An investigation on LQR based optimal terminal states control for exo-atmospheric flight of super heavy hypersonic intercontinental ballistic missiles using Gauss pseudospectral method

Using 3-DOF model for the launch vehicle, the governing dynamic differential equations for the flight of intercontinental ballistic missiles (ICBMs) in a rotating Earth model has been employed to form the optimal control problem (OCP). Based on the Gauss pseudospectral method (GPM) for the OCP, the state and control variables are discretized using the Legendre-Gauss (LG) points in which the system dynamics is approximated on these LG points. Based on GPM, the linear quadratic regulator (LQR) controller for objective function of the OCP has been used for the state and control variables when the vacuum flight of the missile is constrained by a fixed terminal state-fixed terminal time condition. Then the objective function of OCP has been discretized in terms of state variables employing the Gauss differential approximation matrix. Using LQR based optimal control in conjunction with the GPM, input control variables are comprised to meet the terminal conditions of the system accordingly. In order to evaluate the performance and applicability of the present LQR based GPM optimal terminal states control in the exo-atmospheric flight of super heavy hypersonic ICBMs, the Titan II and RS-28 Sarmat are used as the launch vehicles. Numerical simulations for terminal errors are examined for the perturbed thrust and control problems. Numerical simulations reveal that the nonzero terminal errors are quite small in which the terminal accuracy of the solution can be regarded as acceptable. Moreover, it is concluded that the applied method effectively able to keep ICBM on to the desired terminal coordinates.