Optimal Covariance Control for Stochastic Systems Under Chance Constraints

Abstract

This work addresses the optimal covariance control problem for stochastic discrete-time linear timevarying systems subject to chance constraints. Covariance steering is a stochastic control problem to steer the system state Gaussian distribution to another Gaussian distribution while minimizing a cost function. To the best of our knowledge, covariance steering problems have never been discussed with probabilistic chance constraints although it is a natural extension. In this work, first we show that, unlike the case with no chance constraints, the covariance steering with chance constraints problem cannot decouple the mean and covariance steering sub-problems. Then we propose an approach to solve the covariance steering with chance constraints problem by converting it to a semidefinite programming problem. The proposed algorithm is verified using two simple numerical simulations.

Model‐Reference Adaptive Moment Control of Uncertain Nonlinear Stochastic Systems

“…where w r is considered the zero mean white Gaussian noise with covariance 0.1. Now, using (16), the state-space model of moments for the stochastic reference model (41) is established where Γ raug (t) ∈ ℝ 9 × 1 and V raug (t) ∈ ℝ 9 × 1 are determined based on (13) and (14) and W aug ≜ [0 0.1] T . Considering a 1 = 5, a 2 = 6, a 3 = 7 and b 1 = b 2 = b 3 = 1 to have a stable and controllable reference model (41), the following matrices A aug ∈ ℝ 9 × 9 , B aug ∈ ℝ 9 × 9 and G aug ∈ ℝ 9 × 2 are obtained as:…”

“…where w r ¼ w 1r w 2r ½ T is considered the zero mean white Gaussian noise with covariance 0.1. Now, using (16), the statespace model of moments for the stochastic reference model (48) is established where Γ raug (t) ∈ ℝ 9 × 1 and V raug (t) ∈ ℝ 5 × 1 are determined based on (13) and (14) and W aug ≜ 0 0 0:1 0 0:1 ½ T . Considering a 1 = 9, a 2 = 6, a 3 = 7 and b 1 = b 2 = 5 to have a stable and controllable reference model (59), the following matrices A aug ∈ ℝ 9 × 9 , B aug ∈ ℝ 9 × 5 and G aug ∈ ℝ 9 × 5 are obtained as: ; also C aug is considered in order to control the first and the third A 1 , B 1 ), which is defined in (20), is controllable, by considering the closed-loop poles at -3, -3.25, -3.5, -3.75, -4, -4.25, -4.5, -5.5, -5.75, -6, -6.25, -6.5, -6.75, -7, using the place(A 1 , B 1 , ploes) function in Matlab, K c and K I are obtained and then V raug (t) is determined.…”

“…Using these augmented vectors (13)(14)(15) and moments equations (11) and (12), a new linear state-space model is established, which represents the state expected and covariance dynamics:…”

“…It was a covariance control method that controlled the second moment of linear system states. Since the 1970s, many studies on moment control of linear stochastic systems have been conducted [8][9][10][11][12][13][14], while the literature on moment control of nonlinear systems is more limited. The moment control of nonlinear stochastic systems is much more difficult than that related to linear systems.…”

Input Hard Constrained Optimal Covariance Steering

“…Such a situation occurs in many real-world scenarios. For example, in aircraft [8] or spacecraft [10] control problems the control command values have physical restrictions such as minimum/maximum thrust.…”