258IEEE TRANSACTIONS ON EDUCATION, VOL. 46, NO. 2, MAY 2003
The Exact and Unique Solution for Phase-Lead and
Fei-Yue Wang, Senior Member, IEEE
Abstract—Phase-lead and -lag compensation is one of the most
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-
N spite of the advances in modern control system design
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  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  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
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: email@example.com).
Digital Object Identifier 10.1109/TE.2002.808279
timal designs. An extensive search of the relevant literature has
not found similar results (see major control textbooks –).
II. THE SOLUTION
Assume that a single-stage compensator is expressed as ,
a phase-lead compensator.
(in dB) and
nitude and phase, to be contributed by
frequency (or any given frequency)
yields a phase-lag compensator, and yields
(in rad) be the desired gain in the mag-
, at a given crossover
which leads to
new variable and parameter
can be assumed here. Introducing
Then from (3) and
the following two equations are found in termsof and
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
The above equation can be further simplified
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-
0018-9359/03$17.00 © 2003 IEEE
WANG: SOLUTION FOR PHASE-LEAD AND PHASE-LAG COMPENSATION259
The above equation presents two possible solutions for
sign andsign, respectively). Now one can prove that there is
only one possible solution, i.e., only
phase-lead or phase-lag compensator must satisfy the following
sign is allowed in (7).
A. Phase-Lead Compensation
for phase-lead compensation.
B. Phase-Lag Compensation
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
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
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
The Lead-Lag Compensation Theorem:
a) A single-stage phase-lead compensation exists if and only
b) A single-stage phase-lag compensation exists if and only
In both cases, the compensation solution is unique and given by
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
andmust be positive.
Thus, the proof is completed for the lead-lag compensation
Note that none of the major textbooks in control theory (see
– 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
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.
III. RELATION TO WAKELAND SOLUTION
In 1976, Wakeland  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
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
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
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).
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 .
Therefore, the author has demonstrated that the only valid
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 , 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 . In [10,
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 compensator to be de-
rad s, the itera-
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
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
speaking, the starting point in [10, Example 11.2] is incorrect;
must be less than 9.78 rad/s in this case). For any
andrad s andaresponse
and less than
rad s, there is no lag com-
in this case (therefore, strictly
resulting in a 15% overshoot and 0.30 s peak time for the com-
andrad s, one has
resulting in a 9.8% overshoot and 0.26 s peak time. Note that
andshould 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 . In
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:
which results in a
response of 21% overshoot and 0.075 s peak time in the time
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
andrad s, and a
) and Fig. 5(b) (
. (In this example,
and rad s, there is no lead com-
than 32.086 rad/s in this case). For any
, a lead compensator can be found easily using
(8). For example, when
must be larger
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
262IEEE TRANSACTIONS ON EDUCATION, VOL. 46, NO. 2, MAY 2003
the design of phase-lead and phase-lag compensation when the
desired gains in the magnitude and phase are known at a given
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 .
The author wishes to thank anonymous reviewers for their
 W. R. Wakeland, “Bode compensator design,” IEEE Trans. Automat.
Contr., pp. 771–773, Oct. 1976.
 J. R. Mitchell, “Comments bode compensator design,” IEEE Trans. Au-
tomat. Contr., pp. 869–870, Oct. 1977.
 R. C. Dorf and R. H. Bishop, Modern Control Systems, 8th ed.
Park, CA: Addison-Wesley, 1998.
 G. F. Franklin, J. D. Powell, and A. Emami-Haeini, Feedback Control of
Dynamic Systems, 2nd ed. Reading, MA: Addison-Wesley, 1994.
 B. C. Kuo, Automatic Control Systems, 5th ed.
 J. L. Melsa and D. G. Shultz, Linear Control Systems.
 W. J. Palm III, Modeling, Analysis and Control of Dynamic Sys-
tems. New York: Wiley, 1983.
 R. T. Stefani, C. J. Savant Jr, B. Shahian, and G. H. Hostetter, Design
of Feedback Control Systems, 3rd ed.
 W. A. Wolovich, Automatic Control Systems: Basic Analysis and De-
sign.New York: Oxford University Press, 1993.
Englewood Cliffs, NJ:
New York: Mc-
New York: Oxford University
 N. S. Nise, Control Systems Engineering, 3rd ed.
2000, pp. 687–701.
Diagram,” Univ. of Arizona, Tucson, AZ, SIE Working Rep. No. 301,
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
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
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.