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. It is also known as the matrix exponential.

Overview

Consider the general linear state space model

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

${\displaystyle \mathbf {y} (t)=\mathbf {C} (t)\mathbf {x} (t)+\mathbf {D} (t)\mathbf {u} (t)}$ 

The general solution is given by

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

The 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

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

While the state transition matrix φ is not completely unknown, it must always satisfy the following relationships:

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

${\displaystyle \phi (\tau ,\tau )=I}$  for all ${\displaystyle \tau }$  and where ${\displaystyle I}$  is the identity matrix.^[1]

And φ also must have the following properties:

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

If the system is time-invariant, we can define φ as:

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

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 0135897637.

1. Brockett, Roger W. (1970). Finite Dimensional Linear Systems. John Wiley & Sons. ISBN 9780471105855.

 

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