Digital self-tuning PID control of pressure plant with closed-loop optimization

Abstract

In this paper we propose a method for optimization of closed-loop parameters and continuous-time sampling period by digital self-tuning PID control of pressure plant. The quality of pressure plant control is expe-rimentally compared between two modifications of digital self-tuning PID controllers. The results of adaptive pressure plant control show that the optimization of closed-loop parameters and sampling period provides significantly im-proved control performance.

Full text

ISSN 1392 – 124X INFORMATION TECHNOLOGY AND CONTROL, 2011, Vol.40, No.3
DIGITAL SELF-TUNING PID CONTROL OF PRESSURE PLANT
WITH CLOSED-LOOP OPTIMIZATION
Gediminas Liaučius, Vytautas Kaminskas, Raimundas Liutkevičius
Vytautas Magnus University, Department of Systems Analysis
Vileikos St. 8, LT-3035 Kaunas, Lithuania
e-mail: g.liaucius@if.vdu.lt, v.kaminskas@if.vdu.lt, r.liutkevicius@if.vdu.lt
Abstract. In this paper we propose a method for optimization of closed-loop parameters and continuous-time
sampling period by digital self-tuning PID control of pressure plant. The quality of pressure plant control is expe-
rimentally compared between two modifications of digital self-tuning PID controllers. The results of adaptive pressure
plant control show that the optimization of closed-loop parameters and sampling period provides significantly im-
proved control performance.
Keywords: pressure plant, self-tuning PID controller, closed-loop parameters, sampling period, optimization.
1. Introduction
Nowadays, modern modelling, simulation, adapta-
tion and intellect methods in control systems of va-
rious technological processes and plants are used [1, 5,
7-9, 11]. Technological processes commonly are conti-
nuous-time plants. For the digital control of conti-
nuous-time plant a sampling period of the signals is
necessary to choose, which impacts the closed-loop
characteristics [2].In digital PID control, the closed-
loop characteristics are commonly decided by two
parameters – the natural frequency of oscillation and
the damping factor. The digital self-tuning PID control
based on direct model identification is associated with
the discrete model building and online identification
of its parameters. Sampling period influences the
location of the roots of model polynomials in the unit
circle. On a short sampling period, the roots of model
polynomials locate close to the limit of stability
domain. In the process of online identification, this
can cause the discrete model to become unstable [6].
All of this affects control error.
In this paper, a new method, as a solution for this
problem, is proposed – the closed-loop parameters and
sampling period are chosen by optimizing the control
quality criterion. The experimental investigation of
method effectiveness for pressure plant is performed
by comparing two digital self-tuning PID controllers
[10].
2. The plant
The scheme of pressure plant is demonstrated in
Figure 1. The plant consists of four main components:
the combined air inlet (no. 1) and outlet (no. 4) tanks,
two air chambers (no. 2) and two tubes (no. 3) with
balls (no. 6) in them. The air from the inlet tank flows
to air channels through air chambers and leaves the
equipment through the upper outlet tank. The distance
to balls is measured using ultrasound distance sensors
(no. 5). The fans (no. 7) are used to create pressure in
the air channels in order to lift the balls in tubes. The
air chambers are utilized for the purpose to stabilize
oscillations of the pressure in each tube.
Figure 1. The scheme of pressure plant
The momentum of the fan, the inductance of the
fan motor, air turbulence in the tube leads to complex
physics governing ball's behaviour. Slightly different
202
http://dx.doi.org/10.5755/j01.itc.40.3.628

Digital Self-Tuning PID Control of Pressure Plant with Closed-Loop Optimization
weights of the balls and the location of air feeding
vent additionally impact the behaviour of ball in the
tubes.
The input signals of the plant are the voltage
values for each fan and the output signals are the dis-
tances between balls and the bottom of their tubes in
centimetres.
The control objective of pressure plant is to regu-
late the speed of a fan supplying the air into a tube so
as to keep a ball suspended at some predetermined le-
vel in the tube.
Such plants exist in air conditioning and cooling
systems.
The elements of control system and technical cha-
racteristics of the plant are as follows: the volume of
air inlet tank is about 7000 cm3, the diameter of air
feeding valve – 7 cm. The volume of each air chamber
is about 1300 cm3. Each tube is 110 cm long with a
diameter of 4 cm. The volume of air outlet tank is
about 2900 cm3, the diameters of air outlet vents - 6.5
cm. The weights of the balls for the first and for the
second tube are 3.62 g. and 3.58 g. respectively. For
each air chamber, two coupled “Zalman PS80252H”
fans are utilized and “Nivelco Microsonar UTP-212-
4” ultrasound sensors are used for sensing the
positions of the balls. The Beckhoff BK9000 PLC is
used for digital control of pressure plant, i.e. reading
output signals from sensors and sending input signals
to the control mechanism of the fans. Controller is
configured and controlled by TwinCat software.
3. Digital self-tuning PID controllers
The tubes of pressure plant are defined by discrete
linear second order models, that is
,)()( )()(1)()(1)( i
t
i
t
ii
t
iuzByzA

  (1)
, (2)
)( 2)(
2
1)(
1
1)(   zbzbzB iii
, (3)
1)( 2)(
2
1)(
1
1)(   zazazA iii
where denotes the tube of the plant,
, – output and input
signals with sampling period , respectively, – a
white noise of the tube with a zero mean and
finite variance and
2,1i
)( 0
)( tTy i
thi
)
0
tT
0
T
)(
yi
t(
)()( uu ii
t
thi
1
)(i
t

z
is the backward-shift operator.
The digital control of pressure plant is modelled by
two types of digital self-tuning PID controllers: PID-A
and PID-B [10]. The PID-A controller is defined as
[10]
,)()( )(1)()(1)( i
t
ii
t
iezRuzS   (4)
),1)(1()( 1)(11)(   zzzS ii

(5)
,)( 2)(
2
1)(
1
)(
0
1)(   zrzrrzR iiii (6)
, (7)
)(
*
)()( i
t
i
t
i
tyye 
where is a reference signal, - a control error
and are the parameters of PID-A
controller of the tube.
*
)(i
t
y
(
0
)( ,i
ir

)(i
t
e
)(
2
)(
1
),, ii rr
thi
In order to calculate the parameters of PID-A cont-
roller the desired polynomial of closed-loop characte-
ristic is needed, which for the tube is formulated
as thi


  d
n
jd
ji
j
inzdzD
1
)(1)( ,4,1)( (8)
and is defined by converting continuous-time second
order system
, (9)
02 22  iii ss

to its discrete analogue by Laplace transformation.
The coefficients of this polynomial are then calculated
by [4]
,0
),2exp(
,
1),1cosh()exp(2
1),1cos()exp(2
)(
4
)(
3
0
)(
2
2
00
2
00
)(
1










ii
ii
i
iiiii
iiiii
i
dd
Td
ifTT
ifTT
d



(10)
where i

is the natural frequency of oscillation, i

is
the damping factor.
The parameters of PID-A controller are computed
by [4]
,
)(
2
)(
2
)(
2
)(
i
i
ii
a
b
r

(11)
),1(
1)()(
1
)(
1
)(
1
)(
0iii
i
iad
b
r

 (12)
),1( )(
2
)(
1
)(
2
)(
1
)(
2
)(
2
)(
2
)(
1 i
i
i
i
i
i
i
i
a
a
b
b
r
b
a
r (13)
)],)([( )(
1
)(
2
)(
2
)(
1
)(
2
)(
1
)(
2
)(
2iiiiiiii bababbar 
 (14)
),( )(
2
)(
1
)(
2
)(
2
)(
1
)(
2
)(
2
)(
2iiiiiiii bdbdbbar 
 (15)
,
])()(][[ 2)(
2
2)(
1
)(
2
)(
2
)(
1
)(
1
)(
2
)(
1
)(
2
)(
2
)(
2iiiiiiii
ii
i
bbabbabb
rr
r



 (16)
The PID-B controller is defined as [10]
, (17)
)()( )(1)()()()(1)( i
t
ii
t
ii
t
iyzReuzS  

),1)(1()( 1)(11)(   zzzS ii

(18)
.)(
)1)(1()(
2)(
2
1)(
2
)(
0
)(
0
1
)(
0
)(
2
1)(
0
1)(




zrzrrr
z
r
r
zrzR
iiii
i
i
ii (19)
The parameters of PID-B controller of the tube
are computed by [4] thi
),1(
1)(
0
)(
1
)()(
1
)(
1
)(
1
)( iiiii
i
irbad
b

(20)
,
)(
2
)(
2
)(
2
)(
i
i
ii
a
b
r

(21)
,)( )(
2
)(
2
)(
2
)(
1
)(
2
)(
1
)(
2
)(
0i
i
i
i
i
i
ii
b
a
a
a
b
b
rr  (22)
203

G. Liaučius, V. Kaminskas, R. Liutkevičius
)],()([ )(
2
)(
2
)(
1
)(
2
)(
1
)(
1
)(
2
)(
2
)(
2iiiiiiiii adbbbabar 
 (23)
],)()1()[( 2)(
1
)(
2
)(
1
2)(
2
)(
2
)(
2iiiiii badbar 
 (24)
.
])()(][[ 2)(
2
2)(
1
)(
2
)(
2
)(
1
)(
1
)(
2
)(
1
)(
2
)(
2
)(
2iiiiiiii
ii
i
bbabbabb
rr
r



 (25)
Since the model parameters of each tube are
unknown, the technique of adaptive control with
indirect adaptation [3, 4] is applied (Figure 2).
Figure 2. The scheme of digital self-tuning PID control
of pressure plant
The unknown model parameters of the tube
are obtained by recursive least squares algorithm with
forgetting factor [4]
thi
,
,
ˆ
1
ˆ
1,
ˆ
ˆ)(
)(
)( 1
)(
)( 1
)()()()( 1
)(











otherwise
zoreif
i
t
i
t
i
t
i
t
i
t
i
j
i
e
i
t
i
t
i
t



φC
Θ
Θ
Θ (26)
],
ˆ
,
ˆ
,
ˆ
,
ˆ
[
ˆ)(
2
)(
1
)(
2
)(
1
)( iiii
T
i
tbbaaΘ (27)
],,,,[ )( 2
)( 1
)( 2
)( 1
)( 1i
t
i
t
i
t
i
t
T
i
tuuyy  φ (28)
,
)( 1
)()( 1
)( i
t
i
t
T
i
t
i
t
φCφ

(29)
,
ˆ
ˆ)( 1
)( 1
)()( i
t
T
i
t
i
t
i
ty
 φΘ

(30)













,
0,
0,
)()( 1
)(
)(
)(
1
)( 1
)( 1
)( 1
)( 1
)( 1
)(
i
t
i
t
i
t
i
t
i
t
i
t
T
i
t
i
t
i
t
i
t
i
t
if
if



C
CφφC
C
C (31)
,
1
)( 1
)(
)()(
i
t
i
ii
t






(32)
where is the estimated vector of model parame-
ters, – a square covariance matrix, – a data
vector, – the prediction error, – the forget-
ting factor, – auxiliary variables, – a
constant that defines the admissible interval of control
error and are the roots of polynomial
of the tube.
)(
ˆi
Θ
)(i
)(
ˆi

)
1
C
(z
)(i
φ
)(i

)()( ,ii

2,1,
)( j
i
j
thi
)(i
e

z

ˆ
Ai
t
In the modification of the algorithm (26), the
estimates of model parameters are updated only if the
value of is outside of the admissible interval de-
fined by and the model after its last estimation
remains stable.
)(i
t
e
)(i
e

4. Closed-loop optimization
The required control response or error of closed-
loop by digital self-tuning PID-A or PID-B controllers
can be achieved by appropriate selection of 0
,, T
ii


parameters of the tube.
thi
The control quality of the tube can be defined
by criterion thi
,])()[(
1
),,(
1
2)( 1
)(2)(
*
)(
0


 N
t
i
t
i
t
i
t
i
tiii uuyy
N
TQ

(33)
where is the number of observations,
N0

is a
weight coefficient.
In such a case, it is reasonable to find parameters
that minimize criterion (33)
*
0
** ,, T
ii

. (34)
)min(),,(:,,
0
,,
0
*
0
**
Tiiiiii
ii
QTQT



This problem is solved using sub-component opti-
mization methodology [6]
0
)min(),,(:
)min(),,(:,
0
)()()(
0
,
)1(
0
)()(
Ti
j
i
j
ii
j
i
j
iii
j
i
j
i
QTQT
QTQ
ii






(35)
,2,1j
Each of those problems is solved as follows. Since
the functions
),,,()( 0
)()(
0
)( TQTJ j
i
j
ii
j
i

 (36)
at the stage in sub-component optimization are
one-variable functions, it is reasonable to use one of
the most effective direct search method – golden sec-
tion method for their minimization [6]. Search algo-
rithm is related with an initial uncertainty interval
thj
],,0[],[ )(max00401 j
TTT  (37)
reduction to the interval
,],,[ )(
0
)(
01
)(
04
)(
04
)(
01 jLLLL TTTifTT  (38)
where its length is not longer than desired , and
with a function (36) minimum inside.
)(
0j
T
For this purpose, at the iteration of the search
procedure two new values of sampling period are
chosen
thj
0
T
,
)(618.0
)(382.0
)(
01
)(
01
)(
04
)(
03
)(
01
)(
01
)(
04
)(
02
llll
llll
TTTT
TTTT

 (39)
and a new uncertainty interval is then defined by the
rule
otherwiseTTTT
TJTJifTTTT
llll
lj
i
lj
i
llll
],,[],[
)()(],,[],[
)(
04
)(
02
)1(
04
)1(
01
)(
03
)()(
02
)()(
03
)(
01
)1(
04
)1(
01



 (40)
204

Digital Self-Tuning PID Control of Pressure Plant with Closed-Loop Optimization
205
Sampling period at the stage in sub-
component optimization is then calculated by
)(
0j
Tthj
.
2
)(
04
)(
01
)(
0
LL
jTT
T
 (41)
The maximum sampling period is obtained
by [2]
)(max0 j
T
.
6.0
)(
)(max0 j
i
j
T

 (42)
The functions of two variables
),,(),( )1(
0
)( 
j
iiiii
j
iTQF

(43)
are also minimized using sub-component optimization
method, where the golden section algorithms, analo-
gous to (37) – (42), are used for the search of the para-
meters i

and i

.
The optimization of closed-loop parameters and
sampling period is performed offline.
The reference signals with representative spectral
density must be applied in order to satisfy closed-loop
identifiability conditions [6]. A step form signal or the
signal with step changes are examples of such refe-
rence signal.
5. Experimental analysis
The pressure plant has been experimentally ana-
lysed using two types of digital self-tuning PID
controllers – PID-A and PID-B. The same closed-loop
characteristics have been used for digital control of
both tubes. The initial parameters of the algorithm
(26) are as follows: the main diagonal of covariance
matrix ) is selected equal to 1000 while the rest
entries are equal to zero, the forgetting factor is
equal to 0.99 and initial values of model parameters’
vector are set to zero. The same reference signal
for both tubes has been applied which has a step form
of repeatable values of 75 and 40, and with equal
to 1. The observation time of each signal is 1000 se-
conds, collecting data from the plant at one-second
intervals. Only the last 800 values of the signals are
included in criterion calculation, thereby eliminating
the impact of the initial controller training process
from it. The weight coefficient
(i
C
)(
0
ˆi
Θ
)(i

)(i
e


of criterion is set to
9 (in general, the control error of the tube is up to 90
centimetres, whereas the differences between two
individual values of input signal can be up to 10
volts).
The results of criterion optimization showed that
optimal closed-loop parameters and sampling period
for PID-A control are
0.17,=
*
i

0.9,=
*
i

0.08,
*
0T21,

i, and
for PID-B control. The minima of (33)
with optimal closed-loop parameters and sampling
period for the first and the second tubes of PID-A
control are 65.82 and 49.40 respectively, while in PID-
B control - 45.36 and 36.46, i.e. the minimal values of
criterion of PID-B control as compared to PID-A are
less than 31% for the first tube and 26% for the
second.
0.15,=
*
i

0
2.1,
0.9,=
*
i

08.0
*
0
T
Figure 3 illustrates the dependency of control
quality on sampling period with optimal closed-
loop parameters Points (circles and
squares) in graphs denote criterion values of each tube
and curves depict the smoothed version of those
values. We can see that the values of criterion of each
tube little vary when sampling period is between
0.06 and 0.125, but significantly increase if the
sampling period is chosen outside of this range.
T
,
*
i

,
*
i

i
0
T
Figure 3. The influence of sampling period on criterion values with optimal closed-loop parameters
0
T
0.9=0.17,= ** ii

for PID-A (left-hand graph) and for PID-B (right-hand graph) 0.9=0.15,= ** ii

The dependency of control quality on closed-loop
parameters ,
i

,
i

2.1,iwith a fixed sampling pe-
riod is demonstrated in Figures 4 and 5. Notice that
the closed-loop selected with natural frequency
between 0.08 and 0.17, and the damping factor
between 0.8 and 1.2 gives close to minimum criterion
values for both controls.

G. Liaučius, V. Kaminskas, R. Liutkevičius
Figure 4. Criterion values of PID-A control of each tube on various closed-loop parameters ,
i

i

with sampling
period 1.0
0

T
Figure 5. Criterion values of PID-B control of each tube on various closed-loop parameters ,
i

i

with sampling
period 1.0
0

T
The process of adaptive pressure plant digital cont-
rol with optimal parameters of closed-loop and opti-
mal sampling period is depicted in Figure 6 and model
parameters of online identification of each pressure
plant tube are depicted in Figure 7.
Figure 6. Control process of pressure plant with optimal closed-loop parameters and optimal sampling period
left-hand graph – PID-A
right-hand graph – PID-B
,08.0
*
0T
49.40),=65.82,=0.9,=0.17,=( *
2
*
1
** QQ
ii

36.46)=45.36,=0.9,=0.15,=( *
2
*
1
** QQ
ii

Figure 7. Online identification of pressure plant with optimal closed-loop parameters and optimal sampling period,
left-hand graph – PID-A, right-hand graph – PID-B
206

Digital Self-Tuning PID Control of Pressure Plant with Closed-Loop Optimization
Figure 8. Control process of pressure plant with optimal closed-loop parameters and sampling period
left-hand graph – PID-A
right-hand graph – PID-B
,1.0
0T
54.55),=85.16,=0.9,=0.17,=( 21
** QQ
ii

43.31)=50.09,=0.9,=0.15,=( 21
** QQ
ii

Figure 9. Control process of pressure plant with optimal closed-loop parameters and sampling period
left-hand graph – PID-A
right-hand graph – PID-B
,05.0
0T
83.51),=130.52,=0.9,=0.17,=( 21
** QQ
ii

304.84)=650.16,=0.9,=0.15,=( 21
** QQ
ii

Figure 10. Online identification of pressure plant with optimal closed-loop parameters and sampling period ,05.0
0

T
left-hand graph – PID-A, right-hand graph – PID-B
Figure 8 shows the control process of the plant
with optimal closed-loop parameters and sampling pe-
riod shifted to Notice that a small increase
of sampling period slightly increases the values of
criterion. The control results of the plant with optimal
closed-loop parameters, but with decreased sampling
period almost twice ( ) from its optimal va-
lue (Figure 9) show that the choice of too small samp-
ling period substantially decreases the quality of adap-
tive control. Model parameters of online identification
of each pressure plant tube are illustrated in Figure 10.
.1.0
0T
0
T
0
T05.0
Åström and Wittenmark recommends to select na-
tural frequency i

and sampling period so that
inequality
0
T
6.01.0 0


T
i

would be valid [2]. Figu-
re 11 illustrates that the selection of sampling period
is unable to improve the performance of adaptive
control with selected closed-loop parameter
0
T
i

further from its optimal value ( ). This
*
i

207

G. Liaučius, V. Kaminskas, R. Liutkevičius
recommendation causes to choose a relatively big i

with a relatively small or vice versa. Figure 12
presents the control process of the plant with a big
value of
0
T
i

and a small , when its model
parameters of online identification of each pressure
plant tube are presented in Figure 13. The control
process of pressure plant with a small value of
0
T
i

and
a big is depicted in Figure 14. In both cases, the
quality of adaptive control is heavily decreased as
compared to the control quality of the plant with
optimal closed-loop parameters ( ) and optimal
sampling period , which are obtained by (33) and
(34).
0
T
**,ii

*
0
T
Figure 11. The influence of sampling period on criterion values with closed-loop parameters
0
T
0.9=1.0,=
i i


for PID-A (left-hand graph) and PID-B (right-hand graph)
Figure 12. Control process of pressure plant with closed-loop parameters 0.91.0,= =
ii


and sampling period ,1.0
0

T
left-hand graph – PID-A (right-hand graph – PID-B 3001.57),=1673.02, 21 Q=Q2951.48)1794.74,=
1=
2
Q(Q
Figure 13. Online identification of pressure plant with closed-loop parameters 0.9=
ii 1.0,=


and sampling period
left-hand graph – PID-A control, right-hand graph – PID-B control ,1.0T0
208

Digital Self-Tuning PID Control of Pressure Plant with Closed-Loop Optimization
Figure 14. Control process of pressure plant with closed-loop parameters 1.0=0.2,= ii


and sampling period ,0.1
0

T
left-hand graph – PID-A right-hand graph – PID-B 180.40),=200.28,=( 21 QQ 161.02)=231.37,=( 21 QQ
6. Conclusions
The optimization method of closed-loop parame-
ters and sampling period for continuous-time plant
digital control has been proposed. The experimental
results showed that the quality of digital self-tuning
PID control of the pressure plant substantially depends
on the right choice of closed-loop parameters and con-
tinuous-time sampling period. Additionally, we have
shown that control quality of the plant is significantly
improved by optimizing those parameters. The opti-
mal values of natural frequency and sampling period
for digital control of pressure plant substantially differ
from those that are chosen using conventional metho-
dology.
References
[1] A.G. Ak, G. Cansever, A. Delibasi. Robot Trajectory
Tracking with Adaptive RBFNN-Based Fuzzy Sliding
Mode Control. Information Technology and Control,
2011, Vol.40, 151-156.
[2] K.J. Åström, B. Wittenmark. Computer-Controlled
Systems: Theory and Design. New Jersey: Prentice
Hall, Third Edition, 1997.
[3] K.J. Åström, B. Wittenmark. Adaptive control. Rea-
ding, Massachusetts: Addison-Wesley, 1989.
[4] V. Bobál, J. Böhm, J. Fessl, J. Macháček. Digital
Self-tuning Controllers. London: Springer-Verlag,
2005.
[5] V. Galvanauskas. Adaptive pH Control System for
Fed-Batch Biochemical Processes. Information Tech-
nology and Control, 2009, Vol.38, 225-231.
[6] V. Kaminskas. Dynamic System Identification via
Discrete-time Observation: Part 1. Statistical Method
Foundation. Estimation in Linear Systems, Vilnius:
Mokslas Publishers, 1982 (in Russian), 245.
[7] V. Kaminskas, R. Liutkevičius. Adaptive Fuzzy
Control of Pressure and Level. Information Technolo-
gy and Control, 2009, Vol.38, 232-236.
[8] R. Koerfer, R. Simutis. Advanced Process Control
for Fluidized Bed Agglomeration. Information Tech-
nology and Control, 2008, Vol.37, 285-293.
[9] D. Levišauskas, V. Galvanauskas, V. Čipinytė, S.
Grigiškis. Optimization of Feed-Rate in Fed-Batch
Culture Enterobacter Aerogenes 17 E13 for Maximi-
zation of Biomass Productivity. Information Technolo-
gy and Control, 2009, Vol. 38, 102-107.
[10] R. Ortega, R. Kelly. PID self-tuners. Some theore-
tical and practical aspects. IEEE Trans. Ind. Electron.,
Vol. 31, 1984, 332-338.
[11] G. Valiulis, R. Simutis. Particle Growth Modelling
and Simulation in Drum Granulator-Dryer. Informa-
tion Technology and Control, 2009, Vol.38, 147-152.
Received March 2011.
209