# Lect_15 - Nonlinear Systems and Control Lecture # 15...

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Nonlinear Systems and Control Lecture # 15 Positive Real Transfer Functions Connection with Lyapunov Stability – p.1/22

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Definition: A p × p proper rational transfer function matrix G ( s ) is positive real if poles of all elements of G ( s ) are in Re [ s ] 0 for all real ω for which is not a pole of any element of G ( s ) , the matrix G ( ) + G T ( ) is positive semidefinite any pure imaginary pole of any element of G ( s ) is a simple pole and the residue matrix lim s ( s ) G ( s ) is positive semidefinite Hermitian G ( s ) is called strictly positive real if G ( s ε ) is positive real for some ε > 0 – p.2/22
Scalar Case ( p = 1 ): G ( ) + G T ( ) = 2 Re [ G ( )] Re [ G ( )] is an even function of ω . The second condition of the definition reduces to Re [ G ( )] 0 , ω [0 , ) which holds when the Nyquist plot of of G ( ) lies in the closed right-half complex plane This is true only if the relative degree of the transfer function is zero or one Note: for G ( s ) = n ( s ) d ( s ) , the relative degree is deg d -deg n . – p.3/22

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G ( ) = 1 + 1 -40 -35 -30 -25 -20 -15 -10 -5 0 Magnitude (dB) 10 -2 10 -1 10 0 10 1 10 2 -90 -45 0 Phase (deg) Bode Diagram Frequency (rad/sec) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Nyquist Diagram Real Axis Imaginary Axis Bode plot Nyquist plot – p.4/22
G ( ) = 1 ( ) 2 + + 1 -80 -60 -40 -20 0 20 Magnitude (dB) 10 -2 10 -1 10 0 10 1 10 2 -180 -135 -90 -45 0 Phase (deg) Bode Diagram Frequency (rad/sec) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1.5 -1 -0.5 0 0.5 1 1.5 Nyquist Diagram Real Axis Imaginary Axis Bode plot Nyquist plot – p.5/22

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Lemma: Suppose det [ G ( s ) + G T ( s )] is not identically zero. Then, G ( s ) is strictly positive real if and only if
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## This note was uploaded on 07/25/2008 for the course ME 859 taught by Professor Choi during the Spring '08 term at Michigan State University.

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Lect_15 - Nonlinear Systems and Control Lecture # 15...

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