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 $\mathbf {x} (t)$ ares the states of the system, $\mathbf {u} (t)$ is the input signal, and $\mathbf {x} _{0}$ is the intial condition at $t_{0}$ . Using the state-transition matrix $\mathbf {\Phi } (t,\tau )$ , the solution is given byRugh, Wilson (1996). Linear System Theory. Upper Saddle River, NJ: Prentice Hall.</ref>

\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

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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

Brogan, W.L. (1991). Modern Control Theory. Prentice Hall. ISBN 0-13-589763-7.