Consequently, x*(t), i(t) and u*(t) all satisfy the differential
equation
(3) + [X22 (.2) + [2X + 2 () + [-'2 X 0
(3-16)
This completes the example. Although a disturbance was not considered, the inclusion of a disturbance would have led to similar results.
Next we consider another requirement which was needed to solve the
nonlinear servomechanism problem and relate it to the linear problem. This requirement is that the closed-loop transient system NCT given by (2-38) must be asymptotically stable. When the linear system is
considered, (2-38) takes the following form
LCT:
BH A Wt) 0](t)
FL(t) = -K 1(t) - K2-(t) (3-17)
Now if conditions (B.1) and (B.2) both hold then it is possible to select K1 and K2 such that (3-17) is asymptotically stable. This is a consequence of the following well known theorem (see [2] or [3]).
Theorem 3.3: If (B.1) and (B.2) both hold and the eigenvalues of A correspond exactly to xi' i = 1, 2, ., r given by condition (B.2)