State-transition matrix: Difference between revisions

Revision as of 15:12, 14 January 2020

Revision as of 15:18, 14 January 2020

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In the [[time-invariant]] case, we can define <math> \mathbf{\Phi}</math>; as <math>\mathbf{\Phi}(t, t_0) = e^{\mathbf{A}(t - t_0)}</math>.

In the time-variant case, the state-transition matrix can be estimated from the solutions of the differential equation <math>\dot{\mathbf{xu}}(t)=\mathbf{\PhiA}(t, \tau)\mathbf{xu}(\taut)</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>. The state-transition matrix must be determined before analysis on the time-varying solution can continue.

== See also ==