In this paper, the aperiodicity condition of a system's characteristic equation is first formulated for both continuous and discrete systems. It states that the roots of the characteristic equation have to be real, distinct, and negative for the continuous case, and real? distinct, and in the interval [0,1) for the discrete case. The coefficients of these equations are perturbed for an initially aperiodic system such that, in the limit, aperiodicity is violated. The values of the perturbed coefficients are determined and thus the robustness conditions are obtained, This approach to the topic is quite different from those that have appeared in the literature which first assume that the initial perturbation limits are known. Such approaches are quite similar to that first discussed by Kharitonov for stability problems. In situations where the initial perturbation limits are not known, the approach presented in this paper is a valid alternative. The solution presented in this paper is based on two formulations. The first is based on the critical aperiodicity conditions while the second is based on a theorem known in old literature which states that if the system is initially aperiodic, then Psi(s) = [F'(s)](2) - F(s)F "(s), where F'(s) = dF(s)/ds and F "(s) = d(2)F(s)/ds(2), should have no real roots. Here F(s) is the continuous system's characteristic equation. Now, in the limit when the above condition is violated, one obtains the robustness limits zeta(i). Both approaches give the same values. To obtain similar robustness limits for the discrete case, we can use either the linear fractional transformation z = s/(s - 1) which transforms the segment [0,1) in the z-plane onto the negative real axis in the s-plane and use the above conditions, or directly formulate the aperiodicity conditions in the z-plane. Both are discussed in this paper. An example is presented to illustrate the above formulations for obtaining the robustness conditions. [References: 24]
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