Discrete optimal terminal control with application to missile guidance

The following aspects of discrete optimal terminal control of a linear system with a quadratic performance index are considered: 1) specification of a performance index that will give satisfactory results from an engineering standpoint at the critical terminal time; 2) closed-form algebraic solutions of the control law that are computationally feasible on-line, without necessitating prior lengthy off-line computation and storage. For a linear plant with discrete control, it is shown how a generalized performance index can be modified slightly in order to provide implicit feedback. The original plant and control problem are replaced by an "equivalent plant" (with eigenvalues that are shifted only if those of the original plant are unsuitable) and an "equivalent problem." The equivalent problem (with a slight restriction) has a performance index whose loss function weights only the control, and is solved for the optimal control in closed form with minimal difficulty from literal matrix inversion. Solution of the optimal control in the original problem is a minor additional step. If the original plant has suitable eigenvalues, as in the case of a missile system with a well-designed autopilot, then the equivalent problem reduces to the original problem because there is no need for implicit feedback via the performance index. For a general missile-guidance problem, the solution of the restricted equivalent problem with discrete control is given. The example is for a missile airframe (without a conventional autopilot) having unsuitable eigenvalues. Performance in terms of rms state variables is sharply improved by making the equivalent plant have eigenvalues that correspond to a fast integral of the squared acceleration error response.