Attitude Control with Analytic Disturbance-Rejection Guarantee

Analytic formulas that guarantee a minimum level of disturbance-rejection performance are derived for a class of attitude control problems consisting of linear proportional-derivative quaternion feedback applied to a rigid-body spacecraft plant model. Specifically, a Lyapunov-based disturbance-rejection assessment tool is derived for a general class of perturbed nonlinear systems, and then it is specialized to the linear attitude control class. Although the tool accepts generic Lyapunov function candidates, this paper demonstrates that existing Lyapunov functions (that readily prove asymptotic stability for attitude control systems) are insufficiently parameterized for purposes of estimating disturbance-rejection capability via the proposed tool. In response to this shortcoming, two new Lyapunov functions are proposed, and they are evaluated in the context of two closed-form disturbance-rejection performance assessment algorithms. Both algorithms demonstrate that the magnitude of the allowable disturbance torque is proportional to 1) the magnitude of the allowable angular accuracy, 2) the square of the control bandwidth, and 3) the minimum eigenvalue of the spacecraft inertia matrix. Moreover, the constant of proportionality (coefficient of rejection) is shown to degrade gradually for larger angles, and it eventually goes to zero as the user-specified angular accuracy is increased to 180 deg (for a high-order Lyapunov function) and 12 deg (for a low-order quadratic Lyapunov function).