Page 1 Pulse Transfer Function Page 2 Pulse Transfer Function Analysis of Discrete-Time Systems 1. The sampling process 2. z-transform 3. Properties of z-transforms 4. Analysis of open-loop and closed-loop discrete time systems 5. Design of discrete-time controllers Page 3 Pulse Transfer Function Analysis of Discrete-Time Systems 1. The sampling process 2. z-transform 3. Properties of z-transforms 4. Analysis of open-loop and closed-loop discrete time systems 5. Design of discrete-time controllers n Continuous signal and its discrete-time representation with different sampling rates 3 t (sec) y * 6 9 12 3 t (sec) 1 y 7 9 11 5 3 t (sec) 1 y * 6 9 12 y y * Continuous signal Disontinuous signal Dt t = nT t y * Dt t = nT t y * t = nT t y * ( ) nT area Impulse Ñ º n From the response of a real sampler to the response of an ideal impulse sample (a) (b) (c) (a) (b) (c) T = 1 sec T = 3 sec Page 4 Pulse Transfer Function Analysis of Discrete-Time Systems 1. The sampling process 2. z-transform 3. Properties of z-transforms 4. Analysis of open-loop and closed-loop discrete time systems 5. Design of discrete-time controllers n Continuous signal and its discrete-time representation with different sampling rates 3 t (sec) y * 6 9 12 3 t (sec) 1 y 7 9 11 5 3 t (sec) 1 y * 6 9 12 y y * Continuous signal Disontinuous signal Dt t = nT t y * Dt t = nT t y * t = nT t y * ( ) nT area Impulse Ñ º n From the response of a real sampler to the response of an ideal impulse sample (a) (b) (c) (a) (b) (c) T = 1 sec T = 3 sec The Sampling Process 1. At sampling times, strength of impulse is equal to value of input signal. 2. Between sampling times, it is zero. ( ) å ¥ = = + + + + = 0 n * nT) - (t y(nT) ... nT) - (t y(nT) .... T) - (t y(T) (t) y(0) t y d d d d Impulse Sampler y (t) y * nT)] - (t [ y(nT) (s) y 0 n * d L å ¥ = = nTs - 0 n * e y(nT) (s) y å ¥ = = or Laplacing Page 5 Pulse Transfer Function Analysis of Discrete-Time Systems 1. The sampling process 2. z-transform 3. Properties of z-transforms 4. Analysis of open-loop and closed-loop discrete time systems 5. Design of discrete-time controllers n Continuous signal and its discrete-time representation with different sampling rates 3 t (sec) y * 6 9 12 3 t (sec) 1 y 7 9 11 5 3 t (sec) 1 y * 6 9 12 y y * Continuous signal Disontinuous signal Dt t = nT t y * Dt t = nT t y * t = nT t y * ( ) nT area Impulse Ñ º n From the response of a real sampler to the response of an ideal impulse sample (a) (b) (c) (a) (b) (c) T = 1 sec T = 3 sec The Sampling Process 1. At sampling times, strength of impulse is equal to value of input signal. 2. Between sampling times, it is zero. ( ) å ¥ = = + + + + = 0 n * nT) - (t y(nT) ... nT) - (t y(nT) .... T) - (t y(T) (t) y(0) t y d d d d Impulse Sampler y (t) y * nT)] - (t [ y(nT) (s) y 0 n * d L å ¥ = = nTs - 0 n * e y(nT) (s) y å ¥ = = or Laplacing The Hold Process : From Discrete to Continuous Time n Zero – Order Hold : m * (t) m (t) Continuous output discrete impulses Hold Device ( ) t d t m * (t) T m (t) 1 n Transfer Function : Response of an impulse input : d (t) ( ) Ts - -Ts e - 1 s 1 s e - s 1 H(s) = = 1)T (n t nT for (t) m m(t) * + < £ =Read More

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