Tillståndsrumsrepresentation

Tillståndsrumsrepresentation, often abbreviated as state-space representation, is a mathematical framework used to model dynamic systems. It describes a system's behavior using a set of first-order differential equations or difference equations. The core idea is to represent the system's internal state at any given time through a vector of variables, known as the state vector. These state variables capture all the necessary information about the system's past that is required to predict its future behavior, given the current inputs.

The general form of a linear time-invariant (LTI) continuous-time system in state-space representation involves two main equations. The state equation describes how the state vector evolves over time, typically written as $\dot{x}(t) = Ax(t) + Bu(t)$, where $x(t)$ is the state vector, $u(t)$ is the input vector, $A$ is the state matrix, and $B$ is the input matrix. The output equation relates the system's output $y(t)$ to the state vector and the input, often expressed as $y(t) = Cx(t) + Du(t)$, where $C$ is the output matrix and $D$ is the feedforward matrix.

For discrete-time systems, the equations are similar but use difference equations instead of differential equations: $x(k+1) = Ax(k) + Bu(k)$ and $y(k) = Cx(k) + Du(k)$. State-space representation is widely used in control theory, signal processing, and various engineering disciplines due to its flexibility and ability to handle multi-input, multi-output (MIMO) systems. It provides a powerful tool for analysis, simulation, and controller design.