12132008 hsk epm 381 stability 15 the nyquist

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Unformatted text preview: es of the closed- loop system in the RHS. P, corresponds to the number poles of the loop transfer function in the RHS. 12/13/2008 HSK - EPM 381 - Stability 15 The Nyquist Criterion • Normally, P is known. If the loop transfer Normally, is function is stable, then P = 0 and N must be zero function and for a stable closed-loop system. • If P is non zero, then there must be P If is counterclockwise encirclements of the origin. • A slight modification of the process is to map the slight GH(s) function rather than F(s) and then check GH rather and for encirclements of the –1 point in the complex in GH(s) plane. This works since F(s) = 1 + GH(s) . GH plane. GH 12/13/2008 HSK - EPM 381 - Stability 16 Summary: Interpretations and explanations – If the s-plane contour encircles a zero of 1+GH in a certain direction, the image contour will encircle the origin in the same direction. (related to Z) – If the s-plane contour encircles a pole of 1+GH in a certain direction, the image contour will encircle the origin in the opposite direction. (related to P) – The net number of same-direction encirclements, Ncw, equals the difference Ncw = Z- P. – Actually, we should investigate 1+GH and encirclements around the origin, but easier to investigate GH and encirclements around -1. 12/13/2008 HSK - EPM 381 - Stability 17 The Nyquist Criterion: A Simple Example • Consider a system with the loop TF s-plane jω j∞ K GH ( s ) = s(s + a) GH-plane j∞ mapping × R→ ∞ Γ Pole of GH Im{GH} σ Re{GH} -1 -j∞ 12/13/2008 -j∞ HSK - EP...
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