DIGITAL PID CONTROLLER DESIGN FOR DELAYED MULTIVARIABLE SYSTEMS

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Asian Journal of Control, Vol. 6, No. 4, pp. 483-495, December

2004

483

Manuscript received May 27, 2003; revised August 8, 2003;
accepted November 19, 2003.
Yongpeng

Zhang

and

Leang-San

Shieh

are

with

the
Department of Electrical and Computer Engineering, Uni-
versity of Houston, Houston, TX 77204-4005, 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 19-02-1-0321, and Texas Instru-
ments under Grant #410171-03001, and NASA-JSC unde
r
Grant #NNJ04HF32G.
DIGITAL PID CONTROLLER DESIGN FOR DELAYED
MULTIVARIABLE SYSTEMS
Yongpeng Zhang, Leang-San 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 state-feedback and controller state-feedforward
LQR approach, such that satisfactory performance with guaranteed
closed-loop 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 prediction-based digital re-
design method, the digital implementation is obtained based on the
above-proposed analog controller, such that the resulting mixed-signal 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 Proportional-Integral-Derivative (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 Single-Input-Single-Output (SISO) PID con-
trollers and Multi-Input-Multi-Output (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 off-diagonal
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 [10,11,21-23,
28] have been developed in the discrete-time 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 discrete-time domain.
Nevertheless, the direct digital design methods (such as
the deadbeat design method), in general, cannot take
care of the inter-sample 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-
crete-time 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
high-order 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 continuous-time 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., mixed-signal system), which consists of a
continuous-time plant and a discrete-time controller [16].
The digital redesign approach is adopted here based on
the following considerations: (i) Continuous-time 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 inter-sample behavior [8] of the designed
closed-loop mixed-signal (sampled-data) 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 pre-selected. 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-
mum-phase property, low-order 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 unity-feedback
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 delay-free 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 self-tuning [10,21], which will considerably in-
crease the real-time 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
IMC-based 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

IMC-based PID controller
is not necessary to endure a reduced-order 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 prediction-based digital
redesign method [8], such that the mixed-signal 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 state-feedback and state-feed-
forward LQR are discussed in section 3. The IMC-based
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 unity-output 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. Continuous-time 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
pre-compensator 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 non-singular.
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
⎧ ⎫
⎪ ⎪
= +
⎨ ⎬

⎪ ⎪
⎩ ⎭
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 STATE-FEEDBACK AND
STATE-FEEDFORWARD 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 fine-tuning the PID controller parameters in
the context of an LQR formulation. This further tuning
step is necessary to achieve satisfactory closed-loop
performance.
Based on the pre-designed 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 closed-loop 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 state-feedback 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 state-feedback 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 closed-loop 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
state-feedback and state-feedforward control law
u
1
(t)
in
(8) can be indirectly determined from the state-feedback
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 closed-loop 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 closed-loop system stability is guaranteed;
(ii) There are no specific requirements on system stabil-
ity, simplification model and plant decoupling.
IV. IMC-BASED CONTROLLER
STRUCTURE
The first order rational Pade approximation of the
time-delay 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 closed-loop 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 time-delay
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 on-line 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 low-pass 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 mixed-signal (sam-
pled-data) system response closely matches that of the
analogously controlled system.
The prediction-based digital redesign method [8]
can be briefly introduced as follows. Consider a linear
controllable continuous-time 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 state-feedback 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
piecewise-constant reference input vector.
The closed-loop 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 discrete-time state-feedback 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 discrete-time
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 afore-mentioned 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 discrete-time 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 mixed-signal system as
shown below in Fig. 7, where G
c
(z) = −K
d2
(zI –
G
d2
)
−1
H
d2
, z.o.h. represents the zero-order-hold, 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 mixed-signal
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 pilot-scale distillation column that
is used to separate a methanol-water 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




= +





⎩ ⎭
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 time-delay term in the load disturbance is not neces-
sarily considered in the PID controller design, because it
makes no difference for the closed-loop 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 afore-mentioned
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 closed-loop sys-
tem (16a) are





indicating that the system is asymptotically stable. Note
that this is an advantage of the proposed methodology as
the closed-loop 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 feed-flow
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 dash-dot 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 set-point response. The limitation of the
proposed methodology is that its primary focus
is on the set-point 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 closed-loop
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 prediction-based
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 closed-loop 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 time-delay. 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 mixed-signal system design is currently ongoing.
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APPENDIX
The development of the prediction-based digital re-
design method [8].
Consider a linear controllable continuous-time 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 state-feedback 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-
wise-constant 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 piecewise-constant input vector,
satisfying
( ) ( )
d d
u t u kT
=
for
( 1)kT t k T≤ < +
,
and T > 0 is the sampling period. Let the discrete-time
state-feedback controller be
*
( ) ( ) ( )
d d d d
u kT K x kT E r kT
= − +
, (A5)
where K
d
∈ R
m
×
n
is a digital state-feedback gain, E
d

R
m
×
m
is a digital feedforward gain, and r
*
(kT) ∈ R
m
is a
piecewise-constant reference input vector to be deter-
mined in terms of r(t) for tracking purpose. The digitally
controlled closed-loop 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 zero-order-hold 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 closed-loop state x
d
(t) in (A6) can
closely match the closed-loop 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 piecewise-constant 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 prediction-based 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 pre-requisite 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 closed-loop
response (i.e., the matching of the states x
c
(kT) in (A8)
and x
d
(kT) in (A9)) and the stability of the closed-loop
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 post-doctoral 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, mixed-signal systems, and DSP solutions for
motor direct drive.
Y. Zhang et al.: Digital PID Controller Design for Delayed Multivariable Systems

495


Leang-San 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, self-tuning 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
(IEEE-SA), 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 system-based 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.



































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