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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. ! x x u + = , x 0 1 ( ) = . The uncontrolled system is represented by ! x x + = 0 . The uncontrolled response is x t 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 u x = ( ) 2 1 . The controlled response would satisfy the system ! x x x + = ( ) 2 1 , x 0 1 ( ) = , or ! x x + = 3 2 , which has the solution x t e t b g = + 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 u x = − 3 ! . Thus, the controlled response would satisfy the system ! ! x x x + = − 3 , x 0 1 ( ) = , or 4 0 ! x x + = which has the solution x t e t b g = 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 u x x = ( ) − 2 1 3 ! . Thus, the controlled response would satisfy the system ! ! x x x x + = ( ) − 2 1 3 , x 0 1 ( ) = , or 4 3 2 ! x x + = , which has the solution x t e t b g = + 1 3 2 3 3 4 . 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. !! x x u + = 2 , x 0 1 ( ) = . The uncontrolled system is represented by ! x x + = 2 0 , x 0 1 ( ) = , and the uncontrolled response is x t e t ( ) = 2 . (a) Proportional feedback: The proportional feedback is u x = ( ) 2 1 . Thus the controlled response would satisfy the system ! x x x + = ( ) 2 2 1 , x 0 1 ( ) = , or ! x x + = 4 2 , which has the solution x t e t b g = + 1 2 1 2 4 . This response approaches 1 2 . (b) Derivative feedback: The derivative feedback is u x = − 3 ! . Thus the controlled response would satisfy the system ! ! x x x + = − 2 3 , x 0 1 ( ) = , or 4 2 0 ! x x + = , which has the solution x t e t b g = 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 u x x = ( ) − 2 1 3 ! . Thus, the controlled response would sat- isfy the system ! ! x x x x + = ( ) − 2 2 1 3 , x 0 1 ( ) = , or 4 4 2 ! x x + = , which has the solution x t e t b g = + 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. 2 3 ! x x u + = , x 0 1 ( ) = . The uncontrolled system is represented by 2 3 0 ! x x + = , x 0 1 ( ) = and the uncontrolled response is x t e t b g = 3 2 . (a) Proportional feedback: The proportional feedback is u x = ( ) 2 1 Thus, the controlled response would satisfy the system 2 3 2 1 !
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