# Offline state estimation for hybrid systems via nonsmooth variable projection

## TL;DR

This paper introduces an offline method for estimating both discrete and continuous states of hybrid systems by formulating a continuous optimization problem and solving it with a novel nonsmooth variable projection algorithm, demonstrating effectiveness on impact systems.

## Contribution

It presents a new offline estimation algorithm for hybrid systems that relaxes discrete states and employs a nonsmooth optimization approach with convergence guarantees.

## Key findings

- Effective estimation on impact systems
- Algorithm converges to stationary points
- Handles large state changes during switching

## Abstract

A hybrid dynamical system switches between dynamic regimes at time- or state-triggered events. We propose an offline algorithm that simultaneously estimates discrete and continuous components of a hybrid system's state. We formulate state estimation as a continuous optimization problem by relaxing the discrete component and use a robust loss function to accommodate large changes in the continuous component during switching events. Subsequently, we develop a novel nonsmooth variable projection algorithm with Gauss-Newton updates to solve the state estimation problem and prove the algorithm's global convergence to stationary points. We demonstrate the effectiveness of our approach on simple piecewise-linear and -nonlinear mechanical systems undergoing intermittent impact.