In control theory, the state-transition matrix is a matrix whose product with the state vector ${\displaystyle x}$ at an initial time ${\displaystyle t_{0}}$ gives ${\displaystyle x}$ at a later time ${\displaystyle t}$. The state-transition matrix can be used to obtain the general solution of linear dynamical systems.

This article may be too technical for most readers to understand. (December 2018)

The state-transition matrix is used to find the solution to a general state-space representation of a linear system in the following form

${\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t),\;\mathbf {x} (t_{0})=\mathbf {x} _{0}}$,

where ${\displaystyle \mathbf {x} (t)}$ are the states of the system, ${\displaystyle \mathbf {u} (t)}$ is the input signal, ${\displaystyle \mathbf {A} (t)}$ and ${\displaystyle \mathbf {B} (t)}$ are matrix functions, and ${\displaystyle \mathbf {x} _{0}}$ is the initial condition at ${\displaystyle t_{0}}$. Using the state-transition matrix ${\displaystyle \mathbf {\Phi } (t,\tau )}$, the solution is given by:^[1][2]

${\displaystyle \mathbf {x} (t)=\mathbf {\Phi } (t,t_{0})\mathbf {x} (t_{0})+\int _{t_{0}}^{t}\mathbf {\Phi } (t,\tau )\mathbf {B} (\tau )\mathbf {u} (\tau )d\tau }$

The first term is known as the zero-input response and represents how the system's state would evolve in the absence of any input. The second term is known as the zero-state response and defines how the inputs impact the system.

The most general transition matrix is given by the Peano–Baker series

${\displaystyle \mathbf {\Phi } (t,\tau )=\mathbf {I} +\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\,d\sigma _{1}+\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\int _{\tau }^{\sigma _{1}}\mathbf {A} (\sigma _{2})\,d\sigma _{2}\,d\sigma _{1}+\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\int _{\tau }^{\sigma _{1}}\mathbf {A} (\sigma _{2})\int _{\tau }^{\sigma _{2}}\mathbf {A} (\sigma _{3})\,d\sigma _{3}\,d\sigma _{2}\,d\sigma _{1}+...}$

where ${\displaystyle \mathbf {I} }$ is the identity matrix. This matrix converges uniformly and absolutely to a solution that exists and is unique.^[2]

The state transition matrix ${\displaystyle \mathbf {\Phi } }$ satisfies the following relationships:

1. It is continuous and has continuous derivatives.

2, It is never singular; in fact ${\displaystyle \mathbf {\Phi } ^{-1}(t,\tau )=\mathbf {\Phi } (\tau ,t)}$ and ${\displaystyle \mathbf {\Phi } ^{-1}(t,\tau )\mathbf {\Phi } (t,\tau )=I}$, where ${\displaystyle I}$ is the identity matrix.

3. ${\displaystyle \mathbf {\Phi } (t,t)=I}$ for all ${\displaystyle t}$ .^[3]

4. ${\displaystyle \mathbf {\Phi } (t_{2},t_{1})\mathbf {\Phi } (t_{1},t_{0})=\mathbf {\Phi } (t_{2},t_{0})}$ for all ${\displaystyle t_{0}\leq t_{1}\leq t_{2}}$.

5. It satisfies the differential equation ${\displaystyle {\frac {\partial \mathbf {\Phi } (t,t_{0})}{\partial t}}=\mathbf {A} (t)\mathbf {\Phi } (t,t_{0})}$ with initial conditions ${\displaystyle \mathbf {\Phi } (t_{0},t_{0})=I}$.

6. The state-transition matrix ${\displaystyle \mathbf {\Phi } (t,\tau )}$, given by

${\displaystyle \mathbf {\Phi } (t,\tau )\equiv \mathbf {U} (t)\mathbf {U} ^{-1}(\tau )}$

where the ${\displaystyle n\times n}$ matrix ${\displaystyle \mathbf {U} (t)}$ is the fundamental solution matrix that satisfies

${\displaystyle {\dot {\mathbf {U} }}(t)=\mathbf {A} (t)\mathbf {U} (t)}$ with initial condition ${\displaystyle \mathbf {U} (t_{0})=I}$.

7. Given the state ${\displaystyle \mathbf {x} (\tau )}$ at any time ${\displaystyle \tau }$, the state at any other time ${\displaystyle t}$ is given by the mapping

${\displaystyle \mathbf {x} (t)=\mathbf {\Phi } (t,\tau )\mathbf {x} (\tau )}$

In the time-invariant case, we can define ${\displaystyle \mathbf {\Phi } }$, using the matrix exponential, as ${\displaystyle \mathbf {\Phi } (t,t_{0})=e^{\mathbf {A} (t-t_{0})}}$. ^[4]

In the time-variant case, the state-transition matrix ${\displaystyle \mathbf {\Phi } (t,t_{0})}$ can be estimated from the solutions of the differential equation ${\displaystyle {\dot {\mathbf {u} }}(t)=\mathbf {A} (t)\mathbf {u} (t)}$ with initial conditions ${\displaystyle \mathbf {u} (t_{0})}$ given by ${\displaystyle [1,\ 0,\ \ldots ,\ 0]^{T}}$, ${\displaystyle [0,\ 1,\ \ldots ,\ 0]^{T}}$, ..., ${\displaystyle [0,\ 0,\ \ldots ,\ 1]^{T}}$. The corresponding solutions provide the ${\displaystyle n}$ columns of matrix ${\displaystyle \mathbf {\Phi } (t,t_{0})}$. Now, from property 4, ${\displaystyle \mathbf {\Phi } (t,\tau )=\mathbf {\Phi } (t,t_{0})\mathbf {\Phi } (\tau ,t_{0})^{-1}}$ for all ${\displaystyle t_{0}\leq \tau \leq t}$. The state-transition matrix must be determined before analysis on the time-varying solution can continue.

• Magnus expansion
• Liouville's formula