EEL5173 Fall 2009 Lecture _17

EEL5173 Fall 2009 Lecture _17 - Ex(continue U(s u(t 1 s2...

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Ex. (continue) 2 1 s U ( s ) u ( t ) Y ( s ) y ( t ) [] x y t u x dt x d r r r 0 1 ) ( 1 0 0 0 1 0 = + = (a) Design a state feedback controller so that the feedback control system has a time constant of 0.5 second and a damping ratio of 0.707. (b) Design a close-loop observer that is critically damped and with a time constant one half that of the control system.
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(b) Closed-loop Observer: ( ) [] [ ] () [] 16 8 1 0 0 1 1 0 0 1 0 0 16 8 1 1 1 1 = = = W T a Q a T r r β (a) State feedback controller: () [] [] () [] 4 8 0 1 1 0 1 0 0 1 0 0 8 4 1 1 1 1 = = = S T a K a r r
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The Plant with a Closed-loop Observer 2 1 s u ( t ) y ( t ) s 1 s 1 v 1 v 2 8 16 +
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The Plant with a Closed-loop Observer and a State Feedback Controller s 1 s 1 v 1 v 2 2 1 s u ( t ) y ( t ) 8 16 + 8 4 + r ( t )
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dt v d r v r B C A ʃ u ( t ) plant Q y(t) + ) ( ˆ t y D 5. Combined Observer-Controller Compensator + r ( t ) K The controller The observer The complete control system.
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() x QC Bu v QC A dt v d Bu x A dt x d r r r r r + + = + = . ., . , z x v e i v x z Let r r r r r r = = () ( ) () Br z BK x BK A Br z x BK x A v K r B x A dt x d + + = + = + = r r r r r r r r [] z QC A dt v d dt x d dt z d r r r r = = Dr z DK x DK C v K r D x C Du x C y + + = + = + = r r r r r ) ( ) (
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x d r x r B C-DK A-BK ʃ y(t) r ( t ) BK dt z d r z r A-QC ʃ An equivalent state-space model for the complete control system. DK
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EEL5173 Fall 2009 Lecture _17 - Ex(continue U(s u(t 1 s2...

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