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Stochastic LQR Design With Disturbance Preview

Jietian Liu, Laurent Lessard, Peter Seiler

TL;DR

This work develops a stochastic LQR framework with finite disturbance preview in discrete time, providing both finite-horizon and infinite-horizon solutions for LTV/LTI systems under nested information. The authors derive backward Riccati recursions and explicit gain structures that incorporate preview disturbances, and they prove a version of the principle of optimality that relies solely on nested information. A key result is that the finite-preview controller converges geometrically to the optimal noncausal controller as the preview horizon $p$ grows, with clear expressions for the cost reductions and Lyapunov-based characterizations. The approach is illustrated with a Boeing 747 longitudinal model and a time-varying mass–spring–damper system, showing practical gains from disturbance preview and the convergence to noncausal performance as preview increases. Overall, the paper provides a principled, state-dimension-preserving method to exploit disturbance preview in stochastic LQR design with strong theoretical guarantees and concrete numerical demonstrations.

Abstract

This paper considers the discrete-time, stochastic LQR problem with $p$ steps of disturbance preview information where $p$ is finite. We first derive the solution for this problem on a finite horizon with linear, time-varying dynamics and time-varying costs. Next, we derive the solution on the infinite horizon with linear, time-invariant dynamics and time-invariant costs. Our proofs rely on the well-known principle of optimality. We provide an independent proof for the principle of optimality that relies only on nested information structure. Finally, we show that the finite preview controller converges to the optimal noncausal controller as the preview horizon $p$ tends to infinity. We also provide a simple example to illustrate both the finite and infinite horizon results.

Stochastic LQR Design With Disturbance Preview

TL;DR

This work develops a stochastic LQR framework with finite disturbance preview in discrete time, providing both finite-horizon and infinite-horizon solutions for LTV/LTI systems under nested information. The authors derive backward Riccati recursions and explicit gain structures that incorporate preview disturbances, and they prove a version of the principle of optimality that relies solely on nested information. A key result is that the finite-preview controller converges geometrically to the optimal noncausal controller as the preview horizon grows, with clear expressions for the cost reductions and Lyapunov-based characterizations. The approach is illustrated with a Boeing 747 longitudinal model and a time-varying mass–spring–damper system, showing practical gains from disturbance preview and the convergence to noncausal performance as preview increases. Overall, the paper provides a principled, state-dimension-preserving method to exploit disturbance preview in stochastic LQR design with strong theoretical guarantees and concrete numerical demonstrations.

Abstract

This paper considers the discrete-time, stochastic LQR problem with steps of disturbance preview information where is finite. We first derive the solution for this problem on a finite horizon with linear, time-varying dynamics and time-varying costs. Next, we derive the solution on the infinite horizon with linear, time-invariant dynamics and time-invariant costs. Our proofs rely on the well-known principle of optimality. We provide an independent proof for the principle of optimality that relies only on nested information structure. Finally, we show that the finite preview controller converges to the optimal noncausal controller as the preview horizon tends to infinity. We also provide a simple example to illustrate both the finite and infinite horizon results.

Paper Structure

This paper contains 13 sections, 9 theorems, 82 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Formulation
  4. Principle of Optimality
  5. Main Results
  6. FH Stochastic LQR with Preview
  7. IH Stochastic LQR with Preview
  8. Relation to Optimal Noncausal Controller
  9. Example
  10. Boeing 747
  11. Mass-spring-damper
  12. Conclusion
  13. Acknowledgments

Key Result

Theorem 1

Define the value function in eq:ValueFunction and assume the information is nested: $i_0 \subseteq i_1 \subseteq \dots \subseteq i_{T-1}$. Then the value function satisfies the following backwards recursion starting from $t=T$: Moreover, the optimal cost $J^*(i_0)$ is achieved by a deterministic policy defined by selecting $u_t=k_t(i_t)$ at each $t$ to minimize the value function in eq:ValueItera

Figures (4)

  • Figure 1: Comparison of average cost obtained by optimal FH controller and optimal $H_2$ controller computed via an augmented system.
  • Figure 2: Comparison of IH cost versus preview horizon $p$.
  • Figure 3: Comparison of IH relative error on a log scale versus preview horizon $p$.
  • Figure 4: Comparison of average cost obtained by optimal FH controller and optimal noncausal controller in LTV system.

Theorems & Definitions (17)

  • Theorem 1: Principle of Optimality
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • ...and 7 more