Performance specified tuning of modified PID controllers

Transcription

1 20702, CJ Performance pecfe tunng of mofe PID controller It nteretng to notce that the vat majorty of controller n the nutry are proportonalntegralervatve (PID) controller or mofe PID controller [,2]. Preomnantly tune onte by mean of heurtc technque. PID controller have proven ther applcablty even they may not prove optmal control. In partcular when apple to procee P(), where mathematcal moel are abent or naequate ecrbe. A huge amount of tunng rule, orgnate from clac ZeglerNchol metho, can be foun [3], an they erve a guaranty of tablty very well, but leave other ynamc parameter unprecte an eman exctaton of the proce uner coneraton. Th paper are tunng metho, where preetermne ynamc parameter are erable, wthout pecal experment to entfy plant parameter other than DCgan an a qualfe gue of ettlng tme. PID control Coner the bac PID topology hown n Fgure, an followng the equaton U() = ( T )E() A reference change ubject to a tep wll caue an mpule a controller output U(), becaue of the preence of the ervatve term. To avo th etpont kck, the ervatve term upplemente wth a lowpa term, an mplemente a where β omewhere between 0.05 an 0.. β Th wll n general be aume when referrng to the block, an wll often the etpont kck. E() T D() U() P() Fg. Reference nput, Controlle output, Error E(), Controller output U(), Dturbance D() PID control Further attempt to avo th phenomenon are often one by movng the ervatve ntake to the feeback path. Th topology hown n fgure 2, an tratonally egnate PID control.

2 20702, CJ E() T U() D() P() Fg. 2 PID confguraton In the abence of turbance, the cloeloop tranfer functon of the bac PID control ytem, fg., gven by = ( T ) P() ( T ) P() () Smlar reult for the PID control ytem n fg.2 = ( T ) P() ( T ) P() (2) Regarng the turbance D(), t mportant to oberve that the tranfer functon, between D() an the output, n ether cae mlar gven by = P() D() ( T ) P() (3) Oppote to tratonal PIDthnkng, the PID controller n fg.2, together wth the proce P(), evnce a cloeloop tranfer functon (2), that emancpate from the proce P(), a ten to nfnty. C( ) T lm REF ( ) R( ) T T A P() cancel out n equaton (4), a reference tranfer functon REF() can be egne to fulfl preetermne performance pecfcaton, e.g. ettlng tme an overhoot. Th concluon eman Lemma : The econ umman n the enomnator of equaton (2) ha to omnate the leang wthn the banwth uner coneraton. The etale ervaton an applcaton of a reference tranfer functon llutrate n [4]. Wth the choce of a et of ouble real pole n REF(), a 3 % overhoot wll alway be preent, an the ere ettlng tme T wll lea to the recommene ettng n table. (4)

3 K p PID Large T T / 2. 7 T T / 4 Table 20702, CJ In practe noe an tme elay wll lmt the magntue of. A a ruleofthumb, loop gan recommene lower than 20, an the free choce of T preferre aroun the proce P() ettlng tme n openloop. Th n orer, not to volate Lemma n the abence of full proce knowlege. Recommenaton for the PD an PI controller cae can be foun n [4]. Other reference moel REF() may be choen, an wll lea to fferent recommenaton. IPD control To further avo harp re n the control gnal, t often erable to move both the ervatve an the proportonal acton to the feeback loop. Fg. 3 how an IPD controlle ytem E() T D() U() P() T Fg. 3 IPD confguraton In th confguraton the ntegral acton nevtable, an the cloeloop tranfer functon wth no turbance D() gven by = ( T ) P() ( T ) P() (5) Regarng the turbance D() t mportant to oberve that the tranfer functon between D() an the output entcal to that for PID an PID gven by equaton (3). The tranfer functon n (5), agan lea to the concluon, that a ten to nfnty the cloeloop tranfer functon (5) emancpate from the proce P() an a reference tranfer functon REF() a hown n (6) can be egne to fulfl preetermne performance pecfcaton, e.g. ettlng tme an overhoot. C( ) lm R( ) 2 TT T T T REF ( ) (6) A n the cae of equaton (4) th concluon eman Lemma. Dervaton of a pecfc reference moel REF() wll epen on the choce of ynamc charactertc.

4 20702, CJ Compare to the tanar econ orer tranfer functon Equaton (6) lea to ω o 2 G() = 2 2ζω o ω2 o ω o = T an 2ζω o = gvng ζ = 2 T An the ettlng tme can be foun, ung a commonly ue approxmaton [5], T 4 ζω o (2% crteron) Now a pecfc reference moel REF(), for the IPD control loop, can be egne n orer to meet a pecfc ettlng tme T. Wth 2ζω o = the above approxmaton of T lea to Inerte n zeta, th lea to T = 8 T = 2 ζ2 T = T 4 wth a choce of ζ = 2 Wth th choce of zeta a 4 % overhoot wll alway be preent, an the ere ettlng tme T obtane wth the recommene ettng n table 2. IPD Large T T / 4 T / 8 Table 2 A for the ervaton of equaton (4) the choce of T an tll ha to enure that ( T ) P() ( Lemma ) Comment on the magntue of an the free choce of T follow the coneraton aocate wth the PID tunng recommenaton n Table. Alo n the IPD cae, other reference moel REF() may be choen, an wll lea to fferent recommene tunng ettng. A an example, a choce of zeta equal to wll lea to T = T /.7 an T = T / 2.94 )Van Overchee, P. an De Moor, B. (2000). RaPID: the en of heurtc PID tunng, Preprnt of Proceeng of PID 00: IFAC Workhop on gtal control (Terraa, Span), pp

5 20702, CJ 2) O'Dwyer, Aan : Tunng rule for PI an PID control of tme elaye procee: ome recent evelopment. Proceeng of the Irh Sgnal an Sytem Conference, pp , Queen Unverty Belfat, July, ) A. O Dwyer, Hanbook of PI an PID Controller Tunng Rule (3r en). Imperal College Pre, ) Jantzen, J., an Jakoben, C. (206). Turnng PID Controller Tunng Into a Smple Coneraton of Settlng Tme. In Proc 206 European Control Conference, Aalborg, Denmark, IEEE, ISBN (USB): (5) Ogata, K.. Moern Control Engneerng (3r e). (6) Dahln E B. Degnng an tunng gtal controller. Intrument an Control Sytem, 968, 4(6): Lamba tunng.