- Full state feedback
Full state feedback (FSF), or pole placement, is a method employed in
feedback control system theory to place theclosed-loop pole s of a plant in pre-determined locations in thes-plane . Placing poles is desirable because the location of the poles corresponds directly to theeigenvalue s of the system, which control the characteristics of the response of the system.If the closed-loop input-output transfer function can be represented by a state space equation, see
State space (controls) ,:
:
then the poles of the system 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,
:
:
The roots of the FSF system are given by the characteristic equation, . 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.
Example of FSF
Consider a control system given by the following state space equations:
:
The uncontrolled system has closed-loop poles at and . Suppose, for considerations of the response, we wish the controlled system eigenvalues to be located at and . The desired characteristic equation is then .
Following the procedure given above, , and the FSF controlled system characteristic equation is
:.
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.
NOTE: This only works for Single-Input systems. Multiple input systems will have a K matrix that is not unique. Choosing, therefore, the best K values is not trivial. Recommend using Linear Quadratic Regulation for such applications.
Bibliography
*cite book
last = Sontag
first = Eduardo
authorlink = Eduardo D. Sontag
year = 1998
title = Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition
publisher = Springer
id = ISBN 0-387-984895ee also
*
Pole splitting
*Step response
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