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.

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

${\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t)\mathbf {x} (t_{0})=\mathbf {x} _{0}$,

where $\mathbf {x} (t)$ are the states of the system, $\mathbf {u} (t)$ is the input signal, and $\mathbf {x} _{0}$ is the initial condition at $t_{0}$. Using the state-transition matrix $\mathbf {\Phi } (t,\tau )$, the solution is given by

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

Peano-Baker series

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

$\mathbf {\Phi } (t,\tau )=\mathbf {I} +\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\,d\sigma _{1}+\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\int _{\tau }^{t}\mathbf {A} (\sigma _{2})\,d\sigma _{2}\,d\sigma _{1}+\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\int _{\tau }^{t}\mathbf {A} (\sigma _{2})\int _{\tau }^{t}\mathbf {A} (\sigma _{3})\,d\sigma _{3}\,d\sigma _{2}\,d\sigma _{1}+...$

where $\mathbf {I}$ is the identity matrix. This matrix converges uniformly and absolutely to a solution that exists and is unique.

Other properties

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

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

The state transition matrix must always satisfy the following relationships:

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

$\mathbf {\Phi } (\tau ,\tau )=I$ for all $\tau$ and where $I$ is the identity matrix.

And $\mathbf {\Phi }$; also must have the following properties:

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

If the system is time-invariant, we can define $\mathbf {\Phi }$; as:

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

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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