1. ISA Transactions 40 (2001) 133±138
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Analytical design and analysis of mismatched Smith predictor
Weidong Zhang *, Xiaoming Xu
Department of Automation, Shanghai Jiaotong University, Shanghai 200030, PR China
Abstract
In this paper, an analytical design method is developed for the mismatched Smith predictor on the basis of the
internal model control (IMC) method. Design formulas are given and design procedure is signi®cantly simpli®ed. One
important merit of the proposed method is that the response of the closed loop system can be easily adjusted. The
relation between the controller parameter and the system response is monotonous. In addition, necessary and sucient
condition for the robust stability of mismatched Smith predictor is also given. It is shown that the proposed method
can provide good performance for perfectly matched system and a better response for mismatched system. Several
numerical examples are given to illustrate the proposed method. # 2001 Elsevier Science Ltd. All rights reserved.
Keywords: Linear system; Time delay; Smith predictor; Internal model control; Robustness
1. Introduction system. For example, [4] systematically studied the
robust performance of the Smith predictor within
Synthesis and tuning of control structures for the IMC structure using several design methods;
single input±single output (SISO) systems com- [5] presented a simple criterion for the tuning of
prises the bulk of process control problems. In the Smith predictor when the plant time delay is not
past, hardware considerations dictated the use of precisely known; [6] discussed robust PID tuning
PID controllers [1], but through the use of com- problem for the Smith predictor in the presence of
puters, controllers have now advanced to the stage model uncertainty; and [7] developed a two degree-
where virtually any conceivable control policy can of-freedom robust Smith predictor. Recently, Wang
be implemented. Due to the progress, the design of et al. [8] proposed a new scheme. It employs a
Smith predictor [2] is widely studied in the past deliberately mismatched model to enhance perfor-
decades. Though the Smith predictor o€ers mance over a perfectly matched system while
potential improvement in the closed loop perfor- using a simple primary controller.
mance over conventional controllers, it su€ers Based on the IMC theory, this paper will
from a sensitivity problem. In the face of inevi- develop an analytical design procedure for the
table mismatches between the model and actual mismatched Smith predictor proposed by [8]. For
process, the closed loop performance can be very controller design, simplicity, as well as optimality,
poor [3]. A lot of work has been done in relation is important. Compared with the empirical
to the robustness issues of the Smith predictor method of [8] the analytical method can give
design formulas and thus simpli®ed the design
* Corresponding author. Tel.: +86-21-6293-3329; fax: +86- procedure. On the other hand, the result of IMC is
21-6293-2138. suboptimal. This provides possibility of improving
E-mail address: wdzhang@mail.sjtu.edu.cn (W. Zhang). the closed loop performance.
0019-0578/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.
PII: S0019-0578(00)00045-8

2. 134 W. Zhang, X. Xu / ISA Transactions 40 (2001) 133±138
The paper is organized as follows. In the more strict proof was given by [10]. De®ne the
following section the assessment of the achievable IMC controller
performance of classical Smith predictor is
formulated. In the third section an equivalent R…s†
Q…s† ˆ …3†
representation of the mismatched Smith predictor 1 ‡ Gmo …s†R…s†
is given and the design formulas are derived ana-
lytically. The robustness is also discussed. Several Then the two schemes are equivalent to each
examples are provided in the fourth section, in other. The closed loop transfer function can be
which the new controller is compared with that of written as
[8]. Finally, conclusions are given in Section 5.
T…s† ˆ Gm …s†Q…s† …4†
2. Assessment of achievable performance The primary objective of any feedback control
scheme is to make the di€erence between the con-
The structure of classical Smith predictor is trolled outputs y(s) and desired setpoint r(s) as
shown in Fig. 1, where R(s) is the controller, G(s) small as possible. What is meant by ``small'' can
is the actual plant, Gm (s) is the model, and Gmo be de®ned in terms of various performance speci-
(s) is the delay free part of Gm(s). The closed ®cations for the closed loop system. In IMC
loop transfer function between the setpoint r(s) method, the performance objective is de®ned as
„
and the output y(s) is H2 optimal, i.e. min …r…t† À y…t††2 dt. Assume that
the plant is stable. Factor the model
R…s†G…s†
T…s† ˆ …1†
1 ‡ R…s†…Gmo …s† À Gm …s† ‡ G…s†† Gmo …s† ˆ G‡ …s†GÀ …s†
mo mo …5†
In the case of perfect modeling, i.e. such that G‡ …s† contains all right half plane zeros
mo
Gm …s† ˆ G…s†, the closed loop transfer function is and is all pass. The suboptimal IMC controller
given by can be expressed as
R…s†Gm …s† Q…s† ˆ J…s†=GÀ …s†
mo …6†
T…s† ˆ …2†
1 ‡ R…s†Gmo …s†
where J(s) is a user-speci®ed ®lter. Let m be the
This implies that the characteristic equation is free relative degree of GÀ …s†. The ®lter with asympto-
mo
of the time delay so that the primary controller tic tracking ability is usually selected as
R(s) can be designed with respect to Gm(s). The
achievable performance can thus be greatly J…s† ˆ 1=…ls ‡ 1†m …7†
improved over a conventional system without
delay compensation.
The Smith predictor can be related to IMC Here l is a positive real constant. The controller
structure, which has been discussed by [9] and a of Smith predictor can then be obtained through
Q…s†
R…s† ˆ …8†
1 À Gmo Q…s†
By tuning l, one can adjust the nominal perfor-
mance and robust performance monotonously. In
the case of perfect match, the nominal perfor-
mance can arbitrarily approach optimality by
Fig. 1. Structure of Smith predictor. decreasing l.

3. W. Zhang, X. Xu / ISA Transactions 40 (2001) 133±138 135
3. Analytical design of modi®ed Smith predictor which is a PID controller for the default value
of n
Let the delay free part of the model be
…o s ‡ 1†2
Ko R…s† ˆ …15†
Gmo …s† ˆ ; n ˆ 1; 2; 3 …9† l2 s 2‡ …2l À o †s
…o s ‡ 1†n
Furthermore, apart from the introduced certain
where n is speci®ed by the user. The default value uncertainty there always exists other uncertainties
of n is suggested to be 2 by [8]. Factor it into in practice. Assume that
Gmo …s† ˆ Gmo1 …s†Gmo2 …s†; where

6. G…s† À Gm …s†

13. K0

14. G …s†

15. ˆ l…j!† 4l…!†
Gmo1 …s† ˆ …10† m
o s ‡ 1
where l…!† is the bound on the multiplicative
1 uncertainty l…j!† (it consists of certain uncertainty
Gmo2 …s† ˆ …11†
…o s ‡ 1†nÀ1 and other uncertainties). The nominal transfer
function of the closed loop system is
An equivalent of the mismatched Smith predictor
proposed by [8] is show in Fig. 2. Gm …s†R…s†
T…s† ˆ …16†
Though a certain uncertainty is introduced by 1 ‡ Gmo1 …s†R…s†
the simpli®ed model Gm (s) in the scheme, we can
also design it by IMC method. Regard Gm (s) as Then the closed loop system is robustly stable if
the nominal plant. The IMC controller is and only if [9]

18. Gm …j!†R…j!†

19. 1
…o s ‡ 1†n

21. Q…s† ˆ …12†

22. 1 ‡ G …j!†R…j!†

23. 4 ; V! …17†
…ls ‡ 1†n mo1 l…!†
This implies that there is a low bound larger than
On the other hand, zero for l in a uncertainty system.
R…s†
Q…s† ˆ …13†
1 ‡ Gmo1 …s†R…s† 4. Examples
Therefore, the controller is derived analytically Three typical plants from [8] will be used in this
as follows section to illustrate the proposed method. The
perfectly matched system is studied in example 1.
…o s ‡ 1†n For comparison, the mismatched cases are studied
R…s† ˆ …14†
…ls ‡ 1†n À …o s ‡ 1†nÀ1 in example 2 and 3 for both the proposed method
and Wang's method. Since l has a monotonous
relation with the response of the closed loop sys-
tem, one can easily get the required response.
Example 1 Ð Consider the following process [8]
eÀ4s
G…s† ˆ
…s ‡ 1†5
Take l=0.1, l=0.5 and l=1, respectively, for the
Fig. 2. Structure of modi®ed Smith predictor. proposed Smith predictor. When the system is

24. 136 W. Zhang, X. Xu / ISA Transactions 40 (2001) 133±138
perfectly matched, we have 1
Gm …s† ˆ eÀ5:39s
…2:43s ‡ 0:995†2
…s ‡ 1†5
Q…s† ˆ
…ls ‡ 1†5 and Wang's controller [8] is
6:25s ‡ 2:56
…s ‡ 1†5 R…s† ˆ
R…s† ˆ s…s ‡ 2:29†
…ls ‡ 1†5 À …s ‡ 1†4
Take l=0.6667 and l=1 for the proposed Smith
A unit step setpoint change is introduced at predictor. We have
t=0, and a 10% step disturbance is introduced at
the plant input at t=80. It is seen that the …2:43s ‡ 0:995†2
Q…s† ˆ
proposed method can provide arbitrarily good …ls ‡ 1†2
performance by decreasing l (Fig. 3). As a matter
of fact, for perfectly matched case, if l tends to be …2:43s ‡ 0:995†2
zero, the closed loop transfer function has an in®- R…s† ˆ
l2 s 2
‡ …2l À 2:43†s
nity bandwidth and thus the performance tend to
be optimal [9]. When there exists uncertainty, the
sucient and necessary condition (17) shows that A unit step setpoint change is introduced at t=0,
the smaller l is the worse the robustness is. In this and a 10% step disturbance is introduced at the
case the designer should increase l from a very plant input at t=60. The closed loop responses of
small positive real monotonously until a satis®ed the two systems are shown in Fig. 4. It is seen that
trade-o€ between performance and robustness is for l=0.6667 the proposed method provides a
obtained. better response than that of [8] and for l=1 a little
Example 2 Ð Suppose that the process is better response is obtained.
given by Example 3 Ð Consider a non-minimum phase
process with time delay [8]
1
G…s† ˆ
…s ‡ 1†10 Às ‡ 1 À2s
G…s† ˆ e
…s ‡ 1†5
The simpli®ed process model is computed as [8]
Fig. 3. Performance of perfectly matched system (Solid line: Fig. 4. Multiple lag process (Solid line: l=1, dotted line:
l=0.1, dotted line: l=0.5, dashed line l=1). l=0.6667, dashed line Wang's method).

25. W. Zhang, X. Xu / ISA Transactions 40 (2001) 133±138 137
The mismatched model is given by [8] and robust performance by l and thus make a tra-
deo€ between the two con¯icting objectives.
1
Gm …s† ˆ eÀ5:07s
…1:46s ‡ 0:999†2
5. Conclusions
and Wang's controller [8] is
Several problems are discussed in this paper:
5:85s ‡ 4 1. It has been shown that for perfectly matched
R…s† ˆ model, the performance of the closed loop system
s…s ‡ 2:83†
can arbitrarily approach the optimal. Thus, it is
not necessary to utilize the mismatch for improv-
Take l=0.6667 and l=1 for the proposed Smith ing the system performance.
predictor. Therefore, 2. Practical plants are of high order and models
are usually low order. To improve the system per-
…1:46s ‡ 0:999†2 formance, [8] presented a mismatched Smith pre-
Q…s† ˆ
…ls ‡ 1†2 dictor structure. It is found that the structure can
be related to the IMC. A new design procedure is
…1:46s ‡ 0:999†2 then developed in the framework of IMC theory
R…s† ˆ and analytical results are provided. Simulations
l2 s 2
‡ …2l À 1:46†s
show that improved response is obtained.
3. Necessary and sucient condition for robust
A unit step setpoint change is introduced at t=0, stability is provided for the mismatched Smith pre-
and a 10% step disturbance is introduced at the dictor system. If the exact uncertainty pro®le is
plant input at t=40. The closed loop responses for known, the exact controller parameter can be cal-
both cases are presented in Fig. 5. Again a per- culated. In the context of process control, one can
formance improvement is achieved for l=0.6667 adjust the performance and robustness of the closed
with the proposed method and a similar response loop system in such a way: increase the controller
is obtained for l=1. parameter from a small positive real monotonously
The usefulness of the method is not only in per- until a satisfactory response is obtained.
formance enhancement, but also lie in that the
designer can easily adjust the nominal performance
Acknowledgements
Project supported by National Natural Science
Foundation of China (69804007) and Science and
Technology Phosphor Program of Shanghai
(99QD14012).
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Weidong Zhang was born in Xiaoming Xu was born in
Daqing, People's Republic of Shanghai, People's Republic of
China in 1967. He received the China in 1957. He received the
BS, MS and PhD degree from BS degree from Huazhong
Zhejiang University in 1990, University of Science and
1993 and 1996, respectively. He Technology in 1982, and MS
worked in National Key and Ph.D degree from Shang-
Laboratory of Industrial Con- hai Jiaotong University in 1984
trol Technology as a post-doc- and 1987, respectively. From
toral research fellow before 1988 to 1990 he worked in
joining Shanghai Jiaotong Uni- Germany as an Alexander von
versity in 1998 as an associate Homboldt research fellow. He
professor in Automatic Con- joined Shanghai Jiaotong Uni-
trol. Since 1999 he has been the versity in 1990 as a professor.
youngest professor at the Department of Automation, Shang- He was the director of Department of Automation in 1993 and
hai Jiaotong University. His current research interests include the deputy president of Electronics Information College,
process control, robust control, ®eld bus, multi-agent, and Shanghai Jiaotong University in 1996. Since 1997 he has been
digital ®ltering. the deputy principal of Shanghai Jiaotong University. His cur-
rent research interests include control theory, arti®cial intelli-
gence, computer network, and digital signal processing.