Estimation of an attraction domain for nonlinear multivariable control systems

For multivariable Lure systems which are not stable in whole the problem of estimating of the attraction domain is considered. For initial conditions which belong to this domain the asymptotic stability "in large" is guaranteed. This problem has received much attention in the literature (see for example [1], [2] and the references there). The state variable method as well as the frequency domain method were used. In the both cases the problem has been treated by the Lur\´e-Postnikov Lyapnnov function with subsequent usage of the S-procedure. 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 conservative even in the class of criteria based on the Lur\´e-Postnikov function. We use non-conservative extension of the S-procedure ([3], [4]) which makes it possible to derive new absolute stability criterion which is less conservative than multivariable Popov criterion based on the usual S-procedure. Using this frequency criterion, the estimation of the attraction domain is constructed. It is less conservative than one obtained in [1]. The LMI approach to verify whether the initial condition belongs to the estimation of the attraction domain is proposed.