Chapter 10 - CHAPTER 10 Control Theory 10.1 ! Feedback...

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CHAPTER 10 Control Theory 10.1 Feedback Controls ! A Matter of Control In Problems 1–6, note the steady-state error when proportional control is present. Note, too, the additional damping when derivative control is present. 1. ! xxu += , x 01 () = . The uncontrolled system is represented by ! xx 0. The uncontrolled response is xt e t which is shown next. 0 x t 10 1 0 Uncontrolled system 763
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764 CHAPTER 10 Principles of Control Theory (a) Proportional feedback: The propor- tional feedback is ux =− () 21 . The controlled response would satisfy the system ! xx x += − , x 01 () = , or ! += 32 , which has the solution xt e t bg =+ 1 3 2 3 3 , which is shown next. This response approaches 2 3 . 0 x t 10 1 0 Proportional feedback (b) Derivative feedback: The derivative feedback is 3 ! . Thus, the controlled response would satisfy the system !! x 3 , x , or 40 ! which has the solution e t = 4 , which is shown next. This response approaches zero, but more slowly than the uncontrolled response. 0 x t 10 1 0 Derivative feedback
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SECTION 10.1 Feedback Controls 765 (c) Derivative + proportional: The deriva- tive plus proportional feedback is ux x =− () 21 3 ! . Thus, the controlled response would satisfy the system !! xx x x += − 3 , x 01 () = , or 432 ! += , which has the solution xt e t bg =+ 1 3 2 3 34 . 0 x t 10 1 0 Uncontrolled Proportional Derivative Proportional + Derivative Comparison of different controls This response approaches 2 3 , but more slowly than the response of a proportional feedback. All responses are shown in the figure. 2. u 2 , x . The uncontrolled system is represented by ! 20 , x , and the uncontrolled response is e t 2 . (a) Proportional feedback: The proportional feedback is 2 1 . Thus the controlled response would satisfy the system ! x +=− 22 1 , x , or ! 42 , which has the solution e t 1 2 1 2 4 . This response approaches 1 2 . (b) Derivative feedback: The derivative feedback is 3 ! . Thus the controlled response would satisfy the system x 23 , x , or 420 ! , which has the solution e t = 2 . This response approaches zero, but more slowly than the uncontrolled response.
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766 CHAPTER 10 Principles of Control Theory (c) Derivative + proportional: The deriva- tive plus proportional feedback is ux x =− () 21 3 ! . Thus, the controlled response would sat- isfy the system !! xx x x +=− 22 1 3 , x 01 () = , or 442 ! += , which has the solution xt e t bg =+ 1 2 1 2 . 0 x t 10 1 0 Uncontrolled Proportional Derivative Proportional + Derivative Comparison of different controls This response approaches 1 2 . See figure for comparison. 3. 23 ! u , x . The uncontrolled system is represented by 230 ! , x and the uncontrolled response is e t = 32 . (a) Proportional feedback: The proportional feedback is 2 1 Thus, the controlled response would satisfy the system 232 1 ! x , x , or 252 ! , which has the solution e t 3 5 2 5 52 . This response approaches 2 5 . (b) Derivative feedback: The derivative feedback is 3 ! . Thus, the controlled response would satisfy the system 3 x , x , or 530 ! , which has the solution e t = 35 .
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This lab report was uploaded on 04/04/2008 for the course APPM 2360 taught by Professor Williamheuett during the Fall '07 term at Colorado.

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Chapter 10 - CHAPTER 10 Control Theory 10.1 ! Feedback...

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