A Nonlinear Fuel Optimal Reaction Jet Control Law

We derive a nonlinear fuel optimal attitude control system (ACS) that drives the final state to the desired state according to a cost function that weights the final state angular error relative to the angular rate error. Control is achieved by allowing the pulsewidth-modulated (PWM) commands to begin and end anywhere within a control cycle, achieving a pulse width pulse time (PWPT) control. We show through a MATLAB Simulink model that this steadystate condition may be accomplished, in the absence of sensor noise or model uncertainties, with the theoretical minimu m number of actuator cycles. The ability to analytically achieve near-zero drift rates is particularly important in applications such as stationkeeping and sensor imaging. Consideration is also given to the fact that, for relatively small sensor and model errors, the controller requires significantly fewer actuator cycles to reach the final state error than a traditional proportionalintegral-derivative (PID) controller. The optimal PWPT attitude controller may be applicable for a high performance kinetic energy kill vehicle. Introduction In problems related to the attitude control of rigid bodies, it is often advantageous to design an attitude control system that is optimal in some way relative to a set of system constraints. Most often cited are constraints of time and fuel. A constraint pertinent to some systems regards the total number of PWM commands sent to the actuators. In our case that is represented by valve commands sent to the ACS jets. A PID controller is simple to implement (Wie , 1998), takes few CPU cycles, and usually is robust in most applications. Sliding mode controllers are likewise robust and easy to implement (Wertz , 1978). Both of these control systems share a potential common trait, in an environment where sensor noise exists, of requesting numerous PWM commands while dithering about their phase-plane trajectories. These effects can be minimized by introducing deadbands and other nonlinear elements at the expense of system response and accuracy. We derive a single-axis PWPT ACS that minimizes the total number of valve commands while simultaneously minimizes a cost function dependent on the final state errors. We first apply the singleaxis controller to a 1-DOF simulated system, and then to a 3-DOF system, where each axis is controlled by the single-axis controller. This allows us to evaluate its effectiveness in the presence of coupling through the inertia matrix. Phase-plane plots of angle versus angle rate are given as well as plots of the valve commands. Monte Carlo runs are made using Simulink in order to investigate controller sensitivities to gyro noise and camera noise. Published results (Garcia , 1998) are shown in work done previously in support of MicroSat pointing and control development at LLNL. Pulse Width Modulation (PWM) An approximation to an idealized linear command may be represented as in Figure-1, and is shown to start at the beginning of the control cycle and to end within the control cycle, or it may continue into the next cycle if the command is saturated. Figure-1: Ideal linear force command shown with PWM approximation A discrete approximation to the ideal jet turn on time request can be made as follows,