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
-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 Kl(t) and K2(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 ; (t) are themselves, slowly varyi ng.
Another method for obtaining the needed stability is to choose the feedback gains K1(t) and K2(t) in such a way that the eigenvalues of FA(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.,
d -*
11-FA(t)Ili should be suitably small). Obviously, the slowly timevarying 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.
As a final point, note that if the nominal signals (t) and i (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.