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HW3_08S

# HW3_08S - i.e t z K t x t x K t f t x t x z I des p des-=-=...

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University of California at Berkeley Department of Mechanical Engineering ME132 Dynamic Systems and Feedback Spring 2008 Homework Set #3 Assigned: February 19 (Tu) Due: February 28 (Th) [1] Consider a second order ODE described by 0 0 ) 0 ( , ) 0 ( ), ( ) ( y y y y t bu t y = = = a. Let < = 0 1 0 0 ) ( t t t u . Find the particular solution. b. The characteristic equation is 0 2 = λ , which yield the two identical characteristic roots, 0 2 1 = = λ λ . Two independent homogenous solutions are ) ( 1 ot e = and ) ( 0 t te t = . Thus the solution is written as ) ( ) ( 2 1 t y t c c t y P + + = Determine 1 c and 2 c to satisfy the initial conditions. c. Show that the solution to the second order ODE is obtained by directly integrating the original equation twice. [2] 7.10.3 [3] Consider a mass-spring-dashpot system described by ) ( ) ( ) ( ) ( t f t kx t x b t x m = + + The feedback controller is a PI controller:
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Unformatted text preview: i.e. ) ( )) ( ) ( ( ) ( ) ( ) ( t z K t x t x K t f t x t x z I des p des +-=-= Apply the Hurwitz conditions to prove the following points. a. If the controller is a simple P-controller (i.e. = I K ), the closed loop system is stable for all positive p K . b. If the integral controller is activated, the integral control gain must be smaller than the limit set by other parameters for stability. Assume that the feedback gains satisfy the stability condition. The desired position is constant: i.e. const x t x des des = = ) ( . a. Obtain ) ( lim t x x t ∞ → = for P-control only. b. Obtain ) ( lim t x x t ∞ → = for PI-control. [4] 8-1-1...
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