A NOVEL DESIGN APPROACH FOR MULTIVARIABLE QUANTITATIVE FEEDBACK DESIGN WITH TRACKING ERROR SPECIFICATIONS

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1 A OVEL DESIG APPROACH FOR MULTIVARIABLE QUATITATIVE FEEDBACK DESIG WITH TRACKIG ERROR SPECIFICATIOS Seyyed Mohammad Mahd Alav, Al Khak-Sedgh, Batool Labb Department of Electronc and Computer Engneerng, Unversty of Lmerck, Lmerck, Ireland Department of Control, K Toos Unversty of Technology, Tehran, Iran E-mal: Alav_smm@yahoocom, Sedgh@kntuacr, Labb@kntuacr Abstract: In ths paper, a novel Two-Degree-Of-Freedom (DOF) desgn procedure for Mult-Input Mult-Output Quanttatve Feedback Theory (MIMO QFT) problems wth Trackng Error Specfcatons (TES s presented In the proposed procedure, the feedback compensator desgn s separated from the pre-flter desgn, usng the model matchng approach and the unstructured uncertanty modelng concept Ths paper specally deals wth an approprate transformaton of the MIMO system to the euvalent SISO problems, whch allows easy desgn mulaton results have been provded to show the effectveness of the proposed methodology Keywords: Quanttatve Feedback Theory (QFT), Trackng Error Specfcatons (TES, Model-matchng problem I ITRODUCTIO Quanttatve feedback theory (QFT) s a Two- Degree-Of-Freedom (DOF) feedback control desgn method for uncertan plants, whch may nclude structured and unstructured uncertanty The general QFT problem s to desgn the feedback compensator and pre-flter shown n Fg to acheve the desred performance The feedback compensator reduces the effects of uncertantes and pre-flter shfts the response to the desred regon Fg 1 Two-degree-of-freedom feedback system There are two man approaches to descrbe the desred performance n QFT desgn In most QFT problems, the desred specfcatons are assumed to be only on the magntudes of the closed loop transfer functons, termed as standard approaches n ths paper These specfcatons place the response of the closed loop system between two lower and upper bounds, (Borghesan et al, 1994; Cheng, et al, 1996) In the second approach, Trackng Error Specfcatons (TES s employed (Alav et al, 004; Alav et al, 005; Boje, 00; Boje, 003; Yanv and Chat, 199) Constranng the magntude of the dfference between closed loop transfer functon and reference model to le wthn a dsk around a desred response can make a better engneerng sense, see (Boje, 00) The dea of TESs for Mult-Input Mult-Output Quanttatve Feedback Theory (MIMO QFT) problems was proposed by Yanv and Chat, (199) In order to smplfy the desgn procedure, they selected the reference model as the pre-flter and lost one degree of freedom Then, a DOF desgn method has been presented by Boje (00), for multvarable systems based on relatve trackng error concept It has been shown that the zero relatve trackng error assumpton for the nomnal plant could separate the feedback compensator desgn from the pre-flter desgn As n standard QFT approach, the feedback

2 compensator s desgned usng a senstvty constrant for each on-dagonal subsystem A nonlnear search s used to fnd the ndvdual elements of F (, whch causes dffcultes wthout necessarly mprovng the desgned pre-flter The desgned pre-flter may be complex and of hgh order In (Alav et al, 005), a novel DOF desgn technue has been proposed for MIMO QFT problems wth TESs Usng an approprate transformaton of the MIMO system to the euvalent ngle-input, ngle-output (SISO) problems, the feedback compensator desgn has been separated from the pre-flter desgn Ths transformaton resulted n approprate output dsturbance rejecton model for the feedback compensator desgn In (Alav et al, 005), t has been shown how the ndvdual elements of the preflter Transfer Functon Matrx (TFM) can be desgned for MIMO systems, usng the dea suggested by Boje, (003) In ths paper a novel DOF desgn procedure n MIMO QFT problems wth TESs s presented It s shown how the feedback compensator desgn s separated from the pre-flter desgn, usng the modelmatchng approach and the unstructured uncertanty modelng concept The elements of the feedback compensator TFM are desgned usng an approprate output dsturbance rejecton model for each euvalent SISO problems The ndvdual elements of the pre-flter TFM are also desgned va the soluton of the obtaned model-matchng problems, e, the euvalent SISO problems The paper s organzed as follows In secton II, the problem of QFT controller desgn s formulated In secton III, the MIMO problem s transformed to the SISO euvalent problems In secton IV and V, the pre-flter and feedback compensator desgn are dscussed, respectvely In secton VI, the desgn procedure s outlned and fnally n secton VII an llustratve example s carred out to show the effectveness of the proposed methodology II PROBLEM FORMULATIO Consder a gven uncertan plant wth the DOF feedback control system shown n Fg 1 The real TESs E( s defned as the magntude of dfference between the reference model M (, and the TFM of the controlled system T m (, as gven by euaton (1): E( M( Tm E( [ e ], M( [ m ], Tm [ tm ] (1) and j {, } The objectve s to desgn a feedback compensator, G (, and a pre-flter, F (, to meet the desred TESs, E d (, gven by euaton () for all plants n the regon of uncertanty M( T( F( E ( j ) d ω () E [ e ] d d T ( represents the complementary senstvty TFM of the controlled system III THE EQUIVALET SISO PROBLEMS In ths secton, the MIMO problem s transformed to the euvalent SISO problems, (Alav et al, 005) It can be shown that the closed loop TFM T m ( can be decomposed as follows: 1 Tm α T( (3) Where, 1 α1 L α1 1( ) 1 ( ) ( ) α s L α s α s M M O M α 1( α L 1 l T f, 1+ l 1 1 P, α and, l 1 + Substtutng euaton (3) nto euaton (1) results n: e j, k α e k α m t (4) Usng the Schwartz neualty on the rght-hand sde of euaton (4), the part of the dfference between reference model and complementary senstvty functon s separated from the other parts as follow: e j K, α ( e m ) t k s α ( e m ) t k s s (5) Defnng 1 (1 + l ) as the senstvty functon of the on-dagonal closed-loop subsystems and substtutng t nto α (, results n α be rewrtten as: e j L, S t Therefore, euaton (5) can ( e k ) (6) Therefore, n order to acheve the desred specfcaton (), t s suffcent to have:

3 S Or m ( e k t t e e d d ( e k for j, ) for j L, (7) ) (8) As n standard QFT, the desgn specfcatons are used to over-bound any unknown functons on the rght-hand sde of euaton (8) Usng the notaton T l (1 + l ), t ( can be rewrtten as t T f Then, euaton (8) can be expressed as follows: m T f ( γ (9) j, Where, γ ( Ed ( ( ( Ed ( + M ( ) k In euaton (9) M ( and E d ( represent the magntude of desred desgn specfcatons, e, M ( : m and Ed ( : ed Euaton (9) descrbes the MIMO problem n the form of the euvalent SISO problems The rght hand sde of (9) should be postve; therefore the followng constrant on senstvty functon wll be obtaned as the cost pad for transformaton of the MIMO problem to the euvalent SISO problem: K, Ed ( (10) mn j, ( ( ) ( )) Ed ω + M ω k Euaton (10) also ndcates a novel crteron for mnmzng the nteractons resulted from other subsystems In euaton (10), n order to attenuate the nteractons between the subsystems, the worst case of Ed ( ( ( ) ( )) Ed ω + M ω k s consdered IV PRE-FILTER DESIG In the prevous secton, t has been shown that how the MIMO QFT problem () s transformed nto the euvalent SISO problems (9) by approprate cost on the senstvty functon for each solated subsystem In fact, euaton (9) represents a model-matchng problem n the face of uncertanty It seems f M / T s upstandng soluton of the obtaned model matchng problem, but snce T ( s uncertan and unknown, f M / T s mpractcal In addton, we wll lose one degree of freedom by selectng f M / T To solve ths problem, an auxlary functon, J ( s added and f ( s suggested to desgn by the followng euaton: M f J (11) To Where, T o ( s the nomnal closed loop transfer functon The followng ponts are to be noted n usng the auxlary functon, J ( : 1 In secton V, t wll be shown that by J 1 for ω ωh the feedback compensator desgn can be separated from the pre-flter desgn whch DOF structure s preserved ω h s the performance bandwdth In addton, ths assumpton on the auxlary functon at low freuency leads to easy desgn of J ( and then pre-flter To have low order pre-flters, J ( could be desgned to cancel some poles and zeros of M To Furthermore, t s very useful for avodance of mproper realzaton of the desgned pre-flter V FEEDBACK COMPESATOR DESIG Havng the euvalent SISO problems and the structure of f (, ths secton presents the feedback compensator desgn By substtutng the multplcatve uncertanty model of T (, (e, T (1 + To ) To ) and (11) nto (9), we have: M ( 1 (1 + To ) J γ (1) By assumng that J 1for ω ωh, euaton (1) can be rewrtten as (13) and the feedback compensator s separated form the pre-flter desgn M ( To γ (13) By relaton between T o ( and o ( (the multplcatve uncertanty model of o (, e, (1 + o ) o ), n the followng form: To o (14)

4 Euatons (14) mply another constrant on the senstvty functon for each euvalent SISO problem gven by: γ (, ω ωh (15) M ( j, o nce the plant s uncertan, the over-bound of ( o ) o, for the entre uncertanty regon wthn the performance bandwdth, e, o o o for ω ωh, s preferred Thus, euaton (15) s modfed as: γ (, ω ωh (16) j, M ( o Substtutng γ nto euaton (16) and rearrangng t, results n: j, ω ω K h Ed( mn j, M ( ) ( ) + ( ( ) ( )) ω o Ed ω + M ω k (17) In relaton (17), n order to attenuate the nteractons between the subsystems, the worst case of Ed ( M + ( o( ( Ed ( + M ( ) k has been consdered In the desgn procedure, two constrant euatons (10) and (17) have been acheved on the senstvty functon of each euvalent SISO problem Euaton (10) s resulted from the transformaton of the MIMO problem nto the euvalent SISO problems Euaton (17) s resulted from the separaton of preflter desgn and feedback compensator desgn for each euvalent SISO problem wthn the performance bandwdth As n standard QFT desgn, the fnal composte permtted robust performance bounds on the loop functon l are constructed from the ntersecton of the euaton (10) and euaton (17) wthn the performance bandwdth It s obvous that satsfyng euaton (17) results n satsfyng euaton (10) Therefore, euaton (17) s the fnal senstvty constrant to mnmze the nteractons and the effects of uncertantes for each euvalent subsystem whch they are treated as an output dsturbance, D, enterng to the each euvalent SISO problems as shown n Fg From above statements, to assure the achevement of the robust performance, t s suffcent to select output dsturbance rejecton model, T D ( as gven by euaton (18) Fg Control Structure of -th euvalent SISO problem Y < TD for ω ωh (18) D Where, TD K, ωωh Ed( mn j, M + ( o( ( Ed( + M ( ) k Usng MATLAB QFT-Toolbox (Borghesan et al 1994) and based on euaton (18), the permtted robust performance bounds on the loop functon l can be generated to desgn the desred feedback compensator for each euvalent SISO problems In order to acheve robust stablty wthn [ 0, ω h ], t s suffcent to desgn the feedback compensator for each euvalent SISO problem such that the loop functon does not ntersect the crtcal o pont ( 180,0dB) A mum overshoot constrant wth margns: T l (1 + l ) δ (19) for ω ωh s also proposed n order to acheve robust stablty desgn bounds at hgher freuences δ s constant and represents the mum allowable freuency peak In order to have robust stablty at hgher freuences than performance bandwdth, the loop functon should not enter the stablty U-Contours VI DESIG PROCEDURE The desgn procedure s summarzed as follows: 1- Draw ( o ) o, for the entre uncertanty regon and fnd o wthn the performance bandwdth - Select an approprate dsturbance rejecton model such that (18) s satsfed 3- Desgn the compensator for each euvalent SISO subsystem 4- Draw ( T To ) To, for the entre uncertanty regon and fnd T o 5- Select ) (s J s such that the desgned pre-flter for each solated subsystem s (strctly) proper and euaton (1) s satsfed

5 Remarks: 1- An approprate selecton of multplcatve uncertanty model, e, and T o o, reduces the desgn conservatveness - The choce of a sutable nomnal plant, n the sense that the lowest uncertanty model s attaned, leads to lower over-degn 3- In ths approach pre-flter may be hgh order If J ( s exactly desgned such that M / To and J ( have the same poles or zeros, the pre-flter wll be lower order T l 1+ l 13 Usng MATLAB QFT-Toolbox, the permtted desgn bounds become as Fg 4 Fg 4 shows that the followng feedback compensator satsfes the performance and stablty bounds 75( s / )( s / )( s / ) 1, ( s / )( s / )( s / )( s / ) VII ILLUSTRATIVE EXAMPLE Consder the lnear tme nvarant plant: 1 k11 k1 k1 [,6] P, s + 1 k 1 k k1 [05,15] Suppose that the desred specfcatons are to acheve the steady state error less than % and settlng tme 5(sec) Fg 3 The output dsturbance rejecton model for solated subsystems Step1: desred specfcatons wth the Followng M ( and ( are satsfed E d 1 001s ( s 1+ 1)( s 5 + 1) ( 0 0 1) ( ) s + s + M s 001s 1 ( 0 0 1) ( 1 1)( 5 1) s + s + s + s + And, s + 00 ed ( s ) j s + The nomnal plant s chosen by mum value of k and ω h 5( rad / sec) s selected as the closed loop bandwdth Step: o ( s computed as 083( s 5 + 1) ( s 6 + 1) for Step3: Computng the rght hand sde of euaton (14) for each euvalent SISO loop, the output dsturbance rejecton models are selected as: 005( s 1 + s 1+ 1) TD 1, ( s ) s Fg 3 shows the selected output dsturbance rejecton model for solated subsystems If the chols envelope does not ntersect the crtcal pont (-180 o,0db), T D guarantees the robust performance and robust stablty wthn ω 0, ω ] At hgher freuences, e, [ h ω {10,0,50}, robust stablty bound has been consdered usng the constrant on the complementary senstvty matrx as Fg 4 Desgn of g ( n chols Chart usng MATLAB QFT-Toolbox wth robust performance ω { 0,,5}( rad / sec) and robust stablty bounds ω { 10,0,50}( rad / sec) Step4: The multplcatve uncertanty model of T o, s obtaned as: 01s To 1, ( s / 8 + 1) Step5: Selectng ( s ) J, ( s )( s ) J ( s j J s + 479) Euaton (1) s satsfed for each element of the transfer functon matrx Fnally, strctly proper preflter s acheved by: f ( s s + 479) ( s + 418)( s + 5)( s + 1)

6 f j s ( s + 418)( s + 0s + 004) Applyng the desgned controller to the system, fgures 5 and 6 show satsfacton of the error specfcatons and unt step responses of the controlled system for several extreme plant cases They show that the desred trackng error wll be satsfed Fg 5 Error specfcatons of the control system, desred dashed REFERECES Alav SMM, A Khak-Sedgh and B Labb (004) A ovel Desgn Approach for Trackng Error Specfcatons n SISO-QFT: A Mnmum Phase Case, Proc 1 th Medterranean Conference on Control and Automaton, MED 04, Turkey Alav SMM, A Khak-Sedgh and B Labb (005) Pre-flter Desgn for Trackng Error Specfcatons n MIMO-QFT, Proc the Jont Conference IEEE Conference on Decson and Control & European Control Conference, CDC- ECC'05, Span, pp Boje, E, (00) Multvarable Quanttatve Feedback Desgn for Trackng Error Specfcatons, Automatca, 38, Boje, E, (003) Pre-flter desgn for trackng error specfcatons n QFT, Int J Robust and onlnear Control, 13, pp Borghesan C, Y Chat and O Yanv, (1994) Quanttatve Feedback Theory Toolbox User s Gude, The MathWorks Inc Cheng, C-C, Y-K Lao and T-S Wang, (1996) Quanttatve feedback desgn of uncertan multvarable control systems, Int J Control, 65, o 3, Yanv, O and Y Chat, (199) A mplfed Multnput Mult-output Formulaton for the Quanttatve Feedback Theory, J Dynamc Systems, Measurement and Control, 114, pp Fg 6 Unt step responses of the controlled system COCLUSIOS Ths paper presents a smple desgn procedure n MIMO-QFT problems wth TESs It has been shown how the feedback compensator desgn s separated from pre-flter desgn through the concept of model matchng problem and unstructured uncertanty Approprate transformaton of MIMO system to the euvalent SISO problems s the other concept dscussed n ths paper The ndvdual elements of the pre-flter TFM are desgned by employng the model matchng soluton Heurstc algorthms such as genetc algorthm would be useful to fnd approprate pre-flter TFM The elements of the feedback compensator are also desgned usng approprate output dsturbance rejecton model An llustratve example has been provded to show the effectveness of the desgn methodology