State-transition matrix: Difference between revisions

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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> aresare the states of the system, <math>\mathbf{u}(t)</math> is the input signal, and <math>\mathbf{x}_0</math> is the intialinitial condition at <math>t_0</math>. Using the state-transition matrix <math>\mathbf{\Phi}(t, \tau)</math>, the solution is given by<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>