Mechanical Systems Part A: State-Space Systems Lecture AL10. State estimation Compensator design

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1 : 46-4 Mechnicl Stem Prt : Stte-Spce Stem ectre Stte etimtion ompentor deign combintion of control l nd etimtor eprtion principle

2 Stte etimtion We re ble to plce the P rbitrril b feeding bck ll the tte: K. t thee m not ll be vilble for merement. Hence e eek to etimte them b oberving the pln inpt nd otpt, nd e the etimted tte in the control l: Uing or model of the plnt, e cold mke open-loop etimte: Kˆ : Plnt model ˆ ŷ Plnt

3 Hoever or model m not be precie e don't kno the plnt' initil condition the plnt m be bject to nmodelled ditrbnce hence the etimted tte i likel to diverge from the ctl tte We therefore introdce feedbck to tr to get the etimted otpt ŷ to trck the mered otpt Etimtor : ˆ ˆ ŷ Plnt

4 :4 Th, e form or etimte from ( ) ˆ ˆ ˆ gin vector of feedbck i n eqtion to be implemented in etimtor Plnt Etimtor ŷ ˆ ˆˆˆ

5 Etimtor dnmic Given the plnt inpt nd mered otpt, the tte etimte evolve ccording to: ˆ ˆ ( ˆ ) Or plnt model i, Sbtrcting, nd ing the otpt eqtion, give: ˆ ˆ ˆ ( ) ( ) Denote the error in the tte etimte b ~ ˆ Then the error dnmic re: ~ ( ) ~ :5 I.e., given the mption of perfect plnt model, nd no ditrbnce, thi gget tht the etimtion error ~ cn be mde to go to zero in tble nd rpid mnner b itble choice of

6 The etimted tte hold then contine to trck the plnt tte ith zero error In relit there ill be etimtion error: plnt ditrbnce model imperfection Thee error cn be kept mll if the eigenvle of the error tem ~ ( ) ~ re ell dmped nd reltivel ft compred ith the cloed-loop plnt dnmic hrcteritic polnomil for error dnmic: [ I ( )] ( ) det Deired chrcteritic polnomil: α ( ) ( p )( p ) ( e n β pn) n β Provided tem i obervble, e cn olve for o tht () α e (), ing me techniqe for K n :6

7 :7 enefit of oberver c.f. The oltion for the etimtor gin mtri i trivil if the eqtion re in oberver cnonicl form The tem mtri for the error eqtion i then: [ ] o o Thi i till in oberver c.f., nd h the chr. pol ( ) ( ) ( ) ) ( ompre ith deired chr. pol: ) ( β β β α e Hence, b inpection: β, β, β

8 ckermnn' forml gin, ckermnn' forml cn be ed to effectivel implement the proce of: trnformtion of n obervble tem to oberver cnonicl form deign of etimtor gin b inpection trnformtion bck to originl tte :8 α e ( ) O MT: Specif deired etimtor pole, dep» cker(,, dep)» plce(,, dep)

9 Regltor ith etimted tte Plnt: Plnt :9 ontrol l: Kˆ Etimtor: K ˆ ˆ ˆ ( ˆ) Feedbck compentor ŷ Etimtor

10 Stte eqtion for cloed-loop tem Plnt: Etimtor: ontrol l:, ˆ ˆ Kˆ ombining thee eqtion: ( ˆ ) Kˆ ˆ ˆ Kˆ ˆ ( ) Th the cloed-loop tte eqtion re: ˆ [ ] K K ˆ ˆ :

11 Thee eqtion m lo be ritten in term of the etimtion error ~ ˆ. We hd: Kˆ Sbtrcting: ˆ ˆ Kˆ ~ ( ) ~ ( ˆ ) Sbtitting ˆ ~ in the firt eqtion ield: ( K) Kˆ With thi lterntive et of tte vrible the cloed-loop tem eqtion re: : ~ K ~ [ ] K ~

12 K K ~ ~ [ ] ~ The cloed-loop chrcteritic polnomil i th: I K K ( ) det( I c. l. ) det I hich i block tringlr. Hence: ( ) det( K) det( I ) K ( ) ( ) Th, the pole of the tem conit of the pole obtined b fll tte feedbck throgh K, together ith the etimtor pole determined b the election of I.e. the control l nd the etimtor cn be deigned independentl of ech other : Seprtion Principle

13 : Emple: regltion of ter level in to-tnk tem Eqilibrim inflo Q, Q eqil m tnk height H, H Reglte height in econd tnk Ditrbnce flo reqire reglting control flo Stte vrible re devition in tnk height from eqil m ineried model: vlve flo hed diff. / reitnce eqil m flo ctrl flo eqil m flo ditrbnce flo Q Q H R H R Q Q q Q q

14 Eercie: ho tht eqil m flo ctrl flo eqil m flo ditrbnce flo :4 H (R R )Q R Q Q Q H R Q R Q nd tht H R H R R R R R R For R R Q Q q Q q Y ( ) U ( ) (.6)(.8) Y ( ) W ( ) (.6)(.8)

15 :5 Spec: Reqired cloed-loop time contnt re τ., τ.5 ; i.e., deired chr. pol i 5 ) 5)( ( α c ontrol l: Demontrte ckermnn' forml: [ ] [ ] [ ] ; ; - - [ ] ) ( K c α α () c [ ] Hence: K [ 55]

16 :6 ; ; O O O Etimtor: Tr etim. pole controller pole; i.e., deired chr. pol i 4 5 4) )( ( α e Demontrte ckermnn' forml: α ) ( O e [ ] α e() 47 5 Hence

17 omplete tem :7 Eqivlent feedbck compentor: ˆ ( K )ˆ Kˆ Trnfer fnction for compentor: U ( ) H ( ) K Y ( ) 9(.7) 7 5 Plnt K Feedbck compentor ( I K ) ˆ ŷ Etimtor regltor i eentill eqivlent to led compentor ith dditionl filtering

18 licl control vie: root loc W() :8 U() plnt regltor 9(.7) 7 5 Y() 4 ( Dr root loc for.7 6) (5.4) Root oc (.8)(.6) K compentor pole Img i compentor zero cloed-loop pole for K 9 - plnt pole Rel i

19 licl control vie: freqenc repone OTF in horthnd nottion: 5.(.7) [.9,9.] (.8)(.6) :9 Phe (deg) Mgnitde (d) To: Y() To: Y() compented ode Digrm From: GM d ncompented PM increed to 67º Freqenc (rd/ec)

20 Performnce: repone to initil condition..8 Repone to initil condition See tnk.m, im_tnk.mdl ^ ^ ^ : Stte.6.4. Stte etimte converge in bence of ditrbnce Time ()

21 Performnce: repone to tep ditrbnce Step Repone From: : Open loop loed loop mplitde To: ted-tte error: gget need for integrl control -- ee lter Q H Q R H R Q. 5 5 Time (ec)

22 Repone to rndom ple ditrbnce : O tnk level tnk level control flo ditrbnce flo reltivel poor etimte of -- might need to look t choice of etimtor pole i etimted ell