Dynamics and control of limit cycling motions in boosting rockets

Abstract

An investigation concerning the prediction and control of observed limit cycling behavior in a boosting rocket is considered. The suspected source of the nonlinear behavior is the presence of Coulomb friction in the nozzle pivot mechanism. A classical sinusoidal describing function analysis is used to accurately recreate and predict the observed oscillatory characteristic. From this analysis, insight is offered into the limit cycling mechanism and confidence is gained in the closed-loop system design. Nonlinear simulation is used to support and verify the results obtained from describing function theory. Insight into the limit cycling behavior is, in turn, used to adjust control system parameters to indirectly control the oscillatory tendencies. Tradeoffs with the guidance and control system stability/performance are also noted. Finally, active control of the limit cycling behavior, using a novel feedback algorithm to adjust the inherent nozzle sticking-unsticking characteristics, is considered. HE ballistic tactical target vehicle (BTT V) is a suborbital rocket whose re-entry vehicle serves as a target for advanced air defense and theater defense systems under development.1'2 Flight telemetry results from the first two missions indicate the presence of low-amplitude, low-frequency steady oscillations in both the vehicle attitude and actuator piston displacement.3'4 Figure 1 illustrates this characteristic in the vehicle pitch attitude response during second stage boost of the first mission. Note the flight data has been post processed. The impact of the oscillatory motions on the guidance and control objectives appear to have been minor; however, their presence was unexpected and is of concern, especially with regards concerning the origins of the oscillatory behavior and the physical mechanism by which they occur. The hardware in question is the decommissioned Minuteman I Stage 3 system which is used as the second stage for the BTTV vehicle. After discussions with individuals intimately familiar with the Minuteman I Stage 3 characteristics, it became known that this system has a long history of exhibiting small, slow periodic motions during the boost phase. These characteristics are, in fact, so common that routine preflight analysis is used in estimating the oscillatory motion for the specific flight vehicle/control system in question. The results are then compared with an existing database to deter- mine whether the oscillatory motions will impact the guidance and control mission objectives. This database suggests the oscillatory motions are limit cycling behavior due to nonlinearities present in the actuator-nozzle system, a conventional hydraulic actuator driv- ing vectorable nozzles. Coulomb or dry friction present in the nozzle pivot is the primary culprit. Additional nonlinearities are present but are not considered here. The first goal of this research is to provide an accurate prediction, if possible, of the oscillatory behavior experienced by the BTTV flight vehicle.5 This analysis should provide insight as to the ori- gin of and the mechanism by which the characteristics occur. Si- nusoidal describing function theory and nonlinear simulation are to be employed in achieving this goal. A second goal of this re- search is to explore means by which the oscillatory motions may be controlled to lessen the impact upon the guidance and control performance.5 The implemented control systems should also have minimal impact on the performance of the guidance and control feedback loops, while simultaneously providing sufficient robust- ness against reasonably expected modeling uncertainties and errors.