© CREATAS Lead and Lag Compensators with Complex Poles and ... · Figure 1 shows the Bode plot for a lead compensator with a maximum phase lead of 45° at ωm = 1 rad/s. The pole

© CREATAS

44 IEEE CONTROL SYSTEMS MAGAZINE » FEBRUARY 2007 1066-033X/07/$25.00©2007IEEE

Lead and Lag Compensators with Complex

Poles and ZerosDESIGN FORMULAS FOR MODELING AND LOOP SHAPING

WILLIAM C. MESSNER, MARK D. BEDILLION, LU XIA, and DUANE C. KARNS

In classical loop shaping, compensator structures aretypically cascaded to modify the gain and phase char-acteristics of the open-loop frequency response. Thesealterations are used to achieve closed-loop perfor-mance specifications for disturbance rejection, refer-

ence following, noise rejection, and gain and phasemargins. Lead and lag compensators are standard toolsemployed in the loop-shaping process.

A lead compensator has the transfer function

C(s) =√

pz

(s + zs + p

), (1)

whose maximum phase φm occurs at the frequency

ωm = √pz (2)

and p > ωm > z > 0. The gain of (1) is unity at ωm . Themaximum phase lead φm of the lead compensator is

φm = � C( jωm) = � ( jωm + z) − � ( jωm + p)

= arctan(ωm

z

)− arctan

(ωm

p

). (3)

Since 0 < arctan (ωm/p) < arctan (ωm/z) < 90°, the maxi-mum phase lead φm is less than 90°. Furthermore,φm, ωm, p, and z are related by (2) and

zp

= 1 − sin φm

1 + sin φm.(4)

Figure 1 shows the Bode plot for a lead compensator witha maximum phase lead of 45° at ωm = 1 rad/s. The pole isat −p = −2.41, and the zero is at −z = −0.41. The transferfunction of a lag compensator is the reciprocal of a leadcompensator.

Lead compensators are used to increase the phase mar-gin for a given 0-dB crossover frequency. Lag compensatorsare used to increase the low-frequency gain for improveddisturbance rejection or to decrease the high-frequency gainfor improved noise rejection or augmented gain margin.Lead and lag compensators can also be used in modeling tomatch gain and phase features of frequency-response data.

For phase lead greater than or equal to 90°, multiple leadcompensators must be cascaded together. In practice, cascad-ed lead compensators are also used for phase lead less than90° because the high-frequency gain of a single lead compen-sator with maximum phase lead of more than 60° is signifi-cantly larger than the high-frequency gain of n cascaded leadcompensators each contributing 1/n of the same total phaselead. Figure 2 shows the high-frequency gain of a single leadcompensator as a function of the total phase lead.

The simplest cascade is the square of a lead compen-sator, called a double lead compensator. The transfer functionof the double lead compensator, which has maximumphase lead 2φm, is

Cdouble(s) = C2(s)

= pz

(s + zs + p

)2

= ωd

ωn

(s2 + 2ζnωns + ω2

n

s2 + 2ζdωds + ω2d

), (5)

where ωn = z, ωd = p, and ζn = ζd = 1. The poles and zerosof the double lead compensator are real and have multi-plicity of two.

The last expression in (5) suggests a variation of thedouble lead compensator in which the polynomials of thesecond-order biproper transfer function have dampingratios less than one. In particular, the complex lead com-pensator introduced in [1] is a variation of the double leadcompensator in which the poles and zeros are complexconjugates with the same damping ratio. This transferfunction has the form

Ccomplex(s) = ωp

ωz

(s2 + 2ζωzs + ω2

z

s2 + 2ζωps + ω2p

), (6)

where ωp is the undamped natural frequency of the poles,and ωz is the undamped natural frequency of the zeros.The equal damping ratios ζ provide a symmetric phasepeak. The gain of (6) is unity at ωm, the frequency of maxi-

mum phase lead. The maximum phase lead is designatedas 2φm for notational simplicity in the following develop-ment. The transfer function of a complex lag compensatoris the reciprocal of a complex lead compensator.

Figure 3 shows the Bode plot of a complex lead com-pensator with damping ratio ζ = 0.7 and maximum phaselead 90° at ωm = 1 rad/s. For comparison, Figure 3 alsoshows the Bode plot of a double lead compensator withmaximum phase lead 90° at ωm = 1 rad/s.

The Bode plots in Figure 3 illustrate several notable dif-ferences between the double lead compensator and thecomplex lead compensator. First, the phase peak of thecomplex lead compensator is sharper than the phase peakof the double lead compensator. Next, the magnitude tran-sition region of the complex lead compensator is narrowerthan the magnitude transition region of the double lead

FEBRUARY 2007 « IEEE CONTROL SYSTEMS MAGAZINE 45

FIGURE 1 Bode plot of the lead compensator of (1). The maximumphase lead is 45° at ωm = 1 rad/s. The pole is at −p = −2.41, andthe zero is at −z = −0.41.

−10

−5

0

5

10

Mag

nitu

de (

dB)

10−2 10−1 100 101 1020

30

60

Pha

se (

°)

Frequency (rad/s)

FIGURE 2 High-frequency gain √

p/z of the single lead compensator(1) as a function of its maximum phase lead φm. The high-frequencygain increases nonlinearly with increasing phase lead. Lead com-pensators are often cascaded in practice to reduce the high-fre-quency gain for a desired maximum phase lead.

30 40 50 60 70 800

5

10

15

20

25

30

35

40

45

Maximum Phase Lead (°)

Hig

h F

requ

ency

Gai

n (d

B)

compensator. Furthermore, the ratio of the high-frequencygain asymptote to the low-frequency gain asymptote issmaller for the complex lead compensator than for thedouble lead compensator. Finally, the slope of the magni-tude at the frequency ωm is steeper for the complex leadcompensator than for the double lead compensator.

The smaller ratio between the high-frequency and low-frequency gain asymptotes of the complex lead compen-sator is an advantage compared to the double leadcompensator when the complex lead compensator is usedto increase the phase margin for a given 0-dB crossoverfrequency. However, the narrower phase peak of the com-plex lead compensator implies that the phase of the com-pensated open-loop attains −180° crossover at a lowerfrequency when using a complex lead compensator thanwhen using a double lead compensator. Also, the rate atwhich the open-loop magnitude drops above the 0-dBcrossover frequency is smaller for a complex lead compen-sator than for a double lead compensator. These latter twofeatures might result in a smaller gain margin or lowernoise rejection near the 0-dB crossover frequency [2].

For a complex lag compensator, the steeper magnitudeslope and narrower phase notch are advantages compared tothe double lag compensator, while the smaller differencebetween the high-frequency and low-frequency gain is a dis-advantage. The smaller difference in the gain asymptotes ofthe complex lag compensator tends to reduce the low-fre-quency gain and decrease the low-frequency disturbance

rejection compared with the double lag compensator. How-ever, the steeper magnitude slope and the narrower phasenotch mean that the complex lag can be applied to increasethe rate at which the open-loop magnitude drops above the 0-dB crossover frequency, while changing the −180° crossoverfrequency very little. When trading off gain margin for phasemargin, a complex lag compensator can provide a larger gainmargin compared with a double lag compensator. Anotherapplication for which the complex lag has similar advantagesover the double lag compensator is to improve the perfor-mance of single-input, single-output (SISO) systems that havemultiple 0-dB crossover frequencies [3].

Complex lead and complex lag compensators are alsouseful for system modeling. The Bode plots of measuredfrequency response data often exhibit narrow phase peaksand notches with complicated magnitude behavior in thevicinity of the peaks and notches. The complex lead andcomplex lag compensators can be used to replicate theseBode plot features in many cases.

We provide simple closed-form expressions for deter-mining the transfer function parameters of complex leadand lag compensators for a specified phase lead or lag at aparticular frequency of maximum phase lead or lag. Foreach compensator, we characterize the relationshipbetween the damping ratio and the ratio of the asymptoticgains, the slope of the magnitude plot at the frequency ofmaximum phase lead or lag, the width of the phase peak,and the width of the magnitude transition region.

FIGURE 4 The pole-zero plot for the complex lead compensator ofFigure 3. The maximum phase lead is 2φm = 90° at ωm = 1 rad/s.The points −p and −z are the locations of the pole and zero of thelead compensator having maximum phase lead of 45° at ωm. Thepoint cp = −1 is the center of the circle P, which passes through thepoints jωm, − jωm, and −p = −2.41. The conjugate poles p1 andp2 of the complex lead compensator lie at the intersection of P andthe lines corresponding to the damping ratio ζ = 0.7. Likewise, thepoint cz = 1 is the center of the circle Z, which passes through thepoints jωm, − jωm, and −z = −0.41. The conjugate zeros z1 and z2

of the complex lead compensator lie at the intersection of Z and thelines corresponding to the damping ratio.

−2 −1 0 1 2

−1.5

−1

−0.5

0

0.5

1

1.5

Real

Imag

inar

y

p1

p2

z1

z2

cp czO

jwm

−jwm

P Z

−z−p

r r

FIGURE 3 Bode plots of the double lead compensator (5) and thecomplex lead compensator (6). The phase peak and magnitudetransition region of the complex lead compensator are narrowerthan the phase peak and magnitude transition region of the dou-ble lead compensator. The ratio of the high-frequency gainasymptote to the low-frequency gain asymptote is smaller for thecomplex lead compensator than for the double lead compen-sator. The slope of the gain at the frequency of maximum phaseis steeper for the complex lead compensator than for the doublelead compensator.

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90

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Double LeadComplex Lead

46 IEEE CONTROL SYSTEMS MAGAZINE » FEBRUARY 2007

ANALYSIS OF COMPLEXLEAD AND LAG COMPENSATORS

Geometric RelationshipsFigure 4 shows the pole-zero plot of the complex lead com-pensator of Figure 3. A close geometric relationship existsbetween the locations of the poles and zeros of a complexlead compensator with maximum phase lead 2 φm at thefrequency ωm and a lead compensator having maximumlead phase lead φm at the same frequency. The points −pand −z are the locations of the pole and the zero of thelead compensator. The point cp is the center of the circle P,which passes through the points jωm, − jωm and −p. Theconjugate poles p1 and p2 of the complex lead compensatorlie at the intersection of P and the lines corresponding tothe damping ratio ζ . Likewise, the point cz is the center ofthe circle Z, which passes through the points jωm, − jωm,and −z. The conjugate zeros z1 and z2 of the complex leadcompensator lie at the intersection of Z and the lines corre-sponding to the damping ratio.

We begin by deriving properties of the single lead com-pensator. The expression

p = ωm

√1 + sin φm

1 − sin φm= ωm

√(1 + sin φm

1 − sin φm

)(1 + sin φm

1 + sin φm

)

= ωm

(1 + sin φm

cos φm

)= ωm

(1

cos φm+ tan φm

)(7)

is a variation of equation (7.4) in [4]. The point cp is definedas cp ≡ −ωm tan φm . The distance between the point −p andthe point cp is therefore r ≡ ωm/ cos φm. It follows from thePythagorean theorem and the trigonometric identity (SI)(see “Useful Trigonometric Identities”) that the distancebetween jωm and cp is r, while the distance between thepoint − jωm and the point cp is also r. Thus, cp is center ofthe circle P defined by the points jωm, − jωm and −p. Theradius of P is r.

The same procedure for the zeros yields

z = ωm

(1

cos φm− tan φm

). (8)

The point cz ≡ ωm tan φm is the center of the circle Z withradius r defined by the points jωm, − jωm and −z.

The poles and zeros of the complex lead compensatorlie at the intersections of the two circles and the lines corre-sponding to the damping ratio ζ . The undamped naturalfrequency of the poles is ωp, which is the distance from thepoles to the origin O. The law of cosines for the trianglecpOp1 leads to the relation

r2 = c2p + ω2

p − 2|cp|ωp cos(� cpOp1). (9)

Solving this quadratic equation for ωp, substituting for cp andr, applying SI, and using the relation ζ = cos(� cpOp1) implies

ωp = ωm

(ζ tan φm +

√ζ 2 tan2 φm + 1

). (10)

Following the same procedure for the zeros yields

ωz = ωm

(−ζ tan φm +

√ζ 2 tan2 φm + 1

). (11)

Multiplying, subtracting, adding, and summing thesquares of (10) and (11) leads to the relations

ωm = √ωpωz, (12)

ωp − ωz = 2ζωm tan φm, (13)

ωp + ωz = ωm

√ζ 2 tan2 φm + 1, (14)

ω2p + ω2

z = ω2m

(4ζ 2 tan2 φm + 2

). (15)

Equation (12) shows that the frequency of maximum phaselead is the geometric mean of the natural frequencies of thepoles and zeros, which is analogous to (2) for a lead com-pensator with real poles and real zeros. The relations(13)–(15) are useful in the proof below that ωm is the fre-quency of maximum phase lead.

Determining the Frequency of Maximum Phase LeadThe complex lead compensator phase angle � C( jω) isgiven by

� C( jω) = �(

−ω2 + 2ζωzω j+ ω2z

−ω2 + 2ζωpω j+ ω2p

)

= arctan(

2ζωzω

ω2z − ω2

)− arctan

(2ζωpω

ω2p − ω2

), (16)

FEBRUARY 2007 « IEEE CONTROL SYSTEMS MAGAZINE 47

Useful Trigonometric Identities

Several trigonometric identities involving the tangent function

are useful in the derivations. The derivations of the expres-

sions (10) and (11) for ωp and ωz employ

tan2(A) + 1 = sin2(A)

cos2(A)+ cos2(A)

cos2(A)= 1

cos2(A). (S1)

The formula for the tangent of the difference of two angles is

tan(A − B) = tan(A) − tan(B)

1 + tan(A) tan(B). (S2)

Closely related to (S2) is the double angle formula for tangents

tan(2A) = 2 tan(A)

1 − tan2(A). (S3)

The derivation of the frequency of maximum phase lead uses an

identity for the difference of two arctangents given by

arctan(u) − arctan(v) = arctan(tan(arctan(u)

− arctan(v)))

= arctan(

u − v1 + uv

). (S4)

where the subscript complex is suppressed for notationalconvenience. Applying the formula for the difference ofarctangents (S4) and the relations (12)–(15) to (16) and col-lecting terms leads to

� C( jω)=arctan

2ζω

[(ωzωp + ω2

m) (

ωp − ωz)]

ω2zω

2p − ω2

(ω2

z + ω2p

)+ ω4 + 4ζ 2ωzωpω2

=arctan

(4ζ 2ωm tan φm

(ω2

m+ω2)ωω4

m−ω2ω2m(4ζ 2 tan φm+2

)+ω4+4ζ 2ω2mω2

)

=arctan

(4ζ 2ωm tan φm

(ω2

m + ω2)ω(ω2

m − ω2)2 + 4ζ 2ω2

m(1 − tan2 φm

)ω2

).

(17)

Setting ω = ωm in (17) and applying the double angle for-mula for tangents (S3) shows that � C( jωm) = 2φm.

For convenience, we rewrite (17) as

� C( jω) = xy, (18)

where

x = 4ζ 2ωm

(ω2

m + ω2)

ω tan φm (19)

and

y =(ω2

m − ω2)2 + 4ζ 2ω2

m

(1 − tan2 φm

)ω2. (20)

Differentiating (17) with respect to ω leads to

d� C( jω)

dω= d

dωarctan

(xy

)

= 1x2+y2

(y

dxdω

−xdydω

),

(21)

where

dxdω

=4ζ 2ωm tan φm(ω2

m + 3ω2) , (22)

dydω

=4(ω2

m − ω2)ω + 8ζ 2ω2m

× (1 − tan2 φm

)ω. (23)

Evaluating (19)–(23) at ω = ωm shows that (d� C( jωm))/

(dω) = 0. Since (d� C( jω))/

(dω) < 0 for all ω > ωm , and(d� C( jω))/(dω) > 0 for all ω < ωm,the positive phase at ω = ωm mustbe the global maximum.

BODE PLOT FEATURESAND DAMPING RATIOFor modeling and design, it is use-ful to determine the relationshipbetween the parameters of thecomplex lead compensator andfeatures of the compensator Bodeplot. Figure 5 indicates the featuresexamined in this section.

The ratio of the high-frequencymagnitude asymptote to the low-frequency magnitude asymptoteof the complex lead compensatordetermines the reduction in thelow-frequency disturbance rejec-tion, high-frequency noise rejec-tion, and gain margin when thecompensator is used for loop shap-ing. For a complex lag compen-sator, the same ratio determinesthe increase in low-frequency dis-turbance rejection, high-frequencynoise rejection, and gain margin.The expression for this ratio is

FIGURE 5 Some features of the Bode plot of a complex lead compensator. The Bode plot fea-tures of the complex lead compensator are functions of both the maximum phase lead and thedamping ratio. Features of interest on the magnitude plot are the ratio of asymptotic gains, theslope of the magnitude plot at the compensation frequency, and the width of the transitionregion from the intersection of the asymptotes to ωm, the frequency of maximum phase lead.The width of the phase peak is the frequency width from 4° phase lead to the phase-lead peakexpressed in decades or, equivalently, as the full width at half maximum phase lead alsoexpressed in decades.

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10−2 10−1 100 1010

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90

135

Pha

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°)

Frequency (rad/s)

ωm

Ratio of AsymptoticGains (dB)

Slope at Peak (dB/dec)

Intersection ofAsymptotes to Peak (dec)

4° toPeak (dec)

Width at HalfMax (dec)

102

48 IEEE CONTROL SYSTEMS MAGAZINE » FEBRUARY 2007

|C(∞)||C(0)| =

ω2p

ω2z

=(

ζ tan φm +√

ζ 2 tan2 φm + 1

−ζ tan φm +√

ζ 2 tan2 φm + 1

)2

=(

ζ tan φm +√

ζ 2 tan2 φm + 1

−ζ tan φm +√

ζ 2 tan2 φm + 1

)2

×(

ζ tan φm +√

ζ 2 tan2 φm + 1

ζ tan φm +√

ζ 2 tan2 φm + 1

)2

=(

ζ tan φm +√

ζ 2 tan2 φm + 1)4

. (24)

FIGURE 6 Ratios of asymptotic gains. This plot shows the ratio of theasymptotic gains as a function of the damping ratio for four phase-lead values. The ratio in dB is nearly a linear function of the damp-ing ratio, especially for phase-lead values less than 90°.

5

00 0.2 0.4 0.6 0.8 1

10

15

20

25

30

35

40

45

50

Rat

io o

f Asy

mpt

otic

Gai

ns (

dB)

Phase Lead = 30°Phase Lead = 60°Phase Lead = 90°Phase Lead = 120°

Damping Ratio

FIGURE 7 Full width of the phase peak at half of the maximumphase lead for damping ratios between 0.1 and 1. The plot showsthe full width at half maximum as a function of the damping ratio forfive different maximum phase-lead values.

0

0.2

0.1 0.2 0.3 0.4 0.5Damping Ratio

0.6 0.7 0.8 0.9 1

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Ful

l Wid

th a

t Hal

f Max

Pea

k (d

ec) Phase Lead = 30°

Phase Lead = 60°Phase Lead = 90°Phase Lead = 120°Phase Lead = 135°

FIGURE 8 Full width of the phase peak at half of the maximumphase lead for damping ratios between 0.001 and 0.1. The plotshows the full width at half maximum as a function of the dampingratio for five maximum phase-lead values, which is useful for model-ing resonance/antiresonance pairs with low damping.

0.1

Phase Lead = 30°Phase Lead = 60°Phase Lead = 90°Phase Lead = 120°Phase Lead = 135°

0

0.05

0.1

0.15

0.2

0 0.02 0.04Damping Ratio

0.06 0.08

Ful

l Wid

th a

t Hal

f Max

Pea

k (d

ec)

FIGURE 9 Width of the phase peak. This plot shows the ratio ofthe 4° phase-lead frequency to the frequency of maximum phaselead in decades as a function of the damping ratio for severalphase-lead values.

10

0.5

1

1.5

2

0 0.2 0.4Damping Ratio

0.6 0.8

Fou

r D

egre

es to

Pea

k (d

ec)

Phase Lead = 30°Phase Lead = 60°Phase Lead = 90°Phase Lead = 120°

The smaller ratio between the

high-frequency and low-frequency

gain asymptotes of the complex

lead compensator is an advantage

compared to the double

lead compensator.

FEBRUARY 2007 « IEEE CONTROL SYSTEMS MAGAZINE 49

Figure 6 shows the ratio of the asymptotic gains in dB as afunction of the damping ratio.

The width of the phase peak can be quantified in severalways. The full width at half maximum (FWHM) is the fre-

quency span in decades (dec) betweenthe frequencies at which the phase isone-half of its maximum. This measureof the peak width is useful for model-ing since it is easily measured fromexperimental data. Figure 7 shows the

numerically determined FWHM peakwidth for several lead compensators as a function of damp-ing ratio. Figure 8 shows the peak width for damping ratiosbetween 0.001–0.1, which is useful for modeling resonanceand antiresonance pairs with low damping ratio.

50 IEEE CONTROL SYSTEMS MAGAZINE » FEBRUARY 2007

FIGURE 10 The magnitude transition region width. The width is mea-sured by the frequency span between the intersection of the low-frequency and mid-frequency asymptotes and ωm . Note the nearlylinear relationship with respect to the damping ratio.

0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4

Damping Ratio

0.6 0.8

Tra

nsiti

on W

idth

(de

c)

Phase Lead = 30°Phase Lead = 60°Phase Lead = 90°Phase Lead = 120°

FIGURE 12 Magnitude slope at the frequency of maximum phaselead for damping ratios between 0.1 and 0.5. The slope at the fre-quency of maximum phase lead increases dramatically for lowdamping ratios, even for low phase-lead values.

Phase Lead = 30°Phase Lead = 60°Phase Lead = 90°Phase Lead = 120°

200

180

160

140

120

100

80

60

Mag

nitu

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lope

at M

ax P

hase

Lea

d (d

B/d

ec)

40

200.1 0.15 0.2 0.25

Damping Ratio

0.3 0.35 0.4 0.45 0.5

FIGURE 11 Magnitude slope at the frequency of maximum phaselead for damping ratios between 0.5 and 1. The slope at the fre-quency of maximum phase lead decreases by 10–15 dB/dec as thedamping ratio increases from 0.5–1.0.

Phase Lead = 30°Phase Lead = 60°Phase Lead = 90°Phase Lead = 120°

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50

45

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Damping Ratio

0.9 1Mag

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at M

ax P

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Lea

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B/d

ec)

FIGURE 13 Frequency response of an actuated tape guide. Asecond-order model captures the low-frequency response. The twoprominent peaks at approximately 1160 rad/s and 1940 rad/s arisefrom mechanical resonances in the actuator structure.

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Pha

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Second-OrderModel

ExperimentalData

Lead compensators are used to increase the phase

margin for a given 0-dB crossover frequency.

Another way to measure the width of thephase peak is the frequency span from aspecified small phase lead to the maximumphase lead. This quantity is often useful fordesigning lag compensators when the goal isto increase the gain margin with minimaleffect on the phase margin. Figure 9 showsthe numerically determined width from 4° tothe phase peak in decades as a function ofdamping ratio.

The width of the magnitude transitionregion is another important property of leadcompensators because it determines the fre-quency range over which the compensatoraffects the slope of the open-loop magnituderesponse. The width of the magnitude transi-tion region is quantified here by the frequencyspan from the intersection of the asymptotes toωm. The intersection of the low-frequency andmid-frequency asymptotes ωm is related to thedamping ratio and maximum phase lead by

ωm

ωz= ωp

ωm= ζ tan φm +

√ζ 2 tan2 φm + 1. (25)

Figure 10 shows the width of the magnitudetransition region in decades as a function ofdamping ratio.

The maximum slope of the complex leadcompensator magnitude determines howmuch the slope of the open-loop magnituderesponse changes when the compensator isapplied. For lag compensators, a larger magni-tude slope is desirable for increasing low-fre-quency disturbance rejection or increasinghigh-frequency noise rejection near the fre-quency of the maximum phase lag. Converse-ly, a steeper magnitude slope for a leadcompensator limits the design by reducing theslope of the open-loop magnitude at the open-loop 0-dB crossover frequency. The expressionfor the magnitude slope at the compensationfrequency in decibels per decade is

20 log |C( jω)|d log |(ω)|

∣∣∣∣ωm

= 20 ωm

|C( jωm)|1

dω|C( jωm)|. (26)

FEBRUARY 2007 « IEEE CONTROL SYSTEMS MAGAZINE 51

FIGURE 14 Closeup of the first phase peak of Figure 13. The phase peak is 135°above the baseline phase of −180°. The 50% frequencies are the two frequen-cies at which the phase is above the baseline by half of the difference betweenthe baseline and peak values.

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−180

−135

−90

−45P

hase

(°)

Frequency (rad/s)

Frequency: 1080 rad/sPhase: −113° ?

Frequency: 1130 rad/sPhase: −45°

Frequency: 1260 rad/sPhase: −113°

FIGURE 15 Measured and modeled frequency responses in the vicinity of the twophase peaks. The frequency response of the second-order model augmented bythe two complex lead compensators is a good match to the experimental fre-quency response.

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−5

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−135

−90

−45

0

Pha

se (

°)

Frequency (rad/s)

Experimental DataComplex Lead

For a complex lag compensator, the steeper magnitude slope and

narrower phase notch are advantages compared to the double lag compensator.

Figures 11 and 12 show the maximum magnitude slope indecibels per decade obtained numerically for several leadcompensators as a function of damping ratio.

MODELING EXAMPLEWe consider an actuated tape guide used to actively steermoving tape. The tape guide is used in a high-precision tapetransport system [5] for a prototype multiterabyte tape sys-tem under construction at Carnegie Mellon. Figure 13 showsthe frequency response of the device as well as the frequencyresponse of a second-order transfer function manually fit tothe data. The system exhibits two resonance/antiresonancepairs for which finding good transfer function models is oftendifficult without the use of automated fitting techniques.

Figure 14 shows a closeup of the experimental frequencyresponse of the first resonance/antiresonance pair. Theestimated phase lead is 2 φm = 135°. The FWHM is

log10

(ω+

50%

ω−50%

)= log10

(12601080

)

= 0.067 dec, (27)

where ω+50% is the higher of the two frequencies at which

the phase peak is 50% of the maximum and ω−50% is defined

analogously. The estimated damping ratio obtained fromFigure 8 is 0.03. The estimated value of ωm is at the geo-metric mean of the 50% phase-peak frequencies

ωm =√

ω+50%ω−

50% = 1167rad

s. (28)

Figure 15 shows the frequency response in the vicinityof both resonance/antiresonance pairs with the second-order system model augmented by two complex lead com-pensators obtained by the process described above. Theparameters of the second complex lead compensator are2 φm = 117°, ζ = 0.02, and ωm = 1940 rad/s. The completetransfer function model is

P(s) =(

2.19e6s2 + 21.14s + 5625

)(s2 + 65.06s + 1.176e6s2 + 75.21s + 1.555e6

)

×(

s2 + 75.14s + 3.527e6s2 + 80.17s + 4.017e6

). (29)

DESIGN EXAMPLEThis design example applies the complex lag compen-sator to the focus control of a microscope objective lens.

FIGURE 16 Frequency response of the focus control plant. A modelcontaining two resonances matches the experimental response upto 1100 Hz.

−30

−20

−10

0

10

20

Mag

nitu

de (

dB)

101 102 103−540−450−360−270−180

−900

Pha

se (

°)

Frequency (Hz)

Experimental DataTransfer Function Model

FIGURE 17 Frequency responses of the compensated open loop.The large phase margin of the nominal controller allows the use oflag compensators for trading off phase margin for gain margin.

−20

−15

−10

−5

0

5

Mag

nitu

de (

dB)

102 103−315−270−225−180−135−90−45

Pha

se (

°)

Frequency (Hz)

No LagComplex LagLag

52 IEEE CONTROL SYSTEMS MAGAZINE » FEBRUARY 2007

For modeling and design, it is useful to determine the

relationship between the parameters of the complex lead compensator

and features of the compensator Bode plot.

The example is taken from a heat-assisted mag-netic recording (HAMR) media tester at theSeagate Research Center in Pittsburgh, Penn-sylvania. A permanent magnet is used to applya magnetic field to a portion of the disc. Writ-ing is achieved by focusing a laser spot on themagnetized portion of the disc with a micro-scope objective lens. The laser spot size definesthe dimensions of the written data bits. Effec-tive writing requires that the position of themicroscope objective lens relative to the discsurface is maintained to within the depth of focusof the beam. A servo system is used to mitigate distur-bances due to spindle wobble and disc vibration.

The servo system uses a piezo actuator to move theobjective lens and a four-quadrant detector with a knifeedge to measure the spot size. Figure 16 shows the mea-sured and modeled plant responses. Below the resonancefrequency near 1100 Hz, the measured transfer functionclosely matches a fourth-order system with 100-µs delay.Above the resonance frequency, the measurement of plantdynamics is uncertain; hence, the controller must provideadequate robustness at high frequencies.

A proportional-integral (PI) compensator is used toimprove disturbance rejection below 100 Hz, while a19.5-dB notch filter is used to remove the primary reso-nance. A first-order lowpass filter with a 300-Hz cutofffrequency suppresses the high-frequency response. Theblue line in Figure 17 shows the loop shape with the PIcompensator, the notch filter, and the low-pass filterapplied. Figure 17 shows that the gain margin is 4.8 dB at957 Hz, while the phase margin is 86° at 492 Hz. Experi-mental results using this controller indicate that most ofthe error occurs in the frequency range of the sensitivitypeak. Use of the complex lag compensator is intended todecrease sensitivity peaking while improving high-fre-quency robustness.

The complex lag compensator enables a tradeoffbetween gain margin and phase margin that improveshigh-frequency robustness. Figure 18 shows the fre-quency responses of a 25° standard lag compensator at1,100 Hz and of a 25° complex lag compensator at 600Hz with damping ratio 0.3. Both compensators provide19° of phase lag at 500 Hz. The complex lag compen-sator exhibits a small peak and notch in its magnitudebecause its damping ratio is less than 0.7. The phasenotch of the complex lag compensator is much narrowerthan that of the standard lag compensator, while itsmagnitude slope is much steeper than the standard lagcompensator’s, between 500 and 700 Hz. However, thestandard lag compensator has the larger differencebetween asymptotic gains.

Figure 17 also shows the compensated loop shapesemploying the standard lag and complex lag compen-sators. Both compensators result in a phase margin of

FEBRUARY 2007 « IEEE CONTROL SYSTEMS MAGAZINE 53

FIGURE 18 Bode plots for the lag and complex lag compensators.The complex lag compensator magnitude response shows a peakand notch because the compensator damping ratio is below 0.7.The complex lag compensator has a steeper magnitude slope andnarrower phase peak than the lag compensator.

−8

−6

−4

−2

0

2

Mag

nitu

de (

dB)

101 102 103 104 105 106−30

−20

−10

0

Pha

se (

°)

Frequency (Hz)

Complex LagLag

FIGURE 19 Experimental sensitivity function magnitudes for the threeloop-shaping designs. The low-frequency disturbance rejection of thecomplex lag compensator design is similar to that of the nominaldesign. However, the sensitivity peak of the complex lag compen-sator design is smaller than that of the nominal design. The standardlag compensator design has better low-frequency disturbance rejec-tion than the nominal design but has more sensitivity peaking.

101 102 103−25

−20

−15

−10

−5

0

5

10

Mag

nitu

de (

dB)

Frequency (Hz)

No LagComplex LagLag

The complex lag compensator enables

a tradeoff between gain margin and

phase margin that improves

high-frequency robustness.

54° at 500 Hz. However, use of the complex lag com-pensator results in a gain margin of 7.9 dB at 900 Hz,while the gain margin of the system using the standardlag compensator is only 3.6 dB at 807 Hz. The perfor-mance of the complex lag compensator design is similarto the performance of the nominal controller alone atlow frequencies. However, the complex lag compen-sator controller provides about 3.1 dB more gain mar-gin and improved robustness to unmodeledhigh-frequency dynamics compared to the nominal con-troller because its magnitude drops faster between 500and 700 Hz. The standard-lag compensator design has asmaller gain margin than the complex lag compensatordesign and larger phase loss over a wider frequencyrange. However, the standard-lag design has largermagnitude at low frequencies.

Figure 19 shows the magnitudes of the sensitivityfunctions from experimental data. The complex lag com-pensator design has a smaller peak (4.8 dB) relative to thenominal design (7.2 dB) and the design employing thestandard lag compensator (7.9 dB). However, the stan-dard lag compensator design has more disturbance rejec-tion at low frequencies. The complex lag compensatordesign is a better choice in this situation because the lagcompensator is needed only to reduce sensitivity peakingand obtain high-frequency robustness, while its low-fre-quency disturbance rejection is not significantly worsethan that of the nominal controller.

CONCLUSIONS AND FUTURE WORKComplex lead and lag compensators are new additions tothe repertoire of compensator structures for loop shaping.This article facilitates the use of these compensators byproviding explicit formulas that relate the parameters ofthe compensators to features of their frequency responses.Two examples illustrate the utility of these compensatorsfor system modeling and controller design. While theexamples involve low-order plants, the principles ofemploying the complex lead and lag compensators remainthe same for higher-order systems.

We plan to use these compensators as weighting func-tions with automated robust design tools. A weightingfunction is a transfer function whose frequency responsemagnitude is used to bound closed-loop response or mod-eling uncertainty. The complex lead and lag compensatorsprovide new degree of freedom for selecting weightingfunctions. In particular, the steep magnitude slope in thetransition region of these compensators more closelyapproximates an ideal step function than weighting func-tions appearing in the literature [6].

ACKNOWLEDGMENTW. Messner and L. Xia thank Prof. J. Wickert and V. Kartikfor their assistance in designing the actuated tape guide.

AUTHOR INFORMATIONWilliam C. Messner received a B.S. in mathematicsfrom M.I.T. in 1985 and M.S. and Ph.D. degrees inmechanical engineering from the University of Califor-nia at Berkeley in 1989 and 1992, respectively. He is aprofessor of mechanical engineering at Carnegie Mel-lon University. He is the leader of the servo controleffort at Carnegie Mellon’s Data Storage Systems Cen-ter. He is a fellow of the American Society of Mechani-cal Engineers and the American Association for theAdvancement of Science.

Mark D. Bedillion ([email protected])received the B.S.M.E. degree in 1998, the M.S.M.E. degreein 2001, and the Ph.D. degree in 2005 from Carnegie Mel-lon University. He joined Seagate Research in 2002,where he works on the development of novel storagedevices. His research interests include control applica-tions in data storage, distributed manipulation, andhybrid systems. He can be contacted at Seagate Technolo-gy, 1251 Waterfront Pl., Pittsburgh, PA 15222 USA.

Lu Xia received the B.S. degree in industrial automa-tion from Beijing Institute of Technology, Beijing, Chinain 1998 and the M.S.E.E. degree from the University ofNotre Dame, Indiana in 2002. She is currently pursuingthe Ph.D. degree with the department of Electrical andComputer Engineering at Carnegie Mellon University.Her research interests include control applications indata storage systems, robust control, and signal identifi-cation and processing.

Duane C. Karns served in the U.S. Army’s 82nd Air-borne Division from 1986 to 1990. He received a B.S. inengineering science from Pennsylvania State Universityin 1995. He received the M.S. in 1997 and a Ph.D. in 2000,both in electrical and computer engineering fromCarnegie Mellon University. He is currently a researchstaff member at Seagate Research in Pittsburgh, Pennsyl-vania, specializing in the recording physics of magneticand optical systems.

REFERENCES[1] W. Messner, “The development, properties, and application of the com-plex phase lead compensator,” in Proc. 2000 American Control Conf., Chicago,IL, July, 2000, pp. 2621–2626.

[2] W. Messner, “Some advances in loop shaping with applications to diskdrives,” IEEE Trans. Magn., vol. 37, no. 2, pp. 651-656, Mar 2001.

[3] W. Messner and R. Oboe, “Phase stabilized design of a hard disk driveservo using the complex lag compensator, “ in Proc. 2004 American ControlsConf., Boston, MA, 30 June–2 July, 2004, pp. 1165–1170.

[4] G.F. Franklin, J.D. Powell, and A. Emani-Naeini, Feedback Control ofDynamic Systems, 3rd ed. Reading, MA: Addison-Wesley, 1994.

[5] D. Richards, J. Anderson, and L. Erickson, “Dynamic Tape Path Adjust-ment,” US Patent 6 690 531, Feb. 10, 2004.

[6] A. Packard, G. Balas, M. Safonov, R. Chiang, P. Gahinet, and A. Nemirovs-ki, Robust Control Toolbox}, The Mathworks. [Online]. Available: http://www.mathworks.com/access/helpdesk/help/toolbox/robust/

54 IEEE CONTROL SYSTEMS MAGAZINE » FEBRUARY 2007