Asian Journal of Control, Vol. 6, No. 4, pp. 483495, December
2004
483
Manuscript received May 27, 2003; revised August 8, 2003;
accepted November 19, 2003.
Yongpeng
Zhang
and
LeangSan
Shieh
are
with
the
Department of Electrical and Computer Engineering, Uni
versity of Houston, Houston, TX 772044005, U.S.A.
Cajetan M. Akujuobi and Warsame Ali are with
the Cen
ter of Excellence for Communication Systems Technol
ogy Research (CECSTR), Department of Electrical Engi
neering, Prairie View A&M University, Prairie View, TX
77446, U.S.A.
This work was supported in part by the US Army Researc
h
Office under Grant DAAD 190210321, and Texas Instru
ments under Grant #41017103001, and NASAJSC unde
r
Grant #NNJ04HF32G.
DIGITAL PID CONTROLLER DESIGN FOR DELAYED
MULTIVARIABLE SYSTEMS
Yongpeng Zhang, LeangSan Shieh, Cajetan M. Akujuobi, and Warsame Ali
ABSTRACT
A new methodology is proposed to design digital PID controllers for
multivariable systems with time delays. Except for a few parameters that are
preliminarily selected, most of the PID parameters are systematically tuned
using the developed plant statefeedback and controller statefeedforward
LQR approach, such that satisfactory performance with guaranteed
closedloop stability is achieved. In order to deal with the modeling error
owing to the delay time rational approximation, an IMC structure is utilized,
such that robust stability is achieved, without need for an observer, and with
improved online tuning convenience. Using the predictionbased digital re
design method, the digital implementation is obtained based on the
aboveproposed analog controller, such that the resulting mixedsignal sys
tem performance will closely match that of the analog controlled system. An
illustrative example is given for comparison with alternative techniques.
KeyWords: PID, multivariable system, IMC, digital redesign, mixed signal
system.
I. INTRODUCTION
The ProportionalIntegralDerivative (PID) controller
is the most popular form of controller utilized in the
control industry today, due to its simplicity in controller
structure, robustness to constant disturbances and avail
ability of many tuning methods [1]. To unify and en
hance the recent progress in PID control, several special
issues [9,20] and monograph [24] on advances in PID
control have been published over the past few years.
The PID controller structure comes in two forms, as
standard SingleInputSingleOutput (SISO) PID con
trollers and MultiInputMultiOutput (MIMO) PID
controllers. The design, tuning and implementation of
MIMO PID controllers are relatively more complex
compared with those of SISO PID controllers [5],
mostly due to the renowned difficulty arising from loop
interaction (or coupling). One favorable controller
structure in multivariable system design is the decen
tralized controller, which means all the offdiagonal
elements of the transfer function matrix of the controller
are zero. Although this constraint on the controller
structure may lead to performance deterioration when
compared with the centralized controller, the decentral
ized controller remains popular in applications [19]. The
underlying reasons can be summarized as follows: (i)
SISO PID controller design methods can be directly
exploited in decentralized controller design [7]; (ii) The
hardware simplicity considerably reduces the complex
ity and cost in implementing the decentralized controller
[17].
With the rapid progress in microelectronics tech
nology, the digital controller is now widely applied in
industry for better reliability, lower cost, smaller size,
and better performance. In addition, it can provide sub
484
Asian Journal of Control, Vol. 6, No. 4, December 2004
stantial convenience in implementing the controller with
sophisticated interconnected structure, which can be
easily realized or modified through programming [2].
To exploit the advantages of digital controllers over
analog controllers, several design methods [10,11,2123,
28] have been developed in the discretetime domain via
the direct digital design approach, that is to discretize
the analog plant and then determine a digital controller
for the discretized plant in the discretetime domain.
Nevertheless, the direct digital design methods (such as
the deadbeat design method), in general, cannot take
care of the intersample behavior [2], which may lead to
performance deterioration, especially when the sampling
period is relatively long. In addition, if the continuous
time plant ‘A’ matrix has a physical parameter of inter
est in only one or two entries, then these parameters are
transposed to other elements in the corresponding dis
cretetime system matrix ‘exp (AT)’, where T is a sam
pling period. As a result, we may lose physical insight
of these parameters and design for these parameter
variations becomes very difficult, especially for a
highorder multivariable control system.
In this paper, we choose the digital redesign ap
proach [6,16] to design the MIMO PID controller. In
this approach, a centralized analog MIMO PID control
ler is designed in the continuoustime domain, then re
designed to its digital counterpart, while keeping the
essential system performance unchanged. The resulting
digitally controlled analog system becomes a hybrid
system (i.e., mixedsignal system), which consists of a
continuoustime plant and a discretetime controller [16].
The digital redesign approach is adopted here based on
the following considerations: (i) Continuoustime design
theory and tools are well developed, and intuitively
more familiar to most engineers; (ii) Digital redesign
separates the problems of basic controller design from
that of sampling period selection and adjustment [15];
(iii) The digitally redesigned controllers are able to take
care of the intersample behavior [8] of the designed
closedloop mixedsignal (sampleddata) system, and
they can be easily implemented using digital electronics.
However, the analysis of the designed hybrid multivari
able system is not a simple matter in comparison with
the direct digital design approach.
This paper proposes a feedback/feedforward design
methodology, in which the PID tuning problem is
transformed into a Linear Quadratic Regulator (LQR)
design via proper arrangement of the state space equa
tion of the cascaded system. Thus, the tuning of most
PID parameters is automatically carried out by solving
LQR design problems, except for a few parameters that
are preselected. Compared with existing methods, our
proposed method offers the following advantages: (i)
The stability of the MIMO PID controlled system is
guaranteed during the tuning process [30]; (ii) There are
no specific requirements on system stability, mini
mumphase property, loworder model and plant de
coupling [31]; (iii) The designed centralized analog
MIMO PID controllers can be conveniently imple
mented using digital electronics after digital redesign.
The presence of time delays in many industrial
processes is a well recognized phenomenon [23]. The
achievable performance of conventional unityfeedback
control processes can be significantly degraded if a
process has a relatively large time delay compared to the
dominant time constant [12]. Predicted delay time com
pensation is hard to implement since only a causal and
rational controller is realizable. Hence delayed processes
pose a serious challenge to the design of robust feedback
controls for such systems [18]. The most popular
scheme for time delay compensation is possibly the
Smith Predictor Control [22]. Under the Smith predictor
configuration, the control signal is computed on the ba
sis of the predicted system states and current prediction
error. Thus the controller can be designed with respect
to the delayfree portion of the system. Some other
method with the similar Smith structure like GPC (Gen
eralized Predictive Control) has been successfully used
in PID controller design [21,22] for delayed SISO sys
tems. It has been reported that GPC can also be applied
to delayed MIMO system design [10,11]. However, it
has to rely on multivariable system decoupling, that is to
roughly regard the decoupled MIMO system as a series
of SISO subsystems. In addition, in order to reinstate the
performance in the presence of load disturbances, and/or
the effects of modeling error and parameter uncertainty,
some other robustness consideration is needed such as
online selftuning [10,21], which will considerably in
crease the realtime computation burden.
In this paper, the pure delay terms are first ap
proximated by a rational Pade approximation model
[27], then the proposed feedback/feedforward design
method is applied to get the PID controller parameters
based on the approximated system. As some modeling
error is inevitably brought in through the approximation,
an Internal Model Control (IMC) structure is utilized, in
which an approximate model of the plant is constructed
as part of the controller. The control signal is applied to
the approximate plant model, as well as the plant itself.
By comparing the process output with the expected
process response, modeling error and disturbances can
be identified and responded to accordingly [28]. The
advantages of utilizing such an IMC structure in the
proposed design methodology are the following: (i) An
IMCbased control system is more robust, which is
widely recognized in circumstances of imperfect mod
eling [3]; (ii) The proposed IMC structure can provide
direct access to the internal approximate system states
Y. Zhang et al.: Digital PID Controller Design for Delayed Multivariable Systems
485
[22], thus making it unnecessary to design an observer
for implementing the proposed feedback/feedforward
PID controller; (iii) Since it is very convenient for the
digital controller to realize the complicated
intercon
nected
structure,
the
designed
IMCbased PID controller
is not necessary to endure a reducedorder processing
[25]. As a result, some potential performance deteriora
tion is avoided.
To utilize the advantages of digital controllers over
analog controllers, the proposed analog multivariable
PID controller is transformed into the equivalent digital
implementation through the predictionbased digital
redesign method [8], such that the mixedsignal system
performance closely matches that of the analog con
trolled system.
The paper is organized as follows. Section 2 dis
cusses the problem formulation, while the MIMO PID
controller tuning with statefeedback and statefeed
forward LQR are discussed in section 3. The IMCbased
controller structure, the robust stability analysis and the
digital redesign of the analog multivariable PID con
troller are discussed in sections 4, 5, and 6, respectively.
An illustrative example is given in section 7, followed
by conclusions in section 8.
II. PROBLEM FORMULATION
Time delays are common phenomena in many in
dustrial processes [29]. Suppose a multivariable delayed
system is defined as
ˆ
1
G (s)
, in which every component
is described as g
ij
(s)e
−sT
ij
, for i = 1, 2, …, p1 and j = 1, 2,
…, m1. The structure of the delayed multivariable sys
tem is shown in Fig. 1.
In order to obtain a rational transfer function, the
time delay term can be approximated by a first order
Pade approximation model [7,26] as,
2
2
sT
s
T
e
s
T
−
−
≈
+
. (1)
Then, the resulting approximate rational model is de
fined as G
1
(s), in which every component is repre −
a
y
22
)(
22
sT
esg
−
11
)(
11
sT
esg
−
b
y
a
u
b
u
12
)(
12
sT
esg
−
21
)(
21
sT
esg
−
Fig. 1. A delayed multivariable (2
×
2) system structure.
sented as
2
( )
2
ij
ij
ij
s
T
g s
s
T
−
+
, for i = 1, 2, …, p1 and j = 1, 2,
…, m1.
It should be mentioned that the rational approxima
tion inevitably introduces some modeling error [26],
which could lead to performance deterioration, or even
destabilize the entire system, especially when the delay
time is relatively long. For convenience, we skip over
the modeling error problem in sections 2 and 3, and
propose the corresponding solving method for the mod
eling error problem in section 4.
Consider a unityoutput feedback MIMO analog
plant G
1
(s) ∈ R
p1 × m1
, in cascade with a MIMO analog
PID controller G
2
(s) ∈ R
m1 × p1
. Also, suppose output
disturbance d(t) ∈ R
p1
exists at the output point as
shown in Fig. 2.
)(tr
)(ty
PID controller
c
E
)(td
)(
1
ty
plant model
)(
2
sG
)(
1
sG
)(
2
tu
)(
1
tu
)(
2
ty
Fig. 2. Continuoustime cascaded system.
Let the minimal realization of the analog plant G
1
(s)
be
=
+
1 1 1 1 1
x (t) A x (t) B u (t)
,
=
1 10
x (0) x
,
=
1 1 1
y (t) C x (t)
, (2a)
where x
1
(t) ∈ R
n1
, u
1
(t) ∈ R
m1
, y
1
(t) ∈ R
p1
, and A
1
, B
1
,
C
1
are constant matrices of appropriate dimensions.
Let the entire system output be the sum of the plant
output and load disturbance,
=
+
1
y(t) y (t) d(t)
, (2b)
where y(t) ∈ R
p1
, d(t) ∈ R
p1
.
Each component of the MIMO PID controller G
2
(s)
∈ R
m1 × p1
is described as
2
( )
I ij Dij
ij Pij
ij
K
K s
G s K
s s
= + +
+
α
, (3)
for i = 1, 2, …, m1 and j = 1, 2, …, p1. Parameters of
the PID controller, K
Pij
, K
Iij
, K
Dij
, and filter factor α
ij
are
constants to be determined.
Our proposed MIMO controller design methodology
requires a preliminary design procedure to provide an
initial setting for the PID parameters. A subsequent tune
up process then adjusts most parameters based on the
plant and controller properties. Any available method
can be applied in the preliminary design, or the existing
486
Asian Journal of Control, Vol. 6, No. 4, December 2004
PID controllers can be used as the initial setting. Alter
natively, the following initial design method can be
adopted.
The filter factor α for the derivative term is used for
physical implementation of the PID controller, and it
can be chosen by the manufacturer or design specifica
tions [1]. In the presence of load disturbance in the con
trolled system, it is convenient to achieve steady state
compensation by properly selecting the filter factor.
Based on the Internal Model Principle (IMP) [7], steady
state disturbance compensation requires that the distur
bance generating polynomial be included as part of the
controller denominator. Thus, we can determine de
nominators of the controller through setting the filter
factors (i.e., s + α
ij
in (3)) as those of the load distur
bance denominators. When the load disturbance is dif
ferent for each subsystem, we can adjust the filter fac
tors accordingly. Then, the transfer function matrix of
the controller can be rewritten from (3) as follows
1 1 1
1
1
( )
p p
p
j
j
s s s
s
s s
+
+ +
=
+ + + +
=
⋅ +α
∏
1 2 p1 1 p1 2
2
E E E E
G ( )
, (4)
where parameter matrices E
1
, E
2
,
…, E
p1+2
are un
knowns to be determined.
Although our proposed methodology for MIMO
PID controller design does not require that the plant be
of, or can be decoupled into a form of SISO systems, we
let the plant be diagonally dominant, so that the de
signed controller would be near to being decentralized.
The static decoupler [4,7] is widely used as a
precompensator in MIMO system design for achieving
approximate decoupling. In general, this decoupler is
defined as D
2
= G
1
−1
(0), where G
1
(0) is nonsingular.
We can make the plant statically decoupled by setting
E
1
= D
2
= G
1
−1
(0) in (4), which then gives
1 1 1
1
1
( )
p p
p
j
j
s s s
s
s s
−
+ +
=
+ + + +
= +
⋅ +α
∏
2 3 p1 1 p1 2
2 2
F F F F
G ( ) D
, (5a)
where F
2
, F
3
,…, F
p1+2
are constant matrices to be deter
mined. For simplicity, we can first select the MIMO
PID controller as
1
( )
j
diag
s s
⎧ ⎫
⎪ ⎪
= +
⎨ ⎬
+α
⎪ ⎪
⎩ ⎭
2 2
G (s) D
,
for
1,2,...,1j p=
, (5b)
and leave further parameter adjustment to the next tun
ing phase.
Then, the minimal realization of the preliminarily
designed cascaded PID controller G
2
(s) can be written
as
=
+
2 2 2 2 2
x (t) A x (t) B u (t)
,
=
2
x (0) 0
, (6a)
=
+ =
2 2 2 2 2 1
y (t) C x (t) D u (t) u (t)
, (6b)
=
− +
2 c
u (t) y(t) E r(t)
, (6c)
where x
2
(t) ∈ R
n2
, u
2
(t) ∈ R
p1
, y
2
(t) ∈ R
m1
, r(t) ∈ R
p1
,
and A
2
, B
2
, C
2
, D
2
, E
c
, are constant matrices of appro
priate dimensions.
III. THE MIMO PID CONTROLLER TUN
ING WITH STATEFEEDBACK AND
STATEFEEDFORWARD LQR
Having developed a framework of the problem to
be solved in the previous section, along with prelimi
nary controller parameter selection, we now address the
problem of finetuning the PID controller parameters in
the context of an LQR formulation. This further tuning
step is necessary to achieve satisfactory closedloop
performance.
Based on the predesigned controller, the basic idea
of our methodology is to transform the PID tuning
problem to that of optimal design. To achieve this aim,
we formulate the closedloop cascaded systems in Fig. 2
into an augmented system as
=
+ +
e e e e 1 e
x (t) A x (t) B u (t) E r(t)
,
=
=
e 1 e e
y (t) y (t) C x (t)
, (7)
where
⎡
⎤ ⎡ ⎤ ⎡ ⎤
= = =
⎢
⎥ ⎢ ⎥ ⎢ ⎥
⎢
⎥⎢⎥⎢⎥
−
⎣
⎦ ⎣ ⎦ ⎣ ⎦
1 1
e e e
2 1 2 2 c
A 0 B 0
A,B,E,
B C A 0 B E
[ ]
,
⎡ ⎤
= =
⎢ ⎥
⎢ ⎥
⎣ ⎦
1
e e 1
2
x (t)
x,C C 0
x (t)
and
=
− − −
1 1 1 2 2 2 2
u (t) K x (t) K x (t) D u (t)
, (8)
where the state feedback control gains K
1
and K
2
are to
be designed as shown below. For the convenience of
LQR design of u
1
(t), given the existence of the feedfor
ward input term D
2
u
2
(t) in (8), an alternative representa
tion of the augmented system (7) can be described as
ˆ
ˆ
ˆ
= + +
e e e e 1 e
x (t) A x (t) B u (t) E r(t)
, (9a)
Y. Zhang et al.: Digital PID Controller Design for Delayed Multivariable Systems
487
= =
1 e e e
y (t) y (t) C x (t)
, (9b)
where
ˆ
= +
1 1 2 2
u (t) u (t) D u (t)
, (9c)
= − +
2 1 c
u (t) y (t) E r(t)
, (9d)
and
ˆ
+
⎡ ⎤
=
⎢ ⎥
⎢ ⎥
−
⎣ ⎦
1 1 2 1
e
2 1 2
A B D C 0
A
B C A
,
⎡ ⎤
=
⎢ ⎥
⎢ ⎥
⎣ ⎦
1
e
B
B
0
,
ˆ
−
⎡ ⎤
=
⎢ ⎥
⎢ ⎥
⎣ ⎦
1 2 c
e
2 c
B D E
E
B E
,
⎡ ⎤
=
⎢ ⎥
⎢ ⎥
⎣ ⎦
1
e
2
x (t)
x
x (t)
,
[
]
=
e 1
C C 0
.
The statefeedback LQR for the augmented system (9)
can be chosen as
ˆ
= −
1 e e
u (t) K x (t)
, (10)
where
K
e
= [K
1
, K
2
]
with
K
1
∈ R
m1 × n1
and
K
2
∈ R
m1 × n2
.
Let the quadratic cost function for the system (9) be
ˆ ˆ
∞
= +
∫
T T
e e 1 1
0
J [x (t)Qx (t) u (t)Ru (t)]dt
, (11)
where Q ≥ 0, R > 0, (Â
e
, B
e
) is controllable and (Â
e
, Q)
is observable. The optimal statefeedback control gain
K
e
in (10) that minimizes the performance index (11) is
given by
1−
=
T
e e
K R B P
, (12)
in which the matrix P > 0 is the solution of the Riccati
equation [14],
1
ˆ ˆ
T T
−
+ − + =
e e e e
PA A P PB R B P Q 0
. (13)
The resulting closedloop system becomes
ˆ
ˆ
= − +
e e e e e e
x (t) (A B K )x (t) E r(t)
, (14)
which is asymptotically stable due to the property of
LQR design.
Defining
K
1
as the plant state feedback gain,
K
2
as
the controller feedforward gain in (10), then the desired
statefeedback and statefeedforward control law
u
1
(t)
in
(8) can be indirectly determined from the statefeedback
control law (10) and the relationships shown in (9c) and
(9d) as
ˆ
= − −
1 1 2 2
u (t) u (t) D u (t)
[ ]
= − − − −
1 1 2 2 2 c 1
K x (t) K x (t) D E r(t) y (t)
= − − − −
1 2 1 1 2 2 2 c
(K D C )x (t) K x (t) D E r(t)
, (15)
where
ˆ
1
K
= K
1
−D
2
C
1
,
ˆ ˆ
[,]=
e 1 2
K K K
and Ê
c
= −D
2
E
c
.
Substituting the control law in (15) into (7), the de
signed closedloop system becomes
− − −
⎡
⎤ ⎡ ⎤ ⎡ ⎤
=
⎢
⎥ ⎢ ⎥ ⎢ ⎥
⎢
⎥⎢ ⎥⎢ ⎥
−
⎣
⎦ ⎣ ⎦ ⎣ ⎦
1 1 1 1 2 1 1 2 1
2 2 1 2 2
x (t) A B (K D C ) B K x (t)
x (t) B C A x (t)
−
⎡ ⎤
+
⎢ ⎥
⎢ ⎥
⎣ ⎦
1 2 c
2 c
B D E
r(t)
B E
, (16)
=
1 1 1
y
(t) C x (t)
. (17)
The block diagram of the designed augmented system
(7) with the controller (15) is shown in Fig. 3.
)(
2
tx
)(tr
)(
1
ty
)(
1
tu
)()()(
2222
2
tuBtxAtx +=
•
)()()(
1111
1
tuBtxAtx +=
•
2
K
)(
2
tu
2
D
1
K
1
C
plant model
PID controller
c
E
)(
1
tx
Fig. 3. PID controlled system.
Compared with existing methods, advantages of our
method are: (i) MIMO PID parameters are systemati
cally tuned with respect to both plant and controller,
while the closedloop system stability is guaranteed;
(ii) There are no specific requirements on system stabil
ity, simplification model and plant decoupling.
IV. IMCBASED CONTROLLER
STRUCTURE
The first order rational Pade approximation of the
timedelay element discussed in section 2 inevitably
introduces some modeling error [26]. To deal with this,
the Internal Model Control (IMC) structure, which is
recognized as very usefully for handling imperfect mod
eling [3], is introduced below.
The typical IMC structure is shown as Fig. 4, in
which a nominal (approximate) plant model is con
structed as part of the controller. The control signal from
q is applied to the nominal plant model p, as well as the
plant
p
itself. By comparing the process output with
the expected process response, the difference caused by
modeling error and disturbance is fed back as shown.
If the model is exact (
p p
=
), then the above block
diagram of Fig. 4 will become open loop. It can be ar
gued that the lack of model uncertainty (error) is an arti
ficial assumption. However, in any practical situation it
is unacceptable to rely on model uncertainty to construct
the closedloop structure [13]. Besides, some other
transformation is needed to consider the specific struc
488
Asian Journal of Control, Vol. 6, No. 4, December 2004
ture of the proposed control system shown in Fig. 3.
Based on these considerations, we present the im
plementation block diagram for the proposed IMC based
feedback/feedforward PID controlled system in the fol
lowing Fig. 5. Following the definitions given in section
2, the approximate plant model is G
1
(s), and its rational
transfer function matrix is G
1
(s) = C
1
(sI – A
1
)
−1
B
1
+ D
1
.
On the other hand, the true plant model with timedelay
is described as Ĝ
1
(s). Besides, the PID controller trans
fer function matrix is G
c
(s) = − [K
2
(sI – A
2
)
−1
B
2
+ D
2
].
The advantages of such an IMC structure are: (i)
We do not need to design a plant state observer to ob
serve plant states, which are directly accessible in the
internal approximate plant model; (ii) Model uncertain
ties can be detected through comparing process output
and simulation result, thus providing great convenience
for online tuning.
)(tr
)(ty
controller
plant model
)(td
q
p
~
plant
p
Fig. 4. Internal Model Control (IMC) structure.
r
y
PID
1
B
1
C
∫
1
K
1
u
1
x
1
A
d
1
y
)(
ˆ
1
sG
)(sG
c
y
∆
y
plant
Fig. 5. Implementation structure of the proposed
analogously controlled system.
V. ROBUST STABILITY ANALYSIS
Next, we develop the robust stability analysis for
the modeling error. Since the external output distur
bance has previously been considered for steady state
compensation, and it has no effect on the stability
analysis, we assume there is no external disturbance in
the following.
The additive modeling error is defined as the dif
ference between the true plant model and the approxi
mate (nominal) plant model as
ˆ
= −
1 1 1
∆G (s) G (s) G (s)
. (18)
Assume the additive modeling error can be interpreted
as a scalar weight on a normalized perturbation,
( )
L s
=
⋅
1
∆G (s) ∆
, (19)
where ∆ is the normalized perturbation, that is ∆
∞
≤
1,
and
( )
L s
is a scalar weight variable. Then, we get the
modeling uncertainty description shown in Fig. 6 below,
which is transformed from Fig. 5, where G
x
(s) = [I + K
1
(sI − A
1
)
−1
B
1
]
−1
.
r
PID
1
B
1
C
∫
1
A
1
K
1
u
)(sG
c
∫
1
A
1
B
)(sG
x
1
y
a
b
∆
y
∆
)(
1
sG
y
)(sL
Fig. 6. Modeling error structure for robust analysis.
If the approximate model G
1
(s) has the same num
ber of unstable poles as that of the real model Ĝ
1
(s),
then it follows from robust stability analysis [32] that
the above system in Fig. 6 is robustly stable if
   ( )L s
∞
= ⋅ ⋅
ab x c
T (s) G (s) G (s)

−
∞
⋅
+ <
1
P c
[I G (s)G (s)] 1
, (20)
where
1 x
−
= − − + = ⋅
1
P 1 1 1 1 1 1
G (s) C [sI (A B K )] B D G (s) G (s)
.
The modeling error due to the delay time rational ap
proximation will increase with frequency increasing.
The filter factor in the PID controller can be regarded as
a kind of lowpass filter, which if not needed in the
output disturbance steady state compensation, can be
adjusted in the tuning up process until satisfactory per
formance and robust stability are achieved.
Actual modeling error is not limited to the time de
lay approximation; there will always be some mismatch
between the mathematical model and the physical sys
tem. It is still possible for the designed PID controller to
guarantee a zero tracking error for step input. Assume
( )
L
ω
is continuous, the qualitative analysis for the
condition is shown as below. Steady state “perfect con
trol” [13] means
−
⋅
⋅ + =
1
P c P c
G (0) G (0) [I G (0)G (0)] I
, (21)
which, when substituted into (20) gives the following
condition
 (0)   (0) 0  1
L
−
∞ ∞
=
⋅ <
1
ab 1
T G ( )
. (22)
This means the designed system will demonstrate a zero
steady state error despite modeling error, provided con
dition (22) is satisfied.
Y. Zhang et al.: Digital PID Controller Design for Delayed Multivariable Systems
489
VI. DIGITAL REDESIGN OF THE ANALOG
MULTIVARIABLE PID CONTROLLER
Finally, we now transform the obtained multivari
able analog PID controller to its digital implementation,
on the condition that the resulting mixedsignal (sam
pleddata) system response closely matches that of the
analogously controlled system.
The predictionbased digital redesign method [8]
can be briefly introduced as follows. Consider a linear
controllable continuoustime system, described by
= +
c c c
x (t) Ax (t) Bu (t)
,
=
c 0
x (0) x
, (23)
where x
c
(t) ∈ R
n
, u
c
(t) ∈ R
m
, A and B are constant ma
trices of appropriate dimensions. Let the continuous
time statefeedback control law be
= − +
c c c c
u (t) K x (t) E r(t)
, (24)
where K
c
∈ R
m × n
and E
c
∈ R
m × n
have been designed
to satisfy some specified goals, and r(t)
∈ R
m
is a
piecewiseconstant reference input vector.
The closedloop state of the digitally controlled
system will closely match that of the analogously con
trolled system at all the sampling instants, if the equiva
lent discretetime statefeedback control law is defined
as
= − +
*
d d d d
u (kT) K x (kT) E r (kT)
, (25)
where
−
= +
1
d c c
K (I K H) K G
,
−
= +
1
d c c
E (I K H) E
,
= +
*
r (kT) r(kT T)
,
=
AT
G e
,
−
= −
1
H (G I)A B
.
For details, readers are referred to the Appendix. As the
above redesign method cannot be applied directly to the
unity output feedback cascaded system, some transfor
mation as shown below is necessary.
From (16), the analogously controlled system has
the form as:
− − −
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
=
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
−
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
1 1 1 1 2 1 1 2 1
2 2 1 2 2
x (t) A B (K D C ) B K x (t)
x (t) B C A x (t)
−
⎡ ⎤
+
⎢ ⎥
⎢ ⎥
⎣ ⎦
1 2 c
2 c
B D E
r(t)
B E
,
(t)xC(t)y
111
=
. (26)
This can be transformed to the following structure,
⎡ ⎤ ⎡ ⎤
⎛ ⎞
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
= −
⎢ ⎥ ⎢ ⎥
⎜ ⎟
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎜ ⎟
⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎝ ⎠
⎣ ⎦ ⎣ ⎦
1 1
1 1 1 2
2 2
2 2
x (t) x (t)
A 0 B 0 K K
0 A 0 B 0 0
x (t) x (t)
( )
⎡ ⎤
−
⎡ ⎤
+ ⋅ −
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
⎣ ⎦
1
2
2
B 0
D
r(t)
y
(t)
I
0 B
. (27)
Define A
f
= block diag. {A
1
, A
2
}, B
f
= block diag. {B
1
,
B
2
},
⎡
⎤
=
⎢
⎥
⎣
⎦
1 2
f
K K
K
0 0
,
−
⎡ ⎤
=
⎢ ⎥
⎣ ⎦
2
f
D
E
I
. The discretetime
conversion gives the following terms, G
f
= block diag.
{G
d1
, G
d2
}, H
f
= block diag. {H
d1
, H
d2
}, in which G
d1
=
e
A
1
T
, H
d1
= [G
d1
− I]
−
1
1 1
A B
, and G
d2
= e
A
2
T
, H
d2
= [G
d2
− I]
−
1
2 2
A B
. Applying the aforementioned prediction
based digital redesign method, gives the corresponding
digital gains from (27) as
−
−
⎧
⎡ ⎤
= + ⋅ ⋅ =
⎪
⎢ ⎥
⎣ ⎦
⎪
⎪
⎨
⎪
⎡ ⎤
= + ⋅ =
⎪
⎢ ⎥
⎪
⎣ ⎦
⎩
d1 d2
1
d f f f f
d1
1
d f f f
K K
K (I K H ) K G
0 0
E
E (I K H ) E
I
. (28)
Then, we get the discretetime controlled system model
as:
+
⎡ ⎤
⎛
⎡
⎤
=
⎢ ⎥
⎜
⎢
⎥
⎜
⎢ ⎥
+
⎣
⎦
⎝
⎣ ⎦
d1
d1
d2
d2
x (kT T)
G 0
0 G
x (kT T)
⎡ ⎤
⎞
⎡ ⎤ ⎡ ⎤
−
⎢ ⎥
⎟
⎢ ⎥ ⎢ ⎥
⎟
⎢ ⎥
⎣ ⎦ ⎣ ⎦
⎠
⎣ ⎦
d1
d1 d1 d2
d2
d2
x (kT)
H 0 K K
0 H 0 0
x (kT)
( )
⎡ ⎤
⎡ ⎤
+ ⋅ −
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
⎣ ⎦
d1
d1
d2
H 0
E
r(kT)
y
(kT)
I
0 H
, (29)
from which, we can get the corresponding digital con
trol law as
= − −
d1 d1 d1 d2 d2
u (kT) K x (kT) K x (kT)
( )
+
−
d1
E r(kT)
y
(kT)
. (30)
Thus, we get the resulting mixedsignal system as
shown below in Fig. 7, where G
c
(z) = −K
d2
(zI –
G
d2
)
−1
H
d2
, z.o.h. represents the zeroorderhold, and the
part in the dotted line is the digital implementation
structure.
)(kTr
)(ty
1
C
)(
1
kTu
d
)(
1
kTx
d
)(td
)(
1
kTy
)(zG
c
1−
z
1d
K
...hoz
T
)(kTy
∆
)(kTy
1d
E
plant
1d
G
1d
H
Fig. 7. Implementation structure of the proposed mixedsignal
system.
490
Asian Journal of Control, Vol. 6, No. 4, December 2004
VII. AN ILLUSTRATIVE EXAMPLE
Consider the well known Wood and Berry empiri
cal model Ĝ
1
(s) of a pilotscale distillation column that
is used to separate a methanolwater mixture [4],
3
1
7 3
1
12.8 18.9
( )
( )
16.7 1 21 1
( ) ( )
6.6 19.4
10.9 1 14.4 1
s s
a
a
s s
b b
e e
u s
y s
s s
y s u s
e e
s s
− −
− −
⎡ ⎤
−
⎢ ⎥
⎡ ⎤
⎡ ⎤
+ +
⎢ ⎥
⎢ ⎥
=
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
−
⎣ ⎦
⎣ ⎦
⎢ ⎥
⎢ ⎥
+ +
⎣ ⎦
8.1
3.4
3.8
14.9 1
( )
4.9
13.2 1
s
s
e
s
d s
e
s
−
−
⎡ ⎤
⎢ ⎥
+
⎢ ⎥
+
⎢ ⎥
⎢ ⎥
⎢ ⎥
+
⎣ ⎦
, (31)
where y
a
(s) and y
b
(s) are the overhead and bottom com
positions of methanol, respectively; u
1a
(s) is the reflux
flow rate and u
1b
(s) is the steam flow rate to the reboiler;
d(s) is the feed flow rate, a disturbance variable.
Due to the existence of the dead time, a first order
Pade approximation model (1) is introduced. Substitut
ing it into (31), we get the parameter matrices of the
minimal realization of the rational approximation plant
model G
1
(s) as
1 0
1 0
0 1
0 1
1 0
1 0
0 1
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
=
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
1
B
,
The initial design for the PID controller G
2
(s) can be
carried out as (5b) of section 2 to get
1
( )
j
diag
s s
⎧
⎫
⎪
⎪
= +
⎨
⎬
+α
⎪
⎪
⎩ ⎭
2 2
G (s) D
, for
j
= 1, 2,
…
,
p
1,
(33a)
where D
2
is selected as the static decoupler, i.e. D
2
=
−
1
1
G (0)
. The filter factor α
j
can be chosen as the load
disturbance denominator for steady state disturbance
compensation according to IMP. Particularly note that
the timedelay term in the load disturbance is not neces
sarily considered in the PID controller design, because it
makes no difference for the closedloop performance if
a load disturbance enters directly at the plant output or
after passing through a time delay [13]. The preliminar
ily designed PID controller G
2
(s) is given as
1
0
( 0.0671)
0.1572 0.1531
0.0535 0.1036
1
0
( 0.0758)
s s
s s
⎡
⎤
⎢
⎥
+
−
⎡ ⎤
⎢
⎥
= +
⎢ ⎥
⎢
⎥
−
⎣ ⎦
⎢
⎥
⎢
⎥
+
⎣
⎦
2
G (s)
, (33b)
with its corresponding minimal realization as
0 0 0 0
0 0 0 0
0 0 0.0671 0
0 0 0 0.0758
⎡
⎤
⎢
⎥
⎢
⎥
=
⎢
⎥
−
⎢
⎥
−
⎢
⎥
⎣
⎦
2
A
,
1 0
0 1
1 0
0 1
⎡
⎤
⎢
⎥
⎢
⎥
=
⎢
⎥
⎢
⎥
⎢
⎥
⎣
⎦
2
B
,
14.9031 0 14.9031 0
0 13.1926 0 13.1926
−
⎡
⎤
=
⎢
⎥
−
⎣
⎦
2
C
,
0.1572 0.1531
0.0535 0.1036
−
⎡
⎤
=
⎢
⎥
−
⎣
⎦
2
D
. (34)
Based on the preliminarily designed PID
controller, a further tuning process with re
spect to the plant (31) is necessary to guar
antee satisfactory performance. Following
the proposed methodology, the PID tuning
problem has been converted to that of LQR
design. Formulating
the
plant
model
(32)
and
controller
(34) into a cascaded system
as (7), the optimal controller u
1
(t) in (8) can
be obtained through the aforementioned
method shown in section 3, by choosing Q =
diag {3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1}, R = I, as
=
− − −
1 1 1 2 2 2 2
u (t) K x (t) K x (t) D u (t)
, (35)
where
2 0 0 0 0 0 0
0 0.0599 0 0 0 0 0
0 0 0.6667 0 0 0 0
,
0 0 0 0.0476 0 0 0
0 0 0 0 0.2857 0 0
0 0 0 0 0 0.0917 0
0 0 0 0 0 0 0.0694
−
⎡ ⎤
⎢ ⎥
−
⎢ ⎥
⎢ ⎥
−
⎢ ⎥
=
−
⎢ ⎥
⎢ ⎥
−
⎢ ⎥
⎢ ⎥
−
⎢ ⎥
−
⎢ ⎥
⎣ ⎦
1
A
1.5802 0.8138 1.9385 1.0385 0 0 0
.
0 0 3.0078 0 1.7838 1.1783 1.6605
− −
⎡ ⎤
=
⎢ ⎥
− −
⎣ ⎦
1
C
(32)
Y. Zhang et al.: Digital PID Controller Design for Delayed Multivariable Systems
491
0.9640 0.2657 0.3056 0.0757
0.2657 0.9640 0.0860 0.2794
− −
⎡ ⎤
=
⎢ ⎥
⎣ ⎦
2
K
,
0.1572 0.1531
0.0535 0.1036
−
⎡ ⎤
=
⎢ ⎥
−
⎣ ⎦
2
D
.
This indicates the tuned PID controller (A
2
, B
2
, K
2
, D
2
)
has the form of
1
( ) ( )s s
−
= + −
2 2 2 2 2
G D K I A B
Eigenvalues of the optimally designed closedloop sys
tem (16a) are
indicating that the system is asymptotically stable. Note
that this is an advantage of the proposed methodology as
the closedloop system stability is guaranteed during
tuning, so that the designer is free from stability post
check.
Simulation results are shown in Fig. 8,
comparing different PID controller settings. Unit
setpoint changes were introduced in y
a
(t) (at t =
0) and y
b
(t) (at t = 100min), with a unit feedflow
step disturbance occurring at t = 200min.
The solid line represents the output per
formance of the real system (31), with the pro
posed feedback/feedforward PID controller (35),
and the corresponding approximate system re
sponse shown with the dashdot line. The best
performance as cited in reference [4], with PID
setting (β = 1, f = 1), is shown with the dotted
line in Fig. 8. It is observed that the proposed
method successfully achieves a better result for
every setpoint response. The limitation of the
proposed methodology is that its primary focus
is on the setpoint response, with disturbance
compensation only considered at steady state.
Some additional control action has to be added if
the disturbance rejection is needed during the
dynamic process. We also find that performance
mismatch
between
the
real
system and the appro
Fig. 8. Analogously controlled closedloop
system responses.
ximate system, which is ∆y in Fig. 5, is
reflected whenever the setpoint changs,
but that the real system output quickly
returns to the trajectory of the approxi
mate system nevertheless.
Selecting the sampling period as T
= 1min, and digitally redesigning the
analog controller (35) by using the de
veloped cascaded predictionbased
digital redesign method in (30), we get
the digital control law as
=
− −
d1 d1 d1 d2 d2
u (kT) K x (kT) K x (kT)
( )
+
−
d1
E r(kT)
y
(kT)
, (37)
where
0.9037 4.1269 2.1009 4.9944 1.6426 1.3113 2.6740
,
0.3072 0.8485 6.7723 1.3565 6.0612 5.7394 8.8842
− − −
⎡ ⎤
=
⎢ ⎥
− − −
⎣ ⎦
1
K
0.9640 4.5544 0.2657 0.9987
4.3972 0.8456
0.0671 0.0758
.
0.2657 1.2817 0.9640 3.6860
1.3352 3.5824
0.0671 0.0758
s
s
s s s s
s
s
s s s s
− −
⎡ ⎤
− + + + +
⎢ ⎥
+ +
=
⎢ ⎥
− −
⎢ ⎥
+ + + +
⎢ ⎥
+ +
⎣ ⎦
3.7847 2.9582 1.5687 0.6927 0.1945i
,
0.4847 0.0923 0.0396i 0.0380 0.0475 0.0541
− − − − ±
⎧ ⎫
⎪ ⎪
⎨ ⎬
− − ± − − −
⎪ ⎪
⎩ ⎭
(36)
0.0250 0.8095 0.1359 0.9832 0.3882 0.3987 0.7596
,
0.0032 0.0115 0.5623 0.0681 0.8224 0.9382 1.5045
− − −
⎡
⎤
=
⎢
⎥
− − −
⎣
⎦
d1
K
0.1989 0.0831 0.0589 0.0221
,
0.0144 0.1742 0.0046 0.0468
− −
⎡
⎤
=
⎢
⎥
⎣
⎦
d2
K
0.2950 0.1416
.
0.0049 0.2455
−
⎡
⎤
=
⎢
⎥
− −
⎣
⎦
d1
E
The output responses of the proposed system with the analog control
law (35), and those of the proposed system with digital control law (37)
are compared in Fig. 9.
It is observed that the digitally controlled system responses
closely match those of the analogously controlled ones. Obviously,
our proposed digital redesign method gives a smooth transition from
the analog control law to the digital control law. The analog control
law u
c1
(t) in (35), and the corresponding digital control law u
d1
(kT)
in (37) are shown in Fig. 10.
492
Asian Journal of Control, Vol. 6, No. 4, December 2004
VIII. CONCLUSION
In this paper, a new methodology has been pre
sented to design digital MIMO PID controllers for mul
tivariable analog systems with time delay. Compared
with existing methods, the proposed method offers the
following
advantages:
(i)
MIMO
PID
parameters
are
systematically tuned, while the closedloop system sta
bility is guaranteed; (ii) There are no specific require
ments on system stability, simplification model and
plant decoupling; (iii) The proposed system is very ro
bust due to the IMC structure utilized, and it provides
some convenience for online tuning; (iv) Designing a
plant state observer is unnecessary, as the plant states
are directly accessible in the internal approximate plant
model;
(v)
Digital
implementation
can
substantially
Fig. 9. Output responses of analogously controlled system
and digitally controlled system.
Fig. 10. Comparison between analog control law and digital
control law.
reduce the hardware complexity in realization and ad
justment. The simulation results indicate that the pro
posed methodology provides good performance in case
of considerable timedelay. Further research extending
the presented methodology for providing additional
control action for dynamic disturbance rejection, as well
as the physical realization of the proposed methodology
for mixedsignal system design is currently ongoing.
REFERENCES
1. Astrom, K.J. and T. Hagglund, PID Controllers: The
ory, Design, and Tuning, Instrument Society of Amer
ica, Research Triangle Park, NC, U.S.A., pp. 130
(1995).
2. Astrom, K.J. and B. Wittenmark, Computer Controlled
Systems: Theory and Design, Prentice Hall, NJ, U.S.A.,
pp. 129 (1997).
3. Boimond, J.L. and J.L. Ferrier, “Internal Model Con
trol and MaxAlgebra: Controller Design,” IEEE Trans.
Automat. Contr., Vol. 41, No. 3, pp. 457461 (1996).
4. Chen, D. and D.E. Seborg, “Multiloop PI/PID Con
troller Design Based on Gershgorin Bands,” IEE Proc.
Contr. Theory Appl., Vol. 149, No. 1, pp. 6873
(2002).
5. Deshpande, P.B., Multivariable Process Control, In
strument Society of America, Research Triangle Park,
NC, U.S.A., pp. 125 (1991).
6. Fujimoto, H., A. Kawamura, and M. Tomizuka, “Gen
eralized Digital Redesign Method for Linear Feedback
System Based on NDelay Control,” IEEE/ASME
Trans. Mechatron., Vol. 4, No. 2, pp. 101109 (1999).
7. Goodwin, G.C., S.F. Graebe, and M.E. Salgado, Con
trol System Design, Prentice Hall, NJ, U.S.A., pp.
267268 (2001).
8. Guo, S.M., L.S. Shieh, G. Chen, and C. F. Lin, “Effec
tive Chaotic Orbit Tracker: A PredictionBased Digital
Redesign Approach,” IEEE Trans. Circuits Syst. – I,
Fundamen. Theory Appl., Vol. 47, No. 11, pp.
15571570 (2000).
9. Isaksson,
A.
and
T.
Hagglund,
Eds.,
“PID Control,”
IEE Proc. Contr. Theory Appl., Special Issue, Vol. 149,
No. 1, pp. 181 (2002).
10. Katayama, M., T. Yamamoto, and Y. Mada, “A De
sign of Multiloop Predictive SelfTuning PID Control
lers,” Asian J. Contr., Vol. 4, No. 4, pp. 472481
(2002).
11. Moradi, M.H., M.R. Katebi, and M.A. Johnson, “The
MIMO Predictive PID Controller Design,” Asian J.
Contr., Vol. 4, No. 4, pp. 452463 (2002).
12. Marshall, J.E., Control of TimeDelay Systems, Institu
tion of Electrical Engineers, London, UK, pp. 135
(1979).
Y. Zhang et al.: Digital PID Controller Design for Delayed Multivariable Systems
493
13. Morari, M. and E. Zafiriou, Robust Process Control,
Prentice Hall, NJ, U.S.A., pp. 120 (1989).
14. Ogata, K., Modern Control Engineering, Prentice Hall,
NJ, U.S.A., pp. 897902 (1996).
15. Pierre, D.A. and J.W. Pierre, “Digital Controller De
signAlternative Emulation Approaches,” ISA Trans.,
Vol. 34, No. 3, pp. 219228 (1995).
16. Shieh, L.S., W.M. Wang, and M.K. Appu Panicker,
“Design of PAM and PWM Digital Controllers for
Cascaded Analog Systems,” ISA Trans., Vol. 37, No. 3,
pp. 201213 (1998).
17. Sourlas, D.D. and V. Manousiouthakis, “Best Achiev
able Decentralized Performance,” IEEE Trans. Auto
mat. Contr., Vol. 40, No. 11, pp. 18581871 (1995).
18. Szita, G. and C.K. Sanathanan, “Robust Design for
Disturbance Rejection in Time Delay Systems,” J.
Franklin Inst., Vol. 334, No. 4, pp. 611629 (1997).
19. Szita, G. and C.K. Sanathanan, “A Model Matching
Approach for Designing Decentralized MIMO Con
trollers,” J. Franklin Inst., Vol. 337, No. 6, pp.
641660 (2000).
20. Tan, K.K., Ed., “Advances in PID Control,” Asian J.
Contr., Special Issue, Vol. 4, No. 4, pp. 364508
(2002).
21. Tan, K.K., S.N. Huang, and T.H. Lee, “Development
of a GPCBased PID Controller Design for Unstable
Systems with Deadtime,” ISA Trans., Vol. 9, No. 1, pp.
5770 (2000).
22. Tan, K.K., T.H. Lee, and F.M. Leu, “Optimal Smith
Predictor Design Based on a GPC Approach”, Ind.
Eng. Chem. Res., Vol. 41, pp. 12421248 (2002).
23. Tan, K.K., T.H. Lee, and F.M. Leu: “Predictive PI
Versus Smith Control for DeadTime Compensation,”
ISA Trans., Vol. 40, No. 1, pp. 1729 (2001).
24. Tan, K.K., Q.G. Wang, C.C. Hang, and T.J. Hagglund,
Advances in PID Control, Springer, London, UK, pp.
125 (1999).
25. Tan, W., T. Chen, and H.J. Marquez, “Robust Con
troller Design and PID Tuning for Multivariable Proc
esses,” Asian J. Contr., Vol. 4, No. 4, pp. 439451
(2002).
26. Wang, Q.G., C.C. Hang, and X.P. Yang, “SingleLoop
Controller Design via IMC Principles,” Automatica,
Vol. 37, No. 12, pp. 20412048 (2001).
27. Wang, Q.G., Q. Bi, and Y. Zhang, “Partial Internal
Model Control,” IEEE Trans. Ind. Electron., Vol. 48,
No. 5, pp. 976982 (2001).
28. Wassick, J.M. and R.L. Tummala, “Multivariable In
ternal Model Control for a FullScale Industrial Distil
lation Column,” IEEE Contr. Syst. Mag., pp. 9196
(1989).
29. Zhang, W.D., and X.M. Xu, “Optimal Solution, Quan
titative Performance Estimation, and Robust Tuning of
the Simplifying Controller,” ISA Trans., Vol. 41, No. 1,
pp. 3136 (2002).
30. Zhang, Y., Q.G. Wang, and K.J. Astrom, “Dominant
Pole Placement for MultiLoop Control Systems,”
Automatica, Vol. 38, No. 7, pp. 12131220 (2002).
31. Zheng, F., Q.G. Wang, and T.H. Lee, “On the Design
of Multivariable PID Controllers via LMI Approach,”
Automatica, Vol. 38, No. 3, pp. 517526 (2002).
32. Zhou, K. and J.C. Doyle, Essentials of Robust Con
trol, Prentice Hall, NJ, U.S.A., pp. 145146 (1998).
APPENDIX
The development of the predictionbased digital re
design method [8].
Consider a linear controllable continuoustime sys
tem, described by
( ) ( ) ( )
c c c
x
t Ax t Bu t
=
+
,
0
(0)
c
x
x=
, (A1)
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 statefeedback controller be
( ) ( ) ( )
c c c c
u t K x t E r t
=
− +
, (A2)
where K
c
∈ R
m
×
n
and E
c
∈ R
m
×
m
have been designed to
satisfy some specified goals, and r(t) ∈ R
m
is a piece
wiseconstant reference input vector. The controlled
system is
( ) ( ) ( )
c c c c
x
t A x t BE r t
=
+
,
0
(0)
c
x
x=
, (A3)
where A
c
= A − BK
c
. Let the state equation of a corre
sponding hybrid model be
( ) ( ) ( )
d d d
x
t Ax t Bu t
=
+
,
0
(0)
d
x
x=
, (A4)
where u
d
(t) ∈ R
m
is a piecewiseconstant input vector,
satisfying
( ) ( )
d d
u t u kT
=
for
( 1)kT t k T≤ < +
,
and T > 0 is the sampling period. Let the discretetime
statefeedback controller be
*
( ) ( ) ( )
d d d d
u kT K x kT E r kT
= − +
, (A5)
where K
d
∈ R
m
×
n
is a digital statefeedback gain, E
d
∈
R
m
×
m
is a digital feedforward gain, and r
*
(kT) ∈ R
m
is a
piecewiseconstant reference input vector to be deter
mined in terms of r(t) for tracking purpose. The digitally
controlled closedloop system thus becomes
*
( ) ( ) [ ( ) ( )]
d d d d d
x
t Ax t B K x kT E r kT
= + − +
,
0
(0)
d
x
x
=
for
( 1)kT t k T
≤
< +
. (A6)
A zeroorderhold device is used for (A5). The
digital redesign problem is to find digital controller
gains (K
d
, E
d
) in (A5) from the analog gains (K
c
, E
c
) in
494
Asian Journal of Control, Vol. 6, No. 4, December 2004
(A2), so that the closedloop state x
d
(t) in (A6) can
closely match the closedloop state in (A3) at all the
sampling instants, for a given r(t) ≡ r(kT), k = 0, 1, 2,….
The state x
c
(t) in (A1), at t = t
v
= KT + vT for 0 ≤ v
< 1, is found to be
( ) exp( ( )) ( )
c v v c
x
t A t kT x kT
= −
exp( ( )) ( )
v
t
v c
kT
A
t Bu d
+ −τ τ τ
∫
. (A7)
Let u
c
(t
v
) be a piecewiseconstant input. Then equation
(A7) reduces to
( ) exp( ) ( )
c v c
x
t AvT x kT
≈
exp( ( )) ( )
kT vT
c v
kT
A
kT vT Bd u t
+
+ + −τ τ
∫
( ) ( )
( ) ( )
v v
c c v
G x kT H u t
= +
, (A8)
where
( )
exp( ( )) exp( )
v
v
G A t KT AvT
= − =
(exp( )) ( )
v v
A
T G
= =
( )
exp( ( ))
v
t
v
v
kT
H
A t Bd
= −τ τ
∫
( ) 1
0
exp( ) [ ]
vT
v
n
A
Bd G I A B
−
= τ τ = −
∫
.
Here, it must be noted that [G
(v)
− I
n
]
A
−
1
is a shorthand
notation, which is well defined as can be verified by a
cancellation of A
−
1
in the series expansion of the term
[G
(v)
− I
n
]. This convenient notation for an otherwise
long series will be used throughout the paper.
Also, the state x
d
(t) of (A4), at t = t
v
= kT + vT for 0
≤ v ≤ 1, is obtained as
( ) exp( ( )) ( )
d v v d
x
t A t kT x kT
= −
exp( ( )) ( )
v
t
v d
kT
A
t Bd u kT
+ −τ τ
∫
( ) ( )
( ) ( )
v v
d d
G x kT H u kT
= +
. (A9)
Thus, from (A8) and (A9) it follows that to obtain the
state x
c
(t
v
) = x
d
(t
v
), under the assumption of x
c
(kT) =
x
d
(kT), it is necessary to have u
d
(kT) = u
c
(t
v
). This leads
to the following predictionbased digital controller:
( ) ( ) ( ) ( )
d c v c c v c v
u kT u t K x t E r t
= = − +
( ) ( )
c d v c v
K
x t E r t
= − +
, (A10)
where the future state x
d
(t
v
) (denoted as the predicted
state) needs to be predicted based on the available causal
signals x
d
(kT) and u
d
(kT).
Substituting the predicted state x
d
(t
v
) in (A9) into
(A10) and then solving it for u
d
(kT) results in
( ) 1 ( )
( ) ( ) [ ( ) ( )]
v v
d m c c d c v
u kT I K H K G x kT E r t
−
= + − +
.
(A11)
Consequently, the desired predicted digital controller
(A5) is found, from (A11), to be
( ) ( ) *
( ) ( ) ( )
v v
d d
d d
u kT K x kT E r kT
= − +
, (A12)
where, for tracking purpose, r
*
(kT) = r(kT + vT), and
( ) ( ) 1 ( )
( )
v v v
m c c
d
K
I K H K G
−
= +
,
( ) ( ) 1
( )
v v
m c c
d
E I K H E
−
= +
.
In particular, if v = 1, then the prerequisite x
c
(kT) =
x
d
(kT) is ensured. Thus, for any k = 0, 1, 2,…, the con
troller is given by
*
( ) ( ) ( )
d d d d
u kT K x kT E r kT
= − +
, (A13)
where
1
( )
d m c c
K
I K H K G
−
= +
,
1
( )
d m c c
E I K H E
−
= +
,
*
( ) ( )r kT r kT T
=
+
,
in which G = exp(AT) and
H = (G − I
n
)A
−
1
B.
In selecting a suitable sampling period for the digi
tal redesign method, a bisection searching method is
suggested to find an appropriate long sampling period,
so that the reasonable tradeoff between the closedloop
response (i.e., the matching of the states x
c
(kT) in (A8)
and x
d
(kT) in (A9)) and the stability of the closedloop
system can be achieved.
Yongpeng Zhang received his B.S.
degree from Xi’an University of
Technology, Xi’an, China in 1994, and
M.S. and Ph.D. degree from Tianjin
University, Tianjin, China, and Uni
versity of Houston, Houston, Texas,
USA, in 1999 and 2003 respectively,
all in electrical engineering. He was a postdoctoral re
searcher at CECSTR, Prairie View A&M University.
Currently he is an Assistant Professor in Engineering
Technology Department, Prairie View A&M University
Texas, U.S.A. His main research interests are digital
control, mixedsignal systems, and DSP solutions for
motor direct drive.
Y. Zhang et al.: Digital PID Controller Design for Delayed Multivariable Systems
495
LeangSan Shieh
received his B.S.
degree from the National Taiwan Uni
versity, Taiwan in 1958, and his M.S.
and Ph.D. degrees from the University of
Houston, Houston, Texas, in 1968 and
1970, respectively, all in electrical engi
neering. He is a Professor in the De
partment of Electrical and Computer
Engineering and the Director of the Computer and Systems
Engineering. He was the recipient of more than ten College
Outstanding Teacher Awards, the 1973 and 1997 College
Teaching Excellence Awards, and the 1988 College Senior
Faculty Research Excellence Award from the Cullen Col
lege of Engineering, University of Houston, and the 1976
University Teaching Excellence Award and the 2002 El
Paso Faculty Achievement Award from the University of
Houston. He has published more than two hundred articles
in various referred scientific journals. His fields of interest
are digital control, optimal control, selftuning control and
hybrid control of uncertain systems.
Cajetan M. Akujuobi
received the B.S.
degree from Southern University, Baton
Rouge, Louisiana in 1980, and his M.S.
and Ph.D. from Tuskegee University,
Tuskegee, Alabama and George Mason
University, Fairfax, Virginia in 1983 and
1995 respectively, all in Electrical Engi
neering. He has also an MBA Degree from Hampton Uni
versity, Hampton, Virginia, in 1987 in Business Admini
stration. He is an Associate Professor in the Department of
Electrical Engineering and is the founding Director of the
Analog & Mixed Signal (AMS), DSP Solutions and High
Speed (Broadband) Communication Programs at Prairie
View A&M University. He is also the founding Director
of the Center of Excellence for Communication Systems
Technology Research (CECSTR). He has been a member
of the IEEE for over 24 years and is one of the founding
corporate members of the IEEE Standards Association
(IEEESA), Industry Advisory Committee (IAC). He is a
Senior Member of IEEE, Senior Member of ISA, ASEE,
SPIE, and Sigma XI, the Scientific Research Society.
His current research interests include all areas of signal
and image processing and communication systems
(Broadband Telecommunications) using such tools as
wavelet and fractal transforms. Other research interests are
in the areas of DSP Solutions, analog mixed signal system,
control systembased communications. He was a partici
pant and collaborative member of ANSI T1E1.4 Working
Group that had the technical responsibility of developing
T1.413, Issue 2 ADSL Standard. He has been published
extensively and has also written many technical reports. He
was selected as one of U.S. representatives for engineering
educational and consultation mission to Asia in 1989.
Warsame H. Ali
was born in Moga
disho, Somalia, in 1955. He received
the Bsc degree in electrical engineering
from King Saud University of Riyadh,
Saudi Arabia, in 1986. He received MS
from Prairie View A&M University,
Prairie View, Texas, USA in 1988.
Mr. Ali is currently pursuing the PhD
degree in electrical engineering at University of Houston,
Houston, Texas, USA. Presently, he is an instructor of
electrical engineering at Prairie View A&M University
since 1988. His main research interests are concerned with
the application of digital PID controllers, digital methods to
electrical measurements, mixed signals testing techniques,
and hybrid vehicles.
.
Enter the password to open this PDF file:
File name:

File size:

Title:

Author:

Subject:

Keywords:

Creation Date:

Modification Date:

Creator:

PDF Producer:

PDF Version:

Page Count:

Preparing document for printing…
0%
Σχόλια 0
Συνδεθείτε για να κοινοποιήσετε σχόλιο