New approaches named $t^n$-stability of zero effort miss and short-time stability of guidance loop are introduced for analyzing the stability of homing guidance loop which includes the non-ideal missile/autopilot dynamics.
In $t^n$-stability criterion, guidance loop stability is defined as the monotonic convergence of the zero effort miss to zero which is directly related with the miss distance in the intercept engagement. This criterion is applied to the general homing guidance law for which guidance command is proportional to the zero effort miss. Stability conditions for the general homing guidance law show that the guidance loop con be $t^n$-stable until the target interception if the guidance law is proportional to the zero effort miss and the guidance gain increases satisfying the condition derived is this study. This result gives guidelines for selecting the guidance gain of a given homing guidance law. The $t^n$-stability of zero effort miss is also applied to the analysis of a PN guidance loop. Upper bounds and lower bounds of PN guidance gain for $t^n$-stability are derived for the guidance loop with 1st-order missile dynamics.
The short-time stability criterion is extended to accommodate time-varying state weights and time-varying bounds of the state norm, and it is applied to the PN guidance loop stability analysis. An interval during which the guidance loop is stable is defined as stability region and the lower bound of the stability region in the sense of short-time stability is derived. Different from the previous results based on the Popov stability or hyperstability, this condition depends on the total flight time in such a way that the lower bounds of the stability region is reduced.
A comparison study of stability conditions based on the Popov stability criterion, $t^n$-stability theorem, and short-time stability theorem shows that the Popov stability condition is most conservative and the short-time stability condition is least conservative.
To extend the stability region, the time-to-go freezing technique is introduced. Effects of time-to-go freezing on guidance loop stability is analyzed by using the short-time stability theorem developed in this study. Short-time stability theory is used to determine the time of time-to-go freezing. Simulation results of PN guidance loop with time-to-go freezing show that the time to go freezing technique enhances the stability of the guidance loop.