# State-transition matrix

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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.

## Linear systems solutions

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, 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]

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

## Peano-Baker series

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

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

where ${\displaystyle \mathbf {I} }$ is the identity matrix. This matrix converges uniformly and absolutely to a solution that exists and is unique.[1]

## Other properties

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

${\displaystyle {\dot {\mathbf {U} }}(t)=\mathbf {A} (t)\mathbf {U} (t)}$

is a ${\displaystyle n\times n}$ matrix that is a linear mapping onto itself, i.e., with ${\displaystyle \mathbf {u} (t)=0}$, 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

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

The state transition matrix must always satisfy the following relationships:

${\displaystyle {\frac {\partial \mathbf {\Phi } (t,t_{0})}{\partial t}}=\mathbf {A} (t)\mathbf {\Phi } (t,t_{0})}$ and
${\displaystyle \mathbf {\Phi } (\tau ,\tau )=I}$ for all ${\displaystyle \tau }$ and where ${\displaystyle I}$ is the identity matrix.[2]

And ${\displaystyle \mathbf {\Phi } }$; also must have the following properties:

If the system is time-invariant, we can define ${\displaystyle \mathbf {\Phi } }$; as:

${\displaystyle \mathbf {\Phi } (t,t_{0})=e^{\mathbf {A} (t-t_{0})}}$

In the time-variant case, there are many different functions that may satisfy these requirements, and the solution is dependent on the structure of the system. The state-transition matrix must be determined before analysis on the time-varying solution can continue.

## References

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