General Distribution Steering: A Sub-Optimal Solution by Convex Optimization

TL;DR

The paper tackles general distribution steering for a discrete-time first-order linear system by adopting a moment-system representation and formulating an optimization over truncated power moments. It constructs a convex subset of the non-convex feasible space, enabling convex or grid-based optimization to design a moment-control law that steers an initial distribution to a specified terminal moment profile. Control inputs are realized via squared Hellinger distance minimization to yield analytic distributions with matched moments, and the method is contrasted with Gaussian mixture models, highlighting convergence and error-bounding properties as the moment order grows. Numerical results on continuous and discrete distribution steering illustrate tradeoffs among smoothness, energy, and accuracy, validating the sub-optimal yet practical approach for distributional control and paving the way for extensions to nonlinear and multi-dimensional systems.

Abstract

General distribution steering is intrinsically an infinite-dimensional problem, when the continuous distributions to steer are arbitrary. We put forward a moment representation of the primal system for control in [42]. However, the system trajectory was a predetermined one without optimization towards a design criterion, which doesn't always ensure a most satisfactory solution. In this paper, we propose an optimization approach to the general distribution steering problem of the first-order discrete-time linear system, i.e., an optimal control law for the corresponding moment system. The domain of all feasible control inputs is non-convex and has a complex topology. We obtain a subset of it by minimizing a weighted sum of squared integral distances alongside the system trajectory. The feasible domain is then proved convex, and the optimal control problem can be treated as a convex optimization or by exhaustive search, based on the type of the cost function. Algorithms of steering for continuous and discrete distributions are then put forward respectively, by adopting a realization scheme of control inputs. We also provide an explicit advantage of our proposed algorithm by truncated power moments to the prevailing Gaussian Mixture Models. Experiments on different types of cost functions are given to validate the performance of our proposed algorithm. Since the moment system is a dimension-reduced counterpart of the primal system, we call this solution a sub-optimal one to the primal general distribution steering problem.

Figures (25)

• Figure 1: $\mathfrak{X}(k)$ at time steps $k = 0, 1, 2, 3, 4$ with cost function \ref{['costfunc1']}. The upper left figure shows $\mathbb{E}\left[ x(k)\right]$. The upper right one shows $\mathbb{E}\left[ x^{2}(k)\right]$. The lower left one shows $\mathbb{E}\left[ x^{3}(k)\right]$ and the lower right one shows $\mathbb{E}\left[ x^{4}(k)\right]$.
• Figure 2: $\mathfrak{U}(k)$ at time steps $k = 0, 1, 2, 3$ with cost function \ref{['costfunc1']}. The upper left figure shows $\mathbb{E}\left[ u(k)\right]$. The upper right one shows $\mathbb{E}\left[ u^{2}(k)\right]$. The lower left one shows $\mathbb{E}\left[ u^{3}(k)\right]$ and the lower right one shows $\mathbb{E}\left[ u^{4}(k)\right]$.
• Figure 3: Realized control inputs $\nu_{k}(\mathrm{u})$ of $u(k)$ by $\mathfrak{U}(k)$ for $k = 0, 1, 2, 3$ , which are obtained by cost function \ref{['costfunc1']}.
• Figure 4: $\mathfrak{X}(k)$ at time steps $k = 0, 1, 2, 3, 4$ with cost function \ref{['costfunc2']}.
• Figure 5: $\mathfrak{U}(k)$ at time steps $k = 0, 1, 2, 3$ with cost function \ref{['costfunc2']}.

Theorems & Definitions (6)

• Theorem 2.2: Controllability of system \ref{['momentsystem']}
• proof
• Lemma 3.1
• proof
• Proposition 3.2
• proof