Complex sectors and multivariable on-axis and off-axis circle criteria

Earlier work has shown that frequency-dependent real sector conditions, together with a suitable decomposition of the transfer function matrix of a linear time-invariant system G(j¿¿), enable the extension of the circle criterion to the multivariable case. The present paper proposes the use of complex rather than real sectors, and utilises the extra degrees of freedom introduced into the problem to improve the measure of deviation from normality, and thus reduce the diameter of the circles of the criterion. The new circles will not necessarily have their centres on the real axis and thus the approach yields a set of smaller and shifted circles, which provide a useful test for the stability of nonlinear multivariable feedback systems. This approach is also shown to be suitable for the improvement of multivariable off-axis circle type of results.