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Distributionally Robust LQG control under Distributed Uncertainty

Lucia Falconi, Augusto Ferrante, Mattia Zorzi

Abstract

A new paradigm is proposed for the robustification of the LQG controller against distributional uncertainties on the noise process. Our controller optimizes the closed-loop performances in the worst possible scenario under the constraint that the noise distributional aberrance does not exceed a certain threshold limiting the relative entropy pseudo-distance between the actual noise distribution the nominal one. The main novelty is that the bounds on the distributional aberrance can be arbitrarily distributed along the whole disturbance trajectory. We discuss why this can, in principle, be a substantial advantage and we provide simulation results that substantiate such a principle.

Distributionally Robust LQG control under Distributed Uncertainty

Abstract

A new paradigm is proposed for the robustification of the LQG controller against distributional uncertainties on the noise process. Our controller optimizes the closed-loop performances in the worst possible scenario under the constraint that the noise distributional aberrance does not exceed a certain threshold limiting the relative entropy pseudo-distance between the actual noise distribution the nominal one. The main novelty is that the bounds on the distributional aberrance can be arbitrarily distributed along the whole disturbance trajectory. We discuss why this can, in principle, be a substantial advantage and we provide simulation results that substantiate such a principle.
Paper Structure (11 sections, 18 theorems, 81 equations, 4 figures, 2 tables)

This paper contains 11 sections, 18 theorems, 81 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Worst performance analysis
  4. Distributionally robust control policy
  5. Simulations
  6. Extension
  7. Simulations
  8. Conclusions
  9. Relative Entropy and Optimization
  10. Convex Optimization
  11. Integral of multivariate exponential functions

Key Result

Theorem 3.1

The value function of Problem pb:WP has the following form where Here, where ${ S_t} \in \mathbf{S}^n_{++}$ is recursively computed as with terminal conditions ${ S_{N+1} }= Q_{N+1},$ and Moreover, the minimum in eq:wp_value_fcn is attained, meaning that there exists some $(\omega_{t}^\star, .. , \omega_{N}^\star) \in \Omega_t \times \dots \times \Omega_{N}$ such that $\mathcal{V}_{t} (x) =

Figures (4)

  • Figure 1: Inverted pendulum system.
  • Figure 2: LTF uncertain model.
  • Figure 3: Expected closed-loop cost versus the uncertain parameter $\Delta$, with $\Delta = c[1 \; 1 \; 1]$, $c \in [-1/\sqrt{3}, 1/\sqrt{3}].$ We compare the performance of the standard LQG controller (in black), the WDRC controller (in green), the D-LQG with a single-constraint (in blue) and the proposed D$^2$-LQG controller (in red).
  • Figure 4: Expected closed-loop cost versus the initial condition $\bar{x}_0$, with $\bar{x}_0 = d[1 \; 1 \; 1]$, $d \in [-1/\sqrt{3}, 1/\sqrt{3}]$. We compare the performance of the standard LQG controller (in black), the WDRC controller (in green), the D-LQG with a single-constraint (in blue) and the proposed D$^2$-LQG controller (in red).

Theorems & Definitions (23)

  • Remark 1
  • Theorem 3.1
  • Lemma 3.1
  • Theorem 3.2
  • Lemma 3.2
  • Remark 2
  • Theorem 4.1: Optimal policy
  • Lemma 4.1
  • Theorem 4.2
  • Lemma 4.2
  • ...and 13 more