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A Study of a special purpose automatic optimizer Wright, William Lawrence 1965

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A STUDY OP A SPECIAL PURPOSE AUTOMATIC OPTIMIZER by WILLIAM LAWRENCE WRIGHT B.AoSc.p University of Waterloo, 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l Engineering We accept th i s thesis as conforming to the required standard. Members of the Department of E l e c t r i c a l Engineering THE. UNIVERSITY OF BRITISH COLUMBIA August^ 1965 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s , , I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i -c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f »r%?/7£ ^ C ^ T / p The U n i v e r s i t y o f B r i t i s h C olumbia Vancouver 8, Canada Date ABSTRACT Small special purpose d i g i t a l computers (SPDG) c@uld be used to control processes for which the cost of general purpose digital, computers i s pr o h i b i t i v e . This thesis describes a SPDC to optimize a process for which an exact mathematical model does not exist* The SPDC could use any of the empirical or t r i a l and error methods o r i g i n a l l y designed for hand calcu-lations or for use on a large general purpose d i g i t a l computer,. The methods discussed i n this thesis are gradient search P direct search and random search* The overa l l operation of a SPDC i s described i n d e t a i l using logic block symbols* Prom the knowledge gained i n b u i l d -ing and testing the computert improvements i n c i r c u i t r y and search strategy are suggested,. The logic and c i r c u i t r y used i n a SPDC depend? on the nature of the process "b© be controlled,, This i s i l l u s t r a t e d i n the thesis by the description of the optimization of a f l o t a t i o n process« TABLE OF CONTENTS Page LiS"b Of IllUS"fcra"fci.®IlS e a o o o e o o a e a o a a a a a e e a e e e e e a * ;•-,*/»; &"V Acknowledgement 1« INTRODUCTION « a « « n o a a a o o o o o o o o a o o o o o o o o o o a o o a a a a o a a a . * 1 2 a CONTROL STRATEGIES a e a a a a # a a & o o a a a & a : a a a a.« a o a a « « * • « * 3 2 ,« 1 N o n s e q u e n t i a l S t r a t e g i e s a « a a a « a a a 0 o * a o * a a * • » » » • 4 2 » 2 S e q u e n t i a l St jCSitegieS e s a a a e a o a a a e a a o o a e a a o a o a o * 4 2 « 3 D e t a i l s ©f a S p e c i f i c Search S t r a t e g y • «**.««a.<r,*.«< 1 2 3 a THE COMPUTER * » * a * « a f t a a o a o o a e a o a e o o o a o a a a a o e a a c o a a a a a . . « 1 9 3 a 1 C o n t r o l BlifilJfik «> « 4 a a a a a a a a a a a a a a a a a a a a a a o « o a » • 2 1 3 * 2 The SMX and SMI B l o c k s »a » a a » > a a • a a a • a a • • ao 4 .« 2 4 3 * 3 The COX, CCI mxd. GGP B l o c k s • a »a a a « a » a . * a « a • » . < « > 2 5 3 a 4 The SMP BltfiiijSk. • e a a a a a a a a a a a a a a a a a a a o a a a a a a a a a • • o 2 6 4 a TEST OF COMPUTER « * a , a o o e a o a « . a o o o o e a o o o o o o o a o o a > a * a a a , a ' 3 6 4 a 1 Comparators « , > a a a a e a d a a a o a a a o a o a e a a a a a o a a e a a a » a o 3 7 4 a 2 De ad BSfcUcL ,».a > -i» .4 » a a a a a a a a a o o a a a a a .a a a a a a a a a a a a a a a » 4 2 5 a EXAMPLE OF USE OF SPDC o o o o o a o o o o o a e » o o o o a a o e e o a , » a a a 4 8 6« CONCLUSIONS *;^> a f t a v a a p a D o o a o o a e o a o a a a o o o e a o o o o o o e o t t a ; * ; ' * 5 2 APPENDIX A a a a .a,**/',;* o a a o a a a o a a o o a e o a a a a a a e o a p a a a o a a a o o o a 5 3 REFERENCES a o & a f l a a « a a « a a « o o o o o e o o e o o a a o a a a a e o a & a o o a e a a « a . a 5 6 i i i LIST OF ILLUSTRATIONS F i g u r e Page 2—1 B l o c k Diagram ©f D i r e c t O p t i m i z i n g » »» o «<>«o o o o o «•»« • 4 2—2 Flow Diagram ©f Command S u b r o u t i n e >*«.« a a a a a a a 15 2—3 Flow Diagram ®f M@,de 1 a » » » » e e e e a a a a a a e e e a a e a 6 a • • a* 16 2—4 Flow Diagram @ f Mtole 2 and Mode 3 ««<><><>oooo«o«o.««*» 17 2— 5 Flow Diagram #£ Misiie 4 ,*.••.«•«,«• o««o e e«»o«a o ^ .• * <t • i* 18 3— 1 G e n e r a l Cwnputer O u t l i n e * a a a a » a o a a « » o oa a»««a a,»*« 29 3—2 C o n t r o l Blfflsk ^ M<sdes 1 and 4 * • * • o a a o. a a « o a * 30 3—3 C o n t r o l Bl@©k Mf de 2 a a a « a a a a a a a a a a a a a a a a * a * * 31 3—4 BlOCk Diagram SMX a a a * a a « a.a a a a a a a • ..a e,« a a ; e a ..a'.* a,«.,», o 32 3—5 B l o c k Diagram .<&£ GCX ^* *^ *.*•.«. «^ ,4 * « . . 32 3—6 BlOCk DiagX^am © i f CCP a a a • • * aa a a a a «,« a 00 • a a a a * » * a 4 * 33 3—7 BlQCk Diagram SMP a a o a a a a a a a a a a a a a a a a a a a a a « » i > » ^ t 34 3-8 B l o c k Diagram <sf Constraint C i r c u i t * X I — C »*.**:,••* 35 3- 9 C o n s t r a i n t C i r c u i t w i t h No A c c u m l a t i v e E r r o r •»*•»• 28 4— 1 Comparator Output *• a a a a a a a a e a a a e a e e e e a a a e e a a * * * * «.« 38 4—2 Change S t® SMP 4 4 o A a a a a a . e e o e a a e e e a a e e e a e e e a e e a e a ooo 46 4—3 SMP W a V e f i E i r m S « a a « a o o a a a a a a a a o a a a a a o o a u o a a ooooaeaae 47 4— 4 D i g i t a l Qversh.©i<git C i r c u i t a a a a a a o a a o a a a a a a a a a « a 6 a« * 40 5— 1 P e r c e n t a g e M i n e r a l Recovery vs, m [H°*°] * »a * a o *.<».*•„*.+,# 51 5 — 2 Test Se tup • o a * <jy> • p a a a ai a a a a a a a o a o a a « a a a o a a a a a o a a a a ,*; a 51 i v ACKNOWLEDGEMENT Grateful acknowledgement i s given to Northern E l e c t r i c Cob for a fellowship awarded i n 1963 and to the National Research Council for f i n a n c i a l support from Block Term Grant A68* The author would l i k e to thank the supervisor of this project,, Dr B E„ V* Bibhn^ and a l l other s t a f f members and graduate students of the E l e c t r i c a l Engineering Department for the i r assistance during the course of this study* The author i s also indebted to W„ Gowans of The Consolidated Mining and Smelting Co, for suggesting -fche example i n Chapter V B 1 A STUDY OF A SPECIAL PURPOSE AUTOMATIC OPTIMIZER 1. INTRODUCTION In the l a s t few years there has been an increase i n the number of computers used i n i n d u s t r i a l controls,'*' and an 2 accompanying expansion of the l i t e r a t u r e about them» Since many companies are now a c t i v e l y investigating computer use, an even greater increase i s anticipated. Despite t h i s widening inter e s t and a general lowering of computer and computer compon-ent prices, there are s t i l l many processes to which optimal control techniques cannot be applied because of the high cost of available general purpose computers r e l a t i v e to the expected increase i n returns due to optimal c o n t r o l a The object of this thesis i s to outline a special purpose low cost d i g i t a l c o ntroller for on-line optimal controls This type of unit could be used to drive a single loop or small process to a steady state optimum. The controller i s also applicable to the quasi-retatic process where the optimum operating point varies with uncontrolled plant parameters which change-slowly r e l a t i v e to the d,ominant response time of the process* This d i g i t a l c o n t r o l l e r could also be used i n conjunction with a high speed analog computer to control dynamic processes as 3 - 4 suggested by Bohn* * Alone, or i n conjunction with the high speed analog computer, the action of the d i g i t a l controller w i l l be the same. The process^ or plant* would be such that a mathematical model cannot be c o m p l e t e l y p r e p a r e d because;of the c o m p l e x i t y o f the p r o c e s s . Even i f a mat h e m a t i c a l model can be prepared^' i t would be too complex t o be s o l v e d a n a l y t i c a l l y and i s c o m p l i c a t e d f u r t h e r by time v a r y i n g parameters. The optimum i s determined by e x p e r i m e n t i n g w i t h the p r o c e s s , sometimes r e f e r r e d t o as the d i r e c t method o f e s p t i m i z i n g . The changes made by the c o n t r o l l e r i n the c o n t r o l l e d v a r i a b l e s are based on p a s t responses of the performance f u n c t i o n P.* V a r i o u s s t r a t e g i e s have been developed t o f i n d the Optimum under th e s e c o n d i t i o n s * - The d e c i s i o n of what s t r a t e g y t o i n s t r u m e n t would be based on what i s known of the p r o c e s s and performance f u n c t i o n s u r f a c e * The Quarie c o n t r o l l e r and OPCON 5 6 1 8 are commercial c o n t r o l l e r s u s i n g t h i s t e c h n i q u e * * * * An 9 10 a n a l o g computer o f t h i s type has been suggested by Peldbaum. * There are s e v e r a l well-documented s e a r c h s t r a t e g i e s f o r m a x i m i z i n g m a t h e m a t i c a l f u n c t i o n s , u s u a l l y i n t e n d e d f o r use on a l a r g e g e n e r a l purpose computer. T h i s t h e s i s d e s c r i b e s how s t a n d a r d components c o u l d be used to i n s t r u m e n t one such s e a r c h s t r a t e g y f o r d i r e c t p r o c e s s o p t i m i z a t i o n . The g e n e r a l purpose d i g i t a l computer has a l a r g e minimum c o s t because the dame c o n t r o l and a r i t h m e t i c u n i t s are r e q u i r e d r e g a r d l e s s of computer s i z e * As the d i g i t a l c o n t r o l l e r d e s c r i b e d here r e q u i r e s no program f l e x i b i l i t y and works a t speeds compar'able t o the p r o c e s s time c o n s t a n t s , a v e r y i n e x p e n s i v e u n i t s h o u l d be p o s s i b l e T h i s t h e s i s shows how the performance of the c i r c u i t s i n the c o n t r o l l e r may change the b a s i s f o r c h o o s i n g a s e a r c h s t r a t e g y , and concludes w i t h a d e s c r i p t i o n of an i n d u s t r i a l p r o c e s s which might use such a, c o n t r o l l e r . 2. CONTROL STRATEGIES 3 The term "control strategy" i s used to describe the procedure by which the control parameters of a process are varied i n order to operate the process at the desired optimum* For s i m p l i c i t y , the optimum operating point w i l l be csonsidered a minimum of P* The performance function surface w i l l be con-sidered unimodal (one minimum), a r e a l i s t i c assumption as the multimodal case becomes unimodal once the lowest minimum has been found* Some authors suggest handling the multimodal case by repeated use of a control strategy designed f o r the unimodal case using d i f f e r e n t parameter values for starting points»^ The "no—error" eases where the control parameters and P are assumed to be known exactly w i l l be considered here^ as their p r i n c i p l e s are simple and the control strategies based on them can be used even I f the assumption i s not quite true* Process noise and instrumentation error which exist i n most i n d u s t r i a l cases might necessitate use of a va r i a t i o n (using a s t a t i s t i c a l 12 technique ) of the "no-error" strategies o The controlled 3 4 system as suggested by Bohn 9 can be considered error free. Again f o r si m p l i c i t y and ease of descriptian/ the process w i l l be considered as shown i n Figure 2—1 with X and Y as two controlled parameters, v as one uncontrolled parameterj and the performance function P as the output to the control computer6 The control computer must vary the inputs X and Y i n discrete steps as the effect of changing a variable i s not immediately measurable because of time:lags i n the process* Each new control parameter setting must be held u n t i l the st a t i c response to the change can be measured* O r £ | PRO C£SS —/TW COMPUTER CONTROL. j CoMPtrr&fi? | CONTROLLTD Figure 2~d Block Diagram of Direct Optimizing 2«1 Nonsequential Strategies In nonsequential strategies the entire performance function i s systematically explored* Each new setting of X and Y i s independent of past responses. This type of strategy, while useful for multimodal performance surfaces, i s not as applicable to the problem outlined i n the introduction as the sequential . . . 13 strategies. 2.2 Sequential Strategies Sequential strategies are the types of control s t r a t e -gies best suited to a special purpose d i g i t a l computer (SPDC) with limited memory capacity* Sequential strategies are often referred to as "hil l e l i m b i n g " because the physical ascent of a 5 14 h i l l involves similar decisions. In sequential strategies, each new setting of X and Y i s dependent on one or more past responses to step changes i n X and Y, Thus the changes i n X and T cannot be predetermined as i n nonsequential strategies because they depend on the unknown performance surfase and on the starting values of X and Y» There i s extensive l i t e r a t u r e on sequential control strategies and much overlapping of ideas and terms* In general, the sequential strategies may be divided into three main types: gradient, d i r e c t * and random search. Gradient methods explore the variations of P with respect to parameter changes to f i n d the l o c a l gradient or d i r e c t i o n of greatest slope of the P surface in. the neighbourhood of the starting point and move the control parameters i n this d i r e c t i o n . In the univariate gradient method a l l variables except one are held constant and t h i s one i s varied to obtain an improvement i n B# Thus X^ would be held and Y varied u n t i l Y^, the value of Y which gives the largest possible P (X^, Y), i s found. Then Y i s held at Y^ and X i s varied, etc* One of the many possible variations i s to start varying X i f P (X^, Y 2 ) i s not found i n a specified number of steps of Y» The univariate technique works quite well as long as the control parameter " • > • 15 axes are p a r a l l e l to the axes p f the contour surfaces of P. Even with an advantageous alignment of axes, this method requires many more moves to arrive at the minimum than other methods which move a l l parameters simultaneously. Another gradient strategy similar to the univariate 6 method but which makes simult aneous steps i n a l l control para-meters i s called relaxation. The l o c a l P surface i s explored to f i n d the gradient d i r e c t i o n and then a l l control parameters are stepped simultaneously to move the operating point i n this gradient d i r e c t i o n . These steps are continued u n t i l no further improvement i n P (loc a l minimum) i s possible i n this direction and then a new l o c a l gradient i s found. After the f i r s t step there i s no reason to assume the moves are s t i l l i n the dir e c -t i o n of the l o c a l gradient and thi s i s one of the factors that lead to many variations on this general strategy* For example, the change APJJ i n P fori the N step could be monitored and i f A P ^ becomes less than a predetermined percentage of A P ^ i n the par t i c u l a r d i r e c t i o n being moved, a new gradient d i r e c t i o n could be found immediately rather than continuing on to f i n d the l o c a l minimum* A v a r i a t i o n that would not require the storage of A P ^ i s to use either the finding of the l o c a l minimum t h or the N control parameter step as the signal to f i n d a new l o c a l gradient. Perhaps the best known of the gradient methods i s "steepest descent" which l e t s the N of the previous v a r i a t i o n be: 1* Thus a new l o c a l gradient i s found after each step. This enables the maximum advantage to be gained from each step of the parameters once the gradient d i r e c t i o n i s known* If finding the gradient d i r e c t i o n i s very time consuming, the relaxation i method may f i n d the minimum i n a much shorter time than steepest descent. In minimising a mathematical function, the gradient d i r e c t i o n i s found by calculating the derivatives of the function by any s u i t a b l e methoda W i t h the p r o c e s s as o u t l i n e d £h the i n t r o d u c t i o n * , t h e s e d e r i v a t i v e s cannot be c a l c u l a t e d * . The g r a d i e n t i s f o u n d by s e q u e n t i a l l y changing each c o n t r o l a known amount* Sx* •waiting any r e q u i r e d dead time or u n t i l a l l t r a n s i e n t e f f e c t s have s e t t l e d and then measuring the r e s u l t i n g ^P^* Then assuming k * a move can be made i n the g r a d i e n t d i r e c t i o n by s i m u l t a n e o u s l y s t e p p i n g the c o n t r o l parameters a h amount AX *= -k ^ P ^ f A I = - k ^ P ^ a I n most i n d u s t r i a l p r o c e s s e s the r e q u i r e d d e l a y * T * p • between s t e p p i n g &X and the measurement of ^ P ^ i s so much g r e a t e r t h a n any c o m p u t a t i o n time r e q u i r e d by the SJ?PC t h a t , u s i n g a s t e e p e s t d e s c e n t strategy*, a l a r g e p ercentage o f s e a r c h time i s spent f i n d i n g the g r a d i e n t d i r e c t i o n r a t h e r t h a n improv-i n g the o p e r a t i n g c o n d i t i o n of the process*, But on any i r r e g u l a r P s u r f a c e the s t e e p e s t descent method may a r r i v e a i i h e optimum o p e r a t i n g c o n d i t i o n i n s h o r t e r time t h a n the r e l a x a t i o n method* I t s h o u l d be p o i n t e d out t h a t any d i r e c t i o n of s e a r c h w h i c h improves P can be made the s t e e p e s t d e s c e n t d i r e c t i o n by a p r o p e r change o f s c a l e of the c o n t r o l p a r a m e t e r s ^ H t f - ^ j f l T T h i s i n d i c a t e s a b a s i c problem of a l l g r a d i e n t metho&ysj moving i n the g r a d i e n t d i r e c t i o n can be no b e t t e r or no worse th a n moving i n any o t h e r d i r e c t i o n a I f the c o n t o u r l i n e s o f P are c i r c l e s * the g r a d i e n t d i r e c t i o n a t any o p e r a t i n g p o i n t w i l l l e a d d i r e c t l y t o the minimum, b u t , i f the P c o n t o u r s ar6 any shape o t h e r t h a n c i r c u l a r , t h e r e may be v e r y l i t t l e c o r r e l a t i o n between the g r a d i e n t d i r e c t i o n a t an o p e r a t i n g p o i n t and the d i r e c t i o n from the same o p e r a t i n g p o i n t to the minimum*-8 Direct search."^ or tri a l - a n d - e r r o r " ^ strategies are i n no way dependent on a gradient direction,, The central para-meters are stepped i n any dir e c t i o n which the previous responses have shown to improve P.. One type of direct search i s to make simple exploratory moves of each control parameter and use the improvement or lack of improvement of P (no interest i n re l a t i v e sizes of S Pj only i n sign of & P) as a guide to dir e c t i o n of search. This exploration i s followed by a "pattern" step i n the d i r e c t i o n indicated by the exploration. For any P surface having noncircular contours (which would seem to include many i n d u s t r i a l processes)*, the dire c t search strategies should be just as good i f nsjt better than any gradient method* One advantage of d i r e c t search strategies for use i n a SPDC i s that they tend to use repeated i d e n t i c a l arithmetic operations with a simple l o g i c * In random search strategies the control parameters are stepped i n a randomly chosen d i r e c t i o n about a base operating point* If P i s improved at any chosen operating pointy i t becomes the new base point for continued random moves* It might seem obvious to prefer a procedure which makes some use of previous responses i n planning the next move but a random search strategy i s e f f i c i e n t at optimizing very i r r e g u l a r P surfaces. Some tests have shown random strategies better than systematic control strategies for a process with few inputs (less than three) i f the; process can be described iby algebraic equations.^ With a process as outlined i n the introduction where time T i s required after each parameter step before S P 9 can be measured* the random strategies do not perform as well 14 as the gradient methods. A l l the basic sequential strategies so f a r considered w i l l operate^ with various degrees of success t using only a single step size throughout the entire search for the optimum, A much improved performance (fewer steps to converge to the optimum) i s possible i f adjustment of step size i s used. In th i s substrategyj the Usually suggested pattern i s to use coarse steps i n the i n i t i a l stages and f i n e r steps i n the l a t t e r stages of the search t® improve resolution and reduce overshoot and hunting losses* There i s a large number of c r i t e r i a f W chang-ing step size and most modify the i n i t i a l step size depending on the success of the previous step taken ( i f P was improved, the step was a success)* The effectiveness of any search 20 strategy i s dependent on this i n i t i a l step s i z e . The "best" i n i t i a l step size w i l l depend on the p a r t i c u l a r process under consideration. For an unknown process, an adjustment of step size i s very important to correct any i n i t i a l poor choice of step size<. One possible strategy might be to double the step size i f the l a s t step was a success. This could res u l t i n rapid convergence to the ©ptimum i n the i n i t i a l search but would overshoot the optimum. In the gradient strategies the steps are proportional to the previous response, AX = >=*k6p^ * and k can be reduced as the minimum i s approached. This could be done by making k smaller for each negative AP (an overshoot). If k i s only made smaller than the i n i t i a l k, the controller 10 could spend a long time searching i n an area of r e l a t i v e l y f l a t P» Another p o s s i b i l i t y i s to use the frequency of over-12 shoot as a step size criterion.. For example, i f four overshoots occur i n sequence^ the step size i s considered too large; i f four successes occur i n sequence, the step size i s considered too small. This type of step size adjustment i s used i n dir e c t search strategies -where the size of AP i s not measured* The step size i s adjusted by a predetermined amount dependent only on the success of the previous step. Constraint- handling i s another substrategy that supple-ments any of the basic sequential strategies such as gradient search or d i r e c t search. A constraint i s usually a, physical consideration which prohibits eertain areas of the P surface i n which the operating point cannot or should not exist* "Hard constraints" such as maximum temperature or maximum flow obtain-able cannot b£ v i o l a t e d . "Soft constraints" are,those which should not be v i o l a t e d as they represent areas of operation where undesirable effects start to appear. The constraint may be a known function of the system parameters^ such as XY =^  C, or some measure might have to be taken d i r e c t l y to sense constraint v i o l a t i o n . Two possible methods for handling constraints are "hemstitching" and "riding 21 the constraint"* In "hemstitching", as used i n a basic gradient search^ the control parameters are moved i n the d i r e c -t i o n of the gradient of P i f the constraint i s not v i o l a t e d . Once the constraint i s v i o l a t e d the parameters are meved i n the I I d i r e c t i o n of the gradient of the constraint. Like a l l search techniques considered i n this thesis, there are many variations on this idea# hilt b a s i c a l l y the parameters are adjusted to improve P u n t i l the constraint i s viol a t e d and then the parameters are adjusted with the aim of s a t i s f y i n g the constraint* Hemstitch-ing requires the constraint to be v i o l a t e d . This v i o l a t i o n i s not acceptable i n any controller empirically optimizing an in d u s t r i a l pr@c!ess with hard constraints. In "ride the constraint" method the gradient d i r e c t i o n of P i s always found* but i n a zone within one, step of the constraint (assuming the constraint i s a known equation) only a limited parameter step i s allowed so that the operating point w i l l tend to follow the constraint li n e and the constraint i s never violated* This nonviolation of the constraint assumes that the process variables (e*g» tempera-ture) follow closely* without overshoot* the corresponding control set point (s*g* thermostat s e t t i n g ) . This i s part of the over a l l assumption that, to be e f f e c t i v e l y optimizedj a system must be closely controlled* Unlike hemstitching^ the "ride" method does not o s c i l l a t e about the constraint* This would seem to make a search strategy with a "ride" substrategy converge to the optimum i n fewer steps than one with a hem-st i t c h i n g substrategy. For any pa r t i c u l a r process, some control strategies w i l l be better than others. The choice of what basic control strategy (and substrategies) to implement i n a SPDC w i l l depend on the knowledge of the P surface, and on the speed and accuracy required by the control computer. Many detailed comparisons of search strategies over various types of P surfaces are 1 2 a v a i l a b l e , 1 3 * 1 5 * 2 0 2.3 Details of a Specific Search Strategy-Assume that i t i s required to operate a process as in Figure 2*1 at a minimum of P and satisfy the constraint XY — C, and that* based on the knowledge of the process, a relaxation search has-been chosen* The search i s to be divided into two stages^ the i n i t i a l or coarse stage to quickly locate a neighbourhood of the optimum, and a f i n a l or fine stage to obtain and track the optimum operating point. If APg-j the t h change i n P resu l t i n g from the K step i n the coarse search, i s less than AP^/A of the same gradient d i r e c t i o n , a new gradient d i r e c t i o n i s to be found. The step size i s to be made smaller with each overshoot.and, i f the coarse search i s at i t s smallest step si z e * an overshoot causes the change from coarse to fine seareih.*' In the fine search stage, the control para-meters are to be stepped sequentially n times i n the l o c a l gradient d i r e c t i o n using the smallest possible step which w i l l s t i l l produce a recognizable & P. For convenience i n the instrumentation the choice A = 2, n = 7 i s made and only two step sizes k and k/2 w i l l be allowed. While th i s arbitrary choice may not create the best possible gradient search, i t i s suitable to test the a b i l i t y of simple SPDC to carry out a s p e c i f i c search strategy. This search i s very gerieral and should be able to optimize $ to some degree, any type of P surface. It also has the advantage, for t h i s thesis^ <bf being complicated enough that, i n instrumenting 13 a SPDC to carry i t out, the type of problems and considerations of instrumenting any search strategy by simple logic units should be encountered. Once the search strategy i s chosen, the next step in designing a SPDC i s to specify the sequence of operations within the search* The flow diagrams i n Figures 2-2 to 2-4 show one possible organization of operations enabling the control computer to carry out the required search. The search i s shown broken down into subroutines. The command subroutine checks whether to continue i n coarse search* alternating subroutines mode 1 and mode 2, or to change to mode 4* the Subroutine for fine search* When control i s transferred from the fine search subroutine back to the command subroutine, the position of switch #2 (S2) determines whether fine search w i l l be continued or the entire search pro-cedure w i l l be repeated ( i . e . return to coarse search). If at any time during the search the constraint i s vio l a t e d , control i s immediately transferred to mode 3, the constraint handling subroutine. This i s shown by a separate flow diagram labelled " p r i o r i t y " . Mode 1 i s the subroutine which determines the l o c a l gradient d i r e c t i o n * X and Y are sequentially perturbed &X and ^Y; the size and sign of the resul t i n g S P^ and ^>Py are stored. Mode 2 simultaneously changes X and Y amounts AX = - k<£P and AY - - k<5p . The step size, k or k/2, i s chosen by the control, subroutine and &P^and SP^. are provided by mode: 1. These steps are continued u n t i l AP i s negative or 14 Mode 3 i s the c o n s t r a i n t h a n d l i n g s u b r o u t i n e * Since t h e i c o n s t r a i n t i s a known e q u a t i o n of the two c o n t r o l parameters, i t i s p o s s i b l e to handle t h i s c o n s t r a i n t i n the same manner as an e q u a l i t y c o n s t r a i n t when s o l v i n g a s e t of a l g e b r a i c e q u a t i o n s , t h a t i s , t o reduce the number of v a r i a b l e s . As l o n g as XT = C, T i s made a dependent v a r i a b l e . The c o n s t r a i n t e q u a t i o n g i v e s the r e l a t i o n between AX and A l and the r e g u l a r s e a r c h i s con-t i n u e d . T h i s i s a v a r i a t i o n of the " r i d e the c o n s t r a i n t " s u b s t r a t e g y . mentioned i n s e c t i o n 2.2, as the s e a r c h can move a l o n g the c o n s t r a i n t but never c r o s s i t . Mode 4 j the f i n e s e a r c h s t a g e , t a k e s seven s t e p s s e q u e n t i a l l y , s t e p p i n g X and T amounts £ x and S i . The d i r e c -t i o n s of the s t e p s are determined by the p o l a r i t y o f the p r e v i o u s £ p ^ and SPy* These s t e p s i z e s are used i n b o t h mode 1 and mode 4. They s h o u l d be the s m a l l e s t s t e p which produces a r e c o g n i z a b l e change i n P. 15 S t a r t E v a l u a t e P a t i n i t i a l o p e r a t i n g p t , Mode 1 Mode 2 P r i o r i t y X I V XT = V = C Mode 3 6 yes Mode 4 no v Decrease s t e p s i z e F i g u r e 2-2 Flow Diagram of Command S u b r o u t i n e 0 i - 1 16 Set Z. d i r e c t i o n I positive Ste Z 1 P • i / Delay T Store P -ALL Set Z i d i r e c t i o n negative = +X Z2 = -J Exit Figure 2«-«3 Flow Diagram of Mode 1 17 Mode 2 Mode 3 De] Lay T P Y becomes dependent v a r i a b l e AY = — AX/C E x i t E x i t t o Command E x i t t o Command Q F i g u r e 2—4 Flow Diagram of Mode 2 and Mode 3 18 Mode 4 1 w i = 1 Set d i r e c t i o n negative Figure 2-5 Flow Diagram of Mode 4 1 9 3. THE COMPUTER A v e r y inexpensive s p e c i a l purpose d i g i t a l computer can c a r r y out the search d e s c r i b e d i n s e c t i o n 2*3* The computer design e x p l a i n e d i n t h i s s e c t i o n uses standard d i g i t a l l o g i c u n i t s such as gates* monostables and f l i p - f l o p s * The d i s c o n t i n u -ous nature of e m p i r i c a l o p t i m i z a t i o n and the lengthy storage times r e q u i r e d would seem to favour d i g i t a l r a t h e r than analog t e c h -niques as would the c o m p a t i b i l i t y of a SPDC w i t h f u t u r e o v e r a l l p l a n t d i g i t a l c o n t r o l or telemetry* A computer u s i n g a h y b r i d of analog and d i g i t a l techniques could c a r r y out the same search; i t might even be argued t h a t the computer d e s c r i b e d i s a h y b r i d as the outputs of the computer are analog s i g n a l s obtained from potentiometers d r i v e n by step motors. These motorsj r e a d i l y a v a i l a b l e commercially, are d r i v e n by the pulses of the d i g i t a l equipment* The motors are b i d i r e c t i o n a l w i t h each i n p u t pulse causing a 15° angular displacement of the motor shaft*. F i g u r e 3^ -1 shows a v e r y simple o u t l i n e of the computer. The main c o n t r o l u n i t , l a b e l l e d C o n t r o l , c o o r d i n a t e s the a c t i v i t i e s of the a u x i l i a r y u n i t s which are shown as smal l e r b l o c k s * The computer i s termed asynchronous as the c o n t r o l l e r s t a r t s the next o p e r a t i o n when the smaller blocks have s i g n a l l e d completion of the cu r r e n t o p e r a t i o n r a t h e r than w a i t i n g a f i x e d time f o r the s m a l l e r b l o c k s to c a r r y out t h e i r o p e r a t i o n s * P i s a negative v o l t a g e r e p r e s e n t i n g the performance f u n c t i o n f e d i n t o the c o n t r o l computer by the c o n t r o l l e d process* and X and Y are the c o n t r o l l e d parameters of the process* 2 0 Tp s t a r t the s e a r c h f o r the optimum o p e r a t i n g p o i n t , X and X are s e t t o some i n i t i a l v a l u e s w i t h i n the l i m i t s of the c o n s t r a i n t and* when P_, has reached i t s s t e a d y s t a t e Yaluey P + ( F i g u r e 3-1) i s a d j u s t e d t o eq u a l P_. Here P i s an i n t e r n a l p o s i t i v e v o l t a g e of the computer which i s p e r i o d i c a l l y updated to e q u a l the magnitude of P » The SMP b l o c k i s v e r y much l i k e a sample and h o l d u n i t w i t h P as an i n p u t and P_. as an output * — • x As used by the computer, the output of SMP i s the change needed t o update P^» The change I n P + i s e i t h e r 6P^> ^ P y or AP depending on whether i t was Caused by 6X, £l. or AX and AX. I f the change i s i t i s s t o r e d i n the CCX b l o c k ; i f i t i s 6 P-j.* i t i s s t o r e d i n the CCX b l o c k ; and i f i t i s AP, i t i s s t o r e d i n the COP b l o c k . Upon s i g n a l from the C o n t r o l b l o c k * CCX ge n e r a t e s a number of p u l s e s p r o p o r t i o n a l t o £P^to ste p X* CCX gen e r a t e s a number of p u l s e s p r o p o r t i o n a l t o & P^to s t e p X. While the c i r c u i t r y of GCP i s much l i k e t h a t of CCX and CCX> i t s f u n c t i o n i s d i f f e r e n t * COP s t o r e s AP^ i n a c o u n t e r and compares each s u c c e e d i n g AP with AP^ Upon the r e s u l t of t h i s comparison the s e a r c h e i t h e r s t a y s i n mode 2 or r e t u r n s t o mode 1* The b l o c k SMX has X as output and the v a r i o u s changes S~K. and AX as i n p u t s . The £ X i n p u t comes d i r e c t l y from the C o n t r o l b l o c k and AX i s r o u t e d by the C o n t r o l b l o c k t o SMX from CCX. SMX has e x a c t l y the same f u n c t i o n w i t h X i n s t e a d of X* The d e t a i l s o f a l l the b l o c k s are shown i n . F i g u r e s 3-2 to 3-7. The symbols used are e x p l a i n e d i n Appendix A . 21 3.1 Control Block Figures 3~2 and. 3-3 are detail s of the Contrdl block. Figure 3-2 shows the c i r c u i t r y for subroutines mode 1 and mode 4. It can be seen from comparing Figures 2—3 and 2—5 that these two modes are very similar and, as the computer cannfflit be i n both these modes at the same time.* the same equipment i s used for both modes*-' The main components of the Control block and the i r functions are as follows? FFA — selects mode 1 (state l ) or mode 2 (state 2)j FFG — seledts coarse search (state C) or fine search (state F ) j FFB sele"ets> i n mode 1 only, whether to change X by a fixed £ X (state X) or to change T by a fixed S l L (state X); MC sets correct l o g i c a l states of the Control black f l i p -flops to establish the controller i n mode l j MB —» produces time delay of T^ seconds. With X and X at the i r i n i t i a l values and P updated,* the search rputine i s i n i t i a t e d by a pulse on a l l the li n e s marked S i n the figures (start pulse). The compute** must f i r s t carry out mode 1 as shown i n Figure 2—2. In mode 1 the computer i s tos — move X one s,tepy' ^ X , wait T seconds and store —— move X one step, ^X, wait T seconds and store £ P y | p x —- go to mode 2* Referring to Figure 3-2$ the S pulse sets FFG to C and sets MC to 1. The MC pulse sets FFA to 1 and sets f l i p — f l o p s w i t h i n SMX and SMI so t h a t any p u l s e s t o G53 ( F i g u r e 3r-4) and the c o r r e s p o n d i n g gate f o r I , G54,- w i l l s t e p X i n a ' p o s i t i v e d i r e c t i o n and I i n a n e g a t i v e d i r e c t i o n . , The computer i s now i n mode 1* The MC p u l s e a l s o s e t s FFB t o X and MD t a 1* The MD p u l s e passes t h r o u g h G2 to G53 and causes X t o be stepped yby a f i x e d amount <5X* T^ seconds l a t e r the MB-pulse ( s t a r e ) causes SMP t o update P- to the v a l u e of P . The change needed t o update P^ ,£ ^Pj.£' i s s t o r e d by the OCX b l o c k ( F i g u r e 3-5)* The d e c i s i o n t o stis r e t h i s change i n P as £ P^ . r a t h i r ' than £ P T o r AP i s based upon the s t a t e s o f FFA and FFB»' The sequence i s nw r e p e a t e d w i t h I and i s i n i t i a t e d by the end of st o r a g e p u l s e (ES) « The ES p u l s e from SMP s i g n a l s that' P has been updated* T h i s ES p u l s e passes t h r o u g h G31 and s e t s MF* The MF p u l s e : 1 changes the s t a t e @f FFB ( i n t h i s case from X t o I ) * passes t h r o u g h G30 t o s e t MAP and thr o u g h G i l but not Gl» The MA p u l s e now p a s s e s t h r o u g h G6 as FFB has been s e t t«i s t a t e I * The G6 p u l s e causes I t o be changed an amount — £ l * The ES s i g n a l g e n e r a t e d when ^ P y has been s t o r e d passes through G31 and s e t s MF* The MF p u l s e does not pass t h r o u g h G30 t.ti c o n t i n u e the mode 1 s u b r o u t i n e $ i n s t e a d , i t passes t h r o u g h G i l and the n t h r ough G l t o s e t FFA to 2& T h i s s e t t i n g of FFA t o s t a t e 2 s e t s the computer t o mode 2 s u b r o u t i n e ( F i g u r e 3~3)•• C o n t i n u i n g the e x p l a n a t i o n of F i g u r e 3-2, the o p e r a t i o n of the c i r c u i t r y f o r mode 4 i s the same as t h a t g i v e n f o r mode 1 except t h a t the MC p u l s e i s o r i g i n a t e d by G50 (SMP) and FFG must have been s e t t o F by G26 ( F i g u r e 3=3). I n mgde 4 the computer i s t o move a t o t a l of seven f i n e s t e p s i n X and Y. The d i r e c t i o n s of the X and T st e p s are determined by the p o l a r i t y of the p r e v i o u s <5P^  and £ P y 0 The MG p u l s e o r i g i n a t e d by G50 s e t s P P A t o 1^ determines the d i r e c t i e n &f the next X and Y s t e p s , and s e t s FFB t® X and MD t o 1„ The MD p u l s e passes t h r o u g h G2 s t e p p i n g SX and i n mode 4 t h i s G2 p u l s e i s counted on the seven c o u n t e r w h i c h w i l l now have a Cisnant o f one. The MB p u l s e causes SMP tta update P + T seconds a f t e r X has been stepped 6 X „ [The EJS p u l s e passes t h r o u g h G31 and s e t s MF» The MP p u l s e passes t h r o u g h G30 to co n t i n u e mode 4. The G30 p u l s e s e t s MA whi c h causes a s t e p £X but the s i g n (or d i r e c t i o n ) ©f t h i s 6 X i s not p r e s e t by MC as i t was i n mode 1 D The s i g n of t h i s S X i s determined by the response ( s i g n ) of the p r e v i o u s When the co u n t e r has r e c e i v e d seven p u l s e s * the next MP p u l s e does n o t pass t h r o u g h G30 but passes t h r o u g h G34 t o s e t MK. T h i s MK p u l s e can e i t h e r c o n t i n u e the m©de 4 s u b r o u t i n e or s t a r t the e n t i r e s e a r c h r o u t i n e depending on the p o s i t i o n of S2. The c o n t r o l c i r c u i t f o r mode 2 i s shown i n d e t a i l i n F i g u r e 3™3.« I n m©de 2 the computer i s to s i m u l t a n e o u s l y s t e p AX ~-k £ P ^ and AY r^r-k^P^? w a i t T^ seconds and r e c o r d AP« I f 2AP^< AP^ o r i f AP i s n e g a t i v e the computer i s t o r e t u r n t o mode I . When G l s e t s FFA t o 2 ( F i g u r e 3-2) 9 MG i s s e t to 1* The MG p u l s e C l e a r s the ^ X and Sj c o u n t e r s of CCX and CCY and sets MH^ MH y MB, and FFE are arranged to allow c l o c k p u l s e s t h r o u g h G20 f o r Tg seconds., The MB shown i s the same one shown i n F i g u r e 3—2 and i s r e p e a t e d only t o make the diagram l e s s c o m p l i c a t e d * seconds a f t e r MB i s s e t s the MB p u l s e causes SMP to update ( s t o r e ) P. i n o r d e r t o f i n d AP* A fil.o$k r a t e of 100 c/s was used i n t h e computer c o n s t r u c t e d * OCX passes & number of p u l s e s p p r o p o r t i o n a l t o Sv^rp t o G25* These p u l s e s are pa s s e d t h r o u g h G25* o r are d i v i d e d by twa^ depending on t h e s t a t e of FFH„ t o SMX where X i s changed an amount d i r e c t l y p r o p o r t i o n a l t o the number o f p u l s e s passed t h r o u g h G25» E q u i v a l e n t p u l s e s a re s i m u l t a n e o u s l y p a s s e d through G22 t o change I * The f i r s t o v e r s h o o t s i g n a l from SSP d u r i n g mode 2 s e t s ML w h i c h s e t s FFH t o 1* FFH i n s t a t e 1 means t h a t one h a l f of G19 p u l s e s 1 pass t h r o u g h G25 c u t t i n g the s t e p s i z e AX i n h a l f * The second o v e r s h o o t d u r i n g mode 2 passes t h r o u g h G26 t o s e t the C o n t r o l b l o c k to f i n e s e a r c h (FFG t o F ) * ME and G10 are p a r t o f the c i r c u i t r y r e q u i r e d f o r mode 1 but a re shown i n F i g u r e 3-3 because o f l i m i t e d spaee i n F i g u r e 3-2* An o v e r s h o o t s i g n a l d u r i n g mode 1 ( n e g a t i v e 6 P^ ,) i s passed t h r o u g h GIG t o SMX so t h a t the d i r e c t i o n Of AX w i l l be o p p o s i t e t o t h a t o f 6x* 3*2 Th© SMX «M SMI B l o c k s F i g u r e 3—4 shows SMX, and SMI has e x a c t l y the same c i r c u i t i l a y o u t * The f u n c t i o n b l o c k l a b e l l e d " S t e p p i n g Motor" i s e x p l a i n e d > ! i n Appendix A* The change i n X due t o one s t e p o f the s t e p p i n g motor i s 6 X * The sign of the change depends on XFFD* The MC.. pulse at the s t a r t of every mode 1 subroutine sets XFFD to "up", but i f 6p^ i s negative a pulse from G10 passes through G5 and Changes the state of XFFD. 3.3 The CCX. CGI, and CCP Blocks Figure 3-5 shows CCX, arid COJX has exactly the same c i r c u i t layout. The counter and the d i g i t a l to analog converter func-t i o n a l blocks are explained i n Appendix A* A number of pulses* proportional to the magnitude of from G45 passes through. G16 and ^s stared i n the counter. Upon a signal fr&m the Control block the CCX block i s to produce a number of pulses proportional to the stored G16 l i m i t s the number of pulses counted to 15 to prevent th a 16 pulse from setting the counter to zero. In an'Stctual process the maximum AX would be determined by physical l i m i t a -tions* not Counter s i z e . In mode 2* to step AX, G20 of the Control block passes clock pulses which are passed through G19 and then through G25 (Figure 3-3) to SMX and CCX. When the count i s the same i n both counters of CCX* the output of the comparator, C4* becomes 0 and no further pulses pass through G19» Thus, the number of pulses passed by G19 i s exactly equal to the count stared i n the counter* The G19 pulses could be made equal to some rat i o of the c 0 1 1 1 1^ hy adjusting the resistance network at the input to G l * 26 The CCP block i s shown i n Figure 3-6* I t Is very much l i k e CCX* The input to comparator C3 i s arranged so that the output of C3 becomes 0 i f the count on the AP^ . counter i s less than one half the count on the AP^ counter* G14 and G42 both prevent any count over f i f t e e n * The pulses proportional to AP^ and AF K are provided by SMP. I f AP K ^ APj/2 the G48 pulse from SMP passes through G49 to set Mtx (Figure 3<~3) and thus continues the mode 2 subroutine* The two D/A converters of the CCX.CGs: and CCP "blocks have outputs of opposite p o l a r i t y * This sign difference allows the output o f the D/A units to be compared by simple algebraic addition i n a resistance network* with the algebraic stun being amplified by a difference amplifier* The sign difference i s the r e s u l t of d i f f e r e n t f l i p - f l o p s being used i n the two counters* 3*4 The SMP Block The SMP block i s shown i n Figure 3-7* seconds a f t e r either X or I i s changedy the MB pulse sets FFN to 0 which allows clock pulses through G39» G39 pulses pass through G40 or G41*. depending on the sign of P + - P 9 and adjust ^he size of P to equal the size of P » The output of G57 i s a number of pulses proportional to the difference between P, and P when FFN i s set to 0* The G57 pulses pass through G45# G9* G42, or G43* depending on the state of FFA and FFB of the Control block* If P. *•* P i s posi t i v e * an overshoot* the G41 pulses set FFO to 0* This state of FFO stops any G48 pulse from 27 c o n t i n u i n g the mode 2 s u b r o u t i n e . MD r e s e t s F F N t o 1* T s seconds a f t e r i t was s e t to 0. T h i s r e s e t t i n g of. FFN t o 1 i s the ES s i g n a l t o G31 of mode 1 s u b r o u t i n e o r , i f passed through G37 and G48> t h i s r e s e t t i n g w i l l c o n t i n u e the computer i n mode 2. The G31 p u l s e s e t s FFP to 1 so t h a t o n l y the f i r s t P + - P^ (AP^) f o r any p a r t i c u l a r mode 2 s u b r o u t i n e i s passed t h r o u g h G43V The time c o n s t a n t , T , of MO i s a d j u s t e d t o ensure P can be updated f o r any p o s s i b l e P + - P_» Even though t h i s i s a c o n v e n i e n t method f o r o b t a i n i n g an ES s i g n a l , T would be s d i f f i c u l t t o p r e d i c t f o r an unknown P s u r f a c e and t h e G3l p u l s e f o r a r e a l p r o c e s s may have t o be o r i g i n a t e d by t h e c o n d i t i o n of P = P a f t e r t he MB p u l s e . T — The c i r c u i t r y e n a b l i n g the Coti&rol b l o c k t o c a r r y out mode 3, t h e c o n s t r a i n t h a n d l i n g s u b r o u t i n e , was n o t b u i l t and t e s t e d , b u t a suggested c i r c u i t i s shown i n F i g u r e 3«*8* T h i s c i r c u i t would o n l y work f o r the c o n s t r a i n t XT ^  k and i s shown as an a d d i t i o n t o SMX ( F i g u r e 3-4) and SMI* When X I = Gj no G54 p u l s e s c a n move I * I n s t e a d , the G53 p u l s e s s t e p p i n g X are d i v i d e d by C and passe d t o s t e p I * Moving X one s t e p o f the s t e p p i n g motor might c r e a t e problems i f C <C 1, b u t the g e a r i n g i n SMX c o u l d be changed so t h a t Sx becomes a number o f p u l s e s d i v i s i b l e by Ca Any changes i n X arid I c o u l d be k e p t i n o p p o s i t e d i r e c t i o n s by l e t t i n g any ov e r s h o o t i n mode 1 change the s t a t e of XFFD ( F i g u r e 3-4) and TFFD* T h i s would r e q u i r e a co u n t e r and c o u l d r e s u l t i n l o s s of a c c u r a c y due t o a c c u m u l a t i v e e r r o r * A b e t t e r method would be t o d r i v e I as a p o s i t i o n s e r v o . as shown i n Figure 3-»9© 28 Analog Error Driver l o g i c A Step motor X Analog Clock pulses Figure 3—9 Constraint C i r c u i t with No Accumulative Error to F i g u r e 3—1 General Computer Outline Figure 3—2 Control Block - Modes 1 and 4 31 R E S E T LX- CCX 610 FFO ME M L I G26 FFG 620-ccr-^-^ _ G 9 5 MX G 5 5MY FF6 6 2 2 623 G24 Figure 3 i 3 Control Block - Mode 2 32 G2 FIG 3-2 J3S3 MR FIG 3-3 Mc FIG 3-2 star* mode I FFB FIG 3-2 Vary X GIO FIG 3-3 up ^down XFFD G3 C I V CCW ^| G4 STEPPING MOTOR F i g u r e 3-4 B l o c k Diagram of SMX G4S FIG 3-7 c 1 2 A R 6 MC FIG 3-2 $ ? ¥ FIG 3-3 he MG FIG 3-3~*\R I 2 A 8 F i g u r e 3-5 B l o c k Diagram o f CCX ft* 3-3 U> Figure 3-6 Block Diagram of CCP 34 Figure 3-7 Block Diagram of SMP 35 G53 G54 G3 SMX G4 SMX . MG * P i g . 3-2 G7 SMI G8 SMI F i g u r e 3-8 Block Diagram of Constraint C i r c u i t (XI C) 4. TEST OF COMPUTER 36 A computer as d e s c r i b e d i n Chapter 3 was b u i l t w i t h the e x c e p t i o n o f t h e c o n s t r a i n t h a n d l i n g mode* To o b t a i n the l o g i c b l o c k s r e q u i r e d * t r a n s i s t o r monostables and f l i p - f l o p s on p r i n t e d c i r c u i t boards were used a l o n g w i t h r e s i s t o r t r a n s i s -t o r l o g i c gates# b u t any l o g i c system would do j u s t as w e l l * The computer was s e t up as i n F i g u r e 2*-*l w i t h a Pace Ana l o g computer s i m u l a t i n g the c o n t r o l l e d system* V a r i o u s s i m p l e P s u r f a c e s were o b t a i n e d u s i n g the diode f u n ^ t i a n g e n era-t o r o f the Pace* The s u r f a c e s were k e p t s i m p l e as i t was a t e s t o f the SPDC t o c a r r y out a s p e c i f i c s e a r c h s t r a t e g y * n o t a t e s t o f the s e a r c h t o f i n d the minimum o f any p a r t i c u l a r P s u r f a c e * T e s t s on the s e a r c h s t r a t e g y are w e l l documented* * P s u r f a c e s such as P = X I or P = X/X c o u l d have been produced from e x t r a p o t e n t i o m e t e r s on the s h a f t s of the X and X s t e p p i n g motors| t h i s t e c h n i q u e c o u l d be used on a SPDC o p t i m i z i n g a r e a l p r o c e s s where P i s not d i r e c t l y a v a i l a b l e b u t i s a known f u n c t i o n of the v a r i a b l e s * Chapter 5 c o n t a i n s d e t a i l s o f the s e a r c h over a p a r t i c u l a r s u r f a c e * The f i r s t change n e c e s s a r y i n the computer was the a d d i t i o n o f a manual c o n t r o l (push b u t t o n ) to r e p l a c e t h e c l o c k p u l s e s d u r i n g c i r c u i t t e s t s * Perhaps the manual c o n t r o l s h o u l d be i n s t a l l e d permanently on a SPDC as i t i s needed n o t o n l y f o r i n i t i a l t e s t i n g b u t f o r any r e q u i r e d s e r v i c e and maintenance* 4.1. Comparators The two comparators l a b e l l e d C l and C2 i n F i g u r e 3-7 are t o g i v e one o f two p o s s i b l e v o l t a g e l e v e l s depending on the s i g n of the a n a l o g i n p u t ^7 • The p o s s i b l e o u t p u t s are a d j u s t e d t o the 0 and 1 l e v e l s o f the d i g i t a l equipment f o r d i r e c t use. The u n i t s b u i l t i n the computer were D.C. d i f f e r e n c e a m p l i f i e r s ( g a i n ^ 1 2 0 ) w i t h tine i n p u t grounded and the o t h e r i n p u t ^7 • T h i s p a r t i c u l a r c h o i c e of comparator caused a m p l i f i c a t i o n of n o i s e p r e s e n t <&n t&e computer ground l e v e l a l l o w i n g s p u r i o u s p u l s e s t h r o u g h G40 and G4.1 which c o m p l e t e l y u p s e t the computer o p e r a t i o n . The optimum would e v e n t u a l l y be found b u t many unn e c e s s a r y s t e p s wcr.Id be t a k e n . The e x a c t r e a c t i o n t o one of these s t e p s i n P depends upon when i t happens d u r i n g the s e a r c h procedure and t h r o u g h w h i c h gate i t i s r o u t e d . T h i s ground n o i s e problem was overcome on the computer by u s i n g an i s o l a t i n g t r a n s f o r m e r and c a r e f u l l y r e r o u t i n g the ground l i n e s so t h a t o n l y one ground l e v e l e x i s t e d f o r the e n t i r e computers The grounds f o r the v a r i o u s p a r t s o f the computer were r u n i n p a r a l l e l , u s i n g heavy gauge w i r e * t o t h i s common ground l e v e l * . T h i s ground arrangement and t h e i n t e g r a t i o n of the o u t p u t o f C2 and C3, so t h a t v e r y s h o r t p u l s e s ®n the output of C2 and G3 are not f e d i n t o G40 and G41, managed t o make the out p u t o f G2 and C3 r e l a t i v e l y u n d i s t u r b e d lay ground n o i s e . B e f o r e b u i l d i n g any computer of t h i s t y p e * l i t e r a t u r e 21 on n o i s e and ground systems s h o u l d be r e a d c a r e f u l l y * ' The o u t p u t o f the comparator u n i t s * C2 and (23*; was not the d e s i r e d p e r f e c t two l e v e l s as shown i n F i g u r e 4-d.(a) but r a t h e r as shown i n F i g u r e 4 - 1 ( b ) . 38 (a) T h e o r e t i c a l Comparator Output (b) A c t u a l Comparator Output F i g u r e 4-1 Because o f t h i s s l o p i n g zone i n the output ©f the co.mpara.tprs;> G40 sm& G41 c o u l d output p u l s e s w h i c h were unable t o s t e p the P + s t e p p i n g motor s i n c e the motor r e q u i r e s p u l s e s o f 5 v o l t s a m p l i t u d e (minimum) w i t h a r i s e time of »1 t o 2»5 u-s« and a minimum w i d t h o f 10 [xs« As the p u l s e s of G41 have an exponen-t i a l r i s e time*- i t might take a p u l s e w i t h an o v e r a l l a m p l i t u d e of 8 v o l t s t o o b t a i n 5 v o l t s w i t h a r i s e time of 2*5 p,s* A c l o c k p u l s e w i t h f a s t e r r i s e times and f a s t e r t r a n s i s t o r gates c o u l d have mad® t h e e n t i r e edge o f the p u l s e f a s t e r t h a n 5 volts/2«5 {AS* so t h a t the amp l i t u d e of the G40 and G41 p u l s e s i s the o n l y d e c i d i n g V a r i a b l e i n the s t e p p i n g o f t h e s t e p p i n g motor* But w i t h th© components used, i f SMP were t« be b u i l t a c c o r d i n g t o F i g u r e 3—7»; G57 would have t o pass © E l y p u l s e s w h i c h have a s p e c i f i e d r i s e t i m e , w i d t h , and amp l i t u d e t o s t e p the P + s t e p p i n g moter^ Ra t h e r t h a n b u i l d a s p e c i a l g ate* the 3 phases of the s t e p p i n g motor were c a p a c i t i v e l y ' c o u p l e d t o the i n p u t s of a s t a n d a r d AND g a t e , r e s u l t i n g i n the c o r r e c t p u l s e s from G57 ( F i g u r e 4-#)* The s t e p p i n g of P + s h o u l d depend o n l y on the v a l u e of v^7 • However* with, the par t i c u l a r clock and flipr«fl©p c i r c u i t s used* the G39 pulses have two possible r i s e times depen4ing upon the l e v e l of the clock output when FFN i s set to state 0 * (Figure 4-3 (a)) ± The slope i n the comparator output allows only the faster of these r i s e times to step the P + motor f o r a par t i c u l a r raage of ^ 7 * This i s because the stepping motor requires not «aXy a minimum amplitude but also a minimum r i s e time f o r an input pulse to step the motor. This could be over-come i n varices wayS'# the most obvious being to increase the i gain of the compax&tte amplifiers and thus narrow the sloping region i n th e i r output* Unfortunately any increase i n gain w i l l increase the dependence of the amplifier output on the ground l e v e l . One modification which w i l l overcome both the dependence on ground l e v e l and the sloping region of the D.C. amplifier output i s to repllwSe. the D.C. amplifier with a very high gain AoG* amplifier aa.d make ^7 a n •A.C. signal by chopping P + and P^ * Figure 4~»2 shows the block diagram of this suggested SMP block* Because the A>*G. amplifier does not produce an output proportional to the difference between an input voltage and ground as did the D*C* d i f f e r e n t i a l amplifier, the gain may be very .high and ground noise w i l l not cause the serious problems i t did with the D.C. comparator. This much larger gain w i l l produce a compa^atsr output which w i l l overcome thje problem as shown i n Figure 4-i3(b) because C2 can be adjusted to allow no output of G40 when a step i s not required i n the >*;d*wn* dir e c t i o n of the stepping.mpt&r* 40 J u s t as G57 ( F i g u r e 3-7) would have r e q u i r e d d e l i c a t e b i a s i n g t o pass .ttnly p u l s e s w i t h r i s e tim^s fend amplitudes t o s t e p P +* so t h e i n p u t o f FFO ( F i g u r e 3-7) r e q u i r e d s e n s i t i v e •"trimming", so t h a t FFO was s e t t o the 0 state o n l y by a p u l s e w h i c h would s t e p the: motor. I t i s p o s s i b l e t o o b t a i n the o v e r -shoot p u l s e by a c i r c u i t which would r e q u i r e no a d j u s t i n g and be independent o f r i s e times and v o l t a g e l e v e l s . . The motor phases. A, B* G. ( F i g u r e 4-2) have one sequence f o r the *down" d i r e c t i o n and stofctite:r f o r "up". I f the "up" sequence i s A, B* Gj t h e n any c i r c u i t t h a t can i n d i c a t e t h i s sequence can be used as an o v e r s h o o t s i g n a l * Such a c i r c u i t , u s i n g s t a n d a r d d i g i t a l components, i s shown i n F i g u r e 4—4. The motor phase g o i n g from the 1 s t a t e t o t h e 0 s t a t e s e t s o f f the monostable and . t a l y the phase sequence A^ B. C would produce an o v e r s h o o t p u l s e i ' >; 0 B 0 tD X o 3 1 Overshoot F i g u r e 4—4 D i g i t a l Overshoot C i r c u i t Sinfte the comparators, are the o n l y u n i t s #f t h e compu-t e r not made up o f <fcnly s t a n d a r d d i g i t a l l o g i c components, and* i n g e n e r a l * gave m&ar.ef t r o u b l e than any o t h e r u n i t * a l t e r n a t i v e s t o them s h o u l d be c o n s i d e r e d . There are two i n the SMP b l o c k 41 and one i n each <S>f CCX* CCX* and CCP blocks. The two i n SMP can be replaced by A*C* amplifiers as suggested but ©ther possib-i l i t i e s exist f o r CCX* CCI, and CCP* The function of CCP i s to compare AP 1 and AP^k This comparison i s required by the quite arb i t r a r y decision i n section 2.3 to return to mode 1 from mode 2 i f 2APg <^  AP^* Since for an unknown P surface t h i s decision has no inherent value over any of the other possible variations* the comparator of CCP could be eliminated* If the computer returns to mode 1 th from mode 2 at the l o c a l minimum or at the N mode 2 move* CCP can be replaced by a N counter (counting MG (Figure 3-3) pulses) and an arrangement of stanclard gates to route G49 (Figure 3-i6) pulses to MA (Figure 3-2) when the count i s N* This replacing of the amplifier.comparators by a combination of standard d i g i t a l l o g i c c i r c u i t s i s desirable i f consideration i s being given to building the computer using commercial i n t e -grated c i r c u i t s * The small Voltage difference between the logic states of integrated, c i r c u i t s would require very high comparator gain and careful choice of re s i s t o r s i n any simple D/A units (Appendix A)*, This N step v a r i a t i o n would also make the computer more f l e x i b l e as the number of steps which cause return to mode 1 could e a s i l y be changed at a patch panel* While' t h i s patch panel would a.dd very l i t t l e to the cost of the computer, i t would add much t# i t s search e f f i c i e n c y as the search could be varied from steepest ascent (N = l ) to various relaxation strategies* This f l e x i b i l i t y would be useful i f the i r r e g u l a r i t y 42 of the P s u r f a c e i s a f u n c t i o n of an u n c o n t r o l l e d v a r i a b l e . The a m p l i f i e r comparators o f CCX and OCT can be r e p l a c e d by pure d i g i t a l Comparators w i t h a s e t of gates f o r each p o s s i b l e s t a t e o f the c o u n t e r s ^ however, t h i s would i n c r e a s e the number of gates and f l i p - f l o p s used by the computer* The use <>f r i n g c o u n t e r s would make t h i s type o f comparator e a s i e r to design* A n o t h e r s o l u t i o n would be t o a l l o w AX t o be one o f a few p o s s i b l e s i z e s depending on the s i z e o f ^ P ^ * P e r example* AX = & X 0 < P x < L L = Constant AX = 5 6 X L < P x < 2L AX = 10 Sx 2L < P x T h i s s t e p s i z e arrangement c o u l d be i n s t r u m e n t e d by h a v i n g the most s i g n i f i c a n t f l i p - f l o p i n s t a t e 1 o f t h e ^Py. co u n t e r gate MG ( F i g u r e 3-3) p u l s e s t o one of t h r e e p u l s e t r a i n 22 g e n e r a t o r s * T h i s p u l s e t r a i n g e n e r a t o r would s t e p X* The s i z e o f AX would depend on which p u l s e g e n e r a t o r was t r i g g e r e d * The same type of arrangement c o u l d be used f o r s t e p s i z e c hoiqe I n a d i r e c t s e a r c h s t r a t e g y and would i n v o l v e o n l y d i g i t a l l o g i c b l o c k s * 4*2 Dead Band A problem dev e l o p e d w i t h SMP which would be p r e s e n t r e g a r d l e s s o f the comparator used* On the "store** s i g n a l y MB ( F i g u r e 3—2)# t h e s t e p p i n g motor i s d r i v e n u n t i l P =* P „ The 43 motor i s d r i v e n t h r o u g h G40 or G41 depending upon t h e s i g n o f ^7 = P + - P^* I f a t ^ 7 = 0 G40 d r i v e s the motor one s t e p i n the "down" d i r e c t i o n ( P + becomes l e s s p o s i t i v e ) so t h a t ^7 = -A v o l t s * and i f t h i s *k v o l t s i s enough to d r i v e the motor' t h r o u g h G41 back one s t e p t o *\7 — 0* t h e r e w i l l be a c o n t i n u o u s " c h a t t e r " of SMP f o r the e n t i r e MO p u l s e which was o r i g i n a t e d b y the " s t o r e s i g n a l . W i t h the c i r c u i t as i n F i g u r e 3-7* t h i s S h a t t e r " would r e s u l t i n the 6 P X (S^y) c o u n t e r s o f CCX (C C l ) t o be a t maximum f o r any s t e p 6 X ( 6 1 ) r e g a r d l e s s o f the performance s u r f a c e * I t w o u l d a l s o cause FFO t o be s e t t o t h e 0 state*^ the c o n d i t i o n t h a t 6 P i s n e g a t i v e * even i f the & P r e s u l t i n g from a parameter s t e p i s p o s i t i v e . So w i t h t h i s c i r c u i t r y a dead band must be d e s i g n e d i n the comparator c i r c u i t w i t h a, w i d t h o f a t l e a s t A* t h e change i n ^7 caused by one s t e p o f t h e motor s e t t i n g P^» T h i s dead band i n the comparators r e s u l t s i n a hyster.es e f f e c t i n the a b i l i t y o f SMP t o t r a c k P__* I f <\7 = and X i s stepped 6x r e s u l t i n g i n a change i n P_ such t h a t V - 2^A upon the a r r i v a l o f t h e MB p u l s e a t FFN* two s e q u e n t i a l p u l s e s w i l l l e a v e G57 r e s u l t i n g i n <5px = 2» I f S7 = - 4 A when X i s stepped &X* the same change i n P__ w i l l be r e c o r d e d as 6P^ = I s i n c e S7 = 1?A when t h e MB p u l s e a r r i v e s a t FFN. The v a l u e e f ^7 when any parameter i s : stepped c o u l d be any v a l u e w i t h i n t h e comparator dead band* ^ 1 / 2 * T h i s comparator dead band d e c r e a s e s the s e n s i t i v i t y o f the computer s i h c e a £Py> or l e s s t h a n A may or may n o t be r e c o g n i z e d depending upon the s t a r t i n g v a l u e of *v7 « The computer b u i l t u s i n g the c i r c u i t r y of Chapter 4 44 a l s o c o u l d d r i f t away f r o m t h e o p t i m u m i f A s s o o n a s ^ P ^ o r 6 P ^ > A t h e c o m p u t e r w o u l d f u n c t i o n p r o p e r l y b u t , i n s t e a d o f s t a y i n g w i t h i n a f e w s t e p s o f a v a r y i n g o p t i m u m o n c e i t was f o u n d , t h e c o m p u t e r c o u l d d r i f t u n t i l a Sv ^> A was r e c o g n i z a b l e * I f , d u r i n g mode: 1, t h e p e r f o r m a n c e s u r f a c e i s s u c h t h a t <^P^ ~ *C A t h e c o n t r o l p a r a m e t e r s will n o t be m o v e d d u r i n g t h e n e x t mode 2 a s t h e c o u n t e r s h a v e v a l u e s 6 p x = SVj = O* The c o m p u t e r w o u l d t h e n s t a y i i i mode 2 i n d e f i n i t e l y u n l e s s a n u n c o n t r o l l e d variable f o r c e d ^ t o become n e g a t i v e a n d g r e a t e r t h a n A . A s t h i s i s a n u n a c c e p t a b l e c o n d i t i o n , a g a t e s h o u l d be a d d e d t o CCP so t h a t , i f t h e A P ^ c o u n t e r ( F i g u r e 3—6) i s z e r o w h e n G37 g i v e s a p u l s e , t h e c o m p u t e r s h o u l d be r e t u r n e d t o mode 1. I n g e n e r a l , t h e w i d e r t h e d e a d b a n d i n t h e SMP c o m -p a r a t o r t h e p o o r e r i s t h e p e r f o r m a n c e o f t h e c o m p u t e r ? t h e r e f o r e , A , t h e w i d t h , s h o u l d be made a s n a r r o w a s p o s s i b l e by a d j u s t i n g t h e g e a r i n g b e t w e e n t h e s t e p p i n g m o t o r a n d t h e P + p o t e n t i o m e t e r so t h a t o n e s t e p o f t h e m o t o r p r o d u c e s a s m a l l c h a n g e i n P + , t h u s a n a r r o w A« R e d u c i n g t h e v o l t a g e a c r o s s t h e P p o t e n t i o m e t e r w o u l d a l s o r e d u c e A b u t t h i s s h o u l d n o t b e d o n e s i n c e t h e maximum r a n g e o f P i s t h e v o l t a g e d r o p a c r o s s t h e P + p o t e n t i o -m e t e r . A s t h e r a n g e o f P i s made s m a l l e r , a n y p e r c e n t a g e c h a n g e i n P i s a l s o made s m a l l e r a n d m o r e d i f f i c u l t , t o m e a s u r e * The d i r e c t s e a r c h s t r a t e g i e s do n o t u s e t h e s i z e o f &T? a n d so c o u l d u s e a c o m p a r a t o r w i t h no d e a d b a n d b y f i n d i n g t h e s i g n o f Si? f r o m t h e d i r e c t i o n o f t h e f i r s t m o t o r s t e p a f t e r t h e a r r i v a l o f t h e " s t o r e " p u l s e a n d i g n o r i n g a n y l a t e r s t e p s * 45 A m a j o r i t y of these suggestions may make the search t r a t e g y l e s s r e f i n e d than that of s e c t i o n 2.3, but the o v e r a l l erformance, c o s t , and r e l i a b i l i t y of the c o n t r o l computer are f t e n of g r e a t e r importance than the mathematical s o p h i s t i c a t i o n f the search s t r a t e g y . R high gain A.C. amp. TR*MSIST0R CHOPPER P_ clock F F N — * 6 3 9 MOTOR ABC Y w y -3-P 0 T TPQNSlSTbf? CHOPPER G9 T ClOCK + R E F Figure 4-2 Changes to SMP Clock t r FFN,) t G39 C2 G40 t no step t l s t e p C2 (^ C) (a) t G40 Figure 4-3 SMP Waveforms (b) 48 5* EXAMPLE OP USE OF SPDC As an example o f a p r o c e s s which c o u l d be o p t i m i z e d by a low c o s t SPDC, c o n s i d e r the f l o t a t i o n p r o c e s s o f a complex ore* F l o t a t i o n i s the s e p a r a t i o n of m i n e r a l s from p u l v e r i z e d ore by methods w h i c h cause the v a l u a b l e p a r t i c l e s to r i s e t o the s u r f a c e o f a l i q u i d * V a r i o u s c h a m ' i c a l s ^ known as c o l l e c t o r s , are used t o cause s e l e c t i v e . a t t a c h m e n t of p a r t i c l e s o f one m i n e r a l t o a i r bubbles w h i c h f l o a t to the s u r f a c e o f the l i q u i d w h i l e the o t h e r m i n e r a l s remain i n s u s p e n s i o n i n the l i q u i d * There i s an optimum c o n c e n t r a t i o n of the c o l l e c t o r i n the l i q u i d as too h i g h a c o n c e n t r a t i o n w i l l f l o a t too much o f an unwanted m i n e r a l and too s m a l l a c o n c e n t r a t i o n w i l l n o t r e c o v e r a l l t h a t i s p o s s i b l e o f the d e s i r e d m i n e r a l . The c o n t r o l problem i s f i i f c d i n g t h i s optimum c o n c e n t r a -t i o n * The amount o f c o l l e c t o r can be i n c r e a s e d i f the t a i l i n g l o s s e s are h i g h and d e c r e a s e d i f the percentage o f t h e d e s i r e d m i n e r a l i n the f l o a t e d c o n c e n t r a t e i s too low. The d i f f i c u l t y of t h i s problem depends on the v a r i a b i l i t y o f the o r e * I n ore of c o n t i n u a l l y s h i f t i n g c o m p o s i t i o n , the f l o t a t i o n r e s u l t s w i l l change w i t h the c o m p o s i t i o n c a u s i n g f l u c t u a t i o n s i n the grade of the c o n c e n t r a t e * Economic i n c e n t i v e f o r t r a c k i n g the optimum c o l l e c t o r c o n c e n t r a t i o n i n c r e a s e s w i t h the amount o f ore p r o c e s s e d * I n a l a r g e m i l l o p e r a t i o n a s m a l l improvement i n p e r c e n t a g e 24 r e c o v e r y would pay f o r any added c o n t r o l equipment*; U n f o r t u n a t e l y , no simple f o r m u l a i s known l i n k i n g the c o l l e c t o r added t o an ore and the p ercentage o f a m i n e r a l f l o a t e d * 4 9 Laboratory f l o t a t i o n tests have shown that* f.mx at least one ore* f l o t a t i o n results are a function of m H where m = residual concentration of xanth&te JH*J hydrogen ion concentration Curves of the type i n Figure 5-1 show the relationship between m [H and fot the percentage of mineral recovered. As the product m |H"^ J i s increased* the recovery of lead increases u n t i l a plateau of maximum recovery i s reached* Further increases i n m [H ]^ w i l l decrease the grade of lead concentrate because of increasing fI.<o>tati©a. <8>.f zinc. Therefore* the control problem i s not to minimize or maximize a performance function* but to s t a b i l i z e m JH^| at the point at which maximum recovery i s f i r s t reached. If fo i s considered the performance function, the optimum operating point i s at a s p e c i f i c value of the slope S'S/Slog m |k°^ j , A SPDC would therefore have to be designed to track not the point at which 6P — 0 hut rather 6 P = D where D i s a eeast&st deter-mined by the operating personnel of the m i l l . Both m and, jk^ can be controlled with simple,. analog feedback control, loops and % can be considered t© he continuously available. In t h i s process the knowledge of the performance surface would dictate a very simple search strategy as the P surface i s very smooth and unimodal, A single mode search l i k e the mode 4 described i n section 2.3 would probably s u f f i c e as the approximate optimum point i s known from past operating experience and the purpose of the SPDC would be to track the optimum operating point as i t changes with, the characteristics ©f the 50 ore being floated* The SPDC described i n Chapter 3 "was set up as shown i n Figure 5-2 to simulate control of a f l o t a t i o n process*; The SPDC was designed to track a minimum of P. So P = »^ + D (m [k'^ j ), which has. a minimum at the required operating point* wa,s set up on the Pace*s diode function generator. The actual process has a T i n the ©rd©r of many minutes, but a more convenient T of P P .4 seconds was used i n the test of the SPDC. With v^ == v^ = constant the computer would greatly overshoot the optimum i f the i n i t i a l values of m and H were low but thi s i s to; he expected f r o m the gradient s e a r c h o f the SPDC and would n@t happen i n a SPDC designed to optimize this p a r t i c u l a r process;*; For v^ == constant, and the peak to peak value of v^ equal to one t e n t h t h e range of P , the SPDC i n mode 4 could track the optimum f « r sinusoidal, triangular or square wave as long as the period o f v~ was greater than 10 T * ^ p For v„ with a period of 10 T and an amplitude of v, 2 p 1 that varied P ©ne tenth of i t s maximum value* t h e SPDG could track the optimum f@r sinusoidal and triangular wave f®rm v^ with period greater than 10 T . For a square wave t h e period P of v, had to be g r e a t e r than 30 T f o r the SPDC t/s tr;»,^ k the x p optimum. The T ©r phase lag of a process w i l l l i m i t the f r e -quency of change of the input signal that can be tracked for 26 any d i r e c t method of optimization. 51 Figure 5-2 Test Set Up 52 6. CONCLUSIONS It i s possible to b u i l d special purpose d i g i t a l com-puters for process control purposes at much below the minimum cost of large scale general purpose computers. The SPDC can be designed using only "off the shelf" components to handle many control problems such as single loop optimization. The simple logic of direct search makes i t very well suited for SPDCs? and for an unknown performance surface i t should be as capable of finding the optimum as any of the c l a s s i c a l search strategies* A p o s s i b i l i t y for further study is an investigation of how the choice of search strategy affects the size of the computer as the number of control parameters increases, 53 APPENDIX A LOGIC SYMBOLS Gates 6/ ]— AB A — A B A+B. B Gates are g i v e n a number and i n the t e x t the s i g n a l from a gate i s r e f e r r e d t o by t h i s number, e.g. G l i s • ' " 1 " when b o t h A and B are "1", G3 i s " 1 " when A i s "O". F l i p - f l o p RESET B output FFB The " l " l e v e l appears on the B output l i n e i f the d e v i c e i s s e t to the B s t a t e by a p o s i t i v e g o i n g s i g n a l on the "1" i n p u t . A l l f l i p - f l o p s are l a b e l l e d w i t h l e t t e r s , i . e . FFB. Monostables MA The monostable i s n o r m a l l y i n s t a t e 0 u n t i l s e t to 1 by a p u l s e and remains so s e t f o r an a d j u s t a b l e time p e r i o d . Very often combinations of l o g i c a l elements w i l l reoccur. These units w i l l be given a unique symbol—a rectangle with a designated l e t t e r , (see page 55) D/A — D i g i t a l to Analog converter C — Comparator C»- COUIS/T 55 > i R — R e s e t R A A A A A f? 1 v V W V -R ^2 [c 1 2 D/A Counter , 4 r 8 CCW C¥ 1 ^ Mo d r c i t o r i v i n g r e m i t N S t e p p i n g n o t o r i t — i r •cc S t e p p i n g motor T 56 REFERENCES 1* ' P r o c e s s Computer S c o r e c a r d Updated* C o n t r o l Engineering*. March, 1 9 6 5 * P* 5 7 . * 2* Grabbe, E* M*^ ' ^ D i g i t a l Computer C o n t r o l Systems| an Annotated B i b l i o g r a p h y % P r o c e e d i n g s of the F i r s t €®agress of  I n t e r n a t i o n a l F e d e r a t i o n .of. A u t o m a t i c .Control«; ,. Buttetrworth,. Ijondon, I960* 3* Bohn, E* ^The P r a c t i c a l R e a l i z a t i o n o f O p t i m a l C o n t r o l o f M u l t i v a r i a b l e Dynamic Pr o c e s s e s ' 1 , P r o c e e d i n g s  o f t h e Conference on Canadian I n d u s t r i a l R e s e a r c h , Ca r l e : t o ^ l E T n i y e r s i t y , J u l y * 1964* 4. Bohn, E* V*, "The S y n t h e s i s of Real-Time O p t i m a l C o n t r o l l e r s by H y b r i d Computer Techniques", P r o c e e d i n g s o f the  N a t i o n a l Conference on A utomatic C o n t r o l * C a r l e t o n U n i v e r s i t y , September 1965. 5* G i b s o n , J * E*, " M e c h a n i z i n g the A d a p t i v e P r i n c i p l e " , C o n t r o l  E n g i n e e r i n g ^ Octobeg I960, p* 109* 6* White, B*, ^The Quarie O p t i m a l C o n t r o l l e r " , I n s t r u m e n t s and  Automations November. 1956* p. 2213* 7* P r o g r e s s Report en Opcong Dow E v a l u a t e s O p t i m i z i n g C o n t r o l , C o n t r o l E n g i n e e r i n g . November, 1959, p* 1.24* 8* Op con F i l l s a Gap* C o n t r o l . E n g i n e e r i n g , February*l959» p. 27. 9 i Feldbaum^ A* A**V ^Automatic O p t i m a l i z e r " , Auttmat*' Telemech** J u l y i ' i 9 5 8 ^ ( E n g l i s h T r a n s l a t i o n A p r i l , 1959, p* 718.) .10* S t a k h o v s k i j R* I * , "Twin-Channel Automatic O p t i m i z e r ? , Automata Telemec,h* * J u l y , 1958, ( E n g l i s h T r a n s l a t i o n A p r i l , 1959* p. 729.) 11* Boas j A* H*(,' ^ O p t i m i z i n g M u l t i v a r i a b l e Functions'8*1•* Chemical '. .E n g i n e e r i n g ^ V o l * 70, .No* 5, March 4 j 1963* 12ii K u s h n e r | H*; J * ^.. *AHill C l i m b i n g Methods f o r O p t i m i s a t i o n of M u l t i p a r a m e t e r N o i s e D i s t u r b e d Systems*, J*A*6«C* * 1963*. '- ~* "~~ ••• * •• / . 13* B r o o k s , S* H*, >>fA Comparison o f Maximum—Seeking Methods", ^ O p e r a t i o n s Be s e a r c h , Yo1. 7, 1959. 14.* L e v i n e j L*,^ ^Methods f o r S o l v i n g E n g i n e e r i n g Problems U s i n g A n a l o g Computers", O p t i m i z a t i o n Techniques;. MsGraw-H i l l I n c * , T o r o n t o , 1964, Chapter 8. 57 15* T u r n b l a d e , R* C*, "Random O p t i m i z a t i o n & M u l t i p l e A d a p t i v e C o n t r o l " , A d a p t i v e C o n t r o l Systems* Pergamttn P r e s s • New Y o r k , 1963. 16* W i l d e , D» J*,, " O p t i m i z a t i o n Methods", Advances i n Chemical  E n g i n e e r i n g * V o l * 3, Academic P r e s s , New York* 1962* 17* I d e l s o h n | J * M«* "Ten Ways t o F i n d the Optimum'1, C o n t r o l  Engineering^. June* 1964, p* 97* 18* Hooke, R* and J e e v e s * T» A«, " ' D i r e c t S e a r c h ' S o l u t i o n o f N u m e r i c a l and S t a t i s t i c a l P r o b l e m s " , A s s n * f o r  Computing M a c h i n e r y J o u r n a l * The A s s o c i a t i o n f o r Computing M a c h i n e r y , V o l * 8* 1961, p* 212* 19* M i t c h e l l , B* A*, "A H y b r i d A n a l o g - D i g i t a l Parameter O p t i m i z e r f o r AST RAG 1 1 % S i m u l a t i o n * V o l * 4, No* 6, J u n e , 1965* 20* C h e s t n u t , H*£ "Automatic O p t i m i z a t i o n of a P o o r l y D e f i n e d Proce ss J * A. C C . , 1963. 21* K o r n , G* A** and K o r n , T* M«, E l e c t r o n i c A n a l o g and, H y b r i d Computers * M c G r a w - H i l l I h c * , Toronto* 1964* 22* F l o r e s , 1*^ Computer L o g i c . P r e n t i c e — H a l l Inc«, Englewood C l i f f s , New J e r s e y , i 9 6 0 . 23* R o b e r t s , S* M*$ and L y v e r s . H* I * , "The G r a d i e n t Method i n P r o c e s s C o u t r o l " , I n d u s t r i a l and E n g i n e e r i n g C h e m i s t r y * V o l * 53, No* 11, November, 1961, p. 877* 24* B u s h e l l , C*» and M a l n a r i c h , M*, "Reagent C o n t r o l i n F l o t a t i o n ^ M i n i n g E n g i n e e r i n g , J u l y * 1956* 25* B u s h e l l , G«* " B e h a v i o u r o f X a n t h a t e i n F l o t a t i o n " , The CanadJLaji M i n i n g and M e t a l l u r g i c a l B u l l e t i n * March, 1958* 26* L e f k o w i t z ^ , I«, ^Computer C o n t r o l % Handbook o f Automation* Computation* and C o n t r o l . V o l * 3, John W i l e y & Sons, I n o * , New Y o r k , 1961. 

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