### 2.26. STATE TRANSITION MATRIX

The state transition matrix relates the state of a system at *t* = *t*_{0} to its state at a subsequent time *t*, when the input **u**(*t*) = 0. In order to define the state transition matrix of a system, let us consider the general form of the state equation [see Eq. 2.197]:

The Laplace transform of Eq. (2.252) is given by

where **X**(*s*) is the Laplace transform of **x**(*t*) and U(*s*) is the Laplace transform of **u**(*t*). Solving for **X**(*s*), we obtain

The inverse Laplace transform of Eq. (2.254) gives the state transition equation

where the state transition matrix is defined by

The first term on the right-hand side of Eq. (2.255) is known as the homogeneous solution and is due only to the initial conditions; the second term on the right-hand side of Eq. (2.255), the convolution ...

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