83
Now consider the exponential stability of the system (5-13). In
order to achieve this condition with time-varying feedback, it is
necesssary that the pair
F*(t)
0
~G*(t)
-BH
A
*
0
(5-14)
be stabilizable. The feedback gains could then be selected, for
example, by optimal control techniques [20]. Recall, however, that the
rate of variation of the gains K^(t) and ^(t) must be slow if the
quasi-static approach is to remain justifiable. Such a condition is
likely when the nominal signals r (t) and w (t) are themselves, slowly
varying.
Another method for obtaining the needed stability is to choose the
feedback gains K^(t) and ^(t) in such a way that the eigenvalues of
k
F^(t) lie in the left half-plane for all t. This approach is valid
[17] under the assumption of a slowly time-varying system (i.e.,
ll^jrFA(t) ll.. should be suitably small). Obviously, the slowly time-
varying condition is only likely to occur when the nominal signals are
slowly time-varying. Assuming this to be true, the resulting feedback
gains will also be slowly time-varying as required previously.
k k
As a final point, note that if the nominal signals r (t) and w (t)
are constant and if the original system is autonomous, then the matrices
given in (5-14) will be constant. In this circumstance, constant
feedback gains can be employed, thus eliminating any concern that the
slowly-varying feedback gain assumption may not be justified.