Parameterization of state feedback gains for pole placement

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 this paper, a parameterization of state feedback gains for pole placement is characterized with respect to completeness and existence. This parameterization depends on two matrices that can be regarded as design parameters. 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 this paper the case when they are coprime is emphasized. It is shown that for this case every possible feedback gain can be parameterized in this way, and in this sense the parameterization is complete. The parameterization implies that a certain matrix is invertible. Necessary conditions for when this matrix is invertible are given in terms of the two design parameters.