State-transition matrix

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

he state transtion matrix φ is not completely unknown, it must always satisfy the following relationships:

${\frac {\partial \phi (t,t_{0})}{\partial t}}=A(t)\phi (t,t_{0})$

$\phi (\tau ,\tau )=I$

And φ also must have the following properties:

1.	$\phi (t_{2},t_{1})\phi (t_{1},t_{0})=\phi (t_{2},t_{0})$
2.	$\phi ^{-1}(t,\tau )=\phi (\tau ,t)$
3.	$\phi ^{-1}(t,\tau )\phi (t,\tau )=I$
4.	${\frac {d\phi (t,t_{0})}{dt}}=A(t)$

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

$\phi (t,t_{0})=e^{A(t-t_{0})}$

The reader can verify that this solution for a time-invariant system satisfies all the properties listed above. However, in the time-variant case, there are many different functions that may satisfy these requirements, and the solution is dependant on the structure of the system. The state-transition matrix must be determined before analysis on the time-varying solution can continue. We will discuss some of the methods for determining this matrix below.