Internal Model Control

Transcription

1 Internal Model Control Part o a et o leture note on Introdution to Robut Control by Ming T. Tham 2002 The Internal Model Prinile The Internal Model Control hiloohy relie on the Internal Model Prinile, whih tate that ontrol an be ahieved only i the ontrol ytem enaulate, either imliitly or exliitly, ome rereentation o the roe to be ontrolled. In artiular, i the ontrol heme ha been develoed baed on an exat model o the roe, then eret ontrol i theoretially oible. Conider, or examle, the ytem hown in the diagram below: Figure. Oen loo ontrol trategy A ontroller,, i ued to ontrol the roe,. Suoe i a model o. By etting to be the invere o the model o the roe,, and i, the model i an exat rereentation o the roe Then it i lear that the outut will alway be equal to the etoint. Notie that thi ideal ontrol erormane i ahieved without eedbak. What thi tell u i that i we have omlete knowledge about the roe a enaulated in the roe model being ontrolled, we an ahieve eret ontrol. It alo tell u that eedbak ontrol i neeary only when knowledge about the roe i inaurate or inomlete. o 9 Coyright 2002

2 The Strategy In ratie, however, roe-model mimath i ommon; the roe model may not be invertible and the ytem i oten aeted by unknown diturbane. Thu the above oen loo ontrol arrangement will not be able to maintain outut at etoint. Neverthele, it orm the bai or the develoment o a ontrol trategy that ha the otential to ahieve eret ontrol. Thi trategy, known a Internal Model Control ha the general truture deited in Fig.. Figure 2. Shemati o the heme In the diagram, d i an unknown diturbane aeting the ytem. The maniulated inut U i introdued to both the roe and it model. The roe outut, Y, i omared with the outut o the model, reulting in a ignal d ˆ. That i, [ ]! d U d I d i zero or examle, then d ˆ i a meaure o the dierene in behaviour between the roe and it model. I, then d ˆ i equal to the unknown diturbane. 2 o 9 Coyright 2002

3 Thu ˆ d may be regarded a the inormation that i miing in the model,, and an thereore be ued to imrove ontrol. Thi i done by ubtrating d ˆ rom the etoint R, whih i very imilar to eeting a etoint trim. The reulting ontrol ignal i given by, [ ] { [ ] } U R d! R U d Thu, [ ] R d U [ ] Sine Y U d the loed loo traner untion or the heme i thereore [ ] R d Y d [ ] or [ ] [ ] R d Y From thi loed loo exreion, we an ee that i, and i, then eret etoint traking and diturbane rejetion i ahieved. Notie that, theoretially, even i, eret diturbane rejetion an till be realied rovided. Additionally, to imrove robutne, the eet o roe model mimath hould be minimied. Sine direanie between roe and model behaviour uually our at the high requeny end o the ytem' requeny reone, a low-a ilter i uually added to attenuate the eet o roe-model mimath. Thu, the internal model ontroller i uually deigned a the invere o the roe model in erie with a low-a ilter, i.e. 3 o 9 Coyright 2002

4 . The order o the ilter i uually hoen uh that i roer, to revent exeive dierential ontrol ation. The reulting loed loo then beome Y R [ ] [ ] d Pratial Deign o Deigning an internal model ontroller i relatively eay. iven a model o the roe,, irt, ator into "invertible" and "non-invertible" omonent. The non-invertible omonent,, ontain term whih i inverted, will lead to intability and realiability roblem, e.g. term ontaining oitive zero and time-delay. Next, et and then, where i a low-a untion o aroriate order. Worked Examle iven the roe model 2ex 5, i deigned a ollow: 20 Do the atoriation where 2 20 and ex 5. Next, et to be the invere o in erie with a low a ilter, where τ i the ilter arameter and n i the order o the ilter. That i, τ n 4 o 9 Coyright 2002

5 Coyright o 9 n τ 2 20 i roer i n, and a good rule-o-thumb i to hooe τ to be twie a at a the oen loo reone. Hene, in thi examle, τ 0. Let u now look at the loe-loo roertie or the ytem, auming that. Subtituting the relevant inormation into [ ] [ ] d R Y we get [ ] d R Y Hene [ ] d R Y Or 5 ex 5 ex d R Y τ τ Thu, we an ee that the heme ha the ollowing roertie: it rovide time-delay omenation the ilter an be ued to hae both the etoint traking and diturbane rejetion reone at the teady-tate, the ontroller will give oet ree reone Imlementing the within a Conventional ramework The hiloohy an alo be ued to generate etting or onventional PI or ontroller. Note that the blok diagram in Fig. 2 an be redued to a onventional loed loo truture by irt re-arranging to the ollowing orm:

6 and then to: Thu Aume that the roe i a irt-order lu time-delay ytem. Thu, it model ha the ollowing general orm: ex τ At thi oint, we an roeed in one o two way. Serie aroximation or the time-delay We an aroximate the time-delay term uing a t -order erie, i.e. ex. 6 o 9 Coyright 2002

7 ex τ τ Thu, with τ and. and τ τ τ. τ τ Thi an be rearranged to τ τ τ Comaring thi with the ideal ontroller, Td, we an ee that the Ti orreonding roortional gain and integral time are: τ and T i τ τ Pade aroximation or the time-delay Alternatively, we an relae the time-delay with it Pade aroximation, ex In thi ae, 2 τ τ ex o 9 Coyright 2002

8 with τ and. 2 Thu τ τ. 2 τ Simliying, we obtain τ τ Again, by omaring thi againt the ideal ontroller, Td, we get Ti τ / 2 τ / 2 T i τ 2 T d τ 2 τ Senitivity and Comlementary Senitivity Funtion or Finally, let u onider the enitivity untion or the heme and omare thi with thoe or onventional eedbak ontrol. We want to do thi to ee how the hange in ontrol truture ailitate the deign o robut ontrol ytem. Reall that ε E Y R d d For, ine Y R [ ] [ ] d 8 o 9 Coyright 2002

9 Y ε d [ ] Further, uoing that, then ε and η Thereore, in the trategy, the ontroller aear linearly in the reetive untion. Comare thi with the orreonding untion or the onventional ontrol heme, ε and η Sine the enitivity untion determine erormane whilt the omlementary enitivity untion determine robutne, thi imlie that omared to the onventional ontrol heme the rovide a muh eaier ramework or the deign o robut ontrol ytem. 9 o 9 Coyright 2002