State Estimation
Recursive mathematical algorithms estimate the state of a dynamic system by processing a series of noisy measurements observed over a period of time. The kalman filter operates by producing a statistically optimal estimate of the internal variables of a process based on the history of previous states and the current observations. It is used in applications ranging from gps navigation to industrial process control where precise measurements are difficult to obtain.
The algorithm assumes that the system can be modeled by a set of linear equations and that the noise follows a gaussian distribution. By balancing the uncertainty of the model with the uncertainty of the sensors, the filter provides a more accurate value than any single measurement could offer. It is particularly effective for systems that change over time, as it updates its estimate with each new piece of data.
Covariance Update
Mathematical logic within the filter relies on two distinct phases known as the prediction step and the correction step. In the prediction phase, the algorithm uses the physics of the system to project the current state and its uncertainty into the next time interval. This projection includes the process noise, which accounts for the unpredictable forces acting on the system.
During the correction phase, the filter compares this prediction with the actual data received from the sensors. The difference between the predicted and observed values, called the residual, is used to refine the estimate. A weighting factor known as the gain determines how much the filter should trust the new measurement versus the previous model.
If the sensor noise is high, the algorithm places more weight on the prediction, but if the sensor is very accurate, it relies more on the observation.
Gain Optimization
Performance of the filter depends on the tuning of the covariance matrices which define the expected noise levels of the system and the sensors. These parameters, typically labeled q and r, must be carefully selected to achieve the best balance between responsiveness and stability. If the process noise is set too low, the filter becomes sluggish and fails to track rapid changes in the state.
Conversely, if the measurement noise is underestimated, the output becomes jittery and oversensitive to sensor errors. Engineers often use historical data or specialized calibration procedures to find the optimal values for these matrices. The beauty of the kalman filter lies in its ability to handle multiple inputs and outputs simultaneously, providing a coherent picture of a complex system from disparate data sources.
Computational Load
Execution of the algorithm requires a series of matrix multiplications and inversions at each time step, which can be demanding for simple microcontrollers. However, the recursive nature of the filter means that it only needs to store the state from the previous step, making it very memory efficient. For non linear systems, variations such as the extended or unscented versions are used, though they require more processing power.
The filter remains the gold standard for real time estimation in aerospace, robotics and financial modeling. Its ability to provide an optimal estimate in the presence of noise has made it one of the most influential algorithms of the twentieth century.
