Gains, phases and angles in multivariable systems

Of the two standard linear operator decompositions, one - the characteristic-value decomposition - is suitable for studying the stability of a multivariable feedback loop, while the other - the singular-value decomposition - is suitable for studying its performance and robustness. The local and global characteristics of both decompositions are discussed for an operator-valued function of a complex variable. Ways in which phase information may be added to singular values are considered, and a particular way of using a singular-value-related decomposition in design studies for feedback systems is briefly discussed.