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Survey of the Geometric Approach to the Modern Control Theory

The geometric approach is a mathematical concept developed to improve the analysis and synthesis of linear multivariable systems. In this article we summarized the foundations for the geometric approach to both linear and singular systems. A mathematical introduction to the controlled and conditioned invariant subspaces has been presented. Additional characteristic subspaces were introduced, such as maximum-controlled and minimumconditioned ones, as well as output-null invariant subspaces. The solution of the disturbance localization problem with respect to the system stability was described. The mathematical and the geometric tools applied in the control system theory were presented. The geometric approach was systematically applied to the controllability analysis and the pole assignment procedure. The existence and uniqueness of the solutions for singular systems was separately analyzed due to specific system characteristics which are a consequence of the singular matrix in the state space model. The purpose of the article was to give a comprehensive overview of the geometric approach to the control theory.