Multivariable pi-sharing theory and its application on the Lur´e problem

The pi-sharing theory developed by Lawrence and Johnson (1986) is extended to handle square multivariable continuous-time systems, and for the most essential part of finding usable pi-coefficients, LMI formulations are utilized such that for any finite-dimensional LTI systems, time-invariant pi-coefficients can be obtained easily. Furthermore, pi-coefficients are derived for sector-bounded nonlinearities, and the extended pi-sharing theory is applied to the multivariable Lur´e problem. With this approach, not only can a Lur´e system with known sector bounds on nonlinearities be checked for stability, but also the maximal bounds ensuring stability can be found under some conditions