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

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

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

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