IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. AC-32, NO. zyxwvutsrqponmlk 1, JANUARY 1987 53 order reduction in control theory-An overview." zyxwvutsrqpon Automatica. zyxwvutsr vol. zyxwvutsrq 12, pp. 123- 132, 1976. [6] K. Kendig. Elementary Algebraic Geometry. New York: Springer-Verlag. 1977. Frequency Domain Conditions for Strictly Positive Real Functions PETROS IOANNOU AND GANG TAO Absfract-Frequency domain conditionsfor strictly positive real (SPR) functions which appear in literatureareoften only necessary or only sufficient. This point is raised in [l], zyxwvu [Z], where necessary and sufficient conditions in thesdomain aregiven for a transfer functionto be SPR. In this note, thepoints raised in 111, 121 are clarified further by giving necessary and sufficient conditions in the frequency domain for transfer functions to be SPR. These frequency-domain conditions are easier to test than those given in the s-domain or time domain 111, [2]. I. INTRODUCTION The definition of positive real (PR) and SPR transfer functions is motivated from network theory. That is,a PR (SPR) rational transfer function can be realized asthe driving point impedance of a passive (dissipative) network. Conversely, a passive (dissipative) network has a driving point impedance that is rational and PR (SPR). In [I], the following equivalent definitions have been given for PR transfer functions by an appeal to network theory. Definition zyxwvutsrqp 1.1 [I]: A rational function h(s) of the complex variable s = zyxwvutsrq u + zyxwvutsr jw is PR if 1) h(s) is real for real s; 2) Re [h(s)] 2 0 for all Re [s] > 0. 1) h(s) is real for real s; 2) h(s) is analytic in Re [s] > 0 and poles on thejw-axis are simple 3) For any real value of w for which Jw is not a pole of zyxwvut h(jw), Re Definition 1.2 [ Z ] : The rational transfer function h(s) is PR if and such that the associated residue is nonnegative; [h(jo)] 5 0. In most of the literature, h(s) is termed to be SPR if a) h(s) is analytic in Re [s]zO (1.1) and b) Re [h(jw)]>O, Vw E (- 03, 03). (1 4 In many cases, (b) is replaced by e'), Le., b') Re [h(jo)]>6, Vo E [--03, 031 (1 -3) where 6 is a positive constant. As stated by Taylor zyxwvutsrqpo [2] and Narendra and Taylor [3], (a) and (b) are only necessary, whereas (a) and (b') are only sufficient for h(s) to be SPR. For example, supported by the National Science Foundation under Grant ECS-8312233. of Southern California. Los Angela, CA 90089. Manuscript received January 21, 1986: revised September 12. 1986. This work was The authors are with the Department of Electrical Engineering-Systems, University IEEE Log Number 8611557. satisfies (a). (b) but is not SPR. since it cannot be realized as the driving point impedance of a dissipative network. Similarly. h(s)=-, a>o 1 S+CY (1 5) is an SPR transfer function but does not satisfy (b'). Furthermore, h(s) given by (1.4) does not satisfy the Kalman-Yakubovich lemma [3], [4], which is fundamental in the stability analysis of adaptive [5]$ [6] and other nonlinear systems [3] via the Lyapunov direct method. Motivated from network theory. Taylor [2] and Narendra and Taylor [3] proved the following lemma for SPR transfer functions. Lemma 1.1: Assume that h(s) is not identically zero for all s. Then h(s) is SPR if and only if h(s - E) is PR for some E > 0. In 121. Taylor also showed that if h(s) is SPR and strictly proper, then Re [h(jw)] can go to zero as I w I + 03 not faster than w-*. In the following section, we give necessary and sufficient conditions in the frequency domain for proper and improper transfer functions to be SPR. 11. FREQUENCY DOMAIN CONDITIOKS FOR SPR FVKCTIONS Let n* be the relative degree of h(s) = n(s)/d(s), i.e., n* = degree of n(s) - degree of d(s). The following theorem establishes necessary and sufficient conditions for h(s) to be SPR. Theorem 2.1: Assume that h(s) is not identically zero for all s. Then h(s) is SPR if and only if 1) h(s) is analytic in Re Is] 2 0, 2) Re [h(jw)] > 0, V w E (-a, m). and 3) i) Qrn w2 Re [h(jw)]>O when n*=l, or (2.1) ii) lirn Re [h(jw)]>O, lirn - h(jw)>O when n* = - 1. . (2.2) 4A-m dz-p 'Y1-m jw Proof: Necessity: If h (s) is SPR, then from Lemma I,], h (s - E) is PR for some E > 0. Hence, there exists an E* > 0 such that for each E E [0, E*), h(s-E) is analytic in Re Is] < 0. Therefore, there exists a real rational function W(s) such that [I] h(S-t)+h(-s+E)= N'(s-€)W(-s+t) (2.3) where W(s) is analytic and nonzero for all s in Re [s] > -E. Let s = E + jw; then from (2.3) w'e have 2 Re [I~(jw)l=IW(jw)(~>O, Vw E (-00, m). (2.4) Now h(s) can be expressed as (2.5) Ifm= n-l,i.e.,n* = 1,b,-, #O,thenfrom(2.5)itfollowsthatb,-, > 0 and an-lbn-l - bn-? - > 0 for h(s-E) to be PRI and <2-= lim e* Re [h(jw)J=a,_,6,~l-b,_~r~b,_,>0. (2.6) If m = n+ 1. Le., n* = - 1, b,+, # 0, then Since Re [h(jw - E)] z 0 V o E (--03, 03) and then b,,, > 0, b, - b,+la,-l 2 tb,,, > 0, andtherefore(2.2)follows directly. 0018-9286/87/0100-00j3f01.00 0 1987 IEEE