AUTOMATIC SYNTHESIS USING GENETIC PROGRAMMING OF IMPROVED PID TUNING RULES

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1 AUTOMATIC SYNTHESIS USING GENETIC PROGRAMMING OF IMPROVED PID TUNING RULES Matthew J. Streeter 1, Martin A. Keane, and John R. Koza 3 1 Genetic Programming, Inc. Econometrics, Inc. 3 Stanford University Abstract: Astrom and Hagglnd developed tning rles in 1995 for PID controllers that otperform the 194 Ziegler-Nichols rles on an indstrially representative set of plants. In this paper, we se genetic programming to atomatically discover tning rles for PID controllers that otperform the Astrom-Hagglnd rles for the indstrially representative set of plants specified by Astrom and Hagglnd. Copyright 00 IFAC Keywords: Genetic programming, PID control 1 INTRODUCTION The PID controller was patented in 1939 by Albert Callender and Allan Stevenson of Imperial Chemical Limited (Callender and Stevenson 1939). The 194 Ziegler-Nichols tning rles provided a relatively simple and effective method for determining the parameter vales for the proportional, integrative, and derivative blocks of the PID controller (Ziegler and Nichols 194). The Ziegler-Nichols tning rles remain in widespread se today for tning PID controllers. In their 1995 book PID Controllers: Theory, Design, and Tning, Astrom and Hagglnd identified for families of plants "that are representative for the dynamics of typical indstrial processes." The first of the for families of plants in Astrom and Hagglnd 1995 consists of plants represented by transfer fnctions of the form -s e G( s) = (1 st ) where T = 0.1,, 10. (A) The second family consists of the n-lag plants represented by transfer fnctions of the form 1 G( s) = (B) n (1 s) where n = 3, 4, and 8. The third family consists of plants represented by transfer fnctions of the form 1 G( s) = (C) 3 (1 s)(1 α s)(1 α s)(1 α s) where α = 0., 0.5, and 0.7. The forth family consists of plants represented by transfer fnctions of the form 1- αs G ( s) = 3 (D) (s 1) where α = 0.1, 0., 0.5, 1.0, and.0. Astrom and Hagglnd developed a method in their 1995 book for atomatically tning PID controllers for all the plants in all for of these indstrially representative families of plants. In one version of their method, Astrom and Hagglnd se two freqency-domain parameters, namely the ltimate gain, K, and the ltimate period, T. Astrom and Hagglnd describe a procedre for estimating these parameters from the plant's response to a step inpt. The tning rles developed by Astrom and Hagglnd in their 1995 book otperform the widely sed Ziegler-Nichols tning rles on all 16 indstrially representative plants sed by Astrom and Hagglnd. As Astrom- Hagglnd observe, [Or] new methods give sbstantial improvements in control performance while retaining mch of the simplicity of the Ziegler-Nichols rles. The PID tning rles devised by Astrom and Hagglnd in 1995 are given by for eqations. Reference Signal 00 Plant Feedback Eqation 1 Eqation - 50 Eqation Eqation 4 Fig. 1. PID controller 60 1/s 90 s 3 Control Variable 34

2 Referring to figre 1 (which shows a PID controller), eqation 1 implements setpoint weighting, b, for the inpt to the proportional block 30 and is b = 0.5* K e K Eqation (for gain block 30) is the gain, K p, that is associated with proportional block and is K p = * * K K K e Eqation 3 (for gain block 50) is the gain, K i, that is associated with integrative block and is K i = 0.7* K * e 0.59* T * e K K K K Eqation 4 (for gain block 80) is the gain, K d, that is associated with derivative block and is K K K K K d = 0.108* K * T * e * e The transfer fnction for the Astrom-Hagglnd PID controller is given by the eqation: Ki A = K p ( b* R- P) ( R- P) Kd * s* (-P) s where A is the otpt of the Astrom-Hagglnd controller, R is the reference signal, P is the plant otpt, and b, K p, K i, and K d are given above. The tning rles developed by Astrom and Hagglnd in their 1995 book otperform the widely sed Ziegler-Nichols tning rles (Ziegler and Nichols 194) on all 16 indstrially representative plants sed by Astrom and Hagglnd. The qestion arises as to whether there are tning rles that may improve on the Astrom-Hagglnd rles. Boyd and Barratt state in Linear Controller Design: Limits of Performance (Boyd and Barratt 1991), "The challenge for controller design is to prodctively se the enormos compting power available. Many crrent methods of compter-aided controller design simply atomate procedres developed in the 1930's throgh the 1950's, for example, plotting root loci or Bode plots. Even the 'modern' state-space and freqency-domain methods (which reqire the soltion of algebraic Riccati eqations) greatly ndertilize available compting power." In this paper, we se genetic programming to atomatically discover PID tning rles that otperform the atomatic tning rles developed by Astrom and Hagglnd in 1995 for the indstrially representative set of plants sed by Astrom and Hagglnd. Section provides backgrond on genetic programming. Section 3 describes the preparatory steps reqired to apply genetic programming to the problem of atomatically discovering tning rles for PID controllers. Section 4 presents the reslts. GENETIC PROGRAMMING Genetic programming (Koza, Bennett, Andre, and Keane 1999; Koza, Keane, Streeter, Mydlowec, Y, and Lanza 003) is an atomated method for solving problems. Specifically, genetic programming progressively breeds a poplation of compter programs over a series of generations. 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, gene deletion, and certain mechanisms of developmental biology to breed an improved poplation over a series of many generations. Genetic programming breeds compter programs to solve problems by execting the following three steps: (1) Generate an initial poplation of compositions (typically random) of the fnctions and terminals of the problem. () Iteratively perform the following sbsteps (referred to herein as 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 fitness 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 available repertoire of sch operations and create one new offspring program for the new poplation by applying the selected architectre-altering operation to the selected program. (3) Designate the individal program that is identified by reslt designation (e.g., the best-sofar individal) as the reslt of the rn of genetic programming. This reslt may be a soltion (or an approximate soltion) to the problem.

3 3 PREPARATORY STEPS Before applying genetic programming to a problem involving the synthesis of a controller, the hman ser of genetic programming mst perform six major preparatory steps: (1) 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) specify the termination criterion for the rn. Initial experimentation qickly confirmed to s that the Astrom and Hagglnd tning rles are highly effective. Starting from scratch, genetic programming was readily able to evolve for mltidimensional srfaces that were very similar to the Astrom and Hagglnd srfaces and that otperformed the Astrom and Hagglnd tning rles on average. However, the genetically evolved srfaces did not otperform the Astrom and Hagglnd tning rles across the board for every plant. Ths, we decided to approach the problem of discovering improved tning rles by bilding on the Astrom and Hagglnd reslts. Specifically, we sed genetic programming to evolve for mathematical expressions which, when added to the for mathematical expressions devised by Astrom and Hagglnd, yield improved overall performance. Note that any srface can emerge from this additive approach. 3.1 Program Architectre We are seeking for mathematical expressions for for qantities (K p, K i, K d, and b). The for mathematical expressions contain free variables representing the plant s ltimate gain, K, and the plant s ltimate period, T. Ths, the architectre of each program tree in the poplation has for resltprodcing branches (one each for K p, K i, K d, and b). Note that this problem can be viewed as a problem of discovering for mlti-dimensional srfaces. 3. Terminal Set The terminal set, T, or each of the for resltprodcing branches contains T = {R, KU, TU}. where KU represents the plant s ltimate gain, K, and TU represents the plant s ltimate period, T. 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 (between 5.0 and 5.0 for this problem). 3.3 Fnction Set The fnction set, F, for each of the for resltprodcing branches contains fnctions for performing arithmetic, exponential, and logarithmic operations. F = {ADD_NUMERIC, SUB_NUMERIC, MUL_NUMERIC, DIV_NUMERIC, REXP, RLOG, POW}. The first six fnctions perform addition, sbtraction, mltiplication, division, exponential, and logarithm operations, respectively. The two-argment POW fnction retrns the vale of its first argment to the power of the vale of its second argment. All fnctions are protected in the sense that they cannot yield a reslt whose absolte vale is larger than These fnctions (along with the terminals from the previos section) are the ingredients for the for tning rle eqations that are to be prodced by genetic programming. 3.4 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. The fitness measre is a mathematical implementation of the problem's highlevel reqirements. In this paper, genetic programming is being sed to atomatically create for eqations sing a fitness measre bilt from the same elements sed in Astrom and Hagglnd That is, or fitness measre attempts to optimize for the integral of the time-weighted absolte error (ITAE) for a step inpt and also to optimize for maximm sensitivity and sensor noise attenation. We se the SPICE simlator (Qarles, Pederson, Newton, Sangiovanni-Vincentelli 1994) to simlate controllers. A SPICE netlist is constrcted to represent the signal processing fnctions (gain, differentiator, integrator, addition, sbtracting) of a PID controller with setpoint weighting, the plant, and the external feedback loop. This SPICE netlist is wrapped inside an appropriate set of commands to carry ot varios analyses in the time domain and in the freqency domain. We also provide SPICE with sbcircit definitions to implement all the signal processing fnctions necessary to represent the plant and the PID controller with setpoint weighting. The entire control system is then simlated sing or modified version of the original 17,000-line SPICE3 simlator as a sbmodle within or genetic programming system. The genetic programming code then ses SPICE s otpt to calclate the fitness of an individal controller. We se a test bed consisting of 30 different plants for this problem. Each of the 30 plants is from one of the for families (A, B, C, and D) of plants considered by Astrom and Hagglnd in their 1995 book. We se more than the 16 plants sed by Astrom and Hagglnd to redce the likelihood of over-fitting (particlarly for the families for which Astrom and Hagglnd consider only three plants). The 30 plants are identified in colmns 1 and of table. We sbseqently se an additional 18 plants (again from the same for families) to cross-validate the reslts. The 18 additional plants are identified in colmns 1 and of table 3.

4 The fitness of each controller in the poplation is measred by means of eight separate invocations of the SPICE simlator for each plant. Since there are a total of 30 plants, the fitness measre entails 40 separate invocations of the SPICE simlator. This mlti-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, over the eight fitness cases for each of the 30 plants nder consideration, of the detrimental contribtions to fitness. The smaller the sm, the better. Step response and distrbance rejection is measred by means of the first six of the eight SPICE simlations for each plant nder consideration. For each plant nder consideration, six combinations of vales (shown in table TTTTWOFAMILIES1) for the height of the reference signal and distrbance signal are considered. The reference signal is a step fnction that rises from 0 at time t = 0 to a specified height at t = 1 millisecond. The reference signal is one of the inpts to the controller. The distrbance signal is a step fnction that rises from 0 at time t = 10T to a specified height at t = 10T 1 millisecond. The distrbance signal is added to the controller's otpt. Table 1. Six combinations of height of reference and distrbance signals. Reference signal Distrbance signal For each of the 30 plants nder consideration, a transient analysis is performed in the time domain sing the SPICE simlator for each of the six combinations of height of reference and distrbance signals in table 1. The fnction 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 180 elements is based on the sm of two integrals of time-weighted absolte error. The first term of the integral acconts for the controller's step response while the second term acconts for distrbance rejection. 10T 0T t e( t) Bdt ( t 10T ) e( t) Cdt t= 0 t= 10T. 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 the 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 six elements of the fitness measre for each plant, the contribtion to fitness is mltiplied by 0 if the element is greater than that for the Astrom and Hagglnd controller (1995). Stability margin is measred by means of the seventh SPICE simlation for each plant nder consideration. Figre presents a model for the entire system containing the given plant and the to-be-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. 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 plant nder consideration, an AC sweep is performed sing the SPICE simlator from 1/(1000T ) to 1000/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 sensitivity (and therefore maximize the stability margin). The qantity 1/M s is the minimm distance between the Nyqist plot and the point (-1,0) 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 < 1.5; (M s - 1.5) for 1.5 M s.0; and 0(M s -.0) 1 for M s >.0. For each plant nder consideration, the contribtion to fitness is mltiplied by 10 if the element is greater than that for the Astrom and Hagglnd controller (1995). R(s) Q(s) Controller Fig.. Overall model U(s) D(s) Sensor noise attenation is measred by means of the eighth SPICE simlation for each plant nder consideration. 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 plant nder consideration, an AC sweep is performed sing the SPICE simlator from 10/T to 1000/T while holding the reference signal R(s) and the distrbance signal; D(s) at zero. The attenation of 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 N(s) Plant Y(s)

5 sensor noise attenation is 0 if A min > 40 db; (40 - A min )/10 if 0 db A min 40 db; and (0 - A min ) if A min < 0 db. For each plant nder consideration, the contribtion to fitness is mltiplied by 10 if the element is greater than that for the Astrom and Hagglnd controller (1995). A controller that cannot be simlated by SPICE is assigned a high penalty vale of fitness (10 8 ). 3.5 Control Parameters The poplation size, M, is 100,000. The remaining control parameters are in Koza, Keane, Streeter, Mydlowec, Y, and Lanza Termination Criterion We manally monitored and terminated the rn when fitness appeared to reach a platea. 4 RESULTS In the one rn we made on this problem, PID tning rles emerged on generation 76 that otperform the PID tning rles in Astrom and Hagglnd The improved tning rles are obtained by adding the genetically evolved qantities below to the vales of K p, K i, and K d, and b in Astrom and Hagglnd The qantity, K p-adj, that is to be added to K p for the proportional part of the controller is * T *10 The qantity, K i-adj, that is to be added to K i for the integrative part of the controller is K * T The qantity, K d-adj, that is to be added to K d for the derivative part of the controller is ( e T ) log K ( ) log The qantity, b adj, that is to be added to b for setpoint weighting of the proportional block is K K e The athors refer to these evolved tning rles as the Keane-Koza-Streeter (KKS) PID tning rles. Table compares the performance of the best-of-rn genetically evolved PID tning rles from generation 76 as a percentage of the vale for the Astrom and Hagglnd controller for the 30 plants sed by the evoltionary process. An OK appears in the table for the case where both controllers have a vale of 0. As can be seen in the table, all percentages are either below 100% (indicating improvement) or OK. Table 3 compares the performance of the best-of-rn genetically evolved PID tning rles from generation 76 as a percentage of the vale for the Astrom and Hagglnd controller for the additional 18 plants sed to cross-validate the reslts prodced by the evoltionary process. As can be seen in the table, all percentages are either below 100% (indicating improvement) or OK. Averaged over the 30 plants sed by the evoltionary process, the best-of-rn tning rles from generation 76 have 91.6% of the setpoint ITAE of the Astrom- Hagglnd tning rles, 96.% of the distrbance rejection ITAE of the Astrom-Hagglnd tning rles, 99.5% of the reciprocal of minimm attenation of the Astrom-Hagglnd tning rles, and 98.6% of the maximm sensitivity, M s, of the Astrom-Hagglnd tning rles. Averaged over the 18 additional plants, the best-ofrn tning rles from generation 76 have 89.7% of the setpoint ITAE of the Astrom- Hagglnd tning rles, 95.6% of the distrbance rejection ITAE of the Astrom-Hagglnd tning rles, 99.5% of the reciprocal of minimm attenation of the Astrom-Hagglnd tning rles, and 98.5% of the maximm sensitivity, M s, of the Astrom-Hagglnd tning rles. Averaged over the 16 plants originally sed by Astrom and Hagglnd in their 1995 book (a sbset of the 30 plants sed by the evoltionary process here), the best-of-rn tning rles from generation 76 have 90.5% of the setpoint ITAE of the Astrom- Hagglnd tning rles, 96% of the distrbance rejection ITAE of the Astrom-Hagglnd tning rles, 99.3% of the reciprocal of minimm attenation of the Astrom-Hagglnd tning rles, and 98.5% of the maximm sensitivity, M s, of the Astrom-Hagglnd tning rles. 5 CONCLUSION We presented genetically evolved tning rles that otperform the atomatic tning rles developed by Astrom and Hagglnd in 1995 for an indstrially representative set of plants. REFERENCES Astrom, Karl J. and Hagglnd, Tore PID Controllers: Theory, Design, and Tning. Second Edition. Research Triangle Park, NC: Instrment Society of America. Boyd, S. P. and Barratt, C. H Linear Controller Design: Limits of Performance. Englewood Cliffs, NJ: Prentice Hall. Callender, Albert and Stevenson, Allan Brown Atomatic Control of Variable Physical Characteristics. United States Patent,175,985. Filed Febrary 17, 1936 in United States. Filed Febrary 13, 1935 in Great Britain. Issed October 10, 1939 in United States. Koza, John R., Bennett III, Forrest H, Andre, David, and Keane, Martin A Genetic Programming III: Darwinian Invention and

6 Problem Solving. San Francisco, CA: Morgan Kafmann. Koza, John R., Keane, Martin A., Streeter, Matthew J., Mydlowec, William, Y, Jessen, and Lanza, Gido Genetic Programming IV. Genetic Programming IV: Rotine Hman-Competitive Machine Intelligence. Klwer Academic Pblishers. Qarles, Thomas, 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 Ziegler, J. G. and Nichols, N. B Optimm settings for atomatic controllers. Transactions of ASME. (64) Table Comparison of performance of the best-of-rn PID tning rles from generation 76 as a percentage of the vale for the Astrom and Hagglnd controller for 30 plants. Plant Plant ITAE 1 ITAE ITAE 3 ITAE 4 ITAE 5 ITAE 6 Stability Sensitivity parameter vale A A A A A A A A A B OK B OK B OK B OK B OK B OK C OK 93.5 C OK C OK C OK C OK C OK C OK C OK C OK D OK D OK D OK D OK D OK D OK Table 3 Comparison of performance of the best-of-rn PID tning rles from generation 76 as a percentage of the vale for the Astrom and Hagglnd controller for 18 additional plants. Plant Plant ITAE 1 ITAE ITAE 3 ITAE 4 ITAE 5 ITAE 6 Stability Sensitivity parameter vale A A A A A A C OK C OK C OK C OK C OK C OK D OK D OK D OK D OK D OK D OK