Stability A Simple Example

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1 Stbility A Simple Exmple We wt the m to ty t x, but wid gve ome iitil peed (f(t) ). Wht will hppe? f (t) x( ) F( ) x(t) f (t) x(t) x( ) F( ) B f (t) x( ) F( ) x(t) B B f (t) x( ) F( ) x(t) B How to chrcterize differet behvior with TF? 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

2 Stbility Importce The mot bic d importt pecifictio i cotrol lyi d ythei! Utble ytem hve to be tbilized by feedbck. Utble cloed-loop ytem re uele. Wht hppe if ytem i utble? my hit mechicl/electricl top (turtio) my brek dow or bur out 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

3 Stbility Wht Will Hppe to Utble Sytem? Tcom Nrrow Bridge (July -Nov.7, 9) Wid-iduced iduced vibrtio Collped! 8 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

4 Stbility Defiitio BIBO (Bouded-Iput-Bouded-Output) tbility : Ay bouded iput geerte bouded output. u(t) Zero IC BIBO tble ytem y(t) Aymptotic tbility : Ay IC geerte y(t) covergig to zero. u(t) Give y IC Aymptotic tble ytem y(t) 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

5 Stbility Some Termiologie Give the followig trfer fuctio G ( ) ( ) d( ) Ex. ( )( ) G( ) ( )( ) Zero : root of () ( Zero of G( )) ± Pole : root of d() (Pole of G ( )), ± j Chrcteritic polyomil : d() d( ) Chrcteritic equtio : d() 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 5

6 Stbility Domi Stbility For ytem repreeted by trfer fuctio G(), ytem i BIBO tble All the pole of G() re i the ope left hlf of the complex ple. ytem i ymptoticlly tble 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 6

7 Stbility Ide of Stbility Coditio Exmple: Aymptoticl Stbility: (U()) BIBO Stbility: (y()) y &( t) αy( t) u( t), y() y Y ( ) y() αy ( t) U ( ) Y ( ) ( U ( ) y()) α ( ) [ ( )] () α t y t L Y L y e y α if { G( ) U ( ) } g( τ ) u( t τ ) dτ Bouded if Re(α)> (α)> Re( α) > ατ y( t) L [ Y ( )] L e u( t τ ) dτ t ατ ατ y( t) e u( t τ ) dτ e dτ u t t mx t 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 7

8 Stbility Remrk o Stbility Defiitio For geerl ytem (olier etc.), BIBO tbility coditio d ymptotic tbility coditio re differet. For lier time-ivrit (LTI) ytem (to which we c ue Lplce trform d we c obti trfer fuctio), the coditio hppe to be the me. I thi coure, we re itereted i oly LTI ytem, we ue imply tble to me both BIBO d ymptotic tbility. 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 8

9 Stbility Remrk o Stbility Defiitio (cot d) rgilly tble if G() h o pole i the ope RHP (Right Hlf Ple), & G() h t let oe imple pole o jω-xi, & G() h o multiple pole o jω-xi. G( ) ( )( rgilly tble ) G( ) ( ) ( NOT mrgilly tble ) Utble if ytem i either tble or mrgilly tble. G( ) ( ) G( ) ( ) ( ) 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 9

10 9 Sprig E5 - GGZ Pge Week 7-8: Stbility Repeted pole Doe mrgil tbility imply BIBO tbility? No TF: Pick Output Stbility Stbility Exmple Exmple t t U G Y L i ) ( ) ( ) ( ) ( t t L ω ω ω i ) ( t t L ω ω ω co ) ( ) ( ) ( G ) ( ) ( i ) ( U t t u L

11 Stbility Summry Let i be pole of G.. The, G i (BIBO, ymptoticlly) tble if Re( i ) < for ll i. mrgilly tble if Re( i ) for ll i, d imple root for Re( i ) utble if it i either tble or mrgilly tble. 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

12 Stbility Exmple Reviited f (t) x( ) F( ) x(t) f (t) x(t) x( ) F( ) Pole tble? Pole tble? B f (t) x( ) F( ) x(t) B B f (t) x( ) F( ) x(t) B Pole tble? Pole tble? 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

13 Stbility ore Exmple G() Stble mrgilly tble utble 5( ) ( )( ) 5( ) ( )( ) 5 ( )( ) ( )( 5 ( )( ) )????? ( )( )? 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

14 Stbility Summry Stbility for LTI ytem (BIBO d ymptoticlly) tble, mrgilly tble, utble Stbility for G() i determied by pole of G. Next Routh-Hurwitz tbility criterio to determie tbility without explicitly computig the pole of ytem. 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

15 Stbility Routh Hurwitz Criterio Thi i for LTI ytem with polyomil deomitor (without i, co, expoetil etc.) It determie if ll the root of polyomil lie i the ope LHP (left hlf-ple), or equivletly, hve egtive rel prt. It lo determie the umber of root of polyomil i the ope RHP (right hlf-ple). It doe NOT explicitly compute the root. No proof i provided i y cotrol textbook. 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 5

16 Stbility Polyomil d Aumptio Coider polyomil Q( ) L Aume If thi umptio doe ot hold, Q c be fctored where m m m Q ( ) ( ˆ ˆ L ˆ ˆ m m ) ˆ Qˆ ( ) The followig method pplie to the polyomil Q ˆ( ) 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 6

17 9 Sprig E5 - GGZ Pge 7 Week 7-8: Stbility From the give From the give polyomil polyomil Stbility Stbility Routh Arry Routh Arry m l k k c c c b b b L L

18 9 Sprig E5 - GGZ Pge 8 Week 7-8: Stbility Stbility Stbility Routh Arry ( Routh Arry ( rd rd row clcultio) row clcultio) m l k k c c c b b b L L 5 b b

19 9 Sprig E5 - GGZ Pge 9 Week 7-8: Stbility Stbility Stbility Routh Arry ( Routh Arry ( th th row clcultio) row clcultio) m l k k c c c b b b L L 5 b b b c b b b c

20 Stbility Routh-Hurwitz Criterio b c b c b c L L k l m k The umber of root i the ope right hlf-ple i equl to the umber of ig chge i the firt colum of Routh rry. 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

21 Stbility Routh-Hurwitz Exmple (ce ) Q( ) Routh rry 8 ( )( ( 6) 6 Two ig chge Two root i RHP i the firt colum j ± ) 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

22 Stbility Routh-Hurwitz Exmple (ce ) Q( ) 5 Routh rry 5 ε ε 6 ε 6 If pper i the firt colum of ozero row i Routh rry, replce it with mll poitive umber. I thi ce, Q h ome root i RHP. Two ig chge Two root i the firt colum i RHP ε ε ε < 6 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

23 Stbility Routh-Hurwitz Exmple (Ce ) Routh rry Q( ) If zero row pper i Routh rry, Q h root either o the imgiry xi or i RHP. No ig chge i the firt colum Tke derivtive of uxiliry polyomil (which i fctor of Q()) No root i RHP But ome root re o img.. xi. 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

24 Stbility Routh-Hurwitz Exmple Q( ) ( ) Fid the rge of.t. Q() ) h ll root i the left hlf ple. (Here, i deig prmeter.) Routh rry ( ) No ig chge i the firt colum > ( ) > > 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

25 Stbility Simple & Ueful Criteri for Stbility t order polyomil Q ( ) All root re i LHP d hve the me ig d order polyomil Q ( ) All root re i LHP, d hve the me ig Higher order polyomil Q( ) L All root re i LHP k ( k,, L) hve the me ig Necery Coditio 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 5

26 Stbility Routh-Hurwitz Exmple 5 Q() All root i ope LHP? 5 5 Ye / No Ye / No Ye / No ( )( ) Ye / No 5 Ye / No 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 6

27 Stbility Routh-Hurwitz ore Exmple Q( ) Routh rry ( )( ) (Auxiliry poly. i fctor of Q().) d ( ) d Derivtive of uxiliry poly. No ig chge i the firt colum No root i OPEN(!) RHP 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 7

28 Stbility Routh-Hurwitz ore Exmple Q 5 ( ) ( )( Routh rry ) 5 Derivtive of uxiliry poly. d ( ) d d ( ) d No ig chge i the firt colum No root i OPEN(!) RHP 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 8

29 Stbility Routh-Hurwitz ore Exmple Q( ) ( )( )( ) Routh rry ε ε Derivtive of uxiliry poly. d ( ) d Oe ig chge i the firt colum Oe root i OPEN(!) RHP 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 9

30 Stbility Routh-Hurwitz Summry Routh-Hurwitz tbility criterio Routh rry Routh-Hurwitz criterio i pplicble to oly polyomil (o, it i ot poible to del with expoetil, i, co etc.). Next, Routh-Hurwitz criterio i cotrol exmple 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

31 Stbility Summry (Review) Let i be pole of G. The, G i (BIBO, ymptoticlly) tble if Re( i ) < for ll i. mrgilly tble if Re( i ) for ll i, d imple root for Re( i ) utble if it i either tble or mrgilly tble. 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

32 Stbility Routh-Hurwitz Criterio (Review) b c b c b c L L k l m k The umber of root i the ope right hlf-ple i equl to the umber of ig chge i the firt colum of Routh rry. 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

33 Stbility WhyNo Proof of Routh-Hurwitz Criterio? A Elemetry Derivtio of the Routh Hurwitz Criterio ig-tzu Ho, Airuddh Dtt, d S. P. Bhttchryy IEEE Trctio o Automtic Cotrol vol., o., 998, pp mot udergrdute tudet re expoed to the Routh Hurwitz criterio i their firt itroductory cotrol coure. Thi expoure, however, i t the purely lgorithmic level i the ee tht o ttempt i mde whtoever to expli why or how uch lgorithm work. 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

34 Stbility Why Cotiue The pricipl reo for thi i tht the clicl proof of the Routh-Hurwitz criterio relie o the otio of Cuchy idexe d Sturm theorem, both of which re beyod the cope of udergrdute tudet. Routh-Hurwitz criterio h become oe of the few reult i cotrol theory tht mot cotrol egieer re compelled to ccept o fith. 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

35 Stbility Routh-Hurwitz Cotrol Exmple () 5 Deig () tht tbilize the cloed-loop ytem for the followig ce. () (cott) ( ) I P (PI (Proportiol d Itegrl) cotroller) 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 5

36 Stbility Routh-Hurwitz Cotrol Exmple () () Chrcteritic equtio 5 5 ( ) Routh rry 5 ( ) 9 ( ) > > < < 9 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 6

37 9 Sprig E5 - GGZ Pge 7 Week 7-8: Stbility 9 ) ( ) )( (9 9 > < I I P I P I P I P P P Routh rry Stbility Stbility Routh Routh-Hurwitz Cotrol Exmple () Hurwitz Cotrol Exmple () I P ) ( Chrcteritic equtio 5 ) ( I p ) ( 5 I P

38 Stbility Routh-Hurwitz Cotrol Exmple () I ( ) P, Rge of P, I From Routh rry, P I < 9 > ( P )(9 p ) 8 I > I.5 I 8 (9 8 P P ).5 The ytem will tble if be.5 -< < I P < < 8 9 (9 8 P P ) P 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 8

39 Stbility Routh-Hurwitz Cotrol Exmple (5) I ( ) P, Select P, I Routh rry? 5 8 I 8 I I I If < I, Aux. Poly : I < 6 If we elect differet P, the rge of I chge. 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 9

40 Stbility Routh-Hurwitz Cotrol Exmple (6) Wht hppe if P Auxiliry equtio I? 6 ± j () Ocilltio frequecy (rd/ec) Period π. (ec) Uit tep repoe 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

41 Stbility Routh-Hurwitz Cotrol Exmple () ( )( ) Determie the rge of tht tbilize the cloed-loop ytem. 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

42 Stbility Routh-Hurwitz Cotrol Exmple () ( )( ) ( ( )( )( ) ) 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

43 Stbility Routh-Hurwitz Cotrol Exmple () Chrcteritic equtio ( )( ) ( )( ) ( )( ) ( )( ) Sprig E5 - GGZ Week 7-8: Stbility Pge

44 Stbility Routh-Hurwitz Cotrol Exmple () Routh rry of < < 5 If 5, ocilltio frequecy i obtied by the uxiliry equtio 5 5 ± j 7 Frequecy: Period: 7 π 7 9 Sprig E5 - GGZ Week 7-8: Stbility Pge

45 Stbility Routh-Hurwitz Sythei Step : Write the cloed loop trfer fuctio Step : Obti the cloed loop ytem chrcteritic equtio Step : Geerte Routh Arry Step : Let the t colum of Routh Arry greter th zero to fid cotri equtio Step 5: Solve thee cotri equtio for cotrol prmeter 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 5

46 Stbility Routh-Hurwitz Summry Cotrol exmple for Routh-Hurwitz criterio P cotroller gi rge for tbility PI cotroller gi rge for tbility Ocilltio frequecy Chrcteritic equtio Next Root Locu 9 Sprig E5 - GGZ Week 7-8: Stbility Pge 6

Shuli s Math Problem Solving Column

Shuli s Math Problem Solving Column Shuli s Mth Problem Solvig Colum Volume, Issue Jue, 9 Edited d Authored by Shuli Sog Colordo Sprigs, Colordo shuli_sog@yhoocom Cotets Mth Trick: Metl Clcultio: b cd Mth Competitio Skill: Divisibility by