The differential equations governing the behavior of a fixed continuous-timesystem in vector-matrix form are
- q(t) = A q(t) + Bx(t)
- (C-1)
- y(t) = C q(t) + D x(t),
-
(C-2)
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).
-
(C-3)
The solution to this system of equations will be shown to be
-
q(t) = eA(t-t0)q(t0)= Φ(t-t0)q(t0)
-
(C-4)
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).
