Distributionally Robust Linear Quadratic Control

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Authors: Bahar Taskesen, Dan Andrei Iancu, Çağıl Koçyiğit, Daniel Kuhn

Abstract

Linear-Quadratic-Gaussian (LQG) control is a fundamental control paradigm that is studied in various fields such as engineering, computer science, economics, and neuroscience. It involves controlling a system with linear dynamics and imperfect observations, subject to additive noise, with the goal of minimizing a quadratic cost function for the state and control variables. In this work, we consider a generalization of the discrete-time, finite-horizon LQG problem, where the noise distributions are unknown and belong to Wasserstein ambiguity sets centered at nominal (Gaussian) distributions. The objective is to minimize a worst-case cost across all distributions in the ambiguity set, including non-Gaussian distributions. Despite the added complexity, we prove that a control policy that is linear in the observations is optimal for this problem, as in the classic LQG problem. We propose a numerical solution method that efficiently characterizes this optimal control policy. Our method uses the Frank-Wolfe algorithm to identify the least-favorable distributions within the Wasserstein ambiguity sets and computes the controller's optimal policy using Kalman filter estimation under these distributions.

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Experiments

The key testable implication of our theory is the existence of an optimal bottleneck (latent) dimension for the encoder: With too few latent dimensions, the model is not rich enough; with too many, it encodes malignant dimensions that hurt (or simply do not improve) performance: The encoded information "saturates."

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Dependencies

• Numpy 1.24.3
• Pytorch 2.0
• [Pymanopt] https://pymanopt.org/
• [Cvxpy] https://www.cvxpy.org/install/
• [Mosek] https://www.mosek.com/

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Figure 1 (a)

python figure_1a.py

Figure 1 (b)

python figure_1b.py

Plotting the results

We plot the results using the plotter.ipynb notebook.