Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distributionally Robust Model Predictive Control with Mixture of Gaussian Processes

Jingyi Wu, Chao Ning

TL;DR

This work targets the challenge of controlling linear systems under state-dependent, multimodal disturbances where traditional GP-MPC struggles. It introduces MoGP-DR-MPC, which uses a Mixture of Gaussian Processes to model disturbances, constructs a data-driven, state-dependent ambiguity set, and enforces distributionally robust CVaR constraints that are reformulated into second-order cone constraints for tractable optimization. The approach provides recursive feasibility and stability guarantees via an invariant set and Riccati-based design, and demonstrates substantial performance gains in numerical tests and a quadrotor simulation compared with GP-DR-MPC and robust tube MPC. The framework enables robust, less conservative control in robotics settings with heterogeneous disturbances that depend on the state, offering practical applicability and improved closed-loop performance.

Abstract

Despite the success of Gaussian process based Model Predictive Control (MPC) in robotic control, its applicability scope is greatly hindered by multimodal disturbances that are prevalent in real-world settings. Here we propose a novel Mixture of Gaussian Processes based Distributionally Robust MPC (MoGP-DR-MPC) framework for linear time invariant systems subject to potentially multimodal state-dependent disturbances. This framework utilizes MoGP to automatically determine the number of modes from disturbance data. Using the mean and variance information provided by each mode-specific predictive distribution, it constructs a data-driven state-dependent ambiguity set, which allows for flexible and fine-grained disturbance modeling. Based on this ambiguity set, we impose Distributionally Robust Conditional Value-at Risk (DR-CVaR) constraints to effectively achieve distributional robustness against errors in the predictive distributions. To address the computational challenge posed by these constraints in the resulting MPC problem, we equivalently reformulate the DR-CVaR constraints into tractable second-order cone constraints. Furthermore, we provide theoretical guarantees on the recursive feasibility and stability of the proposed framework. The enhanced control performance of MoGP-DR-MPC is validated through both numerical experiments and simulations on a quadrotor system, demonstrating notable reductions in closed-loop cost by 17% and 4% respectively compared against Gaussian process based MPC.

Distributionally Robust Model Predictive Control with Mixture of Gaussian Processes

TL;DR

This work targets the challenge of controlling linear systems under state-dependent, multimodal disturbances where traditional GP-MPC struggles. It introduces MoGP-DR-MPC, which uses a Mixture of Gaussian Processes to model disturbances, constructs a data-driven, state-dependent ambiguity set, and enforces distributionally robust CVaR constraints that are reformulated into second-order cone constraints for tractable optimization. The approach provides recursive feasibility and stability guarantees via an invariant set and Riccati-based design, and demonstrates substantial performance gains in numerical tests and a quadrotor simulation compared with GP-DR-MPC and robust tube MPC. The framework enables robust, less conservative control in robotics settings with heterogeneous disturbances that depend on the state, offering practical applicability and improved closed-loop performance.

Abstract

Despite the success of Gaussian process based Model Predictive Control (MPC) in robotic control, its applicability scope is greatly hindered by multimodal disturbances that are prevalent in real-world settings. Here we propose a novel Mixture of Gaussian Processes based Distributionally Robust MPC (MoGP-DR-MPC) framework for linear time invariant systems subject to potentially multimodal state-dependent disturbances. This framework utilizes MoGP to automatically determine the number of modes from disturbance data. Using the mean and variance information provided by each mode-specific predictive distribution, it constructs a data-driven state-dependent ambiguity set, which allows for flexible and fine-grained disturbance modeling. Based on this ambiguity set, we impose Distributionally Robust Conditional Value-at Risk (DR-CVaR) constraints to effectively achieve distributional robustness against errors in the predictive distributions. To address the computational challenge posed by these constraints in the resulting MPC problem, we equivalently reformulate the DR-CVaR constraints into tractable second-order cone constraints. Furthermore, we provide theoretical guarantees on the recursive feasibility and stability of the proposed framework. The enhanced control performance of MoGP-DR-MPC is validated through both numerical experiments and simulations on a quadrotor system, demonstrating notable reductions in closed-loop cost by 17% and 4% respectively compared against Gaussian process based MPC.

Paper Structure

This paper contains 14 sections, 3 theorems, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. PROBLEM FORMULATION
  3. INFINITE MIXTURE of GPs
  4. Gating Network Based on Dirichlet Process
  5. Local Gaussian Process
  6. CONTROLLER DESIGN
  7. Ambiguity Set Construction
  8. Constraint Formulation
  9. Tractable MPC Formulation
  10. THEORETICAL PROPERTIES
  11. CASE STUDIES
  12. Numerical Experiments
  13. Simulations on a Quadrotor System
  14. CONCLUSIONS

Key Result

Theorem 1

For any $\tau\in\{1,\ldots,n\}$ and $i\in\{1,2\}$, the DR-CVaR constraint (cons_3) is satisfied if and only if where $\eta_{i,t}^{(\tau)}$ is obtained by the following second-order cone program. where Stacking the $2n$ inequalities defined as (ith inequality), the DR-CVaR constraints (cons_3) are equivalent to $s_{1|t}+(A+BK)(x_t-s_{0|t})\in \left[\Tilde{x}_1+\eta_{1,t}, \Tilde{x}_2-\eta_{2,t}\

Figures (2)

  • Figure 1: The closed-loop cost of MoGP-DR-MPC and two baseline methods under 50 realizations of disturbance sequences.
  • Figure 2: The closed-loop trajectories of the proposed MoGP-DR-MPC strategy and two baseline methods under different disturbance realizations.

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2: Recursive feasibility
  • proof
  • Theorem 3: Closed-loop stability
  • proof