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:^[1][2]

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

Peano-Baker series

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

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

where $\mathbf {I}$ is the identity matrix. This matrix converges uniformly and absolutely to a solution that exists and is unique.^[2]

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.^[3]

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

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

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.

Notes

• Baake, M.; Schlaegel, U. (2011). "The Peano Baker Series". Proceedings of the Steklov Institute of Mathematics. 275. pp. 155–159.
• Brogan, W.L. (1991). Modern Control Theory. Prentice Hall. ISBN 0-13-589763-7.

References

1. Baake, Michael; Schlaegel, Ulrike (2011). "The Peano Baker Series". Proceeding of the Steklov Institute of Mathematics. 275: 155–159.
2. Rugh, Wilson (1996). Linear System Theory. Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-441205-2.
3. Brockett, Roger W. (1970). Finite Dimensional Linear Systems. John Wiley & Sons. ISBN 978-0-471-10585-5.