Content deleted Content added
LucasBrown (talk | contribs) Adding short description: "Tool in control theory" |
m bold formatting for identity matrix Tags: Mobile edit Mobile app edit iOS app edit App section source |
||
Line 29:
1. It is continuous and has continuous derivatives.
2, It is never singular; in fact <math>\mathbf{\Phi}^{-1}(t, \tau) = \mathbf{ \Phi}(\tau, t)</math> and <math>\mathbf{\Phi}^{-1}(t, \tau)\mathbf{\Phi}(t, \tau) = \mathbf I</math>, where <math>\mathbf I</math> is the identity matrix.
3. <math>\mathbf{\Phi}(t, t) = \mathbf I</math> for all <math>t</math> .<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>
4. <math>\mathbf{\Phi}(t_2, t_1)\mathbf{\Phi}(t_1, t_0) = \mathbf{\Phi}(t_2, t_0)</math> for all <math>t_0 \leq 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) = \mathbf I</math>.
6. 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>
where the <math>n \times n</math> matrix <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> with initial condition <math>\mathbf{U}(t_0) = \mathbf I</math>.
7. 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
| |||