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hw3 - Homework 3 MAE143B Spring 2010 due Thursday April 22...

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Homework 3 - MAE143B Spring 2010: due Thursday April 22 Question 1: Initial state calculation Consider the state-variable realization of the system described by the n th -order ordinary differential equation y ( n ) ( t ) + a 1 y ( n - 1) ( t ) + · · · + a n - 1 ˙ y ( t ) + a n y ( t ) = b 1 u ( n - 1) ( t ) + · · · + b n - 1 ˙ u ( t ) + b n u ( t ) , with initial conditions y (0) , ˙ y (0) , . . . , y ( n ) (0) ; ˙ x ( t ) = Ax ( t ) + Bu ( t ) , y ( t ) = Cx ( t ) + Du ( t ) . Our aim is to transfer the known initial conditions on output y ( t ) to initial conditions on the state vector x ( t ) . For this, assume that the initial conditions on u ( t ) are all zero. [The argument for this is that u ( t ) is ours to control and we typically do things like step, impulse and sine-wave test, which are preceded by zero input. If the initial conditions of u were important, then – according to our definition of state – we would have to incorporate them into the state and we could do this by redefining the initial conditions on y .] Using the state equations, show that Action!!
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