1. ISA
TRANSACTIONS®
ISA Transactions 43 ͑2004͒ 33–47
Digital redesign of analog Smith predictor for systems with input
time delays
Alex C. Dunn,a Leang-San Shieh,a,* Shu-Mei Guob
a
Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77204-4005, USA
b
Department of Computer Science and Information Engineering, National Cheng-Kung University, Tainan, Taiwan 701, R. O. C.
͑Received 4 November 2002; accepted 9 June 2003͒
Abstract
This paper presents a new methodology for digitally redesigning an existing analog Smith predictor control system,
such that the cascaded analog controller with input delay can be implemented with a digital controller. A traditional
analog Smith predictor system is reformulated into an augmented system, which is then digitally redesigned using the
predicted intersampling states. The paper extends the prediction-based digital redesign method from a delay free
feedback system to an input time-delay cascaded system. A tuning parameter v is optimally determined online such that
in any sampling period, the output response error between the original analogously controlled time-delay system and
the digitally controlled sampled-data time-delay system is significantly reduced. The proposed method gives very good
performance in dealing with systems with delays in excess of several integer sampling periods and shows good
robustness to sampling period selection. © 2004 ISA—The Instrumentation, Systems, and Automation Society.
Keywords: Smith predictor controller; Digital redesign; Optimal digital observer; Input-delay system; Sampled-data system
1. Introduction continuous-time setting, the mathematical models
of time-delay systems are complicated by these
A large class of industrial processes continues to models being infinite dimensional. Discretizing
be primarily continuous-time or sampled-data sys- such systems becomes very attractive, as the re-
tems that are plagued by considerable time delays sulting models now become finite dimensional and
in some cases. These time delays, resulting mainly more amenable to mathematical manipulation.
from transporting material or energy, can lead to While much has appeared in the literature regard-
significant degradation of closed-loop control sys- ing the Smith predictor, most of the research has
tem performance, particularly in cases where the been aimed at addressing some of the more con-
delay is significantly longer than the process troversial issues alluded to previously, while other
dominant time constant ͓1–3͔. efforts have compared the performance of various
Despite some long-standing controversies relat- forms of predictive PI control schemes ͓7,9͔ with
ing to issues of sensitivity and robustness ͓4 –12͔, the Smith predictor. Based on a Newton backward
the Smith predictor ͓13͔ continues to be the most extrapolation formula and the Chebyshev quadra-
quoted method to solve time-delay problems. In a ture formula, Guo et al. ͓14͔ have proposed a con-
troller discretization method that was extended to
*Corresponding author. Tel: ϩ1-713-743-4439; fax: ϩ1- the Smith predictor, for the case where the overall
713-743-4444. E-mail address: Lshieh@uh.edu time delay was either less than or almost equal to
0019-0578/2004/$ - see front matter © 2004 ISA—The Instrumentation, Systems, and Automation Society.

2. 34 A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47
Fig. 1. Continuous-time cascaded delay system.
the discrete-time sampling period. It should also discrete sample points and one additional point be-
be pointed out that while the discretization and tween consecutive discrete sample points.
digital control of an analog system with input time The material presented in this paper is organized
delay can be carried out using the modified z as follows. In Section 2, we first formulate the
transform, the existing methods are mostly devel- standard Smith predictor controller into an aug-
oped for the single-input–single-output analog mented system that is subsequently used for digi-
system in the frequency domain. The present tal redesign. Section 3 presents the development of
method uses a different approach that is capable of the intersampling states for the long and short
handling much longer time delays relative to the input-delay systems, with the short delay case
sampling period and is easily extendable to the shown as a special case of long delay. In Section
multi-input–multi-output analog system with in- 4, the augmented system is digitally redesigned
put delay in the time domain. detailing the necessary relationships for imple-
With the very large installed base of analog con- mentation. For cases where the system states are
trol systems, including Smith predictors, digital not available for measurement, an optimal discrete
redesign is a very attractive approach for design- observer for input-delay systems is developed in
ing controllers for sampled-data systems, that Section 5. To demonstrate the effectiveness of the
avoids the problems of direct digital control, yet proposed scheme, an illustrative example with
enjoys the benefits of flexibility, reduced cost, ease simulation results for short and long delays is pre-
of implementation of complex designs, etc., avail- sented in Section 6. Final observations and conclu-
able in today’s digital systems. While numerous sions are presented in Section 7.
approaches have been proposed for digital rede-
sign, the prediction-based method ͓15͔ that uses an
optimally determined intersample parameter is 2. Augmented reformulated system
very attractive and is used in this development. In
the paper, the prediction-based digital redesign
Consider the unity feedback time-delay system
method that was developed for a delay-free state
shown in Fig. 1 below. According to Smith’s for-
feedback system ͓15͔ is extended to discretize a
mulation, we can redraw Fig. 1 as shown in Fig. 2
time-delay cascaded analog Smith predictor for an
below for the purpose of designing a controller to
input time-delay plant.
be used in the Smith predictor loop in Fig. 3. The
In this paper, a classical analog Smith predictor
original system in Fig. 1 can be redrawn in a
system is reformulated into an augmented system,
Smith predictor formulation as shown in Fig. 3
which is then digitally redesigned using the
below.
prediction-based digital redesign method ͓15͔. In
Consider the state-space representation for the
the prediction-based digital redesign scheme, a
systems in Fig. 3 to be the following:
tuning parameter v is optimally determined such
that the states in the analog system agree very G 1 ͑ s ͒ e ϪsT d : x c1 ͑ t ͒ ϭA 1 x c1 ͑ t ͒ ϩB 1 u c1 ͑ tϪT d ͒ ,
˙
closely with the states of the discrete system at ͑1͒
Fig. 2. System used for Smith controller design.

3. A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 35
Fig. 3. Smith predictor formulation.
y c1 ͑ t ͒ ϭC 1 x c1 ͑ t ͒ , ͑2͒ x ec ͑ t ͒ ϭA e x ec ͑ t ͒ ϩB e u c1 ͑ tϪT d ͒ ϩF ec r ͑ t ͒ ,
˙
͑9͒
G 1 ͑ s ͓͒ 1Ϫe ϪsT d ͔ :
u c1 ͑ t ͒ ϭϪK ec x ec ͑ t ͒ ϩE ec r ͑ t ͒ , ͑10͒
x c3 ͑ t ͒ ϭA 1 x c3 ͑ t ͒ ϩB 1 ͓ u c1 ͑ t ͒
˙
ͫ ͬ
where
Ϫu c1 ͑ tϪT d ͔͒ , ͑3͒
A1 0 0
y c3 ͑ t ͒ ϭC 1 x c3 ͑ t ͒ , ͑4͒ ϪB 2 C 1 A2 ϪB 2 C 1
A eϭ ,
G 2͑ s ͒ : x c2 ͑ t ͒ ϭA 2 x c2 ͑ t ͒ ϩB 2 u c2 ͑ t ͒ ,
˙ ͑5͒ ϪB 1 D 2 C 1 B 1 C 2 A 1 ϪB 1 D 2 C 1
ͫ ͬ ͫ ͬ
y c2 ͑ t ͒ ϭC 2 x c2 ͑ t ͒ ϩD 2 u c2 ͑ t ͒ ϭu c1 ͑ t ͒ , ͑6͒
B1 0
where B eϭ 0 , F ec ϭ B 2E c , ͑11͒
ϪB 1 B 1D 2E c
x c1 ͑ t ͒ ෈R , n1
u c1 ͑ t ͒ ෈R , m1
K ec ϭ ͓ K c1 K c2 K c3 ͔
y c1 ͑ t ͒ ෈Rp1 , x c2 ͑ t ͒ ෈Rn2 ,
ϭ ͓ D 2C 1 ϪC 2 D 2C 1͔ , ͑12͒
u c2 ͑ t ͒ ෈Rp1 , y c2 ͑ t ͒ ෈Rm1 ,
E ec ϭD 2 E c . ͑13͒
x c3 ͑ t ͒ ෈Rn1 , u c3 ͑ t ͒ ෈Rm1 , y c3 ͑ t ͒ ෈Rp1 ,
The system in Eq. ͑9͒ is an input-delay system for
and matrices ( A 1 , B 1 , C 1 , A 2 , B 2 , C 2 , and D 2 )
which an exact method exists for developing a dis-
are of appropriate dimensions with T d the overall
crete model ͓16͔. The input in Eq. ͑10͒ can be
time delay.
viewed as an available state-feedback analog con-
From Fig. 3 and Eq. ͑6͒, we have
trol law, for which the corresponding digital con-
u c2 ͑ t ͒ ϭe ͑ t ͒ Ϫy c3 ͑ t ͒ trol law is developed in the following sections.
ϭE c r ͑ t ͒ Ϫy c1 ͑ t ͒ Ϫy c3 ͑ t ͒
ϭϪC 1 x c1 ͑ t ͒ ϪC 1 x c3 ͑ t ͒ ϩE c r ͑ t ͒ , ͑7͒
3. Evaluation of the predicted intersampling
u c1 ͑ t ͒ ϭC 2 x c2 ͑ t ͒ ϩD 2 u c2 ͑ t ͒ states
ϭC 2 x c2 ͑ t ͒ ϪD 2 C 1 x c1 ͑ t ͒
In order to utilize the prediction-based digital
ϪD 2 C 1 x c3 ͑ t ͒ ϩD 2 E c r ͑ t ͒ , ͑8͒ redesign scheme ͓15͔, also shown in the Appen-
dix, to digitally redesign the control law in Eq.
where ͑10͒, it is necessary to evaluate the predicted inter-
E c ෈Rp1ϫp1 , r ͑ t ͒ ෈Rp1 . sampling states of the input-delay system as
shown in the following sections. Two versions of
From Eqs. ͑1͒–͑8͒, the augmented system can be input time delays shown in Figs. 4 and 5 are con-
written as sidered in this section. These are, namely, the long

4. 36 A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47
Fig. 4. Input-delay system for dϭ1 and v Ͼ ␥ .
and short time delays, with the short time delay sponses of the discrete-time and continuous-time
shown to be a special case of the long input time- systems is minimized. While Figs. 4 and 5 only
delay case. depict the case dϭ1 for simplicity of illustration,
the following derivations are for a general delay
3.1. Long and short input time-delay systems d.
Consider a general input time-delay system with
total integer delay d, then the following condi-
Consider a sampled-data system with long input tions generally hold:
time delay as shown in Fig. 4, in which the input
delay is greater than the sample period of the dis- dу0, 0р ␥ Ͻ1, 0р v р1,
crete system. In Fig. 4, ␥ is the fractional delay Ti
beyond the integer multiples of delay in the over- ␥ϭ ⇒T i ϭ ␥ T,
all delay time. The tuning parameter v is an inter- T
sample parameter used in the prediction-based T d ϭdTϩT i ϭdTϩ ␥ T,
digital redesign scheme, that is determined online,
such that the total error between the output re- t v ϭkTϩ v T. ͑14͒
Fig. 5. Input-delay system for dϭ1, and v Ͻ ␥ .

5. A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 37
Depending on the value of the tuning parameter v , where
which is optimally determined via digital redesign,
two cases need to be considered. G ( v ) ϭe AT v ϭ ͑ e AT ͒ v ,
Case 1
In this first case, the optimal parameter v is such
H ( v ) ϭ ͓ G ( v Ϫ ␥ ) ϪI ͔ A Ϫ1 B,
that v TуT i ⇒ v у ␥ . This is shown in Fig. 4. Con- d
sider a general continuous-time linear system with
input time delay, given by H (dϩ1) ϭG ( v Ϫ ␥ ) ͓ G ( ␥ ) ϪI ͔ A Ϫ1 B.
(v)
x ͑ t ͒ ϭAx ͑ t ͒ ϩBu ͑ tϪT d ͒ .
˙ ͑15͒
Remark 1
Evaluating Eq. ͑15͒ at tϭt v ϭkTϩ v T, we get In cases where matrix A is singular, then a gen-
eral matrix W x ϭ ͓ G ( ␣ ) ϪI ͔ A Ϫ1 B can be evaluated
x ͑ t v ͒ ϭx ͑ kTϩ v T ͒ as
ϭe A v T x ͑ kT ͒ ϩ ͵kT
tv
e A(t v Ϫ␭) Bu ͑ ␭ϪT d ͒ d␭.
W xϭ ͚
ϱ
1
͑ A ␣ T ͒ jϪ1 B ␣ T.
jϭ1 j!
͑16͒
In order to evaluate the integral in Eq. ͑16͒, we Case 2
make the following variable substitution: The second case to be considered is for v Ͻ ␥ , as
shown in Fig. 5. The result for this case is easily
Let ␦ ϭ␭ϪdT⇒␭ϪdTϪT i ϭ ␦ ϪT i , derived from Eq. ͑18͒ and by noting that only the
when ␭ϭkT, ␦ ϭkTϪdTϭ ͑ kϪd ͒ T, ˆ
set of terms Q 1 exists in this case, with v instead
of ␥ in the integration limits. The corresponding
when ␭ϭt v ϭkTϩ v T, equation to Eq. ͑19͒ becomes
␦ ϭkTϩ v TϪdTϭ ͑ kϩ v Ϫd ͒ T. ͑17͒
x ͑ t v ͒ ϭx ͑ kTϩ v T ͒
Eq. ͑16͒ can then be written as
ϭG ( v ) x ͑ kT ͒ ϩH (dϩ1) u ͓͑ kϪdϪ1 ͒ T ͔ ,
(v)
x ͑ t v ͒ ϭx ͑ kTϩ v T ͒ ϭe A v T x ͑ kT ͒ ϩQ 1 ϩQ 2 ,
ˆ ˆ
͑20͒
͑18͒
where where
Q 1 ϭe A(kϩ v Ϫd)T
ˆ ͭ͵ (kϪdϩ ␥ )T
(kϪd)T
e ϪA ␦ Bu ͑ ␦ H (dϩ1) ϭ ͓ G ( v ) ϪI ͔ A Ϫ1 B.
(v)
ͮ
ϪT i ͒ d ␦ , 3.2. Short input time-delay system
ͭ͵
The short input time-delay system is a special
(kϪdϩ v )T
Q 2 ϭe A(kϩ v Ϫd)T
ˆ e ϪA ␦ Bu ͑ ␦ case of the long input time-delay system. Hence
(kϪdϩ ␥ )T the results for this case are easily derived from the
ͮ
previous results by setting dϭ0.
ϪT i ͒ d ␦ . Case 1
Here again, the optimal parameter v is such that
v у ␥ . Substituting dϭ0 in Eq. ͑19͒, we get
This evaluates to
x ͑ t v ͒ ϭx ͑ kTϩ v T ͒ x ͑ t v ͒ ϭx ͑ kTϩ v T ͒
ϭG ( v ) x ͑ kT ͒ ϩH ( v ) u ͓͑ kϪd ͒ T ͔
d ϭG ( v ) x ͑ kT ͒ ϩH ( v ) u ͓͑ kT ͔͒
0
ϩH (dϩ1) u ͓͑ kϪdϪ1 ͒ T ͔ ,
(v)
͑19͒ ϩH ( v ) u ͓͑ kϪ1 ͒ T ͔ ,
1 ͑21͒

6. 38 A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47
Fig. 6. Digitally redesigned Smith predictor for dϭ1.
where The objective now is to digitally redesign this
analog input using the prediction-based digital re-
H ( v ) ϭ ͓ G ( v Ϫ ␥ ) ϪI ͔ A Ϫ1 B,
0 design technique ͓15͔, such that the analog and
H ( v ) ϭG ( v Ϫ ␥ ) ͓ G ( ␥ ) ϪI ͔ A Ϫ1 B. discrete states match very closely even at inter-
1
sample points. For this development, we assume
Case 2 that the continuous-time controller u c1 ( t ) in Eq.
As stated previously, for this case, v Ͻ ␥ . Setting ͑23͒ is approximated by a piecewise-constant
dϭ0 in Eq. ͑20͒, we get discrete-time controller u d1 ( kT ) , yet to be deter-
mined. Also, the choice of the tuning parameter v ,
x ͑ t v ͒ ϭx ͑ kTϩ v T ͒
used in all the subsequent equations, is determined
ϭG ( v ) x ͑ kT ͒ ϩH ( v ) u ͓͑ kϪ1 ͒ T ͔ , ͑22͒ by minimizing the following performance index,
1
in which t f is the finite time of interest,
where
H ( v ) ϭ ͓ G ( v ) ϪI ͔ A Ϫ1 B.
͵ ͉y
1
tf
J͑ v ͒ϭ c1 ͑ t ͒ Ϫy d1 ͑ t ͒ ͉ dt, ͑25͒
4. Prediction-based digital redesign 0
The augmented equations for the Smith predic-
tor presented previously in Eqs. ͑9͒ and ͑10͒ are
where y c1 ( t ) and y d1 ( t ) are shown in Figs. 3 and
x ec ͑ t ͒ ϭA e x ec ͑ t ͒ ϩB e u c1 ͑ tϪT d ͒ ϩF ec r ͑ t ͒ ,
˙ 6, respectively. We again consider two cases de-
͑23͒ pending on the relative values of the parameters v
and ␥ as shown in Figs. 4 and 5.
u c1 ͑ t ͒ ϭϪK ec x ec ͑ t ͒ ϩE ec r ͑ t ͒ , ͑24͒
Case 1
where u c1 ( t ) in Eq. ͑24͒ is the control input before In this case, v у ␥ as shown in Fig. 4. From Eq.
the input time delay of the plant G 1 ( s ) and r ( t ) is ͑19͒, we can write the equivalent discrete expres-
a constant setpoint. sion for x ec ( t ) in Eq. ͑23͒ as

7. A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 39
x ec ͑ kTϩ v T ͒ ϭG ( v ) x ec ͑ kT ͒ ϩH ( v ) u c1 ͓͑ kϪd ͒ T ͔
e ed
which can be written as
ϩH e(dϩ1) u c1 ͓͑ kϪdϪ1 ͒ T ͔
(v) u d1 ͑ kT ͒ ϭϪK ed x ed ͑ kT ͒ ϪJ ed u d1 ͓͑ kϪd ͒ T ͔
ϪJ e(dϩ1) u d1 ͓͑ kϪdϪ1 ͒ T ͔
ϩ ͵ kT
tv
e A(t v Ϫ␭)
F ec r ͑ ␭ ͒ d␭, ͑26͒
ϩE ed r ͑ kT ͒ , ͑31͒
which evaluates to where
x ec ͑ kTϩ v T ͒ ϭG ( v ) x ec ͑ kT ͒ ϩH ( v ) u c1 ͓͑ kϪd ͒ T ͔
e ed
K ed ϭK ec G ( v ) ϭ ͓ K d1
e K d2 K d3 ͔ ,
ϩH e(dϩ1) u c1 ͓͑ kϪdϪ1 ͒ T ͔
(v)
J ed ϭK ec H ( v ) ,
ed
ϩE ( v ) r ͑ kT ͒ ,
e ͑27͒ J e(dϩ1) ϭK ec H e(dϩ1) ,
(v)
where E ed ϭ ͓ E ec ϪK ec E ( v ) ͔ .
e
G ( v ) ϭe A e v T ,
e For the special case when dϭ0, Eq. ͑31͒ becomes
H ( v ) ϭ ͓ G ( v Ϫ ␥ ) ϪI ͔ A Ϫ1 B e ,
ed e e
u d1 ͑ kT ͒ ϭϪK ed x ed ͑ kT ͒ ϪJ e1 u d1 ͓͑ kϪ1 ͒ T ͔
H e(dϩ1) ϭG ( v Ϫ ␥ ) ͓ G ( ␥ ) ϪI ͔ A Ϫ1 B e ,
(v) ϩE ed r ͑ kT ͒ , ͑32͒
e e e
where
E ( v ) ϭ ͓ G ( v ) ϪI ͔ A Ϫ1 F ec .
e e e
H ( v ) ϭ ͓ G ( v Ϫ ␥ ) ϪI ͔ A Ϫ1 B e ,
The discrete model of Eq. ͑23͒ is given ͓16͔ as e0 e e
x ed ͑ kTϩT ͒ ϭG e x ed ͑ kT ͒ ϩH ed u d1 ͓͑ kϪd ͒ T ͔ H ( v ) ϭG ( v Ϫ ␥ ) ͓ G ( ␥ ) ϪI ͔ A Ϫ1 B e ,
e1 e e e
ϩH e(dϩ1) u d1 ͓͑ kϪdϪ1 ͒ T ͔ K ed ϭ ͓ IϩK ec H ( v ) ͔ Ϫ1 K ec G ( v )
e0 e
ϩE e r ͑ kT ͒ , ͑28͒ ϭ ͓ K d1 K d2 K d3 ͔ ,
where J e1 ϭ ͓ IϩK ec H ( v ) ͔ Ϫ1 K ec H ( v ) ,
e0 e1
G e ϭe A e T , E ed ϭ ͓ IϩK ec H ( v ) ͔ Ϫ1 ͓ E ec ϪK ec E ( v ) ͔ .
e0 e
H ed ϭ ͓ G (1Ϫ ␥ ) ϪI ͔ A Ϫ1 B e ,
e e To illustrate the procedure and for use as part of
the subsequent simulation, let dϭ1. Then we get
H e(dϩ1) ϭ ͓ G e ϪG (1Ϫ ␥ ) ͔ A Ϫ1 B e ,
e e from Eqs. ͑28͒ and ͑31͒ that
E e ϭ ͓ G e ϪI ͔ A Ϫ1 F ec .
e
x ed ͑ kTϩT ͒ ϭG e x ed ͑ kT ͒ ϩH e1 u d1 ͓͑ kϪ1 ͒ T ͔
Using the prediction-based digital control law ϩH e2 u d1 ͓͑ kϪ2 ͒ T ͔ ϩE e r ͑ kT ͒ ,
͑A10͒ in the Appendix, we can substitute Eq. ͑27͒
͑33͒
into the following continuous-time control law,
which is equivalent to Eq. ͑24͒: where
u d1 ͑ kT ͒ ϭϪK ec x ec ͑ t v ͒ ϩE ec r ͑ t v ͒ . ͑29͒ G e ϭe A e T ,
As a result, we get H e1 ϭ ͓ G (1Ϫ ␥ ) ϪI ͔ A Ϫ1 B e ,
e e
u d1 ͑ kT ͒ ϭϪK ec ͕ G ( v ) x ed ͑ kT ͒ ϩH ( v ) u d1 ͓͑ k
e ed H e2 ϭ ͓ G e ϪG (1Ϫ ␥ ) ͔ A Ϫ1 B e ,
e e
Ϫd ͒ T ͔ ϩH e(dϩ1) u d1 ͓͑ kϪdϪ1 ͒ T ͔
(v)
E e ϭ ͓ G e ϪI ͔ A Ϫ1 F ec .
e
ϩE ( v ) r ͑ kT ͒ ͖ ϩE ec r ͑ kT ͒ ,
e ͑30͒ For the control law, we get

8. 40 A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47
u d1 ͑ kT ͒ ϭϪK ed x ed ͑ kT ͒ ϪJ e1 u d1 ͓͑ kϪ1 ͒ T ͔ which can be written as
ϪJ e2 u d1 ͓͑ kϪ2 ͒ T ͔ ϩE ed r ͑ kT ͒ , u d1 ͑ kT ͒ ϭϪK ed x ed ͑ kT ͒ ϪJ e(dϩ1) u d1 ͓͑ kϪd
͑34͒ Ϫ1 ͒ T ͔ ϩE ed r ͑ kT ͒ , ͑37͒
where where
G ( v ) ϭe A e v T ,
e
K ed ϭK ec G ( v ) ϭ ͓ K d1
e K d2 K d3 ͔ ,
H ( v ) ϭ ͓ G ( v Ϫ ␥ ) ϪI ͔ A Ϫ1 B e , J e(dϩ1) ϭK ec H e(dϩ1) ,
(v)
e1 e e
H ( v ) ϭG ( v Ϫ ␥ ) ͓ G ( ␥ ) ϪI ͔ A Ϫ1 B e , E ed ϭ ͓ E ec ϪK ec E ( v ) ͔ .
e
e2 e e e
Again, to illustrate the procedure and for later
E ( v ) ϭ ͓ G ( v ) ϪI ͔ A Ϫ1 F ec ,
e e e simulation, let dϭ1. Then from Eqs. ͑28͒ and ͑37͒
we have
K ed ϭK ec G ( v ) ϭ ͓ K d1
e K d2 K d3 ͔ ,
x ed ͑ kTϩT ͒ ϭG e x ed ͑ kT ͒ ϩH e1 u d1 ͓͑ kϪ1 ͒ T ͔
J e1 ϭK ec H ( v ) ,
e1
ϩH e2 u d1 ͓͑ kϪ2 ͒ T ͔ ϩE e r ͑ kT ͒ ,
J e2 ϭK ec H ( v ) ,
e2 ͑38͒
E ed ϭ ͓ E ec ϪK ec E ( v ) ͔ .
e where
Eq. ͑33͒ can be rewritten as G e ϭe A e T ,
ͫ x d1 ͑ kTϩT ͒
ͬͫ
G 11 0 0
x d2 ͑ kTϩT ͒ ϭ G 21 G 22 G 23
x d3 ͑ kTϩT ͒ G 31 G 32 G 33
ͬͫ ͬx d1 ͑ kT ͒
x d2 ͑ kT ͒
x d3 ͑ kT ͒
H e1 ϭ ͓ G (1Ϫ ␥ ) ϪI ͔ A Ϫ1 B e ,
e
e
e
H e2 ϭ ͓ G e ϪG (1Ϫ ␥ ) ͔ A Ϫ1 B e ,
e
ͫ ͬ
E e ϭ ͓ G e ϪI ͔ A Ϫ1 F ec .
e
H e11
ϩ H e12 u d1 ͑ kTϪT ͒ For the control law
H e13
u d1 ͑ kT ͒ ϭϪK ed x ed ͑ kT ͒ ϪJ e2 u d1 ͓͑ kϪ2 ͒ T ͔
ͫ ͬ
H e21
ϩ H e22 u d1 ͑ kTϪ2T ͒
H e23 where
ϩE ed r ͑ kT ͒ , ͑39͒
ϩ
0
ͫ ͬ
E e2 r ͑ kT ͒ .
E e3
͑35͒
G ( v ) ϭe A e v T ,
e
H ( v ) ϭ ͓ G ( ␥ ) ϪI ͔ A Ϫ1 B e ,
e2 e
From Eqs. ͑34͒ and ͑35͒, we can develop the K ed ϭK ec G ( v ) ϭ ͓ K d1
e K d2 K d3 ͔ ,
simulation diagram shown in Fig. 6.
Case 2 J e2 ϭK ec H ( v ) ,
e2
In this case, v Ͻ ␥ as shown in Fig. 5. Substitut-
E ed ϭ ͓ E ec ϪK ec E ( v ) ͔ .
ing from Eq. ͑20͒ into Eq. ͑29͒, we get the digital e
control law as Remark 2
Due to the specific structures of the system ma-
u d1 ͑ kT ͒ ϭϪK ec ͕ G ( v ) x ed ͑ kT ͒ ϩH e(dϩ1) u d1 ͓͑ k
(v)
e trix A e , the input vector B e in Eq. ͑11͒ and the
ϪdϪ1 ͒ T ͔ ϩE ( v ) r ͑ kT ͒ ͖ ϩE ec r ͑ kT ͒ , virtual feedback gain K ec in Eq. ͑12͒, all digital
e
gains J ep ͓for pϭd, ( dϩ1 ) ] in Eqs. ͑31͒ and ͑37͒
͑36͒ are zero. To verify this, consider

9. A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 41
A eB eϭ ͫ A1
ϪB 2 C 1
0
A2
ϪB 1 D 2 C 1 B 1 C 2 A 1 ϪB 1 D 2 C 1
0
ϪB 2 C 1
ͬ Let the discrete model of Eqs. ͑40͒ and ͑41͒ be
given by
x c1 ͑ kTϩT ͒ ϭG 1 x c1 ͑ kT ͒ ϩH d u d1 ͓͑ kϪd ͒ T ͔
ϫ
B1
ͫ ͬͫ ͬ
0 ϭ
ϪB 1
A 1B 1
0
ϪA 1 B 1
,
ϩH (dϩ1) u d1 ͓͑ kϪdϪ1 ͒ T ͔ ,
y c1 ͑ kT ͒ ϭC 1 x c1 ͑ kT ͒ ,
͑42͒
͑43͒
A 2 B e ϭA e ͑ A e B e ͒ ϭ
e
1
0ͫ ͬ
A 2B 1
ϪA 2 B 1
1
,
where ␥ is as shown in Figs. 4 and 5, for the long
and short time-delay cases. Also,
T d ϭdTϩ ␥ T,
then
Ti
␥ϭ ,
H ( ␣ ) ϭ ͓ G ( ␣ ) ϪI ͔ A Ϫ1 B e
e e e T
ϱ
␣T G 1 ϭe A 1 T ,
ϭ͚ ͑ ␣ TA e ͒ jϪ1 B e
ͫ ͬ
j!
jϭ1
H d ϭ ͓ G (1Ϫ ␥ ) ϪI ͔ A Ϫ1 B 1 ,
1 1
ϱ
␣T
H (dϩ1) ϭ ͓ G 1 ϪG (1Ϫ ␥ ) ͔ A Ϫ1 B 1 .
ͫ ͬ
͚
jϭ1 j!
͑ ␣ TA 1 ͒ jϪ1 B 1
H (␣)
1 1
e1
The desired digital observer is given by
ϭ 0 ϭ 0 ,
ϱ
␣T ϪH ( ␣ )
e1 x o ͑ kTϩT ͒ ϭG 1 x o ͑ kT ͒ ϩH d u d1 ͓͑ kϪd ͒ T ͔
ˆ ˆ
Ϫ͚ ͑ ␣ TA 1 ͒ jϪ1 B 1
jϭ1 j! ϩH (dϩ1) u d1 ͓͑ kϪdϪ1 ͒ T ͔
K ec H ( ␣ ) ϭ ͓ D 2 C 1
e ϪC 2 D 2C 1͔
H (␣)
0
e1
ͫ ͬ
ϪH ( ␣ )
e1
ϭ0, where
ϪK o e y ͑ kT ͒ ,
e y ͑ kT ͒ ϭC 1 ͓ x o ͑ kT ͒ Ϫx c1 ͑ kT ͔͒
ˆ
͑44͒
which implies
ϭy o ͑ kT ͒ Ϫy c1 ͑ kT ͒ ,
ˆ ͑45͒
J e ϭK ec H ( ␣ ) ϭ ͓ IϩK ec H ( ␣ ) ͔ Ϫ1 K ec H ( ␣ ) ϭ0.
e eo e
y o ͑ kT ͒ ϭC 1 x o ͑ kT ͒ .
ˆ ˆ ͑46͒
5. Digital state observer design
From Eqs. ͑44͒ and ͑42͒,
In order to implement the system in Fig. 6, the x o ͑ kTϩT ͒ Ϫx c1 ͑ kTϩT ͒
ˆ
discrete state x d1 ( kT ) must be available for mea-
surement. In some practical cases, this state may ϭG 1 ͓ x o ͑ kT ͒ Ϫx c1 ͑ kT ͔͒ ϪK o C 1 ͓ x o ͑ kT ͒
ˆ ˆ
not be accessible and an observer will be required
to estimate it. Ϫx c1 ͑ kT ͔͒ . ͑47͒
Consider the following state-space representa- Let
tion of an input time-delay system,
e ͑ kT ͒ ϭx o ͑ kT ͒ Ϫx c1 ͑ kT ͒ ,
ˆ ͑48͒
x c1 ͑ t ͒ ϭA 1 x c1 ͑ t ͒ ϩB 1 u c1 ͑ tϪT d ͒ ,
˙ ͑40͒
then from Eq. ͑47͒, we get
y c1 ͑ t ͒ ϭC 1 x c1 ͑ t ͒ , ͑41͒
e ͑ kTϩT ͒ ϭ ͓ G 1 ϪK o C 1 ͔ e ͑ kT ͒ . ͑49͒
where x c1 ( t ) ෈Rn1 , u c1 ( t ) ෈Rm1 , y c1 ( t ) ෈Rp1
and ( A 1 , B 1 , and C 1 ) are known constant matri- We now need to find the optimal K o in Eq. ͑49͒
ces of appropriate dimensions. such that

10. 42 A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47
lim ͕ e ͑ kT ͒ ͖ →0. ͑50͒ which the PID controller has been pre-designed to
k→ϱ meet the following control specifications ͓14,18͔:
velocity error constant, K v ϭ20, crossover fre-
This optimal K o can be designed from duality.
quency, ␻ c ϭ5 rad/s, damping ratio, ␨ ϭ0.7. Let
Consider the following delay-free system:
the transfer functions for the plant and controller
x ͑ kTϩT ͒ ϭGx ͑ kT ͒ ϩHu ͑ kT ͒ , ͑51͒ in Fig. 2 be given as
u ͑ kT ͒ ϭϪK d x ͑ kT ͒ . ͑52͒ 6000
G 1͑ s ͒ ϭ , ͑61͒
͑ s 2 ϩ32.44sϩ20͒͑ sϩ30͒
Then, for a free final-state closed-loop system with
performance index given by s 2 ϩ10.42sϩ20 Ki K ds
G 2͑ s ͒ ϭ ϭK p ϩ ϩ ,
1 1
NϪ1 s ͑ sϩ10͒ s ͑ sϩ ␣ ͒
Jϭ x T ͑ NT ͒ Px ͑ NT ͒ ϩ
2 2 ͚
kϭ1
͓ x T ͑ kT ͒ Qx ͑ kT ͒ ͑62͒
where the PID controller parameters are K p
ϩu T ͑ kT ͒ Ru ͑ kT ͔͒ , ͑53͒ ϭ0.842, K i ϭ2, K d ϭ0.158, and ␣ ϭ10.
The state-space models for G 1 ( s ) and G 2 ( s ) are
ͫ ͬ ͫͬ
where Qу0, RϾ0, and PϾ0, with ( G,H ) con-
trollable and ( G,Q ) observable, the steady-state Ϫ62.44 Ϫ993.20 Ϫ600
gain is given by the solution to the Riccati equa- 1
tions ͓17͔ as A 1ϭ 1 0 0 , B 1ϭ 0 ,
0 1 0 0
Ϫ1
PϭG ͓ PϪ PH ͑ H PHϩR ͒
T T
H P ͔ GϩQ,
T
͑54͒ C 1 ϭ ͓ 0 0 6000͔ , D 1 ϭ0, ͑63͒
ͫ ͬ
Ϫ1
K d ϭ ͑ H PHϩR ͒
T T
͑55͒
ͫͬ
H PG,
Ϫ10 0 1
giving the closed-loop system as A 2ϭ , B 2ϭ 0 ,
1 0
x ͑ kTϩT ͒ ϭ ͓ GϪHK d ͔ x ͑ kT ͒ . ͑56͒
C 2 ϭ ͓ 0.42 20͔ , D 2 ϭ1. ͑64͒
The dual system of Eqs. ͑51͒ and ͑52͒ is given by
The bandwidth of the delay-free system with the
x ͑ kTϩT ͒ ϭG T x ͑ kT ͒ ϩC T u ͑ kT ͒ ,
1 ͑57͒ above G 1 ( s ) , G 2 ( s ) , is ␻ b ϭ9.56 rads/s. The
sampling period T can be approximately evaluated
u ͑ kT ͒ ϭϪK T x ͑ kT ͒ .
d ͑58͒ ͓16͔ as TХ ␲ / ( 3 – 10) ␻ b ϭ0.033– 0.11 s. In Ref.
From Eqs. ͑54͒ and ͑55͒, we get the desired gain ͓14͔, the sampling period was chosen as 0.035 and
as 0.07 s. In this paper, simulation runs are shown for
sampling periods 0.035 and 0.1 s, with E c ϭ1 and
K o ϭ ͓͑ C 1 PC T ϩR ͒ Ϫ1 C 1 PG T ͔ T
1 1 ͑59͒ r ( t ) being a step input applied at time tϭ0. It
should be noted that the true sampling period of
with the solution for P given by the following this system is closer to 0.035 s and that the simu-
Riccati equation: lations for 0.1 s are only shown to illustrate the
PϭG 1 ͓ PϪ PC T ͑ C 1 PC T ϩR ͒ Ϫ1 C 1 P ͔ G T ϩQ, tolerance of the system to inaccurate sampling pe-
1 1 1
͑60͒ riod selection. In general, in selecting a suitable
sampling period with respect to the dead time, a
where G T in Eq. ͑54͒ equals G 1 in Eq. ͑59͒ and H bisection search method is suggested to find an
in Eq. ͑54͒ equals C T in Eq. ͑59͒.
1 appropriate sampling period, so that a reasonable
tradeoff between the closed-loop response and the
6. Illustrative example stability of the closed-loop response can be
achieved.
Consider the unity output feedback continuous- From the development, we have for the case in
time delay control system shown in Fig. 2, in Eq. ͑9͒ with Tϭ0.1 s and T d ϭ0.17 s, that

11. A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 43
΄ ΅ ΄΅
Ϫ62.4 Ϫ933.2 Ϫ600 0 0 0 0 0
1
1 0 0 0 0 0 0 0
0
0 1 0 0 0 0 0 0 0
0 0 Ϫ6000 Ϫ10 0 0 0 Ϫ6000 0
A eϭ , B eϭ 0 ,
0 0 0 1 0 0 0 0
0 0 Ϫ6000 0.4 20 Ϫ62.4 Ϫ933.2 Ϫ6600 Ϫ1
0
0 0 0 0 0 1 0 0
0
0 0 0 0 0 0 1 0
͑65͒
΄΅
0
0
0
1
F ec ϭ 0 , K ec ϭ ͓ 0 0 6000 Ϫ0.4 Ϫ20 0 0 6000͔ , E ec ϭ1. ͑66͒
1
0
0
The corresponding system matrices for the discrete model in Eq. ͑33͒ with Tϭ0.1 s, dϭ1, and ␥ ϭ0.7
become
΄ ΅
Ϫ0.0976 Ϫ4.5027 Ϫ2.4211 0 0 0 0 0
0.0040 0.1543 Ϫ0.4950 0 0 0 0 0
0.0008 0.0555 0.9737 0 0 0 0 0
Ϫ0.1867 Ϫ14.3057 Ϫ334.3965 0.3629 Ϫ0.1339 Ϫ0.1867 Ϫ14.3057 Ϫ334.3965
G eϭ ,
Ϫ0.0067 Ϫ0.6048 Ϫ20.9555 0.0631 0.9967 Ϫ0.0067 Ϫ0.6048 Ϫ20.9555
Ϫ0.0407 Ϫ2.3603 Ϫ13.6305 0.0077 0.0340 Ϫ0.1383 Ϫ6.8630 Ϫ16.0516
Ϫ0.0023 Ϫ0.1866 Ϫ4.6818 0.0008 0.0150 0.0017 Ϫ0.0323 Ϫ5.1767
Ϫ0.0001 Ϫ0.0069 Ϫ0.2589 0 0.0008 0.0008 0.0487 0.7148
͑67͒
΄ ΅ ΄ ΅ ΄ ΅
0.0117 Ϫ0.0077 0
0.0002 0.0006 0
0 0 0
0 0 0.05640
H e1 ϭ 0 , H e2 ϭ 0 , E e ϭ 0.0035 . ͑68͒
Ϫ0.0177 0.0077 0.0025
Ϫ0.0002 Ϫ0.0006 0.0008
0 0 0

12. 44 A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47
Fig. 7. Continuous-time and discrete-time system outputs Fig. 8. Continuous-time and discrete-time control laws for
for delays, dϭ0, 1, and 2. delays, dϭ0, 1, and 2.
the system was potentially unstable at the inaccu-
Similarly, the digitally redesigned control gains rate sampling period, Tϭ0.1 s, for dϾ5, while
for the digital control law in Eq. ͑34͒ with T stable at the correct sampling period, T
ϭ0.1 s, dϭ1, v ϭ0.95, and ␥ ϭ0.7 are given by ϭ0.035 s. In general, the system also shows good
K ed ϭ ͓ 4.6 308.8 4966.8 Ϫ1.2 robustness to sampling period selection, even in
the case where the chosen sampling period is
Ϫ15.3 4.6 308.8 4966.8͔ , longer than the sampling period chosen in Ref.
͓14͔.
J e1 ϭϪ2.8623e Ϫ16, A representative case for the observer perfor-
J e2 ϭϪ2.3065e Ϫ14, mance is shown in Figs. 10 and 11 for dϭ1, with
Qϭ103 I and RϭI.
E ed ϭ0.8514. ͑69͒
7. Conclusion
Simulating the example as shown in Fig. 6, using
the calculated coefficients above, with initial con- This paper has proposed a digital redesign
ditions x c1 ( 0 ) ϭ ͓ 0 0 0 ͔ T and x o( 0 )
ˆ scheme for the analog Smith predictor, that forces
ϭ ͓ 0.001 0.0001 0.00001͔ T , we get the results
shown in Figs. 7–11 below. The plots in Figs. 7
and 8 show the plant outputs and control laws for
two sampling periods, Tϭ0.035 s and Tϭ0.1 s,
with integer time delays of dϭ0,1, and 2. Fig. 9
shows outputs and control laws for cases dϭ5 and
10. For each case, the optimally determined inter-
sample parameter v is shown with the correspond-
ing error between the continuous-time and
discrete-time systems. As expected, larger errors
in J ( v ) occur as the delay d increases. As the
plots demonstrate, the proposed system provides
good performance even for delays up to ten sam-
pling periods, when the sampling period is suit-
ably selected. Not surprisingly, as the delay in-
creases the need for more careful sampling period
selection becomes crucial. This point was seen Fig. 9. Continuous-time and discrete-time outputs and con-
during the simulation, where it was observed that trol laws for delays, dϭ5 and 10.

13. A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 45
Acknowledgments
This work was supported in part by the US
Army Research Office, under Grant No. DAAD
19-02-1-0321 and the National Science Council of
the Republic of China, under Contract No. NSC-
91-2213-E006-050.
Appendix: Development of the prediction-
based digital redesign method
Consider a linear controllable continuous-time
system described by ͓15͔
Fig. 10. Observer-based continuous-time and discrete-time
x c ͑ t ͒ ϭAx c ͑ t ͒ ϩBu c ͑ t ͒ , x c ͑ 0 ͒ ϭx 0 , ͑A1͒
˙
system outputs for delay, dϭ1. where x c ( t ) ෈R n , u c ( t ) ෈R m , and A and B are
constant matrices of appropriate dimensions. Let
the continuous-time state-feedback controller be
u c ͑ t ͒ ϭϪK c x c ͑ t ͒ ϩE c r ͑ t ͒ , ͑A2͒
state matching between the continuous-time and
discrete-time system states, even at intersample where K c ෈R mϫn and E c ෈R mϫm have been de-
points. The proposed method reformulates a tradi- signed to satisfy some specified goals, and r ( t )
tional analog Smith predictor into an augmented ෈R m is a piecewise-constant reference input vec-
system, which is then digitally redesigned using tor. The controlled system is
the predicted intersampling states. As evidenced
by the simulation plots, the proposed system is x c ͑ t ͒ ϭA c x c ͑ t ͒ ϩBE c r ͑ t ͒ , x c ͑ 0 ͒ ϭx 0 ,
˙
capable of dealing with long delays, several times ͑A3͒
greater than the sampling period. The method also where A c ϭAϪBK c . Let the state equation of a
demonstrates good robustness in relation to the corresponding hybrid model be
discrete-time sampling period selection for moder-
ate time delays. x d ͑ t ͒ ϭAx d ͑ t ͒ ϩBu d ͑ t ͒ , x d ͑ 0 ͒ ϭx 0 , ͑A4͒
˙
where u d ( t ) ෈R m is a piecewise-constant input
vector, satisfying
u d ͑ t ͒ ϭu d ͑ kT ͒ for kTрtϽ ͑ kϩ1 ͒ T
and TϾ0 is the sampling period. Let the discrete-
time state-feedback controller be
u d ͑ kT ͒ ϭϪK d x d ͑ kT ͒ ϩE d r * ͑ kT ͒ , ͑A5͒
where K d ෈R mϫn is a digital state-feedback gain,
E d ෈R mϫm is a digital feedforward gain, and
r * ( kT ) ෈R m is a piecewise-constant reference in-
put vector to be determined in terms of r ( t ) for
tracking purposes. The digitally controlled closed-
loop system thus becomes
x d ͑ t ͒ ϭAx d ͑ t ͒ ϩB ͓ ϪK d x d ͑ kT ͒
˙
ϩE d r * ͑ kT ͔͒ , x d ͑ 0 ͒ ϭx 0 ͑A6͒
Fig. 11. Observer-based continuous-time and discrete-time
control laws for delay, dϭ1. for kTрtϽ ( kϩ1 ) T.

14. 46 A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47
A zero-order-hold device is used for Eq. ͑A5͒. Thus from Eqs. ͑A8͒ and ͑A9͒ it follows that to
The digital redesign problem is to find digital con- obtain the state x c ( t v ) ϭx d ( t v ) , under the assump-
troller gains ( K d ,E d ) in Eq. ͑A5͒ from the analog tion of x c ( kT ) ϭx d ( kT ) , it is necessary to have
gains ( K c ,E c ) in Eq. ͑A2͒, so that the closed-loop u d ( kT ) ϭu c ( t v ) . This leads to the following
state x d ( t ) in Eq. ͑A6͒ can closely match the prediction-based digital controller:
closed-loop state in Eq. ͑A3͒ at all the sampling
instants, for a given r ( t ) ϵr ( kT ) , kϭ0,1,2,... . u d ͑ kT ͒ ϭu c ͑ t v ͒
The state x c ( t ) in Eq. ͑A1͒, at tϭt v ϭKTϩ v T ϭϪK c x c ͑ t v ͒ ϩE c r ͑ t v ͒
for 0р v Ͻ1, is found to be
ϭϪK c x d ͑ t v ͒ ϩE c r ͑ t v ͒ , ͑A10͒
x c ͑ t v ͒ ϭexp͓ A ͑ t v ϪkT ͔͒ x c ͑ kT ͒ ϩ ͵
kT
tv
exp͓ A ͑ t v where the future state x d ( t v ) ͑denoted as the pre-
dicted state͒ needs to be predicted based on the
Ϫ ␶ ͔͒ Bu c ͑ ␶ ͒ d ␶ . ͑A7͒ available causal signals x d ( kT ) and u d ( kT ) .
Substituting the predicted state x d ( t v ) in Eq.
Let u c ( t v ) be a piecewise-constant input. Then Eq. ͑A9͒ into Eq. ͑A10͒ and then solving it for u d ( kT )
͑A7͒ reduces to results in
x c ͑ t v ͒ Ϸexp͑ A v T ͒ x c ͑ kT ͒ ϩ ͵ kT
kTϩ v T
exp͓ A ͑ kT
u d ͑ kT ͒ ϭ ͑ I m ϩK c H ( v ) ͒ Ϫ1 ͓ ϪK c G ( v ) x d ͑ kT ͒
ϩE c r ͑ t v ͔͒ . ͑A11͒
ϩ v TϪ ␶ ͔͒ Bd ␶ u c ͑ t v ͒ Consequently, the desired predicted digital con-
ϭG (v)
x c ͑ kT ͒ ϩH (v)
u c͑ t v ͒ , ͑A8͒ troller ͑A5͒ is found, from Eq. ͑A11͒, to be
u d ͑ kT ͒ ϭϪK ( v ) x d ͑ kT ͒ ϩE ( v ) r * ͑ kT ͒ ,
d d
where
͑A12͒
G (v)
ϭexp͓ A ͑ t v ϪKT ͔͒ where, for tracking purposes, r * ( kT ) ϭr ( kT
ϭexp͑ A v T ͒ ϭ ͓ exp͑ AT ͔͒ vϭ
͑G͒ v, ϩ v T ) , and
K ( v ) ϭ ͑ I m ϩK c H ( v ) ͒ Ϫ1 K c G ( v ) ,
͵ tv d
H (v)ϭ exp͓ A ͑ t v Ϫ ␶ ͔͒ Bd ␶
kT E ( v ) ϭ ͑ I m ϩK c H ( v ) ͒ Ϫ1 E c .
d
ϭ͵
vT In particular, if v ϭ1 then the prerequisite
exp͑ A ␶ ͒ Bd ␶ x c ( kT ) ϭx d ( kT ) is ensured. Thus, for any k
0
ϭ0,1,2,..., the controller is given by
ϭ ͓ G ( v ) ϪI n ͔ A Ϫ1 B.
u d ͑ kT ͒ ϭϪK d x d ͑ kT ͒ ϩE d r * ͑ kT ͒ , ͑A13͒
Ϫ1
Here, it must be noted that ͓ G ϪI n ͔ A (v)
is a where
shorthand notation, which is well defined as can
be verified by cancellation of A Ϫ1 in the series K d ϭ ͑ I m ϩK c H ͒ Ϫ1 K c G,
expansion of the term ͓ G ( v ) ϪI n ͔ . This convenient
notation for an otherwise long series is used E d ϭ ͑ I m ϩK c H ͒ Ϫ1 E c ,
throughout this appendix. Also, the state x d ( t ) of
Eq. ͑A4͒, at tϭt v ϭkTϩ v T for 0р v р1, is ob-
r * ͑ kT ͒ ϭr ͑ kTϩT ͒ ,
tained as in which
x d ͑ t v ͒ ϭexp͓ A ͑ t v ϪkT ͔͒ x d ͑ kT ͒ ϩ ͵
kT
tv
exp͓ A ͑ t v
Gϭexp͑ AT ͒ and Hϭ ͑ GϪI n ͒ A Ϫ1 B.
In selecting a suitable sampling period for the
Ϫ ␶ ͔͒ Bd ␶ u d ͑ kT ͒ digital redesign method, a bisection searching
method is suggested to find an appropriate long
ϭG ( v ) x d ͑ kT ͒ ϩH ( v ) u d ͑ kT ͒ . ͑A9͒ sampling period, so that the reasonable tradeoff

15. A. C. Dunn, L. S. Shieh, S. M. Guo / ISA Transactions 43 (2004) 33–47 47
between the closed-loop response ͓i.e., matching ͓18͔ Chen, C. F. and Shieh, L. S., An algebraic method for
of the states x c ( kT ) in Eq. ͑A8͒ and x d ( kT ) in Eq. control systems design. Int. J. Control 11, 717–739
͑1970͒.
͑A9͔͒ and the stability of the closed-loop system
can be achieved.
References
Alex C. Dunn received B.S.,
͓1͔ Astrom, K. J. and Hagglund, T., PID Controllers: M.S., and Ph.D. degrees in
Theory, Design and Tuning. Instrument Society of electrical engineering from the
America, Research Triangle Park, NC, 1995. University of Sierra Leone ͑Si-
erra Leone͒, 1976, the Univer-
͓2͔ Tan, K. K., Wang, Q. G., and Hang, C. C., Advances
sity of Aston ͑UK͒, 1982, and
in PID Control. Springer-Verlag, London, 1999. the University of Houston
͓3͔ Morari, M. and Zafirriou, E., Robust Process Control. ͑USA͒, 2003, respectively.
Prentice-Hall, Englewood Cliffs, NJ, 1989. Alex has an extensive industry
͓4͔ Marshall, J. E., Gorecki, H., and Walton, K., Time background in control systems
Delay Systems: Stability and Performance Criteria and information technology,
having worked for companies
With Applications, 1st ed. Ellis Horwood, New York,
such as Shell Oil, Honeywell,
1992. Inc., and Setpoint Inc./Aspen
͓5͔ Laughlin, D. L., Rivera, D. E., and Morari, M., Smith Tech. His research interests include multivariable control of industrial
predictor design for robust performance. Int. J. Control plants, intelligent controls via soft computing techniques, and digital
46, 477–504 ͑1987͒. control of input time-delay and constrained nonlinear systems.
͓6͔ Astrom, K. J., Hang, C. C., and Lim, B. C., A new
Smith predictor for controlling a process with an inte-
grator and long dead-time. IEEE Trans. Autom. Con- Leang-San Shieh received his
trol 39, 343–345 ͑1994͒. B.S. degree from the National
͓7͔ Hagglund, T., A predictive PI controller for processes Taiwan University, Taiwan in
with long dead times. IEEE Control Syst. Mag. 12, 1958, and his M.S. and Ph.D.
degrees from the University of
57– 60 ͑1992͒. Houston, Houston, Texas, in
͓8͔ Huang, J. J. and DeBra, D. B., Automatic Smith- 1968 and 1970, respectively,
predictor tuning using optimal parameter mismatch. all in electrical engineering.
IEEE Trans. Control Syst. Technol. 10, 447– 459 He is a professor in the Depart-
͑2002͒. ment of Electrical and Com-
͓9͔ Tan, K. K., Lee, T. H., and Leu, F. M., Predictive PI puter Engineering and the di-
rector of the Computer and
versus Smith control for dead-time compensation. ISA
Systems Engineering. He was
Trans. 40, 17–29 ͑2000͒. the recipient of more than ten
͓10͔ Fliess, M., Marquez, R., and Mounier, H., An exten- College Outstanding Teacher Awards, the 1973 and 1997 College
sion of predictive control, PID regulation and Smith Teaching Excellence Awards, and the 1988 College Senior Faculty
predictors to some linear delay systems. Int. J. Control Research Excellence Award from the Cullen College of Engineering,
75, 728 –743 ͑2002͒. University of Houston, and the 1976 University Teaching Excellence
Award and the 2002 El Paso Faculty Achievement Award from the
͓11͔ Vrecko, D., Vrancic, D., Juricic, D., and Strmcnik, S.,
University of Houston. He has published more than two hundred ar-
A new modified Smith predictor: The concept, design ticles in various referred scientific journals. His fields of interest are
and tuning. ISA Trans. 40, 111–121 ͑2001͒. digital control, optimal control, self-tuning control, and hybrid control
͓12͔ Zhang, W. and Xu, X., Analytical design and analysis of uncertain systems.
of mismatched Smith predictor. ISA Trans. 40, 133–
138 ͑2001͒.
͓13͔ Smith, O. J. M., Closer control of loops with dead Shu-Mei Guo received the
time. Chem. Eng. Prog. 53, 217–219 ͑1957͒. M.S. degree in Department of
͓14͔ Guo, S. M., Wang, W., and Shieh, L. S., Discretization Computer and Information
of two degree-of-freedom controller and system with Science from the New Jersey
state, input and output delays. IEE Proc.: Control Institute of Technology, USA
in 1987. She received the
Theory Appl. 147, 87–96 ͑2000͒. Ph.D. degree in Computer and
͓15͔ Guo, S. M., Shieh, L. S., Chen, G., and Lin, C. F., Systems Engineering from the
Effective chaotic orbit tracker: A prediction-based University of Houston, USA in
digital redesign approach. IEEE Trans. Circuits Syst., May 2000. Since June 2000,
I: Fundam. Theory Appl. 47, 1557–1570 ͑2000͒. she has been an assistant pro-
͓16͔ Astrom, K. J. and Wittenmark, B., Computer Con- fessor in the Department of
Computer System and Infor-
trolled Systems. Prentice-Hall, Upper Saddle River,
mation Engineering, National
NJ, 1997. Cheng-Kung University, Taiwan. Her research interests include vari-
͓17͔ Goodwin, G. C., Graebe, S. F., and Salgado, M. E., ous applications on evolutionary programming, chaos systems, Kal-
Control System Design. Prentice-Hall, Upper Saddle man filtering, fuzzy methodology, sampled-data systems, image pro-
River, NJ, 2001. cessing, and computer and systems engineering.