EE611

HW#22

Due Friday, October 31

1. a) Prove that by our new definition for observability for time-varying systems, the system:

will be observable iff

is non-singular (W0 is called the Observability Grammanian) ( means complex conjugate transpose).

b) Use your above proof to find an expression for x(t0) in terms of y and w.

c) For time-invariant systems, why don't we use your above expression to implement an exact observer instead of the "exponential Luenberger" observer we learned about earlier?

2a) Determine the controllability of the following time varying system over the period [0,t]:

b) For the linear, time-invariant single input system, show that the system will be controllable by the definition learned today iff M=[b Ab ... An-1b] is nonsingular (you are proving that our new definition of controllability is equivalent to our old defintion when the system is time-invariant).

c) For the time invariant system of part 2b), find an expression for the control, w(t), which will drive any initial x(t0) to any x(t1), assuming that the system is controllable.

d) Use the results of part c) to find w(t) which will drive x(0)=1 to x(1)=2 for the system:

f) Finally, why don't we use this exact control instead of the control/regulator architecture we learned previously (hint: think about feedback control and robustness vs. open-loop control)?