On the stability in oscillations in a class of nonlinear feedback systems containing numerator dynamics

Abstract

I. INTRODUCTION 64 II. RELATED RESULTS 66 III. STATEMENT OF MAIN RESULT 71 IV. ANALYSIS OF THE FEEDBACK SYSTEM 78 V. EXAMPLES 93 VI. CONCLUDING REMARKS 101 VII. REFERENCES 110 ill CONCLUSION REFERENCES ACKNOWLEDGEMENTS g APPENDIX A: THE INTEGRAL MANIFOLD THEOREM APPENDIX B |(j)(tQ)-y^l <6, the solution ^^t.tg.y^) exists for t > tg and satisfies |*(t)-il/Ct.tQ.y^) ] < e, for t > t^. The solution $(t) is said to be asymptotically stable if it is stable and if there exists 6 > 0 such that whenever j^Ctg)-y^j <6, the solution ipCtjl^.y^) approaches the solution 4(t) as t ^ ®. A T-periodic solution <}>(t) is said to be orbitally stable if there is a 6 > 0 such that any solution 'J'(t,tQ .yg), with jfCtg)-y^j <6, tends to the orbit {*(t): t^ < t < t^+T}. The solution $(t) is said to be unstable in the sense of Lyapunov, if it is not stable. The Describing Function Originally, the single-input sinusoidal describing function, simply referred to as the describing function, was physically motivated. The following heuristic derivation can be found in [15, pp. 49-52, 110-120]: Assume that n(») is an odd nonlinearity, i.e., n(-y) =-n(y), and that y = A sin u)t. Then, the output of n(A sin lot) can be represented by the Fourier series n(A sin ojt) = I A