The state-transition equation is defined as the solution of the linear homogeneous state equation. The linear time-invariant state equation given by
with state vector x, control vector u, vector w of additive disturbances, and fixed matrices A, B, and E, can be solved by using either the classical method of solving linear differential equations or the Laplace transform method. The Laplace transform solution is presented in the following equations. The Laplace transform of the above equation yields
where x(0) denotes initial-state vector evaluated at
So, the state-transition equation can be obtained by taking inverse Laplace transform as
The state-transition equation as derived above is useful only when the initial time is defined to be at
Once the state-transition equation is determined, the output vector can be expressed as a function of the initial state.