Synthesis of nonlinear guidance laws for missiles with uncertain dynamics

This technical memorandum describes a nonlinear guidance law for a single-missile single-target engagement. The guidance relies on the concepts of Lyapunov stability and backstepping, which are constructive methods in nonlinear control theory. The design of the guidance law allows taking into account the nonlinear relative kinematics between the missile and the target, and ensuring ultimate boundedness of the missile-target system trajectories provided the estimation error of the target acceleration is bounded in magnitude. In other words, despite the nonlinear kinematics between the missile and the target, the guidance scheme is guaranteed to result in a relatively small miss distance between the missile and the target. There are two steps in designing the nonlinear guidance law. In the first step, using the fully nonlinear missile-target engagement kinematics, an appropriate Lyapunov function candidate is selected and a state-feedback law is obtained. Closed-loop pole placement using linear matrix inequalities provides an ultimate bound to the maximum allowable miss distance, assuming idealized flight control dynamics; that is, infinitely fast reaction times for the missile. In the second step of the guidance law, the control law is robustified by means of a high-gain backstepping approach, taking into account the uncertain flight control dynamics of the pursuer missile as an uncertain although bounded time constant. Numerical simulations of the nonlinear guidance in closed-loop with a missile modeled as an interval second-order transfer function and a maneuvering target demonstrate satisfactory performances when compared to modern and classical guidance laws, such as proportional navigation guidance. Despite the uncertainty on the missile flight control system, which is usually the case in practice, the guidance law achieves small miss distances against highly maneuverable targets. Uncertainties may arise due to a variety of reasons. For instance, uncertainties may be due to variations in the missile aerodynamics over the flight envelope, unexactly known aerodynamic performance, or low-order approximate modeling of the flight control system. However, it is important to note that the satisfactory performance of the nonlinear guidance comes at the expense of potentially large acceleration demands in the early part of the terminal phase of the engagement, when the guidance law is applied. There is therefore a trade-off to be made between the reduction of the miss distance and the acceleration demands. DRDC Valcartier TM 2006—606 i