Convex Approach to Covariance Control with Application to Stochastic Low-Thrust Trajectory Optimization

Constrained minimum variance and covariance steering based on affine disturbance feedback control parameterization

“…While the equivalence between the state history feedback control policy in (44) and the disturbance feedback policy (9) has been established in Reference 32, we extend this result by showing that the auxiliary variable parametrization (51) is also equivalent to the other parametrizations, and we provide a bijective transformation between the parameters of the different policies. Proposition 7 summarizes the necessary and sufficient conditions for the control policies to generate the same input under the same w. Proposition 7.…”

“…An additional important observation is that the auxiliary variable feedback policy provided in (51) with 𝛾 = 1 is identical to the state feedback policy when there is no noise, i.e., when W t = 0 for all t. In contrast, the disturbance feedback control policy with 𝛾 = 1 corresponds to the open-loop control policy in this noise-free scenario, and consequently, the covariance cannot be steered by the disturbance feedback policy in this instance. The following proposition formally articulates the first assertion regarding the auxiliary feedback policy.…”

Covariance Steering for Systems Subject to Unknown Parameters

“…order to ensure that only the additive term v k appears in the mean dynamics and overcome the bilinearities in the covariance constraint by utilizing the symmetry of the covariance and performing a change of variables to create a semidefinite program [9], [11], [21]. However, due to the state-dependent nature of the multiplicative disturbances, it is not possible to remove the feedback policy from all realizations of (13) because the state mean is not independent of the disturbances.…”

“…Note that γ * , ζ * is therefore a solution of Problem ( 21) for a particular linearization point, which we denote by γ * , ζ * . We introduce the following theorem regarding the validity of this solution, which is found using the convex local approximate problem (21), in relation to the original covariance steering problem (10). (21), it is also a stationary point of Problem (16).…”

“…We introduce the following theorem regarding the validity of this solution, which is found using the convex local approximate problem (21), in relation to the original covariance steering problem (10). (21), it is also a stationary point of Problem (16). For the second statement, Problem ( 16) is equivalent to Problem (10) except for the chance constraints (16g)-(16h), which are conservative approximations of (10e)-(10f) due to the use of Cantelli's inequality in (16g)-(16h) and (21g)-(21h).…”