The theorem that we present in this section further develops the relation- ship between real and imaginary components of analytic functions. More specifically, it provides an expression for the behavior of the imaginary part at high frequencies, in terms of the integral of the real part, of a func- tionHthat satisfies the following conditions:
(i) H(jω) = P(ω) +jQ(ω) =H(−jω), wherePandQare real-valued functions ofω;
(ii) H(s)is analytic ats=∞and in the CRHP except possibly for singu- laritiess0 =jω0on the finite imaginary axis that satisfy lims s0(s− s0)H(s) =0.
3.1 Bode Integral Formulae 49 Bode called the real and imaginary parts,PandQ,attenuationandphase, respectively, since he studied functions of the formH = logà, whereà represented characteristics of an electrical network. This suggested the name of the theorem.
Before presenting Bode’s Attenuation Integral Theorem, we discuss a different interpretation of Theorem 2.3.3 in Chapter 2 that serves as mo- tivation. Recall that Theorem 2.3.31 gave an expression of the imaginary part of Hat any frequency ω0 in terms of the integral of the complete frequency response of the real part ofH. In the proof of Theorem 2.3.3, the functionHwas manipulated in order to create poles ats= jω0 and s = −jω0 in the integrand of (2.27). This led to the result that the inte- grand had residues of valuejQ(ω0)at each of these poles, and thus the real-imaginary parts relation (2.26) can be alternatively interpreted as an expression for the residue at a finite frequency. There is no reason why the same procedure cannot be applied to evaluate the residue at infinity ofH, denoted by Ress= H(s)(see §A.9.1 in Chapter A). This is the essential format of Bode’s Attenuation Integral Theorem.
Theorem 3.1.1 (Bode’s Attenuation Integral Theorem). LetHbe a func- tion satisfying conditions (i) and (ii). Then, forH(jω) =P(ω) +jQ(ω),
Z
0
[P(ω) −P(∞)]dω= −π 2 Res
s= H(s). (3.2)
Proof. SinceHis analytic at infinity, it follows from Example A.8.4 of Ap- pendix A that it has a Laurent series expansion of the form
H(s) =ã ã ã+ c−k
sk +ã ã ã+ c−1
s +c0, (3.3)
which is uniformly convergent inR ≤|s| < ∞, for someR > 0. In view of the assumption (i) onH, it follows that the coefficients{ck}are real and thatc0=P(∞).
Consider next the contour of integration shown in Figure 2.3 of Chap- ter 2, where the small indentations correspond to possible singularities of Hon the imaginary axis, and where the large semicircle has radius larger thanR. Denote this semicircle byCR. Then, according to Cauchy’s integral
theorem I
[H(s) −P(∞)]ds=0 . (3.4) The contribution of the integrals on the small indentations on the imag- inary axis can be shown (see the proof of Lemma A.6.2 in Appendix A) to tend to zero as the indentations vanish, due to the particular type of
1Theorem 2.3.3 was proven for proper and stable transfer functions, but the result also holds for any function satisfying conditions (i) and (ii).
50 3. SISO Control singularities characterized in (ii). The integral on the large semicircle of the terms insk,k≤2, of (3.3) tends to zero as the radius becomes infinite (see Example A.4.1 in Appendix A). Hence, taking limits asR→ ∞, (3.4) reduces to
0= Z
−
[P(ω) +jQ(ω) −P(∞)]jdω+ lim
R
Z
CR
c−1
s ds
=2j Z
0
[P(ω) −P(∞)]dω−jπc−1,
where the second line is obtained using the symmetry properties ofPand Q, and Example A.5.1 of Appendix A. Equation (3.2) then follows using Ress= H(s) = −c−1(see (A.80) in Appendix A).
Note that, from (3.3),Q(ω) = ImH(jω) = −c−1/ω+c−3/ω3 +ã ã ã. Hence (3.2) can also be written as
Z
0
[P(ω) −P(∞)]dω= −π 2 lim
ω ω Q(ω).
This gives an expression for the behavior at infinity of the imaginary com- ponent ofH.
A formula parallel to (3.2) but applicable at zero frequency is easily ob- tained from the above result using the fact thatHis analytic at zero. This is established in the following corollary.
Corollary 3.1.2. LetHbe a function such thatH(1/ξ)satisfies conditions (i) and (ii). Then
Z
0
[P(ω) −P(0)]dω ω2 = π
2 lim
s 0
dH(s)
ds . (3.5)
Proof. SinceHis analytic at zero it has a power series expansion of the form
H(s) =a0+a1s+ã ã ã+aksk+ã ã ã , (3.6) which is uniformly convergent in|s| ≤ r, for somer > 0. Consider the functionH(1/ξ), which is analytic at infinity and admits — from (3.6) — a Laurent series expansion of the form
H(1/ξ) =ã ã ã+ak
ξk +ã ã ã+a1
ξ +a0 ,
which is uniformly convergent in|ξ|≥1/r. Then, using (3.2), we have Z
0
[P(1/ν) −P(0)]dν= π 2 a1 .
The proof is completed on making the change of variable of integration ω=1/νand noting thata1=lims 0[dH(s)/ds].
3.1 Bode Integral Formulae 51 The theorem and corollary given above, as well as Theorem 2.3.3 in Chapter 2, give examples of the great variety of formulae that can be obtained via a judicious choice of contour of integration and integrand in Cauchy’s theorems. Bode collected a number of such formulae in his book; see, for example, the table at the end of Chapter 13 of Bode (1945).
Our choice of the relations presented so far is justified by their immediate application in systems theory. Otherwise, to quote Bode,
“. . . it is extremely difficult to organize all the possible relations in any very coherent way. In a purely mathematical sense most of the formulae are related to one another by such obvious transformations and changes of variable that there is no good reason for picking out any particular set as independent. Basi- cally they are merely reflections of Cauchy’s theorem. Thus the expressions which one chooses to regard as distinctive must be selected for their physical meaning for the particular problem in hand.”