Branch-point placement
One possible way of controller design for a multivariable linear system with a square transfer matrix G(z) - or G(s) for a continuous time system - is based upon the eigenfunctions \(n_ i(z)\) and eigenvectors \(w_ i(z)\). In practical circumstances it is not very convenient to use the functions \(n_ i(z)\) and \(w_ i(z)\) but rather approximations for the power series in \(z^{-1}\) of the functions \(n_ i(z)\) and \(w_ i(z)\) are used. Truncations of these power series exhibit an appropriate approximate description of the plant G(z) provided that the so called branch points \(\phi\) are outside the unit circle in the complex plane. Here the branch points are defined as those points in \({\mathbb{C}}\) where eigenfunctions \(n_ i(z)\) meet. The above observation makes it desirable to introduce a compensator K (with constant entries) such that the resulting transfer matrix G(s)K has all branch points outside the unit circle. The paper contains suitable algorithms for obtaining this property and two illustrative examples conclude the work.
eigenvalue and eigenvector approximation
scientific article; zbMATH DE number 4172846