The problem of stabilizing the chain of four integrators under the addi- tional condition of asymptotic tracking a target trajectory when approach- ing the equilibrium state is considered. The target trajectory is defined implicitly as that of a simpler second-order reference system extended to the four-dimensional space. For the reference system, we consider the chain of two integrators closed by a feedback in the form of nested sat- urators. The feedback coefficients in the reference system controller are selected so that to ensure desired characteristics of the target trajectory. The desired stabilizing control is obtained as a discontinuous function of the reference system feedback and its derivatives. The application of the control obtained results in an affine switched system, which is shown to be locally stable. An ellipsoidal estimate of the attraction domain is con- structed by applying results of absolute stability theory. The problem of finding the best estimate is discussed.