Definitions

# 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.

## Overview

Consider the general linear state space model
$dot\left\{mathbf\left\{x\right\}\right\}\left(t\right) = mathbf\left\{A\right\}\left(t\right) mathbf\left\{x\right\}\left(t\right) + mathbf\left\{B\right\}\left(t\right) mathbf\left\{u\right\}\left(t\right)$
$mathbf\left\{y\right\}\left(t\right) = mathbf\left\{C\right\}\left(t\right) mathbf\left\{x\right\}\left(t\right) + mathbf\left\{D\right\}\left(t\right) mathbf\left\{u\right\}\left(t\right)$
The general solution is given by
$mathbf\left\{x\right\}\left(t\right)= mathbf\left\{Phi\right\} \left(t, t_0\right)mathbf\left\{x\right\}\left(t_0\right)+int_\left\{t_0\right\}^t mathbf\left\{Phi\right\}\left(t, tau\right)mathbf\left\{B\right\}\left(tau\right)mathbf\left\{u\right\}\left(tau\right)dtau$
The state-transition matrix $mathbf\left\{Phi\right\}\left(t, tau\right)$, given by
$mathbf\left\{Phi\right\}\left(t, tau\right)equivmathbf\left\{U\right\}\left(t\right)mathbf\left\{U\right\}^\left\{-1\right\}\left(tau\right)$
where $mathbf\left\{U\right\}\left(t\right)$ is the fundamental solution matrix that satisfies
$dot\left\{mathbf\left\{U\right\}\right\}\left(t\right)=mathbf\left\{A\right\}\left(t\right)mathbf\left\{U\right\}\left(t\right)$
is a $n times n$ matrix that is a linear mapping onto itself, i.e., with $mathbf\left\{u\right\}\left(t\right)=0$, given the state $mathbf\left\{x\right\}\left(tau\right)$ at any time $tau$, the state at any other time $t$ is given by the maping
$mathbf\left\{x\right\}\left(t\right)=mathbf\left\{Phi\right\}\left(t, tau\right)mathbf\left\{x\right\}\left(tau\right)$

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

$frac\left\{partial phi\left(t, t_0\right)\right\}\left\{partial t\right\} = A\left(t\right)phi\left(t, t_0\right)$

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

And φ also must have the following properties:

1. $phi\left(t_2, t_1\right)phi\left(t_1, t_0\right) = phi\left(t_2, t_0\right)$
2. $phi^\left\{-1\right\}\left(t, tau\right) = phi\left(tau, t\right)$
3. $phi^\left\{-1\right\}\left(t, tau\right)phi\left(t, tau\right) = I$
4. $frac\left\{dphi\left(t, t_0\right)\right\}\left\{dt\right\} = A\left(t\right)$

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

$phi\left(t, t_0\right) = e^\left\{A\left(t - t_0\right)\right\}$

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.

## References

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