72 research outputs found
Gradient Flows in Filtering and Fisher-Rao Geometry
Uncertainty propagation and filtering can be interpreted as gradient flows
with respect to suitable metrics in the infinite dimensional manifold of
probability density functions. Such a viewpoint has been put forth in recent
literature, and a systematic way to formulate and solve the same for linear
Gaussian systems has appeared in our previous work where the gradient flows
were realized via proximal operators with respect to Wasserstein metric arising
in optimal mass transport. In this paper, we derive the evolution equations as
proximal operators with respect to Fisher-Rao metric arising in information
geometry. We develop the linear Gaussian case in detail and show that a
template two step optimization procedure proposed earlier by the authors still
applies. Our objective is to provide new geometric interpretations of known
equations in filtering, and to clarify the implication of different choices of
metric
A Polynomial Chaos Approach to Stochastic LQ Optimal Control: Error Bounds and Infinite-Horizon Results
The stochastic linear-quadratic regulator problem subject to Gaussian
disturbances is well known and usually addressed via a moment-based
reformulation. Here, we leverage polynomial chaos expansions, which model
random variables via series expansions in a suitable
probability space, to tackle the non-Gaussian case. We present the optimal
solutions for finite and infinite horizons. Moreover, we quantify the
truncation error and we analyze the infinite-horizon asymptotics. We show that
the limit of the optimal trajectory is the unique solution to a stationary
optimization problem in the sense of probability measures. A numerical example
illustrates our findings
Covariance Steering of Discrete-Time Linear Systems with Mixed Multiplicative and Additive Noise
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 convex
semi-definite program (SDP) after applying suitable variable transformations
and constraint relaxations. Furthermore, we propose a two-step solution
procedure for the exact CS problem based on the relaxed problem formulation
which returns a feasible solution, if there exists one. Finally, results from
numerical experiments are provided to show the efficacy of the proposed
solution methods
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