Synthesis of constant feedback controllers for an unstable and nonminimum-phase missile with a large flight envelope

Abstract

The problem of controller synthesis is considered for a class of missiles which are unstable and non-minimum-phase plants over a large flight envelope. This class includes, for example, ballistic, surface-to-air and surface-to-surface missiles. Because of the large flight envelope, the gain scheduling method is commonly used to update the parameters of the designed controllers at different operating points in the flight envelope. To overcome these problems, we propose to employ the Quantitative Feedback Theory (QFT) to find, if it exists, a single, linear time-invariant, robust feedback controller for a specified set of quantitative performance specifications. Specifically, the QFT is applied to a Vanguard missile in order to study feasibility and design tradeoffs related to the controller design. 1. NOMENCLATURE Mach number flight altitude missile dynamic pressure reference missile radius reference missile area moment lift coefficient moment fin coefficient force lift coefficient force fin coefficient aerodynamic dam ping coefficient pitch angle angular velocity angular acceleration fin angle angle of attack path angle servo delay vertical acceleration linear position missile mass moment of inertia total velocity right half plane zero left half plane zero right half plane pole left half plane pole 2. INTRODUCTION An extremely challenging task for the control engineer is to design an automatic autopilot for aerospace vehicles that guarantee stability margin and specified performance over a large flight envelope (altitude vs. Mach number). This is particularly the case if the missile is tail-controlled, Le., non-minimum-phase (nmp), where meeting the performance requirements may not always be possible The common approach used to design an autopilot for this class of missiles is the gain scheduling method [l]. It is used to update during the missile flight, the parameters of the controllers designed for a chosen set of operating points. Given a set of performance specifications, one * on leave from Rafael, Israel h i s t a n t Professor. *I can only hope that a constant topology controller can indeed be found for the whole flight envelope. Moreover, implementation of gain scheduling controllers may be limited by the on-board computer capacity. Updating the controller parameters can potentially introduce an undesirable input to the plant and thus reduce the system's stability during and after the updating (nonlinear effect). An alternative method for control design is the Quantative Feedback Theory (QlT), which was developed by Horowitz [ 2 ] . A certain Mach number and altitude are chosen as a nominal operating point, and the range of the missile's transfer function over the whole flight envelope is then defined as the model's uncertainty. Given this uncertainty and the desired performance specifications, Q l T can be used to find a constant controller that solves the corresponding robust stability and robust performance problem over the whole flight envelope, if sucka controller exists. To illustrate the advantages of QI;T for robust design for this class of missiles. it was applied to a Ballistic-type missile (Vanguard) which is inherently unstable and nmp over the whole flight envelope. Naturally, much of the missile's data are classified but the model described in [3], is sufficient for the illustration of QFT. The work addresses, in the same order, the missile longitudinal dynamics model, its problem statement, and finally the control design using QFT. 3. MISSILE LONGITUDINAL DYNAMICS MODEL The airframe longitudinal dynamics model consists of two modes: a slow mode called phugoid mode and a fast mode called short period. The existence of the two mode in the airframe longitudinal plane is well known [3]. The following assumptions were made in the derivation of the equations of motion: (1) the missile can be modeled as a rigid body; (2) the earth is used as an inertial reference; (3) the missile's mass is constant; and (4) the perturbations from equilibrium are small. In general, in the case of a missile the assumption of small perturbation is more valid than in the case of an airplane, therefore the short period approximation equation can be used to model the longitudinal dynamics. The Missile Transfer Functions: The transfer function relating the fin angle and the angular velocity is given by: The transfer function relating the fin angle and the linear acceleration is given by: The transfer function relating the angular position and linear position is given by: