Pole placement design

The pole placement problem has been a subject for research for a long time. It is well known that state feedback control is an efficient technique for the pole placement problem. For single-input systems this problem is well understood but for multi-input systems the pole placement problem is more complex. In [1] a parameterization of state feedback gains for pole placement was presented. In [2] this parameterization was developed further and characterized with respect to completeness and existence. In [3] some of the results from [2] are summarized.

This parameterization depends on two matrices that can be regarded as design parameters. In [2] it is shown how the degree of freedom in the pole placement problem for multi-input systems is characterized by these two matrices. It turns out that the properties of the parameterization depend on whether the characteristic polynomials of the open and the closed loop systems are coprime or not. In the case when the open loop and the closed loop characteristic polynomials are coprime the parameterization is complete in the sense that every possible feedback gain can be parameterized in this way. However, this is not the case when the open loop and the closed loop characteristic polynomials are not coprime.

References:
[1] K. Nordström and H. Norlander. On the multi input pole placement control problem. In Proceedings of the 36th IEEE Conference on Decision and Control, Vol. 5, pp 4288-4293, San Diego, California, USA, December 1997.
[2] H. Norlander. Parameterization of state feedback gains for pole assignment. Licentiate thesis, Department of Information Technology, Uppsala University, 2001.
[3] H. Norlander. Parameterization of state feedback gains for pole placement. European Control Conference, Cambridge, UK, September 2003.