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Distributionally Robust Stochastic Data-Driven Predictive Control with Optimized Feedback Gain

Ruiqi Li, John W. Simpson-Porco, Stephen L. Smith

TL;DR

This paper relaxes the assumption of Gaussian process and measurement noise, and enables optimization of the gain matrix within the affine feedback policy, and proves that the proposed data-driven control method yields control inputs identical to those produced by an equivalent model-based stochastic predictive controller.

Abstract

We consider the problem of direct data-driven predictive control for unknown stochastic linear time-invariant (LTI) systems with partial state observation. Building upon our previous research on data-driven stochastic control, this paper (i) relaxes the assumption of Gaussian process and measurement noise, and (ii) enables optimization of the gain matrix within the affine feedback policy. Output safety constraints are modelled using conditional value-at-risk, and enforced in a distributionally robust sense. Under idealized assumptions, we prove that our proposed data-driven control method yields control inputs identical to those produced by an equivalent model-based stochastic predictive controller. A simulation study illustrates the enhanced performance of our approach over previous designs.

Distributionally Robust Stochastic Data-Driven Predictive Control with Optimized Feedback Gain

TL;DR

This paper relaxes the assumption of Gaussian process and measurement noise, and enables optimization of the gain matrix within the affine feedback policy, and proves that the proposed data-driven control method yields control inputs identical to those produced by an equivalent model-based stochastic predictive controller.

Abstract

We consider the problem of direct data-driven predictive control for unknown stochastic linear time-invariant (LTI) systems with partial state observation. Building upon our previous research on data-driven stochastic control, this paper (i) relaxes the assumption of Gaussian process and measurement noise, and (ii) enables optimization of the gain matrix within the affine feedback policy. Output safety constraints are modelled using conditional value-at-risk, and enforced in a distributionally robust sense. Under idealized assumptions, we prove that our proposed data-driven control method yields control inputs identical to those produced by an equivalent model-based stochastic predictive controller. A simulation study illustrates the enhanced performance of our approach over previous designs.
Paper Structure (18 sections, 6 theorems, 43 equations, 1 figure, 2 algorithms)

This paper contains 18 sections, 6 theorems, 43 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Stochastic Model-Based and Data-Driven Predictive Control
  4. A framework of Stochastic Model Predictive Control
  5. State Estimation
  6. Feedback Control Policies
  7. Deterministic Formulation of Cost and Constraint
  8. SMPC Optimization Problem and Algorithm
  9. Stochastic Data-Driven Predictive Control (SDDPC)
  10. Use of Offline Data
  11. Auxiliary State-Space Model
  12. Data-Driven State Estimation, Feedback and Constraint
  13. SDDPC Optimization Problem and Algorithm
  14. Theoretical Equivalence of SMPC and SDDPC
  15. Numerical Case Study
  16. ...and 3 more sections

Key Result

Lemma 1

With $h(u_t, y_t)$ as in Eq:constraint_function, for $t \in \mathbb{Z}_{[k,k+N)}$, Eq:DR_CVaR_Constraint holds iff

Figures (1)

  • Figure 1: The system's first output signal with DR/O-SDDPC.

Theorems & Definitions (10)

  • Lemma 1: SOC Expression of DR-CVaR Constraint
  • proof
  • Lemma 2: Data Representation of $\mathbf{\Gamma}$ and $D$ SDDPC
  • Lemma 3: Auxiliary Model SDDPC
  • Lemma 4: Related Means of $x_k$ and $\mathbf{x}_k$ SDDPC
  • Proposition 5: Equivalence of Optimization Problems
  • proof
  • Theorem 7: Equivalence of SMPC and SDDPC
  • proof
  • proof