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ThM3-2 2:20hxeedings of the 1999 E E EInternational Conference on Control Ap plicat ionsKohala Coast-Island of H awaii, Hawaii, USA August 22-27, 1999Auto-TuningPIDUsing Loop-ShapingIdeasSujit Gaikwad, Sac hi Dash and Gunter Steingaikwad-sujit 0tc.honeywell.comHoneywell Technology Cen terMinneapolis, MN 55418, USA

AbstractIn this paper we present a direct approach for auto-tuningPID controllers. The a pproach is based on loop-shapingprinciples. Auto-tuning is accomplished under closed-loopsystem excitation without fitting a model of the system. Thetuning procedure recursively adapts PID parameters toachieve a target loop-shape. In simulation testing theprocedure works very well for system s with integrators,dead-time, lead dynamics, inverse response, sensor noiseand colored noise disturbances. Here we present anexample of a first order plant with delay operating under acolored noise disturbance,1. IntroductionSimple Proportional Integral D erivative (PID) controllersare still the most popular control algorithms in processindustrie s. All plant information and control systems offe rPID control algorithms as standard equipment in theirhardware. However, tuning PID loops is still a formidabletask for control engineers. Today, there ar e hundreds oftools, methods and theories available for this purpose.However, in practice the bulk of these methods require a lotof engineering effort to get satisfactory tuning. Currently,control engineers use commercially available tools only as astartins point and then play with the PID parameters toget acceptable settings. This is a very time consumingpractice. The idea of an auto-tuning or a self-tuning PIDcontroller is very attractive. Such a controller willdetermine PID settings on its own without much operatorintervention. This idea has tremen dous commercial valueand there are a number of auto-tuners in the market. In factalmost every Distributed Control System (DCS)manufacturer has one. Shinskey (Feed back Controllersforthe Process Industries, McGra w Hill,1994)has observedthat many of the commercial offerings are not particularlyeffectiv e. In his experience m ost of them fall into theCategory of un reliable, if working at all. We are sure thatpracticing engineers wh o have tried some of thesetechniques will recognize that auto-tuning does not work allthe time. However, since there are somany PID loops to betuned the technology has its place in industrial controlapplications.For example, a typical refinery has 3000PIDloops. We believe that with the deployment of field-bus inthe process industry there will be continued attention givento this problem. In this paper we have developed an auto-tuning PID controller based on encouraging ideas fromclassical loop-shaping theory. The auto-tuner determinesoptimum PID gains under closed-loop system excitation byrecursively minimizing a weighted fit of the error between

the control-loop and a target loopshape. The fitting isperform ed on sampled observations of the proportional,integral, de rivative actions and the output of the target-looptransfer function driven by the system input.2. Loop-shap ingOverviewTh e objective of feedback control is to maintain a system ata desired output in the presence of disturbances,uncertainty, system instability and measurement noise.Figure 1 show s the block diagram of a typical feedbackcontr ol system . The closed-loop transfer functions relatingthe tracking error e to the set-point r , he measurementnoise n , he disturbances do nd d i are:

Gn- -d ,GKe =- r d o ) +-e = S ( r - d , -Gdi)+Tnl+GK I+GK l + G Kwhere G is the system being controlled, K s thecontroller, S = 1/(1 +GK) s the sensitivity fun ction andT =GK/(l+ GK) is the complimentary sensitivityfunction.It is desired to keep the tracking error small whichtranslates to the minimization of both S nd T .However,the control system must meet the fundamental constraint ofS+T = 1 . Therefore, we cannot make S andT arbitrarily small at the same time. Realizing, thattypically set-point and disturbances are mainly lowfrequen cy signa ls and measurement noise is a highfrequen cy sign al we will get satisfactory performance bymaking S mall at low frequencies and T small at highfrequencies. Since s and T both depend on the looptransfer function, a target loop is selected such that theclosed-loo p transfer functions have desirable properties.Loop-s haping is the classic frequency based control designmetho dology that achieves this objective by shaping theopen-loo p transfer function L ( j a )= G ( j a ) K ( j a ) .This is done by choosing loop-shapes that have a large gainat low fre quencie s below crossover and a small gain at highfrequen cies above crossover. The controller K is selectedsuch tha t the loop transfer function GK approximates thetarget lo op L. Selection of the target loop is govern ed bybandw idth constraints imposed by uncertainty in the mode l,non-m inimum phase behavior and unstable Doles.0-7803-5446-X/99$10.000 1999 IEEE 589

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3.Auto-TuningPIDWe have developed an auto-tuner for Proportion al IntegralDerivative (PID) controllers based on loops-sh aping ideas.The proportional gain K,,ntegral gain K i and thederivative term K d are selected by directly fitting the looptransfer function to a target loopshape.A PID form withindependent gains and derivative approximation is used:

K. SS as+lK = Kc+I+d-Figure 2 shows a block diagram representation of the auto-tuning procedure. PID param eters are estima ted in theclosed-loop(AUTO ode) without identifying a model forthe system. The o bjective is to auto-tune the PID gains withvery little operator interaction. System excitation isprovided by an external signal that is added to the con troIIeroutput as part of the tuning method. The inpu t is seIected tohave power in the frequency region around the desiredbandwidth. This input could be a se ries of steps, a pseudorandom binary sequence or band-pass filtered and clippedwhite noise. It is important to inject an input that is plantfriendly. i.e. a signal w ith which an operator is comfortable.The target loopshape is determined by the desired closed-loop bandwidth and the nature of the plant. For example,for stable systems we se lect a first-orde r loop-shape

(L =wc/s and integrating plants are tuned for a second-order target loop-shape ( L : ). Here a, s theScrossover frequency and x is a parameter that governs thelow-frequency slope or overshoot in response to a stepsetpoint chang e. The bandw idth is limited by the nature ofthe system and uncertainty represe nted by qu ality of thedata collected during testing. The tuning algorithm has asingle knob (band width) that the operator can adjust to get adesirable loop with acceptable performanc e. Anapproximate range for the bandwidth can be obtained usinga pre-tuning step test. A recursive least squares algorithm isused to fit the PID pa rameters. The fitting is performed tomeet the objective:min 11 uID L GK) (l+GK)I, which is equivalent tomin IILu - Ke , II,Note that sinceG oes not explicitly appear in thisobjective it allows us to directly fi t the controller. For thePID controller this is a solution to a least-squares problem.4. Case StudyWe co nsider the case of a plant with time operating under adrifting disturbance as:

d ( t )=N ( 0 , l )L = wc / s ; w c =1;Ts=0.01

The system ha s a time co nstant of 0.1 min. a delay of 0.1min and the data is sampled at 0.01 min. Figure 3 shows thesystem in open-loop with the colored-noise disturbanceacting on the system. Such systems are difficult to identifyand control because of the drifting disturbance and time-delay. Figure 4shows the closed-loop data collected duringthe auto-tuning procedure . The system output is nicelymaintained betwee n + 2 an d -2 throughout the tuningprocedure. The excitation signal iswhite noise that isclipped and bandpa ss filtered to have power aroundcrossover. A recursive least-squares algorithm with non-negativity co nstraints was used to fit the PID parameters.The least-squares algorithm isdocumented in Goodwin andSin (Adaptive Filtering Prediction a nd Control, PrenticeHall, 1984,pp . 92-94). The perform ance of this algorithmcan be improved using dead-zon es and normalizationschemes. At initial time w hen the excitation signal is smallwe observe drifts in the parameters. The PID parame tersare adapted at every sam pling instant. Figure 5shows theconvergence of the PID parameters and the fitting error. Weobserve that the parameters nicely converge in about 10minutes and the fitting error becomes small very q uickly.Figures 6 and 7 how frequency response analyses for thefinal system. In the frequency ranges around the systembandwidth we observe very good fits of the sensitivity,complementary sensitivity and loop transfer functions.5. ConclusionsIn this paper we have presented a novel algorithm fortuning PID controllers under closed-loo p system excitation.The auto-tuning procedure is a direct approach, whichdetermines optimum PID oefficients without identifying amodel of the system. The PID gains are computed toapproximate a specified target loop transfer function. Insimulation testing we have found that the procedure w orksvery well for systems with integrators, dead-time, leaddynamics, inverse response, sensor noise and colored noisedisturbances. Opportunities for future research include anouter loop for estimating an achievable bandwidth, adaptive(aIways on) tuning, extension to higher-order controllers,and multivariable systems.AcknowledgmentSupport from the Hone ywell Initiative program isappreciated. We w ould like to thank Ward MacA rthur andKostas Tsakalis for their insights into the project.

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