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102_1_Lecture18

102_1_Lecture18 - UCLA Spring 2009-2010 Systems and Signals...

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UCLA Spring 2009-2010 Systems and Signals Lecture 18: Feedback June 01 2010 EE102: Systems and Signals; Spr 09-10, Lee 1
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Feedback We often have systems that don’t do quite what we’d like. They might be Sensitive to parameter variation Sensitive to external interference Nonlinear All of these can be addressed by feedback . EE102: Systems and Signals; Spr 09-10, Lee 2
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Simple Feedback System x ( t ) e ( t ) r ( t ) y ( t ) + + - Plant Sensor Output Input Error Feedback * g ( t ) * h ( t ) The plant is the system we want to control The sensor measures the output The summer subtracts the measured output from the desired input, to produce the error signal. The error signal is the input to the plant. EE102: Systems and Signals; Spr 09-10, Lee 3
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Closed Loop Transfer Function Frequency domain model + + - G ( s ) H ( s ) Plant Sensor Output Input X ( s ) Y ( s ) E ( s ) R ( s ) Error Feedback The output is Y ( s ) = ( X ( s ) - H ( s ) Y ( s )) G ( s ) so Y ( s ) + G ( s ) H ( s ) Y ( s ) = G ( s ) X ( s ) EE102: Systems and Signals; Spr 09-10, Lee 4
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The transfer function of the system is T ( s ) = Y ( s ) X ( s ) = G ( s ) 1 + G ( s ) H ( s ) The term G ( s ) H ( s ) is called the loop gain . The term 1 + G ( s ) H ( s ) is called the return di ff erence + + - G ( s ) H ( s ) Y ( s ) R ( s ) - G ( s ) H ( s ) 1 X ( s ) = 0 If we input “1” to the plant with X ( s ) = 0 , and look at the di ff erence between 1 and the value that comes back around the loop. EE102: Systems and Signals; Spr 09-10, Lee 5
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Simple Feedback Amplifier + + - x ( t ) e ( t ) r ( t ) y ( t ) β A We have a powerful amplifier, with a gain A = 1000 that varies. To reduce the sensitivity to variation, we’ll reduce the overall gain. We introduce feedback, as a constant multiplier β . The transfer function of the system is then T = A 1 + β A EE102: Systems and Signals; Spr 09-10, Lee 6
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Often we want to specify β to produce a given T . Solving for β , β = 1 A A T - 1 If we want T = 10 , then β = 0 . 099 .
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