Dynamic stability conditions for a rolling flight vehicle applying continuous actuator

The conditions for dynamic stability of coning motion for a rolling missile are proposed. Considering dynamics of fin actuator, a Proportional–Derivative (PD) controller is applied to the servomechanism employing a permanent magnet DC motor. The necessary and sufficient conditions of dynamic stability are analytically derived. Both steady state and transient responses of actuator are considered in this paper. The differential equation of a whole system for steady state response of actuator leads to a second order differential equation with complex coefficients. A theorem based on Routh–Hurwitz stability criterion is proposed that gives a triple inequality to determine the stability region of PD controller and consequentially, the stability of coning motion. For transient response, the feasibility of a linear matrix inequality should be checked for each PD coefficient. It is shown that considering the transient response of actuator can change the stability region. The controller coefficients that stabilize the coning motion in steady state response may cause instability in transient response of actuator. Finally, validation of this investigation is examined through simulation. Because of continually rolling motion of missile, the cross coupling that is caused by phase lag angle, will occur, so the effect of PD controller coefficients on static phase lag angle is discussed and some technical notes to reduce the cross coupling effect are explained. Furthermore, the influence of roll rate frequency on the region of PD controller coefficients is presented and a safe region for design is recommended.