Synthesis of Dynamic Output Feedback Controller Using Functions of Matrix Root-Clustering in D-Regions

This article considers an algorithm for the synthesis of dynamic output controller, where the eigenvalues of a closed control system should be located in a given region (D-region) of the complex plane. The main direction of the research is to synthesize dynamic controllers of minimal order when there is no complete controllability at the output. For this purpose, a brief excursion into the external Grassmann algebra is made with the purpose of determining the external product of vectors, on the basis of which the operation of the bialternate product of matrices is explained. The use of the bialternate product of matrices made it possible to introduce the functions of matrix root-clustering of complex eigenvalues located in separate transformable D-regions. For the usual product of matrices, the functions of matrix root-clustering of real eigenvalues located in separate transformable D-domains are introduced. The functions of matrix root-clustering are transforming, respectively, complex or real points of a given bounded or unbounded D-region of the complex plane into a left half-plane. The article considers the main D-domains most widely used in practice (disk, cone, stability margin), presents their matrix root-clustering and functions of matrix root-clustering of real and complex modes. An algorithm for parametric optimization of dynamic output feedback controller over D-domains has been developed. Practical examples of synthesis are considered. For a fourth-order object (a two-mass weakly damped system with two integrators and parametric uncertainty), a secondorder dynamic controller providing robust quality has been synthesized.