EE451_Chapter3_Notes_F09

EE451_Chapter3_Notes_F09 - EE451/551: Digital Control...

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EE451/551: Digital Control Chapter 3: Modeling of Digital Control Systems
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Common Digital Control Configurations snotedinCh1 commond igitalcontrolconfigurat ions As noted in Ch 1, common digital control configurations contain both an ADC and a DAC, as shown below:
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ADC Model Assume an ideal ADC model where: outputs are equal in magnitude to inputs, no quantization error i.e., no quantization error outputs occur without delay i.e., zero conversion time sampling is perfectly uniform at a fixed rate T i.e., no time skew between samples () f t ( ) tk T fk ft = =
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DAC Model ssume an ideal DAC model where: Assume an ideal DAC model where: outputs are equal in magnitude to inputs, i.e., no quantization error outputs occur without delay i.e., zero conversion time outputs are constant over each sampling period i.e., no time skew between samples
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ZOH Model e input utput relationship of the ideal DAC is The input output relationship of the ideal DAC is assumed to be a zero order hold (ZOH), defined as: OH { } ( ) () , for 1 , 0,1,2, ZOH u k ut u kk T t kT k  →= + = "
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Transfer Function of The ZOH iven the impulse response of the ZOH we can Given the impulse response of the ZOH, we can determine the Transfer Function (TF) of the ZOH to be: e frequency response of the ZOH can now be written: ( ) 1 sT ZOH e Gs s = The frequency response of the ZOH can now be written: () 1 sinc jT ZOH eT Gj T e ω  ==  This can be expressed in polar form as: 2 j  sinc 22 ZOH ZOH TT GjT ωω ∠=
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Magnitude Response of The ZOH plot of the ZOH Magnitude response with given below A plot of the ZOH Magnitude response with given below for T = 1 s:
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Effect of Sampling on the TF of a Cascade ontrol systems often contain cascades of analog Control systems often contain cascades of analog subsystems, e.g., an actuator followed by a plant and then a sensor, as shown below for the missile control system of Ch 1:
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Effect of Sampling on the TF of a Cascade e sampled output of an analog system with a sampled The sampled output of an analog system with a sampled input can computed as shown (see pp. 60 61 of text): ) s () Ys y extension the sampled output of an analog cascade () () o r Y s H s U s Yz Hz Uz ∗∗ == By extension, the sampled output of an analog cascade separated by samplers at the input of each subsystem in e cascade can be computed as shown: the cascade can be computed as shown: 12 , o r n H s H s H s Us = " n H z H z H z = "
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Effect of Sampling on the TF of a Cascade e sampled output of an analog cascade not separated The sampled output of an analog cascade not separated by samplers at the input of each subsystem, as illustrated in example below: () Ys ust be computed as shown (see pp 60 1) : must be computed as shown (see pp. 60 61) : 1 2 1 () () and () () () () () () Ys H sXs Xs H sU s Ys H sH sU s ∗∗ == = 21 ) () () h Hs H s Us H s H U H H H H U Z { } { } 1 () (), where eq eq eq n Y z z U z z s s s ⇒= = = " ZZ
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Questions?
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In class Exercise Find the z
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This note was uploaded on 12/20/2009 for the course ECE 451 taught by Professor Staff during the Fall '09 term at Clarkson University .