Modern Control Systems

VOLUME 4, FEBRUARY 2011

By : Maziyah Mat Noh

Pole-Placement Method

The pole placement method is to specify the desired location of all N poles in the closed loop system, and then determine the N elements of the state gain matrix to achieve these poles. If the system is fully state controllable, the equality of the closed loop characteristic equation and the characteristic equation formed from the specified pole locations gives a linearly independent system of N equations and N unknowns. Solution of this system of equations gives the required elements of the gain matrix, K. The method can be summarized as follows:

1. Check that the rank of the controllability matrix is N (full rank).

2. Specify the desired poles of the closed loop system, feb 3a .

3. With the desired poles given, one can develop the desired characteristic equation, feb 3b .

4. Finally one develops the characteristic equation for the closed loop system, which is given by feb 3c and equates the coefficients of like powers of s from the desired characteristic equation. This gives N equations for the N unknown elements of K .

The pole placement structure is shown in figure 1. Consider a linear time-invariant of an aircraft in (1)[4].

feb 3d

feb 3f

angle (input). Using MATLAB, we obtain controllability and observability of feb 3g

and feb 3h respectively which gives both full rank, i.e 3.

feb 3i

Figure 1: Pole placement structure

From figure 1, K = control gain matrix, feb 3j = input, Nbar is a scaling factor. Using MATLAB, we obtain the poles of the system are at feb 3k The desired poles are set to be feb 3l With these desired poles, we obtain feb 3m and Nbar = 7.0711. Figure 2 shows the pitch angle open and closed loop responses.