EE3530-4

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

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 G , then it induces a relative error of the overall system T ( s ) = G ( s ) D ol ( s ) H r ( s ) T ( s ) = G ( s ) D ol ( s ) H r ( s ) T T = G ( s ) D ol ( s ) H r ( s ) G ( s ) D ol ( s ) H r ( s ) = G ( s ) G ( s ) that is the same as the relative error of the original plant. 1 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 cant 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 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 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 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 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 ....
View Full Document

Page1 / 31

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

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online