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

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!!