Auto-Tuning of PID Controllers via Extremum Seeking
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1 25 American Conrol Conference June 8-, 25. Porland, OR, USA ThA7.2 Auo-Tuning of PID Conrollers via Exremum Seeking Nick illingsworh* and Miroslav rsić Deparmen of Mechanical and Aerospace Engineering Universiy of California, San Diego La Jolla, CA , U.S.A. and Absrac The proporional-inegral-derivaive (PID) conroller is widely used in he process indusry, bu o various degrees of effeciveness because i is someimes poorly uned. The goal of his work is o presen a mehod using exremum seeking () o une he PID parameers such ha opimal performance is achieved. is a non-model based mehod which searches on-line for he parameers which minimize a cos funcion; in his case he cos funcion is represenaive of he conrollers performance. Furhermore, his mehod has he advanage ha i can be applied o plans in which here is no knowledge of he model. We demonsrae he uning mehod on a cross secion of plans ypical of hose found in indusrial applicaions. The PID parameers are uned based on he resuls of sep response simulaions o produce a response wih minimal seling ime and overshoo. Addiionally, we have compared hese resuls o hose found using oher uning mehods widely used in indusry. I. INTRODUCTION Conrollers used in indusry, especially in process conrol, are almos exclusively proporional-inegral-derivaive (PID) conrollers. Despie he wide spread use of PID conrollers hey are someimes poorly uned impeding heir effeciveness. Manually uning such a conroller is a edious ask since i requires he opimizaion of hree parameers. To remedy his, much effor has been pu ino devising ways o sysemaically une PID conrollers. Many of hese mehods rely on knowledge of he plan model or require special experimens o be performed o idenify a model of he plan. Reviews of hese mehods are given in he book by Asrom and Hagglund (995) [] and in he survey by Asrom e al. (993) [2]. However, in many siuaions a model for he plan is no known and i is no desirable o open he process loop. Thus a mehod of uning he parameers by means of he closed loop sysem is advanageous. One such mehod is relay feedback uning; however, his mehod requires ha a relay is added o he sysem and is used as a feedback ino he plan bypassing he conroller. For mos plans relay feedback will cause he sysem o oscillae, his allows he deerminaion of one poin on he Nyquis diagram. Based on he locaion of his poin PID parameers can be formulaed o give he closed loop sysem a desired phase and ampliude margin, see for example, [3]-[5]. Anoher non-model based mehod which has been applied o PID uning and does no require opening of *suppored by NSF AGEP HRD and he GEM program /5/$ AACC 225 he loop is ieraive feedback uning (). This echnique ieraively opimizes he conroller parameers wih respec o a cos funcion derived from he oupu signal of he closed loop sysem, and is discussed in deail in [6]. This mehod has been applied o une a PID conroller wih a cos funcion based on he performance of he closed loop sysem during a sep response, [7], [8]. In his work we apply a discree version of exremum seeking () from [9] o he ask of opimizing he sep response of a closed loop sysem in a fashion similar o ha in [7], [8]. However, i should be noed ha oher forms of he cos funcion could easily be adaped o he uning mehod. is a non-model based mehod which ieraively modifies he inpu of a funcion such ha he oupu of he funcion is aken o a local minimum or maximum. achieves his by perurbing he inpu parameers of he sysem and hen making an on-line esimaion of he gradien of he oupu relaive o he inpus. This gradien informaion is hen used o modify he inpu parameers in he nex ieraion owards he minimum or maximum of he oupu. A block diagram of he discree version of is depiced in Fig. and will be discussed furher in he following secion. An overview of he heory as well as many sae of he ar applicaions is provided in []. Fig.. θ(k) γ z - αcos(ωk) f(θ) αcos(ωk - φ) z - z h Block diagram of discree exremum seeking scheme. II. IMPLEMENTATION is used o une he parameers of a PID conroller such ha he racking error of he closed loop sysem wih an unknown plan is minimized. We use a sandard PID conroller, however he derivaive erm only acs on he
2 r u y G C r - r() C r ( k θ ) G y() T f ( )d Jk C y C y ( k θ ) Fig. 2. Block diagram of closed-loop sysem. θk Exremum Seeking Algorihm measured plan oupu and no he reference signal. This reduces large conrol effors during a sep change in he reference signal. A block diagram of he conroller is shown in Fig. 2, where C r =( T i s ) () C y =( T i s T ds). (2) Noe r, u, and y are he reference, conrol, and oupu signal, respecively. The effeciveness of he above conroller is quanified using he following cos funcion evaluaed a he conclusion of a sep response experimen, J(θ k )= T e 2 (θ k,)d, (3) T where he error is defined as he difference beween he reference and he oupu signal of he closed loop sysem, e(θ, ) =r y(θ, ), and θ is a vecor conaining he PID parameers, θ =[, T i,t d ] T. In (3), and T are he imes during he sep response experimen a which we begin and end aking ino accoun he error during calculaion of he cos funcion, respecively. Seing equal o he seling ime of he closed loop sysem effecively pus zero weighs on he ransien porion of he response. This focuses all available degrees of freedom of he conroller on minimizing he error beyond he seling ime and none on he ransien porion. From principles of opimaliy we know ha his will always yield a cos beyond which is equal if no beer han ha of a conroller minimized using a value earlier han. This mehod of using zero weighs in he cos funcion is suggesed by [7] where i has been applied o he mehod of PID uning. The discree algorihm has been implemened using he following equaions, which are equivalen o he sysem represenaion shown in Fig., ξ(k) = hξ(k ) J(k ) (4) ˆθ i (k) = ˆθ i (k) γ i α i cos(ω i k)[j(k) (h)ξ(k)](5) θ i (k) = ˆθ i (k) α i cos(ω i (k)), (6) where ξ(k) is a scalar and he subscrip i indicaes he i h enry of he vecors ˆθ(k ) and θ(k ). γ is he esimaion gain and α is he perurbaion sep size. The modulaion frequency ω should be chosen such ha ω = aπ, < a <, where a is raional, his guaranees ha 2252 Fig. 3. Overall PID uning scheme. he Nyquis crierion is saisfied. Addiionally, he high-pass filer z zh is designed wih <h<. The overall PID uning scheme is presened in Fig. 3. The sep funcion experimen porion of he sysem, which is conained wihin he dashed box is run ieraively. The algorihm uses he discree value of he cos funcion, J k generaed a he compleion of each ieraion o compue he nex se of conroller parameers, θ k. A new sep funcion experimen is hen performed wih he new conroller parameers and he process coninues ieraively in his fashion. III. SIMULATIONS In his secion we demonsrae he mehod of PID uning, as well as, compare i wih he mehod and wo classical PID uning mehods: he Ziegler-Nichols uning rules () and he inernal model conrol () mehod. The mehod consiss of performing an open loop experimen and increasing he sysem gain unil he poin where he Nyquis plo crosses he negaive axis. PID parameers are hen prescribed based on his gain and he frequency a which he sysem oscillaes []. The mehod assumes he plan is of he following form, G(s) = p Ts e Ls ; based on his model PID parameers are are hen deermined o give he closed-loop sysem a desired response. If he acual plan model is unkown a sep response experimen can be used o esimae i in he above form. These four mehods have been applied o he following four models wih dynamics ypical of many indusrial plans. G (s)= 2s e 5s (7) G 2 (s)= 2s e 2s (8) G 3 (s)= ( s) 8 (9) 5s G 4 (s)= ( s)( 2s). () Noice ha plans G and G 2 have ime delays, G 3 has muliple poles, and G 4 is non-minimum phase. These models have been aken from [8] o allow comparison wih he mehod, addiionally conroller parameer values for he and mehods have also been used from his work.
3 TABLE I PID PARAMETERS FROM FOUR TUNING METHODS FOR SIMULATIONS OF G. Tuning mehod T i T d J(k)... Simulaions of he closed loop sysems have been performed using a ime sep of. s and he ime delays have been approximaed using a hird-order Padé approximaion. The values for he PID conroller parameers found from he uning rules have been used as a saring poin for he uning mehod in all simulaions. For all simulaions he parameers, a and h in equaions (4) - (6) were se o.8 and.5, respecively. A. Simulaion of G The PID uning mehod has been applied o he firs ransfer funcion G in (7), which has a ime delay of 5 s. For hese simulaions we se α = [.,,.] T, γ = [2, 2, 2] T, and ω i = a i π where a =.8 as menioned previously. The resuls of hese simulaions are presened in Fig. 4. We can see from he upper wo plos of his figure ha he mehod quickly minimizes he cos funcion (3) and converges in less han ieraions o PID parameers which produce a local minimum. achieves his response by reducing he influence of he inegral porion of he conroller and increasing he value of he inegral ime o almos hree imes ha given by he uning rules. The performance of he PID parameers found via he uning mehod compared o hose of he oher hree mehods is also shown, we can see ha produces a conroller wih performance ha is roughly equivalen o ha found using. This is expeced since boh mehods aemp o minimize he same cos funcion. I can also be seen ha hese wo mehods yield closed loop sysems wih less overshoo as well as smaller seling imes han he oher uning mehods evaluaed in his sudy. B. Simulaion of G 3 Here we have used he PID uning mehod o une he PID parameers of he closed-loop sysem wih G 3 in (9), which has a single pole of order eigh. In conras, for his plan we found ha using a cos funcion based on he inegraed absolue error (IAE) J(θ k )= T e(θ k,) d, () T insead of he one based on he inegraed squared error (ISE) in (3) produced a smooher map and hus he behavior of he algorihm was similarly more well behaved. The following parameers were used in hese simulaions: α =[.,.7,.4] T, γ =[22, 2, 22] T, ω = aπ, Ieraions (k) θ y() u() Td Ieraions (k) (c) (d) Fig. 4. Evoluion of he cos funcion and he PID parameers during uning of he closed-loop sysem wih G (s). The oupu signal (c) and conrol signal (d) during a sep response experimen of he closedloop sysem wih G (s) and PID conroller parameers obained by means of he four differen mehods.
4 TABLE II PID PARAMETERS FROM FOUR TUNING METHODS FOR SIMULATIONS OF G 3. Tuning mehod T i T d J(k) TABLE III PID PARAMETERS FROM FOUR TUNING METHODS FOR SIMULATIONS OF G 4. Tuning mehod T i T d ω 2 = aπ π 2, and ω 3 = a 3 π. The resuls are presened in Fig. 5 and he PID parameers are lised in Table 3. From hese plos i can be seen ha he algorihm vasly improves upon he sep response behavior given by he uning rules, and reurns a response ha is once again similar o ha achieved by he oher wo mehods. In his case finds ha a reducion in boh he inegral ime T i and conroller gain produce a smaller value of he cos funcion. This plan proves more of a challenge and requires roughly 5 ieraions unil he PID parameers converge. C. Simulaion of G 4 The PID conroller of he closed-loop sysem wih nonminimum phase G 4 in () has also been uned using he mehod. For hese simulaions we se α =[.5,.6,.2] T, γ = [2,, 2] T, ω = aπ, ω 2 = aπ π 2 and ω 3 = a 3 π. The resuls can be found in Fig. 6, addiionally he PID parameers are lised in Table 4. These plos show ha once again he uning mehod produces a sep response very similar o ha obained using he mehod boh of which yield no overshoo and a seling ime much smaller han hose conrollers uned using he or mehods. However, he mehod in his case does produce a slighly larger conrol signal iniially han he mehod. Also noe ha larger values of he inegral and derivaive imes yield a beer sysem response over ha found using he mehod. D. Cos funcion comparison As we found for he plan G 3, choosing he righ cos funcion is key o geing good resuls and in his secion we will expand upon his opic. We will use he algorihm o minimize he following cos funcions obained from Fig. 5. simulaions of plan G 2 (s): IAE, which is equal o () wih =, ISE, which is equal o (3) wih =, Window, which is (3) wih se o some posiive window ime, 2254 θ y() u() Ieraions (k) Td Ieraions (k) (c) (d) Evoluion of he cos funcion and he PID parameers during uning of he closed-loop sysem wih G 3 (s). The oupu signal (c) and conrol signal (d) during a sep response experimen of he closedloop sysem wih G 3 (s) and PID conroller parameers obained via he four differen mehods.
5 .. J(k) Ieraions (k) y() IAE ISE.2 ITAE window ITSE θ y() u() Td Ieraions (k) (c) (d) u() IAE ISE ITAE window ITSE Fig. 7. The oupu signal and conrol signal during a sep response experimen of he closed-loop sysem wih G 2 (s) and PID conroller parameers obained using wih various cos funcions. and ITAE = T T T e(θ k,) d (2) ITSE = e 2 (θ k,)d. (3) T Noe he ime-dependen weighing in (2) and (3), which de-emphasizes he ransien porion of he response. Fig. 7 illusraes he imporance of using a cos funcion which correcly characerizes he desired sysem behavior. We can see ha he Window cos funcion produces a response wih he smalles overshoo and fases seling ime. The ITAE and IAE have similar performance, which is slighly worse han ha produced by using he Window cos funcion, whereas ISE and ITSE do no offer much of an improvemen over ha found using he uning rules. E. Conrol sauraion In many applicaions of PID conrol we mus deal wih acuaor sauraion. Acuaor sauraion can resul in inegraor windup, in which case he feedback loop becomes Fig. 6. Evoluion of he cos funcion and he PID parameers during uning of he closed-loop sysem wih G 4 (s). The oupu signal (c) and conrol signal (d) during a sep response experimen of he closedloop sysem wih G 4 (s) and PID conroller parameers obained via he an effec on he sysem. In his secion we invesigae emporarily disconneced since he conroller no longer has four differen mehods. how he uning mehod handles his nonlineariy. We 2255
6 u u P D.5 e Σ s u I Σ Acuaor u Fig. 8. e - Σ Tracking ani-windup scheme. TABLE IV PID PARAMETERS FROM THREE TUNING METHODS FOR SIMULATIONS OF G WITH SATURATION. Tuning mehod T i T d racking will une he same PID conroller using, bu in he presence of conrol sauraion, addiionally we une his PID conroller wih a racking ani-windup scheme. Tracking modifies he inegral conrol signal using a feedback which is proporional o he difference beween he requesed conrol effor and he conrol effor produced by he acuaor e, as depiced in Fig. 8. This scheme is explained in more deail in []. We compare hese resuls o a sandard PID conroller uned using and he same conroller wih racking in Fig. 9 for plan G (s) wih conrol sauraion of.6. We can see from hese plos ha here is quie a bi of overshoo for he sysem uned using, bu ha even wihou racking he sysem uned using has minimal overshoo. is able o do so well because i increases he inegral ime such ha he sysem is more robus o sauraion as can be seen in Table 5. The mehod even finds conroller parameers which ouperform he mehod wih racking. And he conroller does almos as well as he sysem wih racking uned using. This illusraes he downfall of using an open-loop uning scheme such as when acuaor sauraion is no accouned for. IV. CONCLUSIONS We have shown ha provides an effecive and efficien mehod o une PID conrollers ha minimize a cos funcion which characerizes some desired behavior. This uning mehod has been demonsraed on four ypical plans and found o give parameers ha yield performance beer han or comparable o ha of oher popular uning mehods. However i should be noed ha iniial values of he PID parameers are needed, hus uning is bes used as a complemen o anoher PID parameer design mehod. I should also be made clear ha he cos funcion can be modified such ha differen performance aribues are emphasized y() u().5 racking racking racking.2 racking Fig. 9. The oupu signal and conrol signal during a sep response experimen of he closed-loop sysem wih G (s), conrol sauraion of.6, and PID conroller parameers obained using and boh wih and wihou racking. REFERENC []. J. Asrom, and T. Hagglund, PID Conrollers: Theory, Design and Tuning (2nd ed.). Norh Carolina, USA: Insrumen Sociey of America, 995. [2]. J. Asrom, T. Hagglund, C. C. Hang, and W.. Ho, Auomaic uning and adapaion for PID conrollers - A survey, Conrol Eng. Pracice, vol., no. 4, pp , 993. [3]. J. Asrom, and T. Hagglund, Auomaic uning of simple regulaors wih specificaions on phase and ampliude margins, Auomaica, vol. 2, no. 5, pp , 984. [4] A. Leva, PID auouning algorihm based on relay feedback, IEEE Proceedings,vol. 4, no. 5, pp , 993. [5] A. A. Voda, and I. D. Landau, A mehod for he auo-calibraion of PID conrollers, Auomaica, vol. 3, no., pp. 4-53, 995. [6] H. Hjalmarsson, M. Gevers, S. Gunnarsson, and O. Lequin, Ieraive feedback uning: Theory and applicaions, IEEE Conrol Sysems Magazine, vol. 8, no. 4, pp. 26-4, 998. [7] O. Lequin, M. Gevers, and T. Tries, Opimizing he seling ime wih ieraive feedback uning, in Proceeedings of he 4h IFAC world congress, Beijing, P.R. China, pp , 999. [8] O. Lequin, E. Bosmans, and T. Tries, Ieraive feedback uning of PID parameers: Comparision wih classical uning rules, Conrol Eng. Pracice, vol., no. 9, pp , 23. [9] J. Y. Choi, M. rsic,. B. Ariyur, and J. S. Lee Exremum Seeking conrol for discree-ime sysems, IEEE Transacions on Auomaic Conrol,vol. 47, pp , 22. []. B. Ariyur, and M. rsic, Real-me Opimizaion by Exremum Seeking Feedback, Hoboken, N.J. : Wiley-Inerscience, 23.
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