Estimation of an attraction domain for multivariable Lur´e systems using looseless extension of the S-procedure

For multivariable Lur´e systems which are not stable “in whole” the problem of estimating of the attraction domain is considered. This problem has received much attention in the literature. The state variable method as well as the frequency domain method were used. In the both cases the problem has been treated by Lur´e-Postnikov Lyapunov functions and the S-procedure within the absolute stability framework. For the MIMO case the S-procedure is known to obtain only sufficient conditions for existence of the Lur´e-Postnikov function. So, the multivariable extension of the Popov criterion derived on its base is too conservative. We use a non-conservative extension of the S-procedure which makes it possible to derive a new absolute stability criterion which is less conservative than the multivariable Popov criterion. Using this frequency criterion, the estimation of the attraction domain is constructed. The LMI approach to verify whether the initial point belongs to this domain is proposed