Licentiate thesis 2001-007

Parameterization of State Feedback Gains for Pole Assignment

Hans Norlander

May 2001


The pole assignment problem has been subject for research for a long time. For single-input single-output systems this problem is well understood but for multi-input multi-output systems the pole assignment problem is more complex.

In this thesis a parameterization of state feedback gains for pole assignment is characterized with respect to completeness, redundancy and existence. In order to make a systematic examination of this parameterization a number of classes are introduced.

This parameterization depends on two matrices that can be regarded as design parameters. In the thesis it is shown how the degree of freedom in the pole assignment problem for multi-input systems is characterized by these two matrices.

It turns out that the properties of the parameterization depends on whether the characteristic polynomials of the open and the closed loop systems are coprime or not. It is shown in the thesis that if the characteristic polynomials are coprime, every possible feedback gain can be parameterized in this way, and in this sense the parameterization is complete. If the characteristic polynomials have factors in common the parameterization is not complete. In the thesis the shortcomings of the parameterization for this case are characterized.

The design parameters seem to offer a greater degree of freedom than what can be offered in the pole assignment problem. This indicates a certain degree of overparameterization. This redundancy in the design parameters is characterized in the thesis.

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. It is shown that this matrix is invertible for almost every value of the design parameters when the characteristic polynomials are coprime, and hence that the parameterized gains are generally applicable.

The parameterization and its properties are illustrated on a linear model of the military aircraft JAS Gripen.

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