 0] ,

the (N I+l) xCN I+I) matrices

o o

Nil

0 0

v(k) - vCr) I

and the composite ones

(2 . 1)

x(r) =

[xr(r)

C

diag[

A(klr) =

(2.2)

cl'

...

xI (r) eT 2

diag[ AI (klr)

.. .

x~(r)r

c~] ,

A2 (klr)

An(klr)]

we have

an,lv(k)1 bdv(k)[

F,

tl vCr) t'-I

[ v(k) - vCr)

(N I - 1)1

y(k-na+l) u(k) u(k-I)

bn"lv(k)[

-

p[ NI)(r)],

[ v(k) -

where

[)(k) = I adv(k)1 a2Iv(k»)

p[l l (r)

+bn"lv(k)l U(k-nb+l)

U(k-nb+ I)] ,

I

l

p\N j)(r) .

vCk) - vCr)

+an,lv(k)]y(k-na+l) +bdv(k)1 u(k) +

mT(k+l+d)= [ y(k) y(k-I)

t

]'

Defining for 1= 1,2, ... , n the (N I+ I) x I vectors

y(k+l +d) = adv(k)1 y(k) + a2[v(k)1 y(k-I)+ .. +

mT(k+l+d) p(k) ,

V(k)~!V(r)

[v(k) - vCr) (N) (NI-I)! + ... + PI I (r)

We consider a time-val)'ing system similar to (l.l) including an auxiliary variable v(k), which correlates well with the parameter changes. The general S[SO predictor model can be written as

=

pil)(r)+ pi 2)(r)[

pil)(k)=

2. DESCRIPTION OF THE IDENTIFICATION METHOD

+b 2 Iv(k)] u(k-I) +

v(r)

(2 .3)

y(k) and u(k) are the plant output and input espectively, d is the time delay, and the parameters

C x(k)

p(k) x(k)

A(klr) x(r)

x(O)

xo .

(2 .5)

Now, the unknown plant parameters are approximately describcd by a known time-varying state space model (2 .5). C and A(k1r) are the

1034

known observation and state tranSItIon matrix respectively, x(r) is the unknown state and x is the unknown initial state. In order to achieve gigh approximation precision, i.e. to keep I v(k)-v(r) I small, (k-r) should be as small as possible and x(r) must be adjusted on-line. Defining a generalized data vector h (k+l+dlr) = mT(k+l+d) C A(klr),

In (2 .7d), Q(k) ~ mI is a positive semidefinite matrix. I is an identity matrix and m is a positive real number. To prevent P(klr+ I) from going to infinity, when the trace P(kJr+ I)~ m the matrix Q(k) is set to zero. The initial conditions are

the system (2 .5) can be rewritten as

where Po

y(k+ l+d)

h T(k+ l+dlr) x(r)

x(r+l)

A(r+1Ir) x(r)

x(O)

xo .

i(O) = Xo P(O) = Po , ~

m I is a positive definite matrix.

3. DESCRIPTION OF THE TURBOGENERATOR SET AND EXPERIMENTAL RESULTS

(2.6)

Now, the unknown time-varying system (2.1) is transformed into the known time-varying state space model (2.6) with unknown initial state xo' In (2 .6) the observation equation represents the dynamics of the system (2 .1), the state transition equation represents the dynamics of the parameters and the initial equation Xo represents the uncertainty of the parameters. So the problem of identification of the unknown parameters in the system (2.1) is transformed to the problem of observation of the state of the system (2 .6). Let i(k1r) be the estimate of x(r) based on the measurements up to and including time k. We can obtain i(k1r) using a least-squares algorithm, based on t le form of a Kalman filter. The covariances of the system noise and the measurement noise are assigned to be 0 and R(t), r
The turbo-generator set consists of an air pressure turbine driving a synchronous generator (Figure 3.1). The maximum power of the turbine is 240W and it is connected to common user network for pressured air. The maximum pressure in the network is 6 bar. The generator is a I2-pole, 3-phase synchronous one of the type usually used in the lorries: nominal voltage nominal current max. number of revolutions max . exciting current

28 V; 17 A; 12000 r/min;

I.3A.

'.

At time t=kT (T is the sampling time) : •

the new values for u(k), y(k) "l-nd v(k) are obtained and the data vector m (k+ 1+ d) is formed, see (2 .2);

At time kT
i(k-llr) + k(k) E(k) ,

where E(k) = y(k) - y (klk-l-d) ;

(2 .7a)

(2 .7b)

The stationary output voltage u , is proportional to the product of the magnetic fiefd density \1' and the angular velocity £\ :

the estimates i(klr) and the covariance matrix P(k\r) arc adjusted to the new point v(r+ I), around which the continious functions p, (v) are approximated (2 .7c) i(klr+I)= A(r+llr)i(klr)' P(klr+l) =

(3.1 ) Because of the multiplication of the two functions in equation (3.1) the changes of the air pressure Pa affect the dynamics of the main transfer behaviour between u and ie . To describe the variation of the dynamic b~haviour of the plant the air pressure p is introduced as an auxiliary variable. The step responses of both channels are shown in Figure 3.2. The system is described as a MISO system by means of the predictor time-varying model of (2 .1)

A(r+llr)P(klr)A(r+Ilr)T +Q(k) ; (2.7d)

•

The turbo-generator set

The generator supplies a balanced load of 1.80. The main output of the set is the generated voltage u . The controllable input signal is the field cfirrent i of the synchronous generator. The air pressure e Pa in the turbine is assumed to be measurable out not controllable.

the new estimates are calculated : i(klr) =

•

Figure 3.1

the correcting vector k(k+ 1), the matrix P(k\r+ 1) and the predicted value y(k+ I\k- d), needed for the next time interval, are calculated :

P(klr+l) h(k+llr+l) , R(k+l)+ hT(k+llr+l) p(klr+l) h(k+1Jr+1) (2 .7e) P(k+llr+I)=[I- k(k+I) hT(k+ 11r+ 1)] P(klr+I) , (2.7f) T y(k+llk-d)= h (k+llr+l) i(klr+l) (2.7g) k(k+l)=

y(k+d) = a,(v)y(k) + a2(v)y(k-l) + a 3(v)y(k-2) + b,(v)u(k) + b 2 (v)u(k-l) +b/v)u(k-2) + c,v(k) + c2v(k-l) , (3.2) where the output y, the input u and the auxiliary variable v correspond to ug , ie and Pa respectively and the coefficients a j and Dj, i=j= I,

1035

2, 3, are functions of the auxiliary variable v at the time k . The measurement noise and a small hysteresis of the plant are not taken into consideration in the model, which influences slightly the prediction error. The relationship between the dynamical normalized signals and their physical values is given in Table 3.1. In Table 3.2 the input signals for two experiments are described with 1000 samples each. The chosen sampling time is O.ls and the nominal values of the input signals are ie = 0.25 A and PA = 2.7 bar.

The following 6 models are compared : I)

SISO model of 3rd order for channel without any knowledge about Pa;

2)

MISO model of 3rd order for channel uji and 2 nd order of the nominator of the chann~1 • ujp , where all coefficients are treated as constant or slowly time-varying;

3)

MISO model in which the coefficients a and b are expanded in 'Taylof , i=j=I, 2, 3, polynomials at v= v(k-I) and NI 2 345 6= 2 . ... . (see Eq. (2.4) );

4)

the same as previous but 1; 1; 0 ;

5)

the same as previous but v= v(k) ;

6)

the same as previous but v= v(k-2) ;

For the parameter estimation the SISO procedure described in Section 2 is used (Equations (2.7» . In

Table 3.1 The relation between the dynamical normalized signals and their physical values ~

-)

~ )

u (t)

-O.3A ~ i e

-)

~ O.3A

~ vet) ~

)

yet) -<

)

Table 3.3

test 2

ampl.

samp.

Prediction error loss functions

model

type

ampl.

samp. time

PRBS

±O.2

(3+20)k

6= 1; 0; 0;

NI ; 2; 3; 4; 5;

Two tests had been performed. During test 1 the auxiliary variable v had been changed abruptly, whereas during test 2 fast but smooth changes in v

Input signal description

tc::st I

type

~

< pa < - 20Y 1.8bar- 1.8bar -20Y -< ug <

Table 3.2 test

-)

uti.

lest

1

lest

2

J%

Jrnax%

J%

Jrnax%

12.75

128.30

13.55

66.29

11.01

88.11

8.73

59.82

6.76

43 .32

4.86

45.42

7.18

42.60

4.48

47.65

7.79

64.68

5.43

56 .43

10.11

136.29

5.02

42.95

time: PRBS

u

±O.2

(3+20)k

I)SISO model , N*/=O

w.

v

.

±0.45

(6+ 15)k

sin(k)

±0.45

(1O+0)k"

2)MISOmodel" N'F 0

nOIse

3)MISOmodel,

"for one period

'uniform distributed

N*F 2, v=v(k-l)

order to obtain the estimates of the nominator of the transfer function of the second channel (coefficients Cl and c2 ), the dimensions of all matrices and vectors are increaced by 2. Thus, the data and parameter vectors according to (2.2) and (2.3) respectively obtain the form:

4)MISO

model,

N'F 1;0;0; 1;1;0, v=v(k-I)

5)MISO model,

m T(k+l+d) = [y(k) y(k-l) y(k-2) u(k) u(k-l) u(k-2) v(k) v(k-l)] ,

N*Fl;O;O; I; 1;O,v =v(k)

p(k) = [al(v) a2(v) a 3 (v) bl(v) biv) b3 (v) Cl c21T

6) MISO model,

The estimation procedure is started with zero initial conditions. The results of the identification are given in Table 3.3 by means of the following normalized loss functions

N*F 1;0 ;0;1;1 ;0, v=v(k-2)

•N/, 1= 1, 2, .. ., 6 is the chosen order of the polynomial of Gp G 2 , G 3 , bp b2 , b3 respectively.

N

L I y(k) - Yp (klk-I) I 1=

_k_ = _I- ; " ; N - - - - - - -

had been applied. It can be seen from both tests that the introduction of a MISO description, and after that of an auxiliary variable for the coefficients of the main transfer system, decreases significantly the prediction error. In the case of test 2 greater chanses are observed in the mean error loss functIOn, whereas for test 1 the maximum error loss function is more affected. The introduction of a delay in the auxiliary variable v can also improve the results to some extend (see the loss functions for models 4,5,6) .

100% ,

L I y(k)- YI

k= I I~

=

max

where _ 1 Y= N

(IY(k)-yp(k1k-I)1 , k -300, _ 301, ... ,N ) 10(Wo, Yab., ••

N

k~1

_

y(k),

Yabs, ••

I

N

= N -300 k =~OO Iy(k) I,

Figure 3.3 shows the prediction error for the models 1,2 and 4. A sample of the predicted and the true values for the same models is given in Figure 3.4. The process of estimation of the coefficients for model 3 is given in Figures 3.3 to

and Yp is the predicted output by (2.1) and N is the number of the samples (N = 1000). The loss functions I and Imax evaluate the mean and the maximum error of prediction respectively.

1036

3.6. From Figure 3.6 can be seen that the dependence of the coefficients a and b for i=j= 1,2,3, on v is linear, so only thb first two thms of the Taylor expansion are necessary and also that only some of the coefficients depend on v, which agrees with the results given in Table 3.3 . a) for channel ug'\ (y/u) b) for channel ug'Pa (y/v) y~

ym

Figure 3.2 Step responses of the main transfer behaviour between ug and ie at ditTerent values of Pa (a) and between ug 4. CONCLUSIONS An identification method for fast time-varying systems, based on the approximation of the coefficients of the model with polynomials of an auxiliary variable, had been applied to a turbogenerator set. It was shown that by estimating only 3 coefficients more a significant increase in the prediction accuracy was achieved. It can be ascertained that the utili/.ed method copes with static as well as dynamic nonlinearities. The results of the identification can be easily interpreted and used for better understanding of the process. The examined method can be successfully utilized to adaptive control algorithms based on prediction of the plant output.

Li , Z. and H. Unbehauen (l99Ia) , 'Adaptive control of fast time-varying systems'. Proc. 9th IFACIIFORS Symposium on Identification and System Parameter Estimation, Budapest, Hungary, 1,35-40. Li, Z. and H. Unbehauen (199Ib), 'Self-tuning control of time-varying systems via an auxiliary variable'. Proc. European Control Conference, Grenoble, France, 1,666-671. Uosaki, K. and M . Yotsuya (\ 991), 'Adaptive identification for abruptly changing systems'. Proc. 9th IFA O IFORS Symposium, Budapest, Hungary, 2, 17 \3-1718. Yolterra, Y. (1930), Theorv of functionals. Glasgow : Blackie. . Yuchkov, I., K. Yelev and Y. Tsochev (1985), 'Identification of parametrically dependent processes with applications to chemical technology. Proc. 7th IFACIIFORS Symp. ]dent. and Syst. Param. Esl., York, UK, 1, 1089-1094. Yuchkov, L, K. Yclev and Y. Tsochev (1986), and design of 'Parameter estimation experimental plans for parametrically dependent processes'. Prep. IMA CSIIFA C S:vmp. Modelling and Simul. fur Control of Lumped and Distrihuted Parameter S:vstems, Villeneul'e, France, 277-280. Wiener. N. (1958). Nonlinear problems in random theory. New York : Wiley . Xie, X. Y. and R. J. Evans (1984), 'Discrete-time adaptive control for deterministic time-varying systems'. Automatica, 20 , 309-319.

"mmml

-0.:

ACKNOWLEDGEMENT

_0.'

The first author was supported at Ruhr-University by a ECC-Scholarship under the Tempus-Project JEP 253X/92

REFERENCES Diekmann, K. and H. Unbehauen (1985), 'On-line parameter estimation in a class of nonlinear systems via modified least-squares and instrumental variable algorithms'. Proc. 7th IFACIIFORS S:vmposium, York, UK, I, 149153. Haber, R. and L. Keviczky (1985), 'Identification of 'linear' systems having signal-dependent parameters'. Int. J. Systems Sci. , 16, X69-X84. Jcdner, U. and H. Unbehauen (I 9XX), 'Intelligent adaptive control of a class of timevarying systems'. Proc. of 12th IMII C.'" World Congress, Paris, I , 377-3XO. Kolmogorov, D. (1940) , 'Interpolation and extrapolation of stationary series', Bulletin de I'll cademie des Sciences de "USS'R. Kortmann, M. and H. Unbehauen (1988), 'A model structure selection algorithm in the identification of multivariable nonlinear systems with application to a turbogenerator set'. Proc. 12th IMIICS World Congress on SCientific Computations, Paris, 2, 536-539. Li, Z. (19X7), 'Discrete time adaptive control for linear fast time-varying systems' . IU~E " 'ans. lIu/. Con/r., AC-32 , 444-447.

the rediction error for model 2

e

the rediction error for model 4

r-------~~~~~~~~~~~~__,

Figure 3.3 Th.: nonnali/..:d input signal s u(k), v(k) and th.: prdiction ':lTor .:(k) for th.: models 1, 2 and 4 (tcst 2 )

1037

Yp

o"

for model!

,---------~~~~~-------------,

,---------,

s,

y

Yp

,---------~~~~~-------------,

y

'" ,-------------,

for model 4

03

-......

-""

u-

e, ,-------------, "2',------------,

.u!

----....,

V·.

.

)",/ r

-~:U

,_

0., :

-.' ,.., .._...•... _... ... !

"

'-___ .JJJ

•

Figure 3.4 True (- ) and the predicted (- -) output values for models I , 2 and 4 and the corresponding input signals for 2.

al

=

'.

_

IIID

_

__

,.

_

• • _'

Figure 3.5 Estimated values of the coefficients of Equ. (3 .2) for test 2 as functions of time ( model I model 2 , model 3 - - - )

b l = f(v)

f(v)

I -~t

b 2 = f(v)

I ~t

b 3 = f(v)

I -~ t l

Figure 3.6

Estimates of the coefficients

al'~'

a 3 ,bp b2, b3 as functions of v and time (model 3, test 2) 1038