Control Systems -- Final Exam (Spring 2006)
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1 6.5 Conrol Syem -- Final Eam (Spring 6 There are 5 prolem (inluding onu prolem oal poin. (p Given wo marie: (6 Compue A A e e. (6 For he differenial equaion [ ] ; y u A wih ( u( wha i y( for >? (8 For he ame equaion a in wih ( u( wha i y( for >?. (p Conider he following yem a a y u a a Under wha ondiion on a and i he yem onrollale? Under wha ondiion on a and i he yem oervale?. (5p Perform onrollale deompoiion on he following yem u Deign a onroller uk o ailize he yem. 4. (5p Given a ranfer funion ( G Realize G( wih a onrollale anonial form. Deign a ae feedak law o aign he pole a -j --j and -. Ue inegraor and amplifier o uild a model o realize he yem along wih he ae feedak. (draw a lok diagram a in Simulink. A A
2 Bonu prolem (p: Conider Z Y X M wih repreening a full lok having elemen. Aume ha Z i quare and noningular. Show ha rank(mrank(xrank(z.. Given A R n n R n R n. Suppoe ha (A i onrollale and de A. Le. A A L L Ue reul from o how ha (A L L i onrollale.
3 Soluion:. ( For A Δ ( I A 4 4 ( Define f( e g( β β f( g( f '( g'( e β β ( e β e Thu g( ( e e e β β A e f( A g( A ( e e A ( e e e. For A Δ ( I A ( ( ( j j Define f( e g( β β β f( g( e β β β f( g( e β jβ β j j f ( g( e β jβ β A e f A g A e o in e o in β β in β e o in e o in g e o in e o in ( ( in A A e o in e o in in Thu ( in e oin e o in o e oin e oin in. e o in e o in in
4 ( Given he iniial ondiion ( inpu u( he zero-inpu repone i e oin e o in o A e oin e oin y ( Ce ( [ ] in e o in e o in in [o in in o ] ( >. ( Given he iniial ondiion ( inpu u ( he zero-ae repone i A ( τ ( ( τ τ ( y Ce Bu d Du ( τ ( τ e o( τ in( τ e o( τ in( τ o( τ ( τ ( τ e o( τ in( τ e o( τ in( τ [ ] in( τ d τ ( τ ( τ e o( τ in( τ e o( τ in( τ in( τ [o( τ in( τ in( τ o( τ] dτ o( τ dτ in ( >.. ( Sine hi i a Jordan form equaion wih he eigenvalue * A B a If Conider eperaely i LI ielf a i LI only if a or. [ ] [ ] { } If are LI only if de i.e. de a a. a
5 In ummary he ondiion for he yem o e onrollale i a or a ( Similar a ( wih he eigenvalue A a C * a * a If Conider eperaely i LI only if a or ame onluion for. a a If { } are LI only if de[ ] i.e. de a a or. a In ummary he ondiion for he yem o e onrollale i a.. Given A B n p Gn p [ B AB] ρ( Gn p < n ( A B i unonrollale. Conrollale deompoiion: Le Q [ q] hen P Q A PAP A A where A A A [ ]. A B B PB where B.
6 The pair ( A B i onrollale and he equivalen yem i z Az Az Bu where z P. z A z Conroller deign: Sine de( I A ( he onrollale uyem i unale u forunaely he unonrollale uyem i ale o we an deign a feedak o make he yem o e ale. For he uyem ( A B hooe he pole a hu F hooe K AT T F BK 4 4 / T T /9 /4 4/ 4 Thu K KT 4/ 4 4/ 4 a In he original yem le K where a an e any real value ju hooe a 4/ 4 ine hey do no affe he eigenvalue hen K KP 4/ 4 8/ 4 Verify: / 4 A BK 8/ 4 / 4 / / 4 / de( I A BK ( ( (. / 4 A [ ]
7 4. ( Given he ranfer funion 4 5 G ( he ae pae realizaion of G ( i A B C [ 4 5 ] D. Verify: * * I A ( I A * * * * * * CI ( A B D [ 4 5 ] * * * * ( Given he pole j j and Δ d ( ( j( j( 5 5 α 5 α α 5. The haraerii polynomial of A i α α α de( I A de. [ ] Thu k α α k α α k α α 7 k 7 Verify: 5 5 A Bk [ 7 ] 5 5 ( The imulink model i de( I A Bk de 5 5 j j
8 r( u( Syem -4 5 y - Feedak 7 X 5. ( Given M Y Z q q m n q n Denoe he dimenion a Z R X R Y R. Sine Z i noningular rank( Z q rank [ Y Z ] rank Z q ( n q [ ] [ ] [ ] {[ y z ]} are LI. Oviouly ( ( alway hold u Alo Y Z R rank( Y Z min( q n q q n q So rank( Y Z q i. e. for eah row veor yi R zi R i... q Alo we have i i [ ] rank( X rank( X {[ i ]} {[ yi zi] } {[ i ]} n denoed a rie.. i R i... r.. are LI. Now Conider he linear ominaion of and r ai[ i ] i[ yi ] i q i r z a y z q q q i i i i i i i i i i i z (Z i noningular r i i i q a y a a ( are LI i i i i i i i i i i i r {[ yi zi] } {[ i ]} ai i {[ yi zi] } {[ i ]} Thu he linear ominaion of and equal only if and are LI rank( M #of LI row r q rank( X rank( Z.
9 n n n n A ( Given A R R R AL L I A de( I AL de de( I A ( A i.e. he eigenvalue' e of A i ha of A and. Sine L A I M( [ AL I L] For A A A A rank( M ( rank rank rank n due o de For ( A and AI AI rank( M ( rank rank ([ ] ( ([ ] rank A I rank n A I ha full row rank ine ha (A i onrollale Thu for every eigenvalue of A M( ha full row rank ( A i onrollale. L L L
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