Lec_06

Lec_06 - Lecture 6 Closed-Loop Systems Closed-Loop Systems...

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1 Lecture 6: Closed-Loop Systems Closed-Loop Systems Graphical Derivation Method Examples Reading: Chapter 5 Closed-Loop System Based on the open-loop system (feedforward path) Output can be used (feedback path) to modify the input Feedback of continuous output c ( t ) to continuous input e ( t ) Or feedback of discrete output c ( kT ) to discrete input e ( kT ) Signal processing in the feedback path (feedback gain) Sampler Data hold e ( t ) e ( kT ) ¹ m ( t ) Plant c ( t ) Sampler (A/D) c ( kT ) Digital Filter m ( kT )

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2 An Example of Closed-Loop System Reference input r ( t ), typically desired for the output c(t ) to track Continuous-time output c(t) is fed back to form the error signal e ( t ) Assume trivial digital controller D ( z )=1 Relation between the sampled input r ( kT ) and sampled output c ( kT )? Sampler Data hold e ( t ) e ( kT ) ¹ e ( t ) c ( t ) G p ( s ) H ( s ) r ( t ) + ¡ Example: Closed-Loop Transfer Function C ( s ) G p ( s ) H ( s ) R ( s ) + ¡ E ( s ) T 1 ¡ e ¡ T s s ¹ E ( s ) E ¤ ( s ) Denote G ( s ) = 1 ¡ e ¡ sT s G p ( s ) C ( z ) R ( z ) = G ( z ) 1+ GH ( z ) where GH ( z ) = Z [ G ( s ) H ( s )] 6 = G ( z ) ¢ H ( z )!
3 Example: A Failed Attempt C ( s ) G p ( s ) H ( s ) R ( s ) + ¡ E ( s ) T 1 ¡ e ¡ T s s ¹ E ( s ) E ¤ ( s ) E ( s ) = R ( s ) ¡ H ( s ) C ( s ) C ( s ) = G ( s ) E ¤

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