State-space systems representing a set of ODEs as the state equation could be solved using standard mathematical techniques. One simple form is discussed here for the sake of brevity. Pre-multiplying the state space equation 25.1 by the matrix we get:

or

Integrating the above equation from 0 to time t

	(24.5)

The matrix is known as the state transition matrix since it transforms any initial state x (0) to states at any time t, and is denoted as . The state transition matrix is unique for each dynamic system and could be computed by inverse Laplace transformation of the matrix (s I - A) as

(24.6)

The eigen-values of the plant A could be obtained by finding out the roots of the polynomial equation

(24.7)

Eqn. (24.7) is also known as the characteristic equation of the open-loop system. The roots of the characteristic equation could be plotted in a real vs. imaginary-plane ( s -plane). For a stable open-loop system, these roots are always situated in the left side of the imaginary axis in the s -plane.