Sector bounds in stability analysis and control design

Abstract

Summary The sector stability theorem is an intuitive condition for the stability of feedback loops that unifies many lines of research including robust control and the theory of passive and dissipative systems. Applying this theorem typically requires checking or enforcing sector bounds on linear time‐invariant systems. This article discusses practical and efficient numerical methods for performing these tasks. We derive a frequency‐domain test for sector boundedness and use it to develop O ( n 3 ) numerical algorithms for computing the relative or directional indices of sector bounds. We also discuss how such algorithms can be combined with nonsmooth optimization techniques to enforce sector bounds while designing and tuning controllers. Several application examples illustrate the potential of these tools and techniques.