# State-transition matrix

In control theory, the **state-transition matrix** is a matrix whose product with the state vector at an initial time gives at a later time . 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

where are the states of the system, is the input signal, and is the initial condition at . Using the state-transition matrix , the solution is given by^{[1]}

## Peano-Baker series

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

where 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 , given by

where is the fundamental solution matrix that satisfies

is a matrix that is a linear mapping onto itself, i.e., with , given the state at any time , the state at any other time is given by the mapping

The state transition matrix must always satisfy the following relationships:

And ; also must have the following properties:

If the system is time-invariant, we can define ; as:

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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