State-transition matrix: Difference between revisions

Revision as of 17:18, 4 February 2015

Revision as of 08:18, 5 February 2015

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==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

: <math>\dot{\mathbf{x}}(t) = \mathbf{A}(t) \mathbf{x}(t) + \mathbf{B}(t) \mathbf{u}(t) \mathbf{x}(t_0) = \mathbf{x}_0 </math>,

where <math>\mathbf{x}(t)</math> are the states of the system, <math>\mathbf{u}(t)</math> is the input signal, and <math>\mathbf{x}_0</math> is the initial condition at <math>t_0</math>. Using the state-transition matrix <math>\mathbf{\Phi}(t, \tau)</math>, the solution is given by:<ref name=baaschl>{{cite journal|last1=Baake|first1=Michael|last2=Schlaegel|first2=Ulrike|title=The Peano Baker Series|journal=Proceeding of the Steklov Institute of Mathematics|year=2011|volume=275|pages=155-159155–159}}</ref><ref name=rugh>{{cite book|last1=Rugh|first1=Wilson|title=Linear System Theory|date=1996|publisher=Prentice Hall|___location=Upper Saddle River, NJ | isbn = 0-13-441205-2}}</ref>: <math>\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</math>

==Peano-Baker series==

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The state-transition matrix <math>\mathbf{\Phi}(t, \tau)</math>, given by

: <math>\mathbf{\Phi}(t, \tau)\equiv\mathbf{U}(t)\mathbf{U}^{-1}(\tau)</math>

where <math>\mathbf{U}(t)</math> is the [[Fundamental_matrix_Fundamental matrix (linear_differential_equationlinear differential equation)|fundamental solution matrix]] that satisfies

: <math>\dot{\mathbf{U}}(t)=\mathbf{A}(t)\mathbf{U}(t)</math>

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

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==Notes==

* {{cite articlenews

| author = Baake, M.

| author2 = Schlaegel, U.

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| journal = Proceedings of the Steklov Institute of Mathematics

| volume = 275

| pages = 155-159155–159

}}

* {{cite book