Linear Time-invariant Systems The Notion of p hase =0a=/()de Let th 2/ G(ia ) u(iu u (=iu)daw The phase for a given input u can then be defined as Re fGiw)](iu (iw) y u) (Hu u DEFINITION a rat io nal transfer funct io n g wit h real Passivity implies that the phase is in the rang coefficients is positive real(PR)if ≤y )≥0 for Re s≥0 a transfer funct io n g is strictly positive real (SPR)if Gs-e)is positive real for so me real C haracteriz ing p ositive real Ex ample Transfer f unctions Recall THEOREM 2 al transfer funct n G(s)with real Re)lu(iw u(ia)dw A coefficients is Pr if and only if the following co ndit io ns hold Positive real pr .(i The funct io n has no poles in the rig ht ReG(iu)≥0 .(ii If the funct io n has poles on the Input st rictly passive ISP imag inary axis or at infinity, they are ReG(u)≥E> simple poles wit h positive resid (ii) The real part of G Out put st ricky passive OSP along the lw axis, that is ReG(ia)≥e|Giu)2 Re(G(u)≥0 G(s)=s+1 SPR and ISP not OSP A transfer funct io n is sPr if conditions(i and (ii hold and if condition() is replaced by the G(s)=+1 SPR and OSP not ISP co io n that G(s)has no pol G(s)=2 OSP and ISP not OSP I maginary axis 口G(6)= OPS or SP K.. Astrom and B wittenmarkThe Notion of Phase Let the signal space have an inner product The phase for a given input u can then be dened as cos ' = hy j ui kuk kyk = hHu j ui kuk kHuk Passivity implies that the phase is in the range ￾  2  '   2 Linear Time-invariant Systems hy j ui = Z1 0 y(t)u(t) dt = 1 2 Z1 ￾1 Y (i!)U (￾i!) d! = 1 2 Z1 ￾1 G(i!)U (i!)U (￾i!) d! = 1  Z1 0 Re fG(i!)g U (i!)U (￾i!) d! Definition 3 A rational transfer function G with real coecients is positive real (PR) if Re G(s)  0 for Re s  0 A transfer function G is strictly positive real (SPR) if G(s ￾ ") is positive real for some real " > 0. Characterizing Positive Real Transfer Functions Theorem 2 A rational transfer function G(s) with real coecients is PR if and only if the following conditions hold.  (i) The function has no poles in the right half-plane.  (ii) If the function has poles on the imaginary axis or at innity, they are simple poles with positive residues.  (iii) The real part of G is nonnegative along the i! axis, that is, Re (G(i!))  0 A transfer function is SPR if conditions (i) and (iii) hold and if condition (ii) is replaced by the condition that G(s) has no poles or zeros on the imaginary axis. Examples Recall hy j ui = 1  Z1 0 Re fG(i!)g U(i!)U(￾i!) d!  Positive real PR Re G(i!)  0  Input strictly passive ISP Re G(i!)  " > 0  Output stricly passive OSP Re G(i!)  "jG(i!)j2 G(s) = s + 1 SPR and ISP not OSP G(s) = 1 s+1 SPR and OSP not ISP G(s) = s 2+1 (s+1)2 OSP and ISP not OSP G(s) = 1 s PR not SPR, OPS or ISP c K. J. Åström and B. Wittenmark 3