Programming + Errors: State transition Matrix

Wednesday, May 16, 2012

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.^[1]

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:

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

Programming + Errors

Wednesday, May 16, 2012

State transition Matrix

Overview

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