Determination of Nonlinear Optimal Feedback Law for Satellite Injection Problem Using Neighboring Optimal Control

Abstract

An optimal trajectory design of a nonlinear satellite injection problem for transfer to a final target orbit by minimizing the time was investigated. Indeed, this design was an exact solution to the nonlinear two-point boundary value problem which determined optimal control history as well as optimal state trajectories in the open-loop form. Furthermore, the obtained optimal guidance strategy was exerted in the closed-loop form against the environment disturbances using neighboring optimal control method in the exact solution. Neighboring Optimal Control (NOC) law could produce timevariant feedback gains minimizing the performance measure to second order for perturbations from a nominal optimal path. Generally, this law was a function of perturbations appeared in the states and constraints and could be computed utilizing the backward sweep method. The simulation results indicated that the presented methodology was successfully utilized in the real world applications with good robustness to each noise or disturbance produced in each state variable.

Variable–Time–Domain Neighboring Optimal Guidance, Part 1: Algorithm Structure

“…(9), as previously remarked). The corresponding (2n + p + q) scalar equations include the 2n scalar differential equations (4) and (10), which govern the state and costate time evolution, the q boundary conditions (2), and the p parameter conditions (13). The latter can be reformulated as follows.…”

“…Pesch and Kugelmann [7][8][9] developed a numerical method for the real-time computation of neighboring optimal feedback control in constrained optimal control problems involving also internal discontinuities. Afshari et al [10] investigated the problem of satellite injection. Seywald and Cliff [11] developed an algorithm dedicated to the near-optimal ascending path of a launch vehicle.…”

Variable-time-domain neighboring optimal guidance and attitude control of low-thrust lunar orbit transfers

“…Several time-varying gain matrices, referring to the nominal trajectory, are defined, computed offline, and stored in the onboard computer. Only a limited number of works have been devoted to studying neighboring optimal guidance [8][9][10][11][12] . A common difficulty encountered in implementing the NOG consists in the fact that the gain matrices become singular while approaching the final time.…”

“…Therefore, the problem of determining the minimum-fuel path can be simplified by assuming that at any time the out-of-plane variables equal 0, i.e. 0 and 0 [28] Only the state equations [4], [5], [7], and [8], the respective adjoint equations, and Eq. [22] are needed for the purpose of determining the optimal planar ascent path.…”

An analytical guidance law of planetary landing mission by minimizing the control effort expenditure

“…To create a closed-loop guidance policy of the satellite injection problem, Pourtakdoust and Novinzadeh presented a fuzzy algorithm that was augmented to the solution of the timeoptimal guidance strategy [4]. Afshari et al presented some analytic approaches in spacecraft guidance [5][6][7]. An optimal guidance law that minimized the commanded acceleration in three dimensions was obtained by Souza [8].…”