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. It is also known as the matrix exponential.

Overview

Consider the general linear state space model

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

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

The general solution is given by

$\mathbf{x}(t)= \mathbf{\Phi} (t, t_0)\mathbf{x}(t_0)+\int_{t_0}^t \mathbf{\Phi}(t, \tau)\mathbf{B}(\tau)\mathbf{u}(\tau)d\tau$

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

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

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

$\frac{\partial \phi(t, t_0)}{\partial t} = A(t)\phi(t, t_0)$ and

$\phi(\tau, \tau) = I$ for all $\tau$ and where $I$ is the identity matrix.

And φ also must have the following properties:

1.	$\phi(t_2, t_1)\phi(t_1, t_0) = \phi(t_2, t_0)$
2.	$\phi^{-1}(t, \tau) = \phi(\tau, t)$
3.	$\phi^{-1}(t, \tau)\phi(t, \tau) = I$
4.	$\frac{d\phi(t, t_0)}{dt} = A(t)\phi(t, t_0)$

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

$\phi(t, t_0) = e^{A(t - t_0)}$

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 0-13-589763-7.