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Disturbance-adaptive Model Predictive Control for Bounded Average Constraint Violations

Jicheng Shi, Colin N. Jones

TL;DR

This work targets stochastic LTI systems with average-state-constraint violations, without assuming a known disturbance distribution. It introduces Disturbance-adaptive MPC (DAD-MPC), which online-tunes the disturbance bound $W(\alpha_t)$ based on observed violations, using split conformal prediction for data-driven disturbance quantification and a violation-feedback mechanism to steer violations toward the target $\alpha$. To guarantee feasibility and bounds on violations, it incorporates a first-step robust invariance (FRI) auxiliary input constraint, enabling formal equivalence between the adaptive update and the asymptotic or robust violation guarantees. Through simulations, DAD-MPC outperforms state-of-the-art methods while satisfying average-violation constraints, and it operates without requiring precise disturbance distributions or i.i.d. assumptions, pointing to strong potential for data-driven disturbance modeling in MPC.

Abstract

This paper considers stochastic linear time-invariant systems subject to constraints on the average number of state-constraint violations over time without knowing the disturbance distribution. We present a novel disturbance-adaptive model predictive control (DAD-MPC) framework, which adjusts the disturbance model based on measured constraint violations. Using a robust invariance method, DAD-MPC ensures recursive feasibility and guarantees asymptotic or robust bounds on average constraint violations. Additionally, the bounds hold even with an inaccurate disturbance model, which allows for data-driven disturbance quantification methods to be used, such as conformal prediction. Simulation results demonstrate that the proposed approach outperforms state-of-the-art methods while satisfying average violation constraints.

Disturbance-adaptive Model Predictive Control for Bounded Average Constraint Violations

TL;DR

This work targets stochastic LTI systems with average-state-constraint violations, without assuming a known disturbance distribution. It introduces Disturbance-adaptive MPC (DAD-MPC), which online-tunes the disturbance bound based on observed violations, using split conformal prediction for data-driven disturbance quantification and a violation-feedback mechanism to steer violations toward the target . To guarantee feasibility and bounds on violations, it incorporates a first-step robust invariance (FRI) auxiliary input constraint, enabling formal equivalence between the adaptive update and the asymptotic or robust violation guarantees. Through simulations, DAD-MPC outperforms state-of-the-art methods while satisfying average-violation constraints, and it operates without requiring precise disturbance distributions or i.i.d. assumptions, pointing to strong potential for data-driven disturbance modeling in MPC.

Abstract

This paper considers stochastic linear time-invariant systems subject to constraints on the average number of state-constraint violations over time without knowing the disturbance distribution. We present a novel disturbance-adaptive model predictive control (DAD-MPC) framework, which adjusts the disturbance model based on measured constraint violations. Using a robust invariance method, DAD-MPC ensures recursive feasibility and guarantees asymptotic or robust bounds on average constraint violations. Additionally, the bounds hold even with an inaccurate disturbance model, which allows for data-driven disturbance quantification methods to be used, such as conformal prediction. Simulation results demonstrate that the proposed approach outperforms state-of-the-art methods while satisfying average violation constraints.

Paper Structure

This paper contains 12 sections, 4 theorems, 26 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. Problem Statement
  3. Disturbance-adaptive Model Predictive Control
  4. DAD-MPC with a violation feedback loop
  5. An example DAD-MPC control policy
  6. Equivalent statements for violation bounds satisfaction
  7. Guarantee violation bounds in DAD-MPC
  8. First-step robust invariance
  9. Combine FRI with DAD-MPC
  10. Simulation validation
  11. Conclusions
  12. Conformal Prediction

Key Result

Theorem 1

Consider system eqn:LTI_sys controlled by Algorithm alg:dad_mpc with user-defined finite parameters $\alpha_0$ and $\eta$, and assume Assumption assump:feasible holds. The following two statements are equivalent: And the following two statements are equivalent:

Figures (4)

  • Figure 1: Illustration of DAD-MPC: Based on the current violation condition $v_t$, the disturbance bound $W(\alpha_t)$ is adjusted. Then an MPC policy $\pi(x_t, W(\alpha_t))$ leverages $W(\alpha_t)$ and the current state $x_t$ to compute the input $u_t$ which is applied to the real system.
  • Figure 2: Comparison of $x_2$ trajectories by different controllers
  • Figure 3: Comparison of $V_t$ trajectories by different controllers
  • Figure 4: $\alpha_t$ and $r_t$ Trajectories in two DAD-MPC methods.

Theorems & Definitions (10)

  • Theorem 1
  • Lemma 2
  • proof
  • proof : for Theorem \ref{['thm:bound']}
  • Theorem 3
  • proof
  • Remark 1
  • Remark 2
  • Lemma 4
  • proof