Intersection form (4-manifold)

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. It is also known as the matrix exponential.

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

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

The state-transition matrix $\mathbf {\Phi } (t,\tau )$, given by

$\mathbf {\Phi } (t,\tau )\equiv \mathbf {U} (t)\mathbf {U} ^{-1}(\tau )$

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

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

is a $n\times n$ matrix that is a linear mapping onto itself, i.e., with $\mathbf {u} (t)=0$, given the state $\mathbf {x} (\tau )$ at any time $\tau$, the state at any other time $t$ is given by the mapping

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

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

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

$\phi (\tau ,\tau )=I$ for all $\tau$ and where $I$ is the identity matrix.

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.	${\frac {d\phi (t,t_{0})}{dt}}=A(t)\phi (t,t_{0})$

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 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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