Partial pole placement with minimum norm controller

The problem of placing an arbitrary subset (m) of the (n) closed loop eigenvalues of a n^th order continuous time single input linear time invariant(LTI) system, using full state feedback, is considered. The required locations of the remaining (n - m) closed loop eigenvalues are not precisely specified. However, they are required to be placed anywhere inside a pre-defined region in the complex plane. The resulting non-uniqueness is utilized to minimize the controller effort through optimization of the feedback gain vector norm. Using a variant of the boundary crossing theorem, the region constraint on the unspecified (n-m) poles is translated into a quadratic constraint on the characteristic polynomial coefficients. The resulting quadratically constrained quadratic program can be approximated by a quadratic program with linear constraints. The proposed theory is demonstrated for power oscillation damping controller design, where the eigenvalues corresponding to poorly damped electro-mechanical modes are critical for performance and hence are specified precisely by the designer, whereas the remaining eigenvalues are non-critical and need not be specified precisely. Acceptable closed loop pole placement is achieved for this example along with a 51% reduction in controller norm.