54
i
Then, provided and e£ are suitably small, there exist feedback
gains K]^ and K2 so that with the controller given as system NC,
(eq. (2-29)) local tracking of r (t) with disturbance w (t) will occur.
Proof: Our main concern here will be to show asymptotic stability of
the system NCT and observability of the pair (A,!^). With these
conditions verified, the remainder of the proof is immediate from the
results obtained in Chapter Two.
First let us compare (4-18) to (4-6) letting F take the role of
F^t) and [F^(t) ~ F^]XA(t) take the role of f-^t.x). Condition (1)
of the Poincare-Liapunov theorem then requires exponential stability of
the system
x
A
(4-21)
In order to meet this stability requirement, the pair
(4-22)
must be at least stabilizable. Since the matrices given in (4-22) are
constant, Theorem 3.3 can be employed. More specifically, conditions
(iii) and (iv) imply that the pair given in equation (4-22) is
stabilizable and hence, proper selection of the feedback and l<2 will
give exponential stability to the system (4-21).
Assuming that suitable feedback gains have been selected, the state
transition matrix $(t,to) associated with (4-21) will satisfy the
inequality