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Risk-sensitive Affine Control Synthesis for Stationary LTI Systems

Yang Hu, Shahriar Talebi, Na Li

Abstract

To address deviations from expected performance in stochastic systems, we propose a risk-sensitive control synthesis method to minimize certain risk measures over the limiting stationary distribution. Specifically, we extend Worst-case Conditional Value-at-Risk (W-CVaR) optimization for Linear Time-invariant (LTI) systems to handle nonzero-mean noise and affine controllers, using only the first and second moments of noise, which enhances robustness against model uncertainty. Highlighting the strong coupling between the linear and bias terms of the controller, we reformulate the synthesis problem as a Bilinear Matrix Inequality (BMI), and propose an alternating optimization algorithm with guaranteed convergence. Finally, we demonstrate the numerical performance of our approach in two representative settings, which shows that the proposed algorithm successfully synthesizes risk-sensitive controllers that outperform the naïve LQR baseline.

Risk-sensitive Affine Control Synthesis for Stationary LTI Systems

Abstract

To address deviations from expected performance in stochastic systems, we propose a risk-sensitive control synthesis method to minimize certain risk measures over the limiting stationary distribution. Specifically, we extend Worst-case Conditional Value-at-Risk (W-CVaR) optimization for Linear Time-invariant (LTI) systems to handle nonzero-mean noise and affine controllers, using only the first and second moments of noise, which enhances robustness against model uncertainty. Highlighting the strong coupling between the linear and bias terms of the controller, we reformulate the synthesis problem as a Bilinear Matrix Inequality (BMI), and propose an alternating optimization algorithm with guaranteed convergence. Finally, we demonstrate the numerical performance of our approach in two representative settings, which shows that the proposed algorithm successfully synthesizes risk-sensitive controllers that outperform the naïve LQR baseline.

Paper Structure

This paper contains 19 sections, 9 theorems, 38 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Risk Measure: Worst-case CVaR
  4. The Risk-sensitive Affine Control Synthesis Problem
  5. An Example: Naive Affine LQR Controller Is Inferior
  6. Risk-sensitive Affine Control Synthesis
  7. Intrinsic Challenges of the nonzero-mean Problem
  8. Relaxation to a Bilinear Matrix Inequality (BMI) Problem
  9. An Alternating Optimization Algorithm
  10. Convergence Guarantee
  11. Simulations
  12. Inverted Pendulum
  13. The Multi-zone HVAC Thermal Control
  14. Conclusions
  15. Proofs
  16. ...and 4 more sections

Key Result

lemma 1

With a stabilizing affine feedback controller $u = Kx + \ell$ (i.e., $K \in \mathcal{K}$), the closed-loop system $x_{t+1} = (A+BK) x_t + B \ell + w_t$ admits a unique set of first two limiting moments $(\bar{\mu}, \bar{\varSigma})$, which are specified by Further, these limiting moments are invariant under the closed-loop dynamics, i.e.,

Figures (6)

  • Figure 1: Comparison of $(K^{\star}, \ell^{\star})$ and $(K_{\textrm{lqr}}, \ell_{\textrm{lqr}})$.
  • Figure 2: Convergence of the algorithm ($L = 1$)
  • Figure 3: Performance of the solution controller against baselines for Inverted Pendulum with different pole lengths $L$.
  • Figure 4: Performance of the solution controller against baselines for an HVAC grid system of size $d \times d$.
  • Figure : HVAC grid ($d = 3$).
  • ...and 1 more figures

Theorems & Definitions (18)

  • lemma 1
  • definition 1: CVaR and W-CVaR rockafellar2000optimizationzymler2013distributionally
  • theorem 1
  • proof
  • remark 1
  • remark 2: Numerical considerations
  • lemma 2
  • proof
  • theorem 2
  • proof
  • ...and 8 more