Matrix‐Valued Rational Functions Pseudo‐Inverses and Control Systems Applications

ABSTRACT

Recent research has established the existence of ‐pseudo‐inverses for rational multivariable matrix‐valued functions and characterized the set of all ‐pseudo‐inverses in terms of free rational multivariable matrix‐valued function parameters. The current article clarifies and summarizes the properties of the ‐pseudo‐inverses and introduces new properties and necessary and sufficient conditions for the existence of ‐pseudo‐inverses. It also provides necessary and sufficient conditions for the existence (and uniqueness) of a rational Moore–Penrose pseudo‐inverse. As a theoretical application of the ‐pseudo‐inverses, the article offers a constructive proof of the well‐known Quillen–Suslin theorem, which has many applications in control systems. Efficiently checkable conditions and description of all solutions are also provided. The article also provides practical applications for differential time‐delayed linear systems, for non‐regular linear algebraic integro‐differential equations, and for control of multidimensional systems, in terms of structurally stable realizable controllers.