Optimal Estimation with Sensor Delay

TL;DR

This paper develops an optimal state estimation framework for plants with sensor delays using Padé approximations, revealing how delay compensation can be integrated into Kalman filtering and control design.

Contribution

It introduces a novel state-space approach for delay compensation that decomposes the Kalman filter and relates it to Smith predictor strategies.

Findings

Delay compensation via Padé approximation partially inverts the delay.

The optimal estimator decomposes into uncontrollable and reduced-order Kalman filter parts.

Tradeoff in estimation error depends only on the reduced-order Kalman filter.

Abstract

Given a plant subject to delayed sensor measurement, there are several approaches to compensate for the delay. An obvious approach is to address this problem in state space, where the $n$-dimensional plant state is augmented by an $N$-dimensional (Pad\'e) approximation to the delay, affording (optimal) state estimate feedback vis-\`a-vis the separation principle. Using this framework, we show: (1) Feedback of the estimated plant states partially inverts the delay; (2) The optimal (Kalman) estimator decomposes into $N$ (Pad\'e) uncontrollable states, and the remaining $n$ eigenvalues are the solution to a reduced-order Kalman filter problem. Further, we show that the tradeoff of estimation error (of the full state estimator) between plant disturbance and measurement noise, only depends on the reduced-order Kalman filter (that can be constructed independently of the delay); (3) A subtly modified version of this state-estimation-based control scheme bears close resemblance to a Smith predictor. This modified state-space approach shares several limitations with its Smith predictor analog (including the inability to stabilize most unstable plants), limitations that are alleviated when using the unmodified state estimation framework.