modeling8 - Modeling QUESTION 8 x( t ) & x Linearize...

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Modeling Q UESTION 8 Linearize the differential equation ( 29 ( 29 ( 29 ( 29 3 2 x t x t x t x t e - + + = . The two states are ( 29 ( 29 1 x t x t = (1) ( 29 ( 29 2 x t x t = (2) The two first order nonlinear differential equations are ( 29 ( 29 ( 29 1 1 2 f t x t x t = = (3) ( 29 ( 29 ( 29 ( 29 ( 29 1 2 2 2 1 3 2 x t f t x t x t x t e - = = - - + (4) At equilibrium, the derivatives are equal to zero and equation (3) becomes 2 2 0 0 x x = = (5) At equilibrium equation (4) becomes 1 1 2 1 1 0 4 2 2 0 x x x x e x e - - = - - + - = (6) Using graphical techniques or the bisection method to solve equation (6), 1 0.3517 x = . The first incremental state and its first derivative with respect to time are ( 29 ( 29 ( 29 ( 29 1 1 1 1 1 ˆ ˆ x t x t x x t x t = - = (7) The second incremental state and its first derivative with respect to time are ( 29 ( 29 ( 29 ( 29
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modeling8 - Modeling QUESTION 8 x( t ) & x Linearize...

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