F t 4f(4,TA, W)
AtA = (t)
T*
wATA(t)
(t) af (,TA w)
3T A
= 0 1INxN
Fl(t) F2(t)1
0
IJ
w = i(t)
It is not difficult [241 to derive equations which explicitly relate the matrices Fl(t) and F2(t) to the nominal trajectory, however, in the interest of brevity, we omit these derivations.
Now consider the feasibility of selecting the feedback gains to
stabilize the linearized closed-loop system. Assuming that the slowly time-varying approach will apply, the eigenvalues of the linearized
matrix must lie in the open left half-plane for all t. Theorem 3.3 can be used to determine when it is possible to meet this goal using an appropriate feedback law. Interpreting Theorem 3.3 for the slowly timevarying system, the following conditions must hold:
(1) [F*(t), G*(t)] must be controllable at each t.
[i I2Nx2N - F(t)
(2)
must have full rank (rank 3N) for
G t)
each t and for each xi which is an eigenvalue of the internal model matrix A.
(6-12)
(6-13)