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By D. Atherton

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Trans. AIEE, Vol 73(II), pp 383-390. Kalman, RE 1955, Analysis and design principles of second and higher order saturating servomechanisms. Trans. AIEE, Vol74(II), pp 294-308. Minorsky, N 1962, Nonlinear Oscillations, Van Nostrand, New York. Slotine, JJE & Li, W 1991, Applied Nonlinear Control, Prentice Hall, New Jersey. Struble, RA 1962, Nonlinear Differential Equations, McGraw Hill, New York. Thaler, GJ & Pastel, MP 1962, Analysis and Design of Nonlinear Feedback Control Systems, McGraw Hill, New York.

21% of the fundamental. 9) gives the amplitude of the assumed sinusoidal limit cycle a as 5h/r . 20) gives 2 N (a) = 2 1/2 4h (a - D ) 2 a r - j 4h2 D from which a r 2 2 1/2 C (a) = - 1 = - r 6(a - D ) + jD @ . 91. 1/2 - j . 30) gives Nic (a) = 2h/ar for the IDF so that the roots of the characteristic equation 1 + Nic (a) G (s) = 0 show the limit cycle is stable. This agrees with the perturbation approach which also shows that the limit cycle is stable when the relay has hysteresis. 10) 2 in a feedback loop with an ideal relay with output ±1.

Since our initial investigations are concerned with the stability of the autonomous system, that is r(t) = 0, the two linear dynamic blocks are in series and can be represented by the single transfer function G(s). Further the position of the single static nonlinearity although assumed in the forward path could equally well be in the feedback path. 51 An Introduction to Nonlinearity in Control Systems Stability and Limit Cycles using the DF An early contribution to the problem was a conjecture by Aizermann.

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