On transfer functions and transients
In the first part of this paper the concept of the positive real function is generalized so that it is applicable to transfer functions and the functions, satisfying this generalized concept, are arranged into classes. Some tests are then developed which may be used to determine whether a transfer function belongs to a particular class. It is also shown that if transfer functions have certain general forms then they will automatically be members of one of the classes. Finally, several properties of the phase functions for such system functions are developed. The second part of the paper considers the impulse and step responses corresponding to these transfer functions. It is found that these transient responses are bounded and moreover the rise time and settling time of the step response are found to be greater than lower bounds which depend on the amount of the maximum overshoot (or undershoot) . These results are also generalizations of the restrictions developed on the transient responses of positive real system functions and are considerably stronger. As the difference in degree between the numerator and denominator of the transfer function increases, the magnitudes of these lower bounds also increase. Introduction. The question of what restrictions exist on the transient responses of various classes of networks has been treated in a series of papers [1-3]. In particular, bounds on the impulse and step responses have been given when the corresponding frequency responses are restricted to being of constant sign or monotonic in a semiinfinite interval. An example of this is that, if a rational system function, Z(s), is positive real and has one more pole than zero, then the absolute value of the impulse response is never greater than l/C which is the constant multiplier of the system function. These results lead in turn to lower bounds on the rise time or settling time of the step response of such networks when the overshoot and undershoot is specified. Recently, one of these results has been improved by Ovseyevich [4]. The improvement is on the bounds developed on the impulse response of the positive real system functions in the interval (0, T) when the bounds on that response in the interval (T, ») are known. A related problem has been considered by Cutteridge [5] who determines the optimum rise time of passive two-terminal networks under various restrictions. For instance, considering only the step responses which are monotonic or have only a small overshoot, the optimum rise time is determined when the system function has only two zeros and three poles. This paper deals with a generalization of these results which is applicable to transfer functions. In particular, certain classes of functions are defined from the system functions having any number of poles in excess of zeros such that the corresponding transient responses are restricted in a manner similar to the restrictions existing on the transient responses of positive real functions. The strength of the restriction on the transient responses increases as the excess of poles over zeros increases. In the first part of this paper, these classes of system functions are defined. The *Received September 3, 1957. 274 ARMEN H. ZEMANIAN [Vol. XVI, No. 3 definition may be considered a generalization of the concept of positive real system functions which is suitable for application to transfer functions. Various tests are then developed which determine whether a system function is a member of any class. One immediate result is that, if the system function has poles only in the left hand complex frequency plane and no zeros anywhere except at infinityN(that is, if the system function is the reciprocal of a Hurwitz polynomial), then it will be a member of one of the classes. Furthermore, certain properties of the phase functions for such system functions are also developed. In the second section the restrictions on the transient responses for the defined classes of functions are derived. These constitute bounds on the impulse responses and the step responses. From these, lower bounds are developed on the rise time and settling time of the step response when the overshoot and undershoot are given and, conversely, the specification of the rise time or settling time fixes the lower bounds on the maximum overshoot or undershoot. Part I. A generalization of the concept of positive real functions. The systems considered in this paper are lumped, linear, fixed, finite and stable systems so that the system functions1 Z(s) have the following rational form where s is the complex frequency variable a + jut, the coefficients and the constant multiplier K are real numbers, and n and m are positive integers (m > n). 7/„\ _ vs" + Q~-is" 1 + ■ • • + flo _ ^ N(s) . . Z{s) K gm + + ... + bo K D(s)' (1) The term "stable system" is taken to mean a system whose response will eventually become arbitrarily small once the input is removed. Thus the polynomial D(s) is a Hurwitz polynomial all of whose roots have a negative (non-zero) real part. The special classes of functions that are of interest here will be defined after a few preliminary remarks. Let ZQ(s) = (-j)a [ ds„_i f dsQ-2 ■ f Z{s0) ds0 , (2) J CO J CO J CO where the real parts of the complex variables s0 , s, , • • • , , s are all non-negative and q < m — n — 1. Letting st = ) and of Z,(jw) may be obtained from the real and imaginary parts, R(o>) and /(&>), of Z(joj) by (3) and (4). Za(ju) = Rq (w) + jIQ(u) /CO ft 0) q — l /»«t dua-i / dw,-2 • • • / R(w0) do)o (3) CO J — CO J — 00 /CO /» Cd q — 1 /* 0) X do3a-1 J dua-2 J IM dwo . (4) It is evident from the following argument that these successive integrations may be performed m — n — 1 times. Since Z(s) is analytic for  0, the integral of Z(s) between two given points will yield the same result for all paths between these points which do not enter the left half s plane. Furthermore, the inverse power series expansion of Z(s), 'These system functions may be impedances, admittances or transfer functions that are ratios of currents or voltages. 1958] ON TRANSFER FUNCTIONS AND TRANSIENTS 275 which holds for | s | > M, is i ^, (5) H = m-n $ where = K and M is a positive number greater than the distance from the origin to the pole of Z{s) farthest away from the origin. Since this series converges uniformly for | s | > M, it may be integrated term by term yielding a series which also converges uniformly in the same region. This process may be continued q times where q < m — n — 1. Thus the successive integrations of Z(s) given by (2) yield unique functions ZQ(s) which are all analytic for  0. Now the aforementioned classes of functions Z(s) given by (1) are defined as follows. Definition. Z(s) will be called a class k junction, where k = m — n, if one of the following inequalities holds for — «> < « < +00 • For k = 2c + 1 (v = 0, 1, 2, • • •), (-1)'/?*_»(«) > 0 (6) and, for k = 2v (v = 1, 2, 3, • • •)> (-ir.7._,(«) > o. (7) The class k functions have the interesting property that they are related to the positive real functions in the region where they are defined (that is, in the right half s plane and on the imaginary axis). Theorem 1. If the system function Z(s) is a class k function, then ( — Zm_„_1(s) is a positive real function for  0 and the constant multiplier K is positive. Proof. The inequalities (6) and (7) state that the real part of ( — is non-negative for all co. Moreover Zm-»_i(s) is analytic for c > 0 since Z(s) is analytic in this region. Thus by the minimax theorem, the real part of ( —Zm_„_x(s) is non-negative for a > 0. Therefore to prove the first part of the theorem it need only be shown that ( — Zm-n-i(s) is real for s =  0. But 'Z.-.-.W = (-1)-—1 f" d(Tm_n_2 r " 3 ••• r ZMdao J co J oo J oo and the right hand side of this expression is a real function of a real variable. Finally the series expansion of Z(s) given by (5) may be integrated term by term q times according to (2) for q < m — n — 1. The resulting expression for Z„(s), which holds for [ s | > M, is " f t, h-Dfa.Vfc-j?;<8) Thus the first term of the inverse power series expansion of ( — ' Zm^„.1(s) is Km.n (m — n — l)!s ' where Km-n = K. Moreover for s real and sufficiently large, this first term becomes the dominant term. Therefore K must be positive. This completes the proof. A property of the functions Z,(s) which will be used subsequently is the fact that their real and imaginary parts are either odd or even. This is stated by Lemma 2. To prove this however, Lemma 1 will be needed. 276 ARMEN H. ZEMANIAN [Vol. XVI, No. 3 Lemma 1. Let h(u) be even (or odd) and integrable for — m < c < +°°; let q(u) = h(u) du; and let q(°°) = 0. Then q(cc) is odd (or even). Proof. q(co) = / h(u) du+ h(u) du = 0. J — 00 J u