State-transition matrix: Difference between revisions

Revision as of 23:53, 8 December 2019

Revision as of 15:03, 14 January 2020

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==Other properties==

The state- transition matrix <math> \mathbf{\Phi}(t, \tau)</math>, givensatisfies the following byrelationships:

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

where <math>\mathbf{U}(t)</math> is the [[Fundamental matrix (linear differential equation)|fundamental solution matrix]] that satisfies

: <math>\dot{\mathbf{U}}(t)=\mathbf{A}(t)\mathbf{U}(t)</math>

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>

1. It is continuous and has continuous derivatives.

The state transition matrix must always satisfy the following relationships:

:2, It is never singular; in fact <math>\frac{\partial \mathbf{\Phi}^{-1}(t, t_0\tau)} = \mathbf{ \partialPhi}(\tau, t})</math> =and <math>\mathbf{A\Phi}^{-1}(t, \tau)\mathbf{\Phi}(t, t_0\tau) = I</math>, andwhere <math>I</math> is the identity matrix.

:3. <math>\mathbf{\Phi}(\taut, \taut) = I</math> for all <math>\taut</math> and where <math>I</math> is the identity matrix.<ref>{{cite book|first=Roger W.|last=Brockett|title=Finite Dimensional Linear Systems|publisher=John Wiley & Sons|year=1970|isbn=978-0-471-10585-5}}</ref>

And4. <math>\mathbf{\Phi}(t_2, t_1)\mathbf{\Phi}(t_1, t_0) = \mathbf{\Phi}(t_2, t_0)</math> alsofor mustall have<math>t_0 the\leq following properties:t_1 \leq t_2</math>.

5. It satisfies the differential equation <math>\frac{\partial \mathbf{\Phi}(t, t_0)}{\partial t} = \mathbf{A}(t)\mathbf{\Phi}(t, t_0)</math> with initial conditions <math>\mathbf{\Phi}(t_0, t_0) = I</math>.

:{| class="wikitable"

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|16.|| The state-transition matrix <math>\mathbf{\Phi}(t_2t, t_1)\mathbf{\Phi}(t_1, t_0) = \mathbf{\Phi}(t_2, t_0tau)</math>, given by

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

|-

where <math>\mathbf{U}(t)</math> is the [[Fundamental matrix (linear differential equation)|fundamental solution matrix]] that satisfies

|2.||<math>\mathbf{\Phi}^{-1}(t, \tau) = \mathbf{ \Phi}(\tau, t)</math>

: <math>\dot{\mathbf{U}}(t)=\mathbf{A}(t)\mathbf{U}(t)</math>

|-

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

|3.||<math>\mathbf{\Phi}^{-1}(t, \tau)\mathbf{\Phi}(t, \tau) = I</math>

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

|-

|4.||<math>\frac{d\mathbf{\Phi}(t, t_0)}{dt} = \mathbf{A}(t)\mathbf{\Phi}(t, t_0)</math>

|}

7. If the system is [[time-invariant]], we can define <math> \mathbf{\Phi}</math>; as:

:<math>\mathbf{\Phi}(t, t_0) = e^{\mathbf{A}(t - t_0)}</math>