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Risk-Aware Linear-Quadratic Regulation with Temporally Coupled States

Chuanning Wei, Kin Fung Li, Dionysis Kalogerias, Margaret P. Chapman

Abstract

We formulate and solve a discrete-time linear-quadratic regulation (LQR) problem in a finite horizon that penalizes temporal variability and stochastic variability of the state trajectory. Our approach enables the user to strike a balance between regulating the state and reducing temporal variability, with explicit sensitivity to risk. We achieve this by extending a risk measure called predictive variance to a setting with temporally coupled states. Numerical examples demonstrate the effect of temporal coupling in both risk-aware and risk-neutral control settings. Particularly, we observe that explicitly penalizing temporal variability alone can also reduce stochastic variability.

Risk-Aware Linear-Quadratic Regulation with Temporally Coupled States

Abstract

We formulate and solve a discrete-time linear-quadratic regulation (LQR) problem in a finite horizon that penalizes temporal variability and stochastic variability of the state trajectory. Our approach enables the user to strike a balance between regulating the state and reducing temporal variability, with explicit sensitivity to risk. We achieve this by extending a risk measure called predictive variance to a setting with temporally coupled states. Numerical examples demonstrate the effect of temporal coupling in both risk-aware and risk-neutral control settings. Particularly, we observe that explicitly penalizing temporal variability alone can also reduce stochastic variability.
Paper Structure (9 sections, 5 theorems, 26 equations, 2 figures)

This paper contains 9 sections, 5 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Sequential State Energy and its Predictive Variance
  4. Risk-Aware Optimal Control Problem
  5. Reformulation to a Quadratic Form
  6. Risk-Aware Optimal Controller Synthesis
  7. Numerical Examples
  8. Conclusion
  9. Proof of Lemma \ref{['lemma:delta-t-decomposition']}

Key Result

Lemma 1

Let Assumption assu1 hold and $t \in \{1,\dots,N\}$ be given. Define the prediction error of the state energy by $y_t \coloneqq x_t^\top Q_{00} x_t - \mathbb{E}(x_t^\top Q_{00} x_t | \mathcal{F}_{t-1})$. Also, define If $\mathbf{k}_t \ge 1$, then and $\vartheta \coloneqq \delta - 4 \mathrm{tr}((\Sigma Q_{00})^2)$. If $\mathbf{k}_t = 0$, then $\Delta_t^2 = \Delta_{t,1}$ is integrable and $\mathbb

Figures (2)

  • Figure 1: Performance of various controllers. Each row compares a set of controllers. Columns 1--3: Trajectories of the point mass from 5000 simulations under $\mathbf{u}^*$ with some choice of $\theta = (\beta,\mathbf{k},\lambda)$, and the corresponding values of $\bar{\mathcal{D}}_\text{tot}$ and $\bar{\mathcal{U}}_\text{tot}$; for each $t \in \{0,2,4,\dots,100\}$, we show the associated empirical mean position (dot) and interval of length $\ell_{t,1}$ containing 95% of the samples of $p_{t,\mu}$ (centered at the median of $p_{t,\mu}$). Columns 4--5: $\ell_{t,1}$ and $\ell_{t,2}$ versus $t$.
  • Figure 2: $\bar{\mathcal{D}}_\text{tot}$ versus $\bar{\mathcal{P}}_\text{max}$ (blue) and $\bar{\mathcal{U}}_\text{tot}$ versus $\bar{\mathcal{P}}_\text{max}$ (red) calculated from $500$ simulations for each $\theta = (\beta,\mathbf{k},\lambda)$. In each subplot, $\beta \in \{0,0.5,\dots,10\}$ is varied for some fixed $(\mathbf{k},\lambda)$. The square and circle markers indicate $\beta = 0$ and $\beta = 10$, respectively. Note that $\beta=0$ represents the absence of temporal coupling, and corresponds to the controller from tsiamis2020risk if $\lambda > 0$. A star marker corresponds to $\theta_8 = (2,9,0)$ or $\theta_9=(1.5,9,0.2)$.

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Corollary 1