Distribution Steering for Discrete-Time Uncertain Ensemble Systems

TL;DR

The paper addresses distribution steering for first-order discrete-time ensemble systems with uncertain parameters without assuming Gaussianity. It introduces a moment-based reduction to a finite-dimensional moment system, and develops a convex, optimal control framework to steer the first $2n$ moments of the state distribution to a prescribed target, while a realization step via the Hamburger moment problem and KL-based density estimation yields feasible control inputs. A density-steering control law $u_i(k)=-c(k)a_i(k)x_i(k)+ ilde{u}_i(k)$ enables stable and even unstable dynamics to be managed through a group-level control distribution, which is implemented via occupation-measure techniques and mean-field-like aggregation. A practical algorithmic pipeline solves for the moment-control policies and realizes them for large swarms, demonstrated by a numerical example with 5000 agents and non-Gaussian distributions, showing close alignment with the target distribution. The results offer a scalable, non-Gaussian distribution steering approach with potential impact on swarm robotics and uncertainty-aware control in large populations, and point to future work on extending to multi-dimensional and nonlinear systems using real-algebraic geometry tools.

Abstract

Ensemble systems appear frequently in many engineering applications and, as a result, they have become an important research topic in control theory. These systems are best characterized by the evolution of their underlying state distribution. Despite the work to date, few results exist dealing with the problem of directly modifying (i.e., ``steering'') the distribution of an ensemble system. In addition, in most existing results, the distribution of the states of an ensemble of discrete-time systems is assumed to be Gaussian. However, in case the system parameters are uncertain, it is not always realistic to assume that the distribution of the system follows a Gaussian distribution, thus complicating the solution of the overall problem. In this paper, we address the general distribution steering problem for first-order discrete-time ensemble systems, where the distributions of the system parameters and the states are arbitrary with finite first few moments. Linear system dynamics are considered using the method of power moments to transform the original infinite-dimensional problem into a finite-dimensional one. We also propose a control law for the ensuing moment system, which allows us to obtain the power moments of the desired control inputs. Finally, we solve the inverse problem to obtain the feasible control inputs from their corresponding power moments. We provide a numerical example to validate our theoretical developments.

Figures (6)

• Figure 1: An illustration of the probability distribution $\kappa_{k}(\mathrm{x})$ of group system state $x(k)$. Each $x_i(k)$ is a sample from $\chi_k(\mathrm{x})$ in $\kappa_k(\mathrm{x})$.
• Figure 2: $\mathscr{X}(k)$ at time steps $k = 0, 1, 2, 3, 4$.
• Figure 3: $\tilde{\mathscr{U}}(k)$ at time steps $k = 0, 1, 2, 3$.
• Figure 4: Realized distributions of control inputs $\nu_{k}(u)$ by $\tilde{\mathscr{U}}(k)$ for $k = 0, 1, 2$, which are obtained by our proposed control scheme.
• Figure 5: The histograms of $u_{i}(k)$ at time step $k$ for each agent $i$. The upper left and right figures are $u_{i}(0)$ and $u_{i}(1), i = 1, \cdots, 5000$ respectively. The lower left and right figures are $u_{i}(2)$ and $u_{i}(3)$ respectively.

Theorems & Definitions (5)

• Lemma 2.1
• proof
• Remark
• Theorem 3.1
• proof