ON ITERATIVE FEEDBACK TUNING AND DISTURBANCE REJECTION USING SIMPLE NOISE MODELS. Bo Wahlberg

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1 ON ITERATIVE FEEDBACK TUNING AND DISTURBANCE REJECTION USING SIMPLE NOISE MODELS Bo Wahlberg S3 Automatic Control, KTH, SE Stockholm, Sween. Abtract: The objective of thi contribution i to icu three baic control eign metho for iturbance rejection. The key iea i to ue a imple fixe continuou time noie moel a a eign variable to pecify the eirable cloe loop propertie, e.g., the cloe loop banwith. We will tuy three cloely relate controller eign metho bae on the noie moel, namely minimum variance control, weighte minimum variance control an a loop haping eign proceure propoe by Skogeta an Potlethwaite. All thee metho lea to imple controller tructure, where the tranfer function of the input/output relation i the only unknown part. The main motivation of thi tuy ha been tuning of controller for iturbance rejection by mean of iterative feeback tuning (IFT). Here, the controller parameter are foun by minimizing a pecifie cot function uing graient earch technique. The graient are etimate uing ata from iterative experiment. Thi i a rather ifficult problem if no external input excitation i allowe. Import factor for goo reult are the controller parametrization an the choice of cot function. We propoe an IFT cot function bae on a imple output ignal weighting tailore to the fixe noie moel controller tructure. Thi lea to robut proceure for iterative feeback tuning of a controller for iturbance rejection. The approach i illutrate on a imple example. Copyright c 005 IFAC Keywor: control tuning; iterative feeback tuning; iturbance rejection control oriente ientification; ientification; minimum-variance regulator. 1. INTRODUCTION Thi paper i motivate by practical experience uing Iterative Feeback Tuning (IFT) for eigning controller for iturbance rejection. The control performance i meaure by a pecifie cot function, an the objective i to fin the controller that within a given tructure minimize thi cot. Thi i one by graient earch technique, 1 Reearch wa partially upporte by the Sweih Reearch Council. where the graient are etimate uing ata from multiple experiment. See (Hjalmaron, 00) for a thorough overview. IFT work well if it i poible to ue external excitation ignal to obtain reliable graient etimate. It i, however, more ifficult to ue IFT if the only excitation i the external iturbance, which houl be rejecte by the feeback. It i then mot important to have a imple controller tructure with a few free parameter a poible together with a cot

2 function that reflect the control pecification, c.f. (Lequin et al., 003). We will ue continuou time moel to ecribe the unerlying iea, even if the feeback an tuning mot often are one uing ample ata. The reaon i that thi make it eaier to preent the baic concept. Conier the feeback control ytem in Fig. 1. r Σ F (ρ) u 1 Fig. 1. Feeback control ytem. The tranfer function of the ytem i given by G(). The plant input i enote by u(t), the output by y(t) an r(t) i the reference ignal.the controller i pecifie by the tranfer function F (, ρ), where ρ i the controller parameter to be etermine. The objective of the feeback control i to reject the aitive output iturbance v(t). We will aume that the reference ignal i zero. The iturbance v(t) i moelle a the output from a noie filter H(), excite by the ignal (t). The tranfer function from (t) to y(t) will be calle the weighte enitivity function, an equal S()H(), S() = where S() i the enitivity function. G H Σ G()F (), (1) We will make extenive ue of the following imple noie filter moel H() = + ω 0S 0 S 0, ω 0, S 0 > 0. () The noie filter H() behave a an integrator for low frequencie, the amplitue of the frequency repone H(iω) 1 at the frequency ω = ω 0 an the high frequency gain i 1/S 0. Thi mean that it i mot important to have effective feeback in the frequency range [0, ω 0 ], but that there alo i high frequency noie. Thi will avoi large controller gain at high frequencie (ifferentiation). The noie moel parameter houl be viewe a eign variable, where ω 0 etermine the eire banwith. The choice of S 0 i le important, but houl be choen larger than one. The control performance will be meaure by the icrete time cot function v y J(ρ) = 1 N [(W T (q)y(kt, ρ)) + λ ( (q)u(kt, ρ)) ], (3) where T i the ampling interval, W T (q) i a icrete time weighting filter an (q) = 1 q 1 the ifference operator. Here q enote the hift operator qy(kt ) = y((k + 1)T ). The parameter λ > 0 balance the output regulation with the change in the input ignal. We have tree that the input an output are function of the controller parameter ρ. The tak i to fin the controller that minimize the cot function J(ρ). Iterative Feeback Tuning (IFT) i a metho to minimize uch a cot function by uing information from iterative experiment. The paper i organize a follow. Section icue controller eign for iturbance rejection. IFT i then introuce in Section 3, while Section 4 ecribe how to combine the iturbance rejection control eign iea ecribe in Section with IFT to obtain a imple ata riven tuning metho. Thi metho i illutrate on a numerical example in Section 5. Finally, Section 6 conclue the paper.. CONTROLLER DESIGN FOR DISTURBANCE REJECTION Next, we will review three imple moel bae controller eign metho, namely minimum variance control, weighte minimum variance control an a loop haping eign proceure. Aumption: In orer to implify the preentation we will aume that the tranfer function G() i of relative egree one, i aymptotically table an i minimum phae (all zero trictly in the left half plane)..1 Minimum Variance Control Aume that the input (t) to the noie filter H() i continuou time white noie, i.e., ha contant power for all frequencie, ee e.g. (Åtröm, 1970). Thi i mainly a technical aumption to obtaine well-poe continuou time tochatic control problem. However, ince the variance of continuou time white noie i infinite, one ha to be careful when eriving the continuou time minimum variance control law. By auming H( ) 0 (emi-proper) the minimum variance controller equal F MV () = 1 [ ] H() G() H( ) 1. (4) Notice that [H()/H( ) 1] will be trictly proper (more pole than zero). The aumption

3 that G() ha relative egree one will guarantee a proper controller (no irect ifferentiation of y(t)). If the relative egree of G() i higher than one, we have to inclue a low-pa filter in the controller to make it proper. Poible untable pole of G() hare by H() can alo be hanle. The cloe loop relation will be y(t) = H( )(t), i.e., the output will alo be white noie. For the imple noie filter () H() = + ω 0S 0 H() S 0 H( ) 1 = ω 0S 0. Hence, the loop tranfer function G()F MV () i jut an integrator, where the gain etermine the banwith. Here, the weighte enitivity (1) equal S MV ()H() = 1/S 0. The banwith of the enitivity function S MV () will be aroun S 0 ω 0. The input ize increae with S 0, an it i well known that minimum variance controller may have poor robutne if the banwith i choen too high. It houl however be note that minimum variance control uing appropriate value of S 0 give well behave controller. It i even poible to e-tune the controller by chooing S 0 < 1.. Weighte Minimum Variance Control One common way to introuce frequency weighting i to minimize variance of the filtere output, i.e., the variance of W ()y(t), where W () typically i a low pa filter. The optimal controller equal F W MV () = 1 G() [ ] W ()H() W ( )H( ) 1. The correponing weighte enitivity i S W MV ()H() = W ( )H( ). W () The icrete time verion of thi approach i tuie from a control performance uperviion perpective in (Horch, 000), where the ue of a ample verion of the firt orer filter W () = 1 + /b, b >> µ, 1 + /µ i propoe. The correponing icrete time approach i calle e-tune minimum variance control in (Åtröm an Wittenmark, 1995). It houl, however, be note that frequency weighting give more high gain control at low frequencie, ince the low frequency iturbance are further uppree while the gain of S W MV ()H() for high frequencie till i H( )..3 A Loop Shaping Deign Aume that the noie input (t) i cale o that (t) < 1. Let alo the output y(t) be cale in uch a way that the control objective i to make y(t) < 1 for all iturbance (t) < 1. Thi can be tranlate to the frequency omain conition S(iω)H(iω) < 1, ω. (5) In (Skogeta an Potlethwaite, 1996) the imple approximate olution G()F () = H() i propoe. The frequency omain conition (5) i then atifie at the frequencie where H(iω) >> 1. The controller F () = H()/G() may have to be moifie to increae the gain at low frequencie an to obtain acceptable phae an gain margin aroun the cro-over frequency. See Section.6.4 in (Skogeta an Potlethwaite, 1996) for etail. The reulting controller will be of the form F LS () = 1 G() H() + a F LP (), where F LP () i a low-pa filter to make the controller proper. Here it i alo aume that the zero of G() are in the left-half plane. If thi i not the cae, poible untable zero have to be mirrore into the left half plane or omitte when forming 1/G() in the controller. The ame hol for time elay. For the imple noie moel () thi eign give S LS ()H() = + ω 0 S 0 (S 0 + 1) + ω 0 S 0. Thi tranfer function ha tatic gain 1, a pole at = ω 0 (1 + S 0 )/S 0 an a zero at = ω 0 S 0. A typical value of S 0 i, which mean that the banwith of S LS ()H() will be lightly le than ω 0. Thi houl be compare with the minimum variance controller which give a banwith aroun S 0 ω 0. Alo here ω 0 i the mot important tuning parameter. The purpoe of the PI-term i to obtain better iturbance rejection for low frequencie. A reaonable choice i a = ω 0 /5. A typical banwith of F LP () i 10ω ITERATIVE FEEDBACK TUNING The graient of the cot function (3) with repect to the controller parameter i given by ρ J(ρ) = N ( [W T (q)y(kt, ρ)][w T (q) y(kt, ρ)] ρ +λ[ (q)u(kt, ρ)][ (q) ) u(kt, ρ)]. ρ Conier the controller parameter upate ρ(i + 1) = ρ(i) γ i R 1 (i) ρ J(ρ(i)), where γ i i the tep-length an R(i) i a poitive efinite matrix. A goo choice i the Heian etimate

4 R(ρ) = N ( [W T (q) ρ y(kt, ρ)][w T (q) y(kt, ρ)]t ρ +λ[ (q) ) u(kt, ρ)][ (q) u(kt, ρ)]t, ρ ρ which houl be moifie to R(i) = R(ρ i ) + δi, δ > 0, to hanle ill-conitione cae. The teplength γ i i typically aroun 1. The main problem i how to fin the graient ρ y(t, ρ) an ρu(t, ρ). The key iea of IFT i to etimate thee ignal uing iterative experiment. Let u 1 (t, ρ) an y 1 (t, ρ) be the input an output from the feeback ytem in Fig. 1 with the controller F (ρ) an with reference ignal r 1 (t) = 0. Now make a firt experiment to meaure thee ignal. Then perform a econ experiment with reference ignal r (t) = K 1 y 1 (t) (where K 1 1 i a gain factor) an enote the correponing input an output by u (t, ρ) an y (t, ρ), repectively. We then have 1 y 1 (t) = 1 + GF (ρ) v 1(t) GF (ρ) y (t) = 1 + GF (ρ) [ K 1 1y 1 (t)] GF (ρ) v (t) GF (ρ) = 1 + GF (ρ) [ K GF (ρ) v 1(t)] GF (ρ) v (t) GF (ρ) K GF (ρ) [ GF (ρ) v 1(t)]. (6) The lat approximation i bae on the aumption that the gain K 1 i large enough o that the v (t) relate noie contribution can be neglecte. Now the graient (with no reference ignal) equal ρ y 1(t, ρ) = G ρ F (ρ) (1 + GF (ρ)) v 1(t) ρ = F (ρ) GF (ρ) 1 F (ρ) 1 + GF (ρ) 1 + GF (ρ) v 1(t). Comparing with (6) give the graient etimate ρ y(t, ρ) = 1 K 1 ρ F (ρ) F (ρ) y (t, ρ). The graient of the input ignal can be erive in a imilar way. For a more etaile erivation taking the tochatic propertie of the noie into account ee (Hjalmaron, 00). Our experience i that it i quite ifficult to tune controller uing IFT for iturbance rejection. The iturbance i the only excitation ignal an the objective of the controller i reject thi ignal. In the recent work, (Hilebran et al., 004), on IFT for iturbance rejection, the gain K 1 ue in the feeback experiment i optimize o a to minimize the variance of the graient etimate error, ubject to power contraint on the ignal. Typically, the cot function (3) i quite inenitive to the choice of regulator. Thi i of coure goo from a control perpective, but ifficult when tuning the controller. 4. DESIGN ISSUES When uing IFT to tune a controller for iturbance rejection it i important: To have a imple controller tructure with a few free parameter a poible. To have a cot function that give an optimal controller which atifie the pecification without uing too large input ignal. Here we have propoe the ue the imple noie moel H() = + ω 0S 0 S 0 with the two eign parameter ω 0 an S 0. Thi lea to a controller tructure of the form F LS () = 1 G() H() + a F LP (), or F W MV () = 1 G() [ W ()H() [W ( )H( ) 1]F LP (), of which the minimum variance control i a pecial cae (W = 1, F LP = 1). Here we alo have to pecify the low-pa filter F LP () an the parameter a or µ (which are cloely relate). The main unknown part in the controller i the tranfer function G. Hence, it i natural to ue the moel parameter of G a controller parameter, c.f. inirect aaptive control (Åtröm an Wittenmark, 1995). From a control perpective it i often enough to ue very imple moel, a firt or econ orer tranfer function or even jut a gain. To connect the iturbance rejection control eign with IFT an obviou caniate for cot function i J(ρ) = 1 N [(W T (q)y(kt, ρ)) + λ ( (q)u(kt, ρ)) ], with W T a firt orer low pa filter with banwith aroun ω 0. The parameter λ i more ifficult to chooe, but reflect the balance between control performance an input ize. Weighte minimum variance control correpon to λ = 0.

5 5. EXAMPLE Here we will tuy the propertie of the propoe approach for a very imple example, namely G() = b +, b = 1, H() = + 1. We will aume that only the gain b of the tranfer function i unknown. Thi i, of coure, an overimplification, but viewing the moel a only a controller eign variable we have fixe everything in the controller except the gain. The pecification mean that we nee active control up to aroun ω 0 = 1 ra/. Introuce K = 1/b, which will correpon to the controller gain. We will tuy the three controller: F LS () = 1 G() H()F LP () = K( + 1) + F MV () = 1 G() [ H() H( ) , K( + 1) 1] =, F W MV () = 1 G() [ W ()H() W ( )H( ) 1] K(11 + 0) =, W () = 1 + /10 (1 + ), an the variance an weighte variance cot J MV (K) = E{(y(kT, K)) }, J W MV (K) = E{(W T (q)y(kt, K)) }, where W T i the ample verion of W (). Notice that the gain of the weighte minimum variance controller for low frequencie i a factor of ten larger than for the two other cae. Thi i to further uppre low frequency noie an reult in a larger input ignal. In orer to illutrate the propertie of the three controller tructure an the two cot function, we calculate J MV (K) an J W MV (K) a function of K, with the ampling interval T = 0.01 an with {(t)} being white noie with variance 1. The continuou time tranfer function are approximate by Euler forwar. In Fig. the variance i plotte a function of the controller gain K for the three controller. The weighte variance cot i given in Fig. 3, an Fig. 4 contain a zoome verion of JW MV (K) for the controller tructure F W MV. In orer to evaluate the correponing input power, the variance of the elta input ignal Variance Soli: Minimum Variance Dotte: Weighte Minimum Variance Dahot: Loop Shaping Deign Fig.. The variance J MV (K) for the three controller tructure. Weighte Variance Soli: Minimum Variance Dotte: Weighte Minimum Variance Dahot: Loop Shaping Deign Fig. 3. The weighte variance J W MV (K) for the three controller tructure. Weighte Variance Fig. 4. Zoome verion of weighte variance J W MV (K) for F W MV (). Notice the cale of the y-axi. J u (K) = E{( (q)u(kt, K)) } i plotte for the three ifferent controller a a function of K in Fig. 5. The input energy i lowet for the minimum variance controller an highet for the weighte minimum variance controller a explaine in Section. The overall obervation i that all cot function are quite flat aroun the minimum an hence

6 Delta Input Variance Soli: Minimum Variance Dotte: Weighte Minimum Variance Dahot: Loop Shaping Deign be able to preict the performance of the control ytem, e.g., for conition monitoring of control loop. We have icue three very imple metho to eign a controller for iturbance rejection. The eign are bae on two uer choice, namely ω 0 an S 0, of which ω 0 i the mot important one. The moel G() of the input output tranfer function i neee to etermine the controller. One poibility i to etimate the tranfer function uing ytem ientification metho, ee e.g. (Ljung, 1987). An alternative that we have tuie i to ue iterative feeback tuning (IFT). Fig. 5. The elta input ignal variance J u (K) for the three controller tructure. Acknowlegement The author woul like to thank Håkan Hjalmaron for valuable comment on thi work an Peter Uenfelt for inutrial collaboration Variance Controller parameter c Controller parameter c 1 Fig. 6. The variance J mv (K) for the controller tructure F C = (c 1 + c )/. rather ifficult to minimize uing graient earch. The goo new are that they all are quite robut to the choice of K. Alo notice that all three controller tructure give comparable reult. Next, the variance cot function for the uncontraine controller parametrization i plotte in Fig. 6. It i rather ifficult ee the etail of thi 3-imenional plot. However, the norm of econ erivative (Heian) of the cot function i a meaure the flatne aroun the minimum, an i alo irectly relate the performance of correponing numerical optimization technique. Here the norm of the invere Heian at the minimum equal 500. Alo notice that variance i quite inenitive to variation in the controller parameter. 6. CONCLUSIONS.4 REFERENCES Åtröm, K. J. (1970). Introuction to tochatic control theory. Acaemic Pre. Åtröm, K. J. an B. Wittenmark (1995). Aaptive Control. Aion an Weley. Hilebran, R., A. Lecchini, G. Solari an M. Gever (004). Prefiltering in iterative feeback tuning: optimization of the prefilter for accuracy. IEEE Tranaction on Automatic Control 49, Hjalmaron, H. (00). Iterative feeback tuning an overview. International Journal on Aaptive Control an Signal Proceing 16, Horch, A. (000). Conition Monitoring of Control Loop. PhD thei. Department of Signal, Senor an Sytem, KTH. Lequin, O., M. Gever, M. Moberg, E. Boman an L. Triet (003). Iterative feeback tuning of pi parameter: comparion with claical tuning rule. Control Engineering Practice 11, Ljung, L. (1987). Sytem Ientification: Theory for the Uer. Prentice-Hall. Englewoo Cliff, NJ. Skogeta, S. an I. Potlethwaite (1996). Multivariable Feeback Control. John Wiley & Son. The objective of thi contribution ha been to tuy imple controller tructure for iturbance rejection. The work i inpire by inutrial application of tuning controller for iturbance rejection uing IFT. If the controller houl be tune on-line uing aaptive control or iterative feeback tuning, it i mot important to have a few free parameter a poible. It i alo important to