Eigenstructure through Matrix Adjugates and Admissible Pairs
Omar M. E. El-Ghezawi · Electronics · 2022
The traditional eigenstructure assignment method is reconsidered in this paper, shedding light on its nature and some of its properties. The approach is through matrix adjugates which enable a more eloquent decomposition compared to the lumped solution usually obtained through matrix null spaces. Based on the new approach, the admissible pair (w,z) is formalized. The new approach enables an alternative methodology to the determination of the permissible closed loop eigenvector subspaces and the might be termed the companion input-subspace. Such approach renders the method a formulae-based method as w and z are now obtained formula-wise as opposed to being extracted out of a lumped solution. Compared to the traditional method, the study reveals information such as w and z are independently and explicitly determined. Moreover, newly assigned eigenvalues always result in z≠0, the input matrix B does not influence z and a fundamental feature of the open loop characteristic polynomial is exposed. Furthermore, closed loop eigenvectors associated with repeated eigenvalues can be explicitly computed by means of differentiation as opposed to null space determination. A special form of system representation is highlighted, which considerably eases the calculations. The myriad concepts pointed out have been demonstrated and authenticated through carefully selected examples involving real, complex, repeated eigenvalues, and two practical systems of an electrical nature.