State-transition matrix

In control theory, the state-transition matrix is a matrix whose product with the state vector ${\displaystyle x}$ at an initial time ${\displaystyle t_{0}}$ gives ${\displaystyle x}$ at a later time ${\displaystyle t}$. 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

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

where ${\displaystyle \mathbf {x} (t)}$  are the states of the system, ${\displaystyle \mathbf {u} (t)}$  is the input signal, and ${\displaystyle \mathbf {x} _{0}}$  is the initial condition at ${\displaystyle t_{0}}$ . Using the state-transition matrix ${\displaystyle \mathbf {\Phi } (t,\tau )}$ , the solution is given by^[1]

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

Peano-Baker series

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

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

where ${\displaystyle \mathbf {I} }$  is the identity matrix. This matrix converges uniformly and absolutely to a solution that exists and is unique.^[1]

Other properties

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

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

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

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

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

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

The state transition matrix must always satisfy the following relationships:

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

${\displaystyle \mathbf {\Phi } (\tau ,\tau )=I}$  for all ${\displaystyle \tau }$  and where ${\displaystyle I}$  is the identity matrix.^[2]

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

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

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

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

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

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

1. Rugh, Wilson (1996). Linear System Theory. Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-441205-2.
2. Brockett, Roger W. (1970). Finite Dimensional Linear Systems. John Wiley & Sons. ISBN 978-0-471-10585-5.

 

Wikibooks has a book on the topic of: Control Systems/Time Variant System Solutions