A Prefiltering Instrumental Algorithm Using Sinusoidal Test Signals

Copyright © IFAC Identification and System Parameter Estimation 1985. York. UK. 1985

A PREFILTERING INSTRUMENTAL ALGORITHM USING SINUSOIDAL TEST SIGNALS C. Q. Zhang and X. M. Zhang Department of Automatic Control, Shaanxi Institute of Mechanical Engineering, Xian , Shaanxi, People's Republic of China

Abstract.

A novel instrumental variable identification

algorithm based on

special digital time varying filter is introduced in this paper. Using this special filter, the ideal input output data can be obtained. So, even under both deterministic and stochastic disturbances, the instrumental variables are easily able to achieve from the contaminated input output data. The robustness of this algorithm has been shown by simulation results. The simulation results are better than any other identification algorithm. The discussion is limited to single input single output system. Keywords.

1

Identification, filtering, instrumental variable.

INTRODUCTION

The block diagram is shown in Fig. 1, we have

Consider the difference equation model of

<1.2 )

a 5150 discrete stochastic system, A(q-1)y(k). B(q-1)u(k)+v(k)

where w(k) is unmeasurable ideal system

v(k)= C(q-1)e(k)

output, y(k) is corrupted by disturbances but can be measured.

where

y(k) is the output signal, u(k) is

the input signal and e{k) is the white noise with zero mean and variance 0-&. Further in (1.1) the polynomials are:

Fig. 1.

Block diagram

Instrumental variable (IV) method was in-

and q-1 is backward shift operator, i.e.

troducwd by WO"g and Polak 1n 1967 to deal with identification problem, cf. ref.1.

q -1 y{k)- y(k-1)

In ref. 2, Finigan and Rowe proposed a It is assumed that the system is asymtotically stable, all zeros of A(q-1) lie outside the unit circle and also A(q-1) B(q-1) have no common factors. For simplicity all

method to generate a strong instrumental variable in 1974. Soderstrom and Stoica had made a comparison of some IV methods 1n consistency and accuracy aspects, cf. ref. 3. At author's knowledge, all those

polynomials have the same order n.

proposed methods are concentrated in finding a suitable strong instrumental variable

1971

C. Q. Zhang and X. M. Zhang

1972

In this paper, a special digital time

It can be proved that all the compo-

varying filter is introduced so the ideal

nents with frequency other than the tuned

system output w(k) can be picked up from

one will die out as time t

the corrupted data and then the IV method

nity.

tends to infi-

can be applied in a straight forward way. The discrete transfer function of the This paper is organized as follows: In

filter is: (cf. ref. 4 and 5)

section 2 the discrete transfer function and some features of the filter are introduced In section 3 IV configuration using spe-

Ufb(k)1 u(k) • ( l-z -2) I

( 1-2coswiTsz-1 +z -2)

cial filter is discussed. The simulation

(2.3)

results are given in section 4 and a brief conclusion is made in section 5. 2

FILTER

where w is the tuned frequency and Ts i is the sampling period in second.

The filter introduced here is called SEAPA filter developed by Silveira and Doraiswami, cf. ref. 4. SEAPA is the short form of Signal Enhancing And Perturbation Annihilating. Essentially it is a resonance based time varying filter. The block diagram is shown in Fig. 2.1.

SEAPA filter has some features: 1, asymptotically stable; 2, asymptotical tracking Sinusoidal signal with no phase shift and no amplitude deviation; 3, asymptotical

annihil~ting

all the

signals with frequency other than wi.

If the resonance frequency of the filter is tuned to fi Hz. , then the output of the filter ufb(t) is intended to be

A mixture input signal comprising four sinusoids with frequencies 0.06, 0.15, 0.57, 1.31 Hz and zero mean white noise is shown in Fig. 2.2 for K-1800-2000, where K refers to number of samples. The output signal during K=1800-2000 is shown in Fig. 2.3. It can be seen evidently that the asymptotic tracking and asympto-

Equation (2.1) divided by time t, we get

tic annihilating characteristic hold.

the filter output

(2.2) which is exactly as same as the input signal. u( t)

---1

resonance part tuned to w i

11-------1

lit

I.....--~

t

Fig.

2.1

SEAPA f i lter

1973

A Prefiltering Instrumental Algorithm

o I}

~

~

\I N

~

N

,

I

~

{f\

•

11

Fig. 2.2

Fig. 2.3

\

A record of mixture input signal

A record of output signal

3 IV IDENTIFICATI ON SCHEME

J 1I

~

0./

I

"

K=1800-2000

K=1800-2000

fication scheme is shown in Fig. 3.1.

It is necessary to use sinusoidal testing

In Fig. 3.1 only output signal is contami-

signals with different frequency and differ-

nated by both stochastic and deterministic

ent amplitude when we apply SEAPA filter in

disturbances. If input signal is also

the IV identification scheme. Frequency do-

corrupted, it needs prefiltering too. For

main design of input signals has been dis-

the filter is digital oriented, the in-

cussed extensively by many authors, cf. ref.

creasing number of signals to be filtered

6. It was proved that it is always possible

only increase the burden of software and

to find an optimal input comprising a finite

the computer is effective enough to com-

number of sinusoids. Further, it was proved

plete the computation within the sampling

that: Consider the system described by

interval Ts. The block diagram of SEAPA

eq.(l.l), then an optimal power constrained

filters tuned to w , ••• ,w is shown in m 1 Fig. 3.2 • So the output signal compri-

input minimizing J- -log det

M exists

com-

prising not more than 2n sinusoids. (cf.

sing m sinusoids can be easily obtained.

ref. 6 Theorem 6.4.4), where n is system order and

M is

the average Fisher's infor-

mation matrix.

The critical point of using this algorithm is how to determine the different frequencies of test sinusoids and the sampling

A second order system only needs four sinusoids for identification. The IV identi-

interval Ts ( or sampling frequency f ). s fs~ 2f , we can m

By Shannon's formular:

c. Q. Zhang and X. M. Zhang

1974

easily determine fs' where fm is the

white noise

_;;.;.:.~.:.....c~:;';;"':"'--I

highest frequency of test sinusoids which

noise model

J--

depends upon the bandwidth of the system deterministic disturbance

and it should be estimated.

,.....-__---;1

l~--·-~o()(''IC~~---

pia te mode 1

u()c) filter tuned to w

.. (k)

----1

sine generator

y(k)

J

u(k) ISEAPA filter w(k)

filter tuned to w

Fig. 3.1 Fig. 3.2

I

~--~LI~I~v~a:l~go:r~i~t:h:m~J--Y~r___~ u(k) y(kJ IV scheme using SEAPA filter

SEAPA filter for mixture input

4

SIMULATION AND COMPARI SON

System model: y(k)-1.64y(k-l)+0.73y(k-2)= 0.54u(k-l)+ disturbances the bandwidth is 20 rad/sec. A) Parameter estimation using SEAPA IV algorithm with different signal noise ratio(SNR) Simulation results see TABLE 1 under conditions: Iteration =500, w D O.13, w =0.05, 1 2 w3 -O.47, w4 - 0.93 TABLE 1

Results of estimation under different SNR Parameter

SNR a

a

1

2

b

b1

O

b

2

Actual

-1.64

0.73

Estimate

-1.645

0.733

0.02

-1.658

0.745

-0.02

5

-1.65

0.74

0.07

0.402

0.07

3

-1.635

0.741

0.03

0.492

0.006

2

-1.61

0.705

-0.009

0.53

0.04

1

-1.619

0.717

0.17

0.167

0.25

10 7

0.00

0.54

0.00

0.52

-0.009

0.59 2

-0.05

B) Parameter estimation using SEAPA IV algorithm with different sampling interval Ts. Simulation results see TABLE 2 under conditions: Iteration - 500, T

w 1

w 2

1.00

.05

.13

0.25

.40 1.1

s

w3 .47 2.1

w 4 .93 3.1

s

w1

w 2

0.18

.15

1.21

T

0.06 4.67

w 3 2.89

9.32 14.3

w 4 5.81 20.31

1975

A Prefiltering Instrumental Algorithm

TABLE 2

Results of estimation under different Ts

Parameter T

s

Estimated bO

a2

a1

b1

b2

1.00

-1.66

0.745

-0.001

0.58

-0.06

0.25

-1.63

0.72

-0.02

0.56

0.01

0.18

-1.61

0.702

0.02

0.49

0.06

-1.67

0.752

-0.02

0.05

0.577

-0.04

IV algorithm can still give good estima-

C) Comparison Comparison is made between three algorithms,

tion results while LS and IV algorithm

I.E., least square (LS), instrumental var-

are fail to do so (cf. sectio n 4 C). The

iable (IV), prefiltering IV (PIV). The tes-

number of input sinusoids depends upon

ting signal of first two algorithms is PRBS

the order of the system. The frequency of

with unit amplitude. The third one applies

the sinusoids is limited by the bandwidth

4 sinusoids, their amplitude all equal to

of the system and the sampling interval is

one and frequencies are 0.45, 1.1, 3.1, 3.5

selected to match with the maximun input

rad/sec respectively. Comparison results

frequency.

see TABLE 3. If we

do not know the frequency of the

testing signal or the signal drifting a

5

CONCLUSION

little bit, then the filter will be out of tune and fail to estimate the parameters.

As the simulation results have shown, the

Hence adaptive SEAPA filter is needed to

proposed prefiltering IV algorithm is re-

solve this problem.

markably effective. Reasonable results are obtained even the signal noise ratio is relatively low (cf. section 4 A). Due to the

REFERENCES

nature of the filter, the algorithm can deal with both stochastic and determinis-

1, Wong, K. Y. , and E. Polak (1967). Iden-

tic disturbances. The proposed prefiltering

tification of linear discrete time system using the instrumental varia-

TABLE 3

Comparison results between three algorithms Parameter

Algorithm

Disturbance

Estimated

Iteration

LS

white

500

-1.61

0.71

0.00

0.54

0.02

IV

white

500

-1.70

0.83

0.00

0.54

-0.03

PRE-IV

white

500

-1.63

0.72

0.00

0.55

0.01

LS

white+rt

1000

-1.83

0.83

0.00

0.54

-0.10

IV

white+ IT

1000

-1.91

0.91

0.00

0.54

0.14

PRE-IV

white+(t

1000

-1.62

0.64

0.06

0.63

0.05

LS

whi te+t

1500

-2.00

1.00

0.00

0.54

-0.20

IV

white+t

1500

-2.00

1.00

0.00

0.54

-0.20

PRE-IV

white+t

1500

-1.57

0.57

0.06

0.65

0.11

0.73

0.00

0.54

0.00

The ideal parameter is -1.64

IPE-r

C. Q. Zhang and X. M. Zhang

1976

ble method. IEEE Trans. on Automatic Control, AC-12, No.6. 2, Pinigan, B. M. , and I.H. Rowe (1974). Strongly consistent parameter estimation by the introduction of strong instrumental variables. IEEE Trans. on Automatic Control, AC-19. No.6. 3, Soderstrom, T. , and P. Stoica (1979). Comparison of some instrumental variable methods-- consistency and accuracy aspects. 5th Symposium on Identification and Parameter Estimation, Damstart. 4, Silveira, H. M. , and R. Doraiswami'1981). An on line identification scheme using a SEAPA filter. Techanical report. University of New Brunswick, Canada. 5, Necule M. A.(1976). On some Z-transform formula.

I~EE

Trans. on Automatic Con-

trol AC- 21, No.5. 6, Goodwin, G. C. , and R. C. Payne (1977). Dynamic System Identificati o n, Experiment design an d data analysis. Academic Pr e ss.

ADDITIONAL REFERENCES

7,

Young, P. C.

(1965)

in Theory of Self Adaptive Systems, (P. H. Hammond Ed) P lIB - 140. B,

Young, P. C.

(1976)

International Journal of Control, 23, 593 - 612 9,

Young,P. C. and Jakeman, A. J.

(1979)

International Journal of Control, 29. 1 -

30.

10, Soderstrom, T. and Stoica, P (19B3) IV Approach to System Identification Springer - Vetlag: Berlin

APPENDIX

The author acknowledges that the Instrumental Variable method was introduced by young (1965) before Way and Polak (1967) and also would wish to mention Optimal IV Preiiltering methods of Young (1976, 1979) and Soderstrom and Stoica (19B3).