This paper addresses output-feedback covariance steering for stochastic discrete-time linear time-invariant systems with unknown dynamics. We construct a controllable non-minimal state representation from past inputs and outputs, allowing the problem to be formulated in a standard state-feedback setting. The induced disturbance, however, is temporally correlated, requiring propagation of the state-disturbance cross-covariance. Using persistently exciting offline data, we employ an indirect approach for mean steering and a direct approach for covariance steering. The indirect formulation requires estimation of the mean dynamics, whereas the direct formulation requires estimation of the noise realization. We develop estimation methods suitable for temporally correlated noise and formulate the resulting covariance steering problem as a convex semidefinite program. Numerical simulations demonstrate the effectiveness of the proposed framework.
This work investigates the finite-horizon optimal covariance steering problem for discrete-time linear systems subject to both additive and multiplicative uncertainties as well as state and input chance constraints. In particular, a tractable convex approximation of the optimal covariance steering problem is developed by tightening the chance constraints and by introducing a suitable change of variables. The solution of the convex approximation is shown to be a valid (albeit potentially suboptimal) solution to the original chance-constrained covariance steering problem.
In this paper, we study the problem of how to optimally steer the state covariance of a general continuous-time linear stochastic system over a finite time interval subject to additive noise. Optimality here means reaching a target state covariance with minimal control energy. The additive noise may include a combination of white Gaussian noise and abrupt "jump noise" that is discontinuous in time. We first establish the controllability of the state covariance for linear time-varying stochastic systems. We then turn to the derivation of the optimal control, which entails solving two dynamicall…
This work considers the optimal covariance steering problem for systems subject to both additive noise and uncertain parameters which may enter multiplicatively with the state and the control. The unknown parameters are modeled as a constant random variable sampled from a distribution with known moments. The optimal covariance steering problem is formulated using a moment-based representation of the system dynamics, which includes dependence between the unknown parameters and future states, and is solved using sequential convex programming. The proposed approach is demonstrated numerically usi…
We consider the covariance steering problem for nonlinear control-affine systems. Our objective is to find an optimal control strategy to steer the state of a system from an initial distribution to a target one whose mean and covariance are given. Due to the nonlinearity, the existing techniques for linear covariance steering problems are not directly applicable. By leveraging the celebrated Girsanov theorem, we formulate the problem into an optimization over the space path distributions. We then adopt a generalized proximal gradient algorithm to solve this optimization, where each update requ…
This article extends the optimal covariance steering (CS) problem for discrete time linear stochastic systems modeled using moment-based ambiguity sets. To hedge against the uncertainty in the state distributions while performing covariance steering, distributionally robust risk constraints are employed during the optimal allocation of the risk. Specifically, a distributionally robust iterative risk allocation (DR-IRA) formalism is used to solve the optimal risk allocation problem for the CS problem using a two-stage approach. The upper-stage of DR-IRA is a convex problem that optimizes the ri…
In this paper we study the finite-horizon optimal covariance steering problem for a continuous-time linear stochastic system subject to both additive and multiplicative noise. The noise can be continuous or it may contain jumps. Additive noise does not depend on the state or the control, whereas multiplicative noise has a magnitude proportional to the current state. The cost is assumed to be quadratic in both the state and the control. First, the controllability of the state covariance is established under mild assumptions. Then, the optimal control for steering the covariance is provided. Las…
In this work, we consider the problem of steering the first two moments of the uncertain state of an unknown discrete-time stochastic nonlinear system to a given terminal distribution in finite time. Toward that goal, first, a non-parametric predictive model is learned from a set of available training data points using stochastic variational Gaussian process regression: a powerful and scalable machine learning tool for learning distributions over arbitrary nonlinear functions. Second, we formulate a tractable nonlinear covariance steering algorithm that utilizes the Gaussian process predictive…
Georgia Institute of Technology（佐治亚理工学院）；California Institute of Technology（加州理工学院）
A new belief space planning algorithm, called covariance steering Belief RoadMap (CS-BRM), is introduced, which is a multi-query algorithm for motion planning of dynamical systems under simultaneous motion and observation uncertainties. CS-BRM extends the probabilistic roadmap (PRM) approach to belief spaces and is based on the recently developed theory of covariance steering (CS) that enables guaranteed satisfaction of terminal belief constraints in finite-time. The nodes in the CS-BRM are sampled in belief space and represent distributions of the system states. A covariance steering controll…
In this work, we consider the problem of steering the first two moments of the uncertain state of a discrete time nonlinear stochastic system to prescribed goal quantities at a given final time. In principle, the latter problem can be formulated as a density tracking problem, which seeks for a feedback policy that will keep the probability density function of the state of the system close, in terms of an appropriate metric, to the goal density. The solution to the latter infinite-dimensional problem can be, however, a complex and computationally expensive task. Instead, we propose a more tract…
In this paper, we study the covariance steering (CS) problem for discrete-time linear systems subject to multiplicative and additive noise. Specifically, we consider two variants of the so-called CS problem. The goal of the first problem, which is called the exact CS problem, is to steer the mean and the covariance of the state process to their desired values in finite time. In the second one, which is called the ``relaxed'' CS problem, the covariance assignment constraint is relaxed into a positive semi-definite constraint. We show that the relaxed CS problem can be cast as an equivalent conv…
Continuous-Time Covariance Steering with Common Free-Final Time: Finite-Horizon Solutions and Infinite-Horizon Limits
Daniel Guggenheim School of Aerospace, Georgia Institute of Technology（佐治亚理工学院）；Department of Electrical and Computer Engineering, FAMU-FSU College of Engineering（FAMU-FSU工程学院）
The problem of optimizing affine feedback laws that explicitly steer the mean and covariance of an uncertain system state in the presence of a Gaussian random field is considered. Spatially-dependent disturbances are successively approximated with respect to a nominal trajectory by a sequence of jointly Gaussian random vectors. Sequential updates to the nominal control inputs are computed via convex optimization that includes the effect of affine state feedback, the perturbing effects of spatial disturbances, and chance constraints on the closed-loop state and control. The developed method is …
In this paper, we present a new trajectory optimization algorithm for stochastic linear systems which combines Model Predictive Path Integral (MPPI) control with Constrained Covariance Steering (CSS) to achieve high performance with safety guarantees (robustness). Although MPPI can be used to solve complex nonlinear trajectory optimization problems, it may not always handle constraints effectively and its performance may degrade in the presence of unmodeled disturbances. By contrast, CCS can handle probabilistic state and / or input constraints (e.g., chance constraints) and also steer the sta…
The goal of this paper is to address finite-horizon minimum variance and covariance steering problems for discrete-time stochastic (Gaussian) linear systems. On the one hand, the minimum variance problem seeks for a control policy that will steer the state mean of an uncertain system to a prescribed quantity while minimizing the trace of its terminal state covariance (or variance). On the other hand, the covariance steering problem seeks for a control policy that will steer the covariance of the terminal state to a prescribed positive definite matrix. We propose a solution approach that relies…
This paper addresses the problem of steering a discrete-time linear dynamical system from an initial Gaussian distribution to a final distribution in a game-theoretic setting. One of the two players strives to minimize a quadratic payoff, while at the same time tries to meet a given mean and covariance constraint at the final time-step. The other player maximizes the same payoff, but it is assumed to be indifferent to the terminal constraint. At first, the unconstrained version of the game is examined, and the necessary conditions for the existence of a saddle point are obtained. We then show …
We formulate a new class of two-person zero-sum differential games, in a stochastic setting, where a specification on a target terminal state distribution is imposed on the players. We address such added specification by introducing incentives to the game that guides the players to steer the join distribution accordingly. In the present paper, we only address linear quadratic games with Gaussian target distribution. The solution is characterized by a coupled Riccati equations system, resembling that in the standard linear quadratic differential games. Indeed, once the incentive function is cal…
In this paper, we study the feedback synthesis problem for steering the joint state density or ensemble subject to multi-input state feedback linearizable dynamics. This problem is of interest to many practical applications including that of dynamically shaping a robotic swarm. Our results here show that it is possible to exploit the structural nonlinearities to derive the feedback controllers steering the joint density from a prescribed shape to another while minimizing the expected control effort to do so. The developments herein build on our previous work, and extend the theory of the Schrö…
In this paper, we present new results on the covariance steering problem with Wasserstein distance terminal cost. We show that the state history feedback control policy parametrization, which has been used before to solve this class of problems, requires an unnecessarily large number of variables and can be replaced by a randomized state feedback policy which leads to more tractable problem formulations without any performance loss. In particular, we show that under the latter policy, the problem can be equivalently formulated as a semi-definite program (SDP) which is in sharp contrast with ou…
University of Texas at Austin（德克萨斯大学奥斯汀分校）；University of California, Santa Cruz（加州大学圣克鲁兹分校）
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, 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 f…
We consider a class of stochastic optimal control problems for discrete-time stochastic linear systems which seek for control policies that will steer the probability distribution of the terminal state of the system close to a desired Gaussian distribution. In our problem formulation, the closeness between the terminal state distribution and the desired (goal) distribution is measured in terms of the squared Wasserstein distance which is associated with a corresponding terminal cost term. We recast the stochastic optimal control problem as a finite-dimensional nonlinear program and we show tha…
We propose a stochastic prediction-control framework to promote safety in automated driving by directly controlling the joint state probability density functions (PDFs) subject to the vehicle dynamics via trajectory-level state feedback. To illustrate the main ideas, we focus on a multi-lane highway driving scenario although the proposed framework can be adapted to other contexts. The computational pipeline consists of a PDF prediction layer, followed by a PDF control layer. The prediction layer performs moving horizon nonparametric forecasts for the ego and the non-ego vehicles' stochastic st…
This paper proposes a novel active Simultaneous Localization and Mapping (SLAM) method with continuous trajectory optimization over a stochastic robot dynamics model. The problem is formalized as a stochastic optimal control over the continuous robot kinematic model to minimize a cost function that involves the covariance matrix of the landmark states. We tackle the problem by separately obtaining an open-loop control sequence subject to deterministic dynamics by iterative Covariance Regulation (iCR) and a closed-loop feedback control under stochastic robot and covariance dynamics by Linear Qu…
