EEL5173_sam_exam5 - EEL 5173: LINEAR SYSTEM THEORY EXAM 2...

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Unformatted text preview: EEL 5173: LINEAR SYSTEM THEORY EXAM 2 Sp 2000 Write your name id number on your exam. The exam is closed book and closed notes. You can use a two—sided formula sheet. In answering these questions, give a clear, step—by—step analysis. The three problems have equal weight. 2 (1) Given the transfer function H(s) = M , s 2 + 33 — 18 (a) Find a non—minimal state space realization (state equation and output equation). (b) Find a minimal state space realization. (c) Are the realizations of (a) and (b) stable? Controllable? Observable? (2) (3) Given a dynamic system realization ' o 1 [o] X= X+ u -1 -1 1 y= l 0 x Use a Lyapunov method to determine whether the equilibrium point is asymptoically stable, Lyapunov stable, or unstable. (b) Given the state space realization of another dynamic system _ -1 o] X= X 0 —2 Find x2 in terms of x1 . Then draw the phase plane portrait on a set of scaled axes, showing the direction of trajectories; let x1(0) and x2 (0) each take initial values between -2 and 2; also show the trajectories that begin" on the axes of the phase portrait. 3) Given a double integrator plant realization ‘ o 1 [o X: X+ ll 0 0 1J y: l 0 x (a) Assuming that both state variables are available for feedback, determine the feedback gain vector such that the closed loop system eigenvalues are -0 . 707:]‘0 . 707 (b) Explain what the " separation principle" principle? is. What is a consequence of the separation Should the observer eigenvalues be faster or slower than those obtained using pole placement? Explain. 0’ 1yva/X/1'2')(2"/H7(2 g2: A~3¥z+/9x’ > *flrX/ F/KX/ +ZK1—3K7'f/(lt y: 554% 1”“ l f M l-llf— - 5!}.3‘gnw urva——_._,-_.‘___._1~____:_.., '-_- ‘ " v ' ' mu.- \fli-Hu 7\ v! \ A4,! :0 'L "A 4fl+l 4i \—4 'Z— “:44? (M/m/‘fiv/( 1-J7’ X! (t): X, (0761 )0‘: X,(o)€ w:k@€* a )9: (magi? 36210) >92“) d‘e/Sm-éd [.0019 Kgfimeaflkfl/D u ans" 4._L §()V. ...
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EEL5173_sam_exam5 - EEL 5173: LINEAR SYSTEM THEORY EXAM 2...

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