Controls coefficient generalized inversion Lyapunov design for spacecraft attitude control

The paper introduces a controls coefficient generalized inversion attitude tracking design methodology for realization of desired linear spacecraft attitude deviation dynamics. A prescribed stable linear second order time-invariant ordinary differential equation in a spacecraft attitude deviation norm measure is evaluated along the solution trajectories of the spacecraft equations of motion, yielding a linear relation in the control variables. Generalized inversion of the relation results in a control law that consists of particular and auxiliary parts. The particular part resides in the range space of the controls coefficient row vector, and it works to drive the spacecraft attitude variables in order to nullify the attitude deviation norm measure. The auxiliary part resides in the complementary orthogonal complement subspace, and therefore it does not affect realization of the desired trajectory. Nevertheless, the auxiliary part is crucial in the control design, because it provides the necessary spacecraft internal stability by proper design of the null-control vector. The null-control vector construction is made by solving a state dependent Lyapunov equation, yielding global internal stability. The control design utilizes a damped controls coefficient generalized inverse to limit the growth of the controls coefficient generalized inverse as the steady state response is approached. The design provides uniformly ultimately bounded attitude trajectory tracking errors, and reveals the tradeoff between generalized inversion stability and tracking performance.