Naioal Tsig Hua Uiversiy Dearme of Power Mechaical Egieerig Mid-Term Eamiaio 3 November 11.5 Hours Liear Sysem Theory (Secio B o Secio E) [11PME 51] This aer coais eigh quesios You may aswer he quesios i Eglish or Madari. The maimum for his aer is 1 marks A formula shee ad s-rasforms are rovided.
1. Cosider a liear SISO sysem Assume ha he imulse resose mari is g(, τ). Wrie he iu-ouu relaioshi bewee ouu y() ad iu u() for: (a) liear sysem. (4 marks) (b) liear causal sysem. (4 marks) (c) liear, causal, ad relaed a sysem. (4 marks) (d) liear, causal, relaed a, ad ime-ivaria sysem. (4 marks). Give he elecric ework: Node equaios: dv v v i d 4 dv i v i d Loo equaio: c1 c1 i.5 L c.5 L c s dil vc vc 1 d (a) Wrie he sae equaio for he ework. (4 marks) (b) Is his sysem corollable? (5 marks) 3. Cosider a sae equaio as 1 1 u y 1 (1) (a) Is he sae equaio corollable ad observable? (5 marks) (b) Reduce he sysem o a corollable ad observable sae equaio. (5 marks) (c) Prove ha he resul of 3(a) ad Equaio (1) are zero-sae equivalece. (5 marks) (d) Give u =.5 (a ui se wih a amliude.5), deermie he soluio of () of he reduced form i 3(b), wih zero iiial codiio. (5 marks) 4. Give he observabiliy mari, rove ha he observabiliy roery of (A, C) is ivaria uder ay equivalece rasform. (8 marks)
5. Cosider a 3 4 mari A as 1 1 A 1 4 3 () (a) Fid he dimesio of he rage of A ad he ull sace of A. (9 marks) (b) Fid oe se of basis for he ull sace of A. Make sure ad rove ha all he vecors of he se of basis are i he ull sace. (9 marks) 6. Cosider a 3 3 mari A as 1 A 1 (3) Fid a o-sigular mari Q such ha, where is Jorda caoical form of A. (9 marks) 7. If he rasfer fucio of a liear sysem is G s s 1 1 s s 1 1 s s1 Fid a dyamic equaio (sae-sace realisaio) for he sysem. (1 marks) 8. Give he liear ime-ivaria sysem A Bu y CDu (4) usig equivalece rasform ad similariy rasform o fid he eressio for {,,, }: A Bu y C Du (5) such ha (4) ad (5) are equivale. (8 marks) 3
APPENDICES APPENDIX 1 Table of Lalace (s) Trasform Pairs Lalace Trasform s ( s) ( ) e d Fucio of ime () 1 () Ui Imulse 1 / s 1 [ie H()] Ui Se 1/ s [ie H()] Ui Ram 1 / s 1 / 1! a 1 / s a e / s a 1 a e / / s si s / s cos 1 1! 1/ e si 1/ s s s d d ; d 1/ 1/ e si d 1/ s s s / s s 1 ; 1 1 d ; d 1 ; 1; cos 1 1 d e sid ; a ; d d 1 ; 1 s() s () D() where D d / d s s () s() () D () where D d / d 1 () () i i ss s D i1 s s () s / s imes imes D where D d / d D whe all i/c's are zero ( ) d d whe all i/c's are zero d ( s) / d( s) - () (Derivaive of he Lalace Trasform of a Fucio) (s + a) e a () (Firs Shifig Theorem) e as (s) H( a) ( a) (Secod Shifig, or Traslaio, Theorem) s s (Fial Value Theorem) s s s s (Iiial Value Theorem) 4
Aedi Liear Algebra ad Corol Equaios Iverse: 1 A adj( A) / A Adjoi: adj(a) = [moc(a)] T T T 1 Pseudo Iverse: A A AA ; dim A m; m T 1 T Pseudo Iverse: A AA A; dim A m; m Eoeial of a Mari: e Aa ˆ 1a e a e ; a e Ai ˆ, ja ˆ for Aa e i j e for i j A A( ) Soluio for () via he 'Classical' Mehod: e e Bu Soluio for () via Lalace Trasforms: Erzberger s codiios: I BB A A m I BB B m, m d 1 1 1 1 si A si A Bu s Trasfer mari G(s) bewee y(s) ad u(s): y s Gs CsI 1 A B u s Sae-sace realisaio: ˆ 1 1 r1 r Gs s s 1s s r1s r d s d s 1 r 1 r u B r A: rr ˆ y 1 r 1 r G u C: qr D 5
Corollable sae equaio: c Ac A1 c Bc u c A c c y C C A c c c c C A Bu c c c c y C Du c c B Observable sae equaio: o Ao o Bo u A A B o 1 o o o o yco Du o A Bu o o o o y C Du o o 6