3. A Review of Some Existing AW (BT, CT) Algorithms

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1 3. A Revew of Some Exstng AW (BT, CT) Algothms In ths secton, some typcal ant-wndp algothms wll be descbed. As the soltons fo bmpless and condtoned tansfe ae smla to those fo ant-wndp, the pesented algothms can be sed fo bmpless and condtoned tansfe except that the lmtaton (LIM) shold be eplaced by a swtch, as shown n Fges 2.12 and AW Algothms fo PID Contolles Lnea Feedback AW Algothms In lnea feedback ant-wndp algothms, feedback fom the dffeence between and s connected to the npt of the ntegal tem lnealy (thogh a constant gan 1/ a ) (Fge 2.12). The lnea feedback ant-wndp algothm s also known as tackng o back calclaton technqe [Åstöm and Rndqwst, 1989] [Hans et al., 1987] [Rndqwst, 1990], poposed by [Fetk and Ross, 1967]. Among lnea feedback ant-wndp algothms, the so-called obseve appoach, condtonng technqe and ncemental algothm wll be evewed Obseve Appoach Ths appoach was fst pesented n [Åstöm and Wttenmak, 1984]. We evew t now n the famewok of PID. The PID contolle descbed by the eqaton 2.3 can also be descbed by the followng state-space eqatons: Dx = e (3.1) 19

2 = T x + e y d (3.2) = LIM( ) (3.3) whee s Dx the npt of the ntegal tem, y d s the otpt of D-tem descbed by d () = Ys () Y s d st s T d 1+ N (3.4) The ntepetaton of the wndp phenomenon s that the state of the contolle does not coespond to the contol sgnal beng fed to the pocess [Åstöm and Rndqwst, 1989] [Hans, 1989] [Moa, 1993] [Campo et al., ACC, 1989] [Rndqwst, 1990] [Zheng, 1994]. To estmate coectly the state when, an obseve s ntodced. The coecton of the state(s) s popotonal to the dffeence between and thogh a gan L: x = e+ L( ) (3.5) e = T x e y e + d (3.6) = LIM( ) (3.7) Ths s the same as lnea feedback ant-wndp algothm by takng a =1/L. The poblem whch stll exsts s how to choose L. Fg epesents ant-wndp obseve appoach solton. 20

3 Fg Obseve AW appoach Condtonng Technqe Ths appoach was fst pesented n [Hans, 1980]. Eqatons (3.1) to (3.3) can be ewtten as: x = e= w y (3.8) = ( ) T x + w y y d (3.9) = LIM( ) (3.10) To ndestand the condtonng technqe, t s mpotant to explan a concept of the ealsable efeence [Åstöm and Rndqwst, 1989] [Hans, 1980] [Hans 1989] [Hans et al., 1987] [Hans and Peng, 1992] [Henotte, 1989] [Moa, 1993] [Campo et al., C&CE, 1989]. The ealsable efeence (w ) s sch that when appled to the contolle nstead of the efeence (w), t eslts n the contol vaable () whch s eqal to the pocess npt ( ) and ths the lmtaton s not actvated. 21

4 By above defnton, (3.8) and (3.9) yelds Dx= w y (3.11) ( ) = T x + w y y d (3.12) As w s not avalable a po, we have to se w to pdate as (3.9). Howeve, we can se w (compted a posteo) nstead of w to pdate x n ode to make contolle state consstent: Dx= w y (3.13) = ( ) T x + w y y d (3.14) = LIM( ) (3.15) ths, by sbtactng (3.14) fom (3.12), we obtan w w = + (3.16) When nsetng (3.16) nto (3.13) to (3.15), we get x = w y+ (3.17) = ( ) T x + w y y d (3.18) = LIM( ) (3.19) 22

5 Ths s a specal case of lnea feedback ant-wndp algothm wth a = (Fge 3.2). Some extensons of the condtonng technqe ae gven n [Hans and Peng, 1992] [Hans and Peng, 1991] [Walgama et al. 1992] [Walgama and Stenby, 1990]. Fg 3.2. Ant-wndp solton when sng condtonng technqe Incemental Algothm The ncemental algothm s vey often sed to pevent wndp [Åstöm and Rndqwst, 1989] [Hans, 1989]. It s also a elatvely smple method to be ncopoated n a dgtal contolle. Fge 3.3 shows a typcal dscete-tme mplementaton. Hee (k) s pdated as k ( ) = ( k 1) + k ( ) (3.20) The dffeence (k) s nomally small (except at the nstant of efeence change), so tacks vey qckly. 23

6 Fg Incemental algothm (dscete tme wth samplng tme T S ) The dscete tme ealsaton of the ncemental algothm can be expessed n anothe way as n Fg Fg Dscete-tme eqvalent of the ncemental algothm 24

7 It can be tansfomed nto ts contnos-tme eqvalent (Fg. 2.12) by deceasng samplng tme T s : T T z S T T e 1 st S S (3.21) lm T lm sts TS T e = 0 TS 0 S T T S (3.22) Ths, the contnos-tme ealsaton can be epesented as a specal case of the lnea feedback ant-wndp algothm wth a 0 (Fge 3.5). The dffeence between all pesented lnea ant-wndp algothms lays n dffeent vale of chosen a (stength of the feedback fom lmtaton back to ntegato). Obseve appoach does not defne exact vale of a, condtonng technqe gves a = and the eslt of the ncemental algothm s a 0 (small vale of a ). Fg 3.5. Contnos tme eqvalent of the ncemental algothm Hee we have to pont ot that n all fthe expements contnos-tme eqvalent of ncemental algothm s sed (see Fg. 3.5). Vale of a =0.01 s sed except n cases whee dffeent vale of a ae specally gven. Theefoe, expements show eslts of 25

8 contnos-tme ealsaton of the ncemental algothm and n some cases eslts can dvege fom ognal dscete-tme ealsaton Non-Lnea Feedback AW Algothms The epesentatves of ths concept ae condtonal ntegaton methods [Åstöm and Rndqwst, 1989] [Hansson et al. 1993] [Moa, 1993] [Rndqwst, 1990] [Shnskey, 1988], whch can be classfed as follows [Hansson et al. 1993]: a) Stop ntegatng when the contolle satates b) Stop ntegatng when the contol eo s lage c) Stop ntegatng when contolle satates and the contol eo has the same sgn as the contol sgnal d) Lmt the ntegato vale e) Stop ntegatng and assgn a pedetemned o compted vale to the ntegato state when a specfed condton s te. In the famewok of ths wok we chose the case a) whch can be epesented by Fg. 3.6 and eqaton (3.23). In the followng text t wll be efeed to as condtonal ntegaton method. Fg Condtonal ntegaton AW method 26

9 e e; = = 0 ; (3.23) The compaato C swtches the npt of the ntegal tem accodng to and. If the contolle woks n the lnea egon ( =), the npt of the ntegal tem (e ) s connected to e, othewse ( ), the compaato swtches e to 0 (the pdatng of the ntegal tem stops) AW Algothms fo Genealsed PID Contolles Fg and eqaton (3.24) epesent the solton of ant-wndp fo genealsed PID contolles. Note that the dffeence fom lnea ant-wndp algothms fo PID contolles s that ant-wndp feedback does not feed only ntegal tem. Ths s de to the fact that D pat also contans some knd of memoy (flte 1+sT f ). Fg AW method fo genealsed PID contolle 27

10 1 st U st st W d 1 st st st Y d st d = β + + γ γ 1+ f 1 + f a 1 st a2 1+ stf ( U U ) (3.24) Note that a1 and/o a2 can be dynamc tansfe fncton(s) AW fo Contolles Wth Geneal Ratonal Tansfe Fncton Fg and eqaton (3.25) epesent the solton of ant-wndp fo contolles wth geneal atonal tansfe fncton. To the scheme n Fg (secton 2.2) we add an ant-wndp feedback (tansfe fncton N 3 (s)/d 3 (s)) fom the lmtaton to contolle otpt. Fg Ant-wndp scheme fo contolles wth geneal atonal tansfe fncton U () () () () () N s D s W N s 1 2 D s Y N3 s = D s () ( U U ) (3.25) 28

11 3.4. AW fo State-Space Contolles Soltons fo state-space contolles ae geneally coveed by obseve appoach and condtonng technqe (sectons and , whee we made a devaton fo PID contolle). Ognally, both methods descbe a desgn of ant-wndp fo statespace contolles. Fg. 3.9 shows the ealsaton of ant-wndp algothm fo the statespace contolle, whee matx G epesents an ant-wndp feedback fom lmtaton to contolle states. Fg Ant-wndp fo state-space contolles ( ) Cx = Ax+ B Ey G = Cx+ Dw Fy (3.26) 29

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