US 3564221 A Abstract available in Claims available in Description (OCR text may contain errors) United States Patent [72] Inventor Robert F. Wheeling Mullica Hill, NJ. [21] Appl. No. 054,503 [22] Filed Sept. 7, 1960 [45] Patented Feb. 16, 1971 [73] Assignee Mobil Oil Corporation [54] OPTIMIZATION WITI-l RANDOM AND HISTORICAL VECTORS 8 Claims, 5 Drawing Figs. [52] U.S.Cl 235/150.l, 235/15 1 12 [51] Int. Cl. G05b 13/00 [50] Field olSearch 235/151, 184, 185, 193; 23/230,253 (A); 235/151 (E); 235/1 50.1 [56] References Cited UNITED STATES PATENTS 2,980,330 4/1961 Ablow et a1 235/184 3,044,701 7/1962 Kerstukas 235/151 39 ADDr---i 37 3,048,331 8/1962 Van Nice et al. 2,972,447 2/1961 White 235/151 2,972,446 2/1961 White 235/151 OTHER REFERENCES 1 Eckm-anet al. Optimizing Control ofa Chemical Process? Control Engineering-Sept. 1957. pp. 197 204. Munson and Rubin, Optimization by Random Search on the Analog Computer. Oct. 25. 1-958; p. 12 Primary Examiner- Eugene G. Botz Attorney- Donald L. Dickerson and Oswald G. Hayes ABSTRACT: The invention relates to the production of an optimum value of a system output function which is dependent upon a plurality of variables where the effect of changes in the variables upon the function may be determined. An optimum is reached quickly and efficiently by changing each variable repetitively in a random manner initially and thereafter the tendency of the changes to be purely random is modified so that past history in the search for the optimum value tends to weight the-randomness in favor of the most desirable direction toward optimum. 1 SO M n NIEU FEB1 SL971 I $564,221 ROBERT F. WHEEL/N6 IN VEN TOR Y @2h-d%& I ATT RNEY PATENIED FEB 1 6 I97! sum 2 or 3- a w I 5 2 A 2 6 I. M 8 I A 8 W IRIIIIJ 0 3 C m m I 7 -I 4 8 3 3 E 2 7 2 T 3 7 6 w M 5 M 5, E 6 E M Z w m A A M M A A N U U U U U U E O O 0 Q 0 w R S 5 S S S S E w 3 E E E .1 A N m 1...: I v v 2 A A M r 1L .L r: I. r: ri -L D D N A 8 3 3 w Mm 0R E NL H OT 1 2 3 4 5 6 M lflm i AO c E v. E E cw E T mm R J; M .PC R A I 5 5 O E O 5 5 w 6 7 ll 9 w T m m u w m w m m w I W M M M M M 2 I Q. a I 3 E A m FM w m l 2 3 4 5 6 WM X X X X X X A0 E, E E E E 6 RC FIG. 3. ROBERT 1-. WHEELING INVENTOR ATTORNEY ROBERT F. WHEELING sum 3 OF 3 FIG. 4-. FIG. 5. PATENTED FEB 1 6 \sn QBIAS OPTIMIZATION WITH RANDOMAND-I'II STORICAL VECTORS This invention-relates to the production of an optimum value of a function in response to a plurality of dependent variables in which the effect of changes of said variables upon said function may be determined. In many systems often there are present a plurality of functions which may be selectively controlled or varied independently of one another. Such functions may coact in such systems in an unknown or undefinable manner to produce a resultant function or functions. Generally, it is difficult to predict or preset the variable functions to values which will produce an optimum resultant function Representative-of such a problem is that presented in the control ofa chemicalprocessing unit. Materials fed thereto maybe varied over known ranges'both as to rate and quality. Operating condi tions within the unit may also be varied at will within given ranges. Prior art systems have, by various methods, permitted selection of a combination of the variables so that operation is maintained at a selected point within a permissible. range. However, optimum operation often is at a point close to a boundary of permissible values of controllable parameters. Operations generally may not be safely maintained at such point so that less advantageous but safely maintainable sets of operating conditions and material feeds, generally are chosen. The foregoing is illustrative of the type of operation -in which the present invention finds application. The selection of the most desirable operating point for such a processing unit is a problem of optimization. The optimum may be defined in terms of quality of product produced, quantity for a given input, or may be dependent upon weighing functions to reflect market prices of the products. Insofar as the effect of a change in one of the variable functions upon an output function may be determined, an optimum relationship between all such variables may be obtained. The process may be understood in connection with a solution of a problem where a mathematical model or expression can be precisely formulated. However, a more general problem to which the present invention applies is in relation to complex systems where the actual functioning of the systems may not be readily expressed and only the effect on the output of a change in input may be observed. In accordance with the present invention there is provided a method of producing an optimum scalar output from a process unit dependent upon a set of inputs which are variable but which are interrelated and coact to produce said scalar output. The method comprises generating input functions representative of said inputs and applying said input functions to said process unit for generation of said scalar output. A condition function is stored which is representative of said scalar output. Thereafter, at least said one of said input functions is charged by an incremental amount of random character to produce a modified scalar output and successive changes are thereafter made in at least one of said inputs by incremental amounts which are of random character weighted in dependence upon the difference between said scalar output and modified scalar outputs. In accordance with a further aspect of the invention, a method of optimizing a scalar output function is provided where a set of input functions which are independently variable are interrelated and coact to produce the scalar output function. The method includes generating input signals representative of the input functions and an output signal representative of the scalar output function. An intermediate function is stored which is representative of the output signal. The input signals are then successively changed by incremental amounts of random character to produce a modified output signal. Thereafter, at least one of the input signals is changed by an incremental amount, the amount being of random character weighted in dependence upon the difference between the stored intermediate function and the modified output signal. In accordance with a further aspect of the invention, there is provided in combination a system having inputs and at least one scalar output, together with a sensing system responsive to variations in said inputs and said output and having means for producing and storing an intermediate function. Means are further provided for progressivelymodifying the stored intermediate function in response to differences between the scalar output function and the stored intermediate function upon variations in the inputs. Means are provided for altering the inputs randomly but successively dependent upon a weighting factor representative of the rate of'change of the scalar output function. For further objects and advantages of the present invention and for a more complete understanding thereof, reference may now be had to the;following description taken in conjunction with the accompanying drawings in which: FIG. 1 is a plot of time-distance data; FIG.-2 is a plot of an error function related to FIG. 1; FIG. 3 illustrates a system for determining an optimum solution for the data of FIG. 1; FIG. 4 is a modification of the invention in a chemicalprocessing system; and 7 FIG. 5 illustrates in schematic form element 82 of FIGS. The invention will first be described in connection with a relatively simple operation in which optimization is employed. The latter operation is in the processing of seismic signals to obtain an indication of the velocity distribution of earth formations penetrated by such seismic signals. After presenting the description of the present invention in connection with this example, there will then be described an application of the invention to a more complex problem involving the control of a chemical processing unit. In seismic exploration where expanding seismic spreads disclosed by Dix in GEOPHYSICS, Vol. XX, pages 68 et seq.,'are employed, it is desireable to process the resultant timedistance data by means of a function of the form: More particularly, equation (1) is descriptive of line 10 of FIG. 1 which is to be fitted to the time-distance data points plotted on FIG. 1. If the points represent time-distance data for a given reflection and are obtained by employing the above-noted expanding spread techniques and if plotted on scales such that the x scale is in terms of distance squared and the y scale is in terms of seismic record time squared, then the slope of line 10 is numerically equal to the reciprocal of the square of the average velocity of the earth formations through which the reflection traveled from source thereof to points of detection. In accordance with prior art techniques, the coefficients a and b of equation (1) have been determined by the process of least-squares fitting which involves the minimization of: I). E (21- ill) i=1 where y is the ordinate of line 10 at a given abscissa, and y, is the data value at the same abscissa. The process involved determining the sum of the squares of the values Ayn Ay uAye. and then adjusting the values of a and b, either simultaneously or separately, for successive solutions of equation (2) until values of a and b, which give the error function e of minimum value, are determined. The value of the error function is thus completely determined by the values of a and b of equation l By substitution of equation (1 in equation (2), the following expression is obtained: By solving equation (3) for different selected values of a and b, the optimum values for a and b may be determined;-but only by successive approximations. The criterion for such solution is that the summation of equation (3) be a minimum. 1f equation (3) is solved for a plurality of values of a and b, there will result data which may be plotted in the form illustrated in FIG. 2. By changing a and b for successive solutions of equation (3), there would be described a family of ellipses, each-characterized by a constant value of e, and having a common center. The value of a and b describing the common center represents the optimum function for equation (1). The technique for arriving at the optimum function may be likened to mountain climbing, since the data plotted in FIG. 2 may be viewed as a familiar topographic plot of a paraboloid. For data such as would be plotted in FIG. 2, the desirable end or optimum values for a and b are unknown. By trial and error through solutions of equations such as equation (3) an attempt is made to identify the coordinates of the center point. The analogy to mountain climbing" is helpful to an understanding of the mathematical processes through which successive solutions to equation (3) lead to the center point. Prior art methods of optimization have been based upon the philosophy that the best way to determine the direction from a given point 11 to the center point 12 is systematically to test various values of a and b near point 11. A solution of equation (3) for each of the points 13, 14, 15, and 16 will indicate the directions a and b must be changed in order to improve the solutions; that is, approach to point 12. By the foregoing method, point 14 obviously would appear to be the best direction. Sample solutions would then be obtained for each of points 17, 18, and 19. A comparison of the results would then indicate which is the most desirable direction to go and thusby successive computations the entire area represented by FIG. 2 can be sampled to identify the center point or optimum solution represented by point 12. in contrast with the foregoing, the present invention is based upon a novel method of mountain climbing in which the variation froma given data point as in FIG. 2 is predicated not upon an ordinary sampling of the entire area for a solution but upon a random variation in the parameters a and b, to which random variation there is applied a historical weighting factor which serves effectively to limit the probability of following a path in a nonpreferred direction. In the case illustrated in FIG. 2, the desired direction would be along a straight line connecting points 11 and 12. Since the location of and direction to point 12 is unknown, a reduction in number of computations to locate the same is desired. Such reduction is possible through use of the system illustrated in FIG. 3. H6. 3 illustrates a system for automatically providing an optimum solutionto an equation such as equation (3) wherein random variations of parameters a and b with historical weighting are employed. A solution of the relatively simple expression of equation (1) involving but two unknowns is presented primarily by way of example in order to provide an understanding of the invention. The invention is deemed to have principal application to systems and methods where a greater number of variables are involved. Thus, while the method of the present invention is wholly operative in systems of fewer variables, it is of principal advantage where many variable functions are present. The system of FIG. 3 is designed to provide a solution for equation (3) based upon the availability of the coordinates of the six data points plotted in FIG. 1. Equation (3) requires that for the value of x and y at each data point there will be performed a first operation to produce the product ax. The product will then be added to the value of the variable b from which there will be subtracted the value of y. The resultant sum or difference is then squared. The sum of all such squared functions represents the error function e. The system of FIG. 3 includes a multiplier 30, having a first input conductor connected to a source of voltage 31 representative of the value of x,. A second input conductor 32 is provided upon which the variable a appears as a voltage 13,. The output signal from multiplier 30, the product ax is then applied by way of conductor 33 and resistor 34 to the input circuit 35 of a squaring unit 36. Resistor 34 forms a part of an adding unit 37. The variable b is applied as a voltage E, to the adding network 37 by way of conductor 38. A voltage E,, is applied to the adding network 37 from source 39. Thus. the signal on circuit 35 leading to the squaring network 36 is representative of the parenthetical expression of equation (3) for the coordinates x, and y of the first data point. This function is then applied to the squaring network 36. The output of the squaring network 36 is then applied to a summing bus 40 by way of adding resistor 41. identical circuitry is provided for the remaining pairs of data points x ,y ...x ,y More particularly, a voltage E from source 50 is applied to a multiplier 51. The conductor 32 is also connected to multiplier 51 for application ofa voltage E,,. The output of multiplier 51, a voltage 5;, appearing on conductor 38, and a voltage -E,, are applied to adding network 52, the sum being then applied to the squaring network 53 whose output is applied by way of resistor 54 to the summing bus 40. A voltage 1'5, & is applied to multiplier 55 along with the voltage E on bu? 32. The output from multiplier 55, the voltage E,, on bus 38, and a voltage --E,, are applied to adding network 56. The resultant sum is applied to squaring network 57 whose output is applied by way of adding resistor 58 to the summing bus 40. A voltage E is applied to multiplier 60 along with the voltage E from bus 32. The output of multiplier 60, the voltage E,, on bus 38, and a voltage E, 4 are applied to adding network 61. The resultant sum is applied to squaring network 62 whose output is applied by way of adding resistor 63 to the summing bus 40. v A voltage E is applied to multiplier 65 along with the voltage 5,, from bus 32. The output of multiplier 65. the voltage E,, from bus 38, and a voltage E, are applied to adding network 66. The resultant sum is applied to squaring network 67 whose output is applied by way of adding resistor 68 to the summing bus 40. Finally, a voltage E, g is applied to multiplier 70 along with the voltage E from bus 32. The output of multiplier 70, the voltage E,, from bus 38, and a voltage E,, are applied to adding network 71. The resultant sum is applied to squaring network 72 whose output is applied by way of adding resistor 73 to the summing bus 40. The values of x1...x,, and yl...-., are actual data points plotted on FIG. 1 and are constant. In contrast. both the values E,, and E will vary upon operation of the system ultimately to attain a value which represents the optimum solution; i.e., represehtative of point 12 of FIG. 2. For any given initial set of values for E,, and E there will be developed through the operation of the system thus far described an output voltage E, or an error function e on the summing bus 40. This function will then be stored as a voltage in a storing unit such as condenser by the momentary closure of switch 81 which is operated under the control of unit 82 to apply such voltage by way of operational amplifier A Bus 40 is also con nected directly tocomparison unit 82 at a first input circuit 84 thereof. Comparison unit 82 also connected directly to condenser 80 at its second input circuit 85. The unit 82 is also connected by way of linkage 87 to a pair of switches 88 and 89. When switches 88 and 89 are closed. electrical charges are placed on storage condensers 90 and 91 which are representative of the values a and b, respectively. Such charges are then to be applied to conductors 32 and 38, respectively. The conductor 32 is to be connected to a source of voltage such as battery by way of a resistor 98 and a normally open switch 99. Battery 100 is connected across the terminals of a potentiometer 102. A first wiper am 103 on the potentiometer 102 is connected to switch 99. The wiper arm 103 is coupled by driving linkage 104 ton motor 105 cyclically to rotate the wiper arm 103. By this means. a voltage is developed on the wiper arm 103 which varies cyclically from a negative to a positive value. The maximum and minimum values of this voltage are dependent upon the battery 100. A second wiper arm 106 associated with the potentiometer element 102 is positioned initially at the midpoint of the potentiometer element 102. Wiper arm 106 is connected to ground. The upper terminal of condenser 90 may be connected by way of switch 88 and the operational amplifier A to the upper terminal of a condenser 110 and to the conductor 32. The lower terminals of condensers 90 and 110 are connected to ground. The wiper arm 106 of potentiometer 102 is connected by way of driving linkage 111 to a position control unit 112. The control unit 112 is actuated in response to signals from a differentiating unit 113, the input of which is connected to the upper terminal of condenser 90. The upper terminal of condenser 90 may also be connected by way of switch 108 and resistor 109 by way of operational amplifier A to the conductor 32. Switches 99 and 108 are interconnected by way of a switch-actuating linkage 115 leading from a random circuit closing unit 116. The random closure unit 116 is also coupled by way ofa switch-cam unit 117 to the motor 105. Switch-cam 117 is actuated when arm 103 is at the beginning of a sweeping cycle on the potentiometer 102. The random switch control 116 serves as a time delay for applying an actuating function to switches 99 and 108. The time delay relative to closure of the switch-cam vunit 117 may be any fraction of the period of revolution of arm 103. For successive cycles, the time delay is varied in a random manner throughout the period. Switch control unit 116 limits closure of switches 99 and 108 to one closure for each cycle of revolution of the wiper arm 103. The closure is purposely designed to occur at random times during successive cycles so that there will be applied to the conductor 32 a voltage which is the algebraic sum of the voltage, if any, at the upper terminal of condenser 90 and the random voltage at the wiper arm 103. That random voltage may be either positive or negative and of any value between extreme limits determined by the battery 100. During the first cycle of operations of the system thus far described, the wiper arm 103 will rotate, unit 117 will be actuated, and thereafter switches 99 and 108 will be closed to establish a voltage on condenser 110 which will be maintained constant to provide an initial value of the voltage E, on conductor 32. A similar system is provided for production of the voltage E on conductor 38. More particularly, conductor 38 may be connected to a source of voltage such as battery 130 by way of a resistor 131 and a switch 132. Battery 130 is connected across the terminals of a potentiometer 133. A first wiper arm 134 on the potentiometer 133 is connected to the switch 132. Wiper arm 134 is coupled to and driven by motor 105 through linkage 104. By this means, the arm 134 cyclically sweeps the potentiometer 133. A voltage is thus developed on the wiper arm 134 which varies cyclically from a negative to a positive value. The extremes of the range of values of this voltage depend upon the battery 130. A second wiper arm 135 associated with potentiometer 133 is positioned normally at the midpoint thereof. Wiper arm 135 is connected to ground. The upper terminal of condenser 91 may be connected by way of switch 89 and operational amplifier A to the upper terminal of a condenser 140 and to the conductor 38. The lower terminals of condensers 91 and 140 are connected to ground. The wiper arm 135 of potentiometer 133 is connected by way of linkage 136 to a position control unit 141. The control unit 141 is actuated in response to signals from a differentiating unit 142, the input of which is connected to the upper terminal of condenser 91. The upper terminal of condenser 91 may be connected by way of switch 145, resistor 146 and operational amplifier A to conductor 38. Switches 132 and 145 are interconnected by way of linkage 147 leading from a random circuit closing unit 150. A switchcam unit 151 is actuated when arm 134 is at the beginning ofa sweep cycle on the potentiometer 133. The random switch closing unit 150 serves as a time delay following actuation of switch-cam unit 151 for applying actuating functions momentarily to close switches 132 and 145. The time delay will be that time intervaL between actuation of the switch-cam unit 151 and the actuation of switches 132 and 145. The time delay may thus be any fraction of the period of revolution of arm 134. For successive cycles, the time delay is varied in a random manner in said period. While switches 132 and 145 will be closed only once during any given cycle, the closure is purposely designed to occur at random times during successive cycles so that there will be applied to the conductor 38 a voltage which is the algebraic sum of the voltage, if any, at the upper terminal of condenser 91 and the random voltage at the wiper arm 134. That random voltage may be either positive or negative and of any value between extreme limits determined by the battery 130. During the first cycle of operations, the wiper arm 134 will rotate, unit 151 will be actuated, and thereafter switches 132 and 145 will be closed to establish a voltage on condenser 140 which will be maintained constant to provide an initial value of voltage E on conductor 38. V A switch-cam unit 160 is coupled to motor 105 by way of linkage 161. Unit 160 is connected in the control circuit 162 of the comparing unit 82. The switch of unit 160 is adapted to be closed at the end of each cycle of arms 103 and 134. The comparing unit 82 operates to close switches 81, 88, and 89 at the end of any given cycle of operations during which the value of the voltage applied to comparing condenser 163 decreases relative to the value previously stored on the error function condenser 80. Operation of'the system above described for the production of a voltage on conductors 32 and 38, representative of the optimum value of the parameters a and b of equation (3) is as follows. Condensers 80, 90, 91, 110, 140, and 163 initially are completely discharged. Source 31 is adjusted to a value representative of an abscissa x,. The source 39 is adjusted to a voltage representative of the ordinate y,. Similar adjustments are made in the sources for E 2 E, and E, 2 ma Motor 105 is then energized to initiate sweeping action of the arms 103 and 134. At the beginning of the first sweep, switch-cam units 117 and 151 apply a start signal to random control units 116 and 150, respectively. At random times during the first cycle of arms 103 and 134, switches 99 and 108 momentarily are closed by unit 116. Switches 132 and 145 similarly are momentarily closed by unit 150. The voltage between tap 103 and ground is momentarily applied by way of resistor 98 to conductor 32 and thus causes a charge proportional to that voltage to be stored on condenser 110. Momentary closure of switches 132 and 145 apply to conductor 38 a voltage equal to that appearing between tap 134 and ground to store a representative charge on condenser 140. The voltages on condensers 110 and 140 thus will be the first trial values ofthe a and b parameters ofequation (3). Multiplier 30, responsive to the voltage from source 31 and the voltage on conductor 32, applies a product voltage to the adder 37 by way of conductor 33. The voltage on conductor 38 along with the negative value of the voltage from source 39 are also applied to adder 37. The sum of the voltages applied to the adder 37 are then applied by way of circuit 35 to the squaring unit 36. The output of unit 36 is then applied to the summing bus 40. Voltages corresponding to the other five terms of the error function can similarly applied to the summing bus 40. Summing bus 40 thus carries a voltage numerically proportional to the error function 2 corresponding to these trail values of the a and b parameters. Regardless of the comparing network 82, switch 81 is momentarily closed at the end of the first cycle of operations to store a charge on condenser representative of the error function. At the same time, switches 88 and 89 are momentarily closed to transfer the chargeson condensers 110 and 140 respectively to condensers and 91, thus recording, i.e., storing, the values ofa and b that yielded the value ofthe error function that is stored on condenser 80. On the next succeeding cycle of operations, the charges on condensers and are altered depending on the random nature of the voltageslfrom arms 103 and 134. The altered voltages are then retained on condensers 110 and 140 for the production of a new error function on bus 40. The latter error function appears as a voltage across the comparing condenser 163. At the end of the sweep cycle of arms 103 and 134, the switch-cam unit 160 actuates the comparing network 82 so that the voltages across condensers 80 and 163 are compared. Comparing unit 82 is so armed as to actuate switch 81 and switches 88 and 89 only following those operations of units 160 when the voltage on condenser 163 is less than the voltage on condenser 80. Closures of switches 88 and 89 transfer from condensers 110 and 140 respectively to condensers 90 and 91 charges representative of the voltages on conductors 32 and 38 which improve the solution of equation (3) Evidence of such improvement is a decrease in voltage across condenser 163 relative to condenser 80. Closure of switch 81 similarly transfers from condenser 163 to condenser 80 a charge representative of the decreased value of the error function. Following the storage of charges on condensers 90 and 91, the closure of switches 99 and 108 by unit 116 on the next succeeding cycle applies to conductor 32 the sum of the voltage on condenser 90 and the voltage sensed by arm 103. Similarly a voltage is applied to conductor 38 which is representative of the sum of the voltage on condenser 91and the voltage from arm 134. By this means there will be accumulated on condensers 90 and 110 charges which will change in the direction of the optimum value of the parameter a of equation (3). Similarly there will be accumulated on condensers 91 and 140 charges which will change in the direction of the optimum value of the parameter b of equation (3 Succeeding cycles of operations extended sufficiently in time will develop optimum voltages on conductors 32 and 38. This will be so in spite of the fact that any variations in the latter voltage will be dependent upon the random operation of units 116 and 150. Effectively the system thus far described operates to search part of the area represented by the plot of FIG. 2 with a random search vector. The objective is to move from a given point, i.e., voltages on condensers 90 and 91, only if a change is in direction leading toward a minimum value of the error function, i.e., toward point 12 of FIG. 2. The present invention contemplates not only searching as above noted with a random vector but also modifying or weighting the random character of the search in a direction which is dependent upon the experience during past cycles of operation. More particularly, the voltage appearing on condenser 90 is applied by way of conductor 200 to the dif ferentiating network 113. The output of unit 113 is applied to control unit 112 which by way of linkage 111 serves to move the second wiper arm 106 and thus alter the probability of selection of positive and negative values by arm 103. if the change in the voltage on condenser 90 is rapid as sensed by differentiation, then the arm 106 will be moved a substantial distance toward the positive or negative end of potentiometer 102. The magnitude and sense of movement will depend upon the rate and sense of change in voltage on condenser 90. By this means, the random vector utilized for searching an optimum will be weighted in favor of those values of search vector which experience has shown are directed toward optimum. Similar operation is provided for the control of the manner in which the voltage on condenser 91 is modified. The latter voltage is applied by way of conductor 201 to a second differentiating unit 142. Unit 142 actuates the control unit 141 to alter the position of arm 135 to weight the random search vector controlled by unit 150. By the foregoing means the path followed from a given starting point of the values of a and b to an optimum value is made much more direct and requires many fewer cycles of computation to arrive at the optimum value. in short, the comparison circuit 82 through its operation of switches 88 and 89 controls the differentiators 113 and 142 so that the position controls 112 and 141 are effective to move the slide contacts 106 and 135 to provide the historical weighting factor which serves effectively to shorten the search for the optimum value of 5,. Thus the historical weighting function is performed only if the voltages on condensers 90 and 91, upon operation of switches 88 and 89 are changed in directions leading toward a minimum value of the error function 15,. The foregoing description has been concerned with the solution of equation (3), the relationships between the parameters of which are well known and understood. By the operation of the system thereare produced scalar output voltages onconductors 32 and 38. By applying the input functions x x and y, y to the system, there is generated an intermediate function or voltage which is stored on condenser 80. Thereafter the voltages applied to conductors 32 and 38 are changed by incremental amounts of random character to modify the scalar output on condenser 80. Successive changes are then made in the voltages on conductors 32 and 38 which are random in character but at least in part are weighted in dependence upon the difference among the successive scalar outputs resulting from such changes. The foregoing description and the system have been utilized in order to explain the invention in connection with a relatively simple case. The more general case has to do with the operation of systems in which relationships between the variables and the effects one upon the other are not known. All that may be available in order to select an optimum set of conditions for controllable parameters are (1) knowledge of the controllable parameters at the input, (2) the scalar output from the system, and (3) the magnitude and sense of changes in the scalar output in response to a given change in one or more of the controllable inputs. Those familiar with solution of the problem above illustrated will recognize that there are ways to arrive at the desired solution which may be less cumbersome and perhaps as direct as to use the system of FIG. 3. Applicants have found that the use of a random search vector and a historical weighting factor is more useful where more than two variables are involved; though in some two-dimensional cases it is more efficient than conventional computations. In such multidimensional space, the problem of search may become astronomical. The present invention provides a means for optimization in multidimensional space within computational time intervals that renders optimization procedures practical for control purposes. This will permit operation of a given system, for example, at a point near critical value but which point is more efficient or desirable than could otherwise be maintained. A representative system is illustrated in FIG. 4. More particularly, the system of FIG. 4 is designed to carry out a catalytic cracking process wherein charge stock enters the system by way of channel 220, passing through a control valve 221 and a flow telemeter 222. The charge stock then passes through a heat exchanger 223 and thence to a heater 224. The output of heater 224 passes through flow path 225 to a catalytic process vessel or case 226. The output flow path 227 from the case 226 leads to the heat exchange unit 223 and thence to a fractionating tower 228. Light fractions pass from the fractionating tower 228 by way of condenser 229 to a surge tank 230. Noncondensible components pass by way of telemeter 231 to a storage unit 232. Condensate from tank 230 passes to a proportioning valve 239. Part of the condensate is forced by pump 240 through a telemeter 241 into the fractionating tower 228 near the top thereof. The remainder of the condensate passes through a telemeter 245 andthencc to a storage unit 246. An intermediate fraction of the product from the fractionating tower 228 passes through telemeter 250 and thence to a storage unit 251. The heaviest components from the fractionating tower 228 pass to a proportioning valve 255. A portion of such heavy components is fed by way of telemeter 256 back into the charge stock line 220 for recycling. The remainder of the heavy components passes by way oftelemeter 257 to a storage unit 258. While the processes performed in the system thus far described are well known, it is to be understood that various conditions of feed rate, temperature and pressure, reflux ratio, and recycle ratio may be varied. For example, the temperature to which the charge stock is heated in furnace 224 may be varied and controlled by unit 260. The temperature and rate of flow of an eutectic solution passing through the case 226 may be controlled by unit 261. The temperature maintained in the fractionating tower 228 may be varied and controlled by unit 262. Thus, there may be produced in response to operation of this system controlled flow to each of the storage units 232, 246, 251, and 258. The proportions of the charge stock which ultimately are processed and reach a given destination will be determined largely by the conditions maintained in the system. In accordance with the present invention, the parameters of the system at various critical points are sensed and applied to a computer 270. The output of the computer is then utilized to modify the controllable conditions in the processing system so that the products resulting therefrom will be of optimum character. The computer 270 in its preferred form will comprise a digital computer capable of carrying out functions of the type above explained in connection with FIGS. 1-3. Its capability is such that it may operate on the multiplicity of variables present in the system of FIG. 4. An analogue system would be capable of so operating, but the capacity of a digital unit is such that it lends itself more readily to problems of this nature than an analogue system. For this reason, the representation of heaviest digital computer in block form his been adopted for the purpose of the present description. Input data are provided for computer 270 by means of the various telemeters. More particularly, the input flow rate under the control of valve 221 is sensed by the telemeter 222 and applied to the computer by way of the signal channel 271 (shown dotted). The temperature representative of the operation of heater 224 is applied as input data by way of channel 272. The temperature representative of that maintained by the eutectic solution in case 226 is applied as input data by way of channel 273. The reflux ratio of the products from condenser 230 is applied as data by way of channel 274 leading to transducer 241 and channel 275 leading to transducer 245. A signal from control unit 262 representing the temperature in the fractioning tower 228 is applied as input data by way of channel 276. A signal representing light end products flowing to storage 232 are applied as input data by way of channel 277 leading to telemeter 231. A signal representative of the intermediate fraction flowing to storage 251 is applied as input data by way of channel 278 which leads to telemeter 250. The recycle ratio of the heavy fraction flowing from fractionating tower 228 is applied as input data by way of a signal on channel 280 leading from telemeter 257 and a signal on channel 281 leading from telemeter 256. Control linkages are provided between the computer 270 and variouscontrol elements in the system. More particularly, the computer may control operation of the heater unit 260 by way of linkage 300. Control of the eutectic solution may be effected by way of linkage 301. The temperature of the fractioning tower 228 is controlled by way of linkage 303. The reflux ratio from tank 230 is controlled by way of linkage 305 leading to proportioning valve 239. The ratio of recycle of heavy ends from fractionating tower 228 is controlled by way of linkage 304 leading to valve 255. Finally, the flow rate of the charge stock is controlled by linkage 302 leading to the valve 221. With the system operating under the control of computer 270, considerable flexibility is available in establishing optimum operation. The volume of products flowing to any one of storage units 232, 246, 251, or 258 may be required to be maximum. Alternatively, the products may be proportioned in dependence upon market conditions so that the value of the combined products from the system will be a maximum. Such criteria may be established in the computer 270 on the same general basis as the minimization criteria of FIG. 3 and conditions within the system will be automatically adjusted for such optimum operating point. In FIG. 4 a complete mathematical model of the system is not available and is not necessary. All that is necessary is that the values of the various output flow rates be monitored by the computer and that the computer contain an optimizer embodying the principles of this invention. The computer 270 may then search the various controllable conditions with a random vector as was done in the case of FIG. 3 and then apply historical weighting to such random vector which by means of linkages 300-305 produce stepwise changes in the various parameters. Ultimately, an optimum operating condition of the system is attained. Thus, computer 270 is functionally representative of the embodiment shown in FIG. 3. except E E ind E, as well as a and b. The above example illustrates that, insofar as it may be feasible to make measurements directly on the physical system under consideration, mathematical models may be unnecessary. The invention is thus applicable to optimization of systems mathematical, physical or combinations, provided that the output to be optimized is generated responsive to a plurality of inputs. Through the use of applicants optimization procedure. the present invention permits the use of direct control of the process unit even though many variables are involved since the computation time to establish optimum may be so reduced as to be compatible with periods during which control action must be taken in the actual process unit. The processing unit illustrated in FIG. 4 is diagrammatically representative of a catalytic cracking unit in which the case 226 along with cases 2260 and 226b may be sequentially connected between flow lines 225 and 227 and to a source of air 226c to permit periodic regeneration of a catalyst in each processing case while maintaining a continuous flow through the system. It may be found desirable to control parameters or variables in the system other than those specifically described above. However, the foregoing is to be taken as representative of applicants method of optimizing wherein actual control of a functional unit may be provided. The system of FIG. 4 has been illustrated to emphasize functional relationships therein. It will be appreciated that the number of actual input functions applied to computer 270 is greater than necessary because the computer itself may include sensing devices so that it knows" the actual settings of control points. For example, the computer, having set valve 221, already is in possession of data representative of such settings. This being the case, the data input circuit 271 may be eliminated because its function may be carried out by means of an internal connection or function of the computer 270. Other points in the system may have a similar relation with respect to the computer 270. i It will now be appreciated that the invention may be carried out by hand even though by so doing the primary advantage of following applicants invention would be greatly reduced. In manual operation of the system of FIG. 3, for example, the arms 103 and 134 are adjusted manually to'random locations on the potentiometers. Momentary closures of switches 99, 108, 132, and 145 are manually accomplished so that an error function is produced across condenser 163. Manual closure of switch 81 transfers a charge to condenser 80. Thereafter, subsequent cycles of random potentiometer settings and subsequent switching operations manually performed produce comparison error functions on condenser 163. Switches 88 and 89 are then manually closed only when the error function is reduced relative to a preceeding cycle of operation. The potentiometer arms 106 and 135 are then manually moved in such direction as to increase the probability of randomly selecting voltages from batteries and for application to conductors 32 and 38. Ultimately, optimum values will appear on the latter conductors. Having described the invention in connection with the analogue device of FIG. 3 and the computer-process system of FIG. 4, it will be appreciated that many of the components of each system have not'been described in detail because both functionally and structurally they are well known to those skilled in the art. The telemeters such as telemeter 22 of FIG. 4 may be of any one of various systems which generate an electrical or pneumatic signal proportional to and dependent upon the rate of flow of fluid therethrough. In FIG. 3, differentiation as carriedout by unit 113 is well known to those skilled in the art. The position control unit 112 may comprise a servomechanism for positioning the arm 106. The comparison circuit 82 may be of the type known as a cathode-coupled, binary comparator such as illustrated at page 169 of PULSE AND DIGITAL CIRCUITS, Millman and Taub, McGraw Hill, 1956. A circuit of the latter type is illustrated in FIG. wherein the condenser 163 is connected by way of conductor 84 to the input grid of a tube 400. The voltage on condenser 163 will change in response to changes in voltage on the summing bus 40. Condenser 80 is connected by way of conductor 85 to the control grid circuit of tube 401. The output of tube 401 is coupled by way of a blocking oscillator-amplifier unit 402 to a relay coil 403. Switch 81 is actuated by relay coil 403. A circuit including tubes 400 and 401 operates to produce an output signal when the voltage across condenser 163 exceeds the voltage across condenser 80. The latter signal is amplified and a pulse is produced in unit 402 and applied to relay 403 to close a circuit through switch 81 for analyzing the voltages on condensers 163 and 80. Linkage 87 may then extend to switches 88'and 89, FIG. 3, to transfer to condensers 90 and 91 respectively the tentative increment values stored on condensers 110 and 140 respectively. Such transfer will take place only when the voltage on condenser 163 changes relative to the voltage on condenser 80 in one of two senses. The voltage on condenser 80 may be made optimum, either a maximum or a minimum, in accordance with the present invention, depending upon polarities of the voltages employed and the manner in which condensers 163 and 80 are connected into the circuit of FIG. 3. In accordance with the operation of the system of FIG. 3, it is desired to so arrange the circuit that the error voltage appearing on conductor 40 will be minimized. In contrast, in the system as illustrated in FIG. 4, it may be desirable to produce output quantities or products of maximum character. Since the motor 105 of FIG. 3 drives the potentiometer arms 103 and 134 cyclically, it is desirable to alter the condition of charge stored on condensers 163 and 80 only at the end of each cycle of operations of the potentiometers. For this purpose, the cam-operated switch 160 has been provided. Switch 160 is connected to the comparison circuit 82 by way of conductor 162 which is shown in FIG. 5 as leading to one terminal of a second relay 410. The second terminal of relay 410 is connected by way of battery 411 to ground. By this means, when the cam-operated switch 160 is closed, relay 410 will be energized to close the switch 412 which is connected in series with switch 81. Thus, equalization of the conditions on condensers 80 and 163 will be carried out periodically but only if the charge on condenser 80 differs from the charge on condenser 163 in one sense only positive or negative, but not both. By this means, a system is provided having at least two inputs into which scalar input quantities, either electrical signals as in FIG. 1 or materials to be processed in FIG. 4, are fed. The system has at least one output from which a resultant scalar output, the charge on condenser 163, FIG. 1, or products from the system of FIG. 4, is obtained. As to the two input quantities, a first control means is provided for changing from an initial level to a different level a characteristic of the quantity applied to the first of the inputs. Similarly, a second control means is provided for changing from an initial level to a different level a characteristic of the quantity applied to the second of the inputs. A first change selector is provided for ac tuating the first control means in a random fashion both as to sense and magnitude. A second change selector is provided for periodically actuating the second control means both as to sense and magnitude. Storage of a first condition representative of the initial level of the output quantity and storage of a second condition representative of the level of the output quantity following each actuation of the control means are then effected. The stored conditions are then compared and means responsive to differences between the first and second conditions, of one sense only, are provided for establishing a new initial level for each of the input quantities and for the output quantity where the changes in the input quantities are equal to the changes made by the first and second change selectors respectively and where the new initial level for the first stored condition corresponds with the second stored condition. In a further aspect, applicantshave provided a method in which there are generated input functions representative of independent variables at each of a series of states of an operating system. In response to such variable functions, there is produced a signal representative of the value of the output function from the system. The latter signal is then compared for each of a plurality of states of the independent variables by progressively varying the input functions representative of successive states conformably with the following schedule: A. The variation between the initial state and a second state is representable by a random vector, and B. The variation between a given state. subsequent to the second state and the state immediately precedent thereto is the weighted resultant of a random vector and a controllable vector representative of variations prior to the immediately precedent state and weighted to a degree representative of the quantity by which earlier variations shall have resulted in successive states wherein comparison shall have demonstrated changes of said output function in the direction of a desired optimum. Having described the invention in'connection with certain modifications thereof, it is to be understood that further modifications may now suggest themselves to those skilled in the art and it is intended to cover such modifications as fall within the scope of the appended claims. lclaim: 1. ln optimizing a scalar output function of a system where a set of input values to said system some of which are independently variable, are interrelated and coact to produce said scalar output function; the combination of: means for generating input values including those representative of said variables at each of a series of states of said system; means for calculating from said input values the value of said scalar function at each such state; means for storing said value of each said state; means for comparing the value of said scalar function as calculated at each said state with the value thereof as calculated at a precedent state; means for randomly varying said independently variable input values; and means for weighting the random variations of said independently variable input values by amounts related to the magnitude of the earlier variations which produced change of said scalar function toward the desired optimum. 2. In optimizing a scalar output function ofa system where a set of input values to said system some of which are independently variable, are interrelated and coact to produce said scalar output function, the method which comprises: generating in said system input signals representative of said input values, some of which are independently variable corresponding with said independently variable input values; generating from said input signals an output signal representative of said scalar output function; storing in said system an intermediate function representa tive of said output signal; repetitively changing said variable input signals by amounts of random character to produce output signals of differing magnitude some of which represent improved scalar output functions; and in response to said changes of said variable input signals which produced said improved output functions, changing said variable input signals by an amount of random selection which selection is weighted in dependence upon the earlier changes in the values of said variable input signals which produced said improved output functions. 3. In optimizing a scalar output function ofa system where a set of input values to said system some of which are independently variable, are interrelated, and coact to produce said scalar output function, the method which comprises: generating in said system input signals representative of said input values, some of which are independently variable corresponding with said independently variable input values; generating from said input signals an output signal representative of said scalar output function; storing in said system an intermediate function representative of said output signal; periodically changing said variable input signals by amounts of random character to produce output signals of differing magnitude some of which represent improved scalar output functions; and thereafter weighting the random change of said variable input signals by amounts related to the magnitude of the earlier changes which produced said improved output functions. 4. in optimizing a scalar output function of a system where a set of input values to said system some of which are indepen dently variable, are interrelated, and coact to produce said scalar output function, the method which comprises the following steps: generating in the system input signals representative of said input values, some of which are independently variable corresponding with said independently variable input values; generating from said input signals an output signal representative of said scalar output function; storing in said system an intermediate tive of said output signal; changing said variable input signals by amounts of random character to produce output signals of differing magnitude some of which may produce output signals representative of improved scalar output functions; thereafter repetitively changing said variable input signals by amounts dependent upon random selections; and weighting the random selections by amounts related to the magnitude of the earlier changes which produced output signals representing improved scalar output functions. 5. The combination which comprises a process unit in which there is developed a physical output quantity which is to be optimized and which unit has a plurality of control elements, control means connected to said elements for changing control characteristics of said elements from initial levels to incrementally different levels, change selector means'for each said control means for repeatedly actuating the same randomly both as to sense and magnitude, means for storing a first physical quantity responsive to and representative of a scalar characteristic of said output quantity, means for generating a second physical quantity responsive to and representative of said characteristic of said output quantity following each change in said elements, comparison means responsive to differences in magnitude between said first physical quantity and said second physical quantity of one sense only for establishing new initial levels in said control characteristics and for changing the storage of said first physical quantity to a level corresponding with that of said second physical quantity, and means for interconnecting said control elements and said change selector means for modifying the random character of said change selector means in accordance with the rates of changes of said control elements from the respective initial levels thereof to new levels. 6. The combination which comprises a process unit in which there is developed a physical output quantity which is to be optimized and which unit has a plurality of control elements, control means connected to said elements for changing control characteristics of said elements from initial levels to incrementally different levels, change selector means for each said control means for repeatedly actuating the same randomly both as to sense and magnitude, means for storing a first physifunction representacal quantity responsive to and representative of a scalar characteristic of said output quantity, means for generating a second physical quantity responsive to and representative of said characteristic of said output quantity following each change in said elements, comparison means responsive to differences in magnitude between said first physical quantit and said second physical quantity of one sense only for esta lShing new initial levels in said control characteristics and for changing the storage of said first physical quantity to a level corresponding with that of said second physical quantity, means for separately interconnecting each said control element and its associated change selector means for modifying the random character of each said change selector means in accordance with the rate of change of said control elements from the respective initial levels thereof to new levels. 7. A system of optimizing the magnitude of a scalar output function of a system where a set of input values to said system some of which are variable, are interrelated and coact to produce said scalar output function, which comprises: .means for generating in said system input signals representative of said input values, some of which are independently variable corresponding with said independently variable input values; means for generating from said input signals an output signal representative of the magnitude of said scalar output function; means for repeatedly and randomly varying said variable input signals for producing a plurality of values of said output signal some of which represent change in said scalar output function in the optimal direction; means for storing said output signals representing said change in said output function in said optimal direction; comparison means for comparing each new value of said output signal with the stored value of the output signal; means for weighting subsequent random changes of said variable input signals by amounts which increase with the extent to which previous changes of said variable input signals caused said output function to approach its 0ptimum value; and means operative under the control of said comparison means for controlling said weighting means for operation only when said new values of said output signal represent change of said output function toward its optimum value. 8. in optimizing the magnitude of a scalar output function of a system where a set of input values to said system some of which are variable, are interrelated and coact to produce said scalar output function, the method which comprises: generating in said system input signals representative of said input values, some of which are independently variable corresponding with said independently variable input values; generating from said input signals an output signal representative of the magnitude of said scalar output function; repeatedly and randomly varying said variable input signals for producing a plurality of values of said output signal some of which represent change in said scalar output function in the optimal direction; storing said output signals representing said change in said output function in said optimal direction; and in response to said random variations of said variable input signals which produced change of said scalar function in said optimal direction weighting subsequent random changes of said variable input signals by amounts which increase with the extent to which previous changes of said variable input signals caused said output function to change in said optimal direction. 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