Jrl Syst Sci & Complexity (2006) 19: 227–235
MATTER-ELEMENT MODELING OF PARALLEL
STRUCTURE AND APPLICATION ABOUT
EXTENSION PID CONTROL SYSTEM∗
Rongde LU · Zonghai CHEN
Received: 22 March 2005 / Revised: 15 October 2005
Abstract This article describes in detail a new method via the extension predictable algorithm of the
matter-element model of parallel structure tuning the parameters of the extension PID controller. In
comparison with fuzzy and extension PID controllers, the proposed extension PID predictable controller
shows higher control gains when system states are away from equilibrium, and retains a lower profile
of control signals at the same time. Consequently, better control performance is achieved. Through
the proposed tuning formula, the weighting factors of an extension-logic predictable controller can be
systematically selected according to the control plant. An experimental example through industrial field
data and site engineers’ experience demonstrates the superior performance of the proposed controller
over the fuzzy controller.
Key words
rithm.
Extension control system, matter modeling, parallel structure, PID predictable algo-
1 Introduction
In recent years, advance proportional-integral-derivative (PID) controllers have been widely
used for complex industry process systems owing to their heuristic nature associated with
simplicity and effectiveness for both linear and nonlinear systems. At present, fuzzy PID
controllers can associate with nonlinear gains, which possess the potential to improve and
achieve better system performance because of the nonlinear property of control gains. On
the other hand, because of the existence of the nonlinear property, it is usually difficult to
conduct theoretical analyses to explain why advance PID controllers can achieve better system
performance[1−5] . Consequently, it is important, from both the theoretical and practical points
of view, to explore the essential nonlinear control properties of extension PID and develop
appropriate design methods that will assist site engineers to confidently utilize the nonlinear
property of extension PID controllers to improve closed-loop performance. This article deals
with a new tuning method, which is similarly based on the predictable algorithm of the matterelement model to determine the parameters of the proposed extension PID controller. The autotuning formula is applied here to decide the extension PID parameters (the weighing coefficients
for error, the change of error and controller output). Practical application demonstrates better
Rongde LU
Department of Astronomy and Application Physics, University of Science and Technology of China, Hefei
230026, China. Email: [email protected].
Zonghai CHEN
Department of Automation, University of Science and Technology of China, Hefei 230027, China.
∗ This work is supported by Youth Foundation of University of Science and Techonology of China (No. KA0001).
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RONGDE LU · ZONGHAI CHEN
control performance of the proposed extension PID controller that possesses higher control
gains but yields less control efforts than the convention and fuzzy PID controller.
2 Design of an Extension PID Control System
Usually, an extension PID controller is either a PD or a PI type depending on the output
of extension control rules. An extension PID controller may be constructed by introducing the
third information besides error and change in error, which is either rate of change in error or
sum of error, with a three-dimensional rule base. Such an extension PID controller is difficult to
construct[6] . In this article, we propose a parallel combination of extension PI and PD controller
to achieve an extension PID controller. The overall structure is shown in Fig. 1. Simplest
structures are used in each ELC. Three extension labels (positive, zero, and negative) are used
for the extension input variables and three extension labels (positive, zero, and negative) for
the extension output variables. Two main reasons motivate us to choose this type of ELC:
(1) theoretical analysis is possible owing to its simplicity and (2) the nonlinear property of the
simplest extension controller is the strongest one[1,3] . Therefore, we can expect better control
performance from this simplest structure controller as long as we can correctly use its nonlinear
property.
First, the error and the change of error are defined as
e(k) = r(k) − y(k),
∆e(k) = e(k) − e(k − 1).
(1)
The inputs of the extension controller are normalized error we∗ e and normalized change of
error w∆e∗ ∆e where we∗ and w∆e∗ are weighting factors. The notation ∗(∗ = {1, 2}) denotes
different types of ELCS. The relationship functions K(·) of extended inputs are defined in Fig.
2. According to this kind of relationship function, there are four extension labels – Pe , P∆e , Ne
and N∆e – for the two extension input variables, and the corresponding relationship functions
are described as

±1,
we∗ .e < −1,



KPE∗ = − 1 ± 1 we∗ .e, −1 ≤ we∗ e ≤ 1,


 2 2
0,
we∗ e ≥ 1,

±1,
we∗ .e < −1,



KNE∗ = ± 1 − 1 we∗ .e, −1 ≤ we∗ e ≤ 1,


 2 2
0,
we∗ e ≥ 1,

0,
we∗ .∆e, ∆e < −1,


(2)



1 1


−1 ≤ w∆e∗ ∆e ≤ 1,
 + w∆e∗ .∆e,
2 2
KP∆E∗ =

±1,
we∗ .e ≥ 1,






 − 1 − 1 w∆e∗ .∆e, −1 ≤ w∆e∗ ∆e ≤ 1,
2 2

±1,
w∆e∗ .∆e, w∆e ∆e < −1,




1 1
KN∆E∗ = ± − w∆e∗ .∆e, −1 ≤ w∆e∗ ∆e ≤ 1,

2 2



0, w∆e∗ .∆e,
w∆e∗ ∆e ≥ 1.
PARALLEL-EXTENSION MODELING AND APPLICATION
- W∆e1 -
We1
-
- W∆e2 -
We2
-
Knowledge base
(and database)
K(p)
229
Control effect matter (relationship function)
ELC1 W∆u -N
(PI)
6
z −1
ELC2 (PD)
N
?
K(s)
6
Wu
Control
matter
- Industry project
@
R
@
Figure 1
The overall structure of extension PID controller
Figure 2 Extension-dependent function of error e, change ∆e, and change rate in error e
Table 1 Control extension rules (ELC1 and ELC2 ) (N : negative; P : positive; Z: zero)
PI part ELC1
PD part ELC2
Rule
Rule
Rule
Rule
Rule
Rule
Rule
Rule
1
2
3
2
1
2
3
2
If e(N1 ) is N
If e(N1 ) is N
If e(N1 ) is P
If e(N1 ) is P
If e(N1 ) is N
If e(N1 ) is N
If e(N1 ) is P
If e(N1 ) is P
and ∆e(N1 ) is N , ∆u(N1 ) is N
and ∆e(N1 ) is P , ∆u(N1 ) is Z
and ∆e(N1 ) is N , ∆u(N1 ) is Z
and ∆e(N1 ) is P , ∆u(N1 ) is P
and ∆e(N2 ) is N , ∆u(N2 ) is N
and ∆e(N2 ) is P , ∆u(N2 ) is Z
and ∆e(N2 ) is N , ∆u(N2 ) is Z
and ∆e(N2 ) is P , ∆u(N2 ) is P
Consequently, only four simple extension control rules are used in each ELC (see Table 1).
The extension labels of control outputs are defined as P = 1, Z = 0 and N = −1. By using
230
RONGDE LU · ZONGHAI CHEN
product inference methods of Cai Wen[1] with extension logic AND & OR, and for simplicity
choosing we1 = we2 = we and w∆e1 = w∆e2 = w∆e , the control output of each ELC can be
obtained, in the universe of discourse, as
(E)
∆u1
(E)
∆u2
w∆u1
W∆u
(We1 e + W∆e1 ∆e) =
(we e + w∆e e),
4 − 2 max(We1 |e|, We1 |∆e|)
4 − 2α
w∆u2
Wu
=
(We2 e + W∆e2 ∆e) =
(we e + w∆e e),
4 − 2 max(We2 |e|, We2 |∆e|)
4 − 2α
=
(3)
where
α = max(we1 |e|, w∆e1 |∆e|) = max(we2 e, w∆e2 |∆e|) = max(we |e|, w∆e |∆e|).
The overall extension control output will be
(E)
uPID
=
k
X
(E)
∆u1
(E)
+ u2
0
k
X
w∆u w∆e
∆t
wu we
w∆e ∆t∆e
∆e +
e +
e+
. (4)
=
4 − 2α
(w∆e /we )t
4 − 2α
we ∆t
0
If we choose
Kc(E) =
w∆u w∆e
,
4 − 2α
(E)
(5)
(E) T
∆t
(E) ∆e
Kc(E) ∆e + (E) e + Kc(E) d(E) e + Ti
.
∆t
Ti
Ti
(6)
=
w∆e
∆t,
we
Td
wu we
,
4 − 2α
(E)
Ti
Kc(E)
(E)
Ti
=
Then the extension control output in Eq. (8) can be rewritten as
(E)
uPID =
k
X
0
Now assume that the time constants of the plant are sufficiently large compared with the
sampling interval, which is common and reasonable in process control, such that ė ≈ ∆e
∆t , Then
the overall control output can be approximated as
(E)
(E) (E) de
(E) Td
dt
+
K
e
+
T
c
i
(F )
(F )
dt
0
Ti
Ti
Z k∆t
Z k∆t (E)
(E) (E)
Kc Td
Kc
(E)
=
Kc(E) ėdt +
edt
+
(e + Ti ė).
(E)
(E)
0
0
Ti
Ti
uPID =
Z
k∆t
Kc(E) de +
e
(7)
Note that the linear PID controller in series form is
Z t
Z t
Kc
Kc
Kc Td
Gc (s) =
(1 + sTi )(1 + sTd ), or u =
Kc ėdt +
edt +
(e + Ti ė).
Ti s
T
Ti
i
0
0
Comparing Eq. (7) with the above formula we can conclude that the extension PID controller
(4) is a nonlinear PID controller with variable proportional gain.
3 Applications for the Moisture Process Control of Cut Tobacco
The drying process of cut tobacco is an important procedure in the technology of cigarette
products. Through heating, drying cut-tobacco moisture can meet technological requirements,
raise elasticity and filling ability, and improve the quality of cut tobacco. After cut tobacco
PARALLEL-EXTENSION MODELING AND APPLICATION
231
is dried, its moisture parameters will directly exercise a great influence on the internal quality
of cut tobacco. The overall structure of the machine for drying cut tobacco is shown in Fig.
3. Through the tuning vapor valve, the PID controller controls the temperature of the rolling
tube of cut tobacco in the drying machine so as to control cut-tobacco moisture. In this work
the process control system makes use of the above results and builds an extension predictable
algorithm of the matter model of parallel structure according to the moisture control plant.
Air input
Hot wind
heater
Air output
input
Cut tobacco output
Stream input
Drain water
Figure 3 Principle of drying cut tobacco
The process control system may obtain a tobacco moisture model by solving the fourth-step
Runge-Kutta integral (the concrete process is left out).
G=
1.62 × [10.13Ht × T − 101324.8 × HI ]/(662 + Hs )]
,
22308.86 × Ht + 1.62
(8)
T + (25 − T ) × exp{(T − 55)/(T − 25)}
,
12
where T is cut-tobacco temperature; HS , HI , Ho is standard moisture (13%), tobacco input
moisture, cut tobacco output moisture, respectively. The formula can depict the laws relating
to changing cut-tobacco moisture and its mathematical model of the project property.
Our research comes from a moisture control problem in cut tobacco in a drying machine,
which deals with differences between the present situation and expected states. To solve the
problem, we seek ways and channels through extending for expected states in the present
situation of cut tobacco in the drying machine, and think of goal, situation, restriction, and
plant in the process. For setting the matter-element model of the problem, we should first
define goal matter-element, situation matter-element, restriction matter-element, and project
matter-element and so on[7] .
Ho =
3.1 Feedback-Forward Extension Control Model for
Cut Tobacco Drying Rolling Tube I, II
Feedback-forward control about rolling tube I, II of cut tobacco in the drying machine
achieves predictable cut tobacco drying through setting up preceding feedback compensation
laws by the extension model, which fully applies tobacco-moisture control experience. According
to practical control data in the cigarette industry, modeling of rolling tube I, II of cut tobacco
in drying machine depicting matter-element temperature of enactment amount is defined as
For rolling tube I
Ri : if f1 is Aj1 , W1 is Aj2 , Tc1 is Aj3 , then Tc10 (t) = T0j + T1j f1 + T2j W1 + T3j Tc1 ;
(9)
For rolling tube II
Ri : if Tc2 is Aj1 , tc is Aj2 , then Tc20 (t) = T0j + T1j Tc2 + T2j Tc1 ,
(10)
232
RONGDE LU · ZONGHAI CHEN
where Tcj (j = 1, 2) is practical temperature of rolling tube I, II; Tj0 (j = 1, 2) is temperature of
matter-element amount of rolling tube I, II. The f1 , W , Tc , Tcj (j = 1, 2) about the extension
subset is divided into small, middle, and large, and the region about each variable is quantitized
[−1, +1], whose relationship function is defined as

 1,
x > 1,
KL (x) =
(x−aL)2
 e− bL ,
−1 < x < +1;
(x−aM )2
KM (x) = e− bM ;

2
 e− (x−aS)
bS
,
KS (x) =
 1,
(11)
−1 < x < +1,
x < −1,
where x is quantity after each variables is quantitized. The beginning quantity of aL, aM , aS is
respectively given by +0.5, 0, −0.5. Consequently, the rule number is 27 about matter-element
temperature of enactment-amount model in rolling tube I, and the rule number is 9 about
matter-element temperature of the enactment amount in the model of rolling tube II.
Then according to the extension modeling algorithm, the matter-element temperature of
enactment-amount model in rolling tubes I, II is obtained as
T10 =
P27
(12)
T20 =
P9
(13)
where Gi = Xµ Aij (xj0 ), j = 1, 2, 3,
Gi Tc10 (i)
,
P27 i
i=1 G
i=1
Gi Tc20 (i)
,
P9
i
i=1 G
i=1
where Gi = Xµ Aij (xj0 ), j = 1, 2. Each premise and conclusion parameters may be recognized
by the extension algorithm of the extension model.
3.2 Predictable Extension PID Control Model for
Cut Tobacco Drying Tube III
It has been determined that cut-tobacco output moisture is the nonlinear function for cuttobacco temperature, moisture, humidity around air, and there are nonlinear relations between
temperature of cut tobacco and that of the rolling tube in the drying machine, so the plant is
thought of as a fourth-input and single-output nonlinear project. At k − d − 1 second, matterelement amount H(k) about the predictable model output (output moisture of cut tobacco at
k second) is assumed as
H(k) = f [Tc (k − d − 1), T1 (k − d − 1), Hd (k − d − 1), H1 (k − d − 1)],
(14)
where Tc is the matter-element amount of rolling-tube temperature; TI is matter-element
amount of cut-tobacco input temperature; Hd is matter-element amount of humidity around
air; HI is matter-element amount of cut-tobacco input moisture; d is detained time. Using
one-step predictable control algorithm is given as
Hop (k + d + 1) = Howt (k + d + 1) + e(k),
(15)
PARALLEL-EXTENSION MODELING AND APPLICATION
233
where Hop (k + d + 1) is predictable output at k + d + 1 second.
E(k) = Ho (k) − Howt (k),
(16)
where Ho (k) is the practical output of project at k second; e( k) is model output error at k
second. If we choose the reference orbit as a straight line H or
(k + d + 1) = Hop (k + d + 1) = HR ,
(17)
where H or (k + d + 1) is expected output of project. Then
Hor (k + d + 1) = Hom (k + d + 1) + Ho (k) − Hom (k)
= f [Tc (k), tt (k), Hd (k, Ht (k))] + Ho (k) − Hom (k),
(18)
f (Tc (k, tt (k), Hd (k, Ht (k))) = Hor (k + d + 1) − Ho (k) + Hom (k).
(19)
In Eq. (17), Hor (k + d + 1), Ho (k), Hom (k) have been known, parameters in function f [·]
will be defined by recognizing off line and correcting on line, so control amount Tc (k) can be
obtained by Eq. (17).
PID predictable control model gives the future output in industry projections through future
practical output, and we can acquire predictable output error. Moreover, using automatically
correcting function, we can revise parameters of the predictable model on line by practical input–
output data. Therefore, correcting the predictable control system automatically treats the
basic layer that interference and robust are divided into so that control plant about nonlinear,
uncertainty and time-varying may provide superior performance.
For reference orbit of cut tobacco matter-element amount (cut-tobacco moisture) is a straight
line in which expected output of cut tobacco is
Hop (k + i) = HR , i = 1, 2, · · · , p + d.
(20)
For performance target, either in manufacturing technology or an actual manufacturing situation, there is no restriction on matter-element amount (control amount) so that the performance
target is given as
h−d
X
min J(k) =
q[Hr (K + i) − Hp (k + i)]2 ,
(21)
i=1
where HP (k + T ) is model output after correction.
Because cut tobacco drying is a slow time-varying process, using a predictable control pace
completely meets requirements, its performance target can be acquired. Matter-element amount
(control amount) Tc (k) is expressed as
HR = Hop (k + d + 1) = Hom (k + d + 1) + Ho (k) − Hom (k).
(22)
The predictable PID model can be written as
Hom (k + d + 1) = T0 + T1 × tt (k) + T2 × tt (k + 1) + T3 × Ho (k) + T4 × Ht (k),
(23)
where T0 , T1 , T2 , T3 , T4 is the result parameters of extension predictable model (see calculating
formula(27)),which is recognized off line and corrected on line (using the above practical inputoutput data Tc (k−d−1), Tt (k−d−1), Hd(k−d−1), HI (k−d−1) and Ho (k) correct parameters).
The control amount is expressed as
Tc (k) = HR = Hd (k) + Hom (k) − P0 + P2 × tt (k + 1) − P3 × Ho (k) − P4 × Ht (k).
(24)
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RONGDE LU · ZONGHAI CHEN
For the predictable extension PID model
Ri : if Tc3 is Ai1 , tt is Ai2 , Hd is Ai3 , HI is Ai4 .
Then
Hom (i) = T0i + T1i Tc3 + T2i Hd + T4i HI .
(25)
The Tc3 , Tt , Hd , HI about the extension subset is divided into small, middle, large, and the
region about each variable is quantitized [−1, +1], whose relationship function is defined as Eq.
(11), therefore giving the 81 extension control rules.
According to the extension modeling algorithm, the predictable output moisture of cut
tobacco is calculated as
P81
Gi hom (i)
Hom = i=1
.
(26)
P81 i
i=1 G
In formula (23), T0 , T1 , T2 , T3 , T4 is calculated as
P81 i i
P81 i i
G T1
G T0
Pi=1
,
T
=
,
T0 = Pi=1
81
81
1
i
i
i=1
G
P81 i i
G T3
,
T3 = Pi=1
81
i
i=1
G
i=1
G
P81 i i
G T4
T4 = Pi=1
.
81
i
i=1
P81 i i
G T2
T2 = Pi=1
,
81
i
i=1
G
(27)
G
3.3 Experimental Results
Experimental results shown as Figs. 4 and 5 demonstrate that in three kinds of possible disturbances of dry-top, dry-end, and breaking material extension PID controller approaches with
a strong robustness give lower control profile, especially achieving better contouring accuracy
than a fuzzy PID controller.
Figure 4 Fuzzy PID control modeling error fault of cut tobacco moisture
Using the extension PID controller, the system responding produces smaller margin to load
interference and a faster decay rate than the fuzzy PID controller. In conclusion, experimental
results confirm the advantages of the proposed extension PID control system.
PARALLEL-EXTENSION MODELING AND APPLICATION
235
Figure 5 Extension PID control modeling error fault of cut tobacco moisture
4 Conclusions
In this article, a new structure of an extension PID controller is presented. The parallel
combination of extension PI and PD predictable controller shows its simplicity in determining
the control rules and controller parameters. A tuning formula via the extension algorithm of
the matter-element model is introduced. The weighting factors of an extension PID predictable
controller can be selected with respect to the second-order plus dead-time plants such as temperature and pressure system. Both theoretical analysis and practical application confirm the
validity of the proposed extension PID controller and extension algorithm of the matter-element
model, and show that the extension PID predictable controller has the nonlinear properties of
higher control gains when the system is away from its steady point and lower control profile
when set-point changes occur. Therefore, these nonlinear properties provide the extension PID
predictable control system with superior performance over the fuzzy PID control system.
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