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A first harmonic approach (describing function method) is used in this work to study the behavior of linear control systems with a saturating high-gain linear feedback. It is shown that when the eigenvalues of the closed-loop system are located deeper in the left-half complex plane, several unstable periodic orbits shrink to the origin, thus leading to an unstable equilibrium point. This dynamical behavior is interpreted as the vanishing of the region of attraction of the origin when a saturating high-gain feedback is used.

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We consider one-dimensional, time-invariant sampled-data linear systems with constant feedback gain, an arbitrary fixed time delay, which is a multiple of the sampling period and a zero-order hold for reconstructing the sampled signal of the system in the feedback control. We obtain sufficient conditions on the coefficients of the characteristic polynomial associated with the system. We get these conditions by finding both lower and upper bounds on the coefficients. These conditions let us give both an estimation of the maximum value of the sampling period and an interval on the controller gain that guarantees the stabilization of the system.