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][2]

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

The first term is known as the zero-input response and the second term is known as the zero-state response.

Peano–Baker series

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

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

where ${\displaystyle \mathbf {I} }$  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 ${\displaystyle \mathbf {\Phi } (t,\tau )}$ , 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.^[3]

And ${\displaystyle \mathbf {\Phi } }$  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 ${\displaystyle \mathbf {\Phi } }$ ; 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.

See also

• Magnus expansion
• Liouville's formula

References

1. Baake, Michael; Schlaegel, Ulrike (2011). "The Peano Baker Series". Proceedings 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.

Further reading

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

 

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