Robust Kalman Filter for Asymptotic Error Convergence
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Solution Overview
Problem
Existing Kalman filter schemes fail to guarantee asymptotic convergence of the mean estimation error in the presence of persistent excitation or non-asymptotically decaying disturbances, leading to biased estimates due to assumptions of minor system parametric uncertainties or asymptotically decaying disturbances.
Innovation Solution
A robust Kalman filter is developed that provides asymptotic convergence of the mean estimation error by using an estimator input to update state variables with a weighted average, incorporating a process model with unknown dynamics and a measurement model with Gaussian white noise, and solving a linear matrix inequality to ensure the estimator error converges to zero.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If existing Kalman filter schemes are used with assumptions of minor system parametric uncertainties or asymptotically decaying disturbances, then the filter can be implemented with standard algorithms, but the mean estimation error fails to converge asymptotically in the presence of persistent excitation
Solution Approach 1:
The patent modifies the standard Kalman filter by introducing a time-varying parameter transformation that converts the state estimation problem into a form where persistent excitation and non-asymptotically decaying disturbances are handled appropriately. This involves changing the parameterization of the estimation equations to ensure asymptotic convergence of the mean estimation error while maintaining adaptability to various disturbance types.
2Reliability
If robust estimation approaches (H∞ filtering, set-valued estimation, guaranteed-cost filtering) are used to handle system uncertainties, then robustness against uncertainties is improved, but asymptotic convergence of mean estimation error still cannot be guaranteed in the presence of persistent excitation
Solution Approach 1:
The patent employs a feedback mechanism where the estimation error dynamics are explicitly analyzed and fed back into the filter design. By formulating the estimation algorithm to include feedback terms that specifically address persistent excitation, the system achieves both robustness against uncertainties and guaranteed asymptotic convergence of the mean estimation error.
Solution Approach 2:
The patent transforms the estimation problem by introducing specific parameter changes in the filter equations that allow simultaneous achievement of robustness and convergence. This involves modifying the covariance update equations and gain calculations to account for persistent excitation while maintaining robustness properties.
3Reliability
If H∞ filtering is used to minimize the worst case H∞ norm of the transfer function from noise inputs to estimation error output, then worst-case performance is guaranteed, but average filter performance is sacrificed
Solution Approach 1:
The patent introduces parameter changes that allow the filter to adapt between worst-case and average performance optimization. By modifying the cost function and estimation equations to include terms that specifically address persistent excitation, the filter achieves both worst-case performance guarantees and improved average performance in the presence of persistent disturbances.
Data Source
AI summary
An apparatus and method for estimation of a system having state variables representing the state of the system comprising predicting the estimate of the state variables along with the uncertainties in the state variables; observing a measurement of at least one state variable corrupted with some amount of error; updating the estimates of the state variables using a weighted average, with more weight being given to estimates with higher certainty; and providing an estimator input to update the estimates of the state variables, the estimator input operating to provide asymptotic convergence of the mean estimation error in all of the state variables in the presence of persistent excitation or disturbance that is not asymptotically decaying to zero.