Modern Control Systems
VOLUME 4, FEBRUARY 2011
Modern Control Systems
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, 
.
3. With the desired poles given, one can develop the desired characteristic equation, 
.
4. Finally one develops the characteristic equation for the closed loop system, which is given by 
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].
angle (input). Using MATLAB, we obtain controllability and observability of 
and 
respectively which gives both full rank, i.e 3.
Figure 1: Pole placement structure
From figure 1, K = control gain matrix, 
= input, Nbar is a scaling factor. Using MATLAB, we obtain the poles of the system are at 
The desired poles are set to be 
With these desired poles, we obtain 
and Nbar = 7.0711. Figure 2 shows the pitch angle open and closed loop responses.
Figure 2: Response of the system pole placement method
References
[1] Ashish Tewari, Modern Control Design with MATLAB and SIMULINK, John Wiley & Sons Ltd, 2002
[2] Herbert Werner. Control Systems 2 lecture notes. Hamburg University of Technology (TUHH), Germany. 2008
[3] Control Tutorial for Matlab. http://www.engin.umich.edu/group/ctm/examples/pitch/SSpitch.html.