MAE142_L15_Linearize.pdf - Linearization Small Deviations...

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Linearization Small Deviations Reference Trajectory Small Deviation Equations of Motion Dimensional Derivatives MAE 142 DSI/NASA Oblique Wing RPV
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2 MAE 142 Small Deviations Suppose a reference trajectory has been determined for the vehicle (perhaps as a result of state-space simulation techniques). ˙ x r t = f [ x r t ] x t = x r t  x t Define the total system state vector as the sum of the reference trajectory and a vector of small deviations. ˙ x t = ˙ x r t  ˙ x t x t x r t x t
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3 MAE 142 Taylor Series The differential equations for the state vector of small deviations is found by a Taylor Series from the original equations of motion. ˙ x t = ˙ x r t  ˙ x t ˙ x t = f [ x t ] ˙ x r t  ˙ x t = f [ x r t  x t ] x t = x r t  x t ˙ x r t  ˙ x t = f [ x r t ] [ f x ] r x t ⋯ ˙ x r t = f [ x r t ]  ˙ x t ≈ [ f x ] r x t Trajectory of Small Deviations Reference Trajectory Taylor Series
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4 MAE 142 Reference Trajectory The reference trajectory can be any solution of the original equations of motion. Choosing a static equilibrium condition as the reference allows for some useful simplifications. 0
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