Reformulation of Disturbance Observer Design in the Frequency Domain

Transcription

1 roceedig of the 5th Mediterraea Coferece o Cotrol & Automatio, July 7-9, 7, Athe - Greece T9-5 Reformulatio of Diturbace Oberver Deig i the Frequecy Domai M. Garcia-Saz, D. Caaju, J. Elo Automatic Cotrol ad Computer Sciece Departmet. ublic Uiverity of Navarra. Campu Arroadia, 36 amploa, Spai Abtract Thi paper preet a imple cotroller deig methodology i the frequecy domai for diturbace rejectio pecificatio. It capture the mai characteritic at low ad high frequecie of a Diturbace Oberver Baed Cotrol, ad olve ome of their pricipal drawback i term of tability ad performace pecificatio, quatifyig the iheret limitatio for diturbace rejectio by mea of the available phae. The ew techique i applied i the cotrol ytem of two applicatio: a hard dik drive ad a ervo-ytem poitio cotrol. I. INTRODUCTION The Diturbace Oberver Baed Cotrol (DOBC) i a particular implemetatio for diturbace rejectio pecificatio of the well-kow Iteral Model Cotrol []. The DOBC ca be a ueful techique to improve the behaviour of the cotrol ytem whe exteral diturbace ad model ucertaity are preet. I the lat few decade it ha bee widely tudied. Oe of the firt DOBC publicatio applied it to cotrol a bledig proce []. Some year later the DOBC wa aalyed i a cotrol aeropace ceario, with pacecraft or laucher [3-6]; i motio cotrol [7-]; i hard dik drive [- 5]; i robot maipulator [6-]; ad i o-liear ytem [-5]. Thi paper aalye the DOBC tructure ad it procedure i Sectio II. Sectio III reduce that tructure to a two-degree-of-freedom diagram. Sectio IV preet a imple ew methodology. Sectio V how two illutrative example. Sectio VI ummarize the mai cocluio. II. DISTURBANCE OBSERVER BASED CONTROL Fig. preet the block diagram cofiguratio of the DOBC, where i the real plat to be cotrolled, i the plat model without ucertaity (the o-called omial plat), G i the cotroller ad Q i a low pa filter. Fig. how a typical magitude diagram of Q. To make feaible the cotrol cheme, the relative degree of Q (the differece betwee the order of the deomiator ad the umerator) hall be larger tha that of the omial model. [Q - mut be trictly proper]. The DOBC deig procedure uually coit of two tage. Firt the mai cotroller G i deiged to regulate without keepig i mid either the ucertaitie or the exteral diturbace. Later, i a ecod tage the DOBC tructure i deiged. The omial plat ad the Q filter are choe to atteuate the ifluece of both, the diturbace d at the plat iput i the output y, ad the ucertaity i the model. From Fig. the followig equatio for the output y ad the cotrol igal u are obtaied: [ G ] Q [ ] { A r B C d } y where, A G; B [ GQ]; C [ Q]. [ G ] Q [ ] { A r B C d } u where, A G; B [ GQ]; C [ GQ]. Now the expreio ad are aalyed for the limit value of the Q filter. At low frequecie, the filter Q, ad the G G r y where the ifluece of the diturbace d ad the ucertaity i the model diappear. Simultaeouly, at high frequecie, the filter Q, ad the y G G r G G d G where the ifluece of the DOBC diappear. (3) (4)

2 roceedig of the 5th Mediterraea Coferece o Cotrol & Automatio, July 7-9, 7, Athe - Greece T9-5 Referece r Error e Cotroller ug G u Diturbace d hyical plat Output y Referece r refilter F Error e Cotroller u G Diturbace d hyical plat Output y d Filter Q / Diturbace Oberver Seor oie Figure 3. Bode diagram for Filter Q Seor oie Figure. Block diagram of DOBC Q O the other had it i importat to otice that due to Q, which i a low pa filter, the pre-filter F -ee (6)- at low frequecy ha i geeral a value differet from oe. Therefore the DOBC tructure i oly valid for variable trackig pecificatio if [ G] >> whe. - IV. NEW CONTROLLER DESIGN FORMULATION - -3 A. Cotroller deig The DOBC tructure eed a low pa filter Q, whoe typical equatio i Frequecy (rad/ec) Figure. Bode diagram for Filter Q Q ( ) ( ) τ τ Replacig that expreio i (5) it i obtaied, (9) III. REDUCTION OF THE DOBC TO A DOF Every cotrol ytem ca be reduced to a degree of freedom (DOF) diagram. Fig. 3 preet the DOF diagram equivalet to the DOBC tructure of Fig., where the cotroller G ad the pre-filter F have the expreio: G Q [ Q ] G (5) F G G Q (6) From Fig. 3 the followig equatio for the output y ad the cotrol igal u are obtaied: G [ G F r G d ] y G [ G F r G G d ] u Replacig i (7) ad (8) the expreio of G ad F give i (5) ad (6), the equatio ad are obtaied repectively. The aalyi of the ytem repreeted i Fig. 3 i equivalet to the previou aalyi i Sectio II for the ytem of Fig.. (7) (8) G ( τ ) ( ) τ G ( τ ) ( τ ) ( τ ) ( τ ) where the deomiator of the ecod member are -order polyomial i without a idepedet term. Now, if the expreio i aalyed for the limit value of the Q filter i the ame way that i the previou Sectio, the followig reult are obtaied: At low frequecie, Q ad G, ad replacig expreio ad (6) i (7) the equatio (3) i obtaied. At high frequecie, Q ad G G, ad replacig expreio ad (6) i (7) the expreio (4) i obtaied. I thi aalyi it i importat to otice that the lat term i ha ot ay ifluece i the objective at high ad low frequecy ad the ca be removed. If Q i for itace a firt order filter, equatio take the followig imple expreio: τ G G τ where the cacadig term i a uit-gai I (proportioal itegral) cotroller. I a more geeral form: τ i G G Π i τ i where typically or. I additio, for referece trackig pecificatio a prefilter F i alo eeded. I thi cae ome robut cotrol techique uch a Quatitative Feedback Theory (QFT)

3 roceedig of the 5th Mediterraea Coferece o Cotrol & Automatio, July 7-9, 7, Athe - Greece T9-5 ca be ued to deig F [6]. The deig requiremet hould iclude F( ) ad F( ). The at low frequecie, G ad F. Replacig thee reult i (7), it i obtaied: r y (3) where the ifluece i the output y of the diturbace d at the plat iput, ad the ucertaity i the model, have bee removed. Simultaeouly at high frequecie, G G ad F, which agree with the origial DOBC a well. B. Feedback maximizatio To achieve a high rejectio of exteral diturbace the ew cotroller G hould take high gai value at low frequecie, miimizig the eitivity, ad keepig imilar value to the origial G at the ret of frequecie. Fig. 4 how the magitude diagram of (τ)/(τ) which i the mai elemet of the ew formulatio. Fig. 5 preet the magitude diagram of the cotroller G ad G, howig a effect of feedback maximizatio at low frequecy, maitaiig imilar value at medium ad high frequecie. That i to ay: for ω /τ the magitude of G icreae. For /τ ω the magitude of G i imilar tha G. C. hae pealizatio. Limitatio i performace Fig. 6 how the Bode diagram (magitude ad phae) of (τ)/(τ). The itroductio of thi term alo implie a phae pealizatio. Both the phae at low frequecy ad the phae margi of G are lower tha thoe of G. I Table I the phae of the term (τ)/(τ) ad the phae pealizatio itroduced i the cotroller G i repreeted, depedig o differet value of frequecy. Thi mea that the itroductio of (τ)/(τ) elemet i the ew cotroller, which are the oe that improve the diturbace rejectio, i limited for the amout of phae available. Thi tatemet wa brilliatly tudied by Bode [7] ad hi follower Freudeberg ad Looze [8] i the wellkow itegral: I the ew formulatio thee apect are directly cotemplated while i the DOBC techique are omitted, beig able to looe the tability. D. Deig procedure. Deig of the cotroller G for the omial plat with ay claical cotrol method (for example QFT [6]).. Repreetatio of the Bode diagram of G. Calculatio of the gai cro-over frequecy ω gc ad the phae margi. 3. Deig of the cotroller G accordig to. The parameter i i elected depedig o the gai to icreae at low frequecie ad both, the phae at low frequecie, ad the available phae margi. Typically τ i ca be elected o that ω /τ i i a decade lower tha the gai cro-over frequecy ω gc. I thi way, a feedback maximizatio effect i produced at frequecie lower tha ω/τ i. I the frequecy ω gc /τ i there i a pealizatio i the phae margi of oly 5.7º for every (τ i )/(τ i ) elemet. 4. Deig of the pre-filter F with the followig requiremet: F( ), F( ) ad trackig pecificatio. E. Implemetatio iue Havig a previou cotroller G already workig, the ew formulatio ca be implemeted i two way:. By multiplyig the curret cotroller G with the τ i imple tructure Π -thi i uually poible i τ i i ay idutrial cotrol ytem-;. By addig the claical DOBC tructure, with - τ Π. i ad Q i τ i τ ( jω) d log S ω (4) which i the o-called water-bed for table pole or, - ω /τ (τ)/(τ) p ( j ) d ω log S p i i (5) for the p i utable pole, ad u ( jω e ) d log H u φ( ω ) log coth du π du (6) where u log (ω/ω ), H i a proper, table ad miimum phae trafer fuctio, ad S i the eitivity fuctio /τ Frequecy (rad/ec) Figure 4. Magitude diagram for (τ)/(τ) TABLE I. HASE ANALYSIS OF (τ)/(τ). Frequecy hae ealizatio of G /τ 45º 45º /τ 63.4º 6.6º /τ 84.3º 5.7º

4 roceedig of the 5th Mediterraea Coferece o Cotrol & Automatio, July 7-9, 7, Athe - Greece T9-5 hae (deg) G G ω /τ Figure 5. Bode diagram for G ad G - - Frequecy (rad/ec) Figure 6. Bode diagram for (τ)/(τ) V. EXAMLES To illutrate the ew methodology, ad to compare it with the claical DOBC tructure, two example are preeted: a hard dik drive ytem [] ad a ervoytem poitio cotrol [9]. A. Hard dik drive The firt example coider a track-followig cotrol of a hard dik drive []. The goal i to maitai the read/write head o the track i the preece of exteral diturbace. The expreio of the DOBC tructure give i [], accordig to Fig., are the followig: k ω (7) ξ ω ω where k.8;.65 ; 7 Q 3 (.84 ) 585 G (8) (9) The expreio of the real plat iclude parametric ucertaity: k ω ξω ω where k [.7,.9]; [65, 75] ; [.55,.75] By applyig the ew formulatio,. The cotroller G previouly deiged i (8).. Bode diagram of G -ee Fig. 7-. Calculatio of the gai cro-over frequecy ω gc 33 rad/. 3. Deig of the cotroller G accordig to. The parameter τ i elected o that ω /τ i at leat a decade lower tha the gai cro-over frequecy ω gc. The fial value are τ.4 ec ad. Therefore,.4 G G.4 4. Thi tep i ot eceary becaue there i ot trackig pecificatio. Therefore F. The behaviour of the cotrol ytem of the hard dik drive i imulated with the two methodologie: the DOBC preeted i [] -ee (7) to - ad the ew techique -ee -. Fig. 8 how the reult with both methodologie. The ew formulatio improve the rejectio of the exteral diturbace. B. Servo-ytem poitio cotrol The ecod example coider a ervo-ytem poitio cotrol [9]. The mai objective i to reject the exteral diturbace that affect the poitio cotrol ytem. The expreio of the DOBC tructure give i [9], accordig to Fig., are: where ( 8) umg G (3) deg um G x x.848 x de G x x x 8 Q (4)

5 roceedig of the 5th Mediterraea Coferece o Cotrol & Automatio, July 7-9, 7, Athe - Greece T9-5 4 while for the previou DOBC techique the deired referece caot be achieved becaue F( ).5. hae (deg) x -4 Frequecy (rad/ec) Figure 7. G Figure 8. Output repoe y uder a exteral diturbace at t. ec: DOBC (dahed lie), New methodology (olid lie) By applyig the ew formulatio,. The cotroller G previouly deiged i (3).. Bode diagram of G -ee Fig. 9-. Calculatio of the gai cro-over frequecy ω gc 5 rad/. 3. Deig of the cotroller G accordig to. The parameter τ i elected o that ω /τ i at leat a decade lower tha the gai cro-over frequecy ω gc. The fial value are τ. ec ad. Therefore,. G G (5). 4. Thi tep i ot eceary becaue there i ot trackig pecificatio. Therefore F. The behaviour of the cotrol ytem i imulated with the two methodologie: the DOBC preeted i [9] -ee to (4)- ad the ew techique -ee (5)-. Fig. preet the magitude diagram of the cotroller G accordig to -the ew oe-, G accordig to (5) - the previou oe [9]- ad G, howig a effect of feedback maximizatio at low frequecy, maitaiig imilar value at medium ad high frequecie. Fig. how the reult with both methodologie. The ew formulatio improve agai the rejectio of the exteral diturbace. I additio, if a trackig referece problem i tudied, the output repoe y for the ew formulatio ca follow the referece [error (t ) ] VI. CONCLUSIONS Thi paper ha itroduced a imple cotroller deig methodology for diturbace rejectio pecificatio that capture the mai characteritic of a Diturbace Oberver Baed Cotrol at low ad high frequecie. The ew techique reformulate previou DOBC method i the followig apect: - It i impler to deig ad eaier to implemet. - It i ot eceary to get the ivere of the omial plat. Thi tep could caue mathematical or computatioal problem. - The pre-filter F ca be pecifically deiged to achieve trackig requiremet, avoidig ome teady tate drawback of the DOBC. - The relative degree of G ad the pre-filter F are idepedet of the omial plat, while i the DOBC are related. - The ew methodology i doe i the frequecy domai, dealig directly with tability, oie rejectio ad performace. - Fially, with the ew formulatio it i eaier to quatify the relatio betwee the amout of diturbace rejectio required (low gai frequecy) ad the phyical limitatio of the cotrol ytem (available phae). The paper alo applied the method to a hard dik drive ad a ervo-ytem poitio cotrol. hae (deg) Frequecy (rad/ec) Figure 9. G G accordig to (5) G accordig to G Frequecy (rad/ec) Figure. Compeator

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