Tillståndsrummodeller

Tillståndsrummodeller, or state-space models, are a mathematical framework used to describe the behavior of a system using a set of first-order differential or difference equations. These equations represent the system's internal state and how it evolves over time, as well as how the system's output is related to its state. The core components of a state-space model are the state vector, which captures the essential internal variables of the system, and the input and output vectors, which represent external influences and observable quantities, respectively.

The general form of a continuous-time linear time-invariant state-space model involves two matrix equations. The first equation describes the state update, where the derivative of the state vector is a linear function of the current state vector and the input vector, scaled by the system matrix (A) and the input matrix (B). The second equation relates the output vector to the current state vector and the input vector through the output matrix (C) and the feedthrough matrix (D). For discrete-time systems, the derivatives are replaced by state differences.

State-space models offer significant advantages over traditional input-output representations, particularly for complex systems. They can naturally handle systems with multiple inputs and outputs (MIMO), are well-suited for analyzing internal system dynamics, and provide a unified framework for designing controllers and observers. This approach is widely applied in various fields, including control engineering, robotics, economics, and signal processing, for analysis, simulation, and control design.