State-transition matrix

In control theory and dynamical systems theory, the state-transition matrix is a matrix function that describes how the state of a linear system changes over time. Essentially, if the system's state is known at an initial time ${\displaystyle t_{0}}$, the state-transition matrix allows for the calculation of the state at any future time ${\displaystyle t}$.

The matrix is used to find the general solution to the homogeneous linear differential equation ${\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)}$ and is also a key component in finding the full solution for the non-homogeneous (input-driven) case.

For linear time-invariant (LTI) systems, where the matrix ${\displaystyle \mathbf {A} }$ is constant, the state-transition matrix is the matrix exponential ${\displaystyle e^{\mathbf {A} (t-t_{0})}}$. In the more complex time-variant case, where ${\displaystyle \mathbf {A} (t)}$ can change over time, there is no simple formula, and the matrix is typically found by calculating the Peano–Baker series.