Cyclicity of linear multivariable systems

Abstract

The paper discusses the application of cyclicity in the design of linear multivariable feedback systems and establishes a simple criterion for cyclicity of a system. Preliminary results from linear algebra on cyclic subspaces are presented first. It is then shown that cyclicity plays a fundamental role in the design of state and output feedback controllers for linear multivariable systems. A necessary and sufficient condition for cyclicity of a system is shown to be that the rational matrix ϕ(s)=(sI – A)-1 is irreducible, where A is the system matrix. Furthermore, a sufficient condition is that the eigenvalues of A are distinct. Using these conditions, a simple and computationally efficient test for cyclicity is described and is illustrated by a numerical example.