Extended Kalman Filter Covariance Matrix Stabilization
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Solution Overview
Problem
In flight control systems, processors with limited computational power face challenges in maintaining the positive definiteness of the covariance matrix in Kalman filters, leading to potential system instability due to accumulated errors during numerical integration and linearization, especially in fast dynamics relative to the integration time step.
Innovation Solution
Implementing a method that periodically checks the main diagonal elements of the covariance matrix for non-positive values and resets the matrix to a known state to maintain positive definiteness, allowing for stable operation on lower-performance processors.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Device complexity
If a standard Kalman filter is implemented on a processor with limited computational power, then the system can operate with lower hardware requirements, but the covariance matrix may lose positive definiteness leading to system instability
Solution Approach 1:
The patent applies preliminary action by checking the main diagonal elements of the covariance matrix before they can cause instability. The method periodically examines whether diagonal elements are positive, and if not, resets them to positive values before they can propagate errors and cause filter divergence. This preventive approach maintains stability without requiring complex computational resources.
Solution Approach 2:
The patent changes the parameter state of the covariance matrix by resetting diagonal elements from non-positive values to positive values. This parameter correction ensures the matrix maintains positive definiteness, which is crucial for Kalman filter stability. The correction is simple arithmetic operation that maintains reliability without increasing device complexity.
2Productivity
If numerical integration and linearization are performed with large time steps, then computational load is reduced, but accumulated errors increase leading to filter divergence
Solution Approach 1:
The patent implements feedback by continuously monitoring the diagonal elements of the covariance matrix during filter operation. This feedback mechanism detects when numerical errors have accumulated to dangerous levels (when diagonal elements become non-positive), triggering a corrective reset action. This allows the use of larger time steps for computational efficiency while maintaining accuracy through error detection and correction.
Solution Approach 2:
The patent applies beforehand cushioning by preparing a correction mechanism that activates before complete filter divergence occurs. By checking and resetting diagonal elements periodically, the system cushions against the accumulation of numerical integration errors, allowing larger time steps without sacrificing measurement precision.
3Reliability
If complex methods like square-root Kalman filter are used to maintain covariance matrix positive definiteness, then system stability is improved, but computational complexity increases
Solution Approach 1:
The patent extracts only the essential stability-maintaining feature from complex methods like square-root Kalman filter. Instead of implementing the full complex algorithm, it extracts and applies only the critical function of ensuring positive diagonal elements. This simplified approach maintains reliability while avoiding the high computational complexity of complete square-root filter implementations.
Solution Approach 2:
The patent uses a simple, computationally inexpensive correction method that can be applied frequently without burden. The diagonal element reset operation is a simple, low-cost computational operation that can be performed often to maintain stability, unlike complex methods that would be expensive to execute frequently.
Data Source
AI summary
Embodiments include systems and methods of estimating at least one state of a modeled dynamic system. In particular, in one embodiment, an observer such as an extended Kalman filter is used to estimate the state of a modeled dynamic system. A covariance matrix associated with state variables of the observer is periodically checked for compliance with a specified condition, e.g., positive definiteness. If the matrix deviates from the specified condition, the matrix is set to a specified value.