havaintoyhtälön

Havaintoyhtälö, meaning "observation equation" in Finnish, is a fundamental concept in state-space modeling and control theory. It describes the relationship between the unobserved state of a system and the observable measurements or outputs. In essence, it provides a way to infer the internal state of a system by observing its external behavior.

The general form of a discrete-time observation equation is typically represented as y(k) = C * x(k) + D * u(k) + v(k), where y(k) is the vector of measurements at time step k, x(k) is the state vector at time step k, u(k) is the input vector at time step k. The matrix C relates the state to the output, and the matrix D relates the input to the output. The term v(k) represents measurement noise, which is assumed to be a random process.

In continuous-time systems, the observation equation is expressed as y(t) = C * x(t) + D * u(t) + v(t), where t denotes continuous time. The interpretation of the matrices C and D, and the noise term v(t), remains analogous to the discrete-time case.

The observation equation is crucial for state estimation algorithms like the Kalman filter. By combining the system's dynamic model (state equation) with the observation equation and the actual measurements, these algorithms can provide optimal estimates of the system's state, even in the presence of noise and uncertainty. This enables effective control and monitoring of complex systems in various fields, including engineering, economics, and biology.