Modern Control System Theory and Design, 2nd Edition

8.6.  ACKERMANN’S FORMULA FOR DESIGN USING POLE PLACEMENT [5–7]

In addition to the method of matching the coefficients of the desired characteristic equation with the coefficients of det (sI − P_h) as given by Eq (8.19), Ackermann has developed a competing method. The pole placement method using the matching of coefficients of the desired characteristic equation with the coefficients of Eq (8.19) is very useful for control systems which are represented in phase-variable form, where phase variable refers to systems where each subsequent state variable is defined as the derivative of the previous state variable. Some control systems require feedback from state variables which are not phase variables. Such high-order control systems can lead to very complex calculations for the feedback gains. Ackermann’s method simplifies this problem by transforming the control system to phase variables, determining the feedback gains, and transforming the designed control system back to its original state-variable representation.

Let us represent a control system which is not represented in phase-variable form by the following:

We will assume that the controllability matrix [see Eq (8.65)] can be represented by

Subscript y is used to designate the original, non-phase-variable, controllability matrix. ...