An optimal anti-windup strategy for repetitive control systems German

An op imal an i-windup s a egy o epe i i e con ol sys ems Ge m´an A. Ramos, Ramon Cos a-Cas ell´o Abs ac — Repe i i e Con ol includes an In e nal Model wi h high gain and slow ime esponse cha ac e is ics which make i p one o he windup e ec . A solu ion o his p oblem is he inclusion o an An i-Windup compensa o . Al hough he e exis many gene al An i-Windup syn hesis me hods in li e a u e, some p oblems can a ise as a esul o a s aigh - o wa d applica ion o hem in Repe i i e Con ol. This pape p esen s he analysis and adap a ion o he Model Reco e y An i-Windup s a egy in he Repe i i e Con ol ame. Thus, an op imal LQ design is p oposed ha looks o a deadbea eco e beha iou a e sa u a ion and a global asymp o ic s abili y o he closed loop sys em. Th ough simula ion i is shown ha he p opounded scheme achie es be e acking pe o mance han o he simila LQ designs. I. INTRODUCTION As an In e nal Model P inciple (IMP) [1] based con ol s a egy, Repe i i e Con ol (RC) [2], [3] uses an In e nal Model (IM) ha cha ac e izes he signal o be acked o ejec ed. In his way, he IM o he RC p o ides in ini e o e y high gain a a gi en equency an i s ha monics. I is well known ha , in sys ems wi h ac ua o sa u a ion, a con olle wi h hese cha ac e is ics may p oduce a wind- up e ec in which he s a es o he con olle can g ow unbounded. E en i he gain is no in ini e bu high, he s a es can o e g ow signi ican ly making ha de o eco e he sys em o he linea ideal one. Some condi ions ela ed o he boundedness o he s a e o he RC wi h ac ua o sa u a ion a e s a ed in [4]. As i is known, ma ginally s able o uns able con olle s a e p one o o igina e he unbounded g owing o he con- olle s a e. Thus, using he pole analysis, i can be no ed ha he IM used in s anda d RC is ma ginally s able and hose used in High O de Repe i i e Con ol (HORC) [5], [6] ha e poles o e he uni ci cle wi h mul iplici y equal o g ea e han wo which can yield Bounded Inpu Bounded Ou pu (BIBO) uns able IM [7]. Addi ionally, he IM gene ally imposes a slow ansien esponse o he closed loop which wo sens he ac ua o sa u a ion e ec . The e o e, since he linea design o he epe i i e con ol does no include he sa u a ion in he ac ua o , i is con enien o include an An i- Windup (AW) compensa o . A ecen e iew o s anda d AW echniques can be ound in [8] and [9]. This wo k is pa ially suppo ed by he spanish Minis e io de Educaci´on y Ciencia (MEC) unde p ojec DPI2010-15110 G. A. Ramos is wi h he Depa men o Elec ical and Elec onic Enginee ing, Uni e sidad Nacional de Colombia, Bogo ´a DC, Colombia [email p o ec ed] R. Cos a-Cas ell´o is wi h he Ins i u e o Indus ial and Con ol Engi- nee ing, Uni e si a Poli `ecnica de Ca alunya, A . Diagonal, 647, 08028 Ba celona, Spain [email p o ec ed] In [9], mode n AW p oposals ha e been classi ied in wo g oups: Di ec Linea An i-Windup (DLAW) and Model Re- co e y An i-Windup (MRAW). The DLAW app oach seeks o ind an AW compensa o ha assu es speci ic pe o mance and s abili y p ope ies o he closed loop sys em. The MRAW app oach selec s he AW il e in such a way ha i makes in a ian he compensa o -plan sys em. Howe e , due o he cha ac e is ics o he RC, mos o he s anda d AW designs should no be applied s aigh o wa d since some di icul ies migh appea du ing design o implemen a ion. As a esul , i would be necessa y o adap he gene ic AW s a egies in o de o be applied in RC. In he DLAW scheme, some s a egies a e based on sol ing a Linea Ma ix Inequali y (LMI) p oblem [10]; howe e , complica ions usually a ise since he size o his LMI depends mos ly on he IM o de which is usually la ge. Thus, o he RC case, he implemen a ion o his scheme depends on whe he he LMI is compu a ionally sol able o no . Al hough he DLAW scheme allows us o ob ain an AW compensa o o o de 0, he solu ion includes elemen s ha yield a la ge numbe o on-line calcula ions, hus inc easing he compu a ional bu den. The MRAW scheme uses he model o he plan in i s s uc u e. Al hough he o de o he plan could be la ge, i is usually signi ican ly smalle han he IM o de . Fu he mo e, he p ocedu e o ind he eedback gains does no depend on he con olle dynamics, he e o e he ela ed LMI is always sol able. The compu a ional load o he MRAW scheme implemen a ion is he lowes one in compa ison wi h he o he s a egies. The p oposals in [11], [4] and [12], a e h ee examples o AW design o epe i i e con ol. In [11], an AW law is de i ed o I e a i e Lea ning Con ol (ILC) and also a ex ension o RC is b ie ly desc ibed. Howe e , he AW s a egy is de i ed o a speci ic plan and he desc ibed epe i i e con olle does no co espond o he s anda d a chi ec u e since he il e s o s abili y and obus ness a e no included. In [4], he AW scheme cancels ou he dynamics o he IM du ing sa u a ion and adds a s uc u e o shape he ansien s when he sys em sa u a es and ge s back om sa u a ion. Howe e , he IM cancella ion implies ha , in addi ion o he epe i i e con olle o de , i is necessa y o implemen an AW il e which has a leas he o de o he IM. The e o e, his scheme will be cos es ic i e since i depends on a sui able implemen a ion pla o m. The wo k in [12], can be ca ego ized as a DLAW design. The s a egy is de i ed in con inuous ime domain. I is an ex ension o RC o he gene al AW design in [10], whe e he case o delayed sys ems is desc ibed. Also in his app oach, unlike he RC R(z)E(z)U(z)Y(z) + ++ +− − + + W(z)Gx(z) Gp(z) H(z) G (z) Gc(z) D(z) Repe i i e con olle Fig. 1. Block-diag am o he epe i i e con olle plug-in app oach. design ha will be desc ibed he e, he il e s o obus ness and s abili y a e designed oge he wi h he DLAW syn hesis. In iew o he analysis abo e, he MRAW appea s as a good s a egy when aken in o accoun he compu a ional sol abili y and load in he design and implemen a ion o RC. Thus, in his wo k a MRAW scheme is p esen ed in which he eco e y o he sys em is achie ed using he app oxima ion o a deadbea ansi ion (see [13] o he deadbea concep ). The ad an age o selec ing a deadbea o e o he designs is shown as well as he design me hod. The pape is o ganized as ollows, Sec ion II p esen s he basics o he epe i i e con ol including some design issues and he s abili y condi ions. Sec ion III desc ibes he gene al MRAW scheme. Sec ion IV analyses he s abili y o he sys em and p opounds an op imal design. In Sec ion V he expe imen al esul s a e shown and inally he conclusions a e p esen ed in Sec ion VI. II. DIGITAL REPETITIVE CONTROL Digi al epe i i e con ol uses an IM which in oduces in ini e/high gain a a selec ed undamen al equency and i s ha monics [5]. This IM has he ollowing ans e unc ion: G (z) = W(z)H(z) 1−W(z)H(z),(1) whe e W(z) = z−Nand H(z)is a null-phase FIR low-pass il e in cha ge o p o ide obus ness a high equencies. Wi h H(z) = 1, IM (1) p o ides in ini e gain a equencies ω= (2k−1)2π/N, wi h k= 1,2,...,(N/2) + 1, whe e N=Tp Tsis he disc e e pe iod o he signal, Tpbeing he pe iod o he signal o be acked/ ejec ed and Tsbeing he sampling pe iod. Also, in o de o p o ide obus ness agains equency unce ain y/ a ia ion a HORC echnique has been de el- oped. This e sion o RC uses he IM (1) wi h W(z) = PM k=1 wkz−kN and PM k=1 wk= 1 [6]. Besides he IM, which assu es s eady s a e pe o mance, epe i i e con olle s include a s abilizing il e , Gx(z), which assu es closed-loop s abili y. T adi ionally, epe i i e con olle s a e implemen ed in a “plug-in” ashion, i.e. he epe i i e compensa o is used o augmen an exis ing nomi- nal con olle , Gc(z)(Figu e 1). This nominal compensa o is designed o s abilize he plan , Gp(z), and p o ides dis u bance a enua ion ac oss a b oad equency spec um. The closed-loop sys em o Figu e 1, using (1) as he IM, is s able i he ollowing condi ions a e ul illed ([14]): 1) The closed loop sys em wi hou he epe i i e con- olle is s able, i.e. Go(z) = Gc(z)Gp(z) 1+Gc(z)Gp(z)is s able. I is ad isable o design he con olle Gc(z)wi h a high enough obus ness ma gin. 2) kW(z)H(z) (1 −To(z)Gx(z)) k∞<1, whe e H(z)and Gx(z)mus be selec ed o mee his condi ion. A i ial s uc u e1which is o en used o minimum phase sys ems is ([15]): Gx(z) = k .(Go(z))−1. As a gued in [16], k mus be designed looking o a ade-o be ween obus ness and an- sien esponse. The ans e unc ion o he comple e con olle (see Figu e 1) esul s: G c(z) = U(z) E(z)= (1 + G (z)Gx(z))Gc(z)(2) Addi ionally, he ansien o con e gence ime is domi- na ed by he IM dynamics, which is in gene al much slowe han he closed loop only wi h he con olle Gc(z)[17], [18]. III. THE GENERAL MRAW SCHEME Figu e 2 shows he MRAW s uc u e, whe e Gp(z)is he plan , G c(z)is he con olle (2), sa (·)is he sa u a ion unc ion and Caw(z)is he AW compensa o . In he MRAW s a egy, he misma ch be ween he sa u a ed con ol ac ion and he non-sa u a ed one is ed back o he con olle by means o he AW compensa o , which is designed o be he model o he plan , σ1,k being he ou pu ha is used wi h his pu pose. Addi ionally, ano he eedback signal, σ2,k, is added wi h he aim o imp o ing he beha iou o he sys em when i ge s ou om sa u a ion. Thus, he design o his eedback in ol es di e en app oaches. The In e nal Model Con ol (IMC) AW s a egy [19], u ns ou o be he pa icula case whe e σ2,k = 0. This causes ha when ge ing ou o sa u a ion he sys em eco e y elies on he plan poles, which can yield a non app op ia ed pe o mance. A s a egy based on P edic i e Con ol which seeks an l2pe o mance c i e ion can be ound in [20], an op imiza ion p ocedu e using he Linea Quad a ic (LQ) app oach is p oposed in [21] and a ully nonlinea s a egy is desc ibed in [22]. In his wo k, he signal σ2,k is designed o be a linea eedback o he AW compensa o s a e. This is aimed a inding a simple linea solu ion o he AW p oblem in case o RC, also a oiding he algeb aic loop ha can be c ea ed using he eedback o he con ol ac ion misma ch, as in [21]. Fu he mo e, we analyse he bene i s o designing a deadbea beha iou in he AW il e in case o epe i i e con ol. Also, as p e iously men ioned, he ad an age o using he MRAW scheme o he epe i i e con ol case is ha he design does no depend on he IM o de . Addi ionally, as will be desc ibed la e on, he e o and con ol signals a e he ideal ones (as i he sys em had no sa u a ion in he ac ua o ), which isola es he con olle om he sa u a ion e ec s. 1The e is no p oblem wi h he imp ope ness o Gx(z)because he IM p o ides he epe i i e con olle wi h a high posi i e ela i e deg ee. +_ + + ++ +_ Caw (z) G c (z)sa () Gp(z) kekuk¯ukˆukyk σ2,k σ1,k ηk Fig. 2. The MRAW scheme in RC. A. Selec ed MRAW scheme Conside he MRAW scheme depic ed in Figu e 2. Le he disc e e- ime and asymp o ically s able linea plan Gp(z)be xk+1 =Axk+Bsa (¯uk) yk=Cxk (3) whe e sa (¯uk) =    umin ¯uk< umin ¯ukumin ≤¯uk≤umax umax ¯uk> umax (4) wi h umin <0and umax >0. The s a e-space ep esen a ion o he epe i i e con olle G c(z)is: ¯xk+1 =A c ¯xk+B cek uk=C c ¯xk+D cek (5) The AW il e Caw(z)is de ined om he plan model (3) as: χk+1 =Aχk+B(uk−sa (uk+σ2,k)) σ1,k =Cχk (6) and σ2,k =Kχk(7) whe e Kis he design pa ame e o he AW il e . I can be no iced ha while he inpu in sys em (3) is he sa u a ed con ol ac ion, he inpu in sys em (6) is he di e ence be ween he sa u a ed and non-sa u a ed con ol ac ion. This ac , oge he wi h ηk=yk+σ1,k,(8) helps o de e mine he sys em in a iance. Thus, de ining ξk=xk+χk, no icing ha ¯uk=uk+σ2,k and adding equa ions (3) wi h (6) we ha e: ξk+1 =Aξk+Buk ηk=Cξk (9) In his way, om he inpu uk o he ou pu ηk, he sys em in Figu e 3 can be seen as a Linea Time In a ian (LTI) one wi h he dynamics o he plan . This means ha ηkis he ideal plan ou pu in he sense ha i would be he plan ou pu in a sys em wi hou ac ua o sa u a ion. Fu he mo e, in he closed loop o Figu e 2, he con ol ac ion ac ion ukis he ideal con ol ac ion, i.e. uk is he same con ol signal as he one in a sys em wi hou ac ua o sa u a ion. This ac isola es he con olle om he sa u a ion e ec s, allowing us o educe he analysis o he +_ + + ++ +_ Caw(z) G c(z)sa () Gp(z) kekuk¯ukˆukyk σ2,k σ1,k ηk Fig. 3. The in a ian pa o he MRAW scheme. beha iou o he in a ian pa shown in Figu e 3, including i s in e nal s abili y. Rema k 1: In his scheme he de ia ion om he ideal pe o mance can be measu ed ough σ1,k, since σ1,k is he di e ence be ween he ideal beha iou and he plan ou pu σ1,k =ηk−yk. P oposal 1: Gi en a RC design, he smalles possible σ1,k co esponds o he bes possible pe o mance in case o sa u a ion ( he smalles de ia ion om he ideal beha iou ). The e o e, he p oblem o mula ion is o ind he design pa ame e Ksuch ha σ1,k is small enough o ob ain a good acking pe o mance. I is impo an ha he AW design aims a achie ing good acking pe o mance since RC is a echnique which is in ended o ob ain null s eady-s a e acking e o . Also due o his RC ea u e, we a e in e es ed in he sa u a ion e ec p oduced in s eady s a e e en hough i also can occu in ansien s a e. IV. MRAW PROPOSAL FOR RC: DESIGN AND STABILITY The p oposal is based on he idea o ha ing a deadbea eco e once he sys em ge s back om sa u a ion. The goal is o ob ain a Caw(z)AW il e such ha du ing sa u a ion akes he o m o he plan model, and addi ionally, when he con ol ac ion ge s back om sa u a ion, he ou pu s o Caw(z) anish in a ini e numbe o samples. To ob ain a deadbea beha iou du ing eco e y i is needed ha he eedback loop c ea ed by σk,2 eloca es all he poles o Caw(z) o z= 0, which can be done using he pole placemen p ocedu e, hus ob aining he gains ec o K. Howe e , he in e nal s abili y o he sys em mus be e i ied. A. S abili y Rema k 2: The closed loop s abili y o he sys em in Fig- u e 2 is es ablished by he design o he RC and addi ionally by he in e nal s abili y o he sys em in Figu e 3. Mo eo e , om he ac s ha : 1) we a e assuming an asymp o ically s able plan and 2) om inpu uk o ou pu ηk he sys em in Figu e 3 can be seen as a LTI one wi h he dynamics o he plan , we ha e ha he in e nal s abili y o his sys em can be es ablished analysing only he s abili y o he in e connec ion be ween he sa u a ion block and Caw(z). As a esul , in o de o check he in e nal s abili y o he sys em in Figu e 3 he ollowing condi ion should be e i ied o he sys em in equa ions (6) and (7): V(χk+1)−V(χk) + Ψ <0,(10) V(χk) = χT kPχkbeing he candida e Lyapuno unc ion and Ψ = 2(uk−sa (uk))W(sa (uk)) he sec o condi ion2 (see [23]), P > 0and W > 0being symme ic ma ices o be ound. Since Ψ≥0, ollowing he S-p ocedu e we can ind ha e i ying condi ion (10) implies V(χk+1)−V(χk)<0 and as a consequence he in e nal s abili y is es ablished. B. Op imal design In o de o ob ain a design as close as possible o he deadbea beha iou explained p e iously and also assu ing global asymp o ic s abili y, condi ion (10) can be pu oge he wi h an op imal LQ design in a LMI o m. Rema k 3: In his case he LQ design is used o ind he deadbea gain Kwhich is shown o be an op imal when he weigh ma ix Qp=TTT,Tbeing he linea ans o ma ion o he sys em (6) in o he con ollable canonical o m (see [24]). O he esul s ela ed wi h he deadbea design as an op imal LQ solu ion can be ound in [25]. Thus, he p oblem o mula ion is o ind Ksuch ha he s abili y o he in e connec ion be ween Caw, equa ions (6) and (7), and he sa u a ion block is p ese ed and addi ionally, sol e he cons ained LQ p oblem: minKP∞ k=0 χT kQpχk subjec o χk+1 =Aχk+Bσ1,k σ1,k =Kχk. The comple e p oblem can be o mula ed as an LMI minimiza ion p oblem: min γ s. .   −Q AQ BU ⋆−Q XT 1 ⋆ ⋆ −2U  <0 γI I I−Q>0   −Q ⋆ ⋆ AQ +BX10⋆ QpQ0−Qp  <0 whe e Q=QT>0,U=UT>0,γ > 0and X1=KQ. Is i wo h o say ha he e exis s some conse a i eness in he sec o condi ion Ψwhich is applied o non-linea i ies belonging o he sec o [0,1]. In gene al, his ac yields a gain K ha is an app oxima ion o he deadbea solu ion. V. SIMULATIONS RESULTS This sec ion shows he esul s ound by simula ion using he AW design p esen ed p e iously and a compa ison wi h o he op imal LQ design oge he wi h he IMC AW s a egy. 2In his case he memo yless unc ion sa () is said o belong o he sec o [0,1] since sa ( , u) [u−sa ( , u)] ≥0, which is called he sec o condi ion. A. Simula ion se up Wi h he pu pose o compa ing he AW s a egies de- sc ibed in his wo k, a linea epe i i e con olle design will be gi en. Thus, conside he ollowing disc e e- ime linea s able plan : Gp(z) = 2.146z+ 0.7585 z2−0.9945z+ 0.03498 (11) The con olle is cons uc ed om model (11), o N= 100 and sampling pe iod o Ts= 5 ms. Acco ding o Sec ion II, he ollowing design issues ha e been aken in o accoun : •Gc(z) = 0.5p o ides a e y obus inne loop. •The i s o de linea -phase FIR il e H(z) = 0.02z+ 0.96 + 0.02z−1 p o ides su icien obus ness in he p esen case. •The ac ha Gp(z)is minimum-phase allows Gx(z) = k G−1 0(z), wi h k = 0.75. Also, a second o de HORC has been designed o com- pa ison pu poses. Thus, M= 2,w1= 2 and w2=−1ha e been selec ed. Gi en he s a e space disc e e- ime sys em: (A, B, C, D), and i s eachabili y ma ix WA=B AB ··· An−1B, hen he ma ix T= T n T nA··· T nAn−1wi h T n he las ow o W−1 A. Thus, o his example: A=0.0244 −0.1251 0.0903 0.9701 , B =0.0903 0.0216, C=0 99.5450, D = 0, T=−8.1858 34.2736 2.8936 34.2736. Using he op imal MRAW app oach desc ibed in Sec ion IV-B, he pa ame e s ha ha e been ound o be a easible solu ion a e: Kdb =3.1823 24.8267, using Qp=TTT o a deadbea app oxima ion and Ks =1.0747 8.1270, using Qp= 105Ias he LQ design used o compa ison pu poses. The idea behind he las LQ design is o ind a solu ion ha keeps small he s a e o he AW compensa o . B. Simula ion esul s This sec ion analyses he sa u a ion in s eady s a e. The sa u a ion limi s ha e been chosen o be umin =−5and umax = 2. The e e ence signal kis depic ed in Figu e 4 oge he wi h he sys em plan ou pu ykand con ol signal ukwhen he se ings o HORC ha e been applied wi hou ac ua o sa u a ion. As can be seen, in his case, he epe i i e con olle success ully acks he e e ence signal. Figu e 5 shows he con ol ac ion uk, i.e. he con ol ac ion p o ided by he epe i i e con olle . The sys em wi h ac ua o sa u a ion bu wi hou AW mechanism is called SAT and he sys em wi hou sa u a ion is called Ideal. Fo he SAT scheme wo op ions a e depic ed, SAT RC and SAT HORC, o s anda d and HORC espec i ely. As can be seen, bo h SAT RC and SAT HORC con ol signals p esen an undesi able wind-up e ec , he SAT HORC being he wo se case. This phenomena is due o he pole mul iplici y in he IM o HORC which also makes i slowe han he RC one. As has been poin ed ou , when he MRAW scheme is included, he con ol signal ukco esponds wi h he ideal one. The es o he examples will be ca ied ou using only he second o de HORC. 9 9.2 9.4 9.6 9.8 10 −100 −50 0 50 100 Ou pu and e e ence Time (s) 9 9.2 9.4 9.6 9.8 10 −5 0 5 Con ol signal Time (s) k yk Fig. 4. S eady s a e e e ence k, ou pu ykand con ol signal ukwi hou sa u a ion. 012345678910 0 500 1000 1500 2000 2500 3000 3500 Con ol ac ion Con ol signal uk Time (s) 1 1.2 1.4 1.6 1.8 0 50 100 150 200 SAT RC SAT HORC Ideal Fig. 5. Con ol ac ion uk o RC and HORC wi h ac ua o sa u a ion and wi hou AW il e . Figu e 6 depic s he plan ou pu ykand sa u a ed con ol signal ˆukusing he p oposed AW design. Thus, he op imal design p oposed he e deno ed by Kdb a e compa ed wi h Ks and K= 0 which co esponds o he esponse using a s anda d LQ design and K= 0 (IMC AW) espec i ely, as desc ibed in he p e ious sec ion. I is shown ha o K= 0 he sys em eco e is oo much slow; in ac , in his example, he ideal ou pu is ha dly eached again be o e ge ing in o sa u a ion again. On he o he hand, he sa u a ed con ol ac ion app oxima es e y well he ideal one, excep when uk> umax, bu as can be seen, his is no he desi able beha iou . Fo Ks , he plan ou pu is close o he ideal one and also i is seen ha i s co esponding con ol ac ion emains sa u a ed longe han in he p e ious case. Finally, using Kdb o app oxima e a deadbea beha iou , i can be no iced ha he plan ou pu ge s close o he ideal ou pu and i s con ol ac ion emains sa u a ed longe . Figu e 7, shows he ou pu o he AW compensa o σ1,k, which, as men ioned be o e, can be seen as he de ia ion om he ideal esponse. I is shown ha du ing he ime he h ee sys ems a e in sa u a ion he de ia ion is simila ; howe e he esponse is qui e di e en once he sys ems ge ou o sa u a ion, he esponse o Kdb being he smalles one. I is wo h o no ice ha , since o his example he RC and he HORC design ha e he same acking pe o mance, σ1,k in Figu e 7 is he same in bo h cases. 9 9.2 9.4 9.6 9.8 10 −100 −50 0 50 100 Plan Ou pu yk Time (s) 9 9.2 9.4 9.6 9.8 10 −4 −2 0 2 4 Time (s) Ideal Kdb Ks K=0 Con ol signal ˆuk Fig. 6. S eady s a e sa u a ion beha iou . 9 9.2 9.4 9.6 9.8 10 0 10 20 30 40 50 60 Time (s) Kdb Ks K=0 σ1,k Fig. 7. E olu ion o σ1,k ( he de ia ion om he ideal beha iou ). VI. CONCLUSIONS In his pape , he Model Reco e y An i-Windup scheme is s udied and adap ed o he Repe i i e Con ol case. An op imal LQ design has been p oposed aimed a inding a deadbea eco e beha iou and assu ing he global asymp- o ic s abili y o he closed loop sys em. 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