Havaintoyhtälö

Havaintoyhtälö, meaning "observation equation" in Finnish, is a fundamental concept in state estimation and control theory, particularly within the context of Kalman filtering and its variants. It mathematically describes the relationship between the true, unobserved state of a system and the measurements that are actually acquired by sensors.

In essence, the observation equation links the system's state vector, often denoted by 'x', to the measurement vector, denoted by 'y'. This relationship is typically modeled as a linear transformation, where the measurement vector is a linear combination of the state vector, plus some measurement noise. The equation can be represented as y_k = H_k * x_k + v_k, where 'y_k' is the measurement at time step 'k', 'H_k' is the observation matrix at time step 'k', 'x_k' is the true state at time step 'k', and 'v_k' is the measurement noise at time step 'k'.

The observation matrix 'H' plays a crucial role, as it dictates which components of the state vector are directly measured and how they are combined into the observed quantity. For instance, if a system's state includes position and velocity, and a sensor only measures position, the 'H' matrix would be designed to extract only the position component from the state vector. The measurement noise 'v' accounts for inaccuracies and uncertainties inherent in the sensing process.

Understanding and correctly formulating the observation equation is vital for accurate state estimation. If the observation model does not accurately reflect the true relationship between the state and the measurements, the resulting state estimates will be biased or inaccurate, potentially leading to suboptimal control decisions. Variants of the Kalman filter, such as the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF), are designed to handle non-linear observation equations as well.