In control theory, the state-transition matrix is a matrix whose product with the state vector at an initial time gives at a later time . The state-transition matrix can be used to obtain the general solution of linear dynamical systems. It is also known as the matrix exponential.
Consider the general linear state space model
The general solution is given by
The state-transition matrix , given by
where is the fundamental solution matrix that satisfies
is a matrix that is a linear mapping onto itself, i.e., with , given the state at any time , the state at any other time is given by the mapping
While the state transition matrix φ is not completely unknown, it must always satisfy the following relationships:
And φ also must have the following properties:
1. 2. 3. 4.
If the system is time-invariant, we can define φ as:
In the time-variant case, there are many different functions that may satisfy these requirements, and the solution is dependent on the structure of the system. The state-transition matrix must be determined before analysis on the time-varying solution can continue.
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