Covariance Steering of Discrete-Time Stochastic Linear Systems Based on Wasserstein Distance Terminal Cost

Balci, Isin M.; Bakolas, Efstathios

doi:10.23919/acc50511.2021.9483289

Cited by 4 publications

(16 citation statements)

References 44 publications

(56 reference statements)

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“…We use the inequalities and in the sense of Löwner partial order. Kronecker product, Kronecker sum, and the vec operator: The basic properties of Kronecker product will be useful in the sequel, including (1) and that matrix transpose and inverse are distributive w.r.t. the Kronecker product.…”

Section: Preliminariesmentioning

confidence: 99%

“…It was shown in [1] that the problem of discrete time covariance steering with Wasserstein terminal cost subject to (9) (or equivalently (10)), can be reduced to a difference of convex functions program, provided the control policy is parameterized as…”

Section: Problem Set Upmentioning

confidence: 99%

“…This instance of stochastic optimal control problem seeks for a feedback control policy that will steer the probability distribution of the state of the uncertain system, close to a goal multivariate normal distribution over a finite time horizon, where the closeness of the two distributions is measured in terms of the squared Wasserstein distance between them. In our previous work [1], we have shown that the latter problem can be reduced into a difference of convex functions program (DCP) provided that the control policy conforms to the so-called state feedback control parametrization according to which the control input can be expressed as an affine function of the current state and all past states visited by the system. Whereas the focus in [1] was on the control design problem, in this work we focus on the analysis of the problem and in particular, addressing Isin M. Balci and Efstathios Bakolas are with the Department of Aerospace Engineering and Engineering Mechanics, University of Texas at Austin, TX 78712-1221, USA; Abhishek Halder is with the Department of Applied Mathematics, University of California, Santa Cruz, CA 95064, USA; Emails: isinmertbalci@utexas.edu, ahalder@ucsc.edu, bakolas@austin.utexas.edu questions about the existence and uniqueness of solutions and the convexity (or lack thereof) of the performance index.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, finitehorizon covariance control problems for Gaussian linear systems have received significant attention; the reader may refer to [5]- [7] for the continuous-time case and [8]- [14] for the discrete-time case. The covariance steering problem for continuous-time Gaussian linear systems with a Wasserstein distance terminal cost was first studied in [15] whereas the same problem but for the discrete-time case was studied in [1]. Both of these references present numerical algorithms (shooting method in [15] and convex-concave procedure in [1]) for control design but do not address theoretical questions regarding the existence and uniqueness of solution, or investigate convexity properties of the performance index.…”

Section: Introductionmentioning

confidence: 99%

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On the Convexity of Discrete Time Covariance Steering in Stochastic Linear Systems with Wasserstein Terminal Cost

Balci¹,

Halder²,

Bakolas³

2021

Preprint

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In this work, we analyze the properties of the solution to the covariance steering problem for discrete time Gaussian linear systems with a squared Wasserstein distance terminal cost. In our previous work [1], we have shown that by utilizing the state feedback control policy parametrization, this stochastic optimal control problem can be associated with a difference of convex functions program. Here, we revisit the same covariance control problem but this time we focus on the analysis of the problem. Specifically, we establish the existence of solutions to the optimization problem and derive the first and second order conditions for optimality. We provide analytic expressions for the gradient and the Hessian of the performance index by utilizing specialized tools from matrix calculus. Subsequently, we prove that the optimization problem always admits a global minimizer, and finally, we provide a sufficient condition for the performance index to be a strictly convex function (under the latter condition, the problem admits a unique global minimizer). In particular, we show that when the terminal state covariance is upper bounded, with respect to the Löwner partial order, by the covariance matrix of the desired terminal normal distribution, then our problem admits a unique global minimizing state feedback gain. The results of this paper set the stage for the development of specialized control design tools that exploit the structure of the solution to the covariance steering problem with a squared Wasserstein distance terminal cost.

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Section: Preliminariesmentioning

confidence: 99%

Section: Problem Set Upmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

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mentioning

confidence: 99%

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On the Convexity of Discrete Time Covariance Steering in Stochastic Linear Systems with Wasserstein Terminal Cost

Balci¹,

Halder²,

Bakolas³

2021

Preprint

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Multi-Agent Distributed Optimal Control for Tracking Large-Scale Multi-Target Systems in Dynamic Environments

Abdulghafoor¹,

Bakolas²

2022

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<p>This paper considers the problem of motion coordination for a multi-agent network whose goal is to track a large-scale multi-target system in a region populated by dynamic obstacles. We first characterize a density path which corresponds to the expected evolution of the macroscopic state of the multi-target system, which is represented by the probability density function (PDF) of a time-varying Gaussian mixture (GM). We compute this density path by using an adaptive optimal control method which accounts for the distribution of the (possibly moving) obstacles over the environment described by a time-varying obstacle map function. We show that each target of the multi-target system can find microscopic inputs that can collectively realize the density path while guaranteeing obstacle avoidance at all times. Subsequently, we propose a Voronoi distributed motion coordination algorithm which determines the individual control inputs of each agent of the multi-agent network so that the latter can track the multi-target system while avoiding collisions with obstacles and their teammates. The proposed algorithm relies on a distributed move-to-centroid control law in which the density over the Voronoi cell of each agent is determined by the estimated macroscopic state evolution of the multi-target system. Finally, simulation results are presented to showcase the effectiveness of our proposed approach.</p>

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Covariance Control of Discrete-Time Gaussian Linear Systems Using Affine Disturbance Feedback Control Policies

Balci¹,

Bakolas²

2021

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In this paper, we present a new control policy parametrization for the finite-horizon covariance steering problem for discrete-time Gaussian linear systems (DTGLS) which can reduce the latter stochastic optimal control problem to a tractable optimization problem. The covariance steering problem seeks for a feedback control policy that will steer the state covariance of a DTGLS to a desired positive definite matrix in finite time. We consider two different formulations of the covariance steering problem, one with hard terminal LMI constraints (hard-constrained covariance steering) and another one with soft terminal constraints in the form of a terminal cost which corresponds to the squared Wasserstein distance between the actual terminal state (Gaussian) distribution and the desired one (soft-constrained covariance steering). We propose a solution approach that relies on the affine disturbance feedback parametrization for both problem formulations. We show that this particular parametrization allows us to reduce the hard-constrained covariance steering problem into a semidefinite program (SDP) and the soft-constrained covariance steering problem into a difference of convex functions program (DCP). Finally, we show the advantages of our approach over other covariance steering algorithms in terms of computational complexity and computation time by means of theoretical analysis and numerical simulations.

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Covariance Steering of Discrete-Time Stochastic Linear Systems Based on Wasserstein Distance Terminal Cost

Cited by 4 publications

References 44 publications

On the Convexity of Discrete Time Covariance Steering in Stochastic Linear Systems with Wasserstein Terminal Cost

On the Convexity of Discrete Time Covariance Steering in Stochastic Linear Systems with Wasserstein Terminal Cost

Multi-Agent Distributed Optimal Control for Tracking Large-Scale Multi-Target Systems in Dynamic Environments

Covariance Control of Discrete-Time Gaussian Linear Systems Using Affine Disturbance Feedback Control Policies

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