2.3 Two Applications of Complex Integration
2.3.2 Bode Gain-Phase Relationships
As a further application of analytic function theory, we will review the gain-phase relationships originally developed by Bode (1945), which es- tablish that, for a stable minimum phase transfer function, the phase of the frequency response is uniquely determined by the magnitude of the frequency response and vice versa.
We begin by showing that the real and imaginary parts of a proper sta- ble rational function with real coefficients are dependent of each other. We consider this dependence at points of the imaginary axis.
Theorem 2.3.3 (Bode’s Real-imaginary Parts Relationship). 11LetHbe a proper stable transfer function, and suppose that, ats =jω,H(s)can be written asH(jω) = U(ω) +jV(ω), whereUandV are real valued. Then for anyω0
V(ω0) = 2ω0
π Z
0
U(ω) −U(ω0)
ω2−ω20 dω . (2.26) Proof. LetCbe the clockwise oriented contour shown in Figure 2.4, and consisting of the imaginary axis, with infinitely small indentations at the points+jω0 and−jω0 (C2 andC3 in Figure 2.4), and the semicircleC1, which has infinite radius in the ORHP12.
Then the functions
H(s) −U(ω0) s−jω0
and H(s) −U(ω0) s+jω0
are analytic on and insideC. Hence, applying Cauchy’s integral theorem (see §A.5.2 in Appendix A) to both functions and subtracting yields
I
C
H(s) −U(ω0) s−jω0
−H(s) −U(ω0) s+jω0
ds=0 . (2.27) The integral above may be decomposed as
2ω0
Z
−
H(jω) −U(ω0)
ω2−ω20 dω+I1+I2+I3=0 , (2.28)
11This is, in fact, one of the many real-imaginary relationships derived by Bode .
12This should be understood as follows: the indentations on the imaginary axis have radii , and the large semicircle in the ORHP has finite radius . These are then considered in the limit as and . In fact, the large semicircle is nothing but an indentation around .
2.3 Two Applications of Complex Integration 41
FIGURE 2.4. Contour used in Theorem 2.3.3.
whereI1,I2, andI3are the integrals overC1,C2, andC3respectively. The integral on the imaginary axis can be written as
2ω0
Z
−
H(jω) −U(ω0)
ω2−ω20 dω=4ω0
Z
0
U(ω) −U(ω0) ω2−ω20 dω , since the real and imaginary parts of a transfer function evaluated ats= jωare even and odd functions ofωrespectively.
Next note that the integralI1vanishes becauseHis proper.
Now consider the integralI2. Since the radius ofC2is infinitely small, we can approximateH(s)onC2 by the constantH(jω0). We can also ne- glect the contribution of the fraction1/(s+jω)in comparison with that of 1/(s−jω). Then
I2= Z
C2
H(jω0) −U(ω0) s−jω0
ds
=jV(ω0) Z
C2
1 s−jω0
ds
= −πV(ω0),
where the last step follows from (A.32) in Appendix A.
Similarly, we can show that the integralI3equals−πV(ω0). Finally, sub- stituting into (2.28) and rearranging gives the desired expression.
The above theorem shows that values on thejω-axis of the imaginary part of a stable and proper transfer function can be reconstructed from knowledge of the real part on the entirejω-axis. Conversely, under the assumption that the transfer function isstrictlyproper and stable, it can be shown that the real part can be obtained from the imaginary part, i.e.,
U(ω0) = −2 π Z
0
ω[V(ω) −V(ω0)]
ω2−ω20 dω , (2.29)
42 2. Review of General Concepts which follows similarly by adding instead of subtracting in (2.27).
It is shown in §A.6 of Appendix A that the values of a function analytic in a given region can be reconstructed from its values on the boundary.
Combining this with relations (2.26) and (2.29), we deduce that the val- ues of a stable and strictly proper transfer function on the CRHP are com- pletely determined by the real or imaginary part of its frequency response.
We will next show that the gain and the phase of the frequency response of a stable, minimum phase transfer function are dependent on each other.
Theorem 2.3.4 (Bode’s Gain-phase Relationship). LetHbe a proper, sta- ble, and minimum phase transfer function, such thatH(0) > 0. Then, at any frequencyω0, the phaseφ(ω0),argH(jω0)satisfies
φ(ω0) = 1 π Z
−
dlog|H(jω0eu)|
du log coth
u 2
du , (2.30) whereu=log(ω/ω0).
Proof. Consider
H(jω0) =U(ω0) +jV(ω0),|H(jω0)|ejφ(ω0). (2.31) SinceHhas no zeros in the CRHP, taking logarithms in (2.31) gives
logH(jω0) =log|H(jω0)|+jφ(ω0),m(ω0) +jφ(ω0). (2.32) Comparing (2.31) and (2.32) shows that the magnitude characteristicm(ω0) and the phaseφ(ω0)are related to logH(jω0)and to each other in the same way thatU(ω0)and V(ω0) are related toH(jω0)and each other.
Hence (2.29) and (2.26) immediately imply m(ω0) = −2
π Z
0
ω[φ(ω) −φ(ω0)]
ω2−ω20 dω , (2.33) φ(ω0) = 2 ω0
π Z
0
m(ω) −m(ω0)
ω2−ω20 dω . (2.34) Note that the assumption thatHhas no zeros or poles in the CRHP guar- antees the validity of the integrals above, since logHis analytic in the finite CRHP. The singularity at∞arising from a strictly properHis ruled out in the chosen contour of integration (see footnote 12 on page 40). The fact that|logH(s)|/|s|→0when|s|→ ∞eliminates the integral along the large semicircle in the ORHP.
Next, consider (2.34). Changing variables tou=log(ω/ω0)and denot- ing ˜m(u) =m(ω)gives
φ(ω0) = 2 π
Z
−
˜
m(u) −m(ω0) eu−e−u du
= 1 π
Z
−
˜
m(u) −m(ω0) sinh(u) du .
2.3 Two Applications of Complex Integration 43 Dividing the complete range of integration into separate ranges above and belowu=0and integrating by parts yields
φ(ω0) =1 π
Z0
−
˜
m(u) −m(ω0) sinh(u) du+ 1
π Z
0
˜
m(u) −m(ω0) sinh(u) du
= − 1
π[m(u) ư˜ m(ω0)]log cothưu 2
0
−
+ 1 π
Z0
−
dm(u)˜
du log coth −u
2
du
− 1
π[m(u) −˜ m(ω0)]log cothu 2
0
+ 1 π
Z
0
dm(u)˜
du log cothu 2
du .
(2.35)
Near u = 0, the quantity ˜m(u) −m(ω0) behaves proportionally to u, whilst log coth(u/2)will vary as−log(u/2). Thus, at the limitu→0, the integrated portions of (2.35) behave asulogu, which is known to van- ish as u → 0. As for the other limits, we have that limu − m(u) =˜ limω 0m(ω) = m(0), which is finite since H is stable and minimum phase; also limu m(u)˜ log coth(u/2) = limω m(ω)2(ω0/ω) = 0.
Hence both integrated portions in (2.35) are equal to zero. The result then follows on combining the remaining two integrals in (2.35).
The implications of (2.30) can be easily appreciated using properties of the weighting function appearing in (2.30), namely
log coth u 2 =log
ω+ω0 ω−ω0
. (2.36)
This function is plotted in Figure 2.5.
As we can see from this figure, the weighting function becomes logarith- mically infinite at the pointω= ω0. Thus, we conclude from (2.30) that the slope of the magnitude curve in the vicinity ofω0, sayc, determines the phaseφ(ω0):
φ(ω0)≈ c π
Z
−
log coth u
2
du= c π
π2 2 =cπ
2 .
Hence, for stable and minimum phase transfer functions, a slope of20c dB/decade in the gain in the vicinity ofω0 implies a phase angle of ap- proximatelyc π/2rad sec−1.
The above arguments lead classical designers to conclude that, to ensure closed-loop stability, the slope of the open-loop gain characteristic,|L(jω)|, should be in the range -20 to -30 dB/decade at the gain cross-over point
44 2. Review of General Concepts
10−1 100 101
0 0.5 1 1.5 2 2.5 3
log (coth |u / 2| )
eu
FIGURE 2.5. Weighting function of the Bode gain-phase relationship.
(i.e., at the frequency at which|L|=1or0dB), since this would imply the phase would be less than180◦i.e. the Nyquist plot would not encircle the
‘-1‘ point.
Example 2.3.2. Let us consider the X-29 aircraft design example discussed in §1.1 of Chapter 1. This system is (approximately) modelled by a strictly proper transfer function, G, which is unstable and nonminimum phase.
For one flight condition, the unstable pole is at6 and the nonminimum phase zero at26. It is desired to use a stable, minimum phase controller in series withG, such that the closed loop is stable and has a phase margin ofπ/4. Consider the open-loop system formed by the cascade of plant and controller and modeled by a transfer function,Lsay. This system is neither stable nor minimum phase in open loop. However, we can associate with Lanother transfer function, ˜L, defined as follows
L(s) = −L(s)˜ s+p
s−p
s−q s+q
, (2.37)
where ˜Lis stable, minimum phase, and such that ˜L(0) > 0, and wherep andq correspond to the (real) ORHP pole and zero ofGrespectively.13 The negative sign in (2.37) is necessary to guarantee a stable closed loop (see Example 2.3.1). We note that
|L(jω)|˜ =|L(jω)|. (2.38) Now from (2.37) we have thatφ(ω0),argL(jω0)is given by
φ(ω0) =arg ˜L(jω0) −argp−jω0
p+jω0 −argjω0+q jω0−q .
13Actually, the device used above to associate a stable, minimum phase transfer function with an unstable, nonminimum phase one will be used repeatedly in subsequent chapters.