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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.
==Overview==
he state transtion matrix φ is not completely unknown, it must always satisfy the following relationships:▼
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)</math>
: <math>\mathbf{y}(t) = \mathbf{C}(t) \mathbf{x}(t) + \mathbf{D}(t) \mathbf{u}(t)</math>
The general solution is given by
: <math>\mathbf{x}(t)= \mathbf{\Phi} (t, t_0)+\int_{t_0}^t \mathbf{\Phi}(t, \tau)\mathbf{B}(\tau)\mathbf{u}(\tau)d\tau</math>
▲
:<math>\frac{\partial \phi(t, t_0)}{\partial t} = A(t)\phi(t, t_0)</math>
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:<math>\phi(t, t_0) = e^{A(t - t_0)}</math>
The reader can verify that this solution for a time-invariant system satisfies all the properties listed above. However, in the time-variant case, there are many different functions that may satisfy these requirements, and the solution is dependant on the structure of the system. The state-transition matrix must be determined before analysis on the time-varying solution can continue.
==References==
* {{cite book
| author = Brogan, W.L.
| year = 1991
| title = Modern Control Theory
| publisher = Prentice Hall
| isbn = 0135897637
}}
[[Category:Control theory]]
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