A new method for dynamic hunting extremum control : based on comparison of measured and estimated value

(1)

A new method for dynamic hunting extremum control : based

on comparison of measured and estimated value

Citation for published version (APA):

Liu Wen-Jiang, N. V., & Ye Dau-Hua, N. V. (1988). A new method for dynamic hunting extremum control : based on comparison of measured and estimated value. (EUT report. E, Fac. of Electrical Engineering; Vol. 88-E-197). Eindhoven University of Technology.

Document status and date: Published: 01/01/1988

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Hunting Extremum Control,

Based on Comparison of

Measured and Estimated Value

by

Liu Wen-Jiang and

Ye Dau-Hua

EUT Report 88-E-197 ISBN 90-6144-197-8 June 1988

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ISSN 0167- 9708

EINDHOVEN UNIVERSITY OF TECHNOLOGY

Faculty of Electrical Engineering Eindhoven The Netherlands

Coden: TEUEDE

A NEW METHOD FOR DYNAMIC HUNTING EXTREMUM CONTROL, BASED ON COMPARISON OF MEASURED AND ESTIMATED VALUE

by

Liu Wen-Jiang and

Ye Dau-Hua

EUT Report 88-E-197 ISBN 90-6144-197-8

Eindhoven June 1988

(4)

~ ~'t~11~#~ ~

CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG

Liu Wen-Jiang

A new method for dynamic hunting extremum control, based on comparison of measured and estimated value / by Liu Wen-Jiang and Ye Dau-Hua. - Eindhoven: Eindhoven University of Technology, Faculty of Electrical Engineering. - Fig.

-(EUT report, IS5N 0167-9708; 88-E-197)

Met lit. opg., reg.

ISBN 90-6144-197-8

5I50 656.2 UDC 519.71.001.26 NUGI832

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ABSTRACT

A new dynamic hunting method is proposed for an extremum control sys-tem.The method is based on comparison of the estimated value and measured value.

A controller using a microprocessor in accordance with this method is ex-plained. The controller can be used in process control systems. In these control systems, there are a lot of inertial and other dynamic ef-fects. The output linear group of the system is approximately expressed as a second-order or nth-order system with a large time constant or with time delay. Moreover, the relation between input drive reversal of the logic circuit and dynamic output of the system is discussed in detail. The calculating formulae for this drive reversal are given for different transfer functions, and through tests they prove correct.

In accordance with the layout mentioned above a controller using a micro-processor and a testing laboratory plant has been designed and implement-ed. The results of experimental test show that the performance of the controller satisfies the technical requirements in design and that it can search the optimum point rapidly.

Liu Wen-Jiang and Ye Dau-Hua

A NEW METHOD FOR DYNAMIC HUNTING EXTREMUM CONTROL, BASED ON COMPARISON OF MEASURED AND ESTIMATED VALUE.

Faculty of Electrical Engineering, Eindhoven University of Technology,

The Netherlands, 1988. EUT Report 88-E-l97

Present address of the authors: Professor Liu Wen-Jiang,

Department of Information and Control En'3:ineerin,g', Xi'an Jiao-tong University,

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CONTENTS

1. Introduction 2. Principles

(1) Principles of comparison of estimated value and measured value

(2) Extremum plant with second-order inertia

1 1

1

components 4

(3) The calculation method for the control plant with nth-order inertial components in the

linear parts 8

(4) Computer digital system simulation and

verification 14

(S) The control model of a plant with pure time

~lQ 15

3. Microprocessor controller 16

4. The result of the test 16

5. Conclusion 19

Acknowledgment 19

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1. I~DUCTION

An extremum control system, which is based on the common hunting methods, cannot work well for industrial processes. Because of the system iner-tial and time delay influences, the system will become oscillating, work in wrong steps or will be hunting for a long time for the working pro-cess. This phenomenon is shown in Fig. 1. In this paper, we propose a dynamic hunting method - which can be called difference pre-estimation and comparison method. This method can be used in second-order or high order systems with large time constants. Together, we provide a scheme of microprocessor controller .

...

,

•

..• ---.

~~-~:::"

...•

~

!!- _~ L __ 'e:§ "r'-- -i_'-~ ,.. ~- ~, r .-' ~

r

.~ ~! ;

_{• •}

\

, ,

\ _,

Fig. 1 The searching process of the extremum control system, which is designed by the common hunting method. 2 . PRINCIPLES

(1) Principles of comparison of estimated value and measured value. In the extremum control system a stepping controller is used;

the step period To and step span ~X are are taken to be constant.

Suppose the control plant can be transformed into a nonlinear component and a linear component connected in series as shown in Fig. 2.

Fig. 2 Control plant transformed into a nonlinear component and a linear

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The linear parts are composed of inertial components; the response to a unit-step function is h(t) and the characteristic equation has negative

real roots, a monotonous response.

During the hunting periods, the output response caused by the K-th input step is described as follows:

with:

y(k)

tlTo

°

t

< k

{y(k) - y(k-l)} h(t-k)

t

> k

output variable of the linear part caused by the k-th step input.

output of the nonlinear part

relative time

a k h(t-k) k=O n-2

L

a k h(t-k) + an_1 h[t-(n-l)] k=O (3)

The comparison values are given by equation (3) minus equation (2):

z

(t) z (t) - z (t) = a

n- l h[t- (n-I)]

(y(n-I) - y(n-2)) h[t-(n-I)

1

(4)

ZIt) is the system output response to the step input carried in at (n-I)-th step. Because hIt) is monotonous, ZIt) and (y(n-I) - y(n-2)) have the same sign.

The starting point of the estimation and comparison principle is that the direction of carry increment can be decided by the sign of Z(t).

Deriving on Method Difference Expression:

In order to simplify the calculating formulae, we can use a different ex-pression, improve sampling precision of the computer and improve the working quality in Fig. 3.

llt.i >-, 1, (~)

_--""1

l,lt .~ I

. --=

I :.ll1f.J I ~(iJ~ I :_ I I I I , I I I I I i n-t r, II

(10)

The real output of the system is described by the solid line and if there is no testing step at the (n-I)th moment, then the output is

des-cribed by the dotted line. At time t _l , t2 (n-2 S tIS t ₂), from Eq. (2) we obtain

n-2

L

k=O

(5)

If the difference of point t

3(n-l < t3< n) and n-l is compared, then we obtain the estimation value:

.

n-2

.1Z(t

3) = _k=O

L

ak (h(t3-k)

-

h(n-I-k)}

The real value is n-l .1Z(t

3) _k=O

L

ak (h(t3-k)

-

h(n-l-k)}

The comparison value is:

.

.1Z(t

3) - .1Z(t3)= an- I h[t3-(n-I)] Compare eq. (8) with eq. (4)

This means that the step direction can be decided by the sign of .1z (t)

(2) Extremum Plant with Second-order Inertia Components.

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(8)

The plant is shown in Fig. 4a; the transfer function of the linear part

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G (s) 1 ;

(T₁s+l) (T₂S+l) Unit step response:

- t

hIt)

Tl

Time is quantized by To and we obtain: - ( l t

hIt) ~ 1 - A e 1

1

relative time constant: t =

1- ;

To is step period. To

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The system output is shown in Fig. 4b. The signal is sampled in the mid-dle of every step period, so we obtain:

dZ 1 z(n-t) - Z(n-2) dZ₂ Z(n-1) - Z(n-t)

We first get h(t

2-k)-h(t1-k), then substitute i t into eq. (5) - ( l

(t

e --2 n-2

_L

k=O (e 2 -1) - 0:₁ - e - - 2 -1X 2 2 (10) (ll)

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z

,...,

Al, I

~",:- }~

:

4Z'z I

At]

I I I I b) I I I t

n-z

n·

1 z 1/-/

n--;:'

1, T2 and step period To are known; the values of a1, a2 can be decided.

LlX(n-1)

(3) The Calculating Method for the Control Plant with nth-order Inertial Components in the Linear Parts.

Suppose the transfer function of the linear parts is:

(14)

1

G(s) = '(~~'I')~"(~~'l'}---(~T~s~+~l'); T1 _TIS+ _T > T2 > T_{3 · · ·} > Tn > 0

2s+ n

The unit step response is: =-t N T. hIt) Ao +

L

Ai e J. i=l G(s)

I

= 1 _Ai (s+i. ) l G (s)

I

1 s=o J. S s=-T In eq. Ao

and after the time is quantized by To we obtain:

N -a.t To

hIt) 1 +

_L

A. e J. _a.

T. (i=l, 2, . . . N) i=l J. J. J.

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I ! t

n-2.,.0 n-z,.l.,.. It-}' K-I h-/

" N N

n~/+ I

N Fig. 5 The system output of n-th order extremum control

t

For the n-th order inertial component, we divide the step period To into N equal periods and sample at the dividing point, so we can get n+1 sample values. The estimation comparing point value we can get at the first sampling point n-I+

~

after the testing step, and the carry direction of step n can be decided by calculating the sampling values.

Differences can be obtained by the subtraction as follows:

Z(n-2+~)

- Z(n-2) I..

.i-I

Z(n-2+_N) - Z(n-2+~) N N-I Z(n-2+

_{N) -}

Z(n-2+~)

Z(n-2+~+I)

- Z(n-2+*)

The last part of eq. (5) is calculated first, we obtain

N -C/..

no:

-K)

L

Ai e

~

2 i=1

-C/..

(t

-K) e ~ I

then substitute i t into eq. (5) n-2

L

k=O

[

.i

.i-I

J

The equations are obtained by the relationship of some output response curve so that we have a group of unique roots, because i t is an nth-order system and there are n linear independent variations in the equations. The last equation is obtained from estimation values. So the number of equations is N+l. The last equation is the linear combination of other equations.

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1 1 1

~l

~2

~lN-l ~2N-l

~/

For the reason mentioned above, this matrix has a rank N and its N+l order determinant is zero.

1 1 1

a

~lN-l ~NN-l

~/

This determinant can be expanded into:

Then we obtain the estimation value:

In the equation 0N+l' ON' elements ~Z ~Z . ~Z N 2 1

°

N-l ~Z 0N+1 N-l + ... +

°1

+

-0-

~zlJ

N+l (16) ... 02' 01 are the remainder formulae of the in the last column of the determinant:

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1 1 1

~1

~2

~N

DN+1

=

(_1)N+1+N+1 ~/-1 ~2N-1 ~/-1 1 1 1 (_1)N+1+1 (_1)N+1+1

The comparison value can be obtained by the estimated value:

aZ N+1

=

_{aZ N+1 - aZ N+1}

(19)

2

6.Z_N + ... +

DN+_l

The logic discriminant of the step direction:

Sign {[ZN+l + D _N_ 6.z DN+_l N (18 ) + ... (19 )

Then expand the matrix (15) into a Vandermonde determinant. Due to the uniuueness of the polynomial

in the equations are equal.

expansion, the correspondence coefficients So we obtain the equations as follows:

(2 0) In the equation 01' 02' ... are the elementary symmetry polynomials; they

are: N

L

l3

_i i=l N

L

l3

_i

l3

_J· i, j=l (i< j)

(20)

(4) Computer Digital System Simulation and Verification.

Take the control plant of the third-order inertia component in linear parts as an example: G (s)

=

where: 60s ; T2 240s ; .1x

iiI

= e 1 300s 0.5 V 0.875 l50s -To 3T₂

e

= 0.766 1.634

The logic discriminant of the step increment is: Sign ( .1X{n» = Sign ( .1Z4.1X(n-l)

The results of computer simulation is shown in Fig. 6.

0.587

(21)

"

Y,Z

Fig. 6 The computer simulation result

(5) The Control Model of a Plant with Pure Time Delay.

According to the principle of estimation and comparison, the comparing point is to be chosen as the point between (n-l) and n.

The step direction is calculated and decided by the microcomputer until the time n when a waiting time period ~T exist; see Fig. 7.

If there were pure time delay ~ in the plant and ~T > ~, the step can be put in at time n-~ to compensate for the plant time delay and the effect of the time delay can be overcome.

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3 . MICROPROCESSOR CONTROLLER

The microprocessor controller consists of Z-80 CPU, RAM, EPROM.PIO.CTC. and DAC 0809 IC. Difference sample unit, analog output interrupting and time unit are also in the microprocessor controller, which is shown in Fig. 8. Fig. 8 r - - - , ,', "-1 I 15) 'P1o:l'lt I pte-Her (1) EA-lSO ?~tte,..

,

lL'J:==f:~:c~-,

, I

1-'

,

~ ____________ J r---~---_/.0) _{E ...}_uc-t,«, _&'''r

...

_I

%

,

,~g3>""'...J: L _ _ _ _ _ _ _ _ J

Block diagram of the microprocessor controller

The working process of the controller is that when time unit gives an interrupting signal at the sample time, the difference sample unit is started by the microprocessor and sends the data to the memory.

At the proper time the data will be compared and the direction of the step increment can be decided; a signal is sent to the executive body through D/A. This process is repeated again and again in order to reach the optimal point by dynamic hunting. The controller parameter can be changed for different plant conditions.

The main program block diagram is shown in Fig. 9.

4 • TBB RESULT OF TBB TEST

The control system is shown in Fig. 7 and the testing laboratory plant consists of a second-order inertial component G(s) =

2

0.6/(600s + 1) (300s + 1) and a nonlinear component y = 2X-0.2X .

The step period To = 120. ~X = 0.4, the process is controlled by the microprocessor controller.

(23)

(1) Fig. lOa, lOb show the process output curve of hunting the optimal point in the dynamic variation period and the phase diagram of hun-ting process.

(2) Fig. lla, lIb show the hunting process and phase diagram of extreme value characteristic curve drift continuously in both horizontal and vertical directions. NO

D.t.

pro .... ,

N'

D'ff.,e"te .1Z,

y"

!Jl,=/i,-z,

-A~AZf yes No

(24)

J I I fJ Z

/

/ /

/

I

/

I

t

Fig. 11

,

a)

Fig. 10

f

J

J I I

\

, ,

1

,

\ X

••

t ...

_

.. _.-. . .

_...

The hunting process and phase diagram

.

..

_ .. -..

-.

'/.,<1

-

..

.. - - ' . : '.: .. :: ::.::'::':" 3 . -". l b)

The hunting process and phase diagram of extremum

value; characteristic curve drifts continuously.

(25)

5. CONCLUSION

A new method of dynamic hunting for extremum control systems has been de-veloped. Results of computer simulation and a physical model test have shown that the optimizing calculating formulae according to the prin-ciples of difference estimation and comparison can be used in extremum control systems for high order processes.

The controller with microprocessor can perform dynamic hunting under sud-den change or drift of the extremum characteristics, and the results are satisfactory.

ACKNOWLEDGMENT

The first author wishes to thank Prof. P. Eykhoff and ER Group of

Eindhoven University of Technology for their encouragement and valuable help; he would also like to thank the Dutch Ministry of Education and Science for providing the scholarship.

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MARTINU5 VAN MARUM: A Dutch scientist in a revolutionary time.

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EUT Report 88-E-197. 1988. ISBN 90-6144-197-8

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~EXTREMUM HUNTING METHOD USING PSEUDO RANDOM BINARY SIGNAL. EUT Report 88-E-198. 1988. ISBN 90-6144-198-6

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THE FULL DECOMPOSITION OF SEQUENTIAL MACHINES WITH THE OUTPUT BEHAVIOUR REALIZATION.

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(200) Huis in It Veld, R.J.

A FORMALISM TO DESCRIBE CONCURRENT NON-DETERMINISTIC SYSTEMS AND AN APPLICATION OF IT 8Y ANALYSING SYSTEMS FOR DANGER OF DEADLOCK. EUT Report 88-E-20D. 1988. ISBN 90-6144-200-1

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THE LINEAR REGRESSION MODEL: Model structure selection and biased estimators.

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(204) Kalasek, V.K.I.

COMPARISON OF AN ANALYTICAL STUDY AND EMTP IMPLEMENTATIDN OF COMPLICATED THREE-PHASE SCHEMES FDR REACTDR INTERRUPTION.

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FiNITE WORDLENGTH EFFECTS IN DiGITAL FILTERS:~iew.

EUT Report 88-E-205. 1988. ISBN 90-6144-205-2

(206) Bollen, M.H.J. and G.A.P. Jacobs

~IVE TESTING OF AN AL~M FOR TRAVELLING-WAVE-BASEO DIRECTIONAL DETECTION AND PHASE-SELECTION BY USING TWONFIL AND EMTP.

EUT Report 88-E-206. 1988. ISBN 90-6144-206-0

(207) Schuurman, W. and M.P.H. Weenink

StABILITY OF A TAYLOR-RELAXED CYLINDRICAL PLASMA SEPARATED FROM THE WALL BY A VACUUM LAYER.

EUT Report 88-E-207. 1988. ISBN 90-6144-207-9

(20B) Lucassen, F.H.R. and H.H. van de Ven

A NOtAtioN CONVENTION IN RIGID ROBOT MODELLING. EUT Report 88-E-208. 1988. ISBN 90-6144-208-7

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MINIMAL REALIZATION OF SEQUENTIAL MACHINES: The method of maximal adjacencies.

EUT Report 88-E-209. 1988. ISBN 90-6144-209-5

(210) Lucassen, F.H.R. and H.H. van de Ven

OPTIMAL BODY FIXED COORDINATE SYSTEMS IN NEWTON/EULER MODELLING. EUT Report 88-E-210. 1988. ISBN 90-6144-210-9

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