Homework #12

1.     Problem 8.1 of Chen’s book.

2.     Problem 8.2 of Chen’s book.

3.     Problem 8.4 of Chen’s book.

4.     Consider the system

where

In addition, suppose that one uses a linear feedback control u = f x + g r, where f is a 3-dimensional row vector and g a constant scalar.

a)     Denoting the closed-loop system by

we have that

where

Show that the zeros of n(s) do not depend on f and g. Compute the zeros.

b)     What must be the zeros of d(s) so that

c)     Compute f and g so that

Hint: Start by computing f to get the desired zeros for d(s).

d)     Is the closed-loop system controllable?  Observable?

Note: This is an example of model matching, i.e., compensating a given system so that it matches the input-output behavior of a desired model.