1. ISA
TRANSACTIONS®
ISA Transactions 41 ͑2002͒ 511–520
Design PID controllers for desired time-domain
or frequency-domain response
Weidong Zhang,* Yugeng Xi, Genke Yang, Xiaoming Xu
Department of Automation, Shanghai Jiaotong University, Shanghai 200030, PRC
͑Received 10 July 2001; accepted 8 February 2002͒
Abstract
Practical requirements on the design of control systems, especially process control systems, are usually specified in
terms of time-domain response, such as overshoot and rise time, or frequency-domain response, such as resonance peak
and stability margin. Although numerous methods have been developed for the design of the proportional-integral-
derivative ͑PID͒ controller, little work has been done in relation to the quantitative time-domain and frequency-domain
responses. In this paper, we study the following problem: Given a nominal stable process with time delay, we design
a suboptimal PID controller to achieve the required time-domain response or frequency-domain response for the
nominal system or the uncertain system. An H ϱ PID controller is developed based on optimal control theory and the
parameters are derived analytically. Its properties are investigated and compared with that of two developed suboptimal
controllers: an H 2 PID controller and a Maclaurin PID controller. It is shown that all three controllers can provide the
quantitative time-domain and frequency-domain responses. © 2002 ISA—The Instrumentation, Systems, and Auto-
mation Society.
Keywords: Linear system; Time delay; PID controller; Optimal control; Time-domain response
1. Introduction the focus of the study is on the optimal or subop-
timal design of PID controllers. ͓7͔ and ͓8͔ pre-
Most industrial processes are controlled by ´
sented a PID controller by Pade approximation in
proportional-integral-derivative ͑PID͒ controllers. the framework of internal model control ͑IMC͒. In
The popularity of PID controllers can be attributed fact, the controller is an H 2 suboptimal controller.
partly to their robust performance in a wide range ͓9͔ provided a PID controller by approximating an
of operating conditions and partly to their func- IMC controller with a Maclaurin series. ͓10͔ pro-
tional simplicity, which allows process engineers vided a suboptimal PID controller for integrator/
to operate them in a simple and straightforward time delay processes. The superiority of these
manner. To implement such a controller, three pa- methods is that the controller is derived analyti-
rameters must be determined for the given pro- cally. The control system can be designed and
cesses, and many papers have been devoted to this tuned by formulas and thus the design and tradeoff
problem. For example, see ͓1– 6͔. Early study on are significantly simplified.
the design and tuning of a PID controller is mainly Although numerous methods have been devel-
based on classical control theory. In recent years, oped, little work has been done in relation to the
quantitative time-domain and frequency-domain
*Corresponding author. Tel: ϩ86.21.62933329; responses. As we know, practical requirements on
fax: ϩ86.21.62826946. E-mail address: the design of control systems, especially on pro-
wdzhang@mail.sjtu.edu.cn cess control systems, are usually specified in terms
0019-0578/2002/$ - see front matter © 2002 ISA—The Instrumentation, Systems, and Automation Society.

2. 512 Weidong Zhang et al. / ISA Transactions 41 (2002) 511–520
Fig. 1. Unity feedback control loop.
Fig. 2. IMC structure.
of time-domain response, such as overshoot and
rise time, or frequency-domain response, such as
resonance peak and stability margin. It can be sim- C͑ s ͒
ply formulated as follows: Given a nominal plant Q͑ s ͒ϭ
1ϩG ͑ s ͒ C ͑ s ͒
and desired time-domain response or frequency-
domain response, we design a PID controller to and we have
meet these indexes. In this paper, an H ϱ PID con-
troller is first presented based on optimal control Q͑ s ͒
C͑ s ͒ϭ . ͑2͒
theory. The properties of the controller are then 1ϪG ͑ s ͒ Q ͑ s ͒
investigated and compared with an H 2 PID con-
troller and a Maclaurin PID controller. It is shown In the nominal case, the unity feedback loop can
that these three controllers can provide quantita- be equivalent to the well-known IMC structure de-
tive time-domain response and frequency-domain picted in Fig. 2. The sensitivity transfer function
response. The work is of significance in that it can of the closed-loop system can be written as
establish a relationship between the classical de- 1
sign requirement and that of optimal design meth- S͑ s ͒ϭ ϭ1ϪG ͑ s ͒ Q ͑ s ͒ ͑3͒
1ϩG ͑ s ͒ C ͑ s ͒
ods, and provide insight into control system de-
sign. and the complementary sensitivity transfer func-
The paper is organized as follows. In Section 2 tion as
the suboptimal H ϱ PID controller is derived ana-
lytically. The H 2 PID controller and Maclaurin G͑ s ͒C͑ s ͒
T͑ s ͒ϭ ϭG ͑ s ͒ Q ͑ s ͒ . ͑4͒
PID controller are briefly introduced. The perfor- 1ϩG ͑ s ͒ C ͑ s ͒
mance degree is defined and the quantitative time-
T ( s ) is just the closed-loop transfer function.
domain response and frequency-domain response
It is well known that any model obtained by
are discussed in Section 3. Finally, the paper is
using experimental procedures involves uncer-
concluded in Section 4 with a brief discussion of
tainty in its parameters. For the low-order models
the results.
used in many practical applications, the uncer-
tainty is expressed in a parametric fashion. The
2. Controller design parameters of the simplified model used to ap-
proximate a usually high-order dynamics are
Consider the unity feedback control system known to lie in an interval. The mean value of
˜ each parameter is chosen to represent its nominal
shown in Fig. 1, where G ( s ) denotes the plant and
value, often used to obtain nominal plant and con-
C ( s ) denotes the controller. In the context of pro-
troller parameters ͓8,9͔. For example, the plant is
˜
cess control, the process G ( s ) is usually described
by the following model ͑or the nominal plant͒ ˜ ˜
K e Ϫ␪s
G(s): G͑ s ͒ϭ
˜ ͑5͒
␶
˜ sϩ1
Ke Ϫ ␪ s
G͑ s ͒ϭ , ͑1͒ with
␶ sϩ1
K ෈ ͓ K Ϫ ,K ϩ ͔ ,
˜ ˜ ෈ ͓ ␪ Ϫ, ␪ ϩ͔ ,
␪ ˜ ෈ ͓ ␶ Ϫ, ␶ ϩ͔ .
␶
where K is the gain, ␶ is the time constant, and ␪ is
the time delay. We define the transfer function The midrange plant is chosen as the model:

3. Weidong Zhang et al. / ISA Transactions 41 (2002) 511–520 513
Ke Ϫ ␪ s In other words, we must guarantee that S ( s )
G͑ s ͒ϭ , ͑6͒ ϭ ( 1ϪG ( s ) Q ( s )) has a zero at sϭ0 to cancel the
␶ sϩ1
pole of W ( s ) . With the constraint, the unique op-
where timal Q im ( s ) is obtained as follows:
K Ϫ ϩK ϩ ␪ Ϫϩ ␪ ϩ ␶ Ϫϩ ␶ ϩ ͑ ␶ sϩ1 ͒͑ 1ϩ ␪ s/2͒
Kϭ , ␪ϭ , ␶ϭ . Q im ͑ s ͒ ϭ .
2 2 2 K
A central objective in automatic control is that a Obviously Q im ( s ) is improper. Now we use the
physical quantity is made to behave in a pre- following low-pass filter to roll Q im ( s ) off at high
scribed way by using the error between the system frequency:
output and the setpoint input. This gives rise to the 1
optimal control. Assume that the optimal perfor- J͑ s ͒ϭ , ␭Ͼ0.
mance index is H ϱ optimal, i.e., minʈW(s)S(s)ʈϱ , ͑ ␭sϩ1 ͒ 2
where W ( s ) is a weighting function. W ( s ) should Then
be selected such that the 2-norm boundary of the
system input is normalized by unity. That is, ͑ ␶ sϩ1 ͒͑ 1ϩ ␪ s/2͒
W ( s ) ϭ1/s for a unit step setpoint. The perfor- Q ͑ s ͒ ϭQ im ͑ s ͒ J ͑ s ͒ ϭ .
K ͑ ␭sϩ1 ͒ 2
mance index implies that the controller is designed ͑11͒
to minimize the worst error resulting from system
´
inputs. With Pade approximation, we have As ␭ tends to be zero, the controller tends to be
optimal. The corresponding controller of the unity
1Ϫ ␪ s/2 feedback loop is
G ͑ s ͒ ϭK , ͑7͒
͑ ␶ sϩ1 ͒͑ 1ϩ ␪ s/2͒ 1 ͑ ␶ sϩ1 ͒͑ 1ϩ ␪ s/2͒
C͑ s ͒ϭ . ͑12͒
which will be regarded as the nominal plant uti- Ks ␭ 2 sϩ2␭ϩ ␪ /2
lized to derive the H ϱ PID controller. The error
introduced by the approximation is included in un- The suboptimal H ϱ PID controller is derived ana-
certainty, which will be discussed later. lytically. Comparing the above controller with the
Instead of a numerical method, an analytical de- following practical PID controller,
sign procedure is developed for the given plant. It
is seen that W ( s ) S ( s ) is analytical in the open
right half plane. According to the well-known
ͩ
C ͑ s ͒ ϭK C 1ϩ
1
T Is
ϩT D s ͪ 1
T F sϩ1
. ͑13͒
maximum modulus theorem, a fundamental fact The parameters of the new PID controller are
concerning complex functions, ͉ W ( s ) S ( s ) ͉ does
not attain its maximum value at an interior point ␭2 ␪ ␪␶
T Fϭ , T Iϭ ϩ ␶ , T Dϭ ,
of the open right half plane. On the other hand, the 2␭ϩ ␪ /2 2 2T I
G ( s ) has a zero at sϭ2/␪ in the open right half
plane. Thus, for all Q ( s ) ’s, TI
K Cϭ .
K ͑ 2␭ϩ ␪ /2͒
ʈ W ͑ s ͒͑ 1ϪG ͑ s ͒ Q ͑ s ͒͒ ʈ ϱ у ͉ W ͑ 2/␪ ͒ ͉ . ͑8͒
If the practical PID controller is in the form of
ͩ ͪ
Consequently we have
1 T D sϩ1
minʈ W ͑ s ͒ S ͑ s ͒ ʈ ϱ ϭminʈ W ͑ s ͒͑ 1ϪG ͑ s ͒ Q ͑ s ͒͒ ʈ ϱ C ͑ s ͒ ϭK C 1ϩ , ͑14͒
T I s T F sϩ1
ϭ ␪ /2. ͑9͒ the parameters of the PID controller are
However, W ( s ) has a pole on the imaginary axis. ␭2 ␪
To obtain a finite infinity norm, a constraint will T Fϭ , T Iϭ ␶ , T Dϭ ,
2␭ϩ ␪ /2 2
be imposed on the design procedure:
lim ͑ 1ϪG ͑ s ͒ Q ͑ s ͒͒ ϭ0. ͑10͒ TI
K Cϭ .
s→0 K ͑ 2␭ϩ ␪ /2͒

4. 514 Weidong Zhang et al. / ISA Transactions 41 (2002) 511–520
If the practical PID controller is in the form of ␭␪ ␪ ␪␶
T Fϭ T Iϭ ϩ ␶ , T Dϭ
ͩ ͪ
, ,
1 T Ds 2 ͑ ␭ϩ ␪ ͒ 2 2T I
C ͑ s ͒ ϭK C 1ϩ ϩ , ͑15͒
T I s T F sϩ1
TI
K Cϭ .
the parameters of the PID controller are K ͑ ␭ϩ ␪ ͒
␭2 ␪
T Fϭ , T I ϭ ϩ ␶ ϪT F , Let C ( s ) ϭ ( ␶ sϩ1 ) / ( ␭sϩ1Ϫe Ϫ ␪ s ) ϭ f ( s ) /s.
2␭ϩ ␪ /2 2 The controller can be expanded in a Maclaurin
series as
␪␶ TI
ͫ ͬ
T Dϭ ϪT F , K Cϭ .
2T I K ͑ 2␭ϩ ␪ /2͒ 1 f Љ͑ 0 ͒ 2
C͑ s ͒ϭ f ͑ 0 ͒ ϩ f Ј ͑ 0 ͒ sϩ s ϩ¯ .
Since Eq. ͑13͒ is used in many papers, for com- s 2!
parison, only this form is considered in the later ͑18͒
part of this paper.
Taking the first three terms, we obtain the Maclau-
If a conventional PID controller is installed, the
rin PID controller, of which the parameters are ͓9͔
parameter T F has been determined. Usually T F
ϭ0.1T D ͓4͔. In this case, one can also use the
␪2
above rules by simply omitting the new T F . Then, T F ϭ0, T Iϭ ϩ␶,
similar results will be obtained. 2 ͑ ␭ϩ ␪ ͒
ͫ ͬ
Both the H 2 PID controller and Maclaurin PID
controller are based on the result of IMC. In IMC, ␪2 ␪ TI
T Dϭ 1Ϫ , K Cϭ .
the plant is factored as 2 ͑ ␭ϩ ␪ ͒ 3T I K ͑ ␭ϩ ␪ ͒
G ͑ s ͒ ϭG ϩ ͑ s ͒ G Ϫ ͑ s ͒ , Sometimes, one may use a second-order model.
where G ϩ ( s ) contains all nonminimum phase fac- The above methods can be directly extended to
tors, G ϩ ( 0 ) ϭ1, and G Ϫ ( s ) is the minimum phase this case. Since the design procedure is almost the
portion of the model. The IMC controller is then same, only the results are given here. Suppose that
given by the following formula ͓8͔: the model is
1 Ke Ϫ ␪ s
Q͑ s ͒ϭ , G͑ s ͒ϭ . ͑19͒
͑ ␭sϩ1 ͒ n G Ϫ ͑ s ͒ ͑ ␶ 1 sϩ1 ͒͑ ␶ 2 sϩ1 ͒
where n is chosen so that the controller is bi- Utilizing the Taylor series, the PID controller pa-
proper, that is, both Q ( s ) and 1/Q ( s ) are proper. rameters designed by the proposed method are
For the given plant we have
␭2 ␶ 1␶ 2
␶ sϩ1 T Fϭ , T Iϭ ␶ 1ϩ ␶ 2 , T Dϭ ,
C͑ s ͒ϭ , ͑16͒ 2␭ϩ ␪ TI
␭sϩ1Ϫe Ϫ ␪ s
Since a time delay is included in the controller, TI
K Cϭ . ͑20͒
it cannot be directly implemented. Many methods K ͑ 2␭ϩ ␪ ͒
have been presented to approximate the control-
ler by a rational transfer function, such as a nu- The H 2 PID controller parameters are
´
merical method, Pade approximation, Taylor se-
ries, Maclaurin series, and so on ͓11͔. The H 2 sub- ␭␪ ␶ 1␶ 2
optimal PID controller given by ͓7͔ utilizes a Pade´ T Fϭ , T Iϭ ,
␭ϩ2 ␪ ␶ 1ϩ ␶ 2
approximation. The resultant controller is
1 ͑ ␶ sϩ1 ͒͑ 1ϩ ␪ s/2͒ TI
C͑ s ͒ϭ ͑17͒ T Dϭ ␶ 1ϩ ␶ 2 , K Cϭ . ͑21͒
Ks ␭ ␪ s/2ϩ␭ϩ ␪ K ͑ ␭ϩ2 ␪ ͒
and the PID controller parameters are The Maclaurin PID controller parameters are

5. Weidong Zhang et al. / ISA Transactions 41 (2002) 511–520 515
2␭ 2 Ϫ ␪ 2 and
T F ϭ0, T Iϭ ␶ 1ϩ ␶ 2Ϫ ,
2 ͑ 2␭ϩ ␪ ͒ 1Ϫ0.5s
T͑ s ͒ϭ .
2␭ 2 Ϫ ␪ 2 ␶ 1 ␶ 2 Ϫ ␪ 3 / ͑ 12␭ϩ6 ␪ ͒ ͑ ␭sϩ1 ͒͑ 1ϩ0.5s ͒
T D ϭϪ ϩ ,
2 ͑ 2␭ϩ ␪ ͒ TI Only one pole of the nominal plant is canceled.
Note that there is an adjustable parameter ␭ in
TI the three PID controllers. It has been shown by
K Cϭ . ͑22͒
K ͑ 2␭ϩ ␪ ͒ some researchers that ␭ relates directly to the
nominal performance and robustness of the sys-
3. Discussion tem. This paper will show that the quantitative
time-domain response and frequency-domain re-
All three PID controllers are suboptimal. The sponse can be gained by adjusting ␭.
design procedure shows that they relate closely Consider the H ϱ PID controller again. We re-
and each has its own features. The difference be- gard the error introduced by the approximation as
tween the Maclaurin PID controller and the H 2 uncertainty. The actual plant is in the form of a
and H ϱ PID controllers is obvious. The difference first-order plus time delay. Then
between the H 2 PID controller and the H ϱ PID
͑ 1ϩ ␪ s/2͒ e Ϫ ␪ s
controller lies in the manner they cancel the poles T͑ s ͒ϭ
of the plant. The proposed H ϱ controller tends to ␭ 2 s 2 ϩ ͑ 2␭ϩ ␪ /2͒ sϩ ͑ 1ϩ ␪ s/2͒ e Ϫ ␪ s
cancel all poles of the process while the H 2 con- ͑23͒
troller tends to cancel the poles of the minimum and
phase part of the process.
␭ 2 s 2 ϩ ͑ 2␭ϩ ␪ /2͒ s
S͑ s ͒ϭ .
3.1. Example 1 ␭ 2 s 2 ϩ ͑ 2␭ϩ ␪ /2͒ sϩ ͑ 1ϩ ␪ s/2͒ e Ϫ ␪ s
͑24͒
The difference between the H 2 PID controller
and the H ϱ PID controller is illustrated in this ex- It is seen that T ( s ) does not depend on K and ␶.
ample. Consider the plant described by This implies that the setpoint response for the
closed-loop system relates only to ␭ and ␪. The
e Ϫs similar result also exists for the H 2 PID controller.
G͑ s ͒ϭ . Strictly speaking, the setpoint response of the sys-
sϩ1
tem with the Maclaurin PID controller relates to
´
The nominal plant with Pade approximation is not only ␭ and ␪ but ␶. Since the difference is very
1Ϫ0.5s
G͑ s ͒ϭ .
͑ sϩ1 ͒͑ 1ϩ0.5s ͒
The H ϱ method yields that
͑ sϩ1 ͒͑ 1ϩ0.5s ͒
Q͑ s ͒ϭ
͑ ␭sϩ1 ͒ 2
and the nominal complementary sensitivity trans-
fer function is
1Ϫ0.5s
T͑ s ͒ϭ .
͑ ␭sϩ1 ͒ 2
We see that both of the two poles of the nominal
plant are canceled. For the H 2 controller we have
sϩ1 Fig. 3. Responses of system A. ͑Solid line: H ϱ ; dashed
Q͑ s ͒ϭ
␭sϩ1 line: Maclaurin; dotted line: H 2 .͒

6. 516 Weidong Zhang et al. / ISA Transactions 41 (2002) 511–520
Fig. 4. Responses of system B. ͑Solid line: H ϱ ; dashed Fig. 6. Quantitative rise time. ͑Solid line: H ϱ ; dashed line:
line: Maclaurin; dotted line: H 2 .͒ Maclaurin; dotted line: H 2 .͒
small, it can be considered that its setpoint re-
sponse relates only to ␭ and ␪. overshoot. The resultant parameters are ␭ϭ0.41␪
for the H 2 controller, ␭ϭ0.46␪ for the H ϱ PID
3.2. Example 2 controller, and ␭ϭ0.42␪ for the Maclaurin PID
controller. The closed-loop responses are shown in
The three suboptimal PID controllers are com- Fig. 3 and Fig. 4. For the two plants, the H 2 con-
pared in this example. Consider the following two troller and the H ϱ PID controller have almost the
plants: same response. For the plant with a small ratio of
␪ and ␶, A, the H 2 controller and the H ϱ PID
e Ϫ0.5s e Ϫ5s controller have faster rise times and settling times
A: G ͑ s ͒ ϭ , B: G ͑ s ͒ ϭ .
sϩ1 sϩ1 than that of the Maclaurin PID controller, and for
the plant with a large ratio of ␪ and ␶, B, the H 2
A unit step setpoint is added at tϭ0s and a unit controller and the H ϱ PID controller have a slower
step load disturbance at tϭ6s or 60s. The param- settling time and disturbance rejection.
eter in all the three controllers is adjustable. To It is found that the response shapes of the H 2
obtain a fair comparison, the parameter is adjusted controller and the H ϱ PID controller for the two
such that the closed-loop responses have the same plants are almost the same. This validates the
Fig. 5. Quantitative overshoot. ͑Solid line: H ϱ ; dashed Fig. 7. Quantitative ISE. ͑Solid line: H ϱ ; dashed line:
line: Maclaurin; dotted line: H 2 .͒ Maclaurin; dotted line: H 2 .͒

7. Weidong Zhang et al. / ISA Transactions 41 (2002) 511–520 517
above analysis. We define ␭ as the ‘‘performance that the ratio of ISE and ␪ is also determined by
degree.’’ It is seen that the overshoot and the ratio the performance degree ͑Fig. 7͒. This implies that
of rise time and ␪ are determined only by the ratio the relationship between the classical performance
of the performance degree and ␪ ͑Figs. 5 and 6͒. indices and the optimal performance indices is es-
The break in Fig. 6 is caused by the difference of tablished.
the definition of the rise time for systems with In frequency-domain analysis, one important
overshoot and without overshoot. The empirical concept is the resonance peak, that is, the maxi-
formulas for estimating the two indices can also be mum modulus of the closed-loop transfer function.
given. For example, the formulas of the H ϱ PID In ͓14͔, it is referred to as the maximum log
controller are as follows: modulus and utilized to design PID controller. A
commonly used specification for it is 2dB. The
Overshoot
ͭ
quantitative relationship between the resonance
Ϫ0.86͑ ␭/ ␪ ͒ 3 ϩ14.21͑ ␭/ ␪ ͒ 2 peak and the performance degree is shown in Fig.
8.
ϭ Ϫ8.72␭/ ␪ ϩ1.86, 0.1р␭/ ␪ р0.59 Another important concept in frequency domain
0, 0.59р␭/ ␪ р1.2, is stability margin, on which numerous methods
are based ͑see, for example, ͓15͔͒. The magnitude
͑25͒ stability margin and phase stability margin provide
Rise time intuitive tools for control system design and are
ͭ
very familiar to engineers. The quantitative rela-
30.34͑ ␭/ ␪ ͒ 3 Ϫ24.63͑ ␭/ ␪ ͒ 2 tionship between the stability margin and the per-
formance degree is shown in Figs. 9 and 10.
ϭ ϩ8.48␭/ ␪ Ϫ0.45, 0.1р␭/ ␪ р0.59
Recently, ͓16͔ proposed a new frequency-
3.97␭/ ␪ Ϫ1.02, 0.59р␭/ ␪ р1.2. domain index for PID controller design,
͑26͒ 1/max͉ Re͓ G ͑ j ␻ ͒ C ͑ j ␻ ͔͒ ͉ ,
In ͓12,13͔, nonovershoot and monotone nonde- which involves both the magnitude stability mar-
creasing response is studied. For the three control- gin and phase stability margin to a certain extent.
lers it is very easy to get such a response. It is found that there also exists the quantitative
Both the H 2 control and the H ϱ control relate to relationship between the new index and the perfor-
the integral square error ͑ISE͒. The H 2 control mance degree ͑Fig. 11͒.
minimizes the ISE for a particular input and the Unfortunately, the simple quantitative relation-
H ϱ control minimizes the worst ISE resulting ship comes into existence only for setpoint re-
from any two-norm bounded inputs. It is found sponse. The transfer function from the load distur-
Fig. 8. Quantitative resonance peak. ͑Solid line: H ϱ ; Fig. 9. Quantitative magnitude margin. ͑Solid line: H ϱ ;
dashed line: Maclaurin; dotted line: H 2 .͒ dashed line: Maclaurin; dotted line: H 2 .͒

8. 518 Weidong Zhang et al. / ISA Transactions 41 (2002) 511–520
Fig. 12. Quantitative perturbance peak.
Fig. 10. Quantitative phase margin. ͑Solid line: H ϱ ;
dashed line: Maclaurin; dotted line: H 2 .͒
where ⌬ ( ␻ ) is the uncertainty profile. Let S ( s )
and T ( s ) be the sensitivity transfer function and
the complementary sensitivity transfer function of
bance to the system output is G ( s ) S ( s ) . Thus, the the approximated system, respectively. The robust
disturbance response relates not only to ␭ and ␪, stability is met if and only if
but K and ␶. We define the perturbance peak to be
the maximum output caused by the disturbance ʈ ⌬ ͑ ␻ ͒ T ͑ j ␻ ͒ ʈ ϱ Ͻ1, ᭙␻. ͑28͒
͓10͔. For the H ϱ controller the relationship is
shown in Fig. 12. This implies that arbitrary robust stability can be
For the second-order model, the closed-loop re- gained by increasing the performance degree ␭.
sponse relates both ␶ 1 and ␶ 2 . If one of them is Assume that the performance is ʈ W ( s ) S ( s ) ʈ ϱ
determined, a similar quantitative relationship also Ͻ5 ␪ . Then, the robust performance is met if and
exists, but not so clearly. only if
There always exists uncertainty in practice. Sup- ʈ ͉ W ͑ j ␻ ͒ S ͑ j ␻ ͒ /5␪ ͉ ϩ ͉ ⌬ ͑ ␻ ͒ T ͑ j ␻ ͒ ͉ ʈ ϱ Ͻ1, ᭙␻.
pose that the norm-bounded uncertainty is de- ͑29͒
scribed by
ͯ ͯ
The quantitative uncertainty profile that guaran-
G ͑ s ͒ ϪG ͑ s ͒
˜ tees robust performance for an H ϱ PID controller
ϭ ͉ ⌬ m ͑ j ␻ ͒ ͉ р⌬ ͑ ␻ ͒ , ͑27͒ is shown in Fig. 13.
G͑ s ͒
In classical control theory, the controller is usu-
ally designed for the nominal performance speci-
fication and used for the control of the uncertain
plant. If a good model can be obtained, the esti-
Fig. 11. Quantitative new index. ͑Solid line: Hϱ ; dashed
line: Maclaurin; dotted line: H2 .͒ Fig. 13. Quantitative uncertainty profile.

9. Weidong Zhang et al. / ISA Transactions 41 (2002) 511–520 519
mates given by the above study are in good agree- the H ϱ PID controller, the relationship is exact,
ment with the actual time-domain responses. If the and for the Maclaurin PID controller, the relation-
uncertainty scope is large, one may hope to know ship roughly exists. Thus, all three controllers can
whether the required performance is obtained or be designed for desired time-domain response or
not. Eqs. ͑28͒ and ͑29͒ give a perfect estimate. frequency-domain response. The given quantita-
However, they are more mathematical than practi- tive relationship between the classical perfor-
cal and very inconvenient. As a matter of fact, we mance indices and the optimal performance indi-
usually wish to estimate the ‘‘worst case’’ re- ces makes it possible to build a bridge between
sponse ͑i.e., the gain and time delay are at their the classical design method and modern design
upper limits, while the time constant is at its lower methods.
limit͒ or the range of the closed-loop response. For practical purposes, the H 2 PID controller
͓17͔ recently developed a method for designing may be the most convenient one, because it has a
IMC systems with parametric uncertainty. First, a linear relationship with many performance indices.
specified value M p is chosen based on the maxi- Since there always exists uncertainty in practice,
mum desired overshoot, or the ‘‘worst case’’ over- the quantitative design for the system with para-
shoot. Second, the magnitude of the complemen- metric uncertainty is also discussed, and a new
tary sensitivity transfer function is designed to be design method is proposed to estimate the ‘‘worst
equal to or less than M p. For example, for the case’’ response or the range of the closed-loop re-
‘‘worst case’’ overshoot 10%, M pϭ1.05. By solv- sponse of the proposed controller.
ing ͉ T ( j ␻ ) ͉ рM p, a unique solution on ␭ can be It is also shown that the suboptimal PID gives a
obtained. satisfactory approximation to the exact time delay
Here, an alternative method is proposed to esti- compensated scheme. This implies that the PID
mate the ‘‘worst case’’ response or the range of the controller can provide relatively good response for
closed-loop response of the proposed controller. systems with time delay, even when the time delay
Assume that we hope to achieve the ‘‘worst case’’ is very large.
overshoot of 5%. The new procedure is as follows:
͑i͒ Design the controller for the nominal Acknowledgments
plant. For 5% overshoot, ␭ϭ0.5␪.
͑ii͒ Substitute the nominal plant by the This project was supported by the National
ϩ
‘‘worst case’’ plant K ϩ e Ϫ ␪ s / ( ␶ Ϫ sϩ1 ) . Natural Science Foundation of China ͑Grant
͑iii͒ Increase the performance degree ␭ No. 69804007͒ and the National Key Technologies
monotonically until the overshoot is R&D Program in the Tenth Five-Year Plan ͑Grant
equal to 5%. No. 2001BA201A04͒.
For system with time delay, the typical step in References
͑iii͒ is 0.01␪, and for systm without time delay, the
typical step in ͑iii͒ is 0.01␪. This procedure can ͓1͔ Ziegler, J. G. and Nichols, N. B., Optimum settings for
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͓5͔ Sung, S. W., Lee, J., and Lee, I., A new tuning rule and
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The most important contribution of this paper is ͓6͔ Astrom, K. J., Panagopoulos, H., and Hagglund, T.,
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zation. Automatica 34 ͑5͒, 585 ͑1998͒.
and the quantitative relationship between the ͓7͔ Rivera, D. E., Morari, M., and Skogestad, S., Internal
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͓8͔ Morari, M. and Zafiriou E., Robust Process Control.
Prentice-Hall, Englewood Cliffs, NJ, 1989. Yugeng Xi was born in the
People’s Republic of China in
͓9͔ Lee, Y., Park, S., Lee, M., and Brosilow, C., PID con- 1946. He graduated from
troller tuning for desired closed loop response for Haerbin Engineering College
SISO systems. AIChE J. 44 ͑1͒, 106 ͑1998͒. in 1968. From 1979 to 1984 he
was a visiting scholar at the In-
͓10͔ Zhang, W. D., Xu, X. M., and Sun, Y. X., Quantitative stitute of Control Technology,
performance design for integrating processes with Technical University Munich,
time delay. Automatica 35 ͑4͒, 719 ͑1999͒. Germany, where he received
͓11͔ Zhang, W. D., Robust control of systems with time the Dr.-Ing. degree in 1984.
Since then, he has been work-
delay, Ph.D thesis, Zhejiang University, 1996. ing at Shanghai Jiaotong Uni-
͓12͔ Lin, S. K. and Fang, C. J., Nonovershoot and mono- versity. He received awards for
tone nondecreasing step response of a third order Achievement in Science and
SISO linear system. IEEE Trans. Autom. Control 42 Technology six times by the State Commission of Education and the
Shanghai government. Prof. Xi was the vice chair of the IFAC TC
͑9͒, 1299 ͑1997͒. Large Scale Systems and is now the vice president of the Chinese
͓13͔ Dan-Isa, A. and Atherton, D. P., Time-domain method Association of Automation. His current research interests include pre-
for the design of optimal linear controllers. IEE Proc.: dictive control, robust control, robot control, and computer vision.
Control Theory Appl. 144, 287 ͑1997͒.
͓14͔ Luyben, W. L., Simple method for tuning SISO con-
trollers in multivariable systems. Ind. Eng. Chem. Pro- Genke Yang was born in the
People’s Republic of China in
cess Des. Dev. 25, 654 ͑1986͒. 1963. He received his B.Sc.
͓15͔ Ho, W. K., Lim, K. W., Hang, C. C., and Ni, L. Y., and M.Sc. degrees in math-
Getting more phase margin and performance out of ematics, and Ph.D. degree in
systems engineering. He is cur-
PID controllers. Automatica 35, 1579 ͑1999͒.
rently an associate professor in
͓16͔ Wang, Y. G. and Shao, H. H., Optimal tuning for PI the department of automation,
controllers. Automatica 36 ͑1͒, 147 ͑2000͒. Shanghai Jiaotong University.
͓17͔ Stryczek, K., Laiseca, M., Brosilow, C., and Leitman, He is the author of 40 papers.
His current research interests
M., Tuning and design of single-input, single-output include modeling, analysis,
control systems for parametric uncertainty. AIChE J. and control theory of discrete
46 ͑8͒, 1616 ͑2000͒. event dynamic systems and hy-
brid dynamic systems.
Xiaoming Xu was born in the
People’s Republic of China in
Weidong Zhang was born in 1957. He received the BS de-
the People’s Republic of China gree from Huazhong Univer-
in 1967. He received the BS, sity of Science and Technology
MS, and Ph.D degree from in 1982, and the MS and Ph.D
Zhejiang University in 1990, degrees from Shanghai Jiao-
1993, and 1996, respectively. tong University in 1984 and
He worked in National Key 1987, respectively. From 1988
Laboratory of Industrial Con- to 1990 he worked in Germany
trol Technology as a postdoc- as an Alexander von Humboldt
toral research fellow before research fellow. He joined
joining Shanghai Jiaotong Uni- Shanghai Jiaotong University
versity in 1998 as an associate in 1990 as a professor in the
professor. Since 1999 he has department of automation. Since 1997 he has been the deputy princi-
been a professor at the Depart- pal of Shanghai Jiaotong University. He is the head of numerous state
ment of Automation, Shanghai Jiaotong University. He is the author of research projects and the author of numerous papers. His current re-
68 papers. His current research interests include process control, ro- search interests include control theory, artificial intelligence, computer
bust control, field-bus, and digital signal processing. networking, and digital signal processing.