Time-optimal attitude control of a spinning missile

The problem of minimum-time attitude control of a spinning missile is addressed. The missile is modeled as a rigid body which is symmetric about one axis. The missile has a large roll rate about this axis of symmetry. Control is achieved by a single reaction jet which, when fired, provides a constant moment about a transverse axis. Disturbance torques are assumed to be zero. The equations of motion are written under these assumptions. The missile is assumed to have some arbitrary initial transverse angular velocity and it is desired to take it to some final attitude in minimum time while reducing the transverse angular velocity to zero. This problem is formulated as an optimal control problem. The solution of this problem involves integrating ten nonlinear equations with split boundary conditions and an unknown final time. Instead of taking this arduous route which requires solving a two point boundary value problem, two alternative approaches are considered. The first approach deals with the specific case where only two thruster firings are needed to change the attitude of the missile. This approach involves iterations on the switch times and integration of only the state equations. In this way the thruster firing times are obtained for a given set of boundary conditions. The other approach uses the concept of the set of reachable states. It involves transforming the state variables and integrating the transformed state and costate equations backwards in time and generating the control history using the properties of the optimal control. The control history, given in terms of the thruster firing times, is stored as a function of the boundary conditions. A feedback control law based on function generation is devised which uses only the first thruster turn-on and turn-off times. Some examples are included to illustrate the application of the concepts presented.