G2 The State Transition Matrix

Insight into the n x n state transition matrix can be obtained as follows. Let dX(t)

^>i(Mo)

9Xo

^>2 (Mo)

=

9r(i) 9Xo

Note that ^>3 (t, t0) is an m x n matrix of constants partitioned into an m x 6 matrix of zeros on the left and an m x m identity matrix on the right, where m is the dimension of ft and X is of dimension n. Because of this, it is only necessary to solve the upper 6 x n portion of Eq. (G.1.11).

Equation (G.1.11) also can be written in terms of a second-order differential equation. This can be shown by differentiating Eq. (G.2.1):

9Xo

<^2(t,to)

9Xo

.</>3(t,to).

In these equations, 0 represents an to x n null matrix. Notice from the fist of Eq. (G.2.2) that dr

Hence, we could solve this second-order system of differential equations to obtain $(t,to),

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