Absolute Stability Criteria for Nonlinear A ne Systems : A Kalman-Yakubovich-Popov Type Approach

In this paper we present nonlinear versions of the Circle and Popov Criteria of absolute asymptotic stability for nonlinear systems. We use a Kalman-Yakubovich-Popov type approach for nonlinear systems which involves a Hamilton-Jacobi equation and a nonlinear KYP system. Our criteria give su ciency conditions of absolute asymptotic stability as well as uniform estimations of the attraction basin of the origin for the closed loop systems. 1 The Hamilton-Jacobi Equation and The Nonlinear KalmanYakubovich-Popov System Let us consider the nonlinear Popov system composed by a nonlinear a ne dynamics and a quadratic criterion: P _ x = f(x) + g(x) u ; x(0) = x0 J(t1) = R t1 0 [q(x) + 2l(x)u+ u R(x)u]dt (1) where x(t) 2 D R, u(t) 2 R, q : D ! R, l : D ! R ( denotes the transpose), R : D ! R , f; gi : D ! TD are vector elds of class C on D a domain of R with 0 2 D, f(0) = 0 and gi(0) = 0 (we have denoted g(x) u = Pm i=1 gi(x)ui). We denote by P = (f; g; q; l; R) a nonlinear Popov system. We shall suppose the following assumptions on criterion: q(0) = 0, q(0) = 0, l(0) = 0, R(x) > 0, 8x 2 D and they are functions of class C on D. Starting with system (1) we de ne two objects: the HJ equation and the nonlinear KYP system. The Hamilton-Jacobi Equation (The nonlinear Riccati equation) has the form: rV f ( 1 2 rV g + l)R ( 1 2 rV g + l) + q = 0 (2) on leave from University "Politehnica" of Bucharest