ch03 - T H R E E Modeling in the Time Domain SOLUTIONS TO...

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T H R E E Modeling in the Time Domain SOLUTIONS TO CASE STUDIES CHALLENGES Antenna Control: State-Space Representation For the power amplifier, E a (s) V p (s) = 150 s+150 . Taking the inverse Laplace transform, e a . +150e a = 150v p . Thus, the state equation is e a = − 150e a + 150v p For the motor and load, define the state variables as x 1 = θ m and x 2 = θ . m . Therefore, x . 1 = x 2 (1) Using the transfer function of the motor, cross multiplying, taking the inverse Laplace transform, and using the definitions for the state variables, x . 2 = - 1 J m (D m + K t K a R a ) x 2 + K t R a J m e a (2) Using the gear ratio, the output equation is y = 0.2x 1 (3) Also, J m = J a +5( 1 5 ) 2 = 0.05+0.2 = 0.25, D m = D a +3( 1 5 ) 2 = 0.01+0.12 = 0.13, K t R a J m = 1 (5)(0.25) = 0.8, and 1 J m (D m + K t K a R a ) = 1.32. Using Eqs. (1), (2), and (3) along with the previous values, the state and output equations are, x . = 0 1 0 -1.32 x + 0 0.8 e a ; y = 0.2 0 x
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