Consider the single-input, single-output system described by 

Where 

Assume that the input is a linear combination of the states, that is, 

Where r(t) is the reference input. The matrix
K = [K₁ K₂] is known as the gain matrix. Substituting u(t) into the state variable equation gives the
closed-loop system

The design process involves finding K so that the eigenvalues of A-BK are at desired locations in the left-half plane. Compute the characteristic polynomial associated with the closed-loop system and determine values of K so that the closed-loop eigenvalues are in the left-half plane.

x(t) = Ax(t) + Bu(t) y(t) = Cx (t)