Content deleted Content added
No edit summary |
Undid revision 938342761 by Hammadhassan135 (talk). It was correct as it was. |
||
Line 5:
==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=Proceedings 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>
| |||