Statespacemuodot

Statespacemuodot refer to a family of mathematical representations used to model dynamic systems by encapsulating the system’s internal state and its evolution over time. A state-space model describes the system with a state vector x(t) (or x_k in discrete time) and an output vector y(t) that depends on the state and input u(t). In continuous time, the evolution is given by x'(t) = f(x(t), u(t), t) and y(t) = g(x(t), u(t), t). In linear time-invariant form, these become x' = Ax + Bu and y = Cx + Du. In discrete time, the standard form is x_{k+1} = Φ x_k + Γ u_k and y_k = H x_k + D u_k.

The state vector is interpreted as a minimal sufficient statistic for predicting future behavior under a given input, summarizing all relevant information about the system. State-space models accommodate multiple inputs and outputs and can represent both linear and nonlinear dynamics, with linear models often used for analysis and control design. Transformation between different state-space representations leaves input-output behavior unchanged.

Extensions commonly include stochastic components: x_{k+1} = Ax_k + Bu_k + w_k and y_k = Cx_k + Du_k + v_k, where w_k and v_k model process and measurement noise. Estimation and identification methods, such as the Kalman filter and subspace identification, are used to infer parameters from data.

Applications span control engineering, robotics, aerospace, economics, and biological systems. State-space forms are contrasted with transfer-function models, offering advantages in handling initial conditions, multi-input/output configurations, and time-varying or nonlinear dynamics.