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Unformatted text preview: aracteristic equation.
• Also, methods are available for testing the
stability of a closed-loop system based only on
stability of closed
the loop transfer function characteristics.
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
pairs of roots with opposite signs.
– form an auxiliary polynomial from the coefficients in the
– Replace the zero coefficients from the coefficients of the
differentiated auxiliary polynomial.
– If there is not a sign change, the roots of the auxiliary
equation define the roots of the system on the imaginary
12/13/2008 HSK - EPM 381 - Stability 5 The Nyquist Criterion
• The Routh-Hurwitz Criterion provides a check
of absolute stability based on the closed-loop
• The Nyquist Criterion may be used to analyze
the relative stability of the closed-loop system
based on the loop characteristics.
• Relative stability refers to how close the system
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
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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This note was uploaded on 03/03/2010 for the course AUTOMATIC 335 taught by Professor ? during the Winter '10 term at Ain Shams University.
- Winter '10