Frequency domain evaluation of circle criterion, Popov criterion and off-axis circle criterion in the MIMO case

Abstract

Many control applications can be described as non-linear feedback systems whose stability properties can, for example, be analysed with the circle criterion, the Popov criterion or the off-axis circle criterion. In the SISO case these sector criteria became very popular because of their simple graphic interpretations; the criteria's significance for multivariable feedback systems remained rather poor since there are almost no practicable ways to evaluate them. In this paper the MIMO versions of the three stability criteria are derived using Popov's theory of hyperstability. The main contribution is the presentation of new methods for a frequency domain evaluation of the sector criteria in the multivariable case. The new methods are based on graphic elements: the 'positivity curves'.