Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unconstrained learning of networked nonlinear systems via free parametrization of stable interconnected operators

Leonardo Massai, Danilo Saccani, Luca Furieri, Giancarlo Ferrari-Trecate

TL;DR

This paper characterizes a new parametrization of nonlinear networked incrementally $L_{2}$ -bounded operators in discrete time that is free - that is, a sparse large-scale operator with bounded incremental gain is obtained for any choice of the real values of the authors' parameters.

Abstract

This paper characterizes a new parametrization of nonlinear networked incrementally $L_2$-bounded operators in discrete time. The distinctive novelty is that our parametrization is \emph{free} -- that is, a sparse large-scale operator with bounded incremental $L_2$ gain is obtained for any choice of the real values of our parameters. This property allows one to freely search over optimal parameters via unconstrained gradient descent, enabling direct applications in large-scale optimal control and system identification. Further, we can embed prior knowledge about the interconnection topology and stability properties of the system directly into the large-scale distributed operator we design. Our approach is extremely general in that it can seamlessly encapsulate and interconnect state-of-the-art Neural Network (NN) parametrizations of stable dynamical systems. To demonstrate the effectiveness of this approach, we provide a simulation example showcasing the identification of a networked nonlinear system. The results underscore the superiority of our free parametrizations over standard NN-based identification methods where a prior over the system topology and local stability properties are not enforced.

Unconstrained learning of networked nonlinear systems via free parametrization of stable interconnected operators

TL;DR

This paper characterizes a new parametrization of nonlinear networked incrementally -bounded operators in discrete time that is free - that is, a sparse large-scale operator with bounded incremental gain is obtained for any choice of the real values of the authors' parameters.

Abstract

This paper characterizes a new parametrization of nonlinear networked incrementally -bounded operators in discrete time. The distinctive novelty is that our parametrization is \emph{free} -- that is, a sparse large-scale operator with bounded incremental gain is obtained for any choice of the real values of our parameters. This property allows one to freely search over optimal parameters via unconstrained gradient descent, enabling direct applications in large-scale optimal control and system identification. Further, we can embed prior knowledge about the interconnection topology and stability properties of the system directly into the large-scale distributed operator we design. Our approach is extremely general in that it can seamlessly encapsulate and interconnect state-of-the-art Neural Network (NN) parametrizations of stable dynamical systems. To demonstrate the effectiveness of this approach, we provide a simulation example showcasing the identification of a networked nonlinear system. The results underscore the superiority of our free parametrizations over standard NN-based identification methods where a prior over the system topology and local stability properties are not enforced.
Paper Structure (6 sections, 2 theorems, 32 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 6 sections, 2 theorems, 32 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Setting, Models, and Problem formulation
  4. Main results
  5. Numerical example
  6. Conclusions

Key Result

Proposition 1

If there exist $\alpha_i>0$, $i \in \mathcal{N}$, such that eq:conditionL2 holds, then the operator $\Sigma_{\theta}^{M}$ has a finite incremental $L_{i2}$-gain $\gamma_{M}$.

Figures (4)

  • Figure 1: Interconnection of $N$ operators $\Sigma_{\theta_i}$.
  • Figure 2: Triple-tank system with recirculation pump.
  • Figure 3: Comparison of the open-loop prediction of the trained distributed RENs built with the proposed parametrization (blue dashed line) versus ground truth (red continuous line) on an independent validation dataset.
  • Figure 4: Comparison of validation loss as a function of tunable parameters for the proposed approach with three RENs (red "$\circ$"), the REN (blue "$\triangle$") and the RNN (green "$\diamond$").

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Remark 2
  • Proposition 1
  • Theorem 1
  • proof
  • Remark 3
  • Remark 4