Short-Time Stability of Proportional Navigation Guidance Loop DONG-YOUNG REW MIN-JEA TAHK Korea Advanced Institute of Science and Technology HANGJU CHO Agency for Defense Development Korea Stability characteristics of proportional navigation (PN) guidance are analyzed by using the short-time stability criterion which is extended here to accommodate time-varying state weights and time-varying bounds of the state norm. As short-time stability is defined over a specified time interval, its application to the stability analysis of a homing guidance loop that operates up to a finite time gives more accurate results than previous studies. Furthermore, within the framework of short-time stability, zero effort miss and acceleration command, which are the most important variables determining guidance performance, can be directly related with guidance loop stability. An application to a PN guidance loop with a 1st-order missile/autopilot time lag shows that the stability condition based on short-time stability is less conservative than the previous results based on hyperstability and Popov stability. Manuscript received January 30, 1995; revised May 5, 1995. IEEE Log No. T-AES/32/3/05881. Authors’ addresses: D.-Y. Rew and M.-J. Tahk, Dept. of Aerospace Engineering, Korea Advanced Institute of Science and Technology, 373-1 Kusong-Dong, Yusung-Gu, Taejon 305-701, Korea; H. Cho, Agency for Defense Development, P.O. Box 35, Yusong-Gu, Taejeon 305-600, Korea. 0018-9251/96/$10.00 c ° 1996 IEEE I. INTRODUCTION Proportional navigation guidance (PNG) is widely used for terminal homing guidance because of its simplicity, effectiveness, and ease of implementation. However, the nonlinear time-varying characteristic of the governing equation makes it difficult to analyze the stability of PNG in analytic terms. This difficulty may be reduced by assuming an ideal missile that behaves as a point mass. Previous works based on the ideal missile model are as follows. A qualitative study is performed by Murtaugh and Criel [1] on pure PN (PPN) for which missile acceleration is commanded normal to the instantaneous line of sight (LOS). Guelman derived a closed form solution of PPN by assuming constant closing velocity [2]. For true PN (TPN), for which missile acceleration is normal to the missile velocity vector, Guelman [3—5] provided conditions for reaching the target regardless of initial condition and obtained the bounds of missile acceleration for intercepting a constant-acceleration target. Mahapatra and Shukla [6, 7] derived an approximated closed-form solution of the nonlinear differential equation of TPN for laterally maneuvering target with a constant acceleration. Ha, et al. [8] proved the intercept performance of PNG against a randomly maneuvering target. Previous studies show that PNG enables an ideal missile to intercept the target when a suitable guidance gain is adopted. Moreover, the intercept performance of PNG is achieved for any initial heading error and target maneuver with the guidance command remaining bounded through the engagement. Unfortunately, for the PNG loop with a non-ideal missile dynamics, no closed-form solution has been reported. However, it is still possible to investigate the effects of guidance gain and missile dynamics on the stability of guidance loop. One of the benefits of the stability analysis is that guidance loop stability can be predicted without obtaining a closed-form solution. Even though there does not exist a widely accepted definition of guidance loop stability, instability of guidance loop can be described as follows: The magnitudes of the acceleration command and the state variables associated with the guidance loop grow infinitely as time to go (t go ) approaches zero. Guidance loop instability may degrade intercept accuracy because of the saturation of missile acceleration. The simplest way to study guidance loop stability is to apply Routh—Hurwitz criterion to the linearized guidance loop as in [9]. Although the Routh—Hurwitz stability criterion is applicable to a slow time-varying system under the assumption of frozen time, its application to the analysis of PNG is not appropriate since a PNG loop has fast time-varying element such as range. A linearized PNG loop consists of a linear time invariant block (missile/autopilot dynamics) and a IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 32, NO. 3 JULY 1996 1107