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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, <math>\mathbf{A}(t)</math> and <math>\mathbf{B}(t)</math> are [[matrix function]]s, 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|doi=10.1134/S0081543811080098|s2cid=119133539}}</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>
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==Estimation of the state-transition matrix==
In the [[time-invariant]] case, we can define <math> \mathbf{\Phi}</math>, using the [[matrix exponential]], as <math>\mathbf{\Phi}(t, t_0) = e^{\mathbf{A}(t - t_0)}</math>. <ref>{{cite journal |last1=Reyneke |first1=Pieter V. |title=Polynomial Filtering: To any degree on irregularly sampled data |journal=Automatika |date=2012 |volume=53 |issue=4 |pages=
In the [[time-variant]] case, the state-transition matrix <math>\mathbf{\Phi}(t, t_0)</math> can be estimated from the solutions of the differential equation <math>\dot{\mathbf{u}}(t)=\mathbf{A}(t)\mathbf{u}(t)</math> with initial conditions <math>\mathbf{u}(t_0)</math> given by <math>[1,\ 0,\ \ldots,\ 0]^T</math>, <math>[0,\ 1,\ \ldots,\ 0]^T</math>, ..., <math>[0,\ 0,\ \ldots,\ 1]^T</math>. The corresponding solutions provide the <math>n</math> columns of matrix <math>\mathbf{\Phi}(t, t_0)</math>. Now, from property 4,
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