Neighboring optimal control based feedback law for the advanced launch system

Abstract

In this paper a robust feedback algorithm is presented for a near-minimum-fuel ascent of a generic two-stage launch vehicle operating in the equatorial plane. The development of the algorithm is based on the ideas of neighboring optimal control and can be divided into three phases. In phase 1 the formalism of optimal control is employed to calculate fuel-optimal ascent trajectories for a simple point-mass model. In phase 2 these trajectories are used to numerically calculate gain functions of time for the control(s), for the total flight time, and possibly for other variables of interest. In phase 3 these gains are used to determine feedback expressions for the controls associated with a more realistic model of a launch vehicle. With the advanced launch system in mind, all calculations in this paper are performed on a two-stage vehicle with fixed thrust history, but this restriction is by no means important for the approach taken. Performance and robustness of the algorithm is found to be excellent. cg y cg * TB yxB JC TC J TC T B T c ft Nomenclature = x coordinate of center of gravity = y coordinate of center of gravity = x coordinate of point from where booster thrust emanates = y coordinate of point from where booster thrust emanates = x coordinate of point from where core thrust emanates = y coordinate of point from where booster thrust emanates = thrust magnitude of booster = thrust magnitude of core = gimbal angle for core thrust S re f = reference length J re f = reference area

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

“…(9)) yields the control variables as functions of the adjoint variables and the state variables; Eq. (10) is the adjoint (or costate) equation, together with the related boundary conditions (11) and (12); Eq. (13) is equivalent to p algebraic scalar equations.…”

“…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. Yan et al [12] proposed a method that avoids solving the Riccati differential equation and instead adopts a pseudospectral discretization approach.…”

Neighboring optimal control for fixed-time multi-burn orbital transfers

“…Based on this method, an increasing number of literatures, including Refs. [36][37][38][39][40] and the references therein, on the topic of the NOG for orbital transfer problems have been published. More recently, a variable-timedomain NOG was proposed by Pontani et al [33,34] to avoid the numerical difficulties arising from the singularity of the gain matrices while approaching the final time and it was then applied to a continuous thrust space trajectories [35].…”

Mathematical Models for Aircraft Trajectory Design: A Survey

“…As a result, the solution of trajectory optimization problem in real-time is still elusive. A common line of attack for solving trajectory optimization problems in real time (or near real time) is to divide the problem into two phases: an offline phase and an online phase [74,48,44,82]. The offline phase consists of solving the optimal control problem for various reference trajectories and storing these reference trajectories onboard for later online use.…”