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258IEEE TRANSACTIONS ON EDUCATION, VOL. 46, NO. 2, MAY 2003

The Exact and Unique Solution for Phase-Lead and

Phase-Lag Compensation

Fei-Yue Wang, Senior Member, IEEE

Abstract—Phase-lead and -lag compensation is one of the most

commonlyusedtechniquesfordesigningcontrolsystemsinthefre-

quency domain, especially when the Bode diagram or root locus is

used. Inmost cases, the graphic-based approximation or trial-and-

error approach has been utilized in the design process. This paper

presents the exact and unique solution to the design of phase-lead

and phase-lag compensation when the desired gains in the magni-

tude and phase are known at a given frequency. It also gives the

concise condition for determining the existence of single-stage lead

or lag compensation.

Index Terms—Bode diagram, control systems, frequency design

methods, lag compensation, lead compensation, phase compensa-

tion.

I. INTRODUCTION

I

N spite of the advances in modern control system design

techniques,suchasstate-basedoptimalandfrequency-based

controls, much design, especially industrial design is still

conducted using classical frequency-domain procedures.

Because of its simplicity and easy implementation, cascade

or series compensation is the most popular method in the

frequency-domain design of feedback control systems. This

method is particularly valuable when the plant to be controlled

is unknown and only experimental data are available. Two

of the most commonly used series compensation strategies

are PID (proportional-integral-derivative) control and phase

lead/lag compensation. Since cascade compensation was first

introduced, the determination of the lead or lag compensator

has been taught as a trial-and-error procedure based on graphic

and approximated information. Usually, an appropriate com-

pensator comes only after trial and error.

In 1976, Wakeland [1] found an analytical solution to the de-

sign of single-stage phase-lead compensation. His solution is of

the quadratic form in terms of compensation gain. In the fol-

lowing year, Mitchell [2] improved Wakeland’s solution and

pointed out that the improved solution can also be used to solve

the design problem of phase-lag compensation.

In this paper, we present a simple and unique solution to

both phase lead/lag compensation problems. The solution has

beendiscoveredindependentlyofbothWakelandandMitchell’s

work. Its procedure is much simpler and uniform for both lead

and lag design. It should be useful for developing analytical

Manuscript received December 19, 2001; revised April 9, 2002. This work

was supported in part by a Grant from the IBM Corporation.

The author is with the Intelligent Control and Systems Engineering Center,

The Institute of Automation, Chinese Academy of Sciences, Beijing 100080,

China, and also with the Program for Advanced Research in Complex Systems,

Department of Systems and Industrial Engineering, The University of Arizona,

Tucson, AZ 85721 USA (e-mail: feiyue@sie.arizona.edu).

Digital Object Identifier 10.1109/TE.2002.808279

proceduresforotherfrequency-basedtechniques,especiallyop-

timal designs. An extensive search of the relevant literature has

not found similar results (see major control textbooks [3]–[9]).

II. THE SOLUTION

Assume that a single-stage compensator is expressed as [3],

(1)

where

a phase-lead compensator.

Let

(in dB) and

nitude and phase, to be contributed by

frequency (or any given frequency)

yields a phase-lag compensator, andyields

(in rad) be the desired gain in the mag-

, at a given crossover

. Then

(2)

which leads to

(3)

Clearly,

new variable and parameter

can be assumed here. Introducing

(4)

Then from (3) and

the following two equations are found in termsof and

(5)

The second equation in (5) is obtained by taking the tangent

of both sides of the second equation in (3). Eliminating

from the first equation using the second equation, one gets an

equation expressed in terms of the unknown variable

only

The above equation can be further simplified

(6)

Since

phase-lag or phase-lead compensator),

itive, therefore, the only possible valid solution to (6) is

, , andare all real and positive (as required by a

is always real and pos-

(7)

0018-9359/03$17.00 © 2003 IEEE

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WANG: SOLUTION FOR PHASE-LEAD AND PHASE-LAG COMPENSATION259

The above equation presents two possible solutions for

sign and sign, respectively). Now one can prove that there is

only one possible solution, i.e., only

Fromthedefinitionsofphase-lag(

)compensationandthefirstequationin(3),onecanshowthata

phase-lead or phase-lag compensator must satisfy the following

conditions.

(

sign is allowed in (7).

)orphase–lead(

A. Phase-Lead Compensation

Since

therefore

for phase-lead compensation.

B. Phase-Lag Compensation

Since

therefore

for phase-lag compensation.

Thus, only the positive sign is allowed for phase-lead com-

pensation in (7) (otherwise there would be a negative value for

). In the case of phase-lag compensation, the negative sign

in (7) seems to be valid, however, a further analysis using (5)

and (7) shows that the negative sign will lead to the following

conclusions

which is invalid since

positive sign is valid in (7) for both phase-lead and phase-lag

compensation. Hence, from the second equation of (5) and the

first equation in definition (4), one has

must be positive. Therefore, only the

(8)

for both phase-lead and phase-lag compensators.

The equations in (8) present the only possible solutions to

phase compensation; those solutions are unique, but may not be

valid. However, based on (8), it is easy to show the following

theorem.

The Lead-Lag Compensation Theorem:

a) A single-stage phase-lead compensation exists if and only

if

and (9)

b) A single-stage phase-lag compensation exists if and only

if

and(10)

In both cases, the compensation solution is unique and given by

(8).

Proof: Clearly, for any

and , one has

and the first equation of (8) can be rewritten as

Then when (9) is true, one finds

a phase-lead compensation exists and is unique. It is easy to

show from (8) that, if a phase-lead compensation exists, then

condition (9) must be true since

Similarly, when (10) is true, one has

from the two inequalities above, and a phase-lag compensation

exists and is unique. One can easily show from (8) that if a

phase-lag compensation exists then condition (10) must be true

since

andmust be positive.

Thus, the proof is completed for the lead-lag compensation

theorem.

Note that none of the major textbooks in control theory (see

[3]–[10] for example), nor any of the literature on compensator

design known to the author, explicitly mentioned condition (9)

or (10) as the necessary and sufficient conditions for lead or lag

solutions.However,Mitchell[2]didmentionthetwoconditions

as the sufficient conditions.

Figs. 1 and 2 show the two conditions in normal and loga-

rithmic scales, respectively. The absolute value of the required

phase contribution ( ) is used in both figures. Note that phase

lead and phase lag are symmetric in decibel.

, and , and

andmust be positive.

, and

III. RELATION TO WAKELAND SOLUTION

In 1976, Wakeland [1] solved the compensation problem for

lead networks in the following form:

For achieving a gain (dB) and phase contribution ( ) at the

desired frequency (

), Wakeland’s solution is

(11)

where

or

Wakeland did not go further with (11), but one can show that

it will lead to the results given in (8), although the derivation is

different and simpler than Wakeland’s. To show this fact, first,

in terms of the notation defined in this paper

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260IEEE TRANSACTIONS ON EDUCATION, VOL. 46, NO. 2, MAY 2003

Fig. 1.Admissible magnitude-phase relationship in decimal.

Fig. 2. Admissible magnitude-phase relationship in decibel.

Equation (11) becomes

(12)

After some manipulations, its determinant can be found as

therefore, the two solutions of (12) are

and thus for

One can show easily that the positivesign will lead toa negative

, thus invalid since must be positive, while the minus sign

leads to (8).

For

In this case, the positive sign leads to (8), while the minus sign

leads to a negative

, hence, an invalid solution.

Fig. 3. A position control system in [10].

Therefore, the author has demonstrated that the only valid

designofcompensationfromWakeland’sequationistheunique

solution given in (8). Wakeland’s solution is just one step away

from theauthor’ssolution. However,thederivation here is more

straightforward and simple than Wakeland’s.

IV. NUMERICAL DESIGN EXAMPLES

Fig. 3 shows a simplified position control system used in de-

sign examples for lag and lead compensation in a control text

book [10], where

uncompensated system, and

signed. (For details, see [10, Fig. 11.2 and pp. 687 to 701].)

For phase-lag compensation, a four-step iterative design pro-

cedure, based on the Bode diagram, is outlined in [10]. In [10,

Example 11.2],

and the required phase margin

for a 9.5% overshoot. Starting with

(10 is added to compensate for the phase angle contribution of

the lag compensator, guesswork) and

tive procedure leads to the following final solution:

is the

is the compensator to be de-

rad s, the itera-

whichresultsina

of 10% overshoot and 0.25 s peak time in the time domain.

To use the results given in this paper, one first finds all

possible

under a given PM for lag compensation using the

lead-lag compensation theorem in Section II. As shown in

Fig. 4(a) (

versus) and Fig. 4(b) (

must be less than 19.795 rad/s for

9.395 rad/s for

. (In this example, Fig. 4(a) has to

be used to determine the admissible

For

and

pensator since

speaking, the starting point in [10, Example 11.2] is incorrect;

must be less than 9.78 rad/s in this case). For any

and,alagcompensatorcanbefoundeasily

using(8).Forexample,when

one has

and rad s andaresponse

vs.),

and less than

.)

rad s, there is no lag com-

in this case (therefore, strictly

with

andrad s,

resulting in a 15% overshoot and 0.30 s peak time for the com-

pensatedsystem. When

and rad s, one has

resulting in a 9.8% overshoot and 0.26 s peak time. Note that

and should be identical. The difference is a result

of the graphic approximation introduced in the iterative design

procedure based on the Bode diagram.

For phase-lead compensation, a twelve-step iterative design

procedure based on the Bode diagram is outlined in [10]. In

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WANG: SOLUTION FOR PHASE-LEAD AND PHASE-LAG COMPENSATION 261

Fig. 4. Admissible phase margin frequency for [10, Example 11.2].

Fig. 5. Admissible phase margin frequency for [10, Example 11.3].

[10, Example 11.3], , and the required phase margin

for a 20% overshoot. Starting with

(10 is added to compensate for the phase angle contribution of

the lead compensator, guesswork again) and

the iterative procedure leads to the following final solution:

rad s,

which results in a

response of 21% overshoot and 0.075 s peak time in the time

domain.

As shown in Fig. 5(a) ( versus

vs. ),must be larger than 29.743 rad/s for

and larger than 30.557 rad/s for

Fig. 5(b) has to be used to determine the admissible

and rad s, and a

) and Fig. 5(b) (

. (In this example,

.) For

and rad s, there is no lead com-

(

with

pensator since

than 32.086 rad/s in this case). For any

, a lead compensator can be found easily using

(8). For example, when

has

must be larger

and

andrad s, one

resulting in a 22.6% overshoot and 0.072 s peak time for the

compensated system. Note that

identical. As for Example 11.2, the difference is the result of

the graphic approximation in the iterative design procedure.

The advantage of using the current method is obvious, no

guesswork is needed; and all possible solutions can be obtained

analytically.

andshould be

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262IEEE TRANSACTIONS ON EDUCATION, VOL. 46, NO. 2, MAY 2003

V. CONCLUSION

Thispaperpresentsaone-stepanalyticanduniquesolutionto

the design of phase-lead and phase-lag compensation when the

desired gains in the magnitude and phase are known at a given

frequency.Asimpleconditionfordeterminingtheexistenceand

uniqueness of single-stage lead or lag compensation has also

been found. Therefore, no trial-and-error or other guesswork is

needed. Note that in most frequency-based designs, gain and

phase margin as criteria for performance are only approximate;

therefore, it seems that a unique solution is not all that mean-

ingful, other than making the teaching of lag/lead compensa-

tion possibly a little more straightforward and appealing. How-

ever, the uniqueness is very useful in a computer-aided design

process for control systems. The solution should also be useful

for developing analytical procedures for other frequency-based

techniques, especially optimal designs [11].

ACKNOWLEDGMENT

The author wishes to thank anonymous reviewers for their

valuable comments.

REFERENCES

[1] W. R. Wakeland, “Bode compensator design,” IEEE Trans. Automat.

Contr., pp. 771–773, Oct. 1976.

[2] J. R. Mitchell, “Comments bode compensator design,” IEEE Trans. Au-

tomat. Contr., pp. 869–870, Oct. 1977.

[3] R. C. Dorf and R. H. Bishop, Modern Control Systems, 8th ed.

Park, CA: Addison-Wesley, 1998.

[4] G. F. Franklin, J. D. Powell, and A. Emami-Haeini, Feedback Control of

Dynamic Systems, 2nd ed.Reading, MA: Addison-Wesley, 1994.

[5] B. C. Kuo, Automatic Control Systems, 5th ed.

Prentice-Hall, 1987.

[6] J. L. Melsa and D. G. Shultz, Linear Control Systems.

Graw-Hill, 1969.

[7] W. J. Palm III, Modeling, Analysis and Control of Dynamic Sys-

tems. New York: Wiley, 1983.

[8] R. T. Stefani, C. J. Savant Jr, B. Shahian, and G. H. Hostetter, Design

of Feedback Control Systems, 3rd ed.

Press, 1993.

[9] W. A. Wolovich, Automatic Control Systems: Basic Analysis and De-

sign. New York: Oxford University Press, 1993.

Menlo

Englewood Cliffs, NJ:

New York: Mc-

New York: Oxford University

[10] N. S. Nise, Control Systems Engineering, 3rd ed.

2000, pp. 687–701.

[11] F.-YWang,“AnAnalyticalApproachforControlDesignBasedonBode

Diagram,” Univ. of Arizona, Tucson, AZ, SIE Working Rep. No. 301,

2003.

New York: Wiley,

Fei-Yue Wang (S’87–M’89–SM’94) received B.S. degree in chemical engi-

neering from Qingdao University of Science and Technology, Qingdao, China,

in 1982, the M.S. degree in mechanics from Zhejiang University, Hangzhou,

China, in 1984, and the Ph.D. degree in electrical, computer and systems engi-

neering from the Rensselaer Polytechnic Institute, Troy, NY, in 1990.

He joined the University of Arizona, Tucson, in 1990 and became a Full Pro-

fessor of Systems and Industrial Engineering in 1999. In 1999, he founded the

Intelligent Control and Systems Engineering Center at the Institute of Automa-

tion, Chinese Academy of Sciences, Beijing, China, under the support of the

Outstanding Oversea Chinese Talents Program. Currently, he is the Director of

theProgramforAdvancedResearchinComplexSystems.Since2002,heisalso

the Director of the Key Laboratory of Complex Systems and Intelligence Sci-

ence at the Chinese Academy of Sciences. His current research interests include

modeling, analysis, and control mechanism of complex systems; agent-based

controlsystems;intelligentcontrolsystems;real-timeembeddedsystems,appli-

cation-specific operating systems (ASOS); and applications in intelligent trans-

portation systems, intelligent vehicles and telematics, web caching and service

caching, smart appliances and home systems, and network-based automation

systems. He has published more than 200 books, book chapters, and papers in

those areas since 1984 and received more than $20 million from the NSF, DOE,

DOT, NNSF, CAS, Caterpillar, IBM, HP, AT&T, GM, BHP, RVSI, ABB and

Kelon. He was the Editor-in-Chief of the International Journal of Intelligent

Control and Systems from 1995 to 2000 and currently is the Editor-in-Charge

ofthe Series on Intelligent Controland Intelligent Automation andan Associate

Editor of several other international journals. He was the Vice President of the

American Zhu Kezhen Education Foundation, the Chinese Association of Sci-

ence and Technology-USA, and a Member of the Boards of Directors of five

companies in information technology and automation.

Dr. Wang received the Caterpillar Research Invention Award with Dr. P.

J. A. Lever in 1996 and the National Outstanding Young Scientist Research

Award from the National Natural Science Foundation of China in 2001. He

is an Associate Editor of the IEEE TRANSACTIONS ON SYSTEMS, MAN, and

CYBERNETICS, the IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, and

the IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS. He is

an AdCom Member of IEEE Systems, Man and Cybernetics Society (SMC)

and IEEE Intelligent Transportation System Council (ITSC), and the Secretary

of the ITSC. He was the Program Chair of the 1998 IEEE International

Symposium on Intelligent Control, the 2001 IEEE International Conference

on Systems, Man, and Cybernetics, and the General Chair of the 2003 IEEE

International Conference on Intelligent Transportation Systems and will be

Co-Program Chair of the 2004 IEEE International Symposium on Intelligent

Vehicles and the General Chair for the same conference in 2005.