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Chapter 3:
The Fundamental Matrix of a Fixed Continuous-Time System

The differential equations governing the behavior of a fixed continuous-timesystem in vector-matrix form are

    q(t) = A q(t) + Bx(t)

    y(t) = C q(t) + D x(t),

where q is the state, x is the input or forcing function, y is the output behavior of interest, and A, B, C, and D are constant matrices.

The unforced (homogeneous) form of Eq. (C-1) is

    q (t) = A q(t).

The solution to this system of equations will be shown to be

    q(t) = eA(t-t0)q(t0)= Φ(t-t0)q(t0)

where q(t0) denotes the value of q(t) at t = t0 and Φ(t) = eAt is a matrix defined by the series (C-5) and is called the fundamental matrix of the system. In engineering literature, Φ(t - t0) is called the transition matrix because it determines the transition from q(t0) to q(t).

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