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Lec_07

# Lec_07 - Lecture 7 Modified z-Transform Modified...

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1 Lecture 7: Modified z -Transform Modified z -transform Reading: 4.4-4.6 of the textbook Systems with Time Delay Sampler Data hold e ( t ) e ( kT ) ¹ e ( t ) Plant c ( t ) e ¡ t 0 s delay by t 0 E ( s ) E ¤ ( s ) T G ( s ) C ( s ) e ¡ t 0 s Sampler Data hold e ( t ) e ( kT ) ¹ m ( t ) Plant c ( t ) Digital Filter m ( kT ) E ( s ) T 1 ¡ e ¡ T s s G p ( s ) C ( s ) e ¡ t 0 s E ( z ) D ( z ) Delay due to processor computation Delay inherent in plant delay by t 0

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2 Systems with Time Delay Sampler Data hold e ( t ) e ( kT ) ¹ e ( t ) c ( t ) G p ( s ) e ¡ t 0 s Sampler c ( kT ) delay by t 0 E ( s ) E ¤ ( s ) T G ( s ) C ( s ) e ¡ t 0 s C ( s ) = e ¡ t 0 s G ( s ) E ¤ ( s ) ) C ( z ) = Z [ e ¡ t 0 s G ( s )] E ( z ) If t 0 is an integer multiple of sample period: t 0 = nT Z [ e ¡ t 0 s G ( s )] = z ¡ n G ( z ) If t 0 is not an integer multiple of sample period: t 0 = nT + ¢ T , 0 < ¢ < 1 Z [ e ¡ t 0 s G ( s )] = z ¡ n Z £ e ¡ ¢ Ts G ( s ) ¤ delayed z -transform of G ( s ) Delayed z -Transform Given a continuous-time signal e ( t ) with Laplace transform E ( s ), its delayed z-transform is defined as the z -transform of e ( t ) delayed by a fraction of the sample period: (0 · ¢ · 1) E ( z; ¢) = Z £ e
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Lec_07 - Lecture 7 Modified z-Transform Modified...

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