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lec10_print - Example Calculation of Forced Dynamics...

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Unformatted text preview: Example Calculation of Forced Dynamics Srinivas Palanki University of South Alabama Srinivas Palanki (USA) Example Calculation of Forced Dynamics 1 / 8 Forced Dynamics Illustrative Example Consider the following system in deviation form: d dt X 1 X 2 =- 1 1- 1 X 1 X 2 + 1 U Assume that X (0) = 0. Calculate the forced response to the following input: U = t < . 5 0 ≤ t < 1 t ≥ 1 Srinivas Palanki (USA) Example Calculation of Forced Dynamics 2 / 8 Forced Dynamics A =- 1 1- 1 Thus,- A = 1- 1 1 Using the analytical formula for calculating e At , we get: e- At = e t- te t e t Since B = 1 we have e- At B = e t- te t e t 1 = e t- te t Srinivas Palanki (USA) Example Calculation of Forced Dynamics 3 / 8 Forced Dynamics Now, e At = e- t te- t e- t Thus: X 1 X 2 = e- t te- t e- t Z t e t- te t U ( t ) dt Solution for 0 ≤ t < 1 X ( t ) = e At Z t e- At B U ( t ) | {z } U =0 . 5 dt Z t e- At BUdt = Z t . 5 e t dt Z t- . 5 te t dt = . 5 e t- . 5- . 5 te...
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