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PSet_01_solution

# PSet_01_solution - Spring 2003 10.450 Process Dynamics...

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Spring 2003 10.450 Process Dynamics, Operations, and Control Problem Sets - 1 Solve these equations. Viewing them as dynamic systems, plot the disturbances and responses over appropriate time intervals. 1. 0.3 y 0.8 x = − dy y (0) = − 1 x ( t ) = 0 2 t t < 2 2 dt Write in standard form. This first order system is disturbed by a delayed step function. Ultimately it will come to the value of the step (-2) multiplied by the gain (8/3). 1 dy + y = 8 3 x ( t ) 0.3 dt The solution of this equation (1 st order, linear, constant coefficient) is τ t t e y ( t ) = τ e τ t Kx ( t ) dt + y (0) e τ t 0 For the case of a delayed step disturbance x, y ( t ) = Kx 1 e ( t t d τ ) + y (0) e τ t The initial condition determines the response until the disturbance occurs at t d . Substituting the given numbers leads to the plot: -6 -5 -4 -3 -2 -1 0 0 4 6 10 12 14 time disturbance and response 2 8 1

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Spring 2003 10.450 Process Dynamics, Operations, and Control Problem Sets - 1 2.
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PSet_01_solution - Spring 2003 10.450 Process Dynamics...

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