Extended Popov criteria for multivariable Lur´e systems

The stability criterion of Popov is considered for multivariable Lur´e systems. To derive the criterion using a Liapunov function, it is necessary to restrict the class of nonlinearities in the system. The largest class reported so far is that of differentiable nonlinearities with symmetric Jacobian matrices. In this paper, the class is broadened by removing the assumption of differentiability from a part of the nonlinearity. For asymmetric nonlinearities, the Popov criterion is not valid. Therefore, modified criteria are proposed, which are applicable to systems with such nonlinearities. The symmetric part of the nonlinearity is treated as normal and the residual as a perturbation