Homework #10

1.      Problem 7.12 of Chen’s book.

2.      Show that if the system

realizes

and det (s I - A) = s² + 2 s + 1, then (A, b) must be controllable and (A, c) must be observable.

3.      For the transfer function

,

find

a)      an uncontrollable realization,

b)      an unobservable realization,

c)      an uncontrollable and unobservable realization,

d)      a minimal realization.

4.      Consider two systems S₁ and S₂ in series as depicted in the following block diagram:

Suppose that S₁ and S₂ have transfer functions

       and       ,

respectively.

a)      Determine controllable and observable state-space descriptions for the individual systems

b)      Determine a state-space representation of the entire system S by stacking the state vectors of each subsystem to form the state S

c)      What is the transfer function of S?  Is the state-space description of S in b) controllable?  Is it observable?