State-transition matrix: Difference between revisions

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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=155-159}}</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>

<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=155-159}}</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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==Notes==

* {{cite book

| author = Brogan, W.L.

| year = 1991

| title = Modern Control Theory

| publisher = Prentice Hall

| isbn = 0-13-589763-7

}}

* {{cite article

| author = Baake, M.

Line 62 ⟶ 54:

| volume = 275

| pages = 155-159

}}

* {{cite book

| author = Brogan, W.L.

| year = 1991

| title = Modern Control Theory

| publisher = Prentice Hall

| isbn = 0-13-589763-7

}}