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: $dot\{mathbf\{x(t)\; =\; mathbf\{A\}(t)\; mathbf\{x\}(t)\; +\; mathbf\{B\}(t)\; mathbf\{u\}(t)$: $mathbf\{y\}(t)\; =\; mathbf\{C\}(t)\; mathbf\{x\}(t)\; +\; mathbf\{D\}(t)\; mathbf\{u\}(t)$The general solution is given by: $mathbf\{x\}(t)=\; mathbf\{Phi\}\; (t,\; t\_0)mathbf\{x\}(t\_0)+int\_\{t\_0\}^t\; mathbf\{Phi\}(t,\; au)mathbf\{B\}(\; au)mathbf\{u\}(\; au)d\; au$The state-transition matrix $mathbf\{Phi\}(t,\; au)$, given by: $mathbf\{Phi\}(t,\; au)equivmathbf\{U\}(t)mathbf\{U\}^\{-1\}(\; au)$where $mathbf\{U\}(t)$ is the fundamental solution matrix that satisfies: $dot\{mathbf\{U(t)=mathbf\{A\}(t)mathbf\{U\}(t)$is a $n\; imes\; n$ matrix that is a linear mapping onto itself, i.e., with $mathbf\{u\}(t)=0$, given the state $mathbf\{x\}(\; au)$ at any time $au$, the state at any other time $t$ is given by the maping:$mathbf\{x\}(t)=mathbf\{Phi\}(t,\; au)mathbf\{x\}(\; au)$

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

:$frac\{partial\; phi(t,\; t\_0)\}\{partial\; t\}\; =\; A(t)phi(t,\; t\_0)$

:$phi(\; au,\; au)\; =\; I$

And φ also must have the following properties:

:

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

:$phi(t,\; t\_0)\; =\; e^\{A(t\; -\; t\_0)\}$

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

* cite book
author = Brogan, W.L.
year = 1991
title = Modern Control Theory
publisher = Prentice Hall
isbn = 0135897637