Automatic Synthesis of Both the Topology and Tuning of a Common Parameterized Controller for Two Families of Plants using Genetic Programming

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1 Atomatic Synthesis of Both the opology and ning of a Common Parameterized Controller for wo Families of Plants sing Genetic Programming Martin A. Keane Econometrics Inc. Chicago, Illinois makeane@ix.netcom.com Abstract his paper demonstrates that genetic programming can be sed to atomatically create the design for both the topology and parameter vales (tning) for a common parameterized controller for all the plants in two families of plants that are representative of typical indstrial processes. he genetically evolved controller is "general" in the sense that it contains free variables representing the characteristics of the particlar plant. he genetically evolved controller otperforms the controller designed with conventional techniqes. In addition, the genetically evolved controller infringes on an early patented invention in the field of control. Introdction Atomatic controllers are biqitos in the real world. he prpose of a controller is to force, in a meritorios way, the actal response of a system (conventionally called the plant) to match a desired response (the reference signal or setpoint). For example, the crise control device in a car continosly adjsts the engine (the plant) based on the difference between the speed specified by the driver (the reference signal) and the car's actal speed (the plant response). Genetic programming has recently been sed to atomatically create the design for both the topology and parameter vales (tning) for a controller for a particlar two-lag plant and a particlar three-lag plant (Koza, Keane, Y, Bennett, and Mydlowec 000). However, these two (different) evolved controllers applied only to particlar plants (of the same family). he qestion arises as to whether it is possible to evolve a common controller (accessing varios parameters representing the overall characteristics of the plant) that can perform well for an entire family of plants (say, the n-lag plants) and perhaps also for one or more additional families of plants. In their inflential book, Astrom and Hagglnd (995) identified for families of plants "that are representative for the dynamics of typical indstrial Jessen Y Genetic Programming Inc. Los Altos, California jy@cs.stanford.ed John R. Koza Stanford University Stanford, California koza@stanford.ed processes." Astrom and Hagglnd then developed a common method for designing controllers and demonstrated improved performance for their method over the Ziegler-Nichols (94) rles on all the plants in all for families of plants. One of the for families consists of the n-lag plants represented by transfer fnctions of the form G( s) = () n ( s) where n = 3, 4, and 8 and where s is the Laplace transform variable. Another family consists of plants represented by G( s) = () 3 ( s)( α s)( α s)( α s) where α = 0., 0.5, and 0.7. he methods developed by Astrom and Hagglnd se pairs of parameters representing the overall characteristics of a plant. hese parameters are not, of corse, a complete representation of the behavior of the plant; however, they offer the practical advantage of sally being obtainable for a given plant by means of relatively straight-forward testing in the field. In one of their methods, Astrom and Hagglnd se two freqency domain parameters, namely the ltimate gain, K (the minimm vale of the gain that mst be introdced into the feedback path to case a system to oscillate) and the ltimate period, (the period of this lowest freqency oscillation). In another version, Astrom and Hagglnd se the time constant, r, and the dead time, L. Astrom and Hagglnd describe a procedre for estimating these two parameters from the plant's response to a step inpt. hese two parameters are obtained by approximating the plant with the transfer fnction sl e ( sr ) his paper shows that genetic programming can be sed to atomatically create the design for both the topology and tning for a common parameterized controller for all plants belonging to the two families of plants described by eqations () and (). he common parameterized controller is created sing a fitness measre that optimizes step response and distrbance

2 rejection, while simltaneosly constraining maximm sensitivity and sensor noise attenation. he common genetically evolved controller otperforms the controller designed sing the techniqes of Astrom and Hagglnd 995. Section discsses how genetic programming can be sed to atomatically synthesize the design for both the topology and tning of controllers. Section 3 itemizes the preparatory steps necessary to apply genetic programming to the above two families of plants. Section 4 presents the reslts. Genetic Programming and Control In a closed-loop continos-time feedback system consisting of a plant and its controller, the otpt of the controller is inpt to the plant and the otpt of the plant is, in trn, inpt to the controller Reference Control Plant 54 Signal Variable Otpt /s Plant Controller s 570 Figre Block diagram of a plant and a PID controller composed of proportional, integrative, and derivative blocks. Figre is a block diagram for an illstrative control system containing a controller and a plant. he directed lines in a block diagram represent time-domain signals while the blocks represent signal processing fnctions that operate in the time domain. he otpt of the controller 500 is a control variable 590 which is, in trn, the inpt to the plant 59. he plant has one otpt (plant response) 594. he plant response is fed back (externally as signal 596) and becomes one of the controller's two inpts. he controller's second inpt is the reference signal 508. he fed-back plant response 596 and the externally spplied reference signal 508 are compared (by sbtraction here). Notice that the takeoff point 50 of figre provides a way to disseminate a particlar reslt (of the sbtraction 50) to three places in the block diagram (5, 54, and 56). he otpt (i.e., control variable 590) of this controller is the sm of three terms. First, there is a proportional (P) term (the gain block 530 with an amplification factor of 4.0). Second, there is an integrating (I) term (the integrator 560 preceded by the gain block 540 with an amplification factor of,000.0). he integrator is shown in the figre as /s. hird, there is a a differentiating (D) term (the derivative block preceded by the gain block 550 with an amplification factor of 5.5). he derivative is shown in the figre as s. Since the controller's otpt is the sm of a P, I, and D term, this type of controller is called a PID controller. he PID controller was invented and patented by Albert Callender and Allan Stevenson of Imperial Chemical Limited of Northwich, England (Callender and Stevenson 939). Genetic programming (Koza 99; Koza and Rice 99; Koza 994a, 994b; Koza, Bennett, Andre, and Keane 999; Koza, Bennett, Andre, Keane, and Brave 999) is an extension of the genetic algorithm (Holland 975). Additional information on genetic programming can be fond in books sch as Banzhaf, Nordin, Keller, and Francone 998; books sch as Langdon 998, Ryan 999, and Wong and Leng 000 in the series on genetic programming from Klwer Academic Pblishers; in edited collections of papers sch as the Advances in Genetic Programming series of books from the MI Press (Spector, Langdon, O'Reilly, and Angeline 999); in the proceedings of the Genetic Programming Conference (Koza, Banzhaf, Chellapilla, Deb, Dorigo, Fogel, Garzon, Goldberg, Iba, and Riolo 998); in the proceedings of the Ero-GP conference (Poli, Nordin, Langdon, and Fogarty 999); in the proceedings of the Genetic and Evoltionary Comptation Conference (Banzhaf, Daida, Eiben, Garzon, Honavar, Jakiela, and Smith 999); at web sites sch as and in the Genetic Programming and Evolvable Machines jornal (from Klwer Academic Pblishers). Evoltionary comptation has been previosly sed for synthesizing controllers having mtally interacting continos-time signal processing blocks and for system identification problems (Marenbach, Bettenhasen, and Freyer 996). (Extensive references are itemized in Koza, Keane, Y, Bennett, and Mydlowec 000.) here are several different styles that are commonly sed in genetic programming. As one example, genetic programming is often sed as an atomatic method for creating a program tree to solve a problem (Koza 99). he individal programs that are evolved by genetic programming are typically mlti-branch programs consisting of reslt-prodcing branches, atomatically defined fnctions (sbrotines), and other types of branches. In this approach, the program tree is simply exected. he reslt of the exection may be a set of retrned vales, a set of side effects on some other entity (e.g., an external entity sch as a robot or an internal entity sch as compter memory), or a combination of retrned vales and side effects. In this approach, the fnctions in the program are seqentially exected, in time, in accordance with a specified "order of evalation" sch that the reslt of execting one fnction is available at the time when the next fnction is to be exected. Early work on the problem of

3 atomatically creating controllers sed this conventional approach to genetic programming. As a second example, genetic programming is often also sed to atomatically create program trees which can be sed in conjnction with a developmental process to design complex strctres, sch as neral networks (Gra 99) and analog electrical circits (Koza, Bennett, Andre, and Keane 996; Koza, Bennett, Andre, and Keane 999). In this approach, the program tree is interpreted as a set of instrctions for constrcting the desired strctre. he constrction process is implemented by applying the fnctions of a program tree to an embryonic strctre so as to develop the embryo into a flly developed strctre. As in the first approach, the fnctions of the program are exected separately, in time, in accordance with the specified "order of evalation." In this paper, a compter program (i.e., program tree, LISP symbolic expression) will represent the block diagram of a controller. he block diagram consists of signal processing fnctions linked by directed lines representing the flow of information. here is no "order of evalation" of the fnctions and terminals of a program tree representing a controller. Instead, the signal processing blocks of the controller and the to-be-controlled plant interact with one another other as part of a closed system in the manner specified by the topology of the block diagram. ADF0 704 PROGN 700 DEFUN 70 VALUES 790 LIS 706 REF 708 VALUES 7-70 PLAN OUPU GAIN /s 760 ADF0 734 GAIN ADF s GAIN ADF0 754 Figre Program tree representation of the PID controller of figre. Figre presents the block diagram for the PID controller of figre as a program tree. he internal points of this program tree represent the signal processing blocks contained in the block diagram of figre (i.e., derivative, integrator, gain, sbtraction, addition). he external points (leaves) of this program tree represent nmerical constants and time-domain signals, sch as the reference signal and plant otpt. Notice that atomatically defined fnction (sbrotine) ADF0 in the left branch prodces a time-domain signal that eqals the reslt of sbtracting the plant otpt from the reference signal. he three references to ADF0 in the reslt-prodcing (right) branch of this program tree disseminate the reslt of sbtracting the plant otpt from the reference signal and correspond to the takeoff point 50 of figre. In the style of ordinary compter programming, a reference to a sbrotine ADF0 from inside ADF0 wold be considered to be a recrsive reference. However, in the context of genetic programming and control systems, a sbrotine that references itself corresponds to a loop in the controller's block diagram (i.e., internal feedback inside the controller). 3 Preparatory Steps 3. Program Architectre Since the to-be-synthesized controller has one otpt (control variable), each program tree in the poplation has one reslt-prodcing branch. Each program tree in the initial random poplation (generation 0) has no atomatically defined fnctions. However, after generation 0, the architectre-altering operations may insert (and delete) atomatically defined fnctions. Atomatically defined fnctions may be sed for takeoff points, internal feedback within the controller, and rese of portions of the block diagram. he permitted maximm of five atomatically defined fnctions is more than sfficient for this problem. 3. erminal Set he nmerical parameter vale for each signal processing block possessing a parameter is established by an arithmetic-performing sbtree containing pertrbable nmerical terminals, arithmetic operations, and the for parameters for representing the overall characteristics of a plant. Arithmetic-performing sbtrees may appear in both reslt-prodcing branches and any atomatically defined fnctions that may be created dring the rn by the architectre-altering operations. he vale retrned by an entire arithmeticperforming sbtree is interpreted as a component vale lying in a range of (positive vales) between 0-3 and 0 3. he terminal set for the arithmetic-performing sbtrees is aps = {R, KU, U, L, R}. Here R denotes a pertrbable nmerical vale. In the initial random generation (generation 0) of a rn, each pertrbable nmerical vale is set, individally and separately, to a random vale in a chosen range (from and 3.0 here). In later generations, a pertrbable nmerical vale may be changed by adding or sbtracting a relatively small nmber determined probabilistically by a Gassian probability distribtion. he standard deviation of the Gassian distribtion is.0 here (i.e., one order of magnitde after the vale retrned by an entire arithmetic-performing sbtree is interpreted). he pertrbations are implemented by a genetic operation for mtating the pertrbable nmerical vales. he pertrbable nmerical vales are coded by 30 bits in or system. A constrained syntactic strctre maintains one fnction and terminal set for the arithmetic-performing sbtrees and a different fnction and terminal set (below) for all other parts of the program tree.

4 he remaining terminals are time-domain signals. he terminal set,, for the reslt-prodcing branch and any atomatically defined fnctions (except the arithmetic-performing sbtrees described above) is = {REFERENCE_SIGNAL, CONROLLER_OUPU, PLAN_OUPU}. Space does not permit a detailed description of the varios terminals sed herein (althogh the meaning of the above terminals shold be clear from their names). See Koza, Keane, Y, Bennett, and Mydlowec Fnction Set he fnction set, F aps, for the arithmetic-performing sbtrees is F aps = {ADD_NUMERIC, SUB_NUMERIC, MUL_NUMERIC, DIV_NUMERIC, REXP, RLOG}. he two-argment DIV_NUMERIC fnction divides the first argment by the second argment, except that the qotient is never allowed to exceed 0 5. he oneargment REXP fnction is the exponential fnction and the one-argment RLOG fnction is the natral logarithm of the absolte vale. he fnction set, F, for the reslt-prodcing branch and any atomatically defined fnctions (except the arithmetic-performing sbtrees described above) consists of continos-time signal processing fnctions and atomatically defined fnctions. F = {GAIN, INVERER, LEAD, LAG, LAG, DIFFERENIAL_INPU_INEGRAOR, DIFFERENIAOR, ADD_SIGNAL, SUB_SIGNAL, ADD_3_SIGNAL, MUL_SIGNAL, DIV_SIGNAL, ULIMI, ADF0, ADF, ADF, ADF3, ADF4}. he one-argment ULIMI fnction limits a signal by constraining it between an pper and lower bond. his fnction retrns the vale of its argment (the incoming signal) when its argment lies between -.0 and.0. If the argment is greater than.0, the fnction retrns.0. If the argment is less than -.0, the fnction retrns -.0. ADF0,, ADF4 denote atomatically defined fnctions added dring the rn by the architectre-altering operations. he definitions of the other fnctions above are sggested by their names. See Koza, Keane, Y, Bennett, and Mydlowec Fitness Measre Genetic programming is a probabilistic algorithm that searches the space of compositions of the available fnctions and terminals nder the gidance of a fitness measre. he fitness measre is a mathematical implementation of the problem's high-level reqirements. It is coched in terms of what needs to be done not how to do it. he fitness measre for most problems of controller design is mlti-objective in the sense that there are several different (sally conflicting) reqirements for the controller. he fitness of each individal in the poplation is determined by execting the program tree (i.e., the reslt-prodcing branch pls any atomatically defined fnctions that may have been created dring the rn by the architectre-altering operations). he exection of the program tree prodces an interconnected seqence of signal processing blocks that is, a block diagram for the individal controller. he controller is embedded into a framework containing the (fixed) plant and the (fixed) external feedback loop. A SPICE netlist is then constrcted to represent the block diagram of the controller, the (fixed) plant, and the (fixed) external feedback loop. his SPICE netlist is wrapped inside an appropriate set of SPICE commands to carry ot varios SPICE analyses in the time domain (described below). We also provide SPICE with sbcircit definitions to implement all the signal processing fnctions in the fnction set (described above) and all the signal processing fnctions necessary to represent the plant. he controller is then simlated sing or modified version of the original 7,000-line SPICE3 simlator (Qarles, Newton, Pederson, and Sangiovanni-Vincentelli 994). Or modified version of SPICE is rn as a sbmodle within or genetic programming system. he SPICE simlator retrns tablar otpt (representing the plant otpt in the time domain). An interface commnicates this information to or genetic programming code. See Koza, Keane, Y, Bennett, and Mydlowec 000 for details. he fitness of each controller in the poplation is measred by means of 48 separate invocations of the SPICE simlator. his 48-part fitness measre attempts to optimize step response and distrbance rejection while simltaneosly imposing constraints on maximm sensitivity and sensor noise attenation. he fitness of an individal controller is the sm of the detrimental contribtions of these 48 elements of the fitness measre. he smaller the sm, the better. he first 36 elements of this 48-part fitness measre are time-domain-based elements that together represent the six plants from the two families (i.e., n = 3, 4, and 8 and α = 0., 0.5, and 0.7), in conjnction with six choices of vales for the height of the reference signal and distrbance signal (shown in table ) that sample a range of vales. he reference signal is step fnction that rises from 0 at time t = 0 to the specified height at t = millisecond. he distrbance signal is a step fnction that rises from 0 at time t = 0 to the specified height at t = 0 millisecond. he distrbance signal is added to the controller's otpt. able Six combinations Reference signal Distrbance signal

5 For each of these first 36 elements of the 48-part fitness measre, a transient analysis is performed in the time domain sing the SPICE simlator. e(t) is the difference (error) at time t between the plant otpt and the reference signal. he contribtion to fitness for each of these 36 elements is based on the sm of two integrals of time-weighted absolte error (IAE). he first term of the integral acconts for the controller's step response while the second term acconts for distrbance rejection. 0 0 t e( t) Bdt ( t 0 ) e( t) Cdt t= 0 t= 0. he factor B in the first term of the integral mltiplies each vale of e(t) by the reciprocal of the amplitde of the reference signal (so that all reference signals are eqally inflential). he factor C in the second term of the integral mltiplies vale of e(t) by the reciprocal of the amplitde of the distrbance signals. When the amplitde of either the reference signal or the distrbance signal is zero, the appropriate factor (B or C) is set to zero. he IAE component of fitness is sch that, all other things being eqal, changing the time scale by a factor of F changes the IAE by F. he division of the integral by is an attempt to eliminate this artifact of the time scale and eqalize the inflence of each of the plants in the overall fitness measre. For these 36 elements of the fitness measre, the contribtion to fitness is mltiplied by 0 if the element is greater than for the Astrom and Hagglnd (995). R(s) Q(s) Controller U(s) D(s) Figre 3 Overall model. he 37 th throgh 4 nd elements of the 48-part fitness measre are freqency-domain-based elements that measre stability margin. Figre 3 presents a model for the entire system containing the given plant and the tobe-evolved controller. In this figre, R(s) is the reference signal; Y(s) is the plant otpt; and U(s) is the controller's otpt (control variable). Distrbance D(s) may be added to the controller's otpt U(s). Sensor noise N(s) may be added to the plant's otpt Y(s) yielding Q(s). Here N(s) is an AC signal. For each of N(s) Plant Y(s) these six elements of the fitness measre, an AC sweep is performed sing the SPICE simlator from /(000 ) to 000/ while holding the reference signal R(s) and the distrbance signal D(s) at zero. he maximm sensitivity, M s, is a measre of the stability margin. It is desirable to minimize the maximm sensitivity (and therefore maximize the stability margin). he qantity /M s is the minimm distance between the Nyqist plot and the point (-,0) and is the stability margin incorporating both gain and phase margin. he maximm sensitivity is the maximm amplitde of Q(s). he contribtion to fitness is 0 if M s <.5; (M s -.5) for.5 M s.0; and 0(M s -.0) for M s >.0. For these six elements of the fitness measre (as well as the six elements below), the contribtion to fitness is mltiplied by 0 if the element is greater than for the Astrom and Hagglnd controller (995). he 43 rd throgh 48 th elements of this 48-part fitness measre are freqency-domain-based elements measring the sensor noise attenation. Achieving favorable sensor noise attenation is often in direct conflict with the goal of achieving a rapid response to setpoint changes and rejection of plant distrbances. For each of these six elements of the fitness measre, an AC sweep is performed sing the SPICE simlator from 0/ to 000/ while holding the reference signal R(s) and the distrbance signal; D(s) at zero. he attenation of the sensor noise is measred at plant otpt at Y(s). A min is the minimm attenation in decibels within this freqency range. It is desirable to maximize the minimm attenation. he contribtion to fitness for sensor noise attenation is 0 if A min > 40 db; (40 - A min )/0 if 0 db A min 40 db; and (0 - A min ) if A min < 0 db. A controller that cannot be simlated by SPICE is assigned a high penalty vale of fitness (0 8 ). 3.5 Control Parameters he poplation size, M, was 00,000. A (generos) maximm size of 50 points (for fnctions and terminals) was established for each reslt-prodcing branch and a (generos) maximm size of 00 points was established for each atomatically defined fnction. he percentages of the genetic operations for each generation are 46% one-offspring crossover on internal points of the program tree other than nmerical constant terminals, 9% one-offspring crossover on points of the program tree other than nmerical constant terminals, 9% one-offspring crossover on nmerical constant terminals, % mtation on points of the program tree other than nmerical constant terminals, 0% mtation on nmerical constant terminals, 9% reprodction, % sbrotine creation, % sbrotine dplication, and % sbrotine deletion. he other parameters are the same defalt vales that

6 we have sed on many other problems (Koza, Bennett, Andre, Keane 999). 3.6 ermination he rn was manally monitored and manally terminated when the fitness of many sccessive best-ofgeneration individals appeared to have reached a platea. he best-so-far individal was harvested and designated as the reslt of the rn. 3.7 Parallel Implementation his problem was rn on a home-bilt Beowlf-style (Sterling, Salmon, Becker, and Savarese 999) parallel clster compter system consisting of, MHz Pentim II processors (each accompanied by 64 megabytes of RAM). he system has a 350 MHz Pentim II compter as host. he processing nodes are connected with a 00 megabit-per-second Ethernet. he processing nodes and the host se the Linx operating system. he distribted genetic algorithm with nsynchronized generations and semi-isolated sbpoplations was sed with a sbpoplation size of Q = 00 at each of D =,000 demes. As each processor (asynchronosly) completes a generation, for boatloads of emigrants from each sbpoplation are dispatched to each of the for toroidally adjacent processors. he,000 processors are hierarchically organized. here are 5 5 = 5 high-level grops (each containing 40 processors). If the adjacent node belongs to a different grop, the migration rate is % and emigrants are selected based on fitness. If the adjacent node belongs to the same grop, emigrants are selected randomly with a 5% migration rate (0% if the adjacent node is in the same physical box). 4 Reslts he initial random generation is a blind random search of the search space of the problem. he best-ofgeneration circit from generation 0 has a fitness of 4, he best-of-rn controller (figre 4) appears in generation 7. his genetically evolved controller has an overall fitness of he program tree has one reslt-prodcing branch with 0 points and five atomatically defined fnctions (with, 38, 3, 9, and 3 points, respectively). he reslt-prodcing branch refers to ADF0. Also, ADF0 hierarchically refers to ADF. he other three atomatically defined fnctions are not referenced. Note that the controller's otpt is fed back internally into the controller. Figre 5 compares the time-domain response of the best-of-rn controller (triangles) from generation 7 and the Astrom and Hagglnd controller (sqares) to a -volt reference signal for the three-lag plant. he comparisons for other reference signals, distrbance signals, and plants from the two families are similarly sperior (and are not shown for reasons of space). Plant Otpt ime Figre 5 Comparison of time-domain responses. ln( K ) s r ln( K ) r s ln( K ) Reference Signal - s Control Variable Plant Otpt ( ) K ln L r K ln K K e - ( ) - Figre 4 Block diagram of best-of-rn controller from generation 7. able Control signal, U(s), for the best-of-rn controller from generation 7 U ( s) = ( s)( s) ln( K ) ( s) ln( K ) r r K ( ) ln( K ( e ) ln( K.33449L) s ( ) r

7 able 3 hree coefficients of PID controller eqivalent to the best-of-rn controller from generation 7 ln ( K )( r ( ln( K ))) K = K K d i = ln ln = ( K ) r ( K ) ln( K ) K ( ) ( ln( K ) ln( e ) ln( K.3349L) ( ) r able 4 Comparison of characteristics of the controller and the Astrom and Hagglnd controller for all six plants Plant Plant Genetically evolved Controller Astrom and Hagglnd Controller IAE IAE M s A min IAE IAE M s A min Step Distrb Step Distrb ( s) ( s) 4 8 ( s) ( s) ( s) ( s) 3 ( 0.s)( 0. s)( 0. s) 3 ( 0.5s)( 0.5 s)( 0.5 s) 3 ( 0.7s)( 0.7 s)( 0.7 s) Figre 6 compares the time-domain response of the best-of-rn controller (triangles) from generation 7 and the Astrom and Hagglnd controller (sqares) to a -volt distrbance signal for the threelag plant. he comparisons for other reference signals, distrbance signals, and plants from the two families are similarly sperior. Plant Otpt ime Figre 6 Comparison of the distrbance responses. able presents the control signal, U(s), for the best-of-rn controller from generation 7. Note that all for parameters (K,, r, and L) appear. When simplified, it can be seen that the best-ofrn controller from generation 7 is a PID controller whose three coefficients are as shown in table 3. able 4 compares the characteristics of the bestof-rn controller from generation 7 with those of the Astrom and Hagglnd (995) controller for all six plants. As can be seen, the genetically evolved controller is sperior to the Astrom and Hagglnd controller for all six plants for the integral of the time-weighted absolte error (IAE) for the step inpt, the IAE for distrbance rejection, and the maximm sensitivity, M s. All vales of A min are above the reqired minimm 40 (except for plant 4). Averaged over the six plants, the IAE for the step inpt for the genetically evolved controller is only 58% of the vale for the Astrom and Hagglnd controller; the IAE for distrbance rejection is 9% of the vale for the Astrom and Hagglnd controller; and the maximm sensitivity, M s. for the genetically evolved controller is only 85% of the vale for the Astrom and Hagglnd controller.

8 he PID controller was a significant improvement over previos approaches to control. As Callender and Stevenson state in their 939 patent, "A specific object of the invention is to provide a system which will prodce a compensating effect governed by factors proportional to the total extent of the deviation, the rate of the deviation, and the smmation of the deviation dring a given period " Claim of Callender and Stevenson (939) covers what is now called the PI controller, "A system for the atomatic control of a variable characteristic comprising means proportionally responsive to deviations of the characteristic from a desired vale, compensating means for adjsting the vale of the characteristic, and electrical means associated with and actated by responsive variations in said responsive means, for operating the compensating means to correct sch deviations in conformity with the sm of the extent of the deviation and the smmation of the deviation." Claim 3 of Callender and Stevenson (939) covers what is now called the PID controller, "A system as set forth in claim in which said operation is additionally controlled in conformity with the rate of sch deviation." he legal criteria for obtaining a U. S. patent are that the proposed invention is "new and sefl" and "... the differences between the sbject matter soght to be patented and the prior art are sch that the sbject matter as a whole wold [not] have been obvios at the time the invention was made to a person having ordinary skill in the art to which said sbject matter pertains." (35 United States Code 03a). Patents are only issed if an arms-length examiner is convinced that the proposed invention is novel, sefl, and satisfies the stattory test for nobviosness. Since filing for a patent entails the expenditre of a considerable amont of time and money, patents are generally soght only if the invention is likely to prove sefl in the real world. Certainly the PID controller has proved sefl since PD, PI, and PID controllers are in widespread se in indstry throghot the world. he fact that genetic programming rediscovered both the topology and sizing of a controller that was nobvios "to a person having ordinary skill in the art" establishes that this evolved reslt satisfies Arthr Samel's criterion (983) for artificial intelligence and machine learning, namely he aim [is]... to get machines to exhibit behavior, which if done by hmans, wold be assmed to involve the se of intelligence. he vales of the PID coefficients of the controller created by genetic programming are very close to those of the Astrom and Hagglnd (995) controller. Most of the compter time was consmed by the fitness evalation of candidate individals in the poplation. he fitness evalation (involving 36 time-consming time-domain SPICE simlations and relatively fast freqency-domain SPICE simlations) averaged abot 6.7 seconds per individal (sing a 350 MHz Pentim II processor). he best-of-rn individal from generation 7 was prodced after evalating individals. his reqired hors on or,000-node parallel compter system that is, the expenditre of compter cycles (abot 5 peta-cycles of compter time). he for parameters (K,, r, and L) in the above atomatically created reslt are free variables. A mathematical formla containing one or more free variables is "general" in the sense that it provides a soltion to an entire category of problems. For example, the familiar formla for solving a qadratic eqation contains free variables representing the coefficients of the eqation. Here genetic programming has atomatically created a "general" soltion to an entire category of problems (i.e., all the plants in the two families) not merely a single instance of the problem (i.e., a particlar single plant). 5 Conclsion his paper demonstrated that genetic programming can be sed to atomatically create the design for both the topology and parameter vales (tning) for a single common controller (containing varios parameters representing the overall characteristics of the plant) for two families of plants. he genetically evolved controller otperforms the controller designed with conventional techniqes. he genetically evolved controller is "general" in the sense that it provides a soltion that is applicable to all the plants in the two families not merely a particlar single plant). References Astrom, Karl J. and Hagglnd, ore PID Controllers: heory, Design, and ning. Second Edition. Research riangle Park, NC: Instrment Society of America.

9 Banzhaf, Wolfgang, Daida, Jason, Eiben, A. E., Garzon, Max H., Honavar, Vasant, Jakiela, Mark, and Smith, Robert E. (editors) GECCO-99: Proceedings of the Genetic and Evoltionary Comptation Conference, Jly 3-7, 999, Orlando, Florida USA. San Francisco, CA: Morgan Kafmann. Banzhaf, Wolfgang, Nordin, Peter, Keller, Robert E., and Francone, Frank D Genetic Programming An Introdction. San Francisco, CA: Morgan Kafmann and Heidelberg: dpnkt. Banzhaf, Wolfgang, Poli, Riccardo, Schoenaer, Marc, and Fogarty, erence C Genetic Programming: First Eropean Workshop. EroGP'98. Paris, France, April 998 Proceedings. Paris, France. April l998. Lectre Notes in Compter Science. Volme 39. Berlin, Germany: Springer-Verlag. Callender, Albert and Stevenson, Allan Brown Atomatic Control of Variable Physical Characteristics. United States Patent,75,985. Filed Febrary 7, 936 in United States. Filed Febrary 3, 935 in Great Britain. Issed October 0, 939 in United States. Gra, Frederic. 99. Genetic synthesis of Boolean neral networks with a cell rewriting developmental process. In Schaffer, J. D. and Whitley, Darrell (editors). Proceedings of the Workshop on Combinations of Genetic Algorithms and Neral Networks 99. Los Alamitos, CA: he IEEE Compter Society Press. Holland, John H Adaptation in Natral and Artificial Systems. Ann Arbor, MI: University of Michigan Press. Koza, John R. 99. Genetic Programming: On the Programming of Compters by Means of Natral Selection. Cambridge, MA: MI Press. Koza, John R. 994a. Genetic Programming II: Atomatic Discovery of Resable Programs. Cambridge, MA: MI Press. Koza, John R. 994b. Genetic Programming II Videotape: he Next Generation. Cambridge, MA: MI Press. Koza, John R., Bennett III, Forrest H, Andre, David, and Keane, Martin A Genetic Programming III: Darwinian Invention and Problem Solving. San Francisco, CA: Morgan Kafmann. Koza, John R., Bennett III, Forrest H, Andre, David, Keane, Martin A., and Brave, Scott Genetic Programming III Videotape: Hman-Competitive Machine Intelligence. San Francisco, CA: Morgan Kafmann. Koza, John R., Keane, Martin A., Y, Jessen, Bennett, Forrest H III, and Mydlowec, William Atomatic creation of hman-competitive programs and controllers by means of genetic programming. Genetic Programming and Evolvable Machines. ( -) Koza, John R., and Rice, James P. 99. Genetic Programming: he Movie. Cambridge, MA: MI Press. Langdon, William B Genetic Programming and Data Strctres: Genetic Programming Data Strctres = Atomatic Programming! Amsterdam: Klwer. Marenbach, Peter, Bettenhasen, Krt D., and Freyer, Stephan Signal path oriented approach for generation of dynamic process models. In Koza, John R., Goldberg, David E., Fogel, David B., and Riolo, Rick L. (editors). Genetic Programming 996: Proceedings of the First Annal Conference, Jly 8-3, 996, Stanford University. Cambridge, MA: MI Press. Pages Poli, Riccardo, Nordin, Peter, Langdon, William B., and Fogarty, erence C Genetic Programming: Second Eropean Workshop. EroGP'99. Proceedings. Lectre Notes in Compter Science. Volme 598. Berlin: Springer-Verlag. Qarles, homas, Newton, A. R., Pederson, D. O., and Sangiovanni-Vincentelli, A SPICE 3 Version 3F5 User's Manal. Department of Electrical Engineering and Compter Science, University of California. Berkeley, CA. March 994. Ryan, Conor Atomatic Re-engineering of Software Using Genetic Programming. Amsterdam: Klwer Academic Pblishers. Samel, Arthr L AI: Where it has been and where it is going. Proceedings of the Eighth International Joint Conference on Artificial Intelligence. Los Altos, CA: Morgan Kafmann. Pages Spector, Lee, Langdon, William B., O'Reilly, Una- May, and Angeline, Peter (editors) Advances in Genetic Programming 3. Cambridge, MA: MI Press. Sterling, homas L., Salmon, John, Becker, Donald J., and Savarese, Daniel F How to Bild a Beowlf: A Gide to Implementation and Application of PC Clsters. Cambridge, MA: MI Press. Wong, Man Leng and Leng, Kwong Sak Data Mining Using Grammar Based Genetic Programming and Applications. Amsterdam: Klwer Academic Pblishers. Ziegler, J. G. and Nichols, N. B. 94. Optimm settings for atomatic controllers. ransactions of ASME. (64)