Ver. 4, 9 th March 7 Dgtal PI Controller Equatons Probably the most common tye of controller n ndustral ower electroncs s the PI (Proortonal - Integral) controller. In feld orented motor control, PI controllers are wdely used for nner current control loos. In dgtal ower suly control, t fnds alcaton n buck, boost, SEPIC, and many other ower toologes. In general, the controller may be desgned to meet secfcatons exressed n ether the tme doman or the frequency doman. me doman secfcatons tycally constran roertes of the transent resonse, such as overshoot, settlng tme, and rse tme. Frequency doman secfcaton nvolves the selecton of a sngle real ero. Ether way, the result s two real numbers corresondng to the gans n the roortonal and ntegral aths. More nformaton on transent tunng usng PI control can be found n the Control heory Fundamentals semnar, and n chater 3 of the accomanyng book []. In ths aer we wll focus on the relatonsh between the gans of contnuous tme (analogue) and dscrete tme (dgtal) PI controllers. We begn by descrbng two common confguratons of controller (seres and arallel), both of whch can be exressed n a smle ero lus ntegrator transfer functon. We then transform ths nto dscrete tme form and comare the dfference equaton wth those of ractcal seres and arallel PI mlementatons. he objectve s to fnd a ar of equatons for each confguraton whch relate the dscrete tme roortonal gans wth those of the corresondng contnuous tme orgnal.. Controller Confguratons he PI controller may be mlemented n ether of two confguratons: seres or arallel. he arallel confguraton s shown below. Fgure In ths confguraton, roortonal and ntegral gans aear n arallel aths. Concetually, the rocess of tunng the controller for transent resonse s straghtforward: one adjusts each gan n turn, blendng together dfferent amounts of roortonal and ntegral control acton, untl the desred secfcatons are met. he arallel PI controller transfer functon s (by nsecton of Fg. ) K K s K s) K () s s An alternatve, but related, confguraton s the seres confguraton (shown below) n whch the roortonal gan aears n seres wth the controller. An attracton of ths structure s that there s less nter-acton between the two gans, slghtly smlfyng the tunng rocess. Note that the seres confguraton cannot be used n alcatons where ero roortonal gan mght be requred.
Fgure he transfer functon of the seres PI controller s (Fg. ) s K K s K K s) K s s () Comarng equatons () and (), we see the relatonsh between seres and arallel controller gans s: K K ; K K K (3) Consequently, once the P & I gans for one confguraton have been found t s a smle matter to comute the gans for the other. In general, both transfer functons have the form of an ntegrator wth a sngle real ero. Adotng a somewhat neutral notaton, we can wrte ether confguraton n the form bs b s) (4) s hs form s the same as the ero lus ntegrator commonly used n ower suly loo comensaton, n whch b = and b s the ero frequency. We wll now examne how the gans are related to the dgtal PI controller.. Dscrete ransformaton here are several methods for convertng a contnuous tme transfer functon nto equvalent dscrete tme form. Among them, the best known s robably the b-lnear, or ustn transform. hs method, named after the Englsh mathematcan whose work on non-lnear systems led to ts ntroducton, can be derved from a numercal aroxmaton of the controller outut (see chater 4 n ref. []). he method nvolves relacement of each nstance of s n the orgnal transfer functon wth the followng term nvolvng and the samlng erod. Alyng the substtuton to equaton (4), we have s b b b b F ( After some re-arrangement, we can wrte the transformed equaton n the form where the numerator coeffcents are c c F ( (6)
c b b ; c b b ustn s method requres that the gans of the orgnal and transformed systems be matched. hs s usually done at =, however the PI controller has nfnte gan there snce t contans an ntegrator. We could match the gans at a dfferent frequency, however n ths case t s robably easer to neglect the ntegrators and match the numerator gans n (4) and (6). b s s b b c c c c herefore the gan of the transformed equaton (6) must be modfed by b A c c whch n ths case turns out to be /. c c F ( A (7) We now have a dscrete tme transfer functon reresentng our PI controller. he corresondng dfference equaton s found by re-arrangement and alcaton of the shftng theorem of the transform []. c ) ( ) u( A c u u( Ac Ac ) ( u u( k ) Ac Ac k ) (8) ( 3. Parallel Controller Gans A reasonable queston s to ask s: what roortonal and ntegral gans do we need to aly n order for the dscrete tme verson to behave smlarly to the contnuous tme orgnal? In the followng, we wll address ths queston to the arallel controller. he seres confguraton s dealt wth n secton 4. In order to roceed, we ll need the dfference equaton of the arallel dscrete tme PI controller. We can then fnd a relatonsh between the gans by matchng coeffcents. he arallel form dscrete tme PI controller structure s shown below. o avod confuson between the orgnal P & I controller gans and those n the dscrete tme structure, we wll refer to the latter as V and V resectvely. 3
Fgure 3 he dfference equaton can be found as follows. Notce that the dscrete ntegrator ntroduces an nternal varable nto the equaton. u( V V ( k ) ( u( V u( V V u( k ) V k ) V V V k ) u( u( k ) (9) he relatonsh between the contnuous and dscrete tme controller gans can be found by matchng coeffcents n (8) and (9). V Ac () V V V Ac A c c () Fnally, substtutng n () and () for c & c, and then for b & b, we fnd the requred dscrete tme controller gans. V A K K () V AK (3) 4. Seres Controller Gans If we are workng wth the seres PI controller, we can correlate dfference equatons coeffcents n a smlar way. o avod ambguty, we wll denote dscrete tme P & I gans W and W resectvely. Fgure 4 Proceedng as before, we have u( W ( ( W W ( k ) ( k ) u( k ) W k ) 4
u( W W W u( k ) W k ) u( u( k ) W W W k ) (4) Comarng equatons (4) and (8), we see W Ac (5) c W (6) c We can now substtute for c & c, b & b, and A, to exress the dscrete tme seres PI gans n terms of the contnuous tme PI gans. W K K (7) Equatons (7) and (8) are related by K W K (8) W K W (9) K Summary Equatons () & (3), and (7) & (8) allow us to comute equvalent dgtal PI controller gans from an analogue rototye. hs has value when a desgner wshes to substtute dgtal control acton for an analogue PI controller. However, the desgner must understand that dgtal control ntroduces a samler (A/D converter) and a reconstructon block (tycally PWM). Both nvolve scalng the nut and outut varables to match the dgtal numerc format. Furthermore, dgtal control ntroduces dynamc effects nto the loo, rncally n the form of hase lags, whch are not resent n the analogue system. In almost all ower electronc control systems, hase lag s detrmental to control and must be accounted for carefully. Further nformaton on these matters can be found n the followng references. References [] Control heory Fundamentals, R. Poley, CreateSace, 3 rd Ed., 5 [] Dgtal Control of Dynamc Systems, Frankln, Powell, & Workman, 3 rd. Ed., 997 5