Lecture 6. Material from Various Sources, Mainly Nise Chapters 5.8 and 12. Similarity Transformations and Introduction to State-Space Control

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1 ETR Advaced Cotrol Lectre 6 aterial from Vario Sorce, ail Nie Chater.8 ad ETR ADVANCED CONTROL SEESTER, Similarit Traformatio ad Itrodctio to State-Sace Cotrol G. Hovlad Z. Dog State-Sace Deig v Freqec Deig I Freqec Domai: Y(/( w T π ζ ETR ADVANCED CONTROL SEESTER, Eamle: % overhoot ζ.9 eak time T. ec ω.9 rad/ec Amtio: w bw i the correct freqec for all tem. We are i ractice ol lacig the domiat d order ole. After the deig, we cro or figer ad hoe that higher-order ole do ot affect the te-reoe.

2 Limitatio of Freqec Domai Techiqe We wat to lace all ole, ot ol the d order domiat I freqec domai, we have three arameter: gai, comeator ole ad comeator ero. ETR ADVANCED CONTROL SEESTER, Three arameter are ot fficiet to lace all ole for tem of high order Eamle: For the th order tem Y ( R( deig a cotroller that ield % overhoot ad eak time. ec for a te-reoe. a a a a a Cotroller Deig via State-Sace (Ch..- ETR ADVANCED CONTROL SEESTER, State-ace formlatio of the cotrolled tem: A C State-ace formlatio of the cotrolled tem: A A ( r K ( A K r C

3 ETR ADVANCED CONTROL SEESTER, Figre. State-ace form for ( 6 ( ( ( ( t c R C Smmar State Sace Form ETR ADVANCED CONTROL SEESTER, hae-variable Form a a a L L O L L A atri Vector ( ( ( a k k a k a K A L L O L L ajor advatage of the haevariable ad cotroller caoical form

4 hae-variable Form ole of cotrolled tem: a L a a tem arameter to adjt ETR ADVANCED CONTROL SEESTER, ole of cotrolled tem: L ( a k ( a k ( a k Eamle.: State-Sace Cotroller Deig Give the lat ( G( ( ( deig the hae-variable feedback gai to ield 9.% overhoot ad a ettlig time T of.7 ec. ETR ADVANCED CONTROL SEESTER, Deired ole: ( 9..6 w T ζ ζw w ( Shold ot iterfere with deig reqiremet!

5 Eamle.: Deired ole ode lot G ( G ( (.( ( ζw ( ζw w w ad ETR ADVANCED CONTROL SEESTER, I geeral: e etra ole to cacel ot ero. If o cacellatio reqired, lace ole far awa from d order ole. N: Watch it atratio - fat ole reqire high feedback gai. Eamle.: Sigal-Flow Diagram G( ( ( ( How? ETR ADVANCED CONTROL SEESTER, A C [ ]

6 Eamle.: Cotroller Gai A K k ( k ( k ETR ADVANCED CONTROL SEESTER, Characteritic eqatio: ( k ( k k t match deig reqiremet: ( ζw w ( Cotroller gai b iectio: k., k.8, k.9 Large ole relt i large feedback gai ad oible atratio. Rle : It cotrait ariig from etoit G G r G G G d d d r G d G d d erfect cotrol, ot oible i ractice With d ad r(jw R ad (jw < ETR ADVANCED CONTROL SEESTER, G ( jw R < G( jw > G( jw > R R where w < w r For erfect cotrol I ractice Note: Thi i a deig reqiremet that alie to all oible cotroller.

7 Eamle.: Fial Trafer Fctio ( ( T ( (.( ζw w R What hae to the trafer fctio if we icreae the cotroller gai? ETR ADVANCED CONTROL SEESTER, How doe thi affect Rle from Lectre b? State-Sace Deig: Smmar o far ETR ADVANCED CONTROL SEESTER, For tem o hae-variable tate-ace form, we ca eail lace all ole b tate feedback. For tem of high order, o ca deig the ole a a erie of d order ole, which all meet the deig reqiremet o overhoot ad ettlig time. Alterativel, o ca e additioal ole to cacel ot ero. If there are left-over ole, lace them at freqecie where the have little iflece o deig reqiremet. Care mt be take to avoid it atratio (ref Skogetad Chater, Lectre b

8 State-ace cotrol for other model form ETR ADVANCED CONTROL SEESTER, If we ca traform the origial model to hae-variable or cotroller caoical form, we ca eail lace all ole for a cotrollable tate-ace model. After the tate feedback gai have bee deiged, the mt be traformed back to the origial tem. I the followig material, imilarit traformatio will be itrodced. Kee i mid the motivatio for thee traform: ea tate-feedback cotrol deig i hae-variable (or cotroller caoical form. Similarit traformatio ETR ADVANCED CONTROL SEESTER, Similar tem: tem of differet tate ace rereetatio, bt have the ame trafer fctio, o the ame ole / eigevale ad the ame reoe. We will td how to traform betwee imilar tem withot ig igal flow grah ad trafer fctio. Thi ca be realied b trafer matri.

9 Ereig a vector i term of aic vector State variable form the ai of the tate ace. The ame oit ca be rereeted b differet rereetatio of tate ace i differet coordiate tem. Coordiate frame ma be rotated, bt ca ot be dilated becae the coordiate tem are baed o it vector, ad the origi i ot allowed to be hifted. ETR ADVANCED CONTROL SEESTER, State-ace traformatio The ame oit: or ca be rereeted b differet coordiate tem: (, or (, : ETR ADVANCED CONTROL SEESTER, ai vector for the ace, :, The ame oit ca be rereeted i, ace a: O

10 ETR ADVANCED CONTROL SEESTER, Vector traformatio The bai vector of ace ca be rereeted b bai vector of ace: btitte Seeig, we have: ( ( : Traformatio matri Alo:, comare coefficiet: ETR ADVANCED CONTROL SEESTER, Fidig the traformatio matri, [,,, ], for -th tem btitte The colm of are the coordiate of the bai vector of the ace ereed a liear combiatio of the bai vector of the ace. t colm of i, d colm of i. We have [, ], or : : Traformatio matri [,,, ], for -th order tem

11 ETR ADVANCED CONTROL SEESTER, Eamle.9 vector traformatio Traform vector [ ] T ereed with it bai vector, [ ] T, [ ] T ad [ ] T ito a vector ereed i [ / / ] T, [ -/ / ] T, [ ] T. SOLTION: ( ( ( ( / / / / Vector i ace traform ito i the ace; ad are the ame vector ereed i differet coordiate tem. ETR ADVANCED CONTROL SEESTER, Traformig the tate eqatio Vector traformatio electio of differet et of tate variable to rereet the ame tem trafer fctio. Now, covert tate ace rereetatio with tate vector ito a tate ace rereetatio with a tate vector. D C A Let D C A D C A ltil - Similarit traformatio:. Same trafer fctio (ame ole ad ero. Same eigevale

12 ETR ADVANCED CONTROL SEESTER, Similarit traformatio o tate eqatio [ ] 7 Give a tem i the hae variable form: Traform it to a ew et of tate variable, which relate to a: SOLTION: D C A A [ ] [ ]. C, [ ] ETR ADVANCED CONTROL SEESTER, Similarit traformatio o tate eqatio [ ] 7 ( * * * (* * * * (* * * * (* * * * (* * * det( ( adj atri iverio

13 Diagoaliig a tem matri decoled tate eqatio. Eigevale ad eigevector: A i λ i i All vector i are the eigevector of A correodig to (cotat eigevale λ i. Solve for i ad λ i : ( λ I A i i ETR ADVANCED CONTROL SEESTER, det( λii A i ( λ I A i adj( λii A det( λ I A i λ i Verif eigevector oltio To be a eigevector, the traformatio A mt be co-liear with ; th i (a, i ot a eigevector; i (b,it i. A i λ i i ETR ADVANCED CONTROL SEESTER,

14 ETR ADVANCED CONTROL SEESTER, Diagoaliig a tem matri. From the tem matri A, fid all eigevale, Λdiag{λ i } ad eigevector, [,, ].. Let the traformatio matri [,, ].. From, A i λ i i we have i matri form: AΛ, where Λdiag{λ i }.. The ew tem matri i Λ - A. Sch traformatio give modal caoical form. D C A λ λ [ ],,,, ETR ADVANCED CONTROL SEESTER, Diagoaliig a tem i tate ace [ ] 8 6 det( λ λ λ λ λ λ λ A I From det(λi-a, the eigevale of A are,. ig A i λ i i, we have Solve for chooe c c For λ -: For λ -: Solve for - chooe c c [ ] Let [ ] D C A / /

15 ETR ADVANCED CONTROL SEESTER, Eamle: Similarit Traformatio [ ] 6 ETR ADVANCED CONTROL SEESTER, Cla Qetio [ ] 6 Fid the traformatio matri ch that [ ] What advatage doe tem ( have that ( doe ot? e: - A - C

16 Smmar Similarit Traformatio ETR ADVANCED CONTROL SEESTER, Give the bae vector of the two tem ad, we ca eail form the traformatio matri. The colm of cotai the bai vector of. The traformatio formed b the eigevector alog the colom of, will diagoalie the tem. For two tem decritio, the traformatio matri ca be cmberome to fid. We eed a better rocedre of fidig, tha olvig A * *A We will develo ch a rocedre, bt firt we eed to defie cotrollabilit of a tem. Cotrollable ad cotrollable tem Cotrollable: the it iflece all the tate, ad. arallel Form ETR ADVANCED CONTROL SEESTER, cotrollable: the it doe ot iflece the tate. Comare tate-cotrollabilit with it-ott cotrollabilit from Skogetad Chater!

17 Cotrollabilit b iectio Form the diagoalied tem b the traformatio matri give b the eigevector. The tem i cotrollable if the -vector of the traformed tem ( - ha o row that are ero (whe the tate are decoled, the it mt iflece all the tate a i the revio eamle. ETR ADVANCED CONTROL SEESTER, Cotrollabilit: Geeral Tet A th -order lat whoe tate eqatio i: A i comletel cotrollable if the matri ETR ADVANCED CONTROL SEESTER, C i of rak. [ A A L A ]

18 Cotrollabilit: Eamle. ETR ADVANCED CONTROL SEESTER, Covert the igal-flow diagram above to the form A C The above form ha a ecial ame Cotrollabilit: Eamle. Cotrollabilit matri C : C [ A A ] ETR ADVANCED CONTROL SEESTER, The tem i cotrollable if the row of C are liearl ideedet. Oe wa to check thi i to fid the determiat. If det(c, the the tem i cotrollable. det( C *( * (* *(* * *(* ( * I atlab: rak(ctrb(a, mt be

19 Smmar o far: tate-ace cotrol ETR ADVANCED CONTROL SEESTER, Firt check if the tem i cotrollable, either b checkig the rak of C or b traformig the tem to arallel form (whichever method i eaier. Secod, if the tem i cotrollable, traform the tem to hae-variable form or cotroller caoical form ad lace the ole b tate-feedback gai. Third, traform the cotroller gai back to the origial tem, ie. K orig K* - from cotrollabilit matri C For the origial tem C Z [ A A L A ] For the traformed tem ETR ADVANCED CONTROL SEESTER, C X [ [ A A Hece, C Z * C X - A ] I atlab: ctrb(a, * iv(ctrb(a, ( A L A L( A - term diaear ]

20 ETR ADVANCED CONTROL SEESTER, -matri that give hae-variable form If we have the origial tem ad the hae-variable form ctrb(a, * iv(ctrb(a, will traform a tem to the hae-variable form!!!! A A a a a L L O L L ETR ADVANCED CONTROL SEESTER, Cotroller deig b traformatio: E. Covert the igal-flow diagram above to the form C A

21 ETR ADVANCED CONTROL SEESTER, E.: Check Cotrollabilit ] [ A A C I thi tem tate cotrollable? Check b iectio ETR ADVANCED CONTROL SEESTER, E.: Traform to hae-variable 7 8 ( ( ( 7 * ] [ 8 7 X Z C C A A C hae-variable form

22 E.: Deired Reoe.8% overhoot ad ettlig time T. ETR ADVANCED CONTROL SEESTER, Deired ole: ( ( ζw w..7 T ζ w ( ( 6 We e the etra ole to cacel the ero State-Feedback Cotroller A K ( k (7 k (8 k ETR ADVANCED CONTROL SEESTER, Deired: ( ( ζw w ( ( 6 iectio: k, k -, k -

23 State-Feedback Cotroller: Origial K orig K * - [ - -] ETR ADVANCED CONTROL SEESTER, Y ( T ( C ( I A D ( Verif deig with deired ole rearatio for Cla Tet: DC otor Cotrol Shaft oitio ETR ADVANCED CONTROL SEESTER, Deig a tate-feedback cotroller to ield.8% overhoot ad ettlig time of T. ecod for a te-reoe Follow the rocedre i Eamle.

The Performance of Feedback Control Systems

The Performance of Feedback Control Systems The Performace of Feedbac Cotrol Sytem Objective:. Secify the meaure of erformace time-domai the firt te i the deig roce Percet overhoot / Settlig time T / Time to rie / Steady-tate error e. ut igal uch