Full state feedback

Full state feedback (FSF), or pole placement, is a method employed in feedback control system theory to place the closed-loop poles of a plant in predetermined locations in the s-plane.

[1] Placing poles is desirable because the location of the poles corresponds directly to the eigenvalues of the system, which control the characteristics of the response of the system.

The system must be considered controllable in order to implement this method.

If the closed-loop dynamics can be represented by the state space equation (see State space (controls)) with output equation then the poles of the system transfer function are the roots of the characteristic equation given by Full state feedback is utilized by commanding the input vector

Consider an input proportional (in the matrix sense) to the state vector, Substituting into the state space equations above, we have The poles of the FSF system are given by the characteristic equation of the matrix

Comparing the terms of this equation with those of the desired characteristic equation yields the values of the feedback matrix

which force the closed-loop eigenvalues to the pole locations specified by the desired characteristic equation.

[2] Consider a system given by the following state space equations: The uncontrolled system has open-loop poles at

Suppose, for considerations of the response, we wish the controlled system eigenvalues to be located at

Following the procedure given above, the FSF controlled system characteristic equation is where Upon setting this characteristic equation equal to the desired characteristic equation, we find Therefore, setting

forces the closed-loop poles to the desired locations, affecting the response as desired.

This only works for Single-Input systems.

Multiple input systems will have a

A linear-quadratic regulator might be used for such applications[citation needed].