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A neural network-based approach to hybrid systems identification for control

Filippo Fabiani, Bartolomeo Stellato, Daniele Masti, Paul J. Goulart

TL;DR

A neural network architecture is adopted that, once suitably trained, yields a hybrid system with continuous piecewise-affine dynamics that is differentiable with respect to the network's parameters, thereby enabling the use of derivative-based training procedures.

Abstract

We consider the problem of designing a machine learning-based model of an unknown dynamical system from a finite number of (state-input)-successor state data points, such that the model obtained is also suitable for optimal control design. We adopt a neural network (NN) architecture that, once suitably trained, yields a hybrid system with continuous piecewise-affine (PWA) dynamics that is differentiable with respect to the network's parameters, thereby enabling the use of derivative-based training procedures. We show that a careful choice of our NN's weights produces a hybrid system model with structural properties that are highly favorable when used as part of a finite horizon optimal control problem (OCP). Specifically, we rely on available results to establish that optimal solutions with strong local optimality guarantees can be computed via nonlinear programming (NLP), in contrast to classical OCPs for general hybrid systems which typically require mixed-integer optimization. Besides being well-suited for optimal control design, numerical simulations illustrate that our NN-based technique enjoys very similar performance to state-of-the-art system identification methods for hybrid systems and it is competitive on nonlinear benchmarks.

A neural network-based approach to hybrid systems identification for control

TL;DR

A neural network architecture is adopted that, once suitably trained, yields a hybrid system with continuous piecewise-affine dynamics that is differentiable with respect to the network's parameters, thereby enabling the use of derivative-based training procedures.

Abstract

We consider the problem of designing a machine learning-based model of an unknown dynamical system from a finite number of (state-input)-successor state data points, such that the model obtained is also suitable for optimal control design. We adopt a neural network (NN) architecture that, once suitably trained, yields a hybrid system with continuous piecewise-affine (PWA) dynamics that is differentiable with respect to the network's parameters, thereby enabling the use of derivative-based training procedures. We show that a careful choice of our NN's weights produces a hybrid system model with structural properties that are highly favorable when used as part of a finite horizon optimal control problem (OCP). Specifically, we rely on available results to establish that optimal solutions with strong local optimality guarantees can be computed via nonlinear programming (NLP), in contrast to classical OCPs for general hybrid systems which typically require mixed-integer optimization. Besides being well-suited for optimal control design, numerical simulations illustrate that our NN-based technique enjoys very similar performance to state-of-the-art system identification methods for hybrid systems and it is competitive on nonlinear benchmarks.
Paper Structure (16 sections, 4 theorems, 15 equations, 3 figures, 3 tables)

This paper contains 16 sections, 4 theorems, 15 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Summary of contribution
  4. Problem formulation
  5. A neural network-based representation of hybrid systems
  6. The in \ref{['eq:OCP']} is a data-based
  7. Mathematical derivation
  8. Prior results on a class of
  9. Main results
  10. Numerical experiments
  11. Comparison with PARC
  12. Piecewise system benchmarks
  13. Linear parameter-varying and nonlinear benchmarks
  14. Sensitivity analysis -- parameters $n_r^\alpha$, $n_r^\beta$
  15. Performance for predictive control
  16. ...and 1 more sections

Key Result

Proposition 1

The $\mathcal{N}_\theta : \mathbb{R}^p \times \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$ in eq:neural_network enjoys the following properties:

Figures (3)

  • Figure 1: Neural network architecture considered in this work, which is composed by the cascade of an OptNet layer with a buffer one, which performs an affine transformation.
  • Figure 2: Proposed -based hybrid system identification for control approach. After choosing hyperparameters $n_r^\alpha$, $n_r^\beta\ge1$ and diagonal $H\in\mathbb{S}^{2n}_{\succ 0}$, one directly obtains elements $Q$, $F$, $W$, and $c$ to be used, along with the available data $\{(x^{(i)},u^{(i)}, x^{+,{(i)}})\}_{i=1}^N$, to train the $\mathcal{N}_\theta$ in \ref{['eq:neural_network']} and \ref{['eq:optnet_PWA']} -- see Theorem \ref{['th:nn_representation']}. Plugging all the elements in \ref{['eq:OCP_LC']} then enables to find some locally optimal triplet $\left(\boldsymbol x^\star, \boldsymbol u^\star, \boldsymbol w^\star\right)$ by solving the system associated to the underlying problem.
  • Figure 3: Comparison between an optimal controller based on SQP solver and one based on the DIRECT global optimization solver when tracking the reference sine-sweep $r_k$. The $y$-axis represents the (normalized) magnitude of the quantities involved, while the $x$-axis denotes the time steps. Both controllers have been parametrized to solve the same optimization problem based on a learnt model of $\Sigma_2$.

Theorems & Definitions (9)

  • Proposition 1
  • proof
  • Remark 1
  • Remark 2
  • Lemma 1
  • Theorem 1
  • proof
  • Corollary 1
  • Remark 3