Characteristic gains, characteristic frequencies and stability

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Abstract
The purpose of this paper is to indicate how characteristic gain loci(or generalized Nyquist diagrams) can be used to count the number of characteristic frequencies (or closed-loop poles) which occur in a given area of the frequency plane for any given feedback gain. The relationship between the gain loci and the characteristic frequency loci is shown to hold for a wide range of linear systems, both sampled and continuous, including those with time delays. The relationship between the frequency loci and the inverse gain loci (or inverse Nyquist diagrams) is also deduced. These results are then used to show how for many systems the frequency loci can be used to determine the only possible ranges of feedback gain to obtain a stable closed-loop system. It is also indicated why in many real systems the closed-loop system should be stable at all but the edges of these possible ranges of feedback gain. The approach used avoids the complex integration which is usually used to justify the Nyquist diagrams (Bode 1945). One example demonstrating the superiority of state-space models over rational polynomial Laplace transfer-function models in the determination of the stability of multivariable systems is given. Some examples of gain loci, inverse gain loci, frequency loci and inverse frequency loci for systems with and without time delays are included.

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