State-transition matrix: Difference between revisions

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In [[control theory]], the '''state-transition matrix''' is a matrix whose product with the state vector <math>x</math> at an initial time <math>t_0</math> gives <math>x</math> at a later time <math>t</math>. The state-transition matrix can be used to obtain the general solution of linear dynamical systems.

==Linear systems solutions==

==Overview==

The state-transition matrix is used to find the solution to a general [state-space representation]] of a [[linear system]] in the following form

Consider the general linear [[state space (controls)|state space]] model

: <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> ares the states of the system, <math>\mathbf{u}(t)</math> is the input signal, and <math>\mathbf{x}_0</math> is the intial condition at <math>t_0</math>. Using the state-transition matrix <math>\mathbf{\Phi}(t, \tau)</math>, the solution is given by{{cite book|last1=Rugh|first1=Wilson|title=Linear System Theory|date=1996|publisher=Prentice Hall|___location=Upper Saddle River, NJ}}</ref>

: <math>\mathbf{yx}(t) = \mathbf{C\Phi} (t), t_0)\mathbf{x}(tt_0) +\int_{t_0}^t \mathbf{D\Phi}(t), \tau)\mathbf{B}(\tau)\mathbf{u}(t\tau)d\tau</math><ref name=rugh>

The general solution is given by

: <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==

The most general transition matrix is given by the Peano-Baker series

:<math> \mathbf{\Phi}(t,\tau) = \mathbf{I} + \int_\tau^t\mathbf{A}(\sigma_1)\,d\sigma_1 + \int_\tau^t\mathbf{A}(\sigma_1)\int_\tau^t\mathbf{A}(\sigma_2)\,d\sigma_2\,d\sigma_1 + \int_\tau^t\mathbf{A}(\sigma_1)\int_\tau^t\mathbf{A}(\sigma_2)\int_\tau^t\mathbf{A}(\sigma_3)\,d\sigma_3\,d\sigma_2\,d\sigma_1 + ...</math>

where <math>\mathbf{I}</math> is the identity matrix. This matrix converges uniformly and absolutely to a solution that exists and is unique.<ref name=rugh />

==Other properties==

The state-transition matrix <math>\mathbf{\Phi}(t, \tau)</math>, given by

: <math>\mathbf{\Phi}(t, \tau)\equiv\mathbf{U}(t)\mathbf{U}^{-1}(\tau)</math>

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is a <math>n \times n</math> matrix that is a linear mapping onto itself, i.e., with <math>\mathbf{u}(t)=0</math>, given the state <math>\mathbf{x}(\tau)</math> at any time <math>\tau</math>, the state at any other time <math>t</math> is given by the mapping

:<math>\mathbf{x}(t)=\mathbf{\Phi}(t, \tau)\mathbf{x}(\tau)</math>

The

While the state transition matrix &phi; is not completely unknown, it must always satisfy the following relationships: