Optimal Timing of Control-Law Updates for Unstable Systems with Continuous Control

Abstract

The optimal control of a linear system is studied relative to a periodic unstable trajectory using continuous control. Gaussian state uncertainties induce a statistical cost of controlling the state over a long period of time. The length of time between control-law updates directly impacts this cost, and in a hyperbolically unstable system, the time between control updates can take an optimal value. If the amount of uncertainty is fixed, there is an optimal distribution between position and velocity uncertainty. We apply these ideas to study the statistical cost of controlling a spacecraft in the vicinity of a relative equilibrium point and a Halo orbit in the Hill three-body problem. Nomenclature At = linear system dynamics matrix a = acceleration component B= linear system control matrix E = expectation operator G = universal gravitational constant, 6:673 10 11 m 3 kg 1 s 2 J = cost function J = optimal cost function l = reference length scale for nondimensionalizing M = mass of secondary body in the Hill three-body problem m = mean vector P = covariance matrix r = distance from origin in the Hill three-body problem r = position vector T = halo orbit period T u = control horizon T u = optimal control horizon t f = control-law planning horizon u = control vector v = velocity vector var = variance operator x = state vector x, y, z = positions of spacecraft in the rotating frame relative to the secondary body in the Hill three-body problem = Lyapunov characteristic exponent = costate vector 2 = variance/covariance value r , = nondimensional uncertainty parameters = reference time scale for nondimensionalizing t 1 ; t 0 = state transition matrix mapping linearized state from time t 0 to time t 1 ! = angular rotation rate of the secondary body about the primary in the Hill three-body problem

Orbital Motion in Strongly Perturbed Environments

“…A smaller value indicates that more frequent control is needed and that the spacecraft must detect its position deviation over a shorter timespan [137,59]. The strength of this instability can be measured by the characteristic time for the equilibrium point.…”

Optimal Trajectory Design of Formation Flying based on Attractive Sets

“…Note that the solutions starting from x 0 = [x, y, 0, 0] become drift motion along y-axis from Eq. (11). Figure 7 shows the contour of the attractive set for different weight parameters.…”

“…However, it is known that the quadratic cost can be used to bound the L 1 -norm. 11) We compare the optimal trajectories for finite-time, fixed final-state problem and infinite-time problem in terms of L 1 -norm and flight time t f . The L 1 -norm of the infinite-time problem approaches its infimum while flight time goes to infinity as Q → 0.…”

Orbital Targeting Using Reduced Eccentric Anomaly Low-Thrust Coefficients

“…Equation (26) can be integrated concurrently with the trajectory to calculate at each time step. Beginning with the zero thrust case, the thrust coefficient vector 0 is then iteratively updated by solving the two simultaneous vector equations…”