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

Transcription

1 Atomatic Design of Both Topology and Tning of a Common Parameterized Controller for Two Families of Plants sing Genetic Programming Jessen Y Genetic Programming Inc., Los Altos, California jy@cs.stanford.ed Martin A. Keane Econometrics Inc., Chicago, Illinois makeane@ix.netcom.com John R. Koza Stanford University, Stanford, California koza@stanford.ed ABSTRACT This paper demonstrates that a techniqe of evoltionary comptation can be sed to atomatically create the design for both the topology and parameter vales (tning) for a common controller (containing varios parameters representing the overall characteristics of the plant) for two families of plants. The atomatically designed controller is created by means of genetic programming sing a fitness measre that attempts to optimize step response and distrbance rejection while simltaneosly imposing constraints on maximm sensitivity and sensor noise attenation. The atomatically designed controller otperforms the controller designed with conventional techniqes. In particlar, the atomatically designed controller is sperior to the Astrom and Hagglnd controller for all plants of both families for the integral of the time-weighted absolte error (ITAE) for a step inpt, the ITAE for distrbance rejection, and maximm sensitivity. Averaged over all plants of both families, the ITAE for the step inpt for the atomatically designed controller is only 58% of the vale for the conventional controller; the ITAE for distrbance rejection is 9% of the vale for the conventional controller; and the maximm sensitivity, M s. for the atomatically designed controller is only 85% of the vale for the conventional controller. The atomatically designed controller is "general" in the sense that it contains free variables and therefore provides a 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). Introdction 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 from a high-level statement of the controller's desired behavior and characteristics (Koza, Keane, Y, Bennett, and Mydlowec 000). However, each of these two (different) atomatically designed controllers applied only to one particlar plant. Moreover, both plants belonged to the same family (the n-lag plants). The qestion arises as to whether it is possible to evolve a single "general" parameterized controller (containing parameters representing the overall characteristics of the plant) that can perform well for an entire family of plants (say, the n-lag plants) and also one or more additional families of plants. In their recent inflential book, Astrom and Hagglnd (995) identified for families of plants "that are representative for the dynamics of typical indstrial processes." Astrom and Hagglnd (995) then develop a common method for designing controllers and demonstrate improved performance for their common method over the Ziegler-Nichols rles (Ziegler and Nichols 94) on all the plants in all for of their families of plants. One of the families of plants in Astrom and Hagglnd 995 consists of the n-lag plants represented by the transfer fnctions of the form G( s) = () n ( s) where n = 3, 4, and 8. Another family consists of plants represented by the transfer fnctions of the form G( s) = 3 ( s)( α s)( α s)( α s) () where α = 0., 0.5, and 0.7.

2 The methods developed by Astrom and Hagglnd se pairs of parameters representing the overall characteristics of a plant. These 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 one version of their method, Astrom and Hagglnd se two freqency domain parameters. They are 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, T (the period of this lowest freqency oscillation). In another version of their method, Astrom and Hagglnd se the time constant, T 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. These two parameters are, in one instance, obtained by approximating the plant with a transfer fnction of the form sl e ( str ) This 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 (). The atomatically designed controller is created sing a fitness measre that attempts to optimize step response and distrbance rejection while simltaneosly imposing constraints on maximm sensitivity and sensor noise attenation. The atomatically designed 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 Genetic programming is an atomatic techniqe for generating compter programs to solve, or approximately solve, problems. In particlar, genetic programming is capable of atomatically creating the design of complex strctres. Genetic programming approaches a program synthesis problem or a design problem in terms of "what needs to be done" as opposed to "how to do it". Genetic programming (Koza 99; Koza and Rice 99; Koza 994a, 994b) is an extension of the genetic algorithm (Holland 975). Genetic programming starts with a primordial ooze of thosands of randomly created compter programs and ses the Darwinian principle of natral selection, recombination (crossover), mtation, gene dplication and deletion to breed a poplation of programs over a series of generations. Genetic programming breeds compter programs to solve problems by execting the following three steps: () Generate an initial poplation of compositions (typically random) of the problem's fnctions and terminals. () Iteratively perform the following sbsteps (a generation) on the poplation of programs ntil the termination criterion has been satisfied: (A) Execte each program in the poplation and assign it a vale sing the fitness measre. (B) Create a new poplation of programs by applying the following operations. The operations are applied to program(s) selected from the poplation with a probability based on fitness (with reselection allowed). (i) Reprodction: Copy the selected program to the new poplation. (ii) Crossover: Create a new offspring program for the new poplation by recombining randomly chosen parts of two selected programs. (iii) Mtation: Create one new offspring program for the new poplation by randomly mtating a randomly chosen part of the selected program. (iv) Architectre-altering operations: Select an architectre-altering operation from the repertoire of sch operations and create one new offspring program for the new poplation by applying the selected operation to the selected program. (3) Designate the individal program that is identified by reslt designation (e.g., the best-so-far individal) as the reslt of the rn of genetic programming. This reslt may be a soltion (or an approximate soltion) to the problem. Genetic programming is capable of evolving resable, parametrized, hierarchically-called atomatically defined fnctions (sbrotines). Architectre-altering operations (Koza, Bennett, Andre, and Keane 999; Koza, Bennett, Andre, Keane, and Brave 999) enable genetic programming to atomatically determine the nmber of atomatically defined fnctions, the nmber of argments that each possesses, and the natre of the hierarchical references, if any, among sch atomatically defined fnctions. Genetic programming is capable of atomatically synthesizing the design of both the topology and sizing for a wide variety of analog electrical circits from a high-level statement of the circit's desired behavior and characteristics (Koza, Bennett, Andre, and Keane 999). Nine of the atomatically designed analog circits in Koza, Bennett, Andre, and Keane 999 were

3 previosly patented. Five of the atomatically designed circits infringe on previosly issed patents. Genetic programming often creates novel designs becase it is a probabilistic process that is not encmbered by the preconceptions that often channel hman thinking down familiar paths. The fact that genetic programming can design both the topology and sizing of circits sggests that it might also be capable of designing other types of complex topological strctres containing parameterized components, sch as controllers. 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. Both genetic algorithms and genetic programming have been previosly sed for synthesizing controllers having mtally interacting continos-time variables and continos-time signal processing blocks (Koza, Keane, Y, Mydlowec, and Bennett 000; Man, Tang, Kwong, and Halang; 997, 999; Crawford, Cheng, and Menon 999; Dewell and Menon 999; Menon, Yosefpor; Lam, and Steinberg 995; Sweridk, Menon, and Steinberg 998, 999). Controller 500 The otpt (i.e., control variable 590) of this controller is the sm of a proportional (P) term (the gain block 530 with an amplification factor of 4.0), an integrating (I) term (the integrator 560 preceded by the gain block 540 with an amplification factor of,000.0), and a differentiating (D) term (the derivative block 570 preceded by the gain block 550 with an amplification factor of 5.5). This type of controller is called a PID controller and was invented and patented in 939 by Albert Callender and Allan Stevenson of Imperial Chemical Limited of Northwich, England. In this paper, a compter program (i.e., program tree, LISP symbolic expression) will represent the block diagram of a controller. The block diagram consists of signal processing fnctions linked by directed lines representing the flow of information. There 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. PROGN 700 DEFUN 70 VALUES Reference Control Plant 54 Signal Variable Otpt /s Plant s 570 Figre Block diagram of a plant and a PID controller composed of proportional, integrative, and derivative blocks. The plant's otpt is fed back to the controller where it is compared to the reference signal. Figre is a block diagram for an illstrative control system containing a controller and a plant. The directed lines in a block diagram represent time-domain signals while the blocks represent signal processing fnctions that operate in the time domain. The otpt of the controller 500 is a control variable 590 which is, in trn, the inpt to the plant 59. The plant has one otpt (plant response) 594. The plant response is fed back (externally as signal 596) and becomes one of the controller's two inpts. The controller's second inpt is the reference signal 508. The 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) ADF0 704 LIST 706 REF 708 VALUES 7-70 PLANT OUTPUT GAIN /s 760 ADF0 734 GAIN ADF s GAIN ADF0 754 Figre Program tree representation of the PID controller of figre. The atomatically defined fnction ADF0 (left) sbtracts the plant otpt from the reference signal and makes the difference available to three points in the reslt-prodcing branch (right). Figre presents the block diagram for the PID controller of figre as a program tree. The 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). The 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. The 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 the fnction definition for itself wold be considered to be a recrsive reference. However, in the context of applying genetic programming to control systems, a sbrotine that references itself corresponds to a loop

4 in the block diagram of the controller (i.e., internal feedback inside the controller). 3 Preparatory Steps Six major preparatory steps are reqired before applying genetic programming to a problem involving the synthesis of a controller: () determine the architectre of the program trees, () identify the terminals, (3) identify the fnctions, (4) define the fitness measre, (5) choose control parameters for the rn, and (6) choose the termination criterion and method of reslt designation. 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. The permitted maximm of five atomatically defined fnctions is more than sfficient for this problem. 3. Terminal Set The 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. The 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. The terminal set for the arithmetic-performing sbtrees is T aps = {R, KU, TU, L, TR}. 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. The 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). The pertrbations are implemented by a genetic operation for mtating the pertrbable nmerical vales. The 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. The remaining terminals are time-domain signals. The terminal set, T, for the reslt-prodcing branch and any atomatically defined fnctions (except the arithmetic-performing sbtrees described above) is T = {REFERENCE_SIGNAL, CONTROLLER_OUTPUT, PLANT_OUTPUT}. 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 The fnction set, F aps, for the arithmetic-performing sbtrees is F aps = {ADD_NUMERIC, SUB_NUMERIC, MUL_NUMERIC, DIV_NUMERIC, REXP, RLOG}. The two-argment DIV_NUMERIC fnction divides the first argment by the second argment, except that the qotient is never allowed to exceed 0 5. The oneargment REXP fnction is the exponential fnction and the one-argment RLOG fnction is the natral logarithm of the absolte vale. The 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, INVERTER, LEAD, LAG, LAG, DIFFERENTIAL_INPUT_INTEGRATOR, DIFFERENTIATOR, ADD_SIGNAL, SUB_SIGNAL, ADD_3_SIGNAL, MUL_SIGNAL, DIV_SIGNAL, ULIMIT, ADF0, ADF, ADF, ADF3, ADF4}. The one-argment ULIMIT fnction limits a signal by constraining it between an pper and lower bond. This 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. The 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

5 measre. The 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. The 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. The 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). The exection of the program tree prodces an interconnected seqence of signal processing blocks that is, a block diagram for the individal controller. The 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. This 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. The 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. The 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. The fitness of each controller in the poplation is measred by means of 48 separate invocations of the SPICE simlator. This 48-part fitness measre attempts to optimize step response and distrbance rejection while simltaneosly imposing constraints on maximm sensitivity and sensor noise attenation. The fitness of an individal controller is the sm of the detrimental contribtions of these 48 elements of the fitness measre. The smaller the sm, the better. Table Six combinations Reference signal Distrbance signal The 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. The reference signal is step fnction that rises from 0 at time t = 0 to the specified height at t = millisecond. The distrbance signal is a step fnction that rises from 0 at time t = 0T to the specified height at t = 0T millisecond. The distrbance signal is added to the controller's otpt. 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. The contribtion to fitness for each of these 36 elements is based on the sm of two integrals of time-weighted absolte error (ITAE). The first term of the integral acconts for the controller's step response while the second term acconts for distrbance rejection. 0T 0T t e( t) Bdt ( t 0T ) e( t) Cdt t= 0 t= 0T. T T The 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). The 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. The ITAE component of fitness is sch that, all other things being eqal, changing the time scale by a factor of F changes the ITAE by F. The division of the integral by T 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). The 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 these six elements of the fitness measre, an AC sweep is performed sing the SPICE simlator from /(000T ) to 000/T while holding the reference signal R(s) and the distrbance signal D(s) at zero. The maximm sensitivity, M s, is a measre of the stability margin. It is desirable to minimize the maximm

6 sensitivity (and therefore maximize the stability margin). The 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. The maximm sensitivity is the maximm amplitde of Q(s). The 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. R(s) Q(s) Controller U(s) D(s) Figre 3 Overall model. The 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/T to 000/T while holding the reference signal R(s) and the distrbance signal, D(s) at zero. The 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. The 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. The SPICE simlator cannot simlate many of the controllers that are randomly created dring a rn of genetic programming. A controller that cannot be simlated by SPICE is assigned a high penalty vale of fitness (0 8 ). 3.5 Control Parameters The 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. The 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 N(s) Plant Y(s) the program tree other than nmerical constant terminals, 0% mtation on nmerical constant terminals, 9% reprodction, % sbrotine creation, % sbrotine dplication, and % sbrotine deletion. The other parameters are the same defalt vales that we have sed on many other problems (Koza, Bennett, Andre, Keane 999). 3.6 Termination The rn was manally monitored and manally terminated when the fitness of many sccessive best-ofgeneration individals appeared to have reached a platea. The best-so-far individal was harvested and designated as the reslt of the rn. 3.7 Parallel Implementation This problem was rn on a home-bilt Beowlf-style (Sterling, Salmon, Becker, and Savarese 999; Bennett, Koza, Shipman, and Stiffelman 999) parallel clster compter system consisting of, MHz Pentim II processors (each accompanied by 64 megabytes of RAM). The system has a 350 MHz Pentim II compter as host. The processing nodes are connected with a 00 megabit-per-second Ethernet. The processing nodes and the host se the Linx operating system. The 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. The,000 processors are hierarchically organized. There 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, the migration rate is 5% (0% if in the same physical box) and emigrants are selected randomly. 4 Reslts The initial random generation is a blind random search of the search space of the problem. The best-ofgeneration circit from generation 0 has a fitness of 4, The best-of-rn controller (figre 4) appears in generation 7. This genetically evolved controller has an overall fitness of The program tree has one reslt-prodcing branch with 0 points and five atomatically defined fnctions (with, 38, 3, 9, and 3 points, respectively). The reslt-prodcing branch refers to ADF0. Also, ADF0 hierarchically refers to ADF. The other three atomatically defined fnctions are not referenced. Note that the controller's otpt is fed back internally into the controller.

7 Table presents the control signal, U(s), for the best-of-rn controller from generation 7. Note that all for parameters (K, T, T r, and L) appear. When simplified, it can be seen that the best-of-rn controller from generation 7 is a PID controller whose three coefficients are as shown in table 3. ln( K ) s T r ln( K ) T r s ln( K ) Reference Signal - T s Control Variable Plant Otpt ( ) K T ln L r K ln K K e - ( ) - Figre 4 Block diagram of best-of-rn controller from generation 7. Table Control signal, U(s), for the best-of-rn controller from generation 7 U ( s) = ( T s)( T s) ln( K ) ( T s) ln( K ) r r T K ( ) ln( K ( e ) ln( K.33449L) T s ( ) r Table 3 Three coefficients of PID controller eqivalent to the best-of-rn controller from generation 7 ln ( K )( T Tr ( ln( K ))) K = T K K d i = ln ln = ( K ) T r ( K ) ln( K ) T K ( ) ( ln( K ) ln( e ) ln( K.3349L) T ( ) r Table 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 ITAE ITAE M s A min ITAE ITAE M s A min Step Distrb Step Distrb ( s) ( s) 4 8 ( s) ( s) ( s) 3 ( 0.s)( 0. s)( 0. s) 3 ( 0.5s)( 0.5 s)( 0.5 s)

8 6 3 ( 0.7s)( 0.7 s)( 0.7 s) ( s) Table 4 compares the characteristics of the best-ofrn 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 timeweighted absolte error (ITAE) for the step inpt, the ITAE 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 ITAE for the step inpt for the genetically evolved controller is only 58% of the vale for the Astrom and Hagglnd controller; the ITAE 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. The vales of the PID coefficients of the controller created by genetic programming are very close to those of the Astrom and Hagglnd (995) controller. The best-of-rn controller from generation 7 is similarly sperior for other reference signals, other distrbance signals, and other plants from the two families (bt are not shown for reasons of space) individals in the poplation. The fitness evalation for each individal in the poplation of 00,000 on each of the 8 generations (generation 0 pls 7 additional generations) in this rn entailed 36 very timeconsming time-domain SPICE simlations and relatively fast freqency-domain SPICE simlations. The fitness evalation for each individal averaged abot 6.7 seconds per individal (sing a 350 MHz Pentim II processor). The best-of-rn individal from generation 7 was prodced after evalating individals. This reqired hors on or,000-node parallel compter system that is, the expenditre of compter cycles (abot 5 peta-cycles of compter time). Plant Otpt Time Figre 6 Comparison of the distrbance responses. Plant Otpt Time Figre 5 Comparison of time-domain responses. 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. Note that the genetically evolved controller (triangles) is sperior based on the fitness measre sed and the measres contained in table 4. 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 three-lag plant. Most of the compter time in rns of genetic programming involving complex simlations is consmed by the fitness evalation of candidate 5 Conclsion This paper demonstrated that genetic programming can be sed to atomatically create the design for both the topology and parameter vales (tning) for a common parameterized controller (containing varios parameters representing the overall characteristics of the plant) for two families of plants. The atomatically designed controller otperforms the controller designed with conventional techniqes. 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. The above reslt created by genetic programming contains for free variables (K, T, T r, and L). That is, genetic programming atomatically created a "general" soltion to an entire category of problems (i.e., all the plants in the two families) not merely a soltion to single instance of the problem (i.e., a particlar single plant).

9 Moreover, genetic programming did not jst atomatically create formlae for the controller's parameter vales (tning) it atomatically created the topology of the controller. Ths, genetic programming can be viewed as a new kind of mathematics in which the reslt consists of not jst general formlae, bt, instead, a combination of a graphical strctre (i.e., the controller's topology) and general formlae for the parameter vales of each block of the controller.

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