EE3530-4 - Chapter 4 Basic Properties of Feedback...

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Chapter 4: Basic Properties of Feedback Sensitivity of Open Loop Control Systems An open loop control system operates without feedback and gen- erates the actuating signal in response of an input signal. y Controller D ol Plant G Input shaping H r U W R If the plant G ( s ) involves variations Δ G , then it induces a relative error of the overall system T ( s ) = G ( s ) D ol ( s ) H r ( s ) Δ T ( s ) = Δ G ( s ) D ol ( s ) H r ( s ) Δ T T = Δ G ( s ) D ol ( s ) H r ( s ) G ( s ) D ol ( s ) H r ( s ) = Δ G ( s ) G ( s ) that is the same as the relative error of the original plant. 1
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Why Use Feedback? Under modeling: The mathematical model can not describe com- plete dynamics of the process, or it is too complex to be useful. Uncertain parameters: Some physical parameters may change with respect to operating conditions, or difficult to measure ac- curately. External disturbance: Physical process can’t be insolated com- pletely from outside world, and disturbance may affect the output performance. The use of feedback suppresses uncertainties due to modeling, parameter variations, and external disturbances. The price: feedback devices. 2
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Example: Operational amplifier. R R 1 2 V V i o V e A + - I I 1 2 Usually A >> 1 and is very uncertain: V o = - AV e . With connection of R 1 and R 2 we have a feedback system. Since A >> 1, V i - V e R 1 V e - V o R 2 . Solving V e gives V e = V i R 1 + V o R 2 1 R 1 + 1 R 2 - 1 = R 1 R 2 R 1 + R 2 V i R 1 + V o R 2 . n 1 R 1 R 1 R 2 R 1 + R 2 - A 1 R 2 - - - - - 6 V i V e V o 3
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Hence the gain from V i to V o is V o V i = 1 R 1 - AR 1 R 2 R 1 + R 2 1 + - AR 2 R 1 + R 2 = - R 2 R 1 + R 1 + R 2 A . If R 2 = 10 R 1 , and A = 1 , 000 10 , 000, then V o V i = - (9 . 881 9 . 989) . The relative error is only 1 . 09% after using feedback. Without feedback, the error is 900%. The use of feedback eliminates significantly the uncertainty. 4
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Basic Equations of Feedback Control General Form H9018 H11001 H11002 Y V Controller D cl Plant G Sensor H y Input shaping H r H9018 H11001 R W H11001 H9018 H11001 H11001 u Unity Feedback: if H y = H r H9018 H11001 H11002 W R H9018 H11001 H11001 Y V H9018 H11001 H11001 u Controller D Plant G can always convert to this form by defining D = D cl H y , ˜ R = H r R/H y Hence Y = DG 1 + DG R + G 1 + DG W - DG 1 + DG V and the error equation is E ( s ) = R ( s ) - Y ( s ) = 1 1 + DG R - G 1 + DG W + DG 1 + DG V Sensitivity Function S ( s ) = 1 1 + D ( s ) G ( s ) Complementary Sensitivity Function T ( s ) = D ( s ) G ( s ) 1 + D ( s ) G ( s ) 5
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If the plant involves small variation, then it induces a relative error of the overall system: T = D ( s )( G ( s ) + ∆ G ( s )) 1 + D ( s )( G ( s ) + ∆ G ( s )) - D ( s ) G ( s ) 1 + D ( s ) G ( s ) = D ( s )∆ G (1 + D ( s )( G + ∆ G ( s )))(1 + D ( s ) G ( s ) D ( s )∆ G ( s ) (1 + D ( s ) G ( s )) 2 . Hence we have a relative error for the overall system as T ( s ) T ( s ) = 1 1 + D ( s ) G ( s ) G ( s ) G ( s ) = S ( s ) G ( s ) G ( s ) . That is, the relative error of the overall system caused by that of the plant is reduced by a factor of S ( s ) = 1 1 + D ( s ) G ( s ) .
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