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Robust Neural IDA-PBC: passivity-based stabilization under approximations

Santiago Sanchez-Escalonilla, Samuele Zoboli, Bayu Jayawardhana

TL;DR

This paper restructure the Neural Interconnection and Damping Assignment - Passivity Based Control (Neural IDA-PBC) design methodology, and formally analyze its closed-loop properties, and introduces a novel optimization objective including stability and robustness constraints issued from the theoretical analysis.

Abstract

In this paper, we restructure the Neural Interconnection and Damping Assignment - Passivity Based Control (Neural IDA-PBC) design methodology, and we formally analyze its closed-loop properties. Neural IDA-PBC redefines the IDA-PBC design approach as an optimization problem by building on the framework of Physics Informed Neural Networks (PINNs). However, the closed-loop stability and robustness properties under Neural IDA-PBC remain unexplored. To address the issue, we study the behavior of classical IDA-PBC under approximations. Our theoretical analysis allows deriving conditions for practical and asymptotic stability of the desired equilibrium point. Moreover, it extends the Neural IDA-PBC applicability to port-Hamiltonian systems where the matching conditions cannot be solved exactly. Our renewed optimization-based design introduces three significant aspects: i) it involves a novel optimization objective including stability and robustness constraints issued from our theoretical analysis; ii) it employs separate Neural Networks (NNs), which can be structured to reduce the search space to relevant functions; iii) it does not require knowledge about the port-Hamiltonian formulation of the system's model. Our methodology is validated with simulations on three standard benchmarks: a double pendulum, a nonlinear mass-spring-damper and a cartpole. Notably, classical IDA-PBC designs cannot be analytically derived for the latter.

Robust Neural IDA-PBC: passivity-based stabilization under approximations

TL;DR

This paper restructure the Neural Interconnection and Damping Assignment - Passivity Based Control (Neural IDA-PBC) design methodology, and formally analyze its closed-loop properties, and introduces a novel optimization objective including stability and robustness constraints issued from the theoretical analysis.

Abstract

In this paper, we restructure the Neural Interconnection and Damping Assignment - Passivity Based Control (Neural IDA-PBC) design methodology, and we formally analyze its closed-loop properties. Neural IDA-PBC redefines the IDA-PBC design approach as an optimization problem by building on the framework of Physics Informed Neural Networks (PINNs). However, the closed-loop stability and robustness properties under Neural IDA-PBC remain unexplored. To address the issue, we study the behavior of classical IDA-PBC under approximations. Our theoretical analysis allows deriving conditions for practical and asymptotic stability of the desired equilibrium point. Moreover, it extends the Neural IDA-PBC applicability to port-Hamiltonian systems where the matching conditions cannot be solved exactly. Our renewed optimization-based design introduces three significant aspects: i) it involves a novel optimization objective including stability and robustness constraints issued from our theoretical analysis; ii) it employs separate Neural Networks (NNs), which can be structured to reduce the search space to relevant functions; iii) it does not require knowledge about the port-Hamiltonian formulation of the system's model. Our methodology is validated with simulations on three standard benchmarks: a double pendulum, a nonlinear mass-spring-damper and a cartpole. Notably, classical IDA-PBC designs cannot be analytically derived for the latter.
Paper Structure (13 sections, 6 theorems, 46 equations, 1 figure, 1 table)

This paper contains 13 sections, 6 theorems, 46 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Port-controlled Hamiltonian systems
  4. Interconnection and Damping Assignment-PBC
  5. Robust IDA-PBC
  6. Neural IDA-PBC
  7. The neural networks design
  8. NN architectures of $\tilde{J}_d, \tilde{R}_d, \tilde{H}_d$
  9. Loss function formulation
  10. Simulations
  11. Benchmarks
  12. Results
  13. Conclusions

Key Result

Proposition 1

Consider the port-Hamiltonian system eq:OLPH, suppose $H_d$ satisfies eq:hamiltonian and assume the triplet $(H_d,J_d,R_d)$ is a solution to the matching equation Let $u=\beta(x)$ with Then, the dynamics of the closed-loop eq:OLPH and eq:u are equivalentNamely, for any initial condition $x^\circ\in{\mathcal{X}}$ the corresponding trajectories $X(t,x^\circ), X_d(t,x^\circ)$ generated by systems e

Figures (1)

  • Figure 1: Simulation results for the different benchmarks: a) and b) are the generalized coordinates and momenta; c) depicts the satisfaction of condition \ref{['eq:damping_assign_relax_tilde']}; d) and e) are the Neural IDA-PBC control signal and closed-loop energy. The gray area in e) determines the region ${\mathcal{E}}$ where the sufficient dissipation condition is not necessarily satisfied. The vertical (dotted) line indicates the time at which the trajectories enter ${\mathcal{E}}$.

Theorems & Definitions (12)

  • Proposition 1: ortega2004survey
  • Theorem 1: becherif2005robustIDA
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Theorem 2
  • ...and 2 more