On efficient computation of rational {1,2}-pseudo-inverses for multivariable rational matrix-valued functions and their applications

In this article we consider the problem of the existence of rational 1,2-pseudo-inverses for rational multivariable matrix-valued functions. We prove that any rational multivariable matrix-valued function has rational 1,2-pseudo-inverse and we describe the set of all 1,2-pseudo-inverses of a given function, in terms of rational free parameters. It is shown that the 1,2-pseudo-inverse can have an effective description relative to the Moore–Penrose pseudo-inverse (that in general is not even rational), thus making it an alternative effective solution to the problem of online matrix inversion of large matrices that depend on real-time measured parameters. The rationality of the 1,2-pseudo-inverse is crucial when it should be realized in the physical real world (e.g. in inverse control), in contrast with realizations in computer algorithms (e.g. in image processing and communication systems) where the rationality is not necessary but the effective description of the pseudo-inverse becomes crucial. The results has applications in control systems, robust control, inverse control, mechanical systems, kinematic chains, kinematic networks, multi-degree-of-freedom systems, image processing, signal processing and communication systems.