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Unformatted text preview: aracteristic equation. actual • Also, methods are available for testing the Also, stability of a closed-loop system based only on stability of closed system the loop transfer function characteristics. the 12/13/2008 HSK - EPM 381 - Stability 4 Routh-Hurwitz Stability Criterion, Special Case 2: • An all zero row in the Routh array which corresponds to An all in pairs of roots with opposite signs. pairs • Remedy: – form an auxiliary polynomial from the coefficients in the form auxiliary from row above. row – Replace the zero coefficients from the coefficients of the Replace differentiated auxiliary polynomial. differentiated – If there is not a sign change, the roots of the auxiliary If equation define the roots of the system on the imaginary axis. axis. 12/13/2008 HSK - EPM 381 - Stability 5 The Nyquist Criterion • The Routh-Hurwitz Criterion provides a check Hurwitz of absolute stability based on the closed-loop of loop characteristic equation. characteristic • The Nyquist Criterion may be used to analyze The the relative stability of the closed-loop system relative of closed system based on the loop characteristics. loop • Relative stability refers to how close the system refers is to the absolute stability boundary. 12/13/2008 HSK - EPM 381 - Stability 6 The Nyquist Criterion • Consider the general feedback system R(s) G(s) + – H(s) C(s) The closed-loop transfer function is C (s) G (s) = R ( s ) 1 + GH ( s ) • The characteristic equation is 1 + GH ( s ) = 0 • Define F(s) as F ( s ) = 1 + GH ( s) 12/13/2008 HSK - EPM 381 - Stability 7 The Nyquist Criterion • F(s) is a rational polynomial in s and can be written generally as ( s + z1 )( s + z2 )( )L F ( s) = ( s + p1 )( s + p2 )( )L where -zi are the z...
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