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On ANN-enhanced positive invariance for nonlinear flat systems

Huu-Thinh Do, Ionela Prodan

Abstract

The concept of positively invariant (PI) sets has proven effective in the formal verification of stability and safety properties for autonomous systems. However, the characterization of such sets is challenging for nonlinear systems in general, especially in the presence of constraints. In this work, we show that, for a class of feedback linearizable systems, called differentially flat systems, a PI set can be derived by leveraging a neural network approximation of the linearizing mapping. More specifically, for the class of flat systems, there exists a linearizing variable transformation that converts the nonlinear system into linear controllable dynamics, albeit at the cost of distorting the constraint set. We show that by approximating the distorted set using a rectified linear unit neural network, we can derive a PI set inside the admissible domain through its set-theoretic description. This offline characterization enables the synthesis of various efficient online control strategies, with different complexities and performances. Numerical simulations are provided to demonstrate the validity of the proposed framework.

On ANN-enhanced positive invariance for nonlinear flat systems

Abstract

The concept of positively invariant (PI) sets has proven effective in the formal verification of stability and safety properties for autonomous systems. However, the characterization of such sets is challenging for nonlinear systems in general, especially in the presence of constraints. In this work, we show that, for a class of feedback linearizable systems, called differentially flat systems, a PI set can be derived by leveraging a neural network approximation of the linearizing mapping. More specifically, for the class of flat systems, there exists a linearizing variable transformation that converts the nonlinear system into linear controllable dynamics, albeit at the cost of distorting the constraint set. We show that by approximating the distorted set using a rectified linear unit neural network, we can derive a PI set inside the admissible domain through its set-theoretic description. This offline characterization enables the synthesis of various efficient online control strategies, with different complexities and performances. Numerical simulations are provided to demonstrate the validity of the proposed framework.

Paper Structure

This paper contains 15 sections, 4 theorems, 37 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and problem formulation
  3. Differentially flat systems
  4. ANN-based constraint description
  5. Invariance characterization and control synthesis
  6. Invariant set characterization from the flat output space
  7. Applications to control design
  8. CLF-based control
  9. Flatness-based MPC with stability guarantee
  10. Explicit Reference Governor
  11. Illustrative examples
  12. Aircraft longitudinal dynamics stabilization
  13. Horizontal 1D quadrotor
  14. Discussion
  15. Conclusion and outlook

Key Result

Proposition 1

Consider a ReLU-ANN approximation of $\phi_u(\boldsymbol{z},v)$ in eq:linearizingMaps, denoted as $\tilde{\phi}_u(\boldsymbol{z},v)$ structured as in eq:singleLayerANN. Let the approximation error be bounded as: for some workspace of interest $\mathcal{Z}_w$. Then, the constraint eq:convolutedConstr is implied by: Furthermore, the constraint eq:tightenedConstr can be represented by mixed-integer

Figures (9)

  • Figure 1: Searching for ellipsoidal PI set in the flat output space for control design in the original space.
  • Figure 2: The approximation $\tilde{\phi}_u(\boldsymbol{z},v)$ of $\phi_u(\boldsymbol{z},v)$ with the structure in \ref{['eq:singleLayerANN']}.
  • Figure 3: Distance from the origin to the set $\partial\mathcal{Z}_K$.
  • Figure 4: The constrained approximated mapping $\tilde{\phi}_u(\boldsymbol{z},v)$ (left) and the corresponding constraint set $\tilde{\mathcal{V}}$ induced from the neural network as in \ref{['eq:cellenum']} (right).
  • Figure 5: PI sets and their volumes (noted with Vol($\cdot$)) with different convergence rates $\kappa$ as in \ref{['eq:CLF_derivative']}.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Definition 1: Differential flatness fliess1995flatness
  • Proposition 1: ANN-based constraint inner approximation DoMILP2024
  • proof
  • Definition 2: Positive invariance blanchini2008set
  • Proposition 2
  • proof
  • Remark 1
  • Remark 2
  • Proposition 3
  • proof
  • ...and 2 more