Then, provided e1 and e2 are suitably small, there exist feedback gains K1 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,K2). 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 0 take the role of
A
Fl(t) and [FA(t) - F ]XA(t) take the role of fl(t,x). Condition (1) of the Poincare-Liapunov theorem then requires exponential stability of the system
=A = FAXA (4-21)
In order to meet this stability requirement, the pair
o] [Go])(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 K1 and K2 will give exponential stability to the system (4-21).
Assuming that suitable feedback gains have been selected, the state transition matrix @°(t,t ) associated with (4-21) will satisfy the inequality