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A Learning- and Scenario-based MPC Design for Nonlinear Systems in LPV Framework with Safety and Stability Guarantees

Yajie Bao, Hossam S. Abbas, Javad Mohammadpour Velni

TL;DR

This paper develops a learning-based SMPC framework for nonlinear systems cast in an LPV representation, using variational Bayesian neural networks (BNN) to identify a probabilistic LPV-SS model from input-output data. It constructs scenario trees via MC sampling of the BNN, reduces them with K-means, and assigns probabilities by moment matching, enabling safe MPC with a robust horizon and non-anticipativity constraints. A novel, data-driven terminal design yields a parameter-dependent terminal cost and controller, derived from an LPV affineization of the BNN model and LMIs, guaranteeing recursive feasibility and asymptotic stability with delta-safety. Numerical results on a double integrator, a MIMO system, and a two-tank plant demonstrate safe constraint satisfaction and competitive performance, while online model updates promise reduced conservativeness and improved control. The approach advances data-driven, safety-critical MPC for general nonlinear systems by integrating uncertainty quantification, scenario-based optimization, and adaptive terminal design in the LPV setting.

Abstract

This paper presents a learning- and scenario-based model predictive control (MPC) design approach for systems modeled in linear parameter-varying (LPV) framework. Using input-output data collected from the system, a state-space LPV model with uncertainty quantification is first learned through the variational Bayesian inference Neural Network (BNN) approach. The learned probabilistic model is assumed to contain the true dynamics of the system with a high probability and used to generate scenarios which ensure safety for a scenario-based MPC. Moreover, to guarantee stability and enhance performance of the closed-loop system, a parameter-dependent terminal cost and controller, as well as a terminal robust positive invariant set are designed. Numerical examples will be used to demonstrate that the proposed control design approach can ensure safety and achieve desired control performance.

A Learning- and Scenario-based MPC Design for Nonlinear Systems in LPV Framework with Safety and Stability Guarantees

TL;DR

This paper develops a learning-based SMPC framework for nonlinear systems cast in an LPV representation, using variational Bayesian neural networks (BNN) to identify a probabilistic LPV-SS model from input-output data. It constructs scenario trees via MC sampling of the BNN, reduces them with K-means, and assigns probabilities by moment matching, enabling safe MPC with a robust horizon and non-anticipativity constraints. A novel, data-driven terminal design yields a parameter-dependent terminal cost and controller, derived from an LPV affineization of the BNN model and LMIs, guaranteeing recursive feasibility and asymptotic stability with delta-safety. Numerical results on a double integrator, a MIMO system, and a two-tank plant demonstrate safe constraint satisfaction and competitive performance, while online model updates promise reduced conservativeness and improved control. The approach advances data-driven, safety-critical MPC for general nonlinear systems by integrating uncertainty quantification, scenario-based optimization, and adaptive terminal design in the LPV setting.

Abstract

This paper presents a learning- and scenario-based model predictive control (MPC) design approach for systems modeled in linear parameter-varying (LPV) framework. Using input-output data collected from the system, a state-space LPV model with uncertainty quantification is first learned through the variational Bayesian inference Neural Network (BNN) approach. The learned probabilistic model is assumed to contain the true dynamics of the system with a high probability and used to generate scenarios which ensure safety for a scenario-based MPC. Moreover, to guarantee stability and enhance performance of the closed-loop system, a parameter-dependent terminal cost and controller, as well as a terminal robust positive invariant set are designed. Numerical examples will be used to demonstrate that the proposed control design approach can ensure safety and achieve desired control performance.
Paper Structure (23 sections, 7 theorems, 32 equations, 16 figures, 2 algorithms)

This paper contains 23 sections, 7 theorems, 32 equations, 16 figures, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Basic Definitions
  4. Problem Formulation
  5. LPV-SS Model Identification Using BNN
  6. Scenario-based MPC Design Approach
  7. Scenario-based MPC Design Using the Learned BNN Models
  8. Proposed Method for Scenario Generation
  9. Probability of Scenarios
  10. Scenario-based MPC Problem
  11. Terminal Ingredients
  12. Recursive Feasibility, Stability and Safety
  13. Numerical Results
  14. Parameter-varying Double Integrator
  15. System Identification
  16. ...and 8 more sections

Key Result

Lemma 2.3

Given $x_{0}$, a scheduling signal $\theta$, a BNN model that satisfies Assumption assp:lpv, and a confidence level $\delta_{c}$, there exists a scalar $N_{\text{MC}}$ such that where $N_{\text{MC}}$ is the number of models drawn from the BNN model using MC methods.

Figures (16)

  • Figure 1: The flow chart of the overall learning-based MPC design procedure.
  • Figure 2: Using a BNN composed of multiple (here, two) DenseVariational layers to represent $A(\cdot)$ with reparameterization trick. Here, the input to the BNN is $\theta$ and the output is the vectorized $A(\theta)$, which once reshaped, provides the full matrix $A$. BNNs use data to learn the parameters $\mu_{w}$ and $\sigma_{w}$ of the posterior density function.
  • Figure 3: Scenario tree representation of the joint uncertainty evolution for MPC. In the figure, $A_{k}^{r(j)}$ refers to the matrix at time $k$ in the $r(j)$-th scenario.
  • Figure 4: The block diagram of the closed-loop learning-based SMPC scheme.
  • Figure 5: Data generated for model identification purposes.
  • ...and 11 more figures

Theorems & Definitions (8)

  • Definition 2.1
  • Lemma 2.3
  • Proposition 3.1
  • Proposition 3.5
  • Lemma 3.8: SMPC stability maiworm2015scenario
  • Lemma 3.9
  • Theorem 3.10: Learning-based SMPC stability and feasibility
  • Theorem 3.11: Learning-based SMPC safety