Auto tuning of cascade control systems

ISA
TRANSACTIONS®
ISA Transactions 42 ͑2003͒ 63–72

Auto-tuning of cascade control systems
Sihai Song, Wenjian Cai, Ya-Gang Wang
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798
͑Received 8 August 2001; accepted 16 April 2002͒

Abstract
In this paper, a novel auto-tuning method for a cascade control system is proposed. By employing a simple relay
feedback test, both inner and outer loop model parameters can be simultaneously identified. Consequently, well-
established proportional-integral-derivative ͑PID͒ tuning rules can be applied to tune both loops. Compared with
existing methods, the new method is simpler and yet more effective. It can be directly integrated into commercially
available industrial auto-tuning systems. Some examples are given to illustrate the effectiveness and robustness of the
proposed method. © 2003 ISA—The Instrumentation, Systems, and Automation Society.

Keywords: Cascade control; Relay feedback; PID auto-tuning; Model matching

1. Introduction controller is tuned. Subsequently, the inner loop
controller is commissioned and the outer loop con-
Proportional-integral-derivative ͑PID͒ control- troller is tuned to complete the tuning process. If
lers are widely used in the process control industry the control performance achieved is unsatisfactory,
due to their relatively simple structure, which can the entire sequence must be repeated. Thus it is a
be easily understood and implemented. In practice, fairly cumbersome and time consuming task to
it has often been integrated into complex control tune a cascade control system, especially for sys-
structures in order to achieve a better control per- tems with large time constant and time delay.
formance. Among those complex control struc- PID auto-tuning relieves the pain of manually
tures, the cascade control scheme is commonly tuning a controller and has been successfully ap-
used for the purpose of reducing both maximum plied in many industry fields ͓1,2͔. However, very
deviation and integral error of disturbance re- little has been reported so far in the literature on
sponses. The advantages of easy implementation the development of auto-tuning techniques for cas-
and potentially large control performance im- cade control systems. Among few of them, Li et
provement have led to widespread applications of al. ͓3͔ made use of fuzzy logic for self-tuning of
cascade control for several decades. It has become cascade controllers. Hang et al. ͓4͔ applied a re-
a standard application provided by industrial pro- newed relay automatic tuning method to tune a
cess controllers. cascade control system where the relay feedback
Cascade control systems are constructed by two test is carried out twice, first to the inner loop and
control loops: an inner loop with fast dynamic to then to the outer loop. While the individual con-
eliminate input disturbances, and an outer loop to troller tuning has been automated, the sequential
regulate output performance. Conventionally, they nature of the tuning process remains unchanged.
are tuned in a sequential manner. First, the outer Tan ͓5͔ proposed a method to carry out the entire
loop controller is put on manual and the inner loop tuning process in one experiment, but the experi-

0019-0578/2003/$ - see front matter © 2003 ISA—The Instrumentation, Systems, and Automation Society.

64 Sihai Song, Wenjian Cai, Ya-Gang Wang / ISA Transactions 42 (2003) 63–72

Fig. 1. Configuration of the cascade control system.

ment requires prior information of the process. the phase lag of the closed inner loop will be much
Furthermore, the ultimate frequency used for outer less than that of the outer loop. This feature leads
loop design is based on initial ultimate frequency to the rationale behind the use of cascade control.
without considering changes in inner loop control The crossover frequency for the inner loop is
parameters. higher than that for the outer loop, which allows
This paper presents a novel auto-tuning method higher gains in the inner loop controller in order to
for the cascade control system. By utilizing the regulate more effectively the effect of a distur-
fundamental characteristic of cascade control sys- bance occurring in the inner loop without endan-
tems, a simple relay feedback test is applied to the gering the stability of the system.
outer loop to identify simultaneously both inner
and outer loop process model parameters. A model
matching the PID controller tuning method based
´
on Pade coefficients and the Markov parameter is 3. Relay feedback test for cascade control
proposed to control the overall system perfor- systems
mance. Two examples are given to illustrate the
effectiveness of the proposed method. The Astrom-Hagglund relay feedback test is
based on the observation: when the process output
2. Fundamentals of cascade control systems lags behind the input by Ϫ␲ radians, the closed-
loop system may generate sustained oscillation
The configuration of the cascade control scheme around the ultimate frequency ͑the frequency
is shown in Fig. 1, where an inner loop is embed- where the phase lag is Ϫ␲͒. The proposed relay
ded within an outer loop and the outer loop output feedback test for the auto-tuning cascade control
variable is to be controlled. The control system system is shown as in Fig. 2. When the relay feed-
consists of two processes and two controllers with back test begins, switch A points to position 2,
outer loop transfer function G p1 , inner loop trans- switch B points to position 4, and switch C points
fer function G p2 , outer loop controller G c1 , and to position 5. After the test, switch A points to
inner loop controller G c2 , respectively. position 1, switch B points to position 3, and
The two controllers of cascade control systems switch C points to position 6. As the inner loop
are standard feedback controllers ͑i.e., P, PI, or process acts much faster than the outer loop pro-
PID͒. Usually, a proportional controller is used for cess, output u of the inner loop process in Fig. 2
the inner loop, integral action is needed when the under the relay feedback test acts as a step re-
inner loop process contains essential time delays, sponse in half of the period of the stationary os-
and the outer process is such that the loop gain in cillation, as shown in Fig. 3. Therefore a single
the inner loop must be limited ͓1͔. relay feedback test can be used to obtain simulta-
To serve the purpose of reducing or eliminating neously both the inner loop and outer loop process
the inner loop disturbance d 2 before its effect can models parameters.
spill over to the outer loop, it is essential that the In practice, the real process model is usually
inner loop exhibit a faster dynamic response than represented by low order plus dead-time model.
that of the outer loop ͑as industry rule of thumb, it Here, the transfer function with the following form
should be at least five times ͓1͔͒. Consequently, ͑first-order plus dead time͒ is adopted:

Sihai Song, Wenjian Cai, Ya-Gang Wang / ISA Transactions 42 (2003) 63–72 65

Fig. 2. Configuration of the proposed identification model.

ki tϭ0 in the process input; the process input and
G pi ͑ s ͒ ϭ e ϪL i s , ͑1͒ output are collected until process enters a new
T i sϩ1
steady state again. The process response after dead
where iϭ1 stands for the outer process model and time tϭL 2 is described by
iϭ2 stands for the inner process model, respec-
tively. This model is characterized by three param- u ͑ t ͒ ϭk 2 ͑ 1Ϫe ͑ tϪL 2 ͒ /T 2 ͒ ϩw ͑ t ͒ , tуL 2 , ͑2͒
eters: the static gain k, the time constant T, and the
where w ( t ) is the white noise in measurement of
dead time L. It describes a linear monotonic pro-
u ( t ) . It follows from the above relation that
cess quite well in many industrial applications and
is often sufficient for PID controller tuning.
ͫ ͬT2
͓ u ͑ t ͒ k 2 ͔ L ϭk 2 tϪA ͑ t ͒ ϩ ␦ ͑ t ͒ , tуL 2 ,
2
3.1. Inner loop process model identification ͑3͒
As the inner loop output u can be considered as where A ( t ) is the area under the process response
a step response in half period of the relay feedback and ␦͑t͒ is the integration of measurement noise;
test, some well-developed step testing methods they are given as following, respectively:
͓2,6,7͔ can be readily applied to identify param-
eters of the inner loop. In this paper, the method
proposed by Ref. ͓2͔ is adopted due to its robust-
A͑ t ͒ϭ ͵ u͑ t ͒dt,
0
t

ness; it is briefly described as follows:
Suppose that the inner process model is repre-
sented by Eq. ͑1͒, and a unit step change occurs at
␦͑ t ͒ϭ ͵ w͑ t ͒dt.
t

0
͑4͒

Fig. 3. Inner loop and outer loop response under the proposed relay feedback test.


The inner process model’s static gain k 2 is com- where T u is the period of the stationary oscillation,
puted from the process steady states of input and and
output, 2␲
␻ uϭ . ͑10͒
⌬u Tu
k 2ϭ , ͑5͒
⌬u d As the overall open loop transfer model function is
where ⌬u denotes the change of process output G p ͑ s ͒ ϭG p1 ͑ s ͒ G p2 ͑ s ͒
and ⌬u d stands for the deviation in the manipu-
lated input. Eq. ͑2͒ falls into a system of linear k 1k 2
ϭ e Ϫ ͑ L 1 ϩL 2 ͒ s ,
equations, ͑ T 1 sϩ1 ͒͑ T 2 sϩ1 ͒
⌿Xϭ⌫ϩ⌬ for tуL 2 , ͑6͒ the outer loop process model transfer function
where k1
G p1 ͑ s ͒ ϭ e ϪL 1 s

ͫ ͬ
T 1 sϩ1
T2
Xϭ L , can be obtained by the following steps.

ͫ ͬ
2
͑1͒ Read off the overall system time delay L
u ͓ mT s ͔ k2 ϭL 1 ϩL 2 of G p in the transfer function from the
u ͓͑ mϩ1 ͒ T s ͔ k2 initial part of the relay feedback test, since the
⌿ϭ , inner loop transfer function delay L 2 is already
] ] available, the time delay L 1 can be computed as
u ͓͑ nϩ1 ͒ T s ͔ k2

ͫ ͬ
L 1 ϭLϪL 2 . ͑11͒
k 2 t ͓ mT s ͔ ϪA ͓ mT s ͔ ͑2͒ Obtain the frequency response of G p1 ( j ␻ )
k 2 t ͓͑ mϩ1 ͒ T s ͔ ϪA ͓͑ mϩ1 ͒ T s ͔ at ␻ ϭ ␻ u from
⌫ϭ ] ,
G p͑ j ␻ u ͒
k 2 t ͓͑ nϩ1 ͒ T s ͔ ϪA ͓͑ nϩ1 ͒ T s ͔ G p1 ͑ j ␻ u ͒ ϭ , ͑12͒

ͫ ͬ
G p2 ͑ j ␻ u ͒
␦ ͓ mT s ͔ and calculate
␦ ͓͑ mϩ1 ͒ T s ͔ G p1 ͑ j ␻ u ͒
⌬ϭ ] . ͑7͒ k1
GЈ ͑ j ␻u͒ϭ ϭ Ϫ jL 1 ␻ u ϭ ␣ ϩ j ␤ ,
p1 jT 1 ␻ u ϩ1 e
␦ ͓͑ nϩ1 ͒ T s ͔ ͑13͒
T s is the sampling interval, and mT s уL 2 . The which is the frequency response for G p1 ( j ␻ )
best estimation X * of X can be obtained using the without delay, where ␣ has positive sign and ␤ has
standard least-square method as negative sign.
X * ϭ ͑ ⌿ T ⌿ ͒ Ϫ1 ⌿ T ⌫. ͑8͒ ͑3͒ Calculate T 1 and k 1 , respectively, by
␤
The best estimation of T 2 and L 2 can then be ob- T 1 ϭϪ , ͑14͒
tained from X * . ␣ϫ␻u
␣ 2ϩ ␤ 2
3.2. Outer loop process model identiﬁcation k 1ϭ . ͑15͒
␣
By relay feedback test, the frequency response
4. Controller design
of overall process model G p ( s ) at the ultimate fre-
quency ␻ u is estimated as
As the main purpose of inner loop control is to

͵ y ͑ t ͒e0
Tu
Ϫ j ␻ut
dt
eliminate the input disturbance, a P or PI control-
ler using widely accepted model based tuning
G ͑ j ␻ ͒ϭ ͑9͒ rules such as Ziegler-Nichols ͓8͔, Chien-Hrones-
͵ u ͑ t ͒e
p u Tu
,
Ϫ j ␻ut
dt Reswick ͑CHR͒ ͓9͔, or Cohen-Coon ͓10͔ tuning
d
0 rules will sufﬁce. This feature makes it very easy


to integrate the tuning method into the existing where ␰ stands for the desired damping ratio, usu-
auto-tuning systems. Without loss of generality, ally selected as 0.707, the natural frequency ␻ n
the PI control structure of the form can be chosen between 0.5 and 1.0 times the ulti-
mate frequency ␻ u from the relay feedback test
K i2 ͓13͔. An alternative of desired closed-loop transfer
G c2 ͑ s ͒ ϭK p2 ϩ
s function for large dead-time systems can be ex-
pressed as ͓14͔
and the Chien-Hrones-Reswick ͑CHR͒ ͓9͔ tuning
rule ͑20% overshoot͒ will be used for comparison ␻2
n
study. The controller parameters are given, respec- H͑ s ͒ϭ e Ϫ ͑ L 1 ϩL 2 ͒ s .
s 2 ϩ2 ␰␻ n sϩ ␻ 2
n
tively, by
0.7T 2 If the control speciﬁcations are not available, de-
K p2 ϭ , ͑16͒ fault settings for the parameter ␨ ϭ0.707 and
k 2L 2 ␻ u ( L 1 ϩL 2 ) ϭ2 can be used, which implies that
the overshoot of the objective step response is
0.304T 2
K i2 ϭ . ͑17͒ about 5%, the phase margin is 60°, and the gain
k 2L 2
2 margin is 2.2. For simplicity, Eq. ͑20͒ can be re-
written as the parametric form:
With the PI controller, the closed-loop transfer
function G 2 ( s ) of inner loop and the open loop d0
transfer function G 1 ( s ) are then obtained as H͑ s ͒ϭ , ͑21͒
e 0 ϩe 1 sϩe 2 s 2
G p2 ͑ s ͒ G c2 ͑ s ͒ where d 0 ϭ ␻ 2 , e 0 ϭ ␻ 2 , e 1 ϭ2 ␰␻ n , and e 2 ϭ1.
G 2͑ s ͒ ϭ n n
1ϩG p2 ͑ s ͒ G c2 ͑ s ͒ ͑2͒ Approximating the time delays in G 1 ( s ) of
Eq. ͑19͒, since the dead time L 2 of the inner loop
k 2 ͑ K p2 sϩK i2 ͒ e ϪL 2 s
ϭ , process model is very small, it is always approxi-
s ͑ 1ϩT 2 s ͒ ϩk 2 ͑ K p2 sϩK i2 ͒ e ϪL 2 s mated as 1 or
͑18͒ e ϪL 2 s Ϸ1ϩ ͑ ϪL 2 s ͒ , ͑22͒
G 1 ͑ s ͒ ϭG 2 ͑ s ͒ G p1 ͑ s ͒ e Ϫ ( L 1 ϩL 2 ) s is approximated as
k 2 ͑ K p2 sϩK i2 ͒ e ϪL 2 s ͑ L 1 ϩL 2 ͒ 2 s 2
ϭ e Ϫ ͑ L 1 ϩL 2 ͒ s Ϸ1Ϫ ͑ L 1 ϩL 2 ͒ sϩ
s ͑ 1ϩT 2 s ͒ ϩk 2 ͑ K p2 sϩK i2 ͒ e ϪL 2 s 2
͑23͒
k1
• e ϪL 1 s . ͑19͒ in order to gain a more accurate approximation.
T 1 sϩ1
Substitute Eqs. ͑22͒ and ͑23͒ to Eq. ͑19͒, and re-
As G 1 ( s ) is not a standard transfer function, it is write G 1 ( s ) as
difﬁcult to directly apply existing tuning rules.
Therefore a model-matching algorithm ͓11,12͔ is g 0 ϩg 1 sϩg 2 s 2 ϩg 3 s 3
G 1͑ s ͒ ϭ , ͑24͒
proposed to obtain the PID control parameters for h 0 ϩh 1 sϩh 2 s 2 ϩh 3 s 3
overall system performance. The brief theoretical
background of a controller design based on model where
matching is attached in the Appendix; for more
g 0 ϭk 1 k 2 K i2 ,
details please refer to Refs. ͓11,12͔. Its application
to this particular problem is given as follows. g 1 ϭk 1 k 2 ͓ K p2 ϪK i2 ͑ L 1 ϩL 2 ͔͒ ,
͑1͒ Assuming that the process dead time is
small, the desired reference model H ( s ) is chosen
as g 2 ϭk 1 k 2 ͫ K i2 ͑ L 1 ϩL 2 ͒ 2
2
ϪK p2 ͑ L 1 ϩL 2 ͒ , ͬ
␻2
n k 1 k 2 K p2 ͑ L 1 ϩL 2 ͒ 2
H͑ s ͒ϭ , ͑20͒ g 3ϭ ,
s 2
ϩ2 ␰␻ n sϩ ␻ 2
n 2


and ͑4͒ The PID parameters can be computed as
h 0 ϭk 2 K i2 , a 1 b 1 Ϫa 0 b 2
K p1 ϭ , ͑27͒
b2
h 1 ϭk 2 T 1 K i2 ϩ1ϩk 2 K p2 Ϫk 2 K i2 L 2 , 1

h 2 ϭT 1 ͑ 1ϩk 2 K p2 Ϫk 2 K i2 L 2 ͒ ϩT 2 Ϫk 2 K p2 L 2 , a0
K i1 ϭ , ͑28͒
b1
h 3 ϭT 1 ͑ T 2 Ϫk 2 K p2 L 2 ͒ .
a 2 b 2 Ϫa 1 b 1 b 2 ϩa 0 b 2
1 2
As indicated in Refs. ͓11,12͔, the Pade coefficients
´ K d1 ϭ , ͑29͒
and the Markov parameters characterize, respec- b3
1
tively, the low- and high-frequency responses of a
b2
system, or the responses of the steady state and T n1 ϭ . ͑30͒
transition state. Pϭ5 and M ϭ0 are selected in b1
order to get a good system response approxima-
tion in a steady state ͓11͔. Pade coefficients of
´ 5. Comparison study
H ( s ) are estimated as
Two examples are presented here to illustrate
c 0 ϭ1, the effectiveness of the proposed tuning method
for cascade control systems. In order to show the
c 1 ϭ ͑ d 1 Ϫe 1 c 0 ͒ /e 0 ,
accuracy of the proposed identification method in
c 2 ϭ ͑ d 2 Ϫe 1 c 1 Ϫe 2 c 0 ͒ /e 0 , a noisy environment, the noise-to-signal ratio
͑NSR͒, defined as ͓15͔ NSRϭmean͓abs͑noise͔͒/
c 3 ϭ ͑ d 3 Ϫe 1 c 2 Ϫe 2 c 1 Ϫe 3 c 0 ͒ /e 0 , mean͓abs͑signal͔͒ is introduced. In this paper, the
proposed identification method is applied to both
c 4 ϭ ͑ d 4 Ϫe 1 c 3 Ϫe 2 c 2 Ϫe 3 c 1 Ϫe 4 c 0 ͒ /e 0 . processes with noise level 10% NSR.
Example 1. Consider a cascade control system
͑3͒ The PID controller with pϭmϭ2 ͓11͔ is discussed by Hang ͓4͔ and Tan ͓5͔ with plant mod-
given as els of
K i1 K d1 s e Ϫs e Ϫ␣s
G c1 ͑ s ͒ ϭK p1 ϩ ϩ G p1 ͑ s ͒ ϭ , G p2 ͑ s ͒ ϭ ,
s 1ϩT n1 s ͑ 1ϩs ͒ 2 1ϩ ␣ s
a 0 ϩa 1 sϩa 2 s 2 where ␣ϭ0.1. After the relay feedback test, the
ϭ , ͑25͒
b 0 ϩb 1 sϩb 2 s 2 parameters for the inner loop process model is ob-
tained as
where b 0 ϭ0 and a 0 , a 1 , a 2 , b 1 , b 2 can be com-
puted by solving the following linear matrix equa- e Ϫ0.115s

ͫ
tions: G p2 ͑ s ͒ ϭ0.9563
1ϩ0.0947s

΅
g 0c 1 0 0 h0 0
and the outer loop process model is estimated as
g 0 c 2 ϩg 2 c 0 g 0c 1 0 h 0 c 1 ϩh 1 h 0c 0
g 0 c 3 ϩg 1 c 2 g 0c 2 g 0c 1 h 0 c 2 ϩh 1 c 1 h 0 c 1 e Ϫ1.63s
ϩg 2 c 1 ϩg 1 c 1 ϩh 2 ϩh 1 G p1 ͑ s ͒ ϭ0.965 .
1ϩ1.908s
g 0 c 4 ϩg 1 c 3 g 0 c 3 ϩg 1 c 2 g 0 c 2 h 0 c 3 ϩh 1 c 2 h 0 c 2
ϩg 2 c 2 ϩg 3 c 1 ϩg 2 c 1 ϩg 1 c 1 ϩh 2 c 1 ϩh 3 ϩh 1 c 1 ϩh 2 The parameters calculated for the inner loop PI

ͫ ͬͫͬ
0 0 g3 0 h3 controller G c2 using the Chien-Hrones-Reswick
͑CHR͒ tuning rule ͑20% overshoot͒ are obtained
a0 0
as K p2 ϭ0.603, K i2 ϭ2.277. The overall system
a1 0 reference model of the cascade control system is
ϫ a2 ϭ 0 . ͑26͒ chosen to be in the form of Eq. ͑20͒ with ␨ϭ0.8
b1 0 and ␻ u ϭ0.6. The outer loop controller’s param-
b2 1 eters are obtained as K p1 ϭ0.6592, K i1 ϭ0.3536,


Fig. 4. Tuning procedure and step response.

K d1 ϭ0.2886, and T n1 ϭ1.4392. The proposed the inner loop are obtained as K p2 ϭ2.895, K i2
auto-tuning procedure and step response in the ϭ0.147. The overall system reference model of
noisy environment are shown in Fig. 4. The result the cascade control system is chosen to be in the
is also compared to the methods proposed by form of Eq. ͑20͒ with ␨ϭ0.707 and ␻ u ϭ0.03, the
Hang and Tan. Fig. 5 shows the closed-loop per- outer loop controller’s parameters are obtained as
formance from tϭ0 s to tϭ80 s; a large step load K p1 ϭϪ4.9225, K i1 ϭϪ0.1105, K d1 ϭϪ51.1205,
disturbance seeps into the process for all cases at and T n1 ϭ6.4596. The control performance com-
tϭ50 s. parison study is carried out from tϭ0 s to
Example 2. Consider a process model of the tϭ2000 s, a large step load disturbance is added
packed-bed reactor provided by Ref. ͓16͔. The into the process at time tϭ1000 s, as shown in
goal is to tightly control the exit concentration Fig. 6.
temperature, and the most significant disturbance From the examples, the improved performance
is the heating medium temperature. The inner and of the proposed tuning method is clearly evident.
outer loop process models are given by
e Ϫ20s 6. Conclusion
G p1 ͑ s ͒ ϭϪ0.19 ,
1ϩ50s
This paper developed a novel auto-tuning
e Ϫ8s method for the cascade control system. As the in-
G p2 ͑ s ͒ ϭ0.57 ,
1ϩ20s ner loop process acts much faster than the outer
loop in the cascade control system, both inner loop
respectively. To reduce the disturbance, a cascade and outer loop process model parameters can be
control strategy is adopted. Using the relay feed- identified using one relay feedback test by utiliz-
back test, the parameters for the inner and outer ing this physical property under the proposed
process models from the experiment are estimated structure. Consequently, well-established model
as based PI tuning rules can be applied to tune the
e Ϫ22.6s inner loop, and a model matching the PID control-
G p1 ͑ s ͒ ϭϪ0.192 , ler design method was proposed to tune the outer
1ϩ47.5s
loop. Finally, two examples were given to show
e Ϫ8.5s the effectiveness of the proposed method. The
G p2 ͑ s ͒ ϭ0.558 . method is very straightforward and has been inte-
1ϩ19.7s
grated into an existing auto-tuning system. It is
The PI controller using the Chien-Hrones- now being tested in a centralized HVAC system
Reswick ͑CHR͒ tuning rule ͑20% overshoot͒ for and the field results will be reported soon.


Fig. 5. Performance comparison.

Appendix and a controller’s transfer function model

Suppose we have a process model a 0 ϩa 1 sϩ¯ϩa p s p
C͑ s ͒ϭ . ͑A2͒
b 0 ϩb 1 sϩ¯ϩb m s m
g 0 ϩg 1 sϩ¯ϩg q s q
G͑ s ͒ϭ , ͑A1͒
h 0 ϩh 1 sϩ¯ϩh n s n It is desired that C ( s ) be obtained in such a way

Fig. 6. Performances of the proposed method ͑solid line͒ and of Ref. ͓16͔ ͑dashed line͒.


´
that the overall system matches a set of Pade co- i
efficients and Markov parameters of a given a ref- x i ϭ ͚ a j g iϪ j , iϭ0,1, . . . ,qϩp, ͑A9͒
jϭ0
erence model
i
d 0 ϩd 1 sϩ¯ϩd r s r
H͑ s ͒ϭ , ͑A3͒ y i ϭ ͚ b j h iϪ j , iϭ0,1, . . . ,nϩm. ͑A10͒
e 0 ϩe 1 sϩ¯ϩe r s r jϭ0

and a simple controller C ( s ) can be found such Using the constraint PϩM ϭpϩmϩ1, where P P
that ´
is the number of Pade coefficients, M is the num-
ber of Markov parameters, p is the numerator’s
C͑ j ␻ ͒G͑ j ␻ ͒ order of C ( s ) , and m is the denominator’s order of
ХH ͑ j ␻ ͒ . ͑A4͒
1ϩC ͑ j ␻ ͒ G ͑ j ␻ ͒ C ( s ) . The parameters a, b i of C ( s ) can be ob-
tained uniquely by solving the set of linear Eqs.
The model matching of time moments and Mar- ͑A5͒–͑A10͒.
kov parameters is very effective in model reduc-
tion for obtaining approximate models, because
´
the Pade coefficients and the Markov parameters References
characterize, respectively, the low- and high- ͓1͔ Astrom, K. J. and Hagglund, T., PID Controller:
´
frequency responses of a system. The Pade coeffi- Theory, Design and Tuning. Instrument Society of
cients of H ( s ) are defined as America, Research Triangle Park, NC, 1995.
͓2͔ Bi, Q., Cai, Wenjian, Lee, E. L., Wang, Qingguo,
c 0 ϭd 0 /e 0 , Hang, C. C., and Zhang, Y., Robust identification of
first order plus dead-time model from step response.

ͫ ͬͲ
k Control Eng. Pract. 1, 71–77 ͑1999͒.
͓3͔ Li, M. X., Bruijn, P. M., and Verbruggen, H. B., Tun-
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͓15͔ Haykin, S., An Introduction to Analog & Digital Com- pany, U. S. A.; Research Scientist at National University of Singapore;
munication. Wiley, New York, 1989. Principal Engineer and R&D Manager at Supersymmetry Services Pte
͓16͔ Marlin, Thomas E., Process Control: Designing Pro- Ltd, Singapore; Senior Research Fellow at Environmental Technology
Institute, Singapore, respectively. He is currently serving as Associate
cesses and Control Systems for Dynamic Perfor- Professor at Nayang Technological University, Singapore. Dr. Cai’s
mance. 2nd ed., McGraw-Hill, New York, 2000. current research interests include multivariable control and HVAC sys-
tem optimization.

Sihai Song was born in 1974,
Zhejiang, P. R. China. He Ya-Gang Wang was born in
graduated from Zhejiang Uni- 1967, Shanxi, P. R. China. He
versity with a bachelor degree received his B.Eng., M.Eng.,
in electrical engineering in and Ph.D. from Department of
1997, and a second bachelor Automation, China University
degree of international com- of Mining and Technology,
modities inspection in 1998. In Taiyuan University of Technol-
2001, he started his postgradu- ogy and Shanghai Jiao Tong
ation studies at Nanyang Tech- University, P. R. China, in
nological University, Sin- 1988, 1991, and 2000, respec-
gapore. He is interested in PID tively. After graduation, he
auto-tuning and computer- worked as a lecturer in Taiyuan
aided control system design. University of Technology, P. R.
China, and a Postdoctoral Re-
search Fellow in Nanyang Technological University, Singapore. He is
interested in process control and instrumentation, PID auto-tuning, and
Wenjian Cai received his
multivariable control.
B.Eng., M.Eng., and Ph.D.
from Department of Precision
Instrumentation Engineering,
Department of Control Engi-
neering, Harbin Institute of
Technology, P. R. China, and
Department of Electrical Engi-
neering, Oakland University,
U. S. A., in 1980, 1983, and
1992, respectively. After
graduation, he worked as a
Postdoctoral Research Fellow
at the Center for Advanced Ro-
botics, Oakland University, U. S. A.; Engineer at CEC Controls Com-

Auto tuning of cascade control systems

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