MAE 281B Nonlinear Control	Prof. M. Krstic
Homework 4	Due: May 28, 2002

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1. Consider the model of the "kinematic" car:

where and are control inputs ("drive" and "steer"). The linearization at the origin is obviously not completely controllable (check this!) By computing and , show that the nonlinear system is completely controllable.

2. Show that the system from Problem 1 fails Brockett's condition, i.e., the origin is not - stabilizable.

3. Consider the model of a spacecraft with thrusters generating torque around two axes:

where is the angular velocity, is the attitude (in modified Rodrigues parameters), are the torque inputs,

the matrix is a skew-symmetric matrix defined as

and the matrix is defined as
.
The linearization at the origin is obviously not completely controllable (check!). By computing , and , show that the nonlinear system is completely controllable whenever .

4. Show that the system from Problem 3 fails Brockett's condition. Note that
.

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