On a modified Lur'e problem

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Abstract
This paper considers the problem of asymptotic stability in the large of an autonomous system containing a single nonlinearity. The nonlinear function is assumed to belong to several subclasses of monotonically increasing functions in the sector(0, K), and the stability criterion is shown to be of the formRe Z(j\omega) [G(j\omega) +\frac{1}{K}] -\frac{\delta'}{K} \geq 0where the constant\delta'is equal toZ(\infty) - Z_{p}(\infty)andZ_{p}(s)is a Popov multiplier. The multiplierZ(s)can, in general, have complex conjugate poles and zeros and is thus more general than the type of multipliers obtained in previous results. The nonlinear functions considered are odd monotonic functions, functions with a power law restriction, and a new class of functions with restricted asymmetry having the property|\frac{f{\theta}{f{\-theta}|\leq cfor all\theta. Unlike in certain earlier publications, no upper bound is placed on the derivative of the nonlinearity here. The results obtained can be used to establish stability in some cases even when the Nyquist plot of the linear part transfer function lies in all four quadrants and the nonlinearity is not necessarily odd. Furthermore, it is shown that the conditions on the multiplier and, consequently, those on the linear part can be relaxed as the feedback function approaches linearity.

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