statespacemodell

Statespacemodell, or state-space model, is a mathematical framework for describing the dynamic behavior of a system using a set of internal state variables that evolve over time. The model combines a state update equation with an observation equation, capturing how inputs and disturbances affect the system and how measurements relate to its hidden state.

In discrete time, the standard form is:

x_k+1 = f(x_k, u_k, w_k)

y_k = h(x_k, v_k)

Here x_k is the state vector, u_k the input vector, y_k the output vector, and w_k and v_k represent process and measurement noise. For linear, time-invariant systems, these equations are often written as:

x_k+1 = A x_k + B u_k + w_k

y_k = C x_k + D u_k + v_k

In continuous time, the corresponding representation uses differential equations:

dx/dt = F x + G u + w

y = H x + v

Key concepts in state-space modeling include observability (whether the internal state can be inferred from outputs) and controllability (whether inputs can drive the state to a desired condition). The framework is well suited for multi-input multi-output systems and can handle nonlinear dynamics and non-Gaussian disturbances through nonlinear functions or extended estimation schemes.

Estimation and filtering play a central role. For linear Gaussian models, the Kalman filter provides optimal state estimates. Extensions such as the Extended Kalman Filter and the Unscented Kalman Filter address nonlinearities. State-space models are widely used in engineering, robotics, economics, signal processing, biology, and other fields, often built by identifying the matrices or functions from physical laws or data.