The describing functions for a constrained-range integration process with bang-bang input and dead zone

Recently, considerable attention has been devoted to the analysis of high-order systems containing severe nonlinearities separated by linear functions, particularly those using bang-bang or on-off controllers. These have proven to be very satisfactory in the attitude control systems of manned or unmanned spacecraft and satellites. This paper develops describing functions for a particularly complicated multiple nonlinearity: a tri-stable (bang-bang with dead zone) characteristic, followed by a linear integrator with a constrained range of integration. It should be noted that constraining range of integration is not equivalent to simple limiting of an integrator´s output. This system of nonlinearities has not previously been treated in the literature, although it is founds for example, in satellite attitude controllers where on-off torques give rise to constrained momentum wheel angular velocities or in guided missile hydraulic actuators with bang-bang power spool and control surface position limits. This frequency-variant nonlinearity has three distinct modes of operation and, therefore, is quite different from the usual single nonlinearity considered for describing function application. The describing function and boundary equation for each mode are derived in the paper, and numerical examples are given. The analytical results were found to agree well with results from an analog computer simulation. These describing functions may be used to size power actuators ands therefores should be very useful for preliminary systems design.