Hybrid Iterative Linear Quadratic Estimation: Optimal Estimation for Hybrid Systems

TL;DR

The saltation matrix is utilized, a first order approximation of the variational update through an event driven hybrid transition, to calculate gradient information through hybrid events in the backward pass of an iterative linear quadratic optimization over state estimates, enabling accurate computation of the value function approximation at each timestep.

Abstract

In this paper we present Hybrid iterative Linear Quadratic Estimation (HiLQE), an optimization based offline state estimation algorithm for hybrid dynamical systems. We utilize the saltation matrix, a first order approximation of the variational update through an event driven hybrid transition, to calculate gradient information through hybrid events in the backward pass of an iterative linear quadratic optimization over state estimates. This enables accurate computation of the value function approximation at each timestep. Additionally, the forward pass in the iterative algorithm is augmented with hybrid dynamics in the rollout. A reference extension method is used to account for varying impact times when comparing states for the feedback gain in noise calculation. The proposed method is demonstrated on an ASLIP hopper system with position measurements. In comparison to the Salted Kalman Filter (SKF), the algorithm presented here achieves a maximum of 63.55% reduction in estimation error magnitude over all state dimensions near impact events.

Figures (4)

• Figure 1: Average error magnitude over 100 trials when simulating a an ASLIP system (inset) which impacts the ground four times during its execution. The error from the proposed HiLQE method is shown as a solid blue line while the error from the SKF is shown as a dashed red line.
• Figure 2: Evolution of a perturbation through a simple 2 mode hybrid system with constant dynamics in each mode. This demonstrates the importance of considering time-to-impact variations with the saltation matrix to obtain correct gradient information through hybrid transitions as the reset of this system is identity and the Jacobian of the reset map fails to capture these effects. First appeared in kong2024saltationmatricesessentialtool.
• Figure 3: Average error magnitude over 1000 trials when simulating a bouncing ball system (inset) which impacts the ground once during its execution. The errors from the proposed method are shown in blue solid lines while the errors from the salted Kalman filter are shown in dashed red lines.
• Figure 4: Average error for each state over 100 trials when simulating an ASLIP system which impacts the ground four times during its execution. The errors from the proposed method are shown in blue solid lines while the errors from the salted Kalman filter are shown in dashed red lines.

Theorems & Definitions (1)

• Definition 1